[    {        "title": "1307.7427v1.Theoretical_Study_of_Spin_Torque_Oscillator_with_Perpendicularly_Magnetized_Free_Layer.pdf",        "content": "arXiv:1307.7427v1  [cond-mat.mes-hall]  29 Jul 20131\nTheoretical Study of Spin-Torque Oscillator with\nPerpendicularly Magnetized Free Layer\nTomohiro Taniguchi, Hiroko Arai, Hitoshi Kubota, and Hiros hi Imamura∗\nSpintronics Research Center, AIST, Tsukuba, Ibaraki 305-8 568, Japan\nAbstract—The magnetization dynamics of spin torque oscilla-\ntor (STO) consisting of a perpendicularly magnetized free l ayer\nand an in-plane magnetized pinned layer was studied by solvi ng\nthe Landau-Lifshitz-Gilbert equation. We derived the anal ytical\nformula of the relation between the current and the oscillat ion\nfrequency of the STO by analyzing the energy balance between\nthe work done by the spin torque and the energy dissipation du e\nto the damping. We also found that the field-like torque break s\nthe energy balance, and change the oscillation frequency.\nIndex Terms —spintronics, spin torque oscillator, perpendicu-\nlarly magnetized free layer, the LLG equation\nI. INTRODUCTION\nSPIN torque oscillator (STO) has attracted much attention\ndue to its potential uses for a microwave generator and a\nrecording head of a high density hard disk drive. The self-\noscillation of the STO was first discovered in an in-plane\nmagnetized giant-magnetoresistive (GMR) system [1]. Afte r\nthat, the self-oscillation of the STO has been observed not\nonly in GMR systems [2]-[6] but also in magnetic tunnel\njunctions (MTJs) [7]-[11]. The different types of STO have\nbeen proposedrecently; for example,a point-contactgeome try\nwith a confinedmagneticdomainwall [12]-[14]whichenables\nusto controlthe frequencyfroma few GHz to a hundredGHz.\nRecently, Kubota et al.experimentally developed the MgO-\nbased MTJ consisting of a perpendicularly magnetized free\nlayer and an in-plane magnetized pinned layer [15],[16]. Th ey\nalso studied the self-oscillation of this type of MTJ, and\nobserved a large power ( ∼0.5µW) with a narrow linewidth\n(∼50MHz) [17].These results are great advancesin realizing\nthe STO device.However,the relationbetweenthe currentan d\nthe oscillation frequency still remains unclear. Since a pr ecise\ncontrol of the oscillation frequency of the STO by the curren t\nis necessary for the application, it is important to clarify the\nrelation between the current and the oscillation frequency .\nIn this paper, we derived the theoretical formula of the\nrelation between the current and the oscillation frequency\nof the STO consisting of the perpendicularly magnetized\nfree layer and the in-plane magnetized pinned layer. The\nderivation is based on the analysis of the energy balance\nbetween the work done by the spin torque and the energy\ndissipation due to the damping. We found that the oscillatio n\nfrequencymonotonicallydecreases with increasing the cur rent\nby keeping the magnetization in one hemisphere of the free\nlayer. The validity of the analytical solution was confirmed\nby numerical simulations. We also found that the field-like\n∗Corresponding author. Email address: h-imamura@aist.go. jppmelectron (I>0)z\nxy\nspin torquedamping damping spin torque\nFig. 1. Schematic view of the system. The directions of the sp in torque and\nthe damping during the precession around the z-axis are indicated.\ntorque breaks the energy balance, and change the oscillatio n\nfrequency. The shift direction of the frequency, high or low ,\nis determined by the sign of the field-like torque.\nThis paper is organized as follows. In Sec. II, the current\ndependence of the oscillation frequency is derived by solvi ng\nthe Landau-Lifshitz-Gilbert (LLG) equation. In Sec. III, t he\neffect of the field-like torque on the oscillation behaviour is\ninvestigated. Section IV is devoted to the conclusions.\nII. LLG STUDY OF SPIN TORQUE OSCILLATION\nThe system we consider is schematically shown in Fig.\n1. We denote the unit vectors pointing in the directions\nof the magnetization of the free and the pinned layers as\nm= (sinθcosϕ,sinθsinϕ,cosθ)andp, respectively. The\nx-axis is parallel to pwhile the z-axis is normal to the film\nplane. The variable θofmis the tilted angle from the z-axis\nwhileϕis the rotation angle from the x-axis. The current I\nflows along the z-axis, where the positive current corresponds\nto the electron flow from the free layer to the pinned layer.\nWe assume that the magnetization dynamics is well de-\nscribed by the following LLG equation:\ndm\ndt=−γm×H−γHsm×(p×m)+αm×dm\ndt.(1)\nThe gyromagneticration and the Gilbert damping constant ar e\ndenotedas γandα, respectively.The magnetic field is defined\nbyH=−∂E/∂(Mm), where the energy density Eis\nE=−MHapplcosθ−M(HK−4πM)\n2cos2θ.(2)\nHere,M,Happl, andHKare the saturation magnetization,the\napplied field along the z-axis, and the crystalline anisotropy\nfield along the z-axis, respectively. Because we are interested\nin the perpendicularly magnetized system, the crystalline\nanisotropy field, HK, should be larger than the demagneti-\nzation field, 4πM. Since the LLG equation conserves the2\nI = 1.2 ~ 2.0 (mA)\nmz\nmxmy1\n-1 0\n-1 \n-1 1\n100(a)\ncurrent (mA)34567(b)\nfrequency (GHz) \n1.2 1.4 1.6 1.8 2.0\nFig. 2. (a) The trajectories of the steady state precession o f the magne-\ntization in the free layer with various currents. (b) The dot s represent the\ndependence of the oscillation frequency obtained by numeri cally solving the\nLLG equation. The solid line is obtained by Eqs. (8) and (9).\nnormofthe magnetization,the magnetizationdynamicscan b e\ndescribed by a trajectory on an unit sphere. The equilibrium\nstates of the free layer correspond to m=±ez. In following,\nthe initial state is taken to be the north pole, i.e., m=ez. It\nshould be noted that a plane normal to the z-axis, in which θ\nis constant, corresponds to the constant energy surface.\nThe spin torque strength, Hsin Eq. (1), is [18]-[20]\nHs=/planckover2pi1ηI\n2e(1+λmx)MSd, (3)\nwhereSanddare the cross section area and the thickness\nof the free layer. Two dimensionless parameters, ηandλ\n(−1< λ <1), determine the magnitude of the spin polariza-\ntion and the angle dependence of the spin torque, respective ly.\nAlthough the relation among η,λ, and the material parameters\ndepends on the theoretical models [20]-[22], the form of Eq.\n(3) is applicable to both GMR system and MTJs. In particular,\nthe angle dependence of the spin torque characterized by λis\na key to induce the self-oscillation in this system.\nFigure 2 (a) shows the steady state precession of the mag-\nnetization in the free layer obtained by numerically solvin g\nEq. (1). The values of the parameters are M= 1313emu/c.c.,\nHK= 17.9kOe,Happl= 1.0kOe,S=π×50×50nm2,\nd= 2.0nm,γ= 17.32MHz/Oe, α= 0.005,η= 0.33,\nandλ= 0.38, respectively [17]. The self-oscillation was\nobservedforthe current I≥1.2mA.Althoughthe spintorque\nbreaks the axial symmetry of the free layer along the z-axis,\nthe magnetization precesses around the z-axis with an almost\nconstanttilted angle. Thetilted angle fromthe z-axisincreases\nwith increasing the current; however, the magnetization st ays\nin the northsemisphere( θ < π/2). The dotsin Fig. 2 (b) show\nthe dependence of the oscillation frequency on the current. As\nshown, the oscillation frequencymonotonicallydecreases with\nincreasing the current magnitude.\nLet us analytically derive the relation between the current\nand the oscillation frequency. Since the self-oscillation occurs\ndue to the energysupply into the free layer by the spin torque ,\nthe energy balance between the spin torque and the damping\nshould be investigated. By using the LLG equation, the time\nderivative of the energy density Eis given by dE/dt=Ws+\nWα, where the work done by spin torque, Ws, and the energydissipation due to the damping, Wα, are respectively given by\nWs=γMHs\n1+α2[p·H−(m·p)(m·H)−αp·(m×H)],\n(4)\nWα=−αγM\n1+α2/bracketleftBig\nH2−(m·H)2/bracketrightBig\n. (5)\nBy assuming a steady precession around the z-axis with a\nconstant tilted angle θ, the time averages of WsandWαover\none precession period are, respectively, given by\nWs=γM\n1+α2/planckover2pi1ηI\n2eλMSd/parenleftBigg\n1/radicalbig\n1−λ2sin2θ−1/parenrightBigg\n×[Happl+(HK−4πM)cosθ]cosθ,(6)\nWα=−αγM\n1+α2[Happl+(HK−4πM)cosθ]2sin2θ.(7)\nThe magnetization can move from the initial state to a point\nat which dE/dt= 0. Then, the current at which a steady\nprecession with the angle θcan be achieved is given by\nI(θ) =2αeλMSd\n/planckover2pi1ηcosθ/parenleftBigg\n1/radicalbig\n1−λ2sin2θ−1/parenrightBigg−1\n×[Happl+(HK−4πM)cosθ]sin2θ.(8)\nThe corresponding oscillation frequency is given by\nf(θ) =γ\n2π[Happl+(HK−4πM)cosθ].(9)\nEquations (8) and (9) are the main results in this section.\nThe solid line in Fig. 2 (b) shows the current dependence of\nthe oscillation frequency obtained by Eqs. (8) and (9), wher e\nthe good agreement with the numerical results confirms the\nvalidity of the analytical solution. The critical current f or the\nself-oscillation, Ic= limθ→0I(θ), is given by\nIc=4αeMSd\n/planckover2pi1ηλ(Happl+HK−4πM).(10)\nThe value of Icestimated by using the aboveparametersis 1.2\nmA, showing a good agreement with the numerical simulation\nshown in Fig. 2 (a). The sign of Icdepends on that of λ,\nand the self-oscillation occurs only for the positive (nega tive)\ncurrent for the positive (negative) λ. This is because a finite\nenergy is supplied to the free layer for λ/negationslash= 0, i.e.,Ws>0. In\nthe case of λ= 0, the average of the work done by the spin\ntorque is zero, and thus, the self-oscillation does not occu r.\nIt should be noted that I(θ)→ ∞in the limit of θ→π/2.\nThis means the magnetization cannot cross over the xy-plane,\nand stays in the north hemisphere ( θ < π/2). The reason\nis as follows. The average of the work done by spin torque\nbecomes zero in the xy-plane (θ=π/2) because the direction\nof the spin torque is parallel to the constant energy surface .\nOn the other hand, the energy dissipation due to the damping\nis finite in the presence of the applied field [21]. Then,\ndE/dt(θ=π/2) =−αγMH2\nappl/(1 +α2)<0, which\nmeansthe dampingmovesthe magnetizationto the northpole.\nThus, the magnetization cannot cross over the xy-plane. The\ncontrollable range of the oscillation frequency by the curr ent\nisf(θ= 0)−f(θ=π/2) =γ(HK−4πM)/(2π), which is\nindependent of the magnitude of the applied field.3\nSince the spin torque breaks the axial symmetry of the\nfree layer along the z-axis, the assumption that the tilted\nangle is constant used above is, in a precise sense, not valid ,\nand thez-component of the magnetization oscillates around\na certain value. Then, the magnetization can reach the xy-\nplane and stops its dynamics when a large current is applied.\nHowever,thevalue ofsuch currentis morethan15mA forour\nparameter values, which is much larger than the maximum of\nthe experimentallyavailable current. Thus, the above form ulas\nwork well in the experimentally conventional current regio n.\nContrary to the system considered here, the oscillation be-\nhaviour of an MTJ with an in-plane magnetized free layer and\na perpendicularly magnetized pinned layer has been widely\ninvestigated [23]-[26]. The differences of the two systems\nare as follows. First, the oscillation frequency decreases with\nincreasing the current in our system while it increases in\nthe latter system. Second, the oscillation frequency in our\nsystem in the large current limit becomes independent of the\nz-component of the magnetization while it is dominated by\nmz= cosθin the latter system. The reasons are as follows.\nIn our system, by increasing the current, the magnetization\nmoves away from the z-axis due to which the effect of the\nanisotropy field on the oscillation frequency decreases, an d\nthe frequency tends to γHappl/(2π), which is independent\nof the anisotropy. On the other hand, in the latter system,\nthe magnetization moves to the out-of-plane direction, due\nto which the oscillation frequency is strongly affected by t he\nanisotropy (demagnetization field).\nThe macrospin model developed above reproduces the ex-\nperimentalresultswith the freelayerof2nmthick[17],for ex-\nample the current-frequency relation, quantitatively. Al though\nonlythezero-temperaturedynamicsisconsideredinthispa per,\nthe macrospin LLG simulation at a finite temperature also\nreproduces other properties, such as the power spectrum and\nits linewidth, well. However, when the free layer thickness\nfurther decreases, an inhomogeneousmagnetization due to t he\nroughnessattheMgOinterfacesmayaffectsthemagnetizati on\ndynamics: for example, a broadening of the linewidth.\nIII. EFFECT OF FIELD -LIKE TORQUE\nThe field-like torque arises from the spin transfer from the\nconductionelectrons to the local magnetizations,as is the spin\ntorque. When the momentum average of the transverse spin of\ntheconductionelectronsrelaxesinthefreelayerveryfast ,only\nthe spin torque acts on the free layer [19]. On the other hand,\nwhen the cancellation of the transverse spin is insufficient ,\nthe field-like torque appears. The field-like torque added to\nthe right hand side of Eq. (1) is\nTFLT=−βγHsm×p, (11)\nwhere the dimensionless parameter βcharacterizes the ratio\nbetween the magnitudes of the spin torque and the field-like\ntorque. The value and the sign of βdepend on the system\nparameters such as the band structure, the thickness, the\nimpurity density, and/or the surface roughness [22],[27]- [29].\nThe magnitude of the field-like torque in MTJ is much larger\nthan that in GMR system [30],[31] because the band selectionmz\nmxmy1\n-1 0\n-1 \n-1 1\n100(a)\nmz\nmxmy1\n-1 0\n-1 \n-1 1\n100(b)\nmz\nmxmy1\n-1 0\n-1 \n-1 1\n100(c) (d)\n0 0.2 0.1\ntime (μs)0\n-0.5 \n-1.00.51.0mzβ=0 β=0.5\nβ=-0.5\nβ=-0.5\nβ=0.5β=0\nFig. 3. The magnetization dynamics from t= 0with (a) β= 0, (b)\nβ= 0.5, andβ=−0.5. The current magnitude is 2.0mA. (d) The time\nevolutions of mzfor various β.\nduringthe tunnelingleads to an insufficient cancellation o f the\ntransverse spin by the momentum average.\nIt should be noted that the effective energy density,\nEeff=E−βM/planckover2pi1ηI\n2eλMSdlog(1+λmx),(12)\nsatisfying −γm×H+TFLT=−γm×[−∂Eeff/(Mm)],\ncan be introduce to describe the field-like torque. The time\nderivative of the effective energy, Eeff, can be obtained by\nreplacing the magnetic field, H, in Eqs. (4) and (5) with\nthe effective field −∂Eeff/∂(Mm) =H+βHsp. Then, the\naverage of dEeff/dtover one precession period around the\nz-axis consists of Eq. (6), (7), and the following two terms:\nW′\ns=βγM\n1+α2/parenleftbigg/planckover2pi1ηI\n2eλMSd/parenrightbigg2/bracketleftbigg1+λ2cos2θ\n(1−λ2sin2θ)3/2−1/bracketrightbigg\n,\n(13)\nW′\nα=−αγM\n1+α2/parenleftbiggβ/planckover2pi1ηI\n2eλMSd/parenrightbigg2/bracketleftbigg1+λ2cos2θ\n(1−λ2sin2θ)3/2−1/bracketrightbigg\n−2αβγM\n1+α2/planckover2pi1ηI\n2eλMSd/parenleftBigg\n1/radicalbig\n1−λ2sin2θ−1/parenrightBigg\n×[Happl+(HK−4πM)cosθ]cosθ.\n(14)\nThe constant energy surface of Eeffshifts from the xy-plane\ndue to a finite |β|(≃1), leading to an inaccuracy of the\ncalculation of the time average with the constant tilted ang le\nassumption. Thus, Eqs. (13) and (14) are quantitatively val id\nfor only |β| ≪1. However, predictions from Eqs. (13) and\n(14) qualitatively show good agreements with the numerical\nsimulations, as shown below.\nFor positive β,W′\nsis also positive, and is finite at θ=π/2.\nThus,dEeff/dt(θ=π/2)can be positive for a sufficiently\nlarge current. This means, the magnetization can cross over\nthexy-plane, and move to the south semisphere ( θ > π/2).\nOn the other hand, for negative β,W′\nsis also negative. Thus,\ntheenergysupplybythespintorqueissuppressedcomparedt o4\ncurrent (mA)34567frequency (GHz) \n1.2 1.4 1.6 1.8 2.02\n1\n0β=0\nβ=0.5β=-0.5\nFig. 4. The dependences of the oscillation frequency on the c urrent for\nβ= 0(red),β= 0.5(orange), and β=−0.5(blue), respectively.\nthecaseof β= 0.Then,arelativelylargecurrentisrequiredto\ninduce the self-oscillation with a certain oscillation fre quency.\nAlso, the magnetization cannot cross over the xy-plane.\nWe confirmed these expectations by the numerical simu-\nlations. Figures 3 (a), (b) and (c) show the trajectories of\nthe magnetization dynamics with β= 0,0.5, and−0.5\nrespectively,while thetime evolutionsof mzareshownin Fig.\n3 (d). The current value is 2.0 mA. The current dependences\nof the oscillation frequency are summarized in Fig. 4.\nIn the case of β= 0.5>0, the oscillation frequency is low\ncompared to that for β= 0because the energy supply by the\nspin torque is enhanced by the field-like torque, and thus, th e\nmagnetization can largely move from the north pole. Above\nI= 1.9mA, the magnetizationmovesto the southhemisphere\n(θ > π/2), and stops near θ≃cos−1[−Happl/(HK−4πM)]\nin the south hemisphere, which corresponds to the zero fre-\nquency in Fig. 4.\nOn the other hand, in the case of β=−0.5<0, the\nmagnetization stays near the north pole compared to the case\nofβ= 0, because the energy supply by the spin torque is\nsuppressed by the field-like torque. The zero frequency in\nFig. 4 indicates the increase of the critical current of the s elf-\noscillation. Compared to the case of β= 0, the oscillation\nfrequency shifts to the high frequency region because the\nmagnetization stays near the north pole.\nIV. CONCLUSIONS\nIn conclusion, we derived the theoretical formula of the\nrelation between the current and the oscillation frequency of\nSTO consisting of the perpendicularly magnetized free laye r\nand the in-plane magnetized pinned layer. The derivation is\nbased on the analysis of the energy balance between the work\ndone by the spin torque and the energy dissipation due to the\ndamping.The validityof the analyticalsolutionwas confirm ed\nby numerical simulation. We also found that the field-like\ntorque breaks the energy balance, and changes the oscillati on\nfrequency. The shift direction of the frequency, high or low ,\ndepends on the sign of the field-like torque ( β).ACKNOWLEDGMENT\nThe authors would like to acknowledge T. 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Phys. , vol.4, pp.67-71, 2008."    },    {        "title": "1102.5384v2.Dynamics_of_Skyrmion_Crystals_in_Metallic_Thin_Films.pdf",        "content": "arXiv:1102.5384v2  [cond-mat.mes-hall]  16 Sep 2011Dynamics of Skyrmion Crystal in Metallic Thin Films\nJiadong Zang1,2,3,∗Maxim Mostovoy4, Jung Hoon Han5, and Naoto Nagaosa2,3†\n1Department of Physics, Fudan University, Shanghai 200433, China\n2Department of Applied Physics, University of Tokyo,\n7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan\n3Cross-Correlated Materials Research Group (CMRG),\nand Correlated Electron Research Group (CERG), RIKEN-ASI, Wako, Saitama 351-0198, Japan\n4Zernike Institute for Advanced Materials, University of Gro ningen,\nNijenborgh 4, 9747 AG Groningen, The Netherlands\n5Department of Physics, BK21 Physics Research Division,\nSungkyunkwan University, Suwon 440-746, Korea\n(Dated: June 6, 2018)\nWe study the collective dynamics of the Skyrmion crystal (Sk X) in thin films of ferromagnetic\nmetals resulting from the nontrivial Skyrmion topology. It is shown that the current-driven motion\nof the crystal reduces the topological Hall effect and the Sky rmion trajectories bend away from the\ndirection of the electric current (the Skyrmion Hall effect) . We find a new dissipation mechanism\nin non-collinear spin textures that can lead to a much faster spin relaxation than Gilbert damping,\ncalculate the dispersion of phonons in the SkX, and discuss e ffects of impurity pinning of Skyrmions.\nPACS numbers: 73.43.Cd,72.25.-b,72.80.-r\nIntroduction: Skyrmion is a topologically nontrivial\nsoliton solution of the nonlinear sigma model. It was\nnoted early on that Skyrmions in three spatial dimen-\nsions have physical properties of baryons and the peri-\nodic Skyrmion crystal (SkX) configurations were used to\nmodel nuclear matter[1, 2]. Skyrmions in two spatial\ndimensions play an important role in condensed matter\nsystems, such as quantum Hall ferromagnets[3, 4]. It\nwas suggested that SkX configurations can be stabilized\nby the Dzyaloshinskii-Moriya (DM) interaction in ferro-\nmagnetswithoutinversionsymmetry[5]. Suchastatewas\nrecently observed in a neutron scattering experiment in\nthe A-phase of the ferromagnetic metal MnSi[6].\nRecent Monte Carlo simulations indicated much\ngreater stability of the SkX when a bulk ferromagnet is\nreplaced by a thin film[7]. This result was corroborated\nby the real-space observation of SkX in Fe 0.5Co0.5Si thin\nfilm in a wide magnetic field and temperature range[8].\nLorentz force microscopy showed that Skyrmions form a\ntriangular lattice with the magnetization vector antipar-\nallel to the applied magnetic field in the Skyrmion center\nand parallel at the periphery, as was also concluded from\nthe neutron experiment[6].\nThe next important step is to explore dynamics of\nSkyrmion crystals and the ways to control them in anal-\nogy to the actively studied current- and field-driven mo-\ntion of ferromagnetic domain walls[9]. Recent observa-\ntion ofthe rotationalmotion ofthe SkX in MnSi suggests\nthat Skyrmionscanbe manipulated bymuchsmallercur-\nrents than domain walls[10].\nIn this Letter we study the coupled dynamics of spins\nand charges in the SkX, focusing on effects of the non-\n∗Electronic address: jdzang@fudan.edu.cn\n†Electronic address: nagaosa@ap.t.u-tokyo.ac.jptrivial Skyrmion topology and effective gauge fields in-\nduced by the adiabatic motion of electrons in the SkX.\nWe derive equation of motion for the collective vari-\nables describing the SkX, calculate its phonon disper-\nsion, and discuss a new form of damping, which can be\nthe dominant spin-relaxation mechanism in half-metals.\nIn addition, we consider new transport phenomena, such\nas the topological Hall effect in a sliding SkX and the\nSkyrmion Hall effect. We also discuss the Skyrmion pin-\nning by charged impurities and estimate the critical cur-\nrent above which the SkX begins to slide.\nLow-energy excitations in Skyrmion crystal: An iso-\nlated Skyrmion has two zero modes corresponding to\ntranslations along the xandydirections. Since an ap-\npliedmagneticfieldopensagapinthecontinuumofspin-\nwave excitations, the low-energy magnetic modes in SkX\nare expected to be superpositions of the Skyrmion dis-\nplacements, or the phonons. The phonon modes, as well\nasthecouplingofSkyrmiondisplacementstotheexternal\ncurrent, can be consistently described in the framework\nof elasticity theory.\nWe begin with the spin Hamiltonian HS=/integraltextd3x/bracketleftbigJ\n2a(∇n)2+D\na2n·[∇×n]−µ\na3H·n/bracketrightbig\n, whereJis\nthe exchange constant and Dis the DM coupling that\nstabilizes the SkX configuration n(x) in some interval of\nthe magnetic field H=Hˆz[5, 11]. We calculate the ‘har-\nmonic lattice energy’ by considering a deformation of the\nSkX,˜n(x,t) =n(x−u(x,t)), where the collective coor-\ndinateu(x,t) varies slowly at the scale of the SkX lattice\nconstant. The result is:\nHS=dηJ/integraldisplayd2x\nξ2[(∇ux)2+(∇uy)2],(1)\nwheredis the film thickness and ξ∼aJ\nDis the\ncharacteristic length scale of SkX[11], with abeing2\nthe lattice spacing. The dimensionless quantity η=\n1\n8π/integraltext\nucd2x(∂in·∂in) encodes the information about D\nandH, and is called shape factor in what follows.\nWhen an electroncurrentis flowingthroughthe metal-\nlic film, the conduction electrons interact with local mag-\nnetic moments through the Hund’s rule coupling HH=\n−JHSψ†σ·nψ, whereψis the electron operator. In\nthe case of small current density and the Skyrmion size\nmuch larger than the Fermi wavelength of conduction\nelectrons, one can apply the adiabatic approximation in\nwhich the electron spins align perfectly with the local\nmoment.ψis projected into the fully polarized state by\nψ=χ|n/angb∇acket∇ightwithσ·n|n/angb∇acket∇ight=|n/angb∇acket∇ight. Then the electron action\nSel=/integraltext\ndtd3x[i¯hψ†˙ψ+¯h2\n2mψ†∇2ψ+JHSψ†σ·nψ] can be\nrewritten as Sel=/integraltext\ndtd3x[i¯hχ†˙χ−ea0−1\n2mχ†(−i¯h∇−\ne\nca)2χ+JHSχ†χ], whereaµ=¯hc\n2e(1−cosθ)∂µϕwithθ\nandϕbeing the spherical angles describing the direc-\ntion of the local magnetization[12, 13]. The gauge po-\ntentialaµgives rise to internal electric and magnetic\nfields,eandh, acting on spin-polarized electrons pass-\ning through the SkX in analogy with the electromag-\nnetic gauge field. Crucially, the internal magnetic field\nb=∇×a=¯hc\n2e(n·∂xn×∂yn)ˆzis intimately re-\nlated to the topological charge Qof Skyrmions by[14]\nQ=1\n4π/integraltext\nucd2x(n·∂xn×∂yn) =±1,where the integra-\ntion goes over the unit cell of the SkX. In the language\nof internal gauge field, this topological feature is nothing\nbut thequantizationofinternalfluxΩ =/integraltexth·dSinunits\nofhc/e. The coupling of the electric current to the inter-\nnal gauge field induced by the SkX, Hint=−1\nc/integraltextd3xj·a,\nhas a simple form in terms of the collective coordinates\nintroduced above:\nHint=d¯hQ\ne/integraldisplayd2x\nξ2(uxjy−uyjx). (2)\nThe crucial difference between the SkX and a con-\nventional crystal is the form of kinetic energy. The\nspin dynamics originates from the Berry phase action,\nSBP=d\nγ/integraltext\ndtd2x(cosθ−1) ˙ϕ. Here,γ=a3\n¯h(S+x/2), where\nxis the filling of the conduction band, and S+x/2 is the\ntotal spin averagely per lattice site. In terms of uthe\nkinetic energy has the form\nSBP=dQ\nγ/integraldisplay\ndtd2x\nξ2(ux˙uy−uy˙ux). (3)\nThis form of the Berry phase shows that the collec-\ntive variables uxanduydescribing local displacements\nof Skyrmions form a pair of canonical conjugate vari-\nables, replacing cos θandϕ. This characteristic prop-\nerty of SkX leads to several unusual responses to ap-\nplied electric currents and fields. It originates from\nthe Skyrmion topology and distinguishes SkX from non-\ntopological spin textures such as spirals and domain wall\narrays.\nUsing Eqs.(1), (2) and (3), we obtain equation of mo-\ntion foru:\n˙u=−e¯hγ\n2j+QγηJ\ne¯hˆz×∇2u. (4)Two consequences follow immediately. First, the dis-\npersion of phonons in the SkX obtained from Eq.(4) is\nquadratic,\n¯hω=ηJa2\n(S+x\n2)k2, (5)\nin contrast to the linear phonon dispersion in usual crys-\ntalsandsimilartothedispersionofmagnonsin auniform\nferromagnet. Since uxanduyplay the role of the coor-\ndinate and momentum, the longitudinal and transverse\nphonon modes in the SkX merge into a single mode cor-\nresponding to the rotational motion of Skyrmions, which\nleads to the quadratic dispersion. Secondly, the SkX can\nmove as a whole driven by the charge current j, with a\nvelocityV/bardbl=˙ u=−e¯hγ\n2j. This rigid motion of SkX leads\nto several interesting results discussed below.\nHall effect due to SkX motion: In such nontrivial spin\ntextures, the external magnetic field (less than 0.2T for\nMnSi) is more than one order of magnitude smaller than\nthe internal one, so that it would be neglected in what\nfollows. As can be seen from Eq. (2), the collective\ncoordinates uxanduyplay the role of electromagnetic\ngauge potentials Ayand−Ax, respectively. It is thus\nexpected that the temporal variation of uinduced by\nthe current leads to a transverse potential drop. This\nHall-type effect can also be intuitively understood using\ntheinternalmagneticfield bintroducedabove. Amoving\nspin texture n(x−V/bardblt) induces an internal electric field\neanalogous to the electric field of a moving magnetic\nflux and related to the internal magnetic field by e=\n−1\nc/bracketleftbig\nV/bardbl×b/bracketrightbig\n.For SkX with b=bzˆz, this electric field\ngenerates an electric current in the direction transverse\ntoV/bardblresulting in the Hall conductivity:\n∆σxy\nσxx≈ −x\n2S+xe/angb∇acketleftbz/angb∇acket∇ightτ\nmc, (6)\nwheremis the electron mass and τis the relaxation\ntime. The average internal magnetic field is /angb∇acketleftbz/angb∇acket∇ight=QΦ0\n2πξ2,\nwhere Φ 0is the elementary flux and 2 πξ2is the area of\nthe unit cell of the SkX. This Hall conductivity has the\nsameorderofmagnitude asthe oneresulting fromthe so-\ncalled topologicalHall effect observedin a staticSkX[15].\nThe latter effect is nothing but the Hall effect induced\nbybviae=1\nc[v×b], andσTop\nxy/σxx≈e/angb∇acketleftbz/angb∇acket∇ightτ/mc,\nwherevis the electron velocity. Our new effect differs\nby the factor of −x\n2S+xfrom the topological Hall effect.\nIts physical origin can be easily understood by noting\nthe total force acting on a single conduction electron is\nF=−e\nc[(v−V/bardbl)×b], i.e. the Lorentz force on elec-\ntrons due to the internal magnetic field of the SkX de-\npendsontherelativevelocityofelectronsandSkyrmions.\nWhen the SkX begins to slide above the threshold elec-\ntric current jc[16], the net topological Hall voltage will\nbe suddenly reduced by the factor2S\n2S+x, which is how\nthe effect of the spin-motive force and the collective shift\nof Skyrmions can be identified experimentally.\nNew damping mechanism and Skyrmion Hall effect:\nPreviously we have systematically discussed the novel3\neffects related to the internal magnetic field. A natu-\nral question thus arises as to whether there is any new\nphenomena associated with the intrinsic internal electric\nfield, which is ei=−∂ia0−1\nc˙ai=¯h\n2e(n·∂in×˙n). Due\nto the time derivative in this expression, its effect is ab-\nsent in the static spin texture. However, in the present\ncase, the motion of SkX makes it nonvanishing, and leads\nto an additional current j′byj′=σewithσthe con-\nductivity of electrons. Substituting this current into the\nLandau-Lifshitz-Gilbert equation[12, 13]\n˙n=¯hγ\n2e[j·∇]n−γ/bracketleftbigg\nn×δHS\nδn/bracketrightbigg\n+α[˙n×n],(7)\nthe time derivative ˙nreceives a correction given by\nδ˙n=¯hγσ\n2e(e·∇)n=α′(n·∂in×˙n)∂in.(8)\nThe corresponding dimensionless damping constant is\nα′=1\n(2S+x)a3σ\nαfsξ2c, whereαfs≈1/137 is the fine struc-\nture constant. The time derivative in the r.h.s. of Eq.(8)\nshows that the current induced by internal electric field\nleads to dissipation. In contrast to Gilbert damping this\nnew mechanism does not require relativistic effects and\nonly involves the Hund’s rule coupling that conserves the\ntotal spin. The relaxation of the uniform magnetization,\ndescribed by Gilbert damping, is clearly impossible with-\nout the spin-orbit coupling, which breaks the conserva-\ntion of the total spin[17]. This argument, however, does\nnot apply to inhomogeneousmagnetic textures where the\nbreaking of the rotational symmetry by noncollinear spin\norders enables the relaxation without the spin-orbit cou-\npling (note that α′vanishes as ξ→ ∞). Despite the\nnon-relativistic origin, α′depends on the DM coupling,\nas the latter determines the Skyrmion size. Estimates of\nα′made below show that in half-metals it can greatly\nexceedα.\nThe effect of this new dissipation can be observed by\ntracing the trajectoryof Skyrmion motion. Including the\nnewdissipationterm, themodifiedequationofmotion(4)\nfor the rigid collective coordinates u(t) has the form\n˙u=−e¯hγ\n2j−Q(αη+α′η′)ˆz×˙u, (9)\nwhere the second shape factor η′is given by η′=\nQ\n4π/integraltext\nucd2x(n·∂xn×∂yn)(∂in·∂in)//integraltext\nucd2x(∂in·∂in).\nThe new dissipation term in Eq.(9) is obtained by mul-\ntiplying Eq.(8) with ∂jn, using ˙n=−(˙u·∇)n, and\nintegrating over one unit cell. The whole damping term\nleads to a transverse motion with velocity\nV⊥≈Q(αη+α′η′)/bracketleftbig\nV/bardbl׈z/bracketrightbig\n. (10)\nThis Skyrmion Hall effect can be observed by real-space\nimages of Lorentz force microscopy. The corresponding\nHall angle is θ= arctan(αη+α′η′).The estimate given\nbelow shows that main contribution to θcomes from the\nnew dissipation mechanism.Pinning of Skyrmion crystal: Next we consider the\npinning of the SkX by charged impurities. The pinning\nresults from spatial fluctuations of the impurity density\nandvariationsofthespindirectionintheSkX.Variations\nof the density of charged impurities δnigive rise to local\nvariations of the electron density neand since the double\nexchangeconstant Jisproportionaltothelatter, wehave\nδJ∼Jδni/ne. The energy per Skyrmion ES∼Jd/a.\nDenote the number of impurities in this volume by N1\nwith/angb∇acketleftN1/angb∇acket∇ight=ni2πξ2dand the variance δN1=√N1, we\nobtain the typical variation of the Skyrmion energy:\nV1=δJd\na∼J\nne2πξ2a/radicalbig\nN1=J\nneaξ/radicalbigg\nnid\n2π.(11)\nThe potential energy density is then V0=V1/(2πξ2).\nSubstituting ni∼(la2)−1, wherelis the electron mean\nfree path, and ne=x\na3, we obtain V0∼J\n(2π)3/2x/radicalBig\nd\nla\nξ3.\nThe pinning regime of the whole SkX depends on the ra-\ntioofthe pinningenergy V1andthe elasticenergy ESofa\nsingle Skyrmion[18]. Let L2be the number of Skyrmions\nin the domain where u∼ξ. The energy gain due to\nthe impurity pinning in the domain is ∼ −V1L, while\nthe elastic energy cost ∼Jd\nais independent of the do-\nmain size. Minimizing the total energy per Skyrmion,\nJd\naL2−V1\nL, we obtain L∼Jd\naV1.L≫1 corresponds to the\ncase of weak (or collective) pinning of SkX, while L∼1\ncorresponds to the strong pinning regime.\nThe pinning potential gives rise to the spin transfer\ntorque−Qγξ2\n2d/bracketleftbigˆz×δV\nδu/bracketrightbig\nin the right-hand side of Eq.(4).\nIn the steady state of moving SkX this torque has to be\ncompensated by the interaction with the electric current.\nThe critical current density is then\njc∼e\n¯hξ2\nd/angbracketleftbigg∂V\n∂u/angbracketrightbigg\nsteady state∼e\n¯hξV0\ndL,(12)\nin the weak pinning regime, while in the strong pinning\ncaseLhas to be substituted by 1. Similarly, one can\nestimate the gap in the spin wave spectrum due to the\npinning:\n¯hωpin∼¯hγξ2\nd/angbracketleftbigg∂2V\n∂u2/angbracketrightbigg\n∼¯hγξ2\ndV0L\nL2ξ2=a3\ndSV0\nL.(13)\nEstimates: For estimates we consider MnSi where Mn\nions form a (distorted) cubic sublattice with a= 2.9˚A.\nThe length of the reciprocal lattice vectors of the SkX\n∼0.035˚A−1correspondsto ξ∼77˚AandD∼0.1J. The\nkinetic energy scales as ¯ h2/mξ2<JH∼1eV, so that the\nadiabaticapproximationisjustified. Theelectrondensity\nne= 3.8·1022cm−3[15] corresponds to x=nea3∼0.9\ncharge carriers per lattice site, while the residual resis-\ntivityρ∼2µΩ·cm[15] gives an estimate of the impurity\nconcentration xi∼5·10−3. From the magnon dispersion\nin the spiral state[19], J∼3 meV.\nUsing these parameters we get α′∼0.1, which shows\nthatthedampingresultingfromtheelectriccurrentsgen-\nerated by non-collinear spin textures can be the domi-\nnant mechanism of spin relaxation in half-metals, where4\nthe Gilbert damping constant αis one-three orders of\nmagnitude smaller[20, 21]. We note, however, that since\nthe typical electron mean free path ξ∼500˚A is larger\nthan the Skyrmion size ξ, the relation between the cur-\nrent and internal electric field is nonlocal. The topo-\nlogical Hall angle, θH∼ehzτ\nmc=1\nαsfneξ2σ\ncatT= 0\nand the change in θHinduced by the sliding Skyrmion\ncrystal [see Eq.(6)] are also ∼0.1. For a 10nm thick\nfilm the parameter L∼xξ√\n2πdl\na2∼103,i.e. the pin-\nning of Skyrmions by charged impurities is exceedingly\nweak and the corresponding values of the pinning fre-\nquency ¯hωpin∼5·10−11meV and the critical current\njc∼0.2A·cm−2. This value is much lower than that\nfor domain walls[9] and smaller than the ultralow thresh-\nold current jc∼102A·cm−2observed in bulk MnSi[10],\nwhich may be attributed to other pinning mechanisms\nand the different dimensionality of the system. This low\ncurrent density also justifies the adiabatic approximation\napplied in this work.\nComparison with vortex dynamics: Finally, we com-\npare the dynamics of SkX with that of the vortex lines\n(VL) in type II superconductors. Asimilarquadraticdis-\npersion was obtained for VL [22]. However, in supercon-\nductors it results from long-ranged interactions between\nvortices, while the interactions between Skyrmions are\nshort-ranged (if one ignores the relatively weak dipole-\ndipole interactions). The absence of long-range interac-\ntions in SkX ensures the stability of the quadratic disper-\nsion. Furthermore, the kinetic terms in VL and SkX are\ncompletely different. For VL uxanduyare two indepen-\ndent variables,while forSkXtheareconjugatedvariablesas in Eq.(3). Therefore VL are massive, while Skymions\nare not. When a supercurrent flows through the type II\nsuperconductor, the charged Cooper pairs are deflected\nby the VL through the Lorentz force, which in turn gives\nrisetothetransversemotionoftheVL.TheVLdynamics\nis usually assumed to be overdamped[23], the kinetic en-\nergy of VL is neglected, and the Lorentz force is assumed\nto be counterbalanced by the friction force. In contrast,\nthe damping of Skyrmions is relatively weak. The spin\ntorque resulting from the strong Hund’s rule coupling re-\nsults in a nearly longitudinal motion Skyrmion motion.\nThe Hall motion of VL results in a longitudinal voltage\ndrop, which is not important for SkX motion due to the\nsmall Hall angle.\nDuring the completion of this paper we became aware\nof a recent paper by Kim and Onoda addressed the dy-\nnamics of Skyrmions in an itinerant double-exchange fer-\nromagnet using a Chern-Simons-Maxwell approach[24].\nTheir focus, however, seems to differ from ours. We are\ngrateful for the insightful discussions with Prof. Yoshi-\nnoriTokura. ThisworkissupportedbyGrant-in-Aidsfor\nScientific Research (No. 17105002, 19019004, 19048008,\n19048015,and21244053)fromtheMinistryofEducation,\nCulture, Sports, Science and Technology of Japan, and\nalso by Funding Program for World-Leading Innovative\nR&D on Science and Technology (FIRST Program). JZ\nis supported by Fudan Research Program on Postgrad-\nuates. MM is supported by the Stichting voor Funda-\nmental Onderzoekder Materie(FOM). HJH issupported\nby Mid-career Researcher Program through NRF grant\nfunded by the MEST Grant No. R01- 2008-000-20586-0.\n[1] T. H. R. Skyrme, Proc. Roy. Soc. London A 260, 127\n(1961); Nuc. Phys. 31, 556 (1962).\n[2] I. Klebanov, Nucl. Phys. B 262, 133 (1985).\n[3] S. L. Sondhi, A. Karlhede, S. A. Kivelson, and E. H.\nRezayi, Phys. Rev. B 47, 16419 (1993).\n[4] C. Timm, S. M. Girvin, and H. A. Fertig, Phys. Rev. B\n58, 10634 (1998).\n[5] A.N. Bogdanov and D. A. Yablonskii, Sov. Phys. JETP\n68, 101-103 (1989); U.K. Rosler, A.N. Bogdanov, and C.\nPfleiderer, Nature 4 42, 797-801 (2006).\n[6] S. M¨ uhlbauer, B. Binz, F. Joinetz, C. Pfleiderer, A.\nRosch, A. Neubauer, R. Georgii, and P. B¨ oni, Science\n323, 915 (2009).\n[7] S. D. Yi, S. Onoda, N. Nagaosa, and J. H. Han, Phys.\nRev. B80, 054416 (2009); Ulrich K. Roeßler, Andrei A.\nLeonov, Alexei N. Bogdanov, arXiv:1009.4849.\n[8] X. Z. Yu et al., Nature (London) 465, 901 (2010).\n[9] A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu,\nand T. Shinjo, Phys. Rev. Lett. 92, 077205 (2004).\n[10] F. Jonietz et al., Science 330, 1648 (2010).\n[11] J. H. Han, J. Zang, Z. Yang, J. H. Park, and N. Nagaosa,\nPhys. Rev. B 82, 094429 (2010).\n[12] Ya. B. Bazaliy, B. A. Jones, and S. C. Zhang, Phys. Rev.\nB57, R3213 (1998).[13] G. Tatara, H. Kohno, J. Shibata, Phys. Rep. 468, 213\n(2008).\n[14] R. Rajaraman, Solitons and Instantons , North-Holland,\n1987.\n[15] M. Lee, W. Kang, Y. Onose, Y. Tokura, and N. P. Ong,\nPhys. Rev. Lett. 102, 186601 (2009); A. Neubauer et al.,\nPhys. Rev. Lett. 102, 186602 (2009).\n[16] S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 95,\n107204 (2005).\n[17] H. Kohno, G. Tatara, and J. Shibata, J. Phys. Soc. Jpn.\n75, 113706 (2006).\n[18] H. FukuyamaandP. A.Lee, Phys.Rev.B 17, 535(1978);\nP. A. Lee and T. M. Rice, Phys. Rev. B 19, 3970 (1979).\n[19] S. V. Grigoriev et al., Phys. Rev. B 74, 214414 (2006).\n[20] T. Kubota et al., Appl. Phys. Lett. 94, 122504 (2009).\n[21] C. Liu et al., Appl. Phys. Lett. 95, 022509 (2009).\n[22] P. G. de Gennes and J. Matricon, Rev. Mod. Phys. 36,\n45 (1964); A. L. Fetter, and P. C. Hohenberg, Phys. Rev.\n159, 330 (1967).\n[23] M. Tinkham, Introduction to Superconductivity ,\nMcGraw-Hill, 1996\n[24] K. S. Kim and S. Onoda, arXiv:1012.0631."    },    {        "title": "2201.04932v1.Enhanced_Planar_Antenna_Efficiency_Through_Magnetic_Thin_Films.pdf",        "content": "> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <  1 \n  Abstract— This work proposes to use magnetic material as the substrate of planar antennas to overcome the platform effect caused by the conducting ground plane. The upper bound of the radiation efficiency of an electric-current-driven low-profile antenna is theoretically derived, which is inversely proportional to the Gilbert damping factor of the magnetic material. Meanwhile, the improvement of radiation due to the use of magnetic material is demonstrated by a three-dimensional (3D) multiphysics and multiscale time-domain model. The simulation results match the theoretical derivation, showing 25% radiation efficiency from a planar antenna backed by a FeGaB thin film with 2.56 µm thickness. Furthermore, for conductive ferromagnetic materials, it is shown that the eddy current loss can be well suppressed by laminating the thin film into multiple layers. The radiation efficiency of the modeled antenna with a conductive ferromagnetic substrate is improved from 2.2% to 11.8% by dividing the substrate into 10 layers, with a ferromagnetic material fill factor of 93%.   Index Terms— ADI, eddy current loss, electromagnetics, antenna, FDTD, ferromagnetic resonance, lamination, magnetic thin films, numerical computation, planar structures, radiation, radiation efficiency, solver  I. INTRODUCTION caling down of circuitry has been a growing trend in modern electronics, enabling miniaturized and interconnected systems. Specifically, conformal devices with very small thicknesses are popular in applications such as wearable devices for law enforcement, military, and civilian emergency services [1], [2]. However, a major challenge that stands in the way of realizing these new technologies is scaling  Manuscript received **, 2021. The work was supported by NSF Nanosystems Engineering Research Center for Translational Applications of Nanoscale Multiferroic Systems (TANMS) Cooperative Agreement Award (No. EEC-1160504), and the Defense Advanced Research Projects Agency (DARPA) Magnetic Miniaturized and Monolithically Integrated Components (M3IC) Program under award W911NF-17-1-0100. Zhi Yao is with Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA (email: jackie_zhiyao@lbl.gov). Sidhant Tiwari is with Sandia National Laboratory, Albuquerque, NM 87123, USA. Joseph Schneider is with Lawrence Livermore National Laboratory, Livermore, CA 94550, USA. Robert N. Candler and Yuanxun Ethan Wang are with the Department of Electrical and Computer Engineering, University of California, Los Angeles, CA 90095, USA. Robert N. Candler is jointly with the California NanoSystems Institute (CNSI), Los Angeles, CA 90095, USA. Gregory P. Carman is with the Mechanical and Aerospace Engineering Department, University of California, Los Angeles, CA 90095, USA. down the antenna. Planar dimensions of traditional antennas must be on par with the electromagnetic (EM) wavelength to transmit efficiently [3]. Moreover, space-saving low-profile antennas require a ground plane for operation, causing poor radiation due to the platform effect. The platform effect is the major issue that conventional current-based antennas suffer when placed at a short distance above a conducting plane. The radiation becomes inefficient because the image current flows in the opposite direction and cancels the original current source. Therefore, antenna scaling is prevented by the excessive reactive energy stored between the radiating element and the conducting plane, thus raising the radiation quality factor (Q factor) and making the antenna difficult to match. In order to alleviate the platform effect and to allow for miniaturization, new materials and technologies need to be implemented.  One of the effective ways to increase efficiency is to replace the regular dielectric substrates with magnetodielectric materials, which provide high values of both relative permittivity 𝜖! and relative permeability 𝜇!\t. In patch antennas with such substrates, the effective EM wavelength is reduced by approximately √𝜖!𝜇! times, leading to a reduction of antenna characteristic length by approximately the same scale factor [4]. Efforts have been made on implementing both natural ferrites [5]–[7] and artificial magnetic materials, such as metamaterials [8]. When the material loss is considered, modeled with a the complex permeability 𝜇!=𝜇!\"−𝑗𝜇!\"\", the common understanding in these works is to avoid large values of 𝜇!\"\", which in turn limits the value of 𝜇!\" to be below a hundred and the operation frequency of the natural magnetic materials to be below hundreds of megahertz [9], i.e. below domain wall resonance frequencies. However, as the frequency increases to the gigahertz range, magnetic materials now exhibit ferromagnetic resonance (FMR), manifesting itself as dramatically high values of  𝜇!\"\"  resulting in large values for the magnitude of 𝜇!. Moreover, there is a causality relation between 𝜇!\" and 𝜇!\"\", meaning these two values cannot be tuned independently of each other. The ferrites/ferromagnets perform as imperfect magnetic conductors, converting the electric current image [10], [11] into one that is parallel to the original source current, enhancing the radiation from the source rather than canceling it. The strong magnetic flux existing in the antenna structure enables the antenna to switch from being electric field dominated to magnetic field dominated. Following this strategy, new types of radiating mechanisms targeting antenna miniaturization have been proposed. These Enhanced Planar Antenna Efficiency Through Magnetic Thin-Films  Zhi Yao, Member, IEEE, Sidhant Tiwari, Member, IEEE, Joseph Schneider, Member, IEEE, Robert N. Candler, Senior Member, IEEE, Gregory P. Carman, and Yuanxun Ethan Wang, Fellow, IEEE \nS > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <  2 include a strain-mediated antenna composed of composite multiferroic materials [12], [13] and mechanical antennas based on physically oscillating magnets [14], [15]. By utilizing the time-varying magnetic flux as the radiating source, these new types of antennas could potentially be immune to the conductive loss and the platform effect. The recent work on electrically small loop antenna with a ferrite substrate has experimentally demonstrated the enhancement of the antenna radiation performance by FMR of the ferrite [16]. It is worth pointing out that the multiferroic antenna and mechanical antenna both rely on high permeability magnetic material to lower the radiation quality factor, which is consistent with the approaches attempted by merely using magnetodielectric substrates [5]–[8].  The magnetic material used in magnetic antennas should possess an FMR frequency in the gigahertz range, large value of permeability, as well as low eddy current loss, i.e. low electric conductive loss. To achieve gigahertz FMR frequency and large permeability at the same time, in-plane biased magnetic thin films should be used. The reason is that, compared to that of bulk materials, the FMR frequency of an in-plane biased thin film is increased by the factor of (𝑀#/𝐻$ , where 𝑀# is the saturation magnetization and 𝐻$ is the magnetic DC bias. As the FMR frequency is increased in the in-plane biased case, lower bias magnetic fields can be used to achieve the same FMR frequencies, easing the requirements on the strength of the electromagnets used to provide the bias magnetic fields. Moreover, rather than ferrites, ferromagnetic materials are preferred. The reason is that in saturated ferromagnetic materials, all the spins are aligned to the bias direction, leading to a large spontaneous magnetization and a higher permeability near FMR. This is in contrast to the saturated ferrites, where adjacent spins are opposite to each other and different in magnitude, leading to a low saturation magnetization. Such spin orientation results in a smaller net magnetization thus lowering the magnitude of the permeability near the FMR frequency. However, even though ferromagnetic materials provide large permeability, they are typically conductive, leading to severe eddy current loss. Therefore, one also needs to resolve the dilemma between having the large magnetic permeability from ferromagnetic materials and requiring low conductive losses. It is proposed in this work to suppress the eddy current loss by segmenting the ferromagnetic film into thin layers, so that the giant eddy current loops are broken, and the conductive loss is reduced. In summary, this work proposes that by inserting in-plane biased, multi-layered, ferromagnetic thin films between the radiating source and the conducting ground plane, one can drastically improve the radiation performance, such as radiation efficiency.  In this manuscript, the elimination of the platform effect is demonstrated by studying a planar antenna. The antenna is composed of a planar electric current backed by a ferromagnetic thin film. At the bottom of the thin film, a perfect electric conducting (PEC) ground plane is assigned. The radiation efficiency of this idealized radiator is derived theoretically, based on the plane wave assumption. Furthermore, a three-dimensional (3D) finite difference time domain (FDTD) algorithm is developed [17], [18], to demonstrate the theory. The algorithm is based on the alternating directional implicit (ADI) method to achieve unconditional stability. In the model, an electric current source on top of an FeGaB substrate is used. The modeling results show that even if the substrate is only several-micrometer-thick, it can boost up the radiation power by six orders of magnitude. Additionally, the simulation results demonstrate the suppression of eddy current loss by laminating the continuous thin film into multiple layers. With laminated ferromagnetic substrate, the radiation efficiency of the current source with the ferromagnetic substrate can be improved from 2.2% to 12% by dividing the substrate into 10 layers. II. THEORY Consider an infinite, uniform current sheet 𝑖%\tthat radiates EM waves into free space, as shown in Fig. 1(a). The current source is placed over an infinite PEC ground plane, with a ferromagnet substrate inserted between them. The PEC-backed current source is the model of the antenna. The thickness of the substrate is electrically small such that 𝑘ℎ≪1, where k is the wave number in the substrate. The ferromagnetic substrate is biased to saturation by an in-plane magnetic DC field 𝐻$ that is parallel to the current source 𝑖%. EM waves are directly radiated into the free space and reflected by the PEC plane, resulting in destructive interference.  Therefore, according to the classical EM theory, the amplitude of the time-varying EM waves in the different regions are: Stored field: 1𝐸&=𝐸$sin(𝑘𝑧)𝐻'=−𝐸$/𝑗𝜂cos(𝑘𝑧),  Radiated field: 1𝐸&=𝐸$sin(𝑘ℎ)exp(−𝑗𝑘$𝑧)𝐻'=−𝐸$/𝜂$sin(𝑘ℎ)exp(−𝑗𝑘$𝑧), (1) where 𝐸$ is the aperture electric field amplitude at the interface between the free space and the substrate. The radiated power into the free space is thus calculated as 𝑃!()=12𝜂$A|𝐸|*#𝑑𝑠≈12𝜂$𝐸$*(𝑘ℎ)*𝑆 =+*,!𝐸$*ℎ*𝑆𝜔*|(𝜇\"−𝑗𝜇′′)𝜖|,    (2) Note that in Equation (2), the off-diagonal permeability terms in the Polder tensor are ignored for mathematical simplicity. In Equation (2), the approximation of a linear distribution along the z-direction for the electric field is applied since the tangential electric field on the PEC is zero. Similarly, an approximation of a uniform magnetic field distribution along the z-direction leads to the stored magnetic energy and magnetic power loss as in Equations (3) and (4), respectively.  𝑊-=+*∭𝜇′|𝐻|*./0𝑑𝑣≈+*𝜇′|𝐻|*ℎ𝑆=+*𝜇′1!\"23#$%&'$%%()\t2\"ℎ𝑆,   (3) > REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <  3 𝑃5=+*𝜔∭𝜇′′|𝐻|*./0𝑑𝑣≈+*𝜔𝜇′′1!\"23#$%&'$%%()\t2\"ℎ,    (4) \n Fig. 1.  Radiation from electric current source. The coordinate system is chosen such that the surface current is in the y direction. The thickness of the substrate satisfies the condition 𝑘ℎ≪1. (a) An infinite, uniform current sheet 𝑖!\tthat radiates EM waves into free space. The current source is grounded by an infinite perfect electrically conducting (PEC) plane, with a ferromagnet substrate inserted between them. The thickness of the substrate is electrically small such that 𝑘ℎ≪1, where k is the wave number in the substrate. The ferromagnetic substrate is biased to saturation by an in-plane magnetic DC field 𝐻\" that is aligned with the current source 𝑖!. Note that in the numerical model, the actual size of the structure is set to be 5×5×10 µm in the x, y, and z directions, respectively. The infinite planar size is realized by periodic boundary conditions applied at the four side walls. (b) The y-z cross section of (a). (c) Analytical permeability spectrum of a ferromagnetic material, FeGaB, under a DC magnetic bias of 60 Oe, with the saturation magnetization 4𝜋𝑀# being 1.2×10$\tGauss. (d) Circuit model of the structure in (a) and (b). The parallel RLC resonator represents the ferromagnetic resonance, and the shunt resistor 𝜂\" represents the intrinsic resistance of the fee space.  In ferromagnetic materials, the electric energy stored in the structure is negligible compared to the magnetic stored energy, thus the total amount of stored energy is approximately equal to the magnetic stored energy, or in the mathematical form, 𝑊67689≈𝑊-. Therefore, the total quality factor of the system is given by: 𝑄:;:(<≈𝜔=*>+,-?>.=+/0\t2!∙$%\"3$%%\"$%?$%%$%=A+\"0B!CA+%\"?A+%%\"D?A+\"\" .  (5) Note that in Equation (5), the electric power loss is neglected by assuming that the conductivity of the material is zero. Similarly, the radiation quality factor is 𝑄!()≈𝜔=*>+,-=+/0\t2!$+%\"3$+%%\"$%=A+\"0B!CA+%\"?A+%%\"D .  (6) Hence, for an antenna working around 2 GHz, 𝑄E8F is on the order of 104 when the thickness of the substrate is 1.5 µm if the material is non-magnetic, or in mathematical form, 𝜇!\"=1 and 𝜇!\"\"=0. Since traditional antennas are mostly made of conductors and rely on conductive current to radiate, the platform effect is an inevitable shortcoming of traditional low-profile antennas. The platform effect results in more energy being stored in the structure instead of being radiated away into free space, raising the antenna quality factor. On the other hand, a magnetic substrate with high relative permeability offers the capability of significantly lowering 𝑄E8F and improving the radiation performance of low-profile antennas. According to the definition, the radiation efficiency can be calculated as  𝜉!()=G454,6G+,-=++?$+%%$+%\"3$+%%\"∙78!/.     (7) In saturated ferromagnetic materials that is biased in-plane, the relative permeability is calculated by (8) [19]. 𝜇!=H9H!H+\"H+\"IH\"?JKH(*H!?H9)+1.     (8) In (8), 𝜔$ is the Larmor frequency, defined as 𝜔$=𝜇$𝛾𝐻$. Similarly, 𝜔N is defined as 𝜇$𝛾𝑀#, where 𝑀# is the saturation magnetization of the ferromagnet. The term 𝜔! stands for ferromagnetic resonance frequency, which is calculated according to Kittel’s equation as 𝜔!=𝜇$𝛾(𝐻$(𝐻$+𝑀#) for in-plane biased films. The term 𝛼 is the Gilbert damping constant of the ferromagnetic material, which is related to the FMR linewidth by the formula Δ𝐻=2𝛼𝜔/𝜇$𝛾. Substituting (8) into (7) yields [16] \n> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <  4 𝜉!()=++?KH∙0!09∙\"0!3090+\"∙78!/ ,    (7’) Note that in the derivation of Equation (9), the latter term of Equation (8) (i.e. the constant number 1) is neglected. This is a valid approximation as near FMR, the value of 𝜇!\"\"* is very large such that 𝜇!\"*+𝜇!\"\"*≈𝜒!\"*+𝜇!\"\"*.  However, as mentioned previously, ferromagnetic materials are electrically conductive. Therefore, heat dissipation will be generated by the oscillating electric field in such materials. With a conductivity 𝜎 in the ferromagnetic film, the electric power dissipation can be calculated as in 𝑃1=12U𝜎|𝐸|*./0𝑑𝑣≈12U𝜎𝐸$*|𝑘𝑧|*./0𝑑𝑣 =+O𝜎𝐸$*𝜔*V((𝜇\"−𝑗𝜇\"\")𝜖V*ℎP𝑆,   (9) Similarly, 𝜉!()=G454,6G+,-=>+,->+,-?>.?>:\t=++?$+%%$+%\"3$+%%\"78!/?;/0$!<8!  . (10) Substituting Equation (8) into Equation (10) yields: 𝜉!()=++?KH∙0!09∙\"0!3090+\"78!/?;/0$!<8! ,  (11) Equation (11) again reveals the previously mentioned conclusion that the electric conductivity of the substrate decreases the radiation performance of the structure. III. MODEL In order to accurately simulate the performance of the low-profile antenna considered in this work, an algorithm that describes both the EM wave propagation and the micromagnetic dynamics is applied [20]. Mathematically, this algorithm simultaneously solves the Maxwell’s Equations (12) and the Landau-Lifshitz-Gilbert (LLG) Equation (13) using finite-difference time-domain (FDTD) method in a coupled fashion simultaneously. 𝛻×𝑯=𝜖Q𝑬Q:+𝑱+𝜎𝑬,\t𝛻×𝑬=−Q𝑩Q: ,  (12) Q𝑴Q:=𝜇$𝛾]𝑴×𝑯UVV_+K|5|𝑴×Q𝑴Q:.   (13) In Equation (12), E represents the electric field, J represents the electric current volume density, H represents the magnetic field intensity, and B represents the magnetic flux density. The material properties are also involved in the Maxwell’s equations, where 𝜎 is the electric conductivity and 𝜖 is the relative permittivity. In Equation (13),  𝜇$ represents the vacuum permeability. 𝛾 is the gyromagnetic ratio, possessing a value of −1.759×10++𝐶/𝑘𝑔. The term 𝛼 is the Gilbert magnetic damping constant defined as 𝛼=𝜇$𝛾𝛥𝐻/4𝜋𝑓:, with 𝛥𝐻 being the FMR linewidth and 𝑓: being the frequency at which the linewidth is measured. As the governing law of micromagnetics, the LLG Equation (13) describes the evolution of magnetization 𝑴, with 𝑯XYY being the total effective magnetic field that drives the magnetic spins [21]. The low-profile antennas considered in this work consist of structures with characteristic dimensions much smaller than the EM wavelength. Conventional FDTD operates under the limit of Courant–Friedrichs–Lewy (CFL) stability condition, which incurs a tremendous amount of calculation, especially with such a small-scale structure. To overcome the stability constraint and reduce the time consumption of the simulation, alternating direction implicit (ADI) methods are used to obtain unconditional stability. The size of the entire simulation space shown in Fig. 1(a) is 5×5×10 µm in the x, y and z directions, respectively. The lower boundary, as mentioned, is set to be PEC. The ferromagnetic film is placed on the PEC, with the thickness being h = 2.56 µm, and the planar dimensions 5×5 µm. Periodic boundary conditions are applied at the four side walls to realize the infinite dimensions of the magnetic film in the planar directions. The upper surface of the space is terminated with absorbing boundary. A uniform electric current excitation is applied on the top surface of the ferromagnetic thin film. The field components are defined such that all the electric field components are along the edges of the spatial cell, and all the magnetic field components are face-centered on the cell surfaces. The spatial resolution is Δ𝑥=1 𝜇𝑚, Δ𝑦=1 𝜇𝑚, Δ𝑧=0.01 𝜇𝑚, and the time step is set as Δ𝑡=2.31´10-13 s, which is 104 times of the CFL limit. The surface current excitation center frequency is 2.4 GHz, and it is in the form of a modified Gaussian pulse, with a bandwidth of ±500 MHz. In this work, FeGaB is used due to its attractive ferromagnetic properties, e.g., low electric conductivity and high saturation magnetization. In the model, the saturation magnetization of FeGaB is 4𝜋𝑀#=12000\tGauss, the FMR linewidth is Δ𝐻=30\tOersted, and the electric conductivity is 𝜎=5×10Z\tSiemens/meter\t[22]. The magnetic DC bias applied in-plane is 60 Oersted so that the FMR frequency and the input signal frequency overlap. Fig. 2 shows the simulated results with the geometry and material properties specified in the last paragraph. To explore the effect of conductive dissipation on the radiation performance, two additional cases have been simulated:  1. Artificial nonconductive ferromagnetic substrate. The setup of this control case is identical to the one previously introduced, except that the conductivity of FeGaB is artificially set to be zero. Therefore, this case is the optimal circumstance with a nonconductive ferromagnetic material.  2. No substrate under the current. In this case, the space between the current source and the PEC ground is filled with air, which is the original platformed antenna with the current source close to the PEC.   (a) \n> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <  5 \n (b) \n Fig. 2.  Simulated radiation efficiency from the electric current source. (a) Antenna structure. The current is placed on the top surface of the ferromagnetic substrate. The size of the entire simulation space is 5×5×10 µm in the x, y and z directions, respectively. The spatial resolution is Δ𝑥=1 𝜇𝑚, Δ𝑦=1 𝜇𝑚, Δ𝑧=0.01 𝜇𝑚. (b) Simulated permeability of the FeGaB thin film, with conductivity of 5×10%\tSiemens/meter, and an artificial conductivity of zero. (c) Simulated radiation efficiency with conductive and nonconductive materials, as well as without substrate.   As can be seen in Fig. 2(b), the simulated permeability is almost independent from the electric conductivity, and it matches the analytical results. At FMR, the imaginary permeability 𝜇′′ is as large as 3000, and close to FMR the real permeability 𝜇′ is as large as 1500. Fig. 2(c) shows the comparison between the radiation efficiency of different substrate materials. Without the ferromagnetic substrate, the simulated radiation efficiency is on the order of 10-7 (as shown by the yellow curve with triangle marks), indicating that almost the entire radiated field is cancelled out by the PEC platform. Compared to the air-filled antenna, the artificial nonconductive ferromagnetic substrate improves the radiation efficiency by 106 times, leading to a radiation efficiency of 25%, shown by the black solid curve with circle marks. The red solid line represents the analytical radiation efficiency calculated by Equation (7’), showing a good match to the simulation. However, this is the ideal case with no eddy current loss. Practically, the conductive FeGaB results in the radiation efficiency being only 2.2%, as shown by the black dashed curve with circle marks in Fig. 2(c). The analytical radiation efficiency corresponding to the FeGaB material is calculated with Equation (11), and plotted as the red dashed curve in Fig. 2(c). Therefore, in order to achieve the full advantage of using a ferromagnetic substrate to improve the radiation performance, modified geometries need to be explored to suppress the eddy current loss. IV. EDDY CURRENT SUPPRESSION It is evident that magnetic materials with high relative permeability help overcome the platform effect. However, unfortunately, most of the materials that have such high permeability are ferromagnetic materials, which are highly conductive and suffer significant eddy current loss. By briefly analyzing Faraday’s law in the integral form ∮𝑬[∙𝑑𝒍=−(𝜕/𝜕𝑡)∬𝑩∙𝑑𝑨#, one can quickly conclude that the eddy current loss can be well suppressed by reducing the magnetic flux by laminating the thin film into multiple layers, as shown in the inset of Fig. 3. Note that the thickness of the laminates should be at least comparable to the skin depth, so that the eddy current loop could be broken into smaller loops and the conductive loss will be reduced. Fig. 3 shows the radiation efficiency of antennas with laminated substrates of various numbers of layers and different layer thicknesses. The gap between the PEC ground and the current source is approximately 2.56 µm for each lamination geometry. Since the planar dimensions of the antenna structure are constant, the thickness ratio represents the volume ratio of the material. The skin depth of FeGaB close to FMR is approximately 0.3 µm. Therefore, laminates with thicknesses smaller than 0.3 µm are effective for the eddy current suppression, such as the 8-layer, 10-layer and 12-layer structures, leading to the peak radiation efficiency of 9.27%, 11.8% and 10.31%, respectively. It is noticed that a dispersive radiation efficiency spectrum is formed, in contrast to the single-layer cases simulated in Fig. 2, where the radiation efficiency is constant over the frequency band under observation. Peaks in the dispersive spectrum 𝜉E8F,]8^(𝑓) are formed around 2.4 GHz, due to the FMR effect influenced by the inductive-capacitive coupling between the laminates. \n> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <  6 \n Fig. 3.  Radiation efficiency of antennas with laminated substrate with various numbers of layers and different layer thicknesses. Inset: geometry of the laminated substrate. tm: thickness of the ferromagnetic layers; tair: gap between adjacent layers. N: number of ferromagnetic layers.   Intuitively, the more layers the ferromagnetic substrate is cut into, the more effectively the eddy current loops will be broken down, and the better suppression effect will be achieved. However, different from the intuition, it is observed from Fig. 3 that a maximum radiation efficiency is achieved with N=10 (N is the number of ferromagnetic layers), instead of the maximum number N=24. This optimum number of laminate layers is attributed to the balance between the interlayer coupling effects and the ferromagnetic material volume fraction. One type of possible interlayer coupling is capacitive coupling between adjacent ferromagnetic layers through the air gap, which is elaborated in Appendix A. To be brief, the electric fields couple to each other through the air gap, leading to an equivalent continuous, giant dielectric eddy current loop, which degrades the radiation efficiency. This effect grows stronger for thinner air gaps. Another type of interlayer coupling captured by the FDTD model is coupling via dipolar magnetic fields generated by the magnetization in each layer. Therefore, competing effects exist between the interlayer coupling, the ferromagnet/air volume ratio, and the change in the eddy current suppression effectiveness as the ferromagnetic layer thickness changes. This multi-factor scenario is captured by the numerical model proposed in this work, while is too complicated to be captured by theoretical analysis. In summary, the numerical model proposed in this work serves as a tool for optimizing the antenna structure for the purpose of suppressing the eddy current loss to the largest extent.  V. CONCLUSION In this work a layered ferromagnetic material is explored as a method to increase radiation efficiency in planar electrically small antennas. Air-filled planar electrically small antennas radiate almost no power into the space due to an image current that is anti-parallel to the source current. Furthermore, the radiation quality factor (Qrad) is large due to the reactive energy associated with this antenna geometry. Inserting a ferromagnetic material between the source and ground plane enhances the radiation efficiency by reversing the direction of the image current in addition to reducing Qrad through the high relative permeability of the material. Radiation efficiency is further improved by dividing the ferromagnetic material into several layers reducing eddy current losses, thus, lowering the negative effects caused by the materials conductivity. An ADI-FDTD model is developed coupling the LLG Equation with Maxwell’s Equations to study the complicated dynamics of a planar antenna with a laminated ferromagnetic substrate. Numerical results show that inserting FeGaB into the airgap increases efficiency from lower than 0.00001% to greater than 2%. Furthermore, dividing the FeGaB into 10 layers further improves the radiation efficiency to approximately 2% to 11.8%. These results show that the radiation efficiency of planar electrically small antennas can be dramatically increased by using a ferromagnetic material inserted between the source plane and the ground plane.  ACKNOWLEDGMENT The work was supported by NSF Nanosystems Engineering Research Center for Translational Applications of Nanoscale Multiferroic Systems (TANMS) Cooperative Agreement Award (No. EEC-1160504), and the Defense Advanced Research Projects Agency (DARPA) Magnetic Miniaturized and Monolithically Integrated Components (M3IC) Program under award W911NF-17-1-0100. \n> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <  7 APPENDIX A: CIRCUIT MODELLING OF DISPLACEMENT EDDY CURRENT FLOW A. Analysis Setup As discussed in the main body of this work, a common method to reduce eddy currents in a ferromagnetic material is to laminate the material into several thin sheets separated by insulators. This has the effect of restricting the possible paths of the eddy currents, limiting the amount of current that flows and thus, limiting the power dissipated. However, as the layers get too thin, interlayer coupling can grow as the layer boundaries become closer together. One type of interlayer coupling is the capacitive coupling between the ferromagnetic laminations through the dielectric gaps. \n Fig. A1. Structure used to study the effect of displacement eddy currents on power dissipation: Two conductive ferromagnetic layers of thickness 𝑡& separated by an air gap of 𝑡'().  A RF magnetic field is applied along the width of the ferromagnetic layers.  To study the effect of the air gap thickness on power loss due to eddy currents, a simple structure of two conductive ferromagnetic layers separated by an air gap (Fig. A1) is used. By Faraday’s Law, the RF magnetic field applied to the sample will generate an electromotive force, which will cause eddy currents to circulate in the ferromagnetic layers. When the gap between the ferromagnetic layers is small, the two layers become strongly capacitively coupled and displacement currents travel between them, leading to an increase in power dissipation as new eddy current paths are created [23]–[26]. If the air gap is too thin, the eddy current suppression effects of the laminations becomes practically nonexistent.   \n Fig. A2. Equivalent circuit model used to intuitively study eddy currents generated in the structure depicted in Fig. A1.  To analyze the structure in Fig. A1, the approach of [27] is used and the structure is modelled by an equivalent circuit (Fig. A2). Here, the resistive elements 𝑅+ and 𝑅*\trepresents the power dissipation by eddy currents traveling along the length and thickness, respectively. The capacitive elements are used to account for the coupling between the layers, and the voltage sources account for the electromotive force generated by the RF magnetic field through Faraday’s Law. The values for the lumped elements and voltage sources used in the circuit model are shown in Table 1. Additional interlayer coupling effects (such as dipolar magnetic fields between the layers), dynamic magnetization effects due to ferromagnetic resonance, or incorporating a large number of layers would be difficult to capture using this approach and is a task better suited for the ADI-FDTD model discussed in the main body of this work.  Variable Formula Description 𝛿 A2/(𝜎𝜇)𝜇\"𝜔) Skin depth in ferromagnetic layer 𝑑* \t𝛿⋅{1−exp[−𝑡&/(2𝛿)\t]} Effective depth of eddy currents traveling along the length of the structure 𝑑+ 𝛿⋅{1−exp[−𝑙/(2𝛿)\t]} Effective depth of eddy currents traveling along the thickness of the structure 𝐶 𝜖\"𝑙𝑤/(2𝑡+) Capacitance coupling the eddy currents between the two ferromagnetic layers 𝑅* 𝑙/(𝜎𝑤𝑑*) Approximate resistance seen by eddy currents traveling along the length of the structure 𝑅+ 𝑡*/(2𝜎𝑤𝑑+) Approximate resistance seen by eddy currents traveling along the thickness of the structure 𝑍, 1/(𝑗𝜔∙𝐶) Impedance due to capacitive coupling of eddy currents 𝑉* 𝜔∙𝜇)𝜇\"∙𝑙∙(𝑡*/2)∙𝐻-. Electromotive force acting on eddy current loops contained solely in the ferromagnetic layer 𝑉+ 𝑉*+(𝜔∙𝜇\"∙𝑙∙𝑡+∙𝐻-.) Electromotive force acting on eddy current loops crossing between the adjacent two layers  Table 1: Formulas for variables used to calculate equivalent circuit parameters. Note that in the formulas for 𝑅* and 𝑅+, crowding of the current due to the skin effect is considered through the parameters 𝑑* and 𝑑+ [27].   B. Determining Unknown Currents  There are six unknown currents in the circuit shown in Fig. A2 that must be solved for to calculate the total power dissipated by the eddy currents. Using Kirchhoff’s voltage and current laws, six equations can be derived to solve for the six \n> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <  8 unknowns. 𝑉+−2𝐼+𝑅*−(𝐼++𝐼P)𝑅+=0 ,   (A1a) 𝑉*−(𝐼*+𝐼_)∙(𝑅*+𝑍`)+(𝐼P+𝐼Z)𝑅+=0,   (A1b) 𝑉+−2𝐼O𝑅*−(𝐼O+𝐼Z)𝑅+=0,   (A1c) 𝐼+−𝐼P=𝐼* ,   (A1d) 𝐼*+𝐼Z=𝐼O ,   (A1e) 𝐼_+𝐼P=𝐼+ ,   (A1f) By adding Equations (A1d) and (A1f), we can solve for 𝐼_.  (𝐼+−𝐼P)+(𝐼_+𝐼P)=𝐼*+𝐼+   𝐼_=𝐼* (A2a) Using Equation (A2a), Equations (A1e) and (A1f) can be subtracted from each other to get the following relation.  (𝐼*+𝐼Z)−(𝐼*+𝐼P)=𝐼O−𝐼+   𝐼Z−𝐼P=𝐼O−𝐼+ (A2b) Rewriting Equations (A1a) and (A1c), solutions for 𝐼P and 𝐼Z can be found.  𝐼P=𝑉+𝑅+−1+2𝑅*𝑅+𝐼+ (A2c)  𝐼Z=𝑉+𝑅+−1+2𝑅*𝑅+𝐼O (A2d) Combining Equations (A2b), (A2c), and (A2d), 𝐼+ and 𝐼O can be solved for.  −1+2𝑅*𝑅+𝐼O+1+2𝑅*𝑅+𝐼+=𝐼O−𝐼+   1+2𝑅*𝑅+(𝐼O−𝐼+)=𝐼O−𝐼+ (A2e) The only way Equation (A2e) can be true for any value of a\"a7 is if 𝐼O−𝐼+ is zero.  Combining this with Equation (A2b), gives the following two equations for 𝐼Z and 𝐼O.  𝐼Z=𝐼P (A2f)  𝐼O=𝐼+ (A2g) With Equations (A2a), (A2f), and (A2g) in mind, the currents in Fig. A2 can be relabeled to reduce the number of unknowns down to three. \n Fig. A3. Equivalent circuit model shown in Fig. A2, but with Equations (A2a), (A2f), and (A2g) applied to reduce the number of unknowns.  Rewriting Equations (A1a) and (A1b), gives the following two equations for the remaining three currents.  𝑉+𝑅+−1+2𝑅*𝑅+𝐼+−𝐼P=0 (A3a)  𝑉*2𝑅+−𝑅*𝑅++𝑍[𝑅+𝐼*+𝐼P=0 (A3b) Combining Equation (A1d), (A3a), and (A3b), 𝐼P can be eliminated and two equations for 𝐼+ and 𝐼* can be found.  𝑉+𝑅+−1+2𝑅*𝑅+𝐼+−𝐼++𝐼*=0   𝑉+𝑅+−21+𝑅*𝑅+𝐼++𝐼*=0 (A3c)  𝑉*2𝑅+−𝑅*𝑅++𝑍[𝑅+𝐼*+𝐼+−𝐼*=0   𝑉*2𝑅+−1+𝑅*𝑅++𝑍[𝑅+𝐼*+𝐼+=0 (A3d) Using Equations (A1d), (A3c), and (A3d), 𝐼+, 𝐼*, and 𝐼P can readily be solved.   𝐷=21+𝑅*𝑅+1+𝑅*𝑅++𝑍`𝑅+−1 (A4a)  𝐼+=𝑉+𝑅+1+𝑅*𝑅++𝑍`𝑅++𝑉*2𝑅+𝐷\t (A4b)  𝐼*=𝑉+𝑅++1+𝑅*𝑅+𝑉*𝑅+𝐷\t (A4c)  𝐼P=𝑉+𝑅+𝑅*𝑅++𝑍`𝑅+−𝑉*𝑅+12+𝑅*𝑅+𝐷 (A4d)  C. Perfect Insulation Limit Assuming 𝑅+≫𝑅*, in the limit of a perfectly insulating air gap (𝑍`→∞), these equations reduce down to the following equations.  𝐼+=𝑉+2𝑅+\t (A5a)  𝐼*=0\t (A5b)  𝐼P=𝑉+2𝑅+ (A5c) In this limit, there is no coupling between the layers and the entirety of the eddy currents are contained within the individual ferromagnetic layers (Fig. A4a). Power dissipation in this scenario will be the minimum possible for the structure shown in Fig. A1. D. No Air Gap Limit Again assuming 𝑅+≫𝑅*,\tin the limit where the air gap goes to zero (𝑡(b!→0, 𝑉*→𝑉+, 𝑍`→0), Equations (A4b), (A4c), and (A4d) reduce down to the following equations.  𝐼+=32𝑉+𝑅+\t (A6a) \n> REPLACE THIS LINE WITH YOUR PAPER IDENTIFICATION NUMBER (DOUBLE-CLICK HERE TO EDIT) <  9  𝐼*=2𝑉+𝑅+\t (A6b)  𝐼P=−𝑉+2𝑅+ (A6c) In this limit, 𝐼P now reverses direction as additional eddy currents are allowed to flow throughout the structure since the two ferromagnetic layers are now shorted together (Fig. A4(b)). Power dissipation in this scenario is now the maximum possible for the structure shown in Fig. A1. \n  (a) (b) Fig. A4. (a) Current flow in the perfect insulation limit. (b) Current flow in the no air gap limit. E. Power Dissipation vs Air Gap Thickness The total power dissipated due to eddy currents is simply the sum of the power dissipated in each of the resistors in the equivalent circuit. Note that because of the reactive contribution from capacitive coupling between the layers, the currents found through Equations (A4b), (A4c), and (A4d) are generally complex.  𝑃)bccbd(:U)=2×12𝐼+𝐼+∗∙(𝑅++2𝑅*)+12𝐼*𝐼*∗∙𝑅*+12𝐼P𝐼P∗∙𝑅+ (A7) Equation (A7) is a strong function of 𝑡(b!, as it not only tunes the capacitive coupling between the layers (𝑍`), but also the electromotive force felt by the central current loop (𝑉*). For small thickness, the power dissipation will increase as it approaches the limit of completely shorted ferromagnetic layers. As the thickness increases, the impedance of the air gap also increases and eventually approaches the limit of the perfectly insulating air gap. Shown below is Equation (A7) plotted as a function of air gap thickness. Here, the structure has in-plane dimensions of 300 μm × 300 μm and the two ferromagnetic layers are 0.24\tµm thick. The ferromagnetic material is FeGaB at ferromagnetic resonance (2.4 GHz for a DC bias of 60 Oe). The plot is normalized to the power dissipated in the no air gap limit.  Fig. A5. Power dissipation for the structure in Fig. A1 as a function of air gap thickness, where the ferromagnetic material is FeGaB at ferromagnetic resonance. REFERENCES [1] M. Orefice, P. Pirinoli, and G. Dassano, “Electrically-small wearable antennas for emergency services applications,” in 2016 International Workshop on Antenna Technology (iWAT), 2016, pp. 131–134. [2] P. Nepa and H. Rogier, “Wearable Antennas for Off-Body Radio Links at VHF and UHF Bands: Challenges, the state of the art, and future trends below 1 GHz.,” IEEE antennas Propag. Mag., vol. 57, no. 5, pp. 30–52, 2015. [3] J. C. E. Sten, A. Hujanen, and P. K. Koivisto, “Quality factor of an electrically small antenna radiating close to a conducting plane,” Antennas Propagation, IEEE Trans., vol. 49, no. 5, pp. 829–837, 2001. [4] P. M. T. Ikonen, K. N. Rozanov, A. V Osipov, P. Alitalo, and S. A. 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"    },    {        "title": "1806.04881v1.Low_magnetic_damping_of_ferrimagnetic_GdFeCo_alloys.pdf",        "content": "1 \n Low magnetic damping of ferrimagnetic GdFeCo alloys \nDuck-Ho Kim1†*, Takaya Okuno1†, Se Kwon Kim2, Se-Hyeok Oh3, Tomoe Nishimura1, \nYuushou Hirata1, Yasuhiro Futakawa4, Hiroki Yoshikawa4, Arata Tsukamoto4, Yaroslav \nTserkovnyak2, Yoichi Shiota1, Takahiro Moriyama1, Kab-Jin Kim5, Kyung-Jin Lee3,6,7, and \nTeruo Ono1,8* \n1Institute for Chemical Research, Kyoto University, Uji, Kyoto 6 11-0011, Japan \n2Department of Physics and Astronomy, University of California, Los Angeles, California \n90095, USA \n3Department of Nano-Semiconductor and Engineering, Korea Univers ity, Seoul 02841, \nRepublic of Korea \n4College of Science and Technology, Nihon University, Funabashi,  Chiba 274-8501, Japan \n5Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon \n34141, Republic of Korea \n6Department of Materials Science & Engineering, Korea University , Seoul 02841, Republic \nof Korea \n7KU-KIST Graduate School of Converging Science and Technology, K orea University, Seoul \n02841, Republic of Korea \n8Center for Spintronics Research Network (CSRN), Graduate School  of Engineering Science, \nOsaka University, Osaka 560-8531, Japan \n \n† These authors contributed equally to this work. \n* E-mail: kim.duckho.23z@st.kyoto-u.ac.jp , ono@scl.kyoto-u.ac.jp  2 \n We investigate the Gilbert damping parameter  for rare earth (RE)–\ntransition metal (TM) ferrimagnets over a wide temperature rang e. Extracted from the \nfield-driven magnetic domain-wall mobility,  was as low as 7.2 × 10-3 and was almost \nconstant across the angular momentum compensation temperature 𝑻𝐀, starkly \ncontrasting previous predictions that  should diverge at 𝑻𝐀 due to vanishing total \nangular momentum. Thus, magnetic damping of RE-TM ferrimagnets is not related to \nthe total angular momentum but is dominated by electron scatter ing at the Fermi level \nwhere the TM has a dominant damping role.  \n  3 \n Magnetic damping, commonly described by the Gilbert damping par ameter, \nrepresents the magnetization relaxation phenomenon, describing how quickly magnetization \nspins reach equilibrium [1–3]. Understanding the fundamental or igin of the damping as well \nas searching for low damping materials has been a central theme  of magnetism research. \nSeveral theoretical models for magnetic damping have been propo sed [4–11] and compared \nwith experiments [12–20]. Ultra-low damping was predicted in fe rromagnetic alloys using a \nlinear response damping model [11] and was demonstrated experim entally for CoFe alloys \n[20]. However, the majority of these studies have focused only on ferromagnetic systems. \nAntiferromagnets, which have alt ernating orientations of their neighboring magnetic \nmoments, have recently received considerable attention because of their potential importance \nfor spintronic applications [21– 30]. Antiferromagnetic spin sys tems can have much faster \nspin dynamics than their ferromagnetic counterparts, which is a dvantageous in spintronic \napplications [21, 25, 31–39]. However, the manipulation and con trol of antiferromagnets is \nchallenging because the net magnetic moment is effectively zero . Recently, antiferromagnetic \nspin dynamics have been successfully demonstrated using the mag netic domain-wall (DW) \ndynamics in ferrimagnets with finite magnetization in the vicin ity of the angular momentum \ncompensation temperature, at which the net angular momentum van ishes [38]. This field-\ndriven antiferromagnetic spin dyn amics is possible because the time evolution of the \nmagnetization is governed by the commutation relation of the an gular momentum rather than \nthe commutation relation of the magnetic moment.  \nMotivated by the aforementioned result, in this letter, we inve stigate the magnetic \ndamping of ferrimagnets across th e angular momentum compensatio n temperature, which \nwill allow us to understand magnetic damping in antiferromagnet ically coupled system. We 4 \n selected rare earth (RE)–transition metal (TM) ferrimagnets for  the material platforms \nbecause they have an angular momentum compensation temperature 𝑇 w h e r e  \nantiferromagnetic spin dynamics are achieved [38, 40, 41]. The magnetic-field-driven DW \nmotion was explored over a wide range of temperatures including  𝑇, and the Gilbert \ndamping parameter was extracted from the measured DW mobility a t each temperature by \nemploying the collective coordina te model initially developed f or ferrimagnetic spin \ndynamics [38]. Contrary to the previous prediction that the Gil bert damping parameter would \ndiverge at 𝑇 due to the vanishing of the total angular momentum [42, 43], w e found that the \nGilbert damping parameter remained nearly constant over a wide range of temperatures \nacross 𝑇 with the estimated value as low as 7.2 × 10-3, which was similar to the reported \nvalues of TM-only ferromagnets [20]. These results suggested th at Gilbert damping was \nmainly governed by electron scattering at the Fermi level, and hence, the 4f electron of the \nR E  e l e m e n t ,  w h i c h  l i e s  f a r  b e l o w  t h e  F e r m i  l e v e l ,  d i d  n o t  p l a y  an important role in the \nmagnetic damping of RE–TM ferrimagnets. \nFor this study, we prepared perpendicularly magnetized ferrimag netic GdFeCo films \nin which the Gd and FeCo moments were coupled antiferromagnetic ally. Specifically, the \nfilms were 5-nm SiN/30-nm Gd 23.5Fe66.9Co9.6/100-nm SiN on an intrinsic Si substrate. The \nGdFeCo films were then patterned  into 5-µm-wide and 500-µm-long  microwires with a Hall \ncross structure using electron beam lithography and Ar ion mill ing. For current injection, \n100-nm Au/5-nm Ti electrodes were stacked on the wire. A Hall b ar was designed to detect \nthe DW velocity via the anomalous Hall effect (AHE). \nWe measured the magnetic DW motion using a real-time DW detecti on technique [38, \n40, 41, 44, 45] [see Fig. 1(a) for a schematic]. We first appli ed a magnetic field of –200 mT 5 \n to saturate the magnetization al ong the –z direction. Subsequen tly, a constant perpendicular \nmagnetic field 𝜇𝐻, which was lower than the coercive field, was applied along +z  direction. \nNext, a d.c. current was applied along the wire to measure the anomalous Hall voltage. Then, \na current pulse (12 V , 100 ns) was injected through the writing  line to nucleate the DW in the \nwire. The created DW was moved along the wire and passed throug h the Hall bar because of \nthe presence of 𝜇𝐻. The DW arrival time was detected by monitoring the change in the Hall \nvoltage using a real-time oscillo scope. The DW velocity could t hen be calculated from the \narrival time and the travel dis tance between the writing line a nd Hall bar (500 µm). \nFigure 1(b) shows the averaged DW velocity 〈𝑣〉 as a function of the perpendicular \nmagnetic field 𝜇𝐻 for several temperatures 𝑇∗. Here, we used the d.c. current density of \n|𝐽|ൌ1.3×1010 A / m2 to measure the AHE change due to DW motion. Note that 𝑇∗ i s  a n  \nelevated temperature that considers Joule heating by d.c. curre nt [46]. To eliminate the \nundesired current-induced spin-transfer-torque effect, we avera ged the DW velocity for 𝐽 \nand –𝐽, i.e., 〈𝑣〉ൌሾ𝑣ሺ𝐽ሻ𝑣ሺെ𝐽ሻሿ/2. Figure 1(b) shows that 〈𝑣〉 increases linearly with \n𝜇𝐻 for all 𝑇∗. Such linear behavior can be described by 〈𝑣〉ൌ𝜇ሾ𝜇𝐻െ𝜇 𝐻ሿ, where 𝜇 \nis the DW mobility and 𝜇𝐻 is the correction field, which generally arises from \nimperfections in the sample or complexities of the internal DW structure [47, 48]. We note \nthat 𝜇𝐻 can also depend on the temperature dependence of the magnetic properties of \nferrimagnets [45]. Figure 1(c) shows 𝜇 as a function of 𝑇∗ at several current densities \n(|𝐽|ൌ1.3, 1.7, and 2.0 ×1010 A / m2). A sharp peak clearly occurs for 𝜇 a t  𝑇∗ൌ241.5 K \nirrespective of |𝐽|. The drastic increase of 𝜇 is evidence of antiferromagnetic spin dynamics \nat 𝑇, as demonstrated in our pre vious report [38, 40, 41]. \nThe obtained DW mobility was theoretically analyzed as follows.  The DW velocity 6 \n of ferrimagnets in the precessional regime is given by [38, 39]  \n          𝑉 ൌ 𝜆𝛼ሺ𝑠ଵ𝑠 ଶሻሺ𝑀ଵെ𝑀 ଶሻ\nሾ𝛼ሺ𝑠ଵ𝑠 ଶሻሿଶሺ𝑠ଵെ𝑠 ଶሻଶ𝜇𝐻,                                                                                ሺ1ሻ  \nwhere 𝑉 is the DW velocity, 𝜆 is the DW width, 𝜇𝐻 is the perpendicular magnetic field, \n𝛼 is the Gilbert damping parameter, 𝑀 and 𝑠 are the magnetization and the spin angular \nmomentum of one sublattice, respectively. The spin angular mome ntum densities are given \nby 𝑠ൌ𝑀 /𝛾 [49], where 𝛾ൌ𝑔 𝜇/ℏ is the gyromagnetic ratio of lattice 𝑖, 𝑔 i s  t h e  \nLandé g factor of lattice 𝑖, 𝜇 is the Bohr magneton, and ℏ is the reduced Plank’s constant. \nThe Gilbert damping is in principle different for two sublattic e s ,  b u t  f o r  s i m p l i c i t y ,  w e  \nassume that it is the same, which can be considered as the aver age value of the damping \nparameters for the two sublattices weighted by the spin angular  momentum density. We note \nthat this assumption does not alter our main conclusion: low da mping and its insensitivity to \nthe temperature. Equation (1) gives the DW mobility 𝜇 a s  𝜆𝛼ሺ𝑠ଵ𝑠 ଶሻሺ𝑀ଵെ𝑀 ଶሻ/\nሼሾ𝛼ሺ𝑠ଵ𝑠 ଶሻሿଶሺ𝑠ଵെ𝑠 ଶሻଶሽ, which can be rearranged as \n          𝜇 ሺ𝑠ଵ𝑠 ଶሻଶ𝛼ଶെ𝜆ሺ𝑠ଵ𝑠 ଶሻሺ𝑀ଵെ𝑀 ଶሻ𝛼𝜇 ሺ𝑠ଵെ𝑠 ଶሻଶൌ 0                                           ሺ2ሻ  \nUsing Eq. (2) to find the solution of 𝛼, we find \n          𝛼 േൌ𝜆ሺ𝑀ଵെ𝑀 ଶሻേඥሾ𝜆ଶሺ𝑀ଵെ𝑀 ଶሻଶെ4 𝜇ଶሺ𝑠ଵെ𝑠 ଶሻଶሿ\n2𝜇ሺ𝑠ଵ𝑠 ଶሻ.                                              ሺ3ሻ  \nEquation (3) allows us to estimate 𝛼 for the given 𝜇. We note that for each value of 𝜇, 𝛼 \nca n h av e t w o v a lu e s,  𝛼ା and 𝛼ି because of the quadratic nature of Eq. (2). Only one of \nthese two solutions is physically sound, which can be obtained using the following energy \ndissipation analysis.  7 \n  The energy dissipation (per unit cross section) through the DW  dynamics is given by \n𝑃ൌ2 𝛼 ሺ 𝑠 ଵ𝑠 ଶሻ𝑉ଶ/𝜆  2𝛼ሺ𝑠 ଵ𝑠 ଶሻ 𝜆Ωଶ [38, 39], where Ω is the angular velocity of the \nDW. The first and the second terms represent the energy dissipa tion through the translational \nand angular motion of the DW, respectively. In the precessional  regime, the angular velocity \nis proportional to the translational velocity: Ωൌ ሺ𝑠ଵെ𝑠 ଶሻ𝑉/𝛼ሺ𝑠 ଵ𝑠 ଶሻ𝜆. Replacing Ω b y  \nthe previous expression yields 𝑃ൌ𝜂 𝑉ଶ w h e r e  𝜂ൌ2 ሺ 𝑀 ଵെ𝑀 ଶሻ/𝜇 is the viscous \ncoefficient for the DW motion:  \n          𝜂 ൌ2\n𝜆ቊ𝛼ሺ𝑠ଵ𝑠 ଶሻ ሺ𝑠ଵെ𝑠 ଶሻଶ\n𝛼ሺ𝑠ଵ𝑠 ଶሻቋ .                                                                                         ሺ4ሻ  \nThe first and the second terms in parenthesis capture the contr ibutions to the energy \ndissipation from the translational and angular dynamics of the DW, respectively. The two \nsolutions for the Gilbert damping parameter,  𝛼ା and 𝛼ି, can yield the same viscous \ncoefficient 𝜂. The case of the equal solutions,  𝛼ାൌ𝛼 ି, corresponds to the situation when \nthe two contributions are identical:  𝛼േൌሺ 𝑠 ଵെ𝑠 ଶሻ/ሺ𝑠ଵ𝑠 ଶሻ. For the larger solution 𝛼ൌ\n𝛼ା, the energy dissipation is dominated by the first term, i.e., through the translational DW \nmotion, which should be the case in the vicinity of 𝑇 where the net spin density ሺ𝑠ଵെ𝑠 ଶሻ \nis small and thus the angular velocity is negligible. For examp le, at exact 𝑇, the larger \nsolution 𝛼ା is the only possible solution because the smaller solution is zero, 𝛼ିൌ0, and \nthus unphysical. For the smaller solution 𝛼ൌ𝛼 ି, the dissipation is dominated by the second \nterm, i.e., through the precessional motion, which should descr ibe cases away from 𝑇. \nTherefore, in the subsequent analysis, we chose the larger solu tion  𝛼ା in the vicinity of 𝑇 \nand the smaller solution 𝛼ି far away from 𝑇 and connected the solution continuously in \nbetween. 8 \n The other material parameters such as 𝑀ଵ, 𝑀ଶ, 𝑠ଵ, and 𝑠ଶ a r e  e s t i m a t e d  b y  \nmeasuring the net magnetic moment of GdFeCo film, |𝑀୬ୣ୲|, for various temperatures. \nBecause 𝑀୬ୣ୲ includes contributions from both  the Gd and FeCo sub-moments, the sub-\nmagnetic moments, 𝑀ଵ a n d  𝑀ଶ, could be decoupled based on the power law criticality [see \ndetails in refs. 38, 40]. The spin angular momentums, 𝑠ଵ and 𝑠ଶ, were calculated using the \nknown Landé g factor of FeCo and Gd (the Landé g factor of FeCo  is 2.2 and that of Gd is \n2.0) [50–52]. \nFigures 2(a)–(c) show the temperature-dependent DW mobility 𝜇, sub-magnetic \nmoment 𝑀, and sub-angular momentum  𝑠, respectively. Here, we used the relative \ntemperature defined as ∆𝑇 ൌ 𝑇∗െ𝑇  to investigate the Gilbert damping near 𝑇. The \nGilbert damping parameter 𝛼 was obtained based on Eq. (3) and the information in Fig. \n2(a)–(c). Figure 2(d) shows the resulting values of 𝛼േ as a function of ∆𝑇. For ∆𝑇ଵ൏\n∆𝑇 ൏ ∆𝑇 ଶ, 𝛼ା is nearly constant, while 𝛼ି varies significantly. For ∆𝑇 ൏ ∆𝑇 ଵ and ∆𝑇 \n∆𝑇ଶ, on the other hand,  𝛼ି is almost constant, while 𝛼ା varies significantly. At ∆𝑇 ൌ ∆𝑇 ଵ \nand ∆𝑇 ൌ ∆𝑇 ଶ, the two solutions are equal, corresponding to the aforementio ned case when \nthe energy dissipation through the translational and angular mo tion of the DW are identical. \nThe proper damping solution can be selected by following the gu ideline obtained \nfrom the above analysis. For ∆𝑇ଵ൏∆ 𝑇൏∆ 𝑇 ଶ, which includes 𝑇, the energy dissipation \nshould be dominated by the translational motion, and thus 𝛼ା is a physical solution. Note \nalso that 𝛼ି becomes zero at 𝑇, which results in infinite DW mobility in contradiction with \nthe experimental observation. For ∆𝑇 ൏ ∆𝑇 ଵ and ∆𝑇  ∆𝑇 ଶ, where the energy dissipation is \ndominated by the angular motion of the DW, 𝛼ି is the physical solution. 9 \n Figure 3 shows the resultant Gil bert damping parameter in all t ested temperature \nranges. The Gilbert damping parameter was almost constant acros s 𝑇 with 𝛼ൌ7.2 × 10-3 \n(see the dotted line in Fig. 3). This result is in stark contra st to the previous prediction. In ref. \n[42], Stanciu et al. investigated the temperature dependence of the effective Gilb ert damping \nparameter based on a ferromagnet-based model and found that the  damping diverged at 𝑇. \nBecause they analyzed the magnetic resonance in ferrimagnetic m aterials based on a \nferromagnet-based model, which cannot describe the antiferromag netic dynamics at 𝑇 a t  \nwhich the angular momentum vanis hes, it exhibits unphysical res ults. However, our \ntheoretical analysis for field-driven ferromagnetic DW motion b ased on the collective \ncoordinate approach can properly describe both the antiferromag netic dynamics in the \nvicinity of 𝑇 and the ferromagnetic dynamics away from 𝑇 [38]. Therefore, the \nunphysical divergence of the Gilbert damping parameter at 𝑇 is absent in our analysis. \nOur results, namely the insensitivity of damping to the compens ation condition and \nits low value, have important implications not only for fundame ntal physics but also for \ntechnological applications. From the viewpoint of fundamental p hysics, nearly constant \ndamping across 𝑇 indicates that the damping is almost independent of the total angular \nmomentum and is mostly determined by electron spin scattering n ear the Fermi level. \nSpecifically, our results suggest that the 4f electrons of RE e lements, which lie in a band far \nbelow the Fermi level, do not play an important role in the mag netic damping of RE-TM \nferrimagnets, whereas the 3d and 4s bands of TM elements have a  governing role in magnetic \ndamping. This result is consistent with the recently reported t heoretical and experimental \nresults in FeCo alloys [20]. From the viewpoint of practical ap plication, we note that the \nestimated damping of 𝛼ൌ7.2 × 10-3 is the upper limit, as the damping estimated from DW 10 \n dynamics is usually overestimated due to disorders [53]. The ob tained value of the Gilbert \ndamping parameter is consistent with our preliminary ferromagne t i c  r e s o n a n c e  ( F M R )  \nmeasurements. The experimental results from FMR measurements an d the corresponding \ntheoretical analysis will be publ ished elsewhere. This low valu e of the Gilbert damping \nparameter suggests that ferrimagne ts can serve as versatile pla t f o r m s  f o r  l o w - d i s s i p a t i o n  \nhigh-speed magnetic devices such as spin-transfer-torque magnet ic random-access memory \nand terahertz magnetic oscillators. \nIn conclusion, we investigated the field-driven magnetic DW mot ion in ferrimagnetic \nG d F e C o  a l l o y s  o v e r  a  w i d e  r a n g e  o f  t e m p e r a t u r e s  a c r o s s  𝑇 and extracted the Gilbert \ndamping parameter from the DW mobility. The estimated Gilbert d amping parameter was as \nlow as 7.2 × 10-3 and almost constant over the temperature range including 𝑇, which is in \nstark contrast to the previous prediction in that the Gilbert d amping parameter would diverge \nat 𝑇 due to the vanishing total angular momentum. 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Shinjo, Science 284, 468 \n(1999). \n[49] In this Letter , the parameters such as the spin angular mo mentum density 𝑠 r e p r e s e n t  \nthe magnitudes of the quantities.  Their directions are separate ly handled through the signs in \nthe equations of motion. \n[50] C. Kittel, Phys. Rev. 76, 743 (1949). \n[51] G. G. Scott, Rev. Mod. Phys. 34, 102 (1962). \n[52] B. I. Min and Y.-R. Jang, J. Phys. Condens. Matter 3, 5131 (1991). \n[53] H. Min, R. D. McMichael, M. J. Donahue, J. Miltat, and M. D. Stiles, Phys. Rev. Lett. \n104, 217201 (2010). \n   16 \n Figure Captions \nFigure 1(a) Schematic illustration of the GdFeCo microwire devi ce. (b) The averaged DW \nvelocity 〈𝑣〉 as a function of the perpendicular magnetic field 𝜇𝐻 for several temperatures \n𝑇∗ (202, 222, 242, 262, and 282 K). The dots indicate the best li n e a r  f i t s .  ( c )  T h e  D W  \nmobility 𝜇 as a function of 𝑇∗ at several current densities ( |𝐽|ൌ1.3, 1.7, and 2.0 ×1010 \nA/m2). \nFigure 2 The temperature-dependent (a) DW mobility 𝜇, (b) sub-magnetic moment 𝑀, and \n(c) sub-angular momentum  𝑠. Here, we use the relative temperature defined as ∆𝑇 ൌ 𝑇∗െ\n𝑇. (d) The Gilbert damping parameter 𝛼േ as a function of ∆𝑇. Here, we use 𝜆ൌ15 nm for \nproper solutions of Eq. (3). \nFigure 3 The resultant Gil bert damping parameter 𝛼 in all tested temperature ranges. \n 17 \n Acknowledgements  \nThis work was supported by the JSPS KAKENHI (Grant Numbers 15H0 5702, 26103002, and \n26103004), Collaborative Research Program of the Institute for Chemical Research, Kyoto \nUniversity, and R & D project for ICT Key Technology of MEXT fr om the Japan Society for \nthe Promotion of Science (JSPS). This work was partly supported  by The Cooperative \nResearch Project Program of the Research Institute of Electrica l Communication, Tohoku \nUniversity. D.H.K. was supported as an Overseas Researcher unde r the Postdoctoral \nFellowship of JSPS (Grant Number P16314). S.H.O. and K.J.L. wer e supported by the \nNational Research Foundation of Korea (NRF-2015M3D1A1070465, 20 17R1A2B2006119) \nand the KIST Institutional Program (Project No. 2V05750). S.K.K . was supported by the \nArmy Research Office under Contract No. W911NF-14-1-0016. K.J.K . was supported by the \nNational Research Foundation of Korea (NRF) grant funded by the  Korea Government \n(MSIP) (No. 2017R1C1B2009686).  \nCompeting financial interests \nThe authors declare no competing financial interests. 200 225 250 275 3000.00.51.01.52.0\n 1.3\n1.7\n2.0\n [104 m/sT]\nT* [K]J [1010 A/m2]0 50 100 1500.00.51.01.5\n  202\n 222\n 242\n 262\n 282<v> [km/s]\n0H [mT]T* [K]\nFigure 1b\nca\nWriting line\n\tܫ\nܸ\nߤܪ\ny xz-60 -40 -20 0 20 40 60 801.52.02.53.0 s1\n s2s [10-6 Js/m3]\nT [K]-60 -40 -20 0 20 40 60 800.00.51.01.52.0\n [104 m/sT]\nT [K]\n-60 -40 -20 0 20 40 60 8010-610-510-410-310-210-1100\n \n +\n -\nT [K]T1T2-60 -40 -20 0 20 40 60 800.30.40.50.6  M1\n M2M [MA/m]\nT [K]a\nb\nc\nd\nFigure 2Figure 3-60 -40 -20 0 20 40 60 8010-610-510-410-310-210-1100\n \nT [K]"    },    {        "title": "1407.0635v1.Spin_Waves_in_Ferromagnetic_Insulators_Coupled_via_a_Normal_Metal.pdf",        "content": "Spin Waves in Ferromagnetic Insulators Coupled via a Normal Metal\nHans Skarsv\u0017 ag,\u0003Andr\u0013 e Kapelrud, and Arne Brataas\nDepartment of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\n(Dated: May 27, 2022)\nHerein, we study the spin-wave dispersion and dissipation in a ferromagnetic insulator{normal\nmetal{ferromagnetic insulator system. Long-range dynamic coupling because of spin pumping and\nspin transfer lead to collective magnetic excitations in the two thin-\flm ferromagnets. In addition,\nthe dynamic dipolar \feld contributes to the interlayer coupling. By solving the Landau-Lifshitz-\nGilbert-Slonczewski equation for macrospin excitations and the exchange-dipole volume as well as\nsurface spin waves, we compute the e\u000bect of the dynamic coupling on the resonance frequencies and\nlinewidths of the various modes. The long-wavelength modes may couple acoustically or optically.\nIn the absence of spin-memory loss in the normal metal, the spin-pumping-induced Gilbert damp-\ning enhancement of the acoustic mode vanishes, whereas the optical mode acquires a signi\fcant\nGilbert damping enhancement, comparable to that of a system attached to a perfect spin sink. The\ndynamic coupling is reduced for short-wavelength spin waves, and there is no synchronization. For\nintermediate wavelengths, the coupling can be increased by the dipolar \feld such that the modes\nin the two ferromagnetic insulators can couple despite possible small frequency asymmetries. The\nsurface waves induced by an easy-axis surface anisotropy exhibit much greater Gilbert damping\nenhancement. These modes also may acoustically or optically couple, but they are una\u000bected by\nthickness asymmetries.\nPACS numbers: 76.50.+g,75.30.Ds,75.70.-i,75.76.+j\nI. INTRODUCTION\nThe dynamic magnetic properties of thin-\flm fer-\nromagnets have been extensively studied for several\ndecades.1,2Thin-\flm ferromagnets exhibit a rich vari-\nety of spin-wave modes because of the intricate inter-\nplay among the exchange and dipole interactions and the\nmaterial anisotropies. In ferromagnetic insulators (FIs),\nthese modes are especially visible; the absence of disturb-\ning electric currents leads to a clear separation of the\nmagnetic behavior. Furthermore, the dissipation rates\nin insulators are orders of magnitude lower than those\nin their metallic counterparts; these low dissipation rates\nenable superior control of travelling spin waves and facil-\nitate the design of magnonic devices.3\nIn spintronics, there has long been considerable in-\nterest in giant magnetoresistance, spin-transfer torques,\nand spin pumping in hybrid systems of normal met-\nals and metallic ferromagnets (MFs).4{7The experimen-\ntal demonstration that spin transfer and spin pumping\nare also active in normal metals in contact with insu-\nlating ferromagnets has generated a renewed interest in\nand refocused attention on insulating ferromagnets, of\nwhich yttrium iron garnet (YIG) continues to be the\nprime example.8{19In ferromagnetic insulators, current-\ninduced spin-transfer torques from a neighboring normal\nmetal (NM) that exhibits out-of-equilibrium spin accu-\nmulation may manipulate the magnetization of the insu-\nlator and excite spin waves.8,20,21The out-of-equilibrium\nspin accumulation of the normal metal may be induced\nvia the spin Hall e\u000bect or by currents passing through\nother adjacent conducting ferromagnets. Conversely, ex-\ncited spin waves pump spins into adjacent NMs, and this\nspin current may be measured in terms of the inverse spinHall voltages or by other conducting ferromagnets.8{14\nThe magnetic state may also be measured via the spin\nHall magnetoresistance.16{19,23,24Because of these devel-\nopments, magnetic information in ferromagnetic insula-\ntors may be electrically injected, manipulated, and de-\ntected. Importantly, an FI-based spintronic device may\ne\u000eciently transport electric information carried by spin\nwaves over long distances15without any excessive heat-\ning. The spin-wave decay length can be as long as cen-\ntimeters in YIG \flms.22These properties make FI{NM\nsystems ideal devices for the exploration of novel spin-\ntronic phenomena and possibly also important for future\nspintronic applications. Magnonic devices also o\u000ber ad-\nvantages such as rapid spin-wave propagation, frequen-\ncies ranging from GHz to THz, and the feasibility of cre-\nating spin-wave logic devices and magnonic crystals with\ntailored spin-wave dispersions.25\nTo utilize the desirable properties of FI{NM systems,\nsuch as the exceptionally low magnetization-damping\nrate of FIs, it is necessary to understand how the mag-\nnetization dynamics couple to spin transport in adjacent\nnormal metals. The e\u000bective damping of the uniform\nmagnetic mode of a thin-\flm FI is known to signi\f-\ncantly increase when the FI is placed in contact with\nan NM. This damping enhancement is caused by the loss\nof angular momentum through spin pumping.26{30Re-\ncent theoretical work has also predicted the manner in\nwhich the Gilbert damping for other spin-wave modes\nshould become renormalized.31For long-wavelength spin\nwaves, the Gilbert damping enhancement is twice as\nlarge for transverse volume waves as for the macrospin\nmode, and for surface modes, the enhancement can be ten\ntimes stronger or more. Spin pumping has been demon-\nstrated, both experimentally9and theoretically,31to be\nsuppressed for short-wavelength exchange spin waves.arXiv:1407.0635v1  [cond-mat.mes-hall]  2 Jul 20142\nA natural next step is to investigate the magnetization\ndynamics of more complicated FI{NM heterostructures.\nIn ferromagnetic metals, it is known that spin pumping\nand spin-transfer torques generate a long-range dynamic\ninteraction between magnetic \flms separated by normal\nmetal layers.32The e\u000bect of this long-range dynamic in-\nteraction on homogeneous macrospin excitations can be\nmeasured by ferromagnetic resonance. The combined ef-\nfects of spin pumping and spin-transfer torque lead to\nan appreciable increase in the resonant linewidth when\nthe resonance \felds of the two \flms are far apart and\nto a dramatic narrowing of the linewidth when the reso-\nnant \felds approach each other.32This behavior occurs\nbecause the excitations in the two \flms couple acous-\ntically (in phase) or optically (out of phase). We will\ndemonstrate that similar, though richer because of the\ncomplex magnetic modes, phenomena exist in magnetic\ninsulators.\nIn the present paper, we investigate the magnetization\ndynamics in a thin-\flm stack consisting of two FIs that\nare in contact via an NM. The macrospin dynamics in\na similar system with metallic ferromagnets have been\nstudied both theoretically and experimentally.32We ex-\npand on that work by focusing on inhomogeneous mag-\nnetization excitations in FIs.\nFor long-wavelength spin waves travelling in-plane in\na ferromagnetic thin \flm, the frequency as a function\nof the in-plane wave number Qstrongly depends on the\ndirection of the external magnetic \feld with respect to\nthe propagation direction. If the external \feld is in-\nplane and the spin waves are travelling parallel to this\ndirection, the waves have a negative group velocity. Be-\ncause the magnetization precession amplitudes are usu-\nally evenly distributed across the \flm in this geometry,\nthese modes are known as backward volume magneto-\nstatic spin waves (BVMSW). Similarly, spin waves that\ncorrespond to out-of-plane external \felds are known as\nforward volume magnetostatic spin waves (FVMSW),\ni.e., the group velocity is positive, and the precession\namplitudes are evenly distributed across the \flm. When\nthe external \feld is in-plane and perpendicular to the\npropagation direction, the precession amplitudes of the\nspin waves become inhomogeneous across the \flm, ex-\nperiencing localization to one of the interfaces. These\nspin waves are thus known as magnetostatic surface spin\nwaves (MSSW).33,34\nWhen two ferromagnetic \flms are coupled via a normal\nmetal, the spin waves in the two \flms become coupled\nthrough two di\u000berent mechanisms. First, the dynamic,\nnonlocal dipole-dipole interaction causes an interlayer\ncoupling to arise that is independent of the properties\nof the normal metal. This coupling is weaker for larger\nthicknesses of the normal metal. Second, spin pumping\nfrom one ferromagnetic insulator induces a spin accu-\nmulation in the normal metal, which in turn gives rise\nto a spin-transfer torque on the other ferromagnetic in-\nsulator, and vice versa. This dynamic coupling, is in\ncontrast to the static exchange coupling35rather long-ranged and is limited only by the spin-di\u000busion length.\nThis type of coupling is known to strongly couple the\nmacrospin modes. When two ferromagnetic \flms become\ncoupled, the characterization of the spin waves in terms\nof FVMSW, BVMSW, and MSSW still holds, but the\ndispersion relations are modi\fed. It is also clear that the\ndamping renormalization caused by spin pumping into\nthe NM may di\u000ber greatly from that in a simpler FI jN\nbilayer system. To understand this phenomenon, we per-\nform a detailed analytical and numerical analysis of a\ntrilayer system, with the hope that our \fndings may be\nused as a guide for experimentalists.\nThis paper is organized as follows. Section II intro-\nduces the model. The details of the dynamic dipolar\n\feld are discussed, and the boundary conditions associ-\nated with spin pumping and spin transfer at the FI jN\ninterfaces are calculated. Sec. III provides the analyti-\ncal solutions of these equations in the long-wavelength\nregime dominated by the dynamic coupling attributable\nto spin pumping and spin transfer. To create a more\ncomplete picture of the dynamic behavior of this system,\nwe perform a numerical analysis for the entire spin-wave\nspectrum of this system, which is presented in Sec. IV.\nWe conclude our work in Sec. V.\nII. EQUATIONS OF MOTION\nConsider a thin-\flm heterostructure composed of two\nferromagnetic insulators (FI1 and FI2) that are in elec-\ntrical contact via an NM layer. The ferromagnetic in-\nsulators FI1 and FI2 may have di\u000berent thicknesses and\nmaterial properties. We denote the thicknesses by L1,\ndN, andL2for the FI1, NM, and FI2 layers, respectively\n(see Fig. 1(a)). The in-plane coordinates are \u0010;\u0011, and the\ntransverse coordinate is \u0018(see Fig. 1(b)). We will \frst\ndiscuss the magnetization dynamics in isolated FIs and\nwill then incorporate the spin-memory losses and the cou-\npling between the FIs via spin currents passing through\nthe NM.\nA. Magnetization Dynamics in Isolated FIs\nThe magnetization dynamics in the ferromagnetic in-\nsulators can be described by using the Landau-Lifshitz-\nGilbert (LLG) equation,\n_Mi=\u0000\rMi\u0002He\u000b+\u000bMi\u0002_Mi; (1)\nwhere Miis the unit vector in the direction of the mag-\nnetization in layer i= 1;2,\ris the gyromagnetic ratio,\n\u000bis the dimensionless damping parameter, and He\u000bis\nthe space-time-dependent e\u000bective magnetic \feld. The\ne\u000bective magnetic \feld is\nHe\u000b=Hint+hex+hd+hsurface; (2)\nwhere Hintis the internal \feld attributable to an external\nmagnetic \feld and the static demagnetization \feld, hex=3\ndN2+L2\ndN2\n-dN2\n-dN2-L1NFI2\nFI1\nSUBx\n(a)\n (b)\nFIG. 1: (Color online) a) A cross section of the FI1 jNjFI2 het-\nerostructure. The ferromagnetic insulators FI1 and FI2 are\nin contact via the normal metal N. The transverse coordinate\n\u0018is indicated along with the thicknesses L1,dN, andL2of\nFI1, N, and FI2, respectively. b) The coordinate system of\nthe internal \feld (blue) with respect to the coordinate system\nof the FI1jNjFI2 structure (red). \u0012denotes the angle between\nthe \flm normal and the internal \feld, and \u001eis the angle be-\ntween the in-plane component of the magnetic \feld and the\nin-plane wave vector.\n2Ar2M=MSis the exchange \feld ( Ais the exchange\nconstant), hdis the dynamic demagnetization \feld, and\nhsurface =2KS\nM2\nS(Mi\u0001^n)\u000e(\u0018\u0000\u0018i)^n (3)\nis the surface anisotropy \feld located at the FI jN in-\nterfaces. In this work, hsurface is assumed to exist only\nat the FIjN interfaces and not at the interfaces between\nthe FIs and the substrate or vacuum. It is straightfor-\nward to generalize the discussion to include these surface\nanisotropies as well. We consider two scenarios: one with\nan easy-axis surface anisotropy ( KS>0) and one with no\nsurface anisotropy ( KS= 0). Note that a negative value\nofKS\u0018 \u0000 0:03 erg=cm2, which implies an easy-plane\nsurface anisotropy, has also been observed for sputtered\nYIGjAu bilayers.36In general, the e\u000bective \feld He\u000bmay\ndi\u000ber in the two FIs. We assume the two FIs consist of\nthe same material and consider external \felds that are\neither in-plane or out-of-plane. Furthermore, we consider\ndevices in which the internal magnetic \felds in the two\nFI layers are aligned and of equal magnitude.\nIn equilibrium, the magnetization inside the FIs is ori-\nented along the internal magnetic \feld, Mi=M0. In the\nlinear response regime, Mi=M0+mi, where the \frst-\norder correction miis small and perpendicular to M0.The magnetization vanishes outside of the FIs. Because\nthe system is translationally invariant in the \u0011and\u0010di-\nrections, we may, without loss of generality, assume that\nmconsists of plane waves travelling in the \u0010direction,\nmi(\u0010;\u0011;\u0018 ) =miQ(\u0018)ei(!t\u0000Q\u0010): (4)\nLinearizing Maxwell's equations in miimplies that the\ndynamic dipolar \feld must be of the same form,\nhd(\u0010;\u0011;\u0018 ) =hdQ(\u0018)ei(!t\u0000Q\u0010): (5)\nFurthermore, the total dipolar \feld (the sum of the static\nand the dynamic dipolar \felds) must satisfy Maxwell's\nequations, which, in the magnetostatic limit, are\nr\u0001(hd+ 4\u0019MSm) = 0; (6a)\nr\u0002hd= 0; (6b)\nwith the boundary equations\n(hd+ 4\u0019MSm)?;in= (hd)?;out; (7a)\n(hd)k;in= (hd)k;out; (7b)\nwhere the subscript in (out) denotes the value on the FI\n(NM, vacuum or substrate) side of the FI interface and ?\n(k) denotes the component(s) perpendicular (parallel) to\nthe FI{NM interfaces. Solving Maxwell's equations (6)\nwith the boundary conditions of Eq. (7) yields33\nhdQ(\u0018) =Z\nd\u00180^G(\u0018\u0000\u00180)mQ(\u00180); (8)\nwhere ^G(r\u0000r0) is a 3\u00023 matrix acting on min the (\u0011;\u0010;\u0018 )\nbasis,\n^G(\u0018) =0\n@GP(\u0018)\u0000\u000e(\u0018) 0\u0000iGQ(\u0018)\n0 0 0\n\u0000iGQ(\u0018) 0\u0000GP(\u0018)1\nA: (9)\nHere,GP(\u0018) =Qe\u0000Qj\u0018j=2, andGQ(\u0018) =\u0000sign(\u0018)GP.\nNote that the dynamic dipolar \feld of Eq. (8) accounts\nfor both the interlayer and intralayer dipole-dipole cou-\nplings because the magnetization varies across the two\nmagnetic insulator bilayers and vanishes outside these\nmaterials.\nIt is now convenient to perform a transformation from\nthe\u0010-\u0011-\u0018coordinate system de\fned by the sample geome-\ntry to thex-y-zcoordinate system de\fned by the internal\n\feld (see Fig. 1(b)). In the linear response regime, the\ndynamic magnetization milies in thex-yplane, and the\nlinearized equations of motion become33\n\u0014\ni!\u0012\n\u000b\u00001\n1\u000b\u0013\n+11\u0012\n!H+2A\nMS\u0014\nQ2\u0000d2\nd\u00182\u0015\u0013\u0015\nmiQxy(\u0018) =2X\ni=1Z\nd\u00180^Gxy(\u0018\u0000\u00180)miQxy(\u00180): (10)4\nN\nm1,QFI1m2,QFI2\nee\nFIG. 2: (Color online) Two coupled spin waves with ampli-\ntudem1Qin ferromagnet FI1 and amplitude m2Qin ferro-\nmagnet FI2. The spin-waves inject a spin current into the nor-\nmal metal (NM) via spin pumping. In the NM, the spins dif-\nfuse and partially relax, inducing a spin accumulation therein.\nIn turn, the spin accumulation causes spin-transfer torques to\narise on FI1 and FI2. The combined e\u000bect of spin transfer and\nspin pumping leads to a dynamic exchange coupling that, to-\ngether with the dynamic demagnetization \feld, couples the\nspin waves in the two FIs.\nHere, miQxy = (miQx;miQy) is the Fourier transform of\nthe dynamic component of the magnetization in the x-\nyplane and ^Gxy(\u0018) is the 2\u00022 matrix that results from\nrotating ^G(\u0018) into thex-y-zcoordinate system (see Ap-\npendix A), and considering only the xx,xy,yxandyy-\ncomponents.\nB. Boundary Conditions and Spin Accumulation\nThe linearized equations of motion (10) must be sup-\nplemented with boundary conditions for the dynamic\nmagnetization at the FI jN interfaces. A precessing mag-\nnetization at the FI jN boundaries injects a spin-polarized\ncurrent, jSP, into the NM, an e\u000bect known as spin\npumping .8,28{30The emitted spin currents at the lower\nand upper interfaces ( i= 1;2) are\njSP\ni=~\neg?Mi\u0002_Mi\f\f\f\f\n\u0018=\u0018i; (11)\nwhere\u0018i=\u0007dN=2 at the lower and upper interfaces,\nrespectively, and g?is the real part of the transverse spin-\nmixing conductance per unit area.37We disregard the\nimaginary part of the spin-mixing conductance because\nit has been found to be small at FI jN interfaces.38The\nreciprocal e\u000bect of spin pumping is spin transfer into the\nFIs because of a spin accumulation \u0016Sin the NM. In the\nnormal metal at the lower and upper interfaces ( i=1,2),the associated spin-accumulation-induced spin current is\njST\ni=\u00001\neg?Mi\u0002(Mi\u0002\u0016S)\f\f\f\f\n\u0018=\u0018i: (12)\nThe signs of the pumped and spin-accumulation-induced\nspin currents in Eqs. (11) and (12) were chosen such that\nthey are positive when there is a \row of spins from the\nNM toward the FIs.\nThe pumped and spin-accumulation-induced spin cur-\nrents of Eqs. (11) and (12) lead to magnetic torques act-\ning on the FI interfaces. The torques that correspond to\nthe spin pumping and spin transfer localized at the FI jN\ninterfaces are\n\u001cSP\ni=\r~2\n2e2g?\u000e(\u0018\u0000\u0018i)Mi\u0002_Mi; (13a)\n\u001cST\ni=\u0000\r~\n2e2g?Mi\u0002(Mi\u0002\u0016S)\u000e(\u0018\u0000\u0018i);(13b)\nrespectively. In the presence of spin currents to and from\nthe normal metal, the magnetization dynamics in the\nFIs is then governed by the modi\fed Landau-Lifshitz-\nGilbert-Slonczewski (LLGS) equation,\n_M=\u0000\rMi\u0002He\u000b+\u000bMi\u0002_Mi+X\ni=1;2\u001cSP\ni+\u001cST\ni:(14)\nBy integrating Eq. (14) over the FI jN interfaces and the\ninterfaces between the FI and vacuum/substrate, we \fnd5\nthatmimust satisfy the boundary conditions21,31\n\u0012\n\u0006Lidmi\nd\u0018+\u001fi\u0014\n_mi\u00001\n~M0\u0002\u0016\u0015\n+LiKS\nAcos (2\u0012)mi\u0013\nx\f\f\f\f\n\u0018=\u0007dN=2= 0;(15a)\n\u0012\n\u0006Lidmi\nd\u0018+\u001fi\u0014\n_mi\u00001\n~M0\u0002\u0016\u0015\n+LiKs\nAcos2(\u0012)mi\u0013\ny\f\f\f\f\f\n\u0018=\u0007dN=2= 0;(15b)\ndm1\nd\u0018\f\f\f\f\n\u0018=\u0000dN=2\u0000L1= 0;dm2\nd\u0018\f\f\f\f\n\u0018=dN=2+L2= 0:(15c)\nHere, we have introduced the timescale \u001fi=\nLi~2g?=4Ae2. The subscripts xandyin Eqs. (15a) and\n(15b) denote the xandycomponents, respectively. In\nour expressions for the boundary conditions (15), we have\nalso accounted for the possibility of a surface anisotropy\narising from the e\u000bective \feld described by Eq. (3),\nwhereKS>0 indicates an easy-axis surface anisotropy\n(EASA). The boundary conditions of Eq. (15), in combi-\nnation with the transport equations in the NM , which we\nwill discuss next, determine the spin accumulation in the\nNM and the subsequent torques caused by spin transfer.\nIn the normal metal, the spins di\u000buse, creating a spa-\ntially dependent spin-accumulation potential \u0016Q, and\nthey relax on the spin-di\u000busion length scale lsf. The\nspin accumulation for an FI jNjFI system has been cal-\nculated in the macrospin model.39The result of this\ncalculation can be directly generalized to the present\nsituation of spatially inhomogeneous spin waves by re-\nplacing the macrospin magnetization in each layer with\nthe interface magnetization and substituting the spin-\ndi\u000busion length with a wave-vector-dependent e\u000bective\nspin-di\u000busion length lsf!~lsf(Q) such that\n\u0016Q=\u0000~\n2M0\u0002[(_mQ(\u00181) +_mQ(\u00182))\u00001(\u0018)\n\u0000(_mQ(\u00181)\u0000_mQ(\u00182))\u00002(\u0018)]:(16)\nSee Appendix B for the details of the functions \u0000 1and\n\u00002. The e\u000bective spin-di\u000busion length is found by Fouriertransforming the spin-di\u000busion equation (see Appendix\nC), resulting in\n~lsf=lsf=p\n1 + (Qlsf)2: (17)\nWe thus have all the necessary equations to de-\nscribe the linear response dynamics of spin waves in the\nFI1jNjFI2 system. We now provide analytical solutions\nof the spin-wave modes in the long-wavelength limit and\nthen complement these solutions with an extensive nu-\nmerical analysis that is valid for any wavelength.\nIII. ANALYTIC SOLUTIONS FOR THE SPIN\nWAVE SPECTRUM\nThe e\u000bect that the exchange and dipolar \felds have\non the spin-wave spectrum depends on the in-plane wave\nnumberQ. WhenQLi\u001c1, the dipolar \feld dominates\nover the exchange \feld. In the opposite regime, when\nQLi\u001d1, the exchange \feld dominates over the dipo-\nlar \feld. The intermediate regime is the dipole-exchange\nregime. Another length scale is set by the spin-di\u000busion\nlength. When Qlsf\u001d1, the e\u000bective spin-relaxation\nlength ~lsfof Eq. (17) becomes small, and the NM acts\nas a perfect spin sink. In this case, only the relatively\nshort-ranged dipolar \feld couples the FIs. We therefore\nfocus our attention on the dipole-dominated regime, in\nwhich the interchange of spin information between the\ntwo FIs remains active.\nIn the limit QLi\u001c1, the magnetization is homoge-\nneous in the in-plane direction. We may then use the\nansatz that the deviation from equilibrium is a sum of\ntransverse travelling waves. Using the boundary condi-\ntions on the outer boundaries of the stack, Eq. (15c), we\n\fnd\nmiQxy(\u0018) =\u0012\nXi\nYi\u0013\ncos\u001a\nki\u0014\n\u0018\u0006(Li+dN\n2)\u0015\u001b\n;(18)\nwherei= 1 when\u0018is inside FI1 and i= 2 when\u0018is inside\nFI2.k1andk2are the out-of-plane wave vectors of the\nlower and upper \flms, respectively. The eigenfrequencies\nof Eq. (10) depend on ki. To \frst order in the damping\nparameter\u000b, we have\n!(ki) =!M\"\n\u0006s\u0012!H\n!M+A\n2\u0019M2\nSk2\ni\u0013\u0012!H\n!M+A\n2\u0019M2\nSk2\ni+ sin2\u0012\u0013\n+i\u000b\u0012!H\n!M+A\n2\u0019M2\nSk2\ni+1\n2sin2\u0012\u0013#\n: (19)\nWe can, without loss of generality, consider only those frequencies that have a positive real part. The eigen-6\nfrequency!is a characteristic feature of the entire sys-\ntem, so we must require !(k1) =!(k2), which implies\nthatk1=\u0006k2. We will discuss the cases of symmetric\n(L1=L2) and asymmetric ( L16=L2) geometries sepa-\nrately.\nA. Symmetric FI \flms without EASA\nConsider a symmetric system in which the FIs are of\nidentical thickness and material properties. We assume\nthat the e\u000bect of the EASA is negligible, which is the\ncase for thin \flms and/or weak surface anisotropy ener-\ngies such that KSL=A\u001c1, whereL=L1=L2. The\nother two boundary conditions, (15a) and (15b), cou-\nple the amplitude vectors\u0000X1Y1\u0001Tand\u0000X2Y2\u0001Tof\nEq. (18). A non-trivial solution implies that the deter-\nminant that contains the coe\u000ecients of the resulting 4 \u00024\nmatrix equation vanishes. Solving the secular equation,\nwe \fnd the following constraints on k,\ni\u001fA!A=kLtan(kL); (20a)\ni\u001fO!O=kLtan(kL); (20b)\nwhere\n\u001fA=\u001f \n1\u0000\u0014\n1 +2g?lsf\n\u001btanh(dN=2lsf)\u0015\u00001!\n;(21a)\n\u001fO=\u001f \n1\u0000\u0014\n1 +2g?lsf\n\u001bcoth(dN=2lsf)\u0015\u00001!\n;(21b)\nand\u001f=L~2g?=4Ae2. The two solutions correspond\nto a symmetric mode (acoustic) and an antisymmetric\nmode (optical). This result can be understood in terms\nof the eigenvectors that correspond to the eigenvalues of\nEqs. (20), which are m1= +m2andm1=\u0000m2for\nthe acoustic and optical modes, respectively. Typically,\nbecause spin pumping only weakly a\u000bects the magne-\ntization dynamics, the timescale \u001fthat is proportional\nto the mixing conductance g?is much smaller than the\nFMR precession period. In this limit, kLtan(kL)\u001c1.\nThis result allows us to expand the secular equations (20)\naroundkL=n\u0019, wherenis an integral number, which\nyields\ni\u001f\u0017!\u0017;n\u0019(kL+\u0019n)kL; (22)\nwhere\u0017= A;O. This result can be reinserted into the\nbulk dispersion relation of Eq. (19), from which we can\ndetermine the renormalization of the Gilbert damping\ncoe\u000ecient attributable to spin pumping, \u0001 \u000b. We de\fne\n\u0001\u000b=\u000b\u0010\nIm[!(SP)]\u0000Im[!(0)]\u0011\n=Im[!(0)] (23)\nas a measure of the spin-pumping-enhanced Gilbert\ndamping, where !(0)and!(SP)are the frequencies of\nthe same system without and with spin pumping, respec-\ntively.Similar to the case of a single-layer ferromagnetic\ninsulator,31we \fnd that all higher transverse volume\nmodes exhibit an enhanced magnetization dissipation\nthat is twice that of the macrospin mode. The enhance-\nment of the Gilbert damping for the macrospin mode\n(n= 0) is\n\u0001\u000b\u0017;macro =\r~2g?\n2LMSe2\u001f\u0017\n\u001f; (24)\nand for the other modes, we obtain\n\u0001\u000b\u0017;n6=0= 2\u0001\u000b\u0017;macro: (25)\nCompared with single-FI systems, the additional fea-\nture of systems with two FIs is that the spin-pumping-\nenhanced Gilbert damping di\u000bers signi\fcantly between\nthe acoustic and optical modes via the mode-dependent\nratio\u001f\u0017=\u001f. This phenomenon has been explored both\nexperimentally and theoretically in Ref. 32 for the\nmacrospin modes n= 0 when there is no loss of spin\ntransfer between the FIs, lsf!1 . Our results repre-\nsented by Eqs. (24) and (25) are generalizations of these\nresults for the case of other transverse volume modes and\naccount for spin-memory loss. Furthermore, in Sec. IV,\nwe present the numerical results for the various spin-wave\nmodes when the in-plane momentum Qis \fnite. When\nthe NM is a perfect spin sink, there is no transfer of spins\nbetween the two FIs, and we recover the result for a sin-\ngle FIjN system with vanishing back \row, \u001f\u0017!\u001f.31\nNaturally, in this case, the FI jNjFI system acts as two\nindependent FIjN systems with respect to magnetiza-\ntion dissipation. The dynamical interlayer dipole cou-\npling is negligible in the considered limit of this section\n(QL\u001c1).\nIn the opposite regime, when the NM \flm is much thin-\nner than the spin-di\u000busion length and the spin conductiv-\nity of the NM is su\u000eciently large such that g?dN=\u001b\u001c1,\nthen\u001fA!0 and\u001fO!\u001f. This result implies that for\nthe optical mode, the damping is the same as for a sin-\ngle FI in contact with a perfect spin sink, even though\nthe spin-di\u000busion length is very large. The reason for\nthis phenomenon is that when the optical mode is ex-\ncited, the magnetizations of the two \flms oscillate out\nof phase such that one layer acts as a perfect spin sink\nfor the other layer. By contrast, there is no enhance-\nment of the Gilbert damping coe\u000ecient for the acoustic\nmode; when the \flm is very thin and the magnetizations\nof the two layers are in phase, there is no net spin \row or\nloss in the NM \flm and no spin-transfer-induced losses\nin the ferromagnets. Finally, when the NM is a poor con-\nductor despite exhibiting low spin-memory loss such that\ng?dN=\u001b\u001d(lsf=dN)\u001d1, then\u001f\u0017!0 because there is no\nexchange of spin information. For the macrospin modes\nin the absence of spin-memory loss, these results are in\nexact agreement with Ref. 32. Beyond these results, we\n\fnd that regardless of how much spin memory is lost, it\nis also the case that in trilayer systems, all higher trans-\nverse modes experience a doubling of the spin-pumping-\ninduced damping. Furthermore, these modes can still7\nbe classi\fed as optical and acoustic modes with di\u000berent\ndamping coe\u000ecients.\nB. Symmetric Films with EASA\nMagnetic surface anisotropy is important when the\nspin-orbit interaction at the interfaces is strong. In this\ncase, the excited mode with the lowest energy becomes\ninhomogeneous in the transverse direction. For a \fnite\nKS, the equations for the xandycomponents of the\nmagnetization in the boundary condition (15) di\u000ber, re-\nsulting in di\u000berent transverse wave vectors for the two\ncomponents, kxandky, respectively. Taking this situa-\ntion into account, we construct the ansatz\nmiQxy(\u0018) =\u0012\nXicos (kx;i\u0018\u0006kx;i(L+dN=2))\nYicos (ky;i\u0018\u0006ky;i(L+dN=2))\u0013\n;(26)\nwhich, when inserted into the boundary conditions of\nEqs. (15a) and (15b), yields\ni\u001f\u0017!\u0017+LKS\nAcos (2\u0012) =kxdtan (kxd);(27a)\ni\u001f\u0017!\u0017+LKS\nAcos2(\u0012) =kydtan (kyd);(27b)\nwhere\u0017continues to denote an acoustic (A) or optical\n(O) mode, \u0017= A;O. Depending on the sign of KSand\nthe angle\u0012, the resulting solutions kxandkycan be-\ncome complex numbers, which implies that the modes\nare evanescent. Let us consider the case of KS>0 and\nan in-plane magnetization ( \u0012=\u0019=2). Although kyis\nunchanged by the EASA, with LKS=A> 1\u001d\u001f\u0017!\u0017,kx\nis almost purely imaginary, \u0014=ik=KS=A\u0000i!\u0017\u001f\u0017, so\nthat\nmiQx(\u0018) =Xcosh(\u0014\u0018\u0006\u0014(d+dN=2)): (28)\nThe magnetization along the xdirection is exponentially\nlocalized at the FI jN surfaces. Following the same proce-\ndure as in Sec. III A for the KS= 0 case, we insert this\nsolution into the dispersion relation (19) and extract the\nrenormalization of the e\u000bective Gilbert damping:\n\u0001\u000bEASA\n\u0017 =\r~2g?\n2LMSe2\u001f\u0017\n\u001f1 +!H\n!M\u0002\n1 +2LKS\nA\u0003\n\u0000K2\nS\n2\u0019M2\nSA\n1 + 2!H\n!M\u0000K2s\n2\u0019M2\nSA:\n(29)\nIn the presence of EASA, the damping coe\u000ecient is a ten-\nsor; thus, the e\u000bective damping of Eq. (29) is an average,\nas de\fned in Eq. (23). This Gilbert damping enhance-\nment may become orders of magnitude larger than the\n\u0001\u000bmacro of Eq. (24). For thick \flms, \u0001 \u000bmacro\u0018L\u00001,\nwhereas \u0001\u000bEASA\n\u0017 reaches a constant value that is in-\nversely proportional to the localization length at the FI jN\ninterface. Note that for large EASA, the equilibrium\nmagnetization is no longer oriented along the external\n\feld, and Eq. (29) for \u0001 \u000bEASA\n\u0017 becomes invalid.C. Asymmetric FI Films\nLet us now consider an asymmetric system in which\nL16=L2. In this con\fguration, we will \frst consider\nKS= 0, but we will also comment on the case of a \f-\nniteKSat the end of the section. Because the analytical\nexpressions for the eigenfrequencies and damping coe\u000e-\ncients are lengthy, we focus on the most interesting case:\nthat in which the spin-relaxation rate is slow.\nAs in the case of the symmetric \flms, the dispersion\nrelation of Eq. (10) dictates that the wave numbers in the\ntwo layers must be the same. To satisfy the boundary\nequations (15), we construct the ansatz\nmiQxy(\u0018) =\u0012\nXicos (k\u0018\u0006k(L+dN=2))\nYicos (k\u0018\u0006k(L+dN=2))\u0013\n: (30)\nThe di\u000berence between this ansatz and the one for the\nsymmetric case represented by Eq. (26) is that the mag-\nnitudes of the amplitudes, XiandYi, of the two layers,\ni= 1;2, that appear in Eq. (30) is no longer expected to\nbe equal.\nWhen the two ferromagnets FI( L1) and FI(L2) are\ncompletely disconnected, the transverse wave vectors\nmust be equivalent to standing waves, qn;1=\u0019n=L 1and\nqm;2=\u0019m=L 2in the two \flms, respectively, where nand\nmmay be any integral numbers. Because spin pumping\nis weak, the eigenfrequencies of the coupled system are\nclose to the eigenfrequencies of the isolated FIs. This\n\fnding implies that the wave vector kof the coupled sys-\ntem is close to either qn;1orqm;2. The solutions of the\nlinearized equations of motion are then\nk=kn;1=qn;1+\u000ekn;1or (31a)\nk=km;2=qm;2+\u000ekm;2; (31b)\nwhere\u000ekn;1and\u000ekm;2are small corrections attributable\nto spin pumping and spin transfer, respectively. Here,\nthe indices 1 and 2 represent the di\u000berent modes rather\nthan the layers. However, one should still expect that\nmode 1(2) is predominantly localized in \flm 1(2). In\nthis manner, we map the solutions of the wave vectors in\nthe coupled system to the solutions of the wave vectors\nin the isolated FIs. Next, we will present solutions that\ncorrespond to the qn;1of Eq. (31a). The other family of\nsolutions, corresponding to qm;2, is determined by inter-\nchangingL1$L2and making the replacement n!m.\nInserting Eq. (31a) into the boundary conditions of\nEq. (15) and linearizing the resulting expression in the\nweak spin-pumping-induced coupling, we \fnd, for the\nmacrospin modes,\ni!~\u001fA,O\n1;macro = (L1\u000ek0;1)2; (32)\nwhere\n~\u001fA\n1;macro\u00191\n2dN\nlsf\u001b\ng?lsfL1\nL1+L2\u001f1; (33a)\n~\u001fO\n1;macro\u00191\n2L1+L2\nL2\u001f1: (33b)8\nHere,\u001f1=L1~2g?=4Ae2. Inserting this parameter into\nthe dispersion relation of Eq. (19), we obtain the follow-\ning damping renormalizations:\n\u0001\u000bA\nmacro =\r~2g?\n2MSe21\n2dN\nlsf\u001b\ng?lsf1\nL1+L2;(34a)\n\u0001\u000bO\nmacro =\r~2g?\n2MSe21\n2\u00121\nL1+1\nL2\u0013\n: (34b)\nThese two solutions correspond to an acoustic mode\nand an optical mode, respectively. The corresponding\neigenvectors are m1=m2for the acoustic mode and\nL1m1=\u0000L2m2for the optical mode. As in the sym-\nmetric case, the damping enhancement of the acoustic\nmode vanishes in the thin-NM limit. In this limit, the\nbehavior of the acoustic mode resembles that of a single\nFI of thickness L1+L2. It is the total thickness that\ndetermines the leading-order contribution of the damp-\ning renormalization. The optical mode, however, experi-\nences substantial damping enhancement. For this mode,\nthe damping renormalization is the average of two sepa-\nrate FIs that are in contact with a perfect spin sink. The\ncause of this result is as follows. When there is no spin-\nmemory loss in the NM, half of the spins that are pumped\nout from one side return and rectify half of the angular-\nmomentum loss attributable to spin pumping. Because\nthe magnetization precessions of the two \flms are com-\npletely out of phase, the other half of the spin current\ncauses a dissipative torque on the opposite layer. In ef-\nfect, spin pumping leads to a loss of angular momentum,\nand the net sum of the spin pumping across the NM and\nthe back \row is zero. The total dissipation is not a\u000bected\nby spin transfer, and thus, the result resembles a system\nin which the NM is a perfect spin sink.\nFor the higher excited transverse modes, there are two\nscenarios, which we treat separately. I. The allowed wave\nnumber for one layer matches a wave number for the\nother layer. Then, for some integer n > 0,qn;1=qm;2\nfor some integer m. In this case, we expect a coupling\nof the two layers. II. The allowed wave number for one\nlayer does not match any of the wave numbers for the\nother layer, and thus, for some integer n > 0, we have\nqn;16=qm;2for all integers m. We then expect that the\ntwo layers will not couple.\nI. In this case, we \fnd two solutions that correspond\nto acoustic and optical modes. These modes behave very\nmuch like the macrospin modes; however, as in the sym-\nmetric case, the damping renormalization is greater by a\nfactor of 2:\n\u0001\u000bA,O\nn6=0= 2\u0001\u000bA,O\nmacro;Case I: (35)\nThe eigenvectors of these coupled modes have the same\nform as for the macrospin modes, such that m1=m2\nandL1m1=\u0000L2m2for the acoustic and optical modes,\nrespectively.\nII. In this case, the two layers are completely decou-pled. To the leading order in dN=lsf, we \fnd\n\u0001\u000bn6=0=\r~2g?\n2L1MSe2;Case II; (36)\nfor all modes that correspond to excitations in FI1.\nThe damping renormalization is thus half that of the\nFI(L1)jN(lsf= 0) system.31This result can be explained\nby the zero loss of spin memory in the NM. Although half\nof the spins are lost to the static FI2, half of the spins\nreturn and rectify half of the dissipation attributable\nto spin pumping. The amplitudes of these modes are\nstrongly suppressed in FI2 (or FI1, upon the interchange\nof FI1$FI2), such thatjm2j=jm1j\u0018!\u001f2.\nFinally, let us discuss the case in which EASA is\npresent. In the limit KSLi=A\u001d1, the excitation en-\nergies of the surface modes are independent of the FI\nthicknesses. However, the surface modes do not behave\nlike the macrospin modes for the asymmetric stack. The\nexcitation volume of these modes is determined by the\ndecay length A=KSin accordance with Eq. (28). This\n\fnding is in contrast to the result for the macrospin\nmodes, where the excitation volume spans the entire FI.\nThus, the surface modes couple in the same manner as in\nthe symmetric case. With a good experimental control\nof surface anisotropy, the coupling of the surface modes\nis thus robust to thickness variations. The higher ex-\ncited transverse modes, in the presence of EASA, have\nthickness-dependent frequencies, which means that these\nmodes behave similarly to the n>0 modes in the KS= 0\ncase.\nIV. NUMERICAL RESULTS\nWhen the spin-wave wavelength becomes comparable\nto the \flm thickness, the dipolar \feld becomes a compli-\ncated function of the wavelength. We study the proper-\nties of the system in this regime by numerically solving\nthe linearized equations of motion (10) with the bound-\nary conditions (15). We use the method presented in\nRef. 31, which solves the spin-wave excitation spectrum\nfor an FIjN system, and extend this approach to the\npresent trilayer system. The physical parameters used\nin the numerical calculations are listed in Table I. We\ninvestigate two geometries: I. the BWMSW geometry, in\nwhich the spin wave propagates parallel to the external\n\feld, and II. the MSSW geometry, in which the spin wave\npropagates perpendicular to the external \feld.\nTo calculate the renormalization of the Gilbert damp-\ning, we perform one computation without spin pumping\nand one computation with spin pumping, in which the\nintrinsic Gilbert damping is excluded. Numerically, the\nrenormalization can then be determined by calculating\n\u0001\u000b=\u000bIm[!(SP)]\u000b=0=Im[!(0)], where!(0)is the eigenfre-\nquency obtained for the computation without spin pump-\ning and!(SP)is the frequency obtained for the compu-\ntation with spin pumping.319\nTABLE I: Physical parameters used in the numerical calcu-\nlations\nConstant Value Units\ng?a3:4\u00011015cm\u00002e2=h\n\u001bb5:4\u00011017s\u00001\n4\u0019MSc1750 G\nAc3:7\u000110\u00007erg=cm\nHint 0:58\u00014\u0019MS\n\u000bc3\u000110\u00004\nKS 0;d0:05 erg=cm2\na) Ref. [47], b) Ref. [48], c) Ref. [34]\nd) Reported to be in the range of 0 :1\u00000:01 erg=cm2in\nRef. [21]\nA. BVMSW\nFIG. 3: (Color online) FI(100nm) jN(50nm)jFI(101nm): a)\nSpin-pumping-enhanced Gilbert damping \u0001 \u000bas a function\nofQL1of the uniform modes and the n= 1 modes. The inset\npresents the corresponding dispersion relation. b) Relative\nphase and c) amplitude between the out-of-plane magnetiza-\ntions along xat the edges of FI1 jN and FI2jN. The apparent\ndiscontinuity in the green line in c) appears because the phase\nis de\fned on the interval \u0000\u0019to\u0019.\nLet us \frst discuss the BVMSW geometry. The cou-\npling of the uniform modes in the two \flms is robust;it is not sensitive to possible thickness asymmetries. In\ncontrast, at Q= 0, the sensitivity to the ratio between\nthe thickness and the rather weak dynamic coupling at-\ntributable to spin pumping implies that the coupling of\nthe higher transverse modes in the two bilayers is fragile.\nSmall asymmetries in the thicknesses destroy the cou-\npling. This e\u000bect can best be observed through the renor-\nmalization of the damping. However, we will demon-\nstrate that a \fnite wave number Qcan compensate for\nthis e\u000bect such that the higher transverse modes also\nbecome coupled. To explicitly demonstrate this result,\nwe numerically compute the real and imaginary parts\nof the eigenfrequencies of a slightly asymmetric system,\nFI(100nm)jN(50nm)jFI(101nm) with lsf= 350 nm. The\nasymmetry between the thicknesses of the ferromagnetic\ninsulators is only 1%. The surface anisotropy is consid-\nered to be small compared with the ratio Li=A, and we\nsetKS= 0.\nIn Fig. 3, the numerical results for the e\u000bective Gilbert\ndamping, the dispersion of the modes, and the relative\nphase and amplitude between the magnetizations in the\ntwo FIs are presented. As observed in the relative phase\nresults depicted in Fig. 3(c), the two uniform modes in\nwidely separated FIs split into an acoustic mode and\nan optical mode when the bilayers are coupled via spin\npumping and spin transfer. Figure 3(a) also demon-\nstrates that the acoustic mode has a very low renor-\nmalization of the Gilbert damping compared with the\noptical mode. Furthermore, there is no phase di\u000berence\nbetween the two modes with a transverse node ( n= 1) in\nFig. 3(a), which indicates that the modes are decoupled.\nThesen= 1 modes are strongly localized in one of the\ntwo \flms; see Fig. 3(b). For small QL1, Fig. 3(a) demon-\nstrates that these modes have approximately the same\nrenormalization as the optical mode, which is in agree-\nment with the analytical results. Because the magnetiza-\ntion in the layer with the smallest amplitude is only a re-\nsponse to the spin current from the other layer, the phase\ndi\u000berence is \u0019=2 (Fig. 3(b)). When Qincreases, the dipo-\nlar and exchange interactions become more signi\fcant.\nThe interlayer coupling is then no longer attributable\nonly to spin pumping but is also caused by the long-range\ndipole-dipole interaction. This additional contribution to\nthe coupling is su\u000ecient to synchronize the n= 1 modes.\nThe relative amplitude between the two layers then be-\ncomes closer to 1 (see Fig. 3(b)). Again, we obtain an\nacoustic mode and an optical n= 1 mode, which can be\nobserved from the phase di\u000berence between the two lay-\ners in Fig. 3(c). The spin-pumping-induced coupling only\noccurs as long as the e\u000bective spin-di\u000busion length ~lsfis\nlarge or on the order of dN. Once this is no longer the\ncase, the modes rapidly decouple, and the system reduces\nto two separate FI jN systems with a relatively weak in-\nterlayer dipole coupling. In the limit of large QL1, the\nexchange interaction becomes dominant. The energy of\nthe wave is then predominantly attributable to the mo-\nmentum in the longitudinal direction, and the dynamic\npart of the magnetization goes to zero at the FI jN inter-10\nfaces, causing the renormalization attributable to spin\npumping to vanish.31\nWe also note that the dispersion relation depicted in\nthe inset of Fig. 3(a) reveals that the acoustic mode (blue\nline) exhibits a dip in energy at lower QL1than does the\noptical mode (red line). We suggest that this feature\ncan be understood as follows: The shift in the position\nof the energy dip can be interpreted as an increase in\nthe e\u000bective FI thickness for the acoustic mode with re-\nspect to that for the optical mode. When ~lsfis larger\nthan the NM thickness, the uniform mode behaves as\nif the NM were absent and the two \flms were joined.\nThis result indicates that the dispersion relation for the\nacoustic mode exhibits frequency behavior as a function\nofQ~L=2, where the e\u000bective total thickness of the \flm is\n~L=L1+L2. The optical mode, however, \\sees\" the NM\nand thus behaves as if ~L=L1. Consequently, the dip in\nthe dispersion occurs at lower QL1for the acoustic mode\nthan for the optical mode.\nB. MSSW\nFinally, let us study the dynamic coupling of mag-\nnetostatic surface spin waves (MSSWs). We now con-\nsider a perfectly symmetric system, FI(1000 nm) jN(200\nnm)jFI(1000 nm), with lsf= 350 nm. For such thick\n\flms, surface anisotropies may play an important role.\nWe therefore discuss a case in which we include a surface\nanisotropy of KS= 0:05 erg=cm2. According to the an-\nalytical result presented in Eq. (28), the lowest-energy\nmodes with QL1\u001c1 are exponentially localized at the\nFIjN surfaces, with a decay length of A=KS\u0018200 nm.\nWe now compute the eigenfrequencies, !, as a function\nof the wave vector in the range 10\u00004< QL 1<103. In\nFig. 4(a), we present the real part of the frequency for\nthe six lowest-energy modes with a positive real part, and\nin Fig. 4(b), we present the corresponding renormaliza-\ntions of the Gilbert damping for the four lowest-energy\nmodes. The dispersion relations indicate that the mode\npairs that are degenerate at QL1\u001c1 rapidly split in\nenergy when QL1approaches 10\u00002. Strong anticrossings\ncan be observed between the n= 1 andn= 2 modes.\nSuch anticrossings are also present between the surface\nmode and the n= 1 mode; they are almost too strong to\nbe recognized as anticrossings. The enhanced damping\nrenormalizations exhibit very di\u000berent behavior for the\ndi\u000berent modes. We recognize the large-\u0001 \u000bmode of one\npair as the surface optical mode and the low-\u0001 \u000bmode\nas the volume n= 1 acoustic mode. Without EASA,\nthe anticrossings in Fig. 4(a) would become crossings.\nThe lowest-energy modes at QL1\u001c1 would then cut\nstraight through the other modes. In the case considered\nhere, this behavior is now observed only as steep lines at\nQL1\u00180:05 and atQL1\u00180:5.\nWhenQis increased, the e\u000bective spin-di\u000busion\nlength decreases (see Eq. (17)), which reduces the spin-\npumping-induced coupling between the modes at largeQ. WhenQL1\u0018100, the coupling becomes so weak\nthat the two FIs decouple. This phenomenon can be ob-\nserved from the behavior of \u0001 \u000bin Fig. 4(b), where the\ndamping of the acoustic modes become the same as for\nthe optical modes.\nFIG. 4: (Color online) FI(1000nm) jN(200nm)jFI(1000nm)\nlsf= 350 nm, KS= 0:05 erg=cm2: a) The dispersion rela-\ntion as a function of QL1for the six lowest positive-real-part\nmodes. b) The renormalization of the damping attributable\nto spin pumping for the four lowest modes with frequencies\nwith positive real parts as a function of QL1. At largeQL1,\nthe computation becomes increasingly demanding, and the\npoint density of the plot becomes sparse. We have therefore\nindividually marked the plotted points in this region.\nIn the MSSW geometry, an isolated FI has magneto-\nstatic waves that are localized near one of the two sur-\nfaces, depending on the direction of propagation with\nrespect to the internal \feld.34Asymmetries in the exci-\ntation volume are therefore also expected for the trilayer\nin this geometry. In Fig. 5, we present the eigenvectors\nof the surface modes as functions of the transverse co-\nordinate\u0018for increasing values of the wave vector Q.\nAtQL1= 0:5, the modes have already begun to ex-\nhibit some asymmetry. Note that the renormalization\nof the damping observed in Fig. 4(b) is approximately\none order of magnitude larger than the intrinsic Gilbert\ndamping for the optical mode and that the damping of\nany one mode may vary by several orders of magnitude\nas a function of QL1.31Therefore, these e\u000bects should\nbe experimentally observable. The greatest damping oc-\ncurs when the two layers are completely decoupled; see\nFigs. Fig. 4(b) and 5. Because the damping of the opti-\ncal mode is equivalent to that of a system with a perfect\nspin sink, one might expect that the greatest damping11\nFIG. 5: (Color online)FI(1000nm) jN(200nm)jFI(1000nm),\nlsf= 350nm, KS= 0:05 erg=cm2: a) and b) present the\nreal parts of the xcomponents of the out-of-equilibrium mag-\nnetization vectors for the acoustic and optical surface modes,\nrespectively, for several values of QL1. For values of QL1&1,\nthe modes decouple and become localized in one of the two\nlayers. For large values of QL1\u0018100, the two modes are\nstrongly localized at one of the two FI jN interfaces, which\ncorrespond to the peaks in the damping that are apparent in\nFig. 4(b).\nshould occur for this mode. However, the large localiza-\ntion, which is achieved only at large QL1, in combination\nwith the vanishing of the e\u000bective spin-di\u000busion length\nleads to damping that is much greater than that of the\nsynchronized optical mode.\nV. CONCLUSIONS\nWe investigated the dynamic coupling of spin-wave ex-\ncitations, which are present in single FI thin \flms, pri-\nmarily through spin pumping and spin transfer but also\nthrough the dynamic demagnetization \feld created when\ntwo FI thin \flms are in contact via an NM layer. Because\nof this coupling, the modes are split into acoustical and\noptical excitations. When the NM is thin compared with\nlsf, the renormalization of the Gilbert damping vanishes\nfor the acoustic modes, whereas for the optical modes,\nthe renormalization is equally as large as for a single-\nFIjN system in which the NM is a perfect spin sink. A\nspin current pumped by a travelling magnetic wave has a\nwavelength of equal magnitude, which leads to traversal\npaths across the NM that are longer than the thickness\nof the NM. Consequently, the spin-memory loss is greater\nfor short-wavelength spin currents. This phenomenonleads to an e\u000bective spin-di\u000busion length in the NM that\ndecreases for increasing values of Q. As a result, the dy-\nnamic coupling strength is reduced for short-wavelength\nspin waves. At some critical value of Q, the coupling be-\ncomes so weak that the acoustic- and optical-mode con-\n\fgurations are lost in favor of modes that are localized\nin one of the two FIs. At these values of Q, the inter-\nlayer dipole coupling is also dominated by the intralayer\nexchange coupling. For these high-wave-number modes,\nthe system behaves similar to two separate FI jN(lsf= 0)\nsystems.\nWhen the two \flms are of di\u000berent thicknesses, the\nexchange energies of the higher-order transverse n > 1\nmodes di\u000ber between the two layers. Because of the rel-\natively small coupling attributable to spin pumping, the\nsynchronization of these modes at QL1\u001c1 requires that\nthe FI thicknesses be very similar. A small asymmetry\nbreaks the synchronization; however, for larger QL1\u00181,\nthe modes can again become coupled through interlayer\ndipole interaction. This coupling arises in addition to\nthe spin-pumping- induced coupling. For even larger Q,\nthe e\u000bective spin-di\u000busion length becomes small, and the\ncoupling attributable to spin pumping vanishes. The rel-\natively small dipole coupling alone is not su\u000ecient to\ncouple the modes when there is a \fnite di\u000berence in \flm\nthickness , and the synchronization breaks down.\nDepending on the quality of the interface between the\nFIs and the strength of the spin-orbit coupling in the\nNM , additional e\u000bective surface \felds may be present\nbecause of surface anisotropy energies. For the EASA\ncase, the lowest-energy modes are localized at the FI jN\nsurfaces. These modes couple in the same manner as the\nmacrospin modes. For \flms that are much thicker than\nthe decay length A=KS, the energies of the surface modes\ndo not depend on the \flm thickness. Consequently, the\ncoupling of these modes is independent of the thickness\nof the two FIs. Similar to the simpler FI jN system, the\ndamping enhancement may attain values as high as an or-\nder of magnitude larger than the intrinsic Gilbert damp-\ning. However, in the trilayer system, the presence of both\nacoustic and optical modes results in large variations in\nthe e\u000bective damping within the same physical sample.\nBecause of this wide range of e\u000bective damping, which\nspans a di\u000berence in \u0001 \u000bof several orders of magnitude\nas a function of Q, we suggest that trilayer modes should\nbe measurable in an experimental setting.\nWith more complicated FI structures in mind, we be-\nlieve that this work may serve as a guide for experimen-\ntalists. 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(9) can be rotated by the xyzcoordinate system\nwith the rotation matrix\nR=0\n@s\u0012\u0000c\u0012s\u0012\u0000c\u0012c\u001e\n0c\u001e\u0000s\u001e\nc\u0012s\u0012s\u001es\u0012c\u001e1\nA; (A1)where we have introduced the shorthand notation s\u0012\u0011\nsin\u0012,c\u0012\u0011cos\u0012and so on. We then get that\n(*\n^Gxyz=R^GRT\n=0\n@s2\n\u0012G\u0018\u0018\u0000c\u001es2\u0012G\u0018\u0010+c2\n\u0012c2\n\u001eG\u0010\u0010\u0000s\u001es\u0012G\u0018\u0010+s\u001ec\u001ec\u0012G\u0010\u0010s\u0012c\u0012G\u0018\u0018\u0000s\u0012c\u0012c2\n\u001eG\u0010\u0010+c\u001e(s2\n\u0012\u0000c2\n\u0012)G\u0018\u0010\n\u0000s\u001es\u0012G\u0018\u0010+s\u001ec\u001ec\u0012G\u0010\u0010 s2\n\u001eG\u0010\u0010 \u0000s\u001ec\u0012G\u0018\u0010+s\u001es\u0012c\u001eG\u0010\u0010\ns\u0012c\u0012G\u0018\u0018\u0000s\u0012c\u0012c2\n\u001eG\u0010\u0010+c\u001e(s2\n\u0012\u0000c2\n\u0012)G\u0018\u0010\u0000s\u001ec\u0012G\u0018\u0010+s\u001es\u0012c\u001eG\u0010\u0010c2\n\u0012G\u0018\u0018+s2\u0012c\u001eG\u0018\u0010+c2\n\u001es2\n\u0012G\u0010\u00101\nA:\n(A2)\nBecause we work in the linear respons regime the equilibrium magnetization should be orthogonal to the dynamic\ndeviation, mi\u0001^z= 0, it is therefor su\u000ecient to only keep the xypart of ^Gxyz. We then \fnd\n^Gxy=\u0012s2\n\u0012G\u0018\u0018\u0000c\u001es2\u0012G\u0018\u0010+c2\n\u0012c2\n\u001eG\u0010\u0010\u0000s\u001es\u0012G\u0018\u0010+s\u001ec\u001ec\u0012G\u0010\u0010\n\u0000s\u001es\u0012G\u0018\u0010+s\u001ec\u001ec\u0012G\u0010\u0010 s2\n\u001eG\u0010\u0010\u0013\n: (A3)\nAppendix B: Spin Accumulation\nThe functions \u0000 1(\u0018) and \u0000 2(\u0018) are taken directly from\nRef.39, and modi\fed to cover the more complicated mag-\nnetic texture model. We then have\n\u00001(\u0018)\u0011cosh\u0010\n\u0018=~lsf\u0011\ncosh\u0010\n\u0018=~lsf\u0011\n+\u001bsinh\u0010\n\u0018=~lsf\u0011\n=2g?~lsf;\n\u00002(\u0018)\u0011sinh\u0010\n\u0018=~lsf\u0011\nsinh\u0010\n\u0018=~lsf\u0011\n+\u001bcosh\u0010\n\u0018=~lsf\u0011\n=2g?~lsf:(B1)\nForQlsf\u001d1 the e\u000bective spin di\u000busion length becomes\nshort, \u0000 1!1 and \u0000 2!0 at the FIjN interfaces.\nAppendix C: E\u000bective spin di\u000busion length\nThe di\u000busion in the NM reads\n@t\u0016S=Dr2\u0016S\u00001\n\u001csf\u0016S; (C1)whereDis the di\u000busion constant and \u001csfis the spin \rip\nrelaxation time. We assume that the FMR frequency is\nmuch smaller than the electron traversal time, D=d2\nN, and\nthe spin-\rip relaxation rate, 1 =\u001csf.39This means the LHS\nof Eq. (C1) can be disregarded. In linear response the\nspin accumulation, which is a direct consequence of spin\npumping, must be proportional to the rate of change of\nmagnetization at the FI jN interfaces. We do the same\nFourier transform, as for the magnetization, so that \u0016\u0018\nexpfi(!t\u0000Q\u0010)g. The spin di\u000busion equation then takes\nthe form\n@2\n\u0018\u0016S=\u0012\nQ2+1\nD\u001csf\u0013\n\u0016S: (C2)\nThe spin di\u000busion length is then lsf=pD\u001csf, and\nby introducing the e\u000bective spin di\u000busion length ~lsf=\nlsf=q\n1 + (Qlsf)2one gets\n@2\n\u0018\u0016S=1\n~l2\nsf\u0016S: (C3)"    },    {        "title": "0812.3184v2.Origin_of_intrinsic_Gilbert_damping.pdf",        "content": "Origin of Intrinsic Gilbert Damping\nM. C. Hickey\u0003and J. S. Moodera\nFrancis Bitter Magnet Laboratory, Massachusetts Institute of Technology,\n150 Albany Street, Cambridge, Massachusetts 02139 USA.\nThe damping of magnetization, represented by the rate at which it relaxes to equilibrium, is\nsuccessfully modeled as a phenomenological extension in the Landau-Lifschitz-Gilbert equation.\nThis is the damping torque term known as Gilbert damping and its direction is given by the vector\nproduct of the magnetization and its time derivative. Here we derive the Gilbert term from \frst\nprinciples by a non-relativistic expansion of the Dirac equation. We \fnd that this term arises when\none calculates the time evolution of the spin observable in the presence of the full spin-orbital\ncoupling terms, while recognizing the relationship between the curl of the electric \feld and the time\nvarying magnetic induction.\nPACS numbers: 76.20.-m, 75.30.-m and 75.45.+j\nThe Gilbert damping torque in magnetic systems de-\nscribes the relaxation of magnetization and it was intro-\nduced into the Laudau-Lifschitz equation [1, 2] for de-\nscribing spin dynamics. Gilbert damping is understood\nto be a non-linear spin relaxation phenomenon and it con-\ntrols the rate at which magnetization spins reach equilib-\nrium. The introduction of this term is phenomenological\nin nature [3] and the question of whether it has an in-\ntrinsic physical origin has not been fully addressed, in\nthe face of rather successful modeling of the relaxation\ndynamics of measured systems. Correlating ferromag-\nnetic resonance spectral line-widths [4, 5] in magnetic\nthin \flms with the change in damping has been success-\nful for con\frming the form of the damping term in the\nunderlying dynamical equations. The intrinsic origin of\nthe damping itself is still an open question. The damping\nconstant,\u000bis often reformulated in terms of a relaxation\ntime, and the dominant relaxation processes are invoked\nto calculate this, but this approach presupposes preces-\nsional damping torque.\nIt has been long thought that intrinsic Gilbert damp-\ning had its origin in spin-orbital coupling because this\nmechanism does not conserve spin, but it has never been\nderived from a coherent framework. Non-local spin re-\nlaxation processes [6] and disorder broadening couple to\nthe spin dynamics and can enhance the Gilbert damp-\ning extrinsically in thin \flms and heterostructures. This\ntype of spin relaxation, which is equivalent to ensemble\ndephasing [7], is modeled as the (S-S 0)/T\u0003\n2decay term\nin the dynamical Bloch equation, where T\u0003\n2is the decay\ntime of the ensemble of spins. Crudely speaking, during\nspin relaxation, some spins lag behind the mean mag-\nnetization vector and the exchange and magnetostatic\n\felds then exert a time dependent torque. Calculations\non relaxation driven damping of this kind presuppose the\nGilbert damping term itself which begs the question.\nThe inhomogeneous damping term can be written as\nM\u0002dr2M=dtwhich gives rise to non-local e\u000bects such\n\u0003Electronic mail : hickey@mit.eduas spin wave dissipation [6, 8]. These non-local theo-\nries are successful in quantifying the enhancement of the\nGilbert damping, but do not derive the intrinsic Gilbert\nterm itself. There are models [9, 10] which deal with\nthe scattering of electron spins from thermal equilibrium\nin the presence of phonon and spin-orbital interactions\nwhich is a dynamic interaction and this allows us to de-\ntermine the strength of the Gilbert damping for itiner-\nant ferromagnetic metals, generalizing the Gilbert damp-\ning response to a tensorial description. Both the s-d\nexchange relaxation models [11, 12] and the Fermi sur-\nface breathing models of Kambersky [9, 13] either pre-\nsuppose a Gilbert damping term in the dynamical equa-\ntion or specify a phenomenological Hamiltonian H = -\n1/(\rMs)^\u000b.dM/dt. While this method is ab initio from\nthe point of view of electronic structure, it already as-\nsumes the Gilbert term ansatz. Hankiewicz et al. [14]\nconstruct the inhomogeneous Gilbert damping by con-\nnecting the spin density-spin current conservation law\nwith the imaginary part of magnetic susceptibility ten-\nsor and show that both electron-electron and impurity\nscattering can enhance the damping through the trans-\nverse spin conductivity for \fnite wavelength excitations\n(q6= 0). In previous work [15], there are derivations\nof the Gilbert constant by comparing the macroscopic\ndamping term with the torque-torque correlations in ho-\nmogeneously magnetized electron gases possessing spin\norbital coupling. For the case of intrinsic, homogeneous\nGilbert damping, it is thought that in the absence of\nspin-orbital scattering, the damping vanishes. We aim to\nfocus on intrinsic, homogeneous damping and its physical\norigin in a \frst-principles framework and the question as\nto whether spin in a homogeneous time-varying magne-\ntization can undergo Gilbert damping is addressed.\nIn this work, we show that Gilbert damping does indeed\narise from spin-orbital coupling, in the sense that it is\ndue to relativistic corrections to the Hamiltonian which\ncouple the spin to the electric \feld and we arrive at the\nGilbert damping term by \frst writing down the Dirac\nequation for electrons in magnetic and electric potentials.\nWe transform the Hamiltonian in such a way as to write\nit in a basis in which the canonical momentum terms arearXiv:0812.3184v2  [cond-mat.other]  1 Apr 20092\neven powers. This is a standard approach in relativistic\nquantum mechanics and we do this in order to calculate\nthe terms which couple the linear momentum to the spin\nin a basis which is diagonal in spin space. This is often\nreferred to as a non-relativistic expansion of the Dirac\nequation. This allows us to formulate the contributions\nas a perturbation to an otherwise non-relativistic parti-\ncle. We then wish to calculate the rate equation for the\nspin observable with all of the spin-orbital corrections in\nmind.\nNow, we start with a purely relativistic particle, a Dirac\nparticle and we write the Dirac-Pauli Hamiltonian, as\nfollows :\nH=c\u000b:(p\u0000eA\nc) +\fm 0c2+e\u001e (1)\n=O+\fm 0c2+\" (2)\nwhere Aand\u001eare the magnetic vector potential and the\nelectrostatic potential, respectively and\n\u000b=\u0012\n0\u001bi\n\u001bi0\u0013\nwhile\n\f=\u0012\n1 0\n0\u00001\u0013\n:\nWe observe immediately that \fO=\u0000O\f.Ois the Dirac\ncanonical momentum , c and e are the speed of light in\na vacuum and the electronic charge, respectively.\nWe now need to rewrite the Hamiltonian in a basis where\nthe odd operators (whose generators are o\u000b diagonal in\nthe Pauli-Dirac basis : \u000bi,\ri,\r5..) and even operators\n(whose generators are diagonal in the Pauli-Dirac basis :\n(1,\f, \u0006,.. ) are decoupled from one another.\nIf we are to \fnd S so that H0does not contain odd powers\nof spin operators, we must chose the operator S, in such\na way as to satisfy the following constraint :\n[S;\f] =\u0000O\nim0c2(3)\nIn order to satisfy cancelation of the odd terms of O\nto \frst order, we require S=\u0000iO\f\n2m0c2and this is known\nas the Foldy-Wouthuysen transformation in relativistic\nquantum mechanics and it is treated in some detail in,\nfor example, reference [16]. We now would like to collect\nall of the terms into the transformed Hamiltonian, and\nthis is written as\nH0=\f\u0012\nm0c2+O2\n2m0c2\u0000O4\n8m3\n0c6\u0013\n+\"\u00001\n8m2\n0c4[O;[O;\"]] +\f\n2m0c2[O;\"]\u0000O3\n3m2\n0c4\nThe expression above contains odd powers of the canon-\nical momentumO, so we rede\fne the canonical momen-\ntum to encapsulate all of these odd power terms. So wenow apply the procedure of eliminating odd powers once\nagain :\nS0=\u0000i\f\n2m0c2O0=\u0000i\f\n2m0c2\u0012\f\n2m0c2[O;\"]\u0000O3\n3m2\n0c4\u0013\n(4)\nH00=eiS0\nH0e\u0000iS0\n=\fm 0c2+\"0+O00; (5)\nwhereO00is now O(1\nm2\n0c4), which can be further elimi-\nnated by applying a third transformation (S00=\u0000i\fO00\n2m0c4),\nwe arrive at the following Hamiltonian :\nH000=eiS00\u0010\nH00\u0011\ne\u0000iS00\n=\fm 0c2+\"0\n=\f\u0012\nm0c2+O2\n2m0c2\u0000O4\n8m3\n0c6\u0013\n+\n\"\u00001\n8m2\n0c4[O;[O;\"]]\nThus we have the fully Foldy-Wouthuysen transformed\nHamiltonian :\nH000=\f\u0012\nm0c2+(p\u0000eA=c)2\n2m0\u0000p4\n8m3\n0c6\u0013\n+e\b\n\u0000e~\n2m0c2\f\u0006:B\u0000ie~2\n8m2\n0c2\u0006:(r\u0002E)\n\u0000e~\n4m2\n0c2\u0006:E\u0002p\u0000e~2\n8m2\n0c2(r:E)\nThe terms which are present in the above Hamiltonian,\nshow us that we have a p4kinetic part which is the rela-\ntivistic expansion of the mass of the particle. The terms\nwhich couple to the spin \u0006 are of importance and we see\nthat these terms correspond to the Zeeman, spin-orbital\n(comprising momentum and electric \feld curl terms) and\nthe Darwin term, respectively. Strictly speaking, the\npresence of the scalar potential \u001ebreaks the gauge invari-\nance in the Pauli-Dirac Hamiltonian and a fully gauge in-\nvariant theory would require that this contain the gauge-\nfree electromagnetic \feld energy. We omit the term\ne2~\n4m2c3\u0006:(A\u0002E) (which establishes gauge invariance in\nthe momentum terms) in this rotated Hamiltonian, as it\nis O(1=m2c3) and we are only interested in calculating\nsemiclassical rate equations for \felds, which are mani-\nfestly gauge-invariant, and not wavefunctions or energy\neigenvalues. We can now de\fne the spin dependent cor-\nrections to a non-relativistic Hamiltonian :\nH\u0006=\u0000e~\n2m0c2\f\u0006:B\u0000e~\n4m2\n0c2\u0006:E\u0002p\u0000ie~2\n8m2\n0c2\u0006:(r\u0002E):\n(6)\nwhere\n\u0006=\u0012\n\u001bi0\n0\u001bi\u0013\n\u0011^Si:3\nand\u001biare the Pauli matrices. Note that the last\ntwo terms in Equation 6 encapsulate the entire spin\norbital coupling in the sense that these terms couple\nthe particle's linear momentum to the spin ^Si. The\n\frst spin-orbital term in the Hamiltonian is well known\nand give rise to momentum dependent magnetic \felds.\nWhen the ensuing dynamics are calculated for this\ncase, it gives rise to spin relaxation terms which are\nlinear in spin [17]. Note that, while neither spin-orbital\nterm is Hermitian, the two terms taken together are\nHermitian and so the particles angular momentum\nis a conserved quantity and the total energy lost in\ngoing from collective spin excitations (spin waves) to\nsingle particles states via spin-orbital coupling is gained\nby the electromagnetic \feld. Recognizing the curl of\nthe electric \feld in the last term, we now rewrite this\nthe time varying magnetic \feld as given by Maxwells\nequations asr\u0002E=\u0000@B\n@t. We now have an explicitly\ntime-dependent perturbation on the non-relativistic\nHamiltonian. We can write the time-varying magnetic\n\feld seen by the spin (in, for example a magnetic\nmaterial) as@B\n@t=@B\n@M\u0001@M\n@t=\u00160(1 +\u001f\u00001\nm)@M\n@t. We now\nhave the spin dependent Hamiltonian :\nHS=\u0000e~\n2m0c2\fS:B\u0000e~\n4m2\n0c2S:E\u0002p\n+ie~2\u00160\n8m2\n0c2S:\u0000\n1 +\u001f\u00001\nm\u0001\n:dM\ndt=\nHS=HS\n0+HS(t):\nWe focus our attention on the explicitly time-dependent\npart of the Hamiltonian HS(t) ;\nHS(t) =ie~2\u00160\n8m2\n0c2S:\u0000\n1 +\u001f\u00001\nm\u0001\n:dM\ndt: (7)\nIn this perturbation scheme, we allow the Hermitian\ncomponents of the Hamiltonian to de\fne the ground sate\nof the system and we treat the explicitly time-dependent\nHamiltonian (containing the spin orbital terms) as a time\ndependent perturbation. In this way, the rate equation is\nestablished from a time dependent perturbation expan-\nsion in the quantum Liouville description. We now de\fne\nthe magnetization observable as ^M=X\n\u000bg\u0016B\nVTr\u001a^S\u000b(t)\nwhere the summation is taken over the site of the magne-\ntization spin \u000b. We now examine the time dependence of\nthis observable by calculating the rate equation according\nto the quantum-Liouville rate equation ;\nd\u001a(t)\ndt+1\ni~[^\u001a;H] = 0 (8)\nThis rate equation governs the time-evolution of the\nmagnetization observable as de\fned above, in the non-\nequilibrium regime. We can write the time derivative ofthe magnetization [18], as follows ;\ndM\ndt=X\nn;\u000bg\u0016b\nVh\tn(t)j1\ni~[\u001aS\u000b;H] +@\u001a\n@tS\u000b+\u001a@S\u000b\n@tj\tn(t)i;\nand we can use the quantum Liouville rate equation as\nde\fned by Equation 8 to simplify this expression and we\narrive at the following rate equation :\ndM\ndt=X\n\u000bg\u0016b\nV1\ni~Trf\u001a[S\u000b;HS(t)]g (9)\nIn the case of the time dependent Hamiltonian derived\nin equation 7, we can assume a \frst order dynamical\nequation of motion given bydM\ndt=\rM\u0002Hand calculate\nthe time evolution for the magnetization observable :\ndM\ndt=X\n\u000b;\fg\u0016B\nV1\ni~Tr\u001a[Si\n\u000b;ie~2\u00160\n8m2c2Sj\n\f]:(1 +\u001f\u00001\nm) !@M\ndt\n=X\n\u000bg\u0016B\nVie~2\u00160\n8m2c21\ni~Tr\u001ai ~\u000fijkSk\n\u000b\u000e\u000b\f(1 +\u001f\u00001\nm)\u000ejl !@ Ml\ndt\n=\u0000ie~\u00160\n8m2c2(1 +\u001f\u00001\nm)M\u0002 !@M\ndt;\nwhere, in the last two steps, we have used the fol-\nlowing commutation relations for magnetization spins :\n[Si\n\u000b;Sj\n\f] =i~\u000fijkSk\n\u000b\u000e\u000b\fwhich implies that the theory pre-\nsented here is that which relates to local dynamics and\nthat the origin of the damping is intrinsic. We now rec-\nognize the last equation as the which describes Gilbert\ndamping, as follows :\ndM\ndt=\u0000\u000b\nMs:M\u0002 !@M\n@t(10)\nwhereby the constant \u000bis de\fned as follows :\n\u000b=ie~\u00160Ms\n8m2\n0c2\u0000\n1 +\u001f\u00001\nm\u0001\n(11)\nThe\u000bde\fned above corresponds with the Gilbert\ndamping found in the phenomenological term in the\nLandau-Lifschitz-Gilbert equation and \u001fmis the mag-\nnetic susceptibility. In general, the inverse of the suscep-\ntibility can be written in the form [19],\n\u001f\u00001\nij(q;!) = ~\u001f\u00001\n?(q;!)\u0000!ex\n\r\u00160M0\u000eij; (12)\nwhere the equilibrium magnetization points along the z-\naxis and!exis the excitation frequency associated with\nthe internal exchange \feld. The \u000eijterm in the in-\nverse susceptibility does not contribute to damping mech-\nanisms as it corresponds to the equilibrium response.4\nIn the basis (M x\u0006iMy,Mz), we have the dimensionless\ntransverse magnetic susceptibility, as follows :\n~\u001fm?(q;!) =\r\u00160M0\u0000i\r\u001b?q2\n!0\u0000!\u0000i\r\u001b?q2!0=M0\nThe \frst term in the dimensionless Gilbert coe\u000ecient\n(Equation 11) is small ( \u001810\u000011) and the higher damp-\ning rate is controlled by the the inverse of the suscep-\ntibility tensor. For uniformly saturated magnetization,\nthe damping is critical and so the system is already at\nequilibrium as far as the Gilbert mechanism is concerned\n(dM/dt = 0 in this scenario). The expression for the\ndimensionless damping constant \u000bin the dc limit ( !=0\n) is :\n\u000b=e~\u00160Ms\n8m2\n0c2Im0\n@!0\n\r\u00160M0\u0000i\u001b?q2!0\n\u00160M2\n0\n1\u0000i\r\u001b?q2=M01\nA; (13)\nand we have the transverse spin conductivity from the\nfollowing relation (in units whereby ~=1) :\n\u001b?=n\n4m\u0003!2\n0\u00121\n\u001cdis\n?+1\n\u001cee\n?\u0013\n;\nwhere\u001cdis\n?and\u001cee\n?are the impurity disorder and electron\nelectron-electron scattering times as de\fned and param-\neterized in Reference [14]. We calculate the extrinsically\nenhanced Gilbert damping using the following set of pa-\nrameters as de\fned in the same reference ; number den-\nsity of the electron gas, n=1.4 \u00021027m\u00003, polarization p,\nequilibrium magnetization M 0=\rpn/2, equilibrium ex-\ncitation frequency !0=EF[(1 +p)2=3\u0000(1\u0000p)2=3] and\nwave-number de\fned as q = 0.1 k F, where E Fand kF\nare the Fermi energy and Fermi wave number, respec-\ntively. m\u0003is taken to be the electronic mass. Using these\nquantities, we evaluate \u000bvalues and these are plotted as\na function of both polarization and disorder scattering\nrate in Figure 1.\nIn general, the inverse susceptibility \u001f\u00001\nmwill deter-\nmine the strength of the damping in real inhomogeneous\nmagnetic systems where spin relaxation takes place, sub-\nbands are populated by spin orbit scattering and spin\nwaves and spin currents are emitted. The susceptibil-\nity term gives the Gilbert damping a tensorial quality,\nagreeing with the analysis in Reference [10]. Further, the\nconnection between the magnetization dynamics and the\nelectric \feld curl provides the mechanism for the energy\nloss to the electromagnetic \feld. The generation of radi-\nation is caused by the rotational spin motion analog of\nelectric charge acceleration and the radiation spin inter-\naction term has the form :\nHS(t) =ie~2\u00160\n8m2\n0c2X\n\u000b\u0000\n1 +\u001f\u00001\nm\u0001\nS\u000b:dM\ndt: (14)In conclusion, we have shown that the Gilbert term,\nheretofore phenomenologically used to describe damping\nFIG. 1: (Color Online) Plot of the dimensionless Gilbert\ndamping constant \u000bin the dc limit ( !=0), as a function of\nelectron spin polarization and disorder scattering rate.\nin magnetization dynamics, is derivable from \frst prin-\nciples and its origin lies in spin-orbital coupling. By a\nnon-relativistic expansion of the Dirac equation, we show\nthat there is a term which contains the curl of the elec-\ntric \feld. By connecting this term with Maxwells equa-\ntion to give the total time-varying magnetic induction,\nwe have found that this damping term can be deduced\nfrom the rate equation for the spin observable, giving the\ncorrect vector product form and sign of Gilberts' origi-\nnal phenomenological model. Crucially, the connection\nof the time-varying magnetic induction and the curl of\nthe electric \feld via the Maxwell relation shows that\nthe damping of magnetization dynamics is commensu-\nrate with the emission of electromagnetic radiation and\nthe radiation-spin interaction is speci\fed from \frst prin-\nciples arguments.\nAcknowledgments\nM. C. Hickey is grateful to the Trinity and the uni-\nformity of nature. We thank the U.S.-U.K. Fulbright\nCommission for \fnancial support. The work was sup-\nported by the ONR (grant no. N00014-09-1-0177), the\nNSF (grant no. DMR 0504158) and the KIST-MIT pro-\ngram. The authors thank David Cory, Marius Costache\nand Carlos Egues for helpful discussions.5\n[1] L. Landau and E. Lifshitz, Phys. Z. Sowiet. Un. 8, 153\n(1935).\n[2] E. M. Lifschitz and L. P. Pitaevskii, Statistical Physics\nPart 2 (Pergamon Press, Oxford, United Kingdom,\n1980).\n[3] T. Gilbert, Magnetics, IEEE Transactions on 40, 3443\n(2004).\n[4] C. E. Patton, C. H. Wilts, and F. B. Humphrey, J. Appl.\nPhys. 38, 1358 (1967).\n[5] M. Oogane, T. Wakitani, S. Yakata, R. Yilgin, Y. Ando,\nA. Sakuma, and T. Miyazaki, Japanese Journal of Ap-\nplied Physics 45, 3889 (2006).\n[6] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n[7] Y. Tserkovnyak, G. A. Fiete, and B. I. Halperin, Appl.\nPhys. Lett. 84, 5234 (2004).\n[8] G. Eilers, M. L uttich, and M. M unzenberg, Phys. Rev.\nB74, 054411 (2006).\n[9] V. Kambersk\u0012 y, Canadian Journal of Physics 48, 2906\n(1970).[10] D. Steiauf and M. F ahnle, Phys. Rev. B 72064450\n(2005).\n[11] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n[12] M. F ahnle, R. Singer, D. Steiauf, and V. P. Antropov,\nPhys. Rev. B 73, 172408 (2006).\n[13] J. Kune\u0014 s and V. Kambersk\u0013 y, Phys. Rev. B 65, 212411\n(2002).\n[14] E. M. Hankiewicz, G. Vignale, and Y. Tserkovnyak,\nPhys. Rev. B 78, 020404 (2008).\n[15] E. M. Hankiewicz, G. Vignale, and Y. Tserkovnyak,\nPhys. Rev. B 75, 174434 (2007).\n[16] W. Greiner, Relativistic Quantum Mechanics (Springer-\nVerlag, Berlin, Germany, 1987).\n[17] H.-A. Engel, E. I. Rashba, and B. I. Halperin, Phys. Rev.\nLett. 98, 036602 (2007).\n[18] J. Ho, F. C. Khanna, and B. C. Choi, Phys. Rev. Lett.\n92, 097601 (2004).\n[19] Z. Qian and G. Vignale, Phys. Rev. Lett. 88, 056404\n(2002)."    },    {        "title": "2206.09969v1.First_principles_calculation_of_the_parameters_used_by_atomistic_magnetic_simulations.pdf",        "content": "arXiv:2206.09969v1  [cond-mat.mtrl-sci]  20 Jun 2022APS/123-QED\nFirst-principles calculation of the parameters used by ato mistic magnetic simulations\nSergiy Mankovsky and Hubert Ebert\nDepartment of Chemistry/Phys. Chemistry, LMU Munich,\nButenandtstrasse 11, D-81377 Munich, Germany\n(Dated: June 22, 2022)\nWhile the ground state of magnetic materials is in general we ll described on the basis of spin den-\nsity functional theory (SDFT), the theoretical descriptio n of finite-temperature and non-equilibrium\nproperties require an extension beyond the standard SDFT. T ime-dependent SDFT (TD-SDFT),\nwhich give for example access to dynamical properties are co mputationally very demanding and can\ncurrently be hardly applied to complex solids. Here we focus on the alternative approach based on\nthe combination of a parameterized phenomenological spin H amiltonian and SDFT-based electronic\nstructure calculations, giving access to the dynamical and finite-temperature properties for example\nvia spin-dynamics simulations using the Landau-Lifshitz- Gilbert (LLG) equation or Monte Carlo\nsimulations. We present an overview on the various methods t o calculate the parameters of the\nvarious phenomenological Hamiltonians with an emphasis on the KKR Green function method as\none of the most flexible band structure methods giving access to practically all relevant parameters.\nConcerning these, it is crucial to account for the spin-orbi t coupling (SOC) by performing rela-\ntivistic SDFT-based calculations as it plays a key role for m agnetic anisotropy and chiral exchange\ninteractions represented by the DMI parameters in the spin H amiltonian. This concerns also the\nGilbert damping parameters characterizing magnetization dissipation in the LLG equation, chiral\nmultispin interaction parameters of the extended Heisenbe rg Hamiltonian, as well as spin-lattice\ninteraction parameters describing the interplay of spin an d lattice dynamics processes, for which an\nefficient computational scheme has been developed recently b y the present authors.\nPACS numbers: 71.15.-m,71.55.Ak, 75.30.Ds\nI. INTRODUCTION\nDensity functional theory (DFT) is a ’formally exact\napproachto the static electronic many-body problem’ for\ntheelectrongasintheequilibrium, whichwasadoptedfor\na huge number of investigations during the last decades\nto describe the ground state of solids, both magnetic and\nnon-magnetic,aswellasvariousgroundstateproperties1.\nHowever, dealing with real systems, the properties\nin an out-of-equilibrium situation are of great interest.\nAn example for this is the presence of external pertur-\nbation varying in time, which could be accounted for\nby performing time-dependent first-principles electronic\nstructure calculations. The time-dependent extension\nof density functional theory (TD-DFT)2is used suc-\ncessfully to study various dynamical processes in atoms\nand molecules, in particular, giving access to the time\nevolution of the electronic structure in a system af-\nfected by a femtosecond laser pulse. However, TD-DFT\ncan be hardly applied to complex solids because of the\nlack of universal parameter-free approximations for the\nexchange-correlation kernel. Because of this, an ap-\nproach based on the combination of simulation methods\nfor spin- and lattice dynamics, using model spin and lat-\ntice Hamiltonians is more popular for the moment. A\ngreat progress with this approach has been achieved dur-\ning last decade due to the availability of parameters for\nthe model Hamiltonians calculated on a first principles\nlevel, that is a central issue of the present contribution.\nAs it was pointed out in Ref. 1, this approach has the ad-\nvantage, that the spin-related many-body effects in thiscase are much simpler to be taken into account when\ncompared to the ab-initio approach. Thus, the isotropic\nexchangecouplingparameters JijfortheclassicalHeisen-\nberg Hamiltonian worked out Liechtenstein et al.3,4have\nbeen successfully used by many authors to predict the\nground state magnetic structure of material and to in-\nvestigateitsfinite-temperatureproperties. Dependingon\nthe materials, the isotropic Jijcan exhibit only spatial\nanisotropy. Extension of the Heisenberg Hamiltonian ac-\ncounting for anisotropy in spin subspace is often done by\nadding the so-called Dzyaloshinskii-Moriya interactions\n(DMI) and the magnetic anisotropy term,\nHH,rel=−/summationdisplay\ni,jJij(ˆei·ˆej)−/summationdisplay\ni,j/vectorDij(ˆei׈ej)+/summationdisplay\niˆeiKiiˆei.\n(1)\nwith ˆei(j)the orientation of the spin magnetic moment at\nsitei(j). Alternatively, one may describe exchange inter-\nactions in the more general tensorial form, Jij, leading\nto:\nHH,rel=−/summationdisplay\ni,jˆeiJijˆej+/summationdisplay\niˆeiKiiˆei,(2)\nIn the second case the DMI is represented as the an-\ntisymmetric part of the exchange tensor, i.e. Dα\nij=\n1\n2(Jβγ\nij−Jγβ\nij)ǫαβγ. It should be stressed, that calcula-\ntions of the spin-anisotropic exchange interaction param-\neters as well as of the magnetic anisotropy parameters\nrequire a relativistic treatment of the electronic struc-\nture in contrast to the case of the isotropic exchange pa-\nrameters which can be calculated on a non-relativistic2\nlevel. Various schemes to map the dependence of the\nelectronicenergyonthemagneticconfigurationweresug-\ngested in the literature to calculate the parameters of the\nspin Hamiltonians5–8, depending of its form given in Eqs.\n(1) or (2).\nDespite of its simplicity, the spin Hamiltonian gives\naccess to a reasonable description of the temperature\ndependence of magnetic properties of materials when\ncombined with Monte Carlo (MC) simulations9, or non-\nequilibrium spin dynamics simulations based on the phe-\nnomenological Landau-Lifshitz-Gilbert equations10,11\n1\nγd/vectorM\ndτ=−/vectorM×/vectorHeff+/vectorM×/bracketleftBigg˜G(/vectorM)\nγ2M2sd/vectorM\ndτ/bracketrightBigg\n.(3)\nHere/vectorHeffis the effective magnetic field defined as /vectorHeff=\n−1\nM∂F\n∂ˆm, whereFis the free energy of the system and\nˆm=/vectorM\n/vectorMswithMsthesaturationmagnetizationtreatedat\nfirst-principles level, and γis the gyromagnetic ratio and\n˜Gis the Gilbert damping parameter. Alternatively, the\neffective magnetic field can be representedin terms ofthe\nspin Hamiltonian in Eq. (2), i.e. /vectorHeff=−1\nM∂/angbracketleftHH,rel/angbracketrightT\n∂ˆm,\nwith/an}b∇acketle{t.../an}b∇acket∇i}htTdenoting the thermal averagefor the extended\nHeisenberg Hamiltonian HH,rel.\nThe first-principles calculation of the parameters for\nthe Heisenberg Hamiltonian as well as for the LLG equa-\ntion for spin dynamics have been reported in the litera-\nture by various groups who applied different approaches\nbased on ab-initio methods. Here we will focus on calcu-\nlations based on the Green function multiple-scattering\nformalism being a rather powerful tool to supply all pa-\nrameters for the extended Heisenberg Hamiltonian as\nwell as for the LLG equation.\nA. Magnetic anisotropy\nLet’s first consider the magnetic anisotropy term in\nspin Hamiltonian, characterized by parameters (written\nintensorialforminEqs.(1)and(2))deducedfromtheto-\ntalenergydependentontheorientationofthemagnetiza-\ntion ˆm. The latter is traditionallysplit into the magneto-\ncrystalline anisotropy(MCA) energy, EMCA(ˆm), induced\nby spin-orbit coupling (SOC) and the shape anisotropy\nenergy,Eshape(ˆm), caused by magnetic dipole interac-\ntions,\nEA(ˆm) =EMCA(ˆm)+Eshape(ˆm). (4)\nAlthough a quantum-mechanical description of the mag-\nneticshapeanisotropydeservesseparatediscussion12this\ncontribution can be reasonably well estimated based on\nclassical magnetic dipole-dipole interactions. Therefore,\nwe will focus on the MCA contribution which is fully\ndetermined by the electronic structure of the considered\nsystem. In the literature the focus is in general on the\nMCA energy of the ground state, which can be estimated\nstraightforwardlyfromthe totalenergycalculatedfordif-\nferent orientations of the magnetization followed by amapping onto a model spin Hamiltonian, given e.g. by\nan expansion in terms of spherical harmonics Ylm(ˆm)13\nEMCA(ˆm) =/summationdisplay\nlevenm=l/summationdisplay\nm=−lκm\nlYlm(ˆm).(5)\nAlternative approach to calculate the MCA parameters\nis based on magnetic torque calculations, using the defi-\nnition\nTˆm(θˆu) =−∂E(ˆm)\n∂θˆu, (6)\navoiding the time-consuming total energy calculations.\nThis scheme is based on the so-called magnetic force the-\noremthatallowstorepresenttheMCAenergyintermsof\na correspondingelectronic single-particleenergies change\nunder rotation of magnetization, as follows14:\n∆ESOC(ˆm,ˆm′) =−/integraldisplayEˆm\nF\ndE/bracketleftBig\nNˆm(E)−Nˆm′(E)/bracketrightBig\n−1\n2nˆm′(Eˆm′\nF)(Eˆm\nF−Eˆm′\nF)2\n+O(Eˆm\nF−Eˆm′\nF)3(7)\nwithNˆm(E) =/integraltextEdE′nˆm(E′) the integrated DOS for\nthe magnetization along the direction ˆ m, andnˆm(E) the\ndensityofstates(DOS) representedin termsofthe Green\nfunction as follows\nnˆm(E) =−1\nπIm TrGˆm(E). (8)\nThis expressioncan be used in a very efficient way within\nthe framework of the multiple-scattering formalism. In\nthis case the Green function is given in terms of the scat-\ntering path operator τ(E)nn′connecting the sites nand\nn′as follows\nG0(/vector r,/vector r′,E) =/summationdisplay\nΛΛ′Zn\nΛ(/vector r,E)τnn′\nΛΛ′(E)Zn′×\nΛ′(/vector r′,E)\n−/summationdisplay\nΛ/bracketleftBig\nZn\nΛ(/vector r,E)Jn×\nΛ(/vector r′,E)Θ(r′−r)\n+Jn\nΛ(/vector r,E)Zn×\nΛ(/vector r′,E)Θ(r−r′)/bracketrightBig\nδnn′,(9)\nwhere the combined index Λ = ( κ,µ) represents the rela-\ntivistic spin-orbit and magnetic quantum numbers κand\nµ, respectively15;Zn\nΛ(/vector r,E) andJn\nΛ(/vector r,E) are the regular\nand irregular solutions of the single-site Dirac equation\n(27)16–18. The scattering path operator is given by the\nexpression\nτ(E) = [m(E)−G0(E)]−1(10)\nwithm(E) =t−1(E) andG0(E) the inverse single-site\nscattering and structure constant matrices, respectively.\nThe double underline used here indicates matrices with\nrespect to site and angular momentum indices17.3\nUsing the Lloyd’s formula that gives the integrated\nDOS in terms of the scattering path operator, Eq. (7)\ncan be transformed to the form\n∆ESOC(ˆm,ˆm′) =−1\nπIm Tr/integraldisplayEF\ndE\n×/parenleftbig\nlnτ(ˆm,E)−lnτ(ˆm′,E)/parenrightbig\n(11)\nwith the scattering path operator evaluated for the mag-\nnetization along ˆ mand ˆm′, respectively.\nWith this, the magnetic torque T(θ) can be expressed\nby means of multiple scattering theory leading for the\ntorque component with respect to a rotation of the mag-\nnetization around an axis ˆ u, to the expression19\nTˆm(θˆu) =−1\nπℑ/integraldisplayEF\ndE∂\n∂θˆu/bracketleftbig\nlndet/parenleftbig\nt(ˆm)−1−G0/parenrightbig/bracketrightbig\n.\n(12)\nMapping the resulting torque onto a corresponding pa-\nrameterized expression as for example Eq. (5), one ob-\ntains the corresponding parameters of the spin Hamilto-\nnian.\nHowever,oneshouldnotethatthemagneticanisotropy\nof materials changes when the temperature increases.\nThis occurs first of all due to the increasing amplitude\nof thermally induced spin fluctuations responsible for a\nmodification of the electronic structure. A correspond-\ning expression for magnetic torque st finite temperature\nwas worked out by Staunton et al.19, on the basis of the\nrelativistic generalization of the disordered local moment\n(RDLM) theory20. To perform the necessary thermal av-\neraging over different orientational configurations of the\nlocal magnetic moments it uses a technique similar to\nthe one used to calculate the configurational average in\nthe case of random metallic alloys, so-called Coherent\nPotential Approximation (CPA) alloy theory21,22. Ac-\ncordingly, the free energy difference for two different ori-\nentations of the magnetization is given by\n∆F(ˆm,ˆm′) =−/integraldisplay\ndEfFD(E,ˆm) (13)\n/bracketleftbigg\n/an}b∇acketle{tNˆm/an}b∇acket∇i}ht(E)−/an}b∇acketle{tNˆm′/an}b∇acket∇i}ht(E)/bracketrightbigg\n.(14)\nBy using in this expression the configurational aver-\naged integrated density of states20,23given by Lloyd’s\nformula, the corresponding expression for the magnetic\ntorque at temperature T\nTˆm,T(θˆu) =−∂\n∂θˆu/parenleftbigg/summationdisplay\ni/integraldisplay\nPˆm\ni(ˆei)/an}b∇acketle{tΩˆm/an}b∇acket∇i}htˆeidˆei/parenrightbigg\n.(15)\ncan be written explicitly as:\nTˆm,T(θˆu) =−1\nπIm/integraldisplayEF\ndEfFD(E,ˆm)\n/parenleftbigg/summationdisplay\ni/integraldisplay∂Pˆm\ni(ˆei)\n∂θˆuln detMˆm\ni(ˆei,E)dˆei/parenrightbigg\n.(16)where\nMˆm\ni(ˆei,E) = 1+([ti(ˆei)]−1−tˆm\ni,c(ˆei)]−1)τˆm\nii,c,(17)\nand\nτˆm\nii,c= ([tˆm\ni,c(ˆei)]−1−G0)−1. (18)\nwhere the index cindicates quantities related to the CPA\nmedium.\nFig. 1 (top) shows as an example the results for\nthe temperature-dependent magnetization ( M(T)) cal-\nculated within the RDLM calculations for L10-ordered\nFePt24. Fig. 1 (bottom) gives the corresponding param-\neterK(T) for a uni-axial magneto-crystalline anisotropy,\nwhich is obviously in good agreement with experiment.\n200 400 600800\nTemperature T (K)00.20.40.60.8M(T)\n0.2 0.4 0.60.8\n(M(T))2-2-1.5-1-0.5∆ESOC (meV)\nFIG. 1. RDLM calculations on FePt. Top: the magneti-\nzationM(T) versus Tfor the magnetization along the easy\n[001] axis (filled squares). The full line shows the mean field\napproximation to a classical Heisenberg model for compar-\nison. Bottom: the magnetic anisotropy energy ∆ ESOCas\na function of the square of the magnetization M(T). The\nfilled circles show the RDLM-based results, the full line giv e\nK(T)∼[M(T)/M(0)]2, and the dashed line is based on the\nsingle-ion model function. All data taken from24.\nB. Inter-atomic bilinear exchange interaction\nparameters\nMost first-principles calculations on the bilinear ex-\nchange coupling parameters reported in the literature,4\nFIG. 2. Adiabatic spin-wave dispersion relations along hig h-\nsymmetry lines of the Brillouin zone for Ni. Broken line:\nfrozen-magnon-torque method, full line: transverse susce pti-\nbility method31. All data are taken from Ref. 31.\nare based on the magnetic force theorem (MFT) by\nevaluating the energy change due to a perturbation on\nthe spin subsystem with respect to a suitable reference\nconfiguration25. Many results are based on calculations\nof the spin-spiral energy ǫ(/vector q), giving access to the ex-\nchange parameters in the momentum space, J/vector q7,26–28,\nfollowed by a Fourier transformation to the real space\nrepresentation Jij. Alternatively, therealspaceexchange\nparameters are calculated directly by evaluating the en-\nergy change due to the tilting of spin moments of inter-\nacting atoms. The corresponding non-relativistic expres-\nsion (so-called Liechtenstein or LKAG formula) has been\nimplemented based on the KKR as well as LMTO Green\nfunction (GF)3,4,25,29band structure methods. It should\nbe noted that the magnetic force theorem provides a rea-\nsonable accuracy for the exchange coupling parameters\nin the case of infinitesimal rotations of the spins close to\nsome equilibrium state, that can be justified only in the\nlong wavelength and strong-coupling limits30. Accord-\ningly, calculations of the exchange coupling parameters\nbeyond the magnetic force theorem, represented in terms\nof the inverse transverse susceptibility, were discussed\nin the literature by various authors25,30–33. Grotheer et\nal., for example, have demonstrated31a deviation of the\nspin-wave dispersion curves away from Γ point in the\nBZ, calculated for fcc Ni using the exchange parameters\nJ/vector q∼χ−1\n/vector q, from the MFT-based results for J/vector q. On the\notherhand, the resultsareclosetoeachotherin the long-\nwavelength limit (see Fig. 2). The calculations beyond\nthestandardDFTaredonebymakinguseoftheso-called\nconstrained-field DFT. The latter theory was also used\nby Bruno33who suggested the ’renormalization’ of the\nexchange coupling parameters expressed in terms of non-\nrelativistic transverse magnetic susceptibility, according\ntoJ=1\n2Mχ−1M=1\n2M(˜χ−1−Ixc)M, with the various\nquantities defined as follows\n˜χ−1\nij=2\nπ/integraldisplayEF\ndE/integraldisplay\nΩid3r/integraldisplay\nΩjd3r′(19)\n×Im[G↑(/vector r,/vector r′,E)G↓(/vector r′,/vector r,E)],(20)Mi=/integraldisplay\nΩid3rm(/vector r), (21)\nand\n˜Ixc\nij=δij∆i\n2Mi, (22)\nwith ∆ i=4\nMi/summationtext\nj˜Jij, where\n˜Jij=1\nπIm/integraldisplayEF\ndE/integraldisplay\nΩid3r/integraldisplay\nΩjd3r′(23)\n×[Bxc(/vector r)G↑(/vector r,/vector r′,E)Bxc(/vector r′)G↓(/vector r′,/vector r,E)].(24)\nThis approach results in a Curie temperature of 634 K\nfor fcc Ni (vs. 350 K based on the MFT) which is in good\nagreement with the experimental value of (621 −631 K).\nAs was pointed out by Solovyev30, such a corrections can\nbe significant only for a certain class of materials, while,\nfor instance, the calculations of spin-wave energies31and\nTC33for bcc Fe demonstrate that these corrections are\nquite small. As most results in the literature were ob-\ntained using the exchange parameters based on the mag-\nnetic force theorem, we restrict below to this approxima-\ntion.\nSimilar to the case of the MCA discussed above, ap-\nplication of the magnetic force theorem gives the energy\nchange due to tilting of two spin moments represented in\nterms of the integrated DOS4. Within the multiple scat-\ntering formalism, this energy can be transformed using\nthe Lloyd’s formula leading to the expression\n∆E=−1\nπIm Tr/integraldisplayEF\ndE/parenleftbig\nlnτ(E)−lnτ0(E)/parenrightbig\n(25)\nwithτ(0)(E) andτ(E) the scattering path operators for\nnon-distorted and distorted systems, respectively.\nAs reported in Ref. 4, the expression for Jijrepresent-\ning the exchange interaction between the spin moments\non sitesiandj, is given by the expression\nJij=−1\n4πImTrL/integraldisplayEF\ndE∆iτ↑\nij∆jτ↓\nji,(26)\nwith ∆i(j)= ([t↑]−1\ni(j)−[t↓]−1\ni(j)), wheret↑\ni(j)andt↓\ni(j)are\nthespin-upandspin-downsingle-sitescatteringmatrices,\nrespectively, while τ↑\nijandτ↓\njiare the spin-up and spin-\ndown, respectively, scattering path operators. As rela-\ntivistic effects are not taken into account, the exchange\ninteractions are isotropic with respect to the orientation\nof the magnetization as well as with respect to the di-\nrection of the spin tilting. On the other hand, spin-orbit\ncoupling gives rise to an anisotropy for exchange inter-\nactions requiring a representation in the form of the ex-\nchange tensor Jijwith its antisymmetric part giving ac-\ncess to the Dzyaloshinskii-Moriya (DM) interaction /vectorDij.\nUdvardi et al.5and later Ebert and Mankovsky6sug-\ngested an extension of the classical Heisenberg Hamilto-\nnianbyaccountingforrelativisticeffects forthe exchange\ncoupling (see also Ref. 25). These calculations are based5\nonafullyrelativistictreatmentoftheelectronicstructure\nobtained by use of of the Dirac Hamiltonian\nHD=−ic/vector α·/vector∇+1\n2c2(β−1)\n+¯V(/vector r)+β/vector σ·/vectorB(/vector r)+e/vector α·/vectorA(/vector r).(27)\nHere,αiandβare the standard Dirac matrices15while\n¯V(/vector r) and/vectorB(/vector r) are the spin independent and spin depen-\ndent parts of the electronic potential.\nConsidering a ferromagnetic (FM) state as a reference\nstate with the magnetization along the zdirection, a tilt-\ning of the magnetic moments on sites iandjleads to a\nmodification ofthe scattering path operatorimplying the\nrelation\nlnτ−lnτ0=−ln/parenleftbig\n1+τ[∆mi+∆mj+...]/parenrightbig\n,(28)\nwithmi=t−1\ni. This allows to write down the expression\nfor the energy change due to a spin tilting on sites iand\njas follows\nEij=−1\nπImTr/integraldisplayEF\ndE∆miτij∆mjτji(29)\nWithin the approach of Udvardi et al.5, the depen-\ndence of the single-site inverse scattering matrix mion\nthe orientation of magnetic moment ˆ eiis accounted for\nby performing a corresponding rotation operation us-\ning the rotation matrix R(θ,φ), i.e., one has mi(θ,φ) =\nR(θ,φ)m0\niR+(θ,φ). The change of the scattering matrix\nmiunder spin rotation, ∆ mi, linearized with respect to\nthe rotation angles, is given by the expression\n∆mi=R(θi,φi)m0\niR+(θi,φi)−m0\ni\n=mθ\niδθi+mφ\niδφi (30)\nwith\nmθ\ni=∂\n∂θmi=∂R\n∂θmiR++Rmi∂R+\n∂θ,\nmφ\ni=∂\n∂φmi=∂R\n∂φmiR++Rmi∂R+\n∂φ.(31)\nTo calculate the derivatives of the rotation matrix, the\ndefinition\nˆR(αˆn,ˆn) =eiαˆn(ˆn·ˆ/vectorJ)(32)\nfor the corresponding operator is used, withˆ/vectorJthe total\nangular momentum operator. ˆR(αˆn,ˆn) describes a rota-\ntion of the magnetic moment ˆ mby the angle αˆnabout\nthe direction ˆ n⊥ˆm, that gives in particular R(θ,ˆn) for\nˆn= ˆyandR(φ,ˆn) for ˆn= ˆz.\nThis leads to the second derivatives of the total energy\nwith respect to the titling angles αi={θi,φi}andβj=\n{θj,φj}\n∂2E\n∂αi∂βj=−1\nπImTr/integraldisplayEF\ndEmα\niτijmβ\njτji(33)As is discussed by Udvardi et al.5, these derivatives give\naccess to all elements Jµν\nijof the exchange tensor, where\nµ(ν) ={x,y,z}. Note, however, that only the tensor el-\nements with µ(ν) ={x,y}can be calculated using the\nmagnetization direction along the ˆ zaxis, giving access to\nthezcomponent Dz\nijof the DMI. In order to obtain all\nother tensor elements, an auxiliary rotation of the mag-\nnetization towards the ˆ xand ˆydirections of the global\nframe of reference is required. For example, the com-\nponentDx\nijif the DMI vector can be evaluated via the\ntensor elements\nJzy\nij=∂2E\n∂θi∂φjandJyz\nij=∂2F\n∂φi∂θj(34)\nforθ=π\n2andφ= 0.\nAn alternative expression within the KKR multiple\nscattering formalism has been worked out by Ebert and\nMankovsky6, by using the alternative convention for the\nelectronicGreenfunction(GF) assuggestedbyDederichs\nand coworkers34. According to this convention, the off-\nsite part of the GF is given by the expression:\nG(/vector ri,/vector rj,E) =/summationdisplay\nΛΛ′Ri\nΛ(/vector ri,E)Gij\nΛΛ′(E)Rj×\nΛ′(/vector rj,E),(35)\nwhereGij\nΛΛ′(E) is the so-called structural Green’s func-\ntion,Ri\nΛisaregularsolutiontothesingle-siteDiracequa-\ntionlabeledbythecombinedquantumnumbersΛ15. The\nenergy change ∆ Eijdue to a spin tilting on sites iandj\n, given by Eq. (29), transformed to the above mentioned\nconvention is expressed as follows\n∆Eij=−1\nπImTr/integraldisplay\ndE∆tiGij∆tjGji,(36)\nwhere the change of the single-site t-matrix ∆ tican be\nrepresented in terms of the perturbation ∆ Vi(/vector r) at site\niusing the expression\n∆ti\nΛ′Λ=/integraldisplay\nd3rRi×\nΛ′(r)∆V(r)Ri\nΛ(r) = ∆V(R)i\nΛ′Λ,(37)\nwherethe perturbation causedby the rotationof the spin\nmagnetic moment ˆ eiis represented by a change of the\nspin-dependent potential in Eq. (27) (in contrast to the\napproach used in Ref. 5)\n∆V(r) =Vˆn(r)−Vˆn0(r) =β/vector σ(ˆn−ˆn0)B(r).(38)\nUsing again the frozen potential approximation implies\nthat the spatial part of the potential Vˆn(r) does not\nchange upon rotation of spin orientation.\nComing back to the convention for the GF used by\nGy¨ orffy and coworkers35according to Eq. (9) the expres-\nsion for the elements of the exchange tensor represented\nin terms of the scattering path operator τij\nΛ′Λ(E) has the\nform\nJαiαj\nij=−1\nπImTr/integraldisplay\ndETαiτijTαjτji,(39)6\nwhere\nTαi\nΛΛ′=/integraldisplay\nd3rZ×\nΛ(/vector r)βσαB(r)ZΛ′(/vector r).(40)\nWhen compared to the approach of Udvardi et al.5,\nthe expression in Eq. (39) is given explicitly in Cartesian\ncoordinates. However, auxiliary rotations of the magne-\ntization are still required to calculate all tensor elements,\nand as a consequence, all components of the DMI vec-\ntor. This can be avoided using the approach reported\nrecently36for DMI calculations.\nIn this case, using the grand-canonical potential in the\noperator form\nK=H−µN, (41)\nwithµthe chemical potential, the variation of single-\nparticle energy density ∆ E(/vector r) caused by a perturbation\nis written in terms of the electronic Green function for\nT= 0 K as follows\n∆E(/vector r) =−1\nπImTr/integraldisplayµ\ndE(E−µ)∆G(/vector r,/vector r,E).(42)\nAssuming the perturbation ∆ Vresponsible for the\nchange of the Green function ∆ G=G−G0(the in-\ndex 0 indicates here the collinear ferromagnetic reference\nstate) to be small, ∆ Gcan be expanded up to any order\nw.r.t. the perturbation\n∆G(E) =G0∆VG0\n+G0∆VG0∆VG0\n+G0∆VG0∆VG0∆VG0\n+G0∆VG0∆VG0∆VG0∆VG0+...,(43)\nleading to a corresponding expansion for the energy\nchange with respect to the perturbation as follows\n∆E= ∆E(1)+∆E(2)+∆E(3)+∆E(4)+...,(44)\nHere and below we drop the energy argument for the\nGreen function G(E) for the sake of convenience. This\nexpression is completely general as it gives the energy\nchange as a response to any type of perturbation. When\n∆Vis associated with tiltings of the spin magnetic mo-\nments, it can be expressed within the frozen potential\napproximation and in line with Eq. (38) as follows\n∆V(/vector r) =/summationdisplay\niβ/parenleftbig\n/vector σ·ˆsi−σz/parenrightbig\nBxc(/vector r).(45)\nWith this, the energy expansion in Eq (44) gives access\nto the bilinear DMI as well as to higher order multispin\ninteractions37. To demonstrate the use of this approach,\nwe start with the xandycomponents of the DMI vector,\nwhich can be obtained by setting the perturbation ∆ Vin\nthe form of a spin-spiral described by the configuration\nof the magnetic moments\nˆmi=/parenleftBig\nsin(/vector q·/vectorRi),0,cos(/vector q·/vectorRi)/parenrightBig\n,(46)with the wave vector /vector q= (0,q,0). As it follows from\nthe spin Hamiltonian, the slope of the spin wave energy\ndispersion at the Γ point is determined by the DMI as\nfollows\nlim\nq→0∂E(1)\nDM\n∂qy= lim\nq→0∂\n∂qy/summationdisplay\nijDy\nijsin(/vector q·(/vectorRj−/vectorRi))\n=/summationdisplay\nijDy\nij(/vectorRj−/vectorRi)y. (47)\nIdentifying this with the corresponding derivative of the\nenergy ∆ E(1)in Eq. 44\n∂∆E(1)\n∂qα/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq→0=∂E(1)\nDM\n∂qα/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq→0, (48)\nand equating the corresponding terms for each atomic\npair (i,j), one obtains the following expression for the y\ncomponent of the DMI vector:\nDy\nij=/parenleftbigg\n−1\n2π/parenrightbigg\nImTr/integraldisplayµ\ndE(E−µ)\n×/bracketleftbigg\nOj(E)τji(E)Ti,x(E)τij(E)\n−Oi(E)τij(E)Tj,x(E)τji(E)/bracketrightbigg\n,(49)\nIn a completely analogous way one can derive the x-\ncomponent of the DMI vector, Dx\nij. The overlap inte-\ngralsOj\nΛΛ′and matrix elements Ti,α\nΛΛ′of the operator\nTi,α=βσαBi\nxc(/vector r) (which are connected with the compo-\nnents of the torque operator β[/vector σ׈m]Bi\nxc(/vector r)) are defined\nas follows:6\nOj\nΛΛ′=/integraldisplay\nΩjd3rZj×\nΛ(/vector r,E)Zj\nΛ′(/vector r,E) (50)\nTi,α\nΛΛ′=/integraldisplay\nΩid3rZi×\nΛ(/vector r,E)/bracketleftBig\nβσαBi\nxc(/vector r)/bracketrightBig\nZi\nΛ′(/vector r,E).(51)\nAs is shown in Ref. 37, the Dz\nijcomponent of the DMI,\nas well isotropic exchange parameter Jijcan also be ob-\ntained on the basis of Eqs. (43) and (44) using the second\norder term w.r.t. the perturbation, for a spin spiral with\nthe form\nˆsi= (sinθcos(/vector q·/vectorR),sinθsin(/vector q·/vectorR),cosθ).(52)\nIn this case case, the DMI component Dz\nijand the\nisotropic exchange interaction are obtained by taking the\nfirst- and second-orderderivatives of the energy ∆ E(2)(/vector q)\n(see Eq. (44)), respectively, with respect to /vector q:\n∂\n∂/vector q∆EH(/vector q)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq→0=−sin2θN/summationdisplay\ni/negationslash=jDz\nijˆq·(/vectorRi−/vectorRj) (53)\nand\n∂2\n∂/vector q2∆EH(/vector q)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq→0= sin2θ/summationdisplay\ni,jJij(ˆq·(/vectorRi−/vectorRj))2(54)7\nwith ˆq=/vector q/|/vector q|the unit vector giving the direction of the\nwave vector /vector q. Identifying these expressions again with\nthe corresponding derivatives of ∆ E(2)(/vector q), one obtains\nthe following relations for Dz\nij\nDz\nij=1\n2(Jxy\nij−Jyx\nij) (55)\nand forJij\nJij=1\n2(Jxx\nij+Jyy\nij), (56)\nwhere the tensor elements Jαβare given by Eqs. (39)\nand (40).\nSimilar to the magnetic anisotropy, the exchange cou-\npling parameters depend on temperature, that should be\ntaken into account within the finite temperature spin dy-\nnamic simulations. An approach that gives access to\ncalculations of exchange coupling parameters for finite\ntemperature has been reported in Ref. 37. It accounts\nfor the electronic structure modification due to temper-\nature induced lattice vibrations by using the alloy anal-\nogy model in the adiabatic approximation. This implies\ncalculations of the thermal average /an}b∇acketle{t.../an}b∇acket∇i}htTas the configu-\nrational average over a set of appropriately chosen set of\natomic displacements, using the CPA alloy theory38–40.\nTo make use of this scheme to account for lattice vi-\nbrations, a discrete set of Nvvectors ∆/vectorRq\nv(T) is intro-\nduced for each atom, with the temperature dependent\namplitude, which characterize a rigid displacement of\nthe atomic potential in the spirit of the rigid muffin-\ntin approximation41,42. The corresponding single-site t-\nmatrix in the common global frame of the solid is given\nby the transformation:\ntq\nv=U(∆/vectorRv)tq,locU(∆/vectorRv)−1, (57)\nwith the so-called U-transformation matrix U(/vector s) given in\nits non-relativistic form by:41,42\nULL′(/vector s) = 4π/summationdisplay\nL′′il+l′′−l′CLL′L′′jl′′(|/vector s|k)YL′′(ˆs).(58)\nHereL= (l,m) represents the non-relativistic angu-\nlar momentum quantum numbers, jl(x) is a spheri-\ncal Bessel function, YL(ˆr) a real spherical harmonics,\nCLL′L′′a corresponding Gaunt number and k=√\nEis\nthe electronic wave vector. The relativistic version of\nthe U-matrix is obtained by a standard Clebsch-Gordan\ntransformation.15\nEvery displacement characterized by a displacement\nvectors ∆/vectorRv(T) can be treated as a pseudo-component\nof a pseudo alloy. Thus, the thermal averaging can be\nperformed as the site diagonal configurational average\nforasubstitutional alloy,bysolvingthe multi-component\nCPA equations within the global frame of reference40.\nThe same idea can be used also to take into account\nthermalspinfluctuations. Asetofrepresentativeorienta-\ntion vectors ˆ ef(withf= 1,...,Nf) for the local magneticmoment is introduced. Using the rigid spin approxima-\ntion, the single-site t-matrix in the global frame, corre-\nsponding to a given orientation vector, is determined by:\ntq\nf=R(ˆef)tq,locR(ˆef)−1, (59)\nwheretq,locis the single-site t-matrix in the local frame.\nHere the transformation from the local to the global\nframe of reference is expressed by the rotation matrices\nR(ˆef) that are determined by the vectors ˆ efor corre-\nsponding Euler angles.15Again, every orientation can be\ntreated as a pseudo-component of a pseudo alloy, that\nallows to use the alloy analogy model to calculate the\nthermal average over all types of spin fluctuations40.\nThe alloy analogy for thermal vibrations applied to\nthe temperature dependent exchange coupling parame-\nters leads to\n¯Jαiαj\nij=−1\n2πℑ/integraldisplay\ndETrace/an}b∇acketle{t∆Vαiτij∆Vαjτji/an}b∇acket∇i}htc,(60)\nwhere/an}b∇acketle{t.../an}b∇acket∇i}htcrepresents the configurational average with\nrespect to the set of displacements. In case of the ex-\nchange coupling parameters one has to distinguish be-\ntween the averaging over thermal lattice vibrations and\nspin fluctuations. In the first case the configurational av-\nerage is approximated as follows /an}b∇acketle{t∆Viτij∆Vjτji/an}b∇acket∇i}htvib≈\n/an}b∇acketle{t∆Viτij/an}b∇acket∇i}htvib/an}b∇acketle{t∆Vjτji/an}b∇acket∇i}htvib, assuming a negligible impact of\nthe so-called vertex corrections43. This averaging ac-\ncounts for the impact of thermally induced phonons on\nthe exchange coupling parameters for every temperature\nbefore their use in MC or spin dynamics simulations that\ndealsubsequentlywith thethermalaveraginginspinsub-\nspace. The impact of spin fluctuations can be incorpo-\nrated as well within the electronic structure calculations.\nFor a non-polarized paramagnetic reference state, this\ncan be done, e.g., by using the so-called disorder local\nmoment (DLM) scheme formulated in general within the\nnon-relativistic (or scalar-relativistic) framework. Mag-\nnetic disorder in this case can be modeled by creating a\npseudo alloy with an occupation of the atomic sites by\ntwo types of atoms with opposite spin moments oriented\nupwards,M↑and downwards M↓, respectively, i.e. con-\nsidering the alloy M↑\n0.5M↓\n0.5. In the relativistic case the\ncorresponding RDLM scheme has to describe the mag-\nnetic disorder by a discrete set of Nforientation vectors,\nand as a consequence, the average /an}b∇acketle{tτij/an}b∇acket∇i}htspinhas to be\ncalculated taking into account all these orientations. A\ncomparison of the results obtained for the isotropic ex-\nchange coupling constants Jijfor bcc Fe using the DLM\nand RDLM schemes is shown in Fig. 3, demonstrating\nclose agreement, with the small differences to be ascribed\nto the different account of relativistic effects, i.e. in par-\nticular the spin-orbit coupling.8\n11.5 22.5 3\nRij/a051015202530Jij (meV)SR-DLM\nRDLMFe (bcc), T = 1500 K\nFIG. 3. Isotropic exchange coupling parameters calculated\nfor the disordered magnetic state of bcc Fe within the scalar -\nrelativistic approach, using the DLM scheme (circles, SR-\nDLM) and within the fully-relativistic approach, using the\nRDLM scheme19,24(squares, RDLM).\nC. Multi-spin expansion of spin Hamiltonian:\nGeneral remarks\nDespite the obvious success of the classical Heisenberg\nmodel for many applications, higher-order multi-spin ex-\npansionHmsof the spin Hamiltonian H, given by the\nexpression\nHms=−1\n3!/summationdisplay\ni,j,kJijkˆsi·(ˆsj׈sk),\n−2\np!/summationdisplay\ni,j,k,lJs\nijkl(ˆsi·ˆsj)(ˆsk·ˆsl)\n−2\np!/summationdisplay\ni,j,k,l/vectorDijkl·(ˆsi׈sj)(ˆsk·ˆsl)+...,\n=H3+H4,s+H4,a+... (61)\ncan be of great importance to describe more subtle prop-\nerties of magnetic materials44–56.\nThis concerns first of all systems with a non-collinear\nground state characterized by finite spin tilting angles,\nthat makes multispin contributions to the energy non-\nnegligible. Inparticular,manyreportspublishedrecently\ndiscuss the impact of the multispin interactions on the\nstabilization of exotic topologically non-trivial magnetic\ntextures, e.g. skyrmions, hopfions, etc.57–59\nCorresponding calculations of the multi-spin exchange\nparameters have been reported by different groups. The\napproach based on the Connolly-Williams scheme has\nbeen used to calculate the four-spin non-chiral (two-site\nand three-site) and chiral interactions for Cr trimers52\nand for a deposited Fe atomic chain60, respectively, for\nthe biquadratic, three-site four spin and four-site four\nspin interaction parameters58,61. The authors discuss\nthe role of these type of interactions for the stabilization\nof different types of non-collinear magnetic structures as\nskyrmions and antiskyrmions.\nA more flexible mapping scheme using perturbation\ntheory within the KKR Green function formalism wasonly reported recently by Brinker et al.62,63, and by the\npresent authors37. Here we discuss the latter approach,\ni.e. the energy expansion w.r.t. ∆ Vin Eq. (44). One\nhas to point out that a spin tilting in a real system has a\nfinite amplitude and therefore the higher order terms in\nthis expansion might become non-negligible and in gen-\neral should be taken into account. Their role obviously\ndepends on the specific materialandshould increasewith\ntemperature that leads to an increasing amplitude of the\nspin fluctuations. As these higher-order terms are di-\nrectly connected to the multispin terms in the extended\nHeisenberg Hamiltonian, one has to expect also a non-\nnegligibleroleofthe multispin interactionsforsomemag-\nnetic properties.\nExtending the spin Hamiltonian to go beyond the clas-\nsical Heisenberg model, we discuss first the four-spin ex-\nchange interaction terms Jijkland/vectorDijkl. They can be\ncalculated using the fourth-order term of the Green func-\ntion expansion ∆ E(4)given by:\n∆E(4)=−1\nπImTr/integraldisplayEF\ndE\n×(E−EF)∆VG∆VG∆VG∆VG\n=−1\nπImTr/integraldisplayEF\ndE∆VG∆VG∆VG∆VG.\n(62)\nwhere the sum rule for the Green functiondG\ndE=−GG\nfollowed by integration by parts was used to get a more\ncompact expression. Using the multiple-scattering repre-\nsentation for the Green function, this leads to:\n∆E(4)=/summationdisplay\ni,j,k,l−1\nπImTr/integraldisplayEF\ndE\n×∆Viiτij∆Vjjτjk∆Vkkτkl∆Vllτli.(63)\nwith the matrix elements ∆ Vii=/an}b∇acketle{tZi|∆V|Zi/an}b∇acket∇i}ht. Using the\nferromagnetic state with /vectorM||ˆzas a reference state, and\ncreating the perturbation ∆ Vin the form of a spin-spiral\naccording to Eq. (52), one obtains the corresponding /vector q-\ndependent energy change ∆ E(4)(/vector q), written here explic-9\nitly as an example\n∆E(4)=−1\nπ/summationdisplay\ni,j,k,lImTr/integraldisplayEF\ndEsin4θ\n×/bracketleftbigg\nIxxxx\nijklcos(/vector q·/vectorRi)cos(/vector q·/vectorRj)cos(/vector q·/vectorRk)cos(/vector q·/vectorRl)\n+Ixxyy\nijklcos(/vector q·/vectorRi)cos(/vector q·/vectorRj)sin(/vector q·/vectorRk)sin/vector q·/vectorRl)\n+Iyyxx\nijklsin(/vector q·/vectorRi)sin(/vector q·/vectorRj)cos(/vector q·/vectorRk)cos(/vector q·/vectorRl)\n+Iyyyy\nijklsin(/vector q·/vectorRi)sin(/vector q·/vectorRj)sin(/vector q·/vectorRk)sin(/vector q·/vectorRl)\n+Ixyxx\nijklcos(/vector q·/vectorRi)sin(/vector q·/vectorRj)cos(/vector q·/vectorRk)cos(/vector q·/vectorRl)\n+Iyxyy\nijklsin(/vector q·/vectorRi)cos(/vector q·/vectorRj)sin(/vector q·/vectorRk)sin/vector q·/vectorRl)\n+Iyxxx\nijklsin(/vector q·/vectorRi)cos(/vector q·/vectorRj)cos(/vector q·/vectorRk)cos(/vector q·/vectorRl)\n+Ixyyy\nijklcos(/vector q·/vectorRi)sin(/vector q·/vectorRj)sin(/vector q·/vectorRk)sin(/vector q·/vectorRl)+.../bracketrightbigg\n(64)\nwhere\nIαβγδ\nijkl=Ti,α(E)τij(E)Tj,β(E)τjk(E)\n×Tk,γ(E)τkl(E)Tl,δ(E)τli(E).(65)\nAs is shownin Ref. 37, the four-spinisotropicexchange\ninteraction Jijklandz-component of the DMI-like in-\nteraction Dz\nijklcan be obtained calculating the energy\nderivatives∂4\n∂q4∆E(4)and∂3\n∂q3∆E(4)in the limit of q= 0,\nand then identified with the corresponding derivatives of\nthe termsH4,sandH4,ain Eq. (61). These interaction\nterms are given by the expressions\nJs\nijkl=1\n4/bracketleftbigg\nJxxxx\nijkl+Jxxyy\nijkl+Jyyxx\nijkl+Jyyyy\nijkl/bracketrightbigg\n(66)\nand\nDz\nijkl=1\n4/bracketleftbigg\nJxyxx\nijkl+Jxyyy\nijkl−Jyxxx\nijkl−Jyxyy\nijkl)/bracketrightbigg\n,(67)\nwhere the following definition is used:\nJαβγδ\nijkl=1\n2πImTr/integraldisplayEF\ndETα\niτijTβ\njτjkTγ\nkτklTδ\nlτli(68)\nThese expressionobviously give also access to a special\ncases, i.e. the four-spin three-site interactions with l=j,\nand the four spin two-site, socalled biquadratic exchange\ninteractions with k=iandl=j.\nThe scalar biquadratic exchange interaction parame-\ntersJs\nijijcalculated on the basis of Eq. (66) for the three\n3dbulk ferromagnetic systems bcc Fe, hcp Co and fcc Ni\nhave been reported in Ref. 37. The results are plotted in\nFig. 4 as a function of the distance Rij+Rjk+Rkl+Rli.\nFor comparison, the insets give the corresponding bilin-\near isotropic exchange interactions for these materials.\nOne can see rather strong first-neighbor interactions for\nbcc Fe, demonstrating the non-negligible characterof the\n✵ \u0000 ✁\n✶✶ \u0000 ✁\n✷✷\n\u0000 ✁\n✸✸\n\u0000 ✁❘✐ ✂\n✴ ✄\n✵\n✵ \u0000 ✁\n✶\n✶ \u0000 ✁\n✷\n✷\n\u0000 ✁\n✸\n✸\n\u0000 ✁\n✹\n✹\n\u0000 ✁\n✁❏\n☎✆☎✆s\n✥✝✞✟✠✡\n① ①✡\n② ②\n☛☛☞ ✌\n✍✍☞ ✌\n✎✎☞ ✌✏✑ ✒\n✓✔\n✕\n✌\n☛✕\n☛✌✖\n✗✘\n✙✚✛✜✢❜ ✣✣ ✤✦\n(a)✶✶\u0000 ✁\n✷✷\u0000 ✁\n✸✸\u0000 ✁❘✐ ✂\n✴ ✄\n✲ ☎ \u0000 ☎✷\n☎\n☎\u0000 ☎ ✷☎\u0000 ☎✵\n☎\u0000 ☎\n✆\n☎\u0000 ☎\n✝\n☎ \u0000\n✶❏\n✞✟✞✟s\n✥✠✡☛☞\n✌\n① ①✌\n② ②\n✍✍✎ ✏\n✑✑✎ ✏\n✒✒✎ ✏✓✔ ✕\n✖✗\n✘\n✏\n✍✘\n✍✏✙\n✚✛\n✜✢✣✤✦❤ ✧★ ✩ ✪\n(b)✵ \u0000 ✁\n✶✶ \u0000 ✁\n✷✷\n\u0000 ✁\n✸❘✐ ✂\n✴ ✄\n✲\n✵ \u0000 ✵✷\n✲\n✵ \u0000 ✵✶ ✁\n✲\n✵ \u0000 ✵✶\n✲\n✵ \u0000 ✵✵ ✁\n✵❏\n☎✆☎✆s\n✥✝✞✟✠✡\n① ①✡\n② ②\n☛☛☞ ✌\n✍✍☞ ✌\n✎✏✑ ✒\n✓✔\n✕\n☛\n✍\n✎✖\n✗✘\n✙✚✛✜✢❢ ✣✣ ✤✦\n(c)\nFIG. 4. Scalar biquadratic exchange interactions Js\nijijin bcc\nFe (a), hcp Co (b) and fcc Ni (Ni). For comparison, the insets\nshow the bilinear exchange interaction parameters calcula ted\nfor the FM state with the magnetization along the ˆ z-axis. All\ndata are taken from Ref. 37.\nbiquadratic interactions. This is of course a material-\nspecific property, and one notes as decrease for the bi-\nquadratic exchange parameters when going to Co and Ni\nas shown in Fig. 4 (b) and (c), respectively.\nIn order to calculate the xandycomponents of\nthe four-spin and as a special case the three-site-DMI\n(TDMI) and biquadratic-DMI (BDMI) type interactions,\nthe scheme suggested in Ref. 37 for the calculation of the\nDMI parameters36,64can be used, which exploited the\nDMI-governed behavior of the spin-wave dispersion hav-\ning a finite slope at the Γ point of the Brillouin zone.10\nNote, however, that a more general form of perturbation\nisrequiredin thiscasedescribedbya2Dspin modulation\nfield according to the expression\nˆsi=/parenleftbig\nsin(/vector q1·/vectorRi) cos(/vector q2·/vectorRi),sin(/vector q2·/vectorRi),\ncos(/vector q1·/vectorRi)cos(/vector q2·/vectorRi)/parenrightbig\n, (69)\nwhere the wave vectors /vector q1and/vector q2are orthogonal to each\nother, as for example /vector q1=q1ˆyand/vector q2=q2ˆx.\nTaking the second-order derivative with respect to the\nwave-vector /vector q2and the first-order derivative with respect\nto the wave-vectors /vector q1and/vector q2, and considering the limit\nq1(2)→0, one obtains\n∂3\n∂q3\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq2=0H4,a=/summationdisplay\ni,j,k,lDx\nijkl(ˆq2·/vectorRij)(ˆq2·/vectorRlk)2,\nand\n∂\n∂q1/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq1=0∂2\n∂q2\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq2=0H4,a=/summationdisplay\ni,j,k,lDy\nijkl(ˆq1·/vectorRij)(ˆq2·/vectorRlk)2,\nwhere/vectorRij=/vectorRj−/vectorRiand/vectorRlk=/vectorRk−/vectorRl.\nThe microscopic expressions for the xandycompo-\nnents of/vectorDijkldescribing the four-spin interactions is de-\nrived on the basis of the third-order term in Eq. (43)\n∆E(3)=−1\nπImTr/integraldisplayEF\ndE(E−EF)\n×G0∆VG0∆VG0∆VG0. (70)\nThe final expression for Dα\nijklis achieved by taking the\nsecond-order derivative with respect to the wave-vector\n/vector q2and the first-orderderivative with respect to the wave-\nvectors/vector q1(2), considering the limit q1(2)→0, i.e. equat-\ning within the ab-initio and model expressions the cor-\nresponding terms proportional to ( /vectorRi−/vectorRj)y(/vectorRk−/vectorRl)2\nx\nand (/vectorRi−/vectorRj)x(/vectorRk−/vectorRl)2\nx(we keep a similar form in both\ncasesforthe sakeofconvenience)givestheelements Dy,x\nijkl\nandDy,y\nijkl, as well as Dx,x\nijklandDx,y\nijkl, respectively, of the\nfour-spin chiral interaction as follows\nDα,β\nijkj=ǫαγ1\n8πImTr/integraldisplayEF\ndE(E−EF)\n/bracketleftBig\nOiτijTj,γτjkTk,βτklTl,βτli\n−Ti,γτijOjτjkTk,βτklTl,βτli/bracketrightBig\n+/bracketleftBig\nOiτijTj,βτjkTk,βτklTl,γτli\n−Ti,γτijTj,βτjkTk,βτklOlτli/bracketrightBig\n(71)\nwithα,β=x,y, andǫαγthe elements of the transverse\nLevi-Civita tensor ǫ=/bracketleftbigg\n0 1\n−1 0/bracketrightbigg\n. The TDMI and BDMI\nparameterscan be obtained as the special cases l=jand\nl=j,k=i, respectively, from Eq. (71).The expression in Eq. (71) gives access to the xandy\ncomponents of the DMI-like three-spin interactions\nDα\nijkj=Dα,x\nijkj+Dα,y\nijkj. (72)\nFinally, three-spin chiral exchange interaction (TCI)\nrepresented by first term in the extended spin Hamilto-\nnianhas been discussedin Ref. 37. As it followsfrom this\nexpression, the contribution due to this type of interac-\ntion is non-zero only in case of a non-co-planar and non-\ncollinear magnetic structure characterized by the scalar\nspatial type product ˆ si·(ˆsj׈sk) involving the spin mo-\nments on three different lattice sites.\nIn order to work out the expression for the Jijkinter-\naction, one has to use a multi-Q spin modulation65–67\nwhich ensure a non-zero scalar spin chirality for every\nthree atoms. The energy contribution due to the TCI,\nis non-zero only if Jijk/ne}ationslash=Jikj, etc. Otherwise, the\ntermsijkandikjcancel each other due to the relation\nˆsi·(ˆsj׈sk) =−ˆsi·(ˆsk׈sj).\nAccordingly, the expression for the TCI is derived us-\ning the 2Q non-collinear spin texture described by Eq.\n(69), which is characterized by two wave vectors oriented\nalong two mutually perpendicular directions, as for ex-\nample/vector q1= (0,qy,0) and/vector q2= (qx,0,0). Applying such a\nspin modulation in Eq. (69) for the term H3associated\nwith the three-spin interaction in the spin Hamiltonian\nin Eq. (61), the second-order derivative of the energy\nE(3)(/vector q1,/vector q2) with respect to the wave vectors q1andq2is\ngiven in the limit q1→0,q2→0 by the expression\n∂2\n∂/vector q1∂/vector q2H(3)\n=−/summationdisplay\ni/negationslash=j/negationslash=kJijk/parenleftbig\nˆz·[(/vectorRi−/vectorRj)×(/vectorRk−/vectorRj)]/parenrightbig\n.(73)\nThe microscopic energy term of the electron system,\ngiving access to the chiral three-spin interaction in the\nspin Hamiltonian is described by the second-order term\n∆E(2)=−1\nπImTr/integraldisplayEF\ndE(E−EF)\nG0∆VG0∆VG0 (74)\nof the free energy expansion. Taking the first-order\nderivative with respect to q1andq2in the limit q1→\n0,q2→0, and equating the terms proportional to/parenleftbig\nˆz·[(/vectorRi−/vectorRj)×(/vectorRk−/vectorRj)]/parenrightbig\nwith the correspondingterms\ninthespinHamiltonian,oneobtainsthefollowingexpres-\nsion for the three-spin interaction parameter\nJijk=1\n8πImTr/integraldisplayEF\ndE(E−EF)\n/bracketleftBig\nTi,xτijTj,yτjkOkτki−Ti,yτijTj,xτjkOkτki\n−Ti,xτijOjτjkTk,yτki+Ti,yτijOjτjkTk,xτki\n+OiτijTi,xτjkTk,yτki−OjτijTi,yτjkTk,xτki/bracketrightBig\n,(75)11\ngiving access to the three-spin chiral interaction deter-\nmined asJ∆=Jijk−Jikj. Its interpretation was dis-\ncussed in Ref. 68, where its dependence on the SOC as\nwell as on the topological orbital susceptibility χTO\n∆=\nχTO\nijk−χTO\nikjwas demonstrated. In fact that the expres-\nsion forχTO\nijkworked out in Ref. 68 has a rather similar\nform asJijk, as that can be seen from the expression\nχTO\nijk=−1\n4πImTr/integraldisplayEF\ndE\n×/bracketleftBig\nTi,xτijTj,yτjklk\nzτki−Ti,yτijTj,xτjklk\nzτki\n−Ti,xτijlj\nzτjkTk,yτki+Ti,yτijlj\nzτjkTk,xτki\n+li\nzτijTj,xτjkTk,yτki−li\nzτijTj,yτjkTk,xτki/bracketrightBig\n.\n(76)\nFor everytrimerofatoms, both quantities, χTO\nijkandJijk,\nare non-zero only in the case of non-zero scalar spin chi-\nrality ˆsi·(ˆsj׈sk) and depend on the orientation of the\ntrimermagneticmomentwith respecttothetrimerplain.\nThis is shown in Fig. 668representing ∆ Jand ∆χTOas\na function of the angle between the magnetization and\nnormal ˆnto the surface, which are calculated for the two\nsmallest trimers, ∆ 1and ∆ 2, centered at the Ir atom and\nthe hole site in the Ir surface layer for 1ML Fe/Ir(111),\nrespectively (Fig. 5).\nFIG. 5. Geometry of the smallest three-atom clusters in the\nmonolayer of 3 d-atoms on M(111) surface ( M= Au, Ir): M-\ncentered triangle ∆ 1and hole-centered triangle ∆ 2.\nThe role of the SOC for the three-site 4-spin DMI-like\ninteraction, Dz\nijik, and the three-spin chiral interaction,\nJ∆is shown in Fig. 7. These quantities are calculated\nfor 1ML Fe on Au (111), for the two smallest triangles\n∆1and ∆ 2centered at an Au atom or a hole site, re-\nspectively (see Fig. 5). Here, setting the SOC scaling\nfactorξSOC= 0 implies a suppression of the SOC, while\nξSOC= 1 corresponds to the fully relativistic case. Fig.\n7 (a) shows the three-site 4-spin DMI-like interaction pa-\nrameter, Dz\nijik(ξSOC) when the SOC scaling parameter\nξSOCapplied to all components in the system, shown by\nfull symbols, and with the SOC scaling applied only to\nthe Au substrate. One can see a dominating role of the\nSOC of substrate atoms for Dz\nijik. Also in Fig. 7 (b), a\nnearly linear variation can be seen for J∆(ξSOC) when\nthe SOC scaling parameter ξSOCis applied to all com-\nponents in the system (full symbols). Similar to Dz\nijik,\n✵ ✷✵ ✹✵✻✵\n✽✵❣ ✥ \u0000 ✁✂ ✄\n✵\n✵ ☎ ✆\n✵ ☎ ✷\n✵ ☎ ✝\n✵ ☎ ✹\n✵ ☎ ✞\n✵ ☎ ✻✲✟\n❉ ❚\n✠\n✡☛☞✌✍\n❏✎✶\n✥ ❣ ✄ ✥ ✮ ✏ ✑✒ ✄❏✎\n✓\n✥ ❣ ✄ ✥ ✔ ✑✕ ✖✄❏✎✶\n✥ ✭ ✄ ✗ ✑✘✥ ❣ ✄❏✎\n✓\n✥ ✭ ✄ ✗ ✑✘✥ ❣ ✄\n(a)✵ ✷✵ ✹✵✻✵\n✽✵❣ ✥ \u0000✁✂ ✄\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✥ ❣ ✄ ✥ ✓\n✏✔ ✕ ✄✌✍ ✶\n✥ ✵ ✄ ✭\n✏✖ ❣✌✍ ✒\n✥ ✵ ✄ ✭\n✏✖ ❣\n(b)\nFIG. 6. (a) Three-spinchiral exchange interaction paramet ers\n−J∆(γ) (’minus’ is used to stress the relation between J∆and\nχTO\n∆), and (bc) topological orbital susceptibility (TOS, for\nSOC = 0), calculated for Fe on Ir (111), as a function of the\nangle between the magnetization and normal ˆ nto the surface,\nfor the smallest triangles ∆ 1and ∆ 2. The dashed lines rep-\nresentJ∆(0) cos(γ) (a) and χTO\n∆(0) cos(γ) (b), respectively.\nAll data are taken from Ref. 68.\nthis shows that the SOC is an ultimate prerequisite for a\nnon-vanishing TCI J∆. When scaling the SOC only for\nAu (open symbols), Fig. 7 (b) show only weak changes\nfortheTCIparameters J∆(ξSOC), demonstratingaminor\nimpact of the SOC of the substrate on these interactions,\nin contrast to the DMI-like interaction shown in Fig. 7\n(a). One can see also that Dz\nijikis about two orders of\nmagnitude smaller than J∆for this particular system.\nThe origin of the TCI parameters have been discussed\nin the literature suggesting a different interpretation of\nthe correspondingterms derivedalsowithin the multiple-\nscattering theory Green function formalism62,69,70. How-\never, the expression worked out in Ref. 69 has obviously\nnot been applied for calculations so far. As pointed out\nin Ref. 68, the different interpretation of this type of in-\nteractions can be explained by their different origin. In\nparticular, one has to stress that the parameters in Refs.\n68 and 69 were derived in a different order of pertur-\nbation theory. On the other hand, the approach used\nfor calculations of the multispin exchange parameters re-\nported in Ref. 62, 69, and 71 is very similar to the one\nused in Refs. 37 and 68. The corresponding expressions\nhave been worked out within the framework of multiple-\nscattering Green function formalism using the magnetic\nforce theorem. In particular, the Lloyd formula has been\nused to express the energy change due to the perturba-\ntion ∆Vleading to the expression\n∆E=−1\nπIm Tr/integraldisplayEF\ndE/summationdisplay\np1\npTr/bracketleftbig\nG(E)∆V/bracketrightbigp.(77)\nUsing the off-site part of the GF in Eq. (35), as defined12\n0 0.2 0.4 0.6 0.8 1\nξSOC-0.004-0.00200.0020.0040.0060.008Dijik  (meV)\nDx(∆1), SOC\nDx(∆1), SOC(Au)Dx(∆2), SOC(Au) Dx(∆2), SOC\n(a)\n0 0.2 0.4 0.6 0.8 1\nξSOC-0.4-0.3-0.2-0.10J∆ (meV)∆1, SOC\n∆2, SOC\n∆1, SOC(Au)\n∆2, SOC(Au)\n(b)\nFIG. 7. (a) Three-site 4-spin DMI-like interaction, Dz\nijkjand\n(c) three-spin chiral exchange interaction (TCI) paramete rs\nJ∆calculated for Fe on Au (111) on the basis of Eq. (75) as\na function of SOC scaling parameter ξSOCfor the smallest\ntriangles ∆ 1and ∆ 2. In figure (b), full symbols represent\nthe results obtained when scaling the SOC for all elements in\nthe system, while open symbols show the results when scaling\nonly the SOC for Au. All data are taken from Ref. 68.\nby Dederichs et al.34, Eq. (77) is transformed to the form\n∆E=−1\nπIm Tr/integraldisplayEF\ndE/summationdisplay\np1\npTr/bracketleftBig\nGstr(E)∆t(E)/bracketrightBigp\n.(78)\nBy splitting the structural Green function Gstr\nijinto a\nspin-dependent ( /vectorBstr\nij) and a spin-independent ( Astr\nij)\nparts according to\nGstr\nij=Astr\nijσ0+/vectorBstr\nij·/vector σ (79)\nand expressing the change of the single-site scattering\nmatrix\n∆ti(E) = (t↑\ni(E)−t↓)δˆsi×/vector σ, (80)\nby means of the rigid spin approximation, the different\nterms in Eq. (78) corresponding to different numbers p\ngive access to corresponding multispin terms, chiral and\nnon-chiral, in the extended spin Hamiltonian. In particu-\nlar, the isotropic six-spin interactions, that are responsi-\nbleforthenon-collinearmagneticstructureofB20-MnGe\naccording to Grytsiuk et al69, is given by the expression\nκ6−spin\nijklmn=1\n3πIm Tr/integraldisplayEF\ndE\n×Aijtσ\njAjktσ\nkAkltσ\nlAlmtσ\nmAmntσ\nnAnitσ\ni.(81)A rather different point of view concerning the multi-\nspin extension of the spin Hamiltonian was adopted by\nStreib et al.72,73, who suggested to distinguish so-called\nlocal and global Hamiltonians. According to that classi-\nfication, a global Hamiltonian implies to include in prin-\nciple all possible spin configurations for the energy map-\nping in orderto calculate exchangeparametersthat char-\nacterize in turn the energy of any spin configuration. On\nthe other hand, a local Hamiltonian is ’designed to de-\nscribe the energetics of spin configurations in the vicinity\nof the ground state or, more generally, in the vicinity of a\npredefined spin configuration’72. This implies that taking\nthe ground state as a reference state, it has to be deter-\nmined first before the calculating the exchange parame-\nters which are in principle applicable only for small spin\ntiltings around the reference state and can be used e.g.\nto investigate spin fluctuations around the ground state\nspin configuration. In Ref. 72, the authors used a con-\nstrainingfieldtostabilizethenon-collinearmagneticcon-\nfiguration. This leads to the effective two-spin exchange\ninteractions corresponding to a non-collinear magnetic\nspin configuration72,73. According to the authors, ’lo-\ncal spin Hamiltonians do not require any spin interac-\ntions beyond the bilinear order (for Heisenberg exchange\nas well as Dzyaloshinskii-Moriya interactions)’ . On the\nother hand, they point out the limitations for these ex-\nchange interactions in the case of non-collinear system in\nthe regime when the standard Heisenberg model is not\nvalid73, and multi-spin interactions get more important.\nII. GILBERT DAMPING\nAnother parameter entering the Landau-Lifshitz-\nGilbert (LLG) equation in Eq. (3) is the Gilbert damping\nparameter ˜Gcharacterizingenergydissipation associated\nwith the magnetization dynamics.\nTheoretical investigations on the Gilbert damping pa-\nrameter have been performed by various groups and ac-\ncordinglythepropertiesofGDisdiscussedindetailinthe\nliterature. Many of these investigations are performed\nassuming a certain dissipation mechanism, like Kamber-\nsky’sbreathingFermisurface(BFS)74,75, ormoregeneral\ntorque-correlationmodels (TCM)76,77to be evaluated on\nthe basis of electronic structure calculations. The earlier\nworks in the field relied on the relaxation time param-\neter that represents scattering processes responsible for\nthe energy dissipation. Only few computational schemes\nfor Gilbert damping parameter account explicitly for dis-\norderin the systems, which is responsible forthe spin-flip\nscatteringprocess. This issuewasaddressedin particular\nby Brataas etal.78who described the Gilbert damping\nmechanism by means of scattering theory. This develop-\nmentsuppliedtheformalbasisforthefirstparameter-free\ninvestigations on disordered alloys38,39,79.\nA formalism for the calculation of the Gilbert damping\nparameter based on linear response theory has been re-\nported in Ref. 39 and implemented using fully relativistic13\nmultiple scattering or Korringa-Kohn-Rostoker (KKR)\nformalism. Considering the FM state as a reference state\nof the system, the energy dissipation can be expressed in\nterms of the GD parameter by:\n˙Emag=/vectorHeff·d/vectorM\ndτ=1\nγ2˙/vector m[˜G(/vector m)˙/vector m].(82)\nOn the other hand, the energy dissipation in the elec-\ntronic system is determined by the underlying Hamilto-\nnianˆH(τ) as follows ˙Edis=/angbracketleftBig\ndˆH\ndτ/angbracketrightBig\n. Assuming a small\ndeviation of the magnetic moment from the equilibrium\n/vector u(τ), the normalized magnetization /vector m(τ) can be written\nin a linearized form /vector m(τ) =/vector m0+/vector u(τ), that in turn leads\nto the linearized time dependent electronic Hamiltonian\nˆH(τ)\nˆH=ˆH0(/vector m0)+/summationdisplay\nµ/vector uµ∂\n∂/vector uµˆH(/vector m0).(83)\nAs shown in Ref. 38, the energy dissipation within the\nlinear response formalism is given by:\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).(84)\nIdentifying it with the corresponding phenomenological\nquantity in Eq. (82), ˙Emag=˙Edisone obtains for the GD\nparameterαa Kubo-Greenwood-like expression:\nαµν=−/planckover2pi1γ\nπMsTrace/angbracketleftBigg\n∂ˆH\n∂uµImG+(EF)∂ˆH\n∂uνImG+(EF)/angbracketrightBigg\nc,\n(85)\nwhereα=˜G/(γMs), and/an}b∇acketle{t.../an}b∇acket∇i}htcindicates a configura-\ntional average required in the presence of chemical or\nthermally induced disorder responsible for the dissipa-\ntion processes. Within the multiple scattering formalism\nwith the representation of the Green function given by\nEq. (9), Eq. (85) leads to\nαµµ=g\nπµtot/summationdisplay\nnTrace/angbracketleftbig\nT0µ˜τ0nTnµ˜τn0/angbracketrightbig\nc(86)\nwith the g-factor 2(1 + µorb/µspin) in terms of the spin\nand orbital moments, µspinandµorb, respectively, the\ntotal magnetic moment µtot=µspin+µorb, and ˜τ0n\nΛΛ′=\n1\n2i(τ0n\nΛΛ′−τ0n\nΛ′Λ) and with the energy argument EFomit-\nted. The matrix elements Tnµare identical to those oc-\ncurring in the context of exchange coupling6and can be\nexpressed in terms of the spin-dependent part Bof the\nelectronic potential with matrix elements:\nTnµ\nΛ′Λ=/integraldisplay\nd3rZn×\nΛ′(/vector r) [βσµBxc(/vector r)]Zn\nΛ(/vector r).(87)\nAs is discussed in Ref. 39, fora system havingchemical\ndisorder, the configurational average is performed using00.10.2 0.3 0.4 0.5\nconcentration xV02040α × 103without vertex corrections\nwith vertex correctionsFe1-xVx\n(a)\nFIG. 8. The Gilbert damping parameter for (a) bcc Fe 1−xVx\n(T= 0 K) as a function of V concentration. Full (open)\nsymbols give results with (without) the vertex corrections .\nAll data are taken from Ref. 39.\nthe scattering path operators evaluated on the basis of\nthe coherent potential approximation (CPA) alloy the-\nory. In the case of thermally induced disorder, the al-\nloy analogy model is used, which was discussed already\nabove. When evaluating Eq. (86), the so-called vertex\ncorrections have to be included43that accounts for the\ndifference between the averages /an}b∇acketle{tTµImG+TνImG+/an}b∇acket∇i}htcand\n/an}b∇acketle{tTµImG+/an}b∇acket∇i}htc/an}b∇acketle{tTνImG+/an}b∇acket∇i}htc. Within the Boltzmann formal-\nism these corrections account for scattering-in processes.\nThe crucial role of these corrections is demonstrated39\nin Fig. 8 representing the Gilbert damping parameter\nfor an Fe 1−xVxdisordered alloy as a function of the con-\ncentrationx, calculated with and without vertex correc-\ntions. As one can see, neglect of the vertex corrections\nmay lead to the nonphysical result α <0. This wrong\nbehavior does not occur when the vertex corrections are\nincluded, that obviously account for energy transfer pro-\ncesses connected with scattering-in processes.\nThe impact of thermal vibrations onto the Gilbert\ndamping can be taken into account within the alloy-\nanalogy model (see above) by averaging over a discrete\nset of thermal atom displacements for a given temper-\natureT. Fig. 9 represents the temperature dependent\nbehavior of the Gilbert damping parameter αfor bcc Fe\nwith 1% and 5% of impurities of Os and Pt38,39. One can\nseeastrongimpactofimpuritiesonGD.Inthecaseof1%\nof Pt in Fig. 9 (a), αdecreases in the low-temperature\nregime much steeper upon increasing the temperature,\nindicating that the breathing Fermi surface mechanism\ndominates. When the concentration of the impurities in-\ncreases up to 5% (Fig. 9 (a)), the spin-flip scattering\nmechanism takes the leading role for the magnetization\ndissipation practically for the whole region of tempera-\ntures under consideration. The different behavior of GD\nforFe with OsandPt isaresult ofthe different densityof\nstates (DOS) of the impurities at the Fermi energy (see\nRef. 39 for a discussion).\nThe role of the electron-phonon scattering for the ul-\ntrafast laser-induced demagnetization was investigated14\n0100200 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. 9. 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.\nAll data are taken from Ref. 39.\nby Carva et al.80based on the Elliott-Yafet theory of\nspin relaxation in metals, that puts the focus on spin-\nflip(SF) transitionsupon theelectron-phononscattering.\nAs the evaluation of the spin-dependent electron-phonon\nmatrix elements entering the expression for the rate of\nthe spin-flip transition is a demanding problem, various\napproximations are used for this. In particular, Carva et\nal.80,81use the so-called Elliott approximation to evalu-\nate a SF probability Pb\nS=τ\nτsfwith the spin lifetime τsf\nand a spin-diagonal lifetime τ:\nPb\nS=τ\nτsf= 4/an}b∇acketle{tb2/an}b∇acket∇i}ht (88)\nwith the Fermi-surface averaged spin mixing of Bloch\nwave eigenstates\n/an}b∇acketle{tb2/an}b∇acket∇i}ht=/summationdisplay\nσ,n/integraldisplay\nd3k|bσ\n/vectorkn|δ(Eσ\n/vectorkn−EF).(89)\nIn the case of a non-collinear magnetic structure, the\ndescription of the Gilbert damping can be extended byadding higher-order non-local contributions. The role of\nnon-local damping contributions has been investigated\nby calculating the precession damping α(/vector q) for magnons\nin FM metals, characterized by a wave vector /vector q. Follow-\ning the same idea, Thonig et al.82used a torque-torque\ncorrelationmodel based on atight binding approach,and\ncalculated the Gilbert damping for the itinerant-electron\nferromagnets Fe, Co and Ni, both in the reciprocal, α(/vector q),\nand realαijspace representations. The important role\nof non-local contributions to the GD for spin dynam-\nics has been demonstrated using atomistic magnetization\ndynamics simulations.\nAformalismforcalculatingthe non-localcontributions\nto the GD has been recently worked out within the KKR\nGreen function formalism83. Using linear response the-\nory for weakly-noncollinear magnetic systems it gives ac-\ncess to the GD parameters represented as a function of\na wave vector /vector q. Using the definition for the spin sus-\nceptibility tensor χαβ(/vector q,ω), the Fourier transformation\nof the real-space Gilbert damping can be represented by\nthe expression84,85\nααβ(/vector q) =γ\nM0Vlim\nω→0∂ℑ[χ−1]αβ(/vector q,ω)\n∂ω.(90)\nHereγ=gµBis the gyromagneticratio, M0=µtotµB/V\nis the equilibrium magnetization and Vis the volume of\nthe system. As is shown in Ref. 83, this expression can\nbe transformed to the form which allows an expansion of\nGD in powers of wave vector /vector q:\nα(/vector q) =α+/summationdisplay\nµαµqµ+1\n2/summationdisplay\nµναµνqµqν+....(91)\nwith the following expansion coefficients:\nα0±±\nαα=g\nπµtot1\nΩBZTr/integraldisplay\nd3k/angbracketleftbigg\nTβτ(/vectork,E±\nF)Tβτ(/vectork,E±\nF)/angbracketrightbigg\nc\nαµ±±\nαα=g\nπµtot1\nΩBZTr/integraldisplay\nd3k/angbracketleftbigg\nTβ∂τ(/vectork,E±\nF)\n∂kµTβτ(/vectork,E±\nF)/angbracketrightbigg\nc\nαµν±±\nαα=−g\n2πµtot1\nΩBZ\n×Tr/integraldisplay\nd3k/angbracketleftbigg\nTβ∂τ(/vectork,E±\nF)\n∂kµTβ∂τ(/vectork,E±\nF)\n∂kν/angbracketrightbigg\nc.(92)\nFor the prototype multilayer system\n(Cu/Fe 1−xCox/Pt)nthe calculated zero-order (uni-\nform) GD parameter αxxand the corresponding\nfirst-order (chiral) αx\nxxcorrection term for /vector q/ba∇dblˆxare\nplotted in Fig. 10 top and bottom, respectively, as a\nfunction of the Co concentration x. Both terms, αxx\nandαx\nxx, increase approaching the pure limits w.r.t. the\nFe1−xCoxalloy subsystem. As is pointed out in Ref. 83,\nthis increase is associated with the dominating so-called\nbreathing Fermi-surface damping mechanism due to the\nmodification of the Fermi surface (FS) induced by the\nSOC, which follows the magnetization direction that\nslowly varies with time. As αis caused for a ferromagnet15\n0 0.2 0.4 0.6 0.8 100.20.4αxx\n0 0.2 0.4 0.6 0.8 1xCo0123αxxx (a.u.)\nFIG. 10. The Gilbert damping parameters αxx(top) and\nαx\nxx(bottom) calculated for the model multilayer system\n(Cu/Fe 1−xCox/Pt)nusing first and second expressions in Eq.\n(92), respectively. All data are taken from Ref. 83.\nexclusively by the SOC one can expect that it vanishes\nfor vanishing SOC. This was indeed demonstrated\nbefore39. The same holds also for αxthat is caused by\nSOC as well.\nAlternatively, a real-space extension for classical\nGilbert dampingtensorwasproposedrecentlybyBrinker\net al.86, by introducing two-site Gilbert damping tensor\nGijentering the site-resolved LLG equation\n1\nγd/vectorMi\ndτ=−γ/vectorMi×/parenleftbigg\n/vectorHi,eff+/summationdisplay\nj/bracketleftBigg\nGij(/vectorM)·d/vectorMi\ndτ/bracketrightBigg/parenrightbigg\n,(93)\nwhich is related to the inverse dynamical susceptibility\nχijvia the expression\nd\ndωIm[χ]αβ\nij=δij/parenleftbigg1\nγMiǫαβγ/parenrightbigg\n+/parenleftbigg\nRiGijRT\nj/parenrightbigg\nαβ,(94)\nwhereRiandRjarethe rotationmatricesto gofromthe\nglobal to the local frames of reference for atoms iandj,\nrespectively, assuming a non-collinear magnetic ground\nstate in the system. Thus, an expression for the GD\ncan be directly obtained using the adiabatic approxima-\ntion for the slow spin-dynamics processes. This justifies\nthe approximation ([ χ]−1(ω))′\nω≈([χ0]−1(ω))′\nω, with the\nun-enhanced dynamical susceptibility given in terms ofelectronic Green function Gij\nχαβ\nij(ω+iη) =−1\nπTr/integraldisplayEF\ndE\n/bracketleftbigg\nσαGij(E+ω+iη)σβImGij(E)\n+σαGij(E)σβImGij(E−ω−iη)/bracketrightbigg\n,(95)\nwith the Green function G(E±iη) = (E− H ±iη)−1\ncorresponding to the Hamiltonian H.\nMoreover, this approach allows a multisite expansion\noftheGDaccountingforhigher-ordernon-localcontribu-\ntions for non-collinearstructures86. For this purpose, the\nHamiltonian His split into the on-site contribution H′\nand the intersite hopping term tij, which is spin depen-\ndent in the general case. The GF can then be expanded\nin a perturbative way using the Dyson equation\nGij=G0\niδij+G0\nitijG0\nj+G0\nitikG0\nktkjG0\nj+....(96)\nAs a result, the authors generalize the LLG equation\nby splitting the Gilbert damping tensor in terms pro-\nportional to scalar, anisotropic, vector-chiral and scalar-\nchiral products of the magnetic moments, i.e. terms like\nˆei·ˆej, (ˆnij·ˆei)(ˆnij·ˆej), ˆnij·(ˆei׈ej), etc.\nIt should be stressed that the Gilbert damping param-\neter accounts for the energy transfer connected with the\nmagnetization dynamics but gives no information on the\nangular momentum transfer that plays an important role\ne.g. for ultrafast demagnetization processes. The formal\nbasis to account simultaneously for the spin and lattice\ndegrees of freedom was considered recently by Aßmann\nand Nowak87and Hellsvik et al.88. Hellsvik et al.88,89re-\nportonanapproachsolvingsimultaneouslytheequations\nfor spin and lattice dynamics, accounting for spin-lattice\ninteractions in the Hamiltonian, calculated on a first-\nprinciples level. These interactions appear as a correc-\ntion to the exchange coupling parameters due to atomic\ndisplacements. As a result, this leads to the three-body\nspin-lattice coupling parameters Γαβµ\nijk=∂Jαβ\nij\n∂uµ\nkand four-\nbody parameters Λαβµν\nijkl=∂Jαβ\nij\n∂uµ\nk∂uν\nlrepresented by rank 3\nand rank 4 tensors, respectively, entering the spin-lattice\nHamiltonian\nHsl=−1\n2/summationdisplay\ni,j,k,αβ,µΓαβµ\nijkeα\nieβ\njuµ\nk\n−1\n4/summationdisplay\ni,j,k,l,αβ,µ,νΛαβµν\nijkleα\nieβ\njuµ\nkuν\nl.(97)\nThe parameters Γαβµ\nijkin Ref. 88 are calculated using a\nfinite difference method, using the exchange coupling pa-\nrametersJijfor the system without displacements ( J0\nij)\nand with a displaced atom k(J∆\nij(/vector uk)), used to estimate\nthe coefficient Γαβµ\nijk≈(J∆\nij(/vector uk)−J0\nij)\nuµ.16\nAlternatively, to describe the coupling of spin and spa-\ntial degrees of freedom the present authors (see Ref. 90)\nadopt an atomistic approach and start with the expan-\nsion of a phenomenological spin-lattice Hamiltonian\nHsl=−/summationdisplay\ni,j,α,β/summationdisplay\nk,µJαβ,µ\nij,keα\nieβ\njuµ\nk\n−/summationdisplay\ni,j/summationdisplay\nk,lJαβ,µν\nij,kleα\nieβ\njuµ\nkuν\nl,(98)\nthat can be seen as a lattice extension of a Heisenberg\nmodel. Accordingly, the spin and lattice degrees of free-\ndom are represented by the orientation vectors ˆ eiof the\nmagnetic moments /vector miand displacement vectors /vector uifor\neach atomic site i. The spin-lattice Hamiltonian in Eq.\n(98) is restricted to three and four-site terms. As rel-\nativistic effects are taken into account, the SLC is de-\nscribed in tensorial form with Jαβ,µ\nij,kandJαβ,µν\nij,klrepre-\nsented by rank 3 and rank 4 tensors, similar to those\ndiscussed by Hellsvik et al.88.\nThesamestrategyasforthe exchangecouplingparam-\netersJij4orJαβ\nij5,6, is used to map the free energy land-\nscapeF({ˆei},{/vector ui}) accounting for its dependence on the\nspin configuration {ˆei}as well as atomic displacements\n{/vector ui}, making use of the magnetic force theorem and the\nLloyd formulato evaluate integrated DOS ∆ N(E). With\nthis, the free energy change due to any perturbation in\nthe system is given by Eq. (25).\nUsing as a reference the ferromagnetically ordered\nstate of the system with a non-distorted lattice, and the\nperturbed state characterized by finite spin tiltings δˆei\nand finite atomic displacements /vector uiat sitei, one can\nwrite the corresponding changes of the inverse t-matrix\nas ∆s\nµmi=mi(δˆeµ\ni)−m0\niand ∆u\nνmi=mi(uν\ni)−m0\ni.\nThis allows to replace the integrand in Eq. (11) by\nlnτ−lnτ0=−ln/parenleftBig\n1+τ[∆s\nµmi+∆u\nνmj+...]/parenrightBig\n,(99)\nwhere all site-dependent changes in the spin configura-\ntion{ˆei}and atomic positions {/vector ui}are accounted for in\na one-to-one manner by the various terms on the right\nhand side. Due to the use of the magnetic force theorem\nthese blocks may be written in terms of the spin tiltings\nδˆeµ\niand atomic displacements of the atoms uν\nitogether\nwith the corresponding auxiliary matrices Tµ\niandUν\ni,\nrespectively, as\n∆s\nµmi=δˆeµ\niTµ\ni, (100)\n∆u\nνmi=uν\niUν\ni. (101)\nInserting these expressionsinto Eq. (99) and the result in\nturnintoEq.(25)allowsustocalculatetheparametersof\nthe spin-lattice Hamiltonian as the derivatives of the free\nenergy with respect to tilting angles and displacements.\nThis way one gets for example for the three-site term:\nJαβ,µ\nij,k=∂3F\n∂eα\ni∂eβ\nj∂uµ\nk=1\n2πIm Tr/integraldisplayEF\ndE\n×/bracketleftBig\nTα\niτijTβ\njτjkUµ\nkτki+Tα\niτikUµ\nkτkjTβ\njτji/bracketrightBig\n(102)\nFIG. 11. The absolute values of site-off-diagonal and site-\ndiagonal SLC parameters: DMI |/vectorDx\nij,j|and isotropic SLC\nJiso,x\nij,j(top), anti-symmetric diagonal components Jdia−a,x\nij,j\nandJdia−a,x\nii,k(middle), and symmetric off-diagonal compo-\nnentsJoff−s,x\nij,jandJoff−s,x\nii,k(bottom) for bcc Fe, as a function\nof the interatomic distance rij\nand for the four-site term:\nJαβ,µν\nij,kl=∂4F\n∂eα\ni∂eβ\nj∂uµ\nk∂uν\nl=1\n4πIm Tr/integraldisplayEF\ndE\n×/bracketleftBigg\nUµ\nkτkiTα\niτijTβ\njτjlUν\nlτlk\n+Tα\niτikUµ\nkτkjTβ\njτjlUν\nlτli\n+Uµ\nkτkiTα\niτilUν\nlτljTβ\njτjk\n+Tα\niτikUµ\nkτklUν\nlτljTβ\njτji/bracketrightBigg\n.(103)\nFig. 11 shows corresponding results for the SLC pa-\nrameters of bcc Fe, plotted as a function of the distance\nrijfori=kwhich implies that a displacement along the\nxdirection is applied for one of the interacting atoms.\nThe absolute values of the DMI-like SLC parameters\n(DSLC) |/vectorD|µ=x\nij,k(note that Dz,µ\nij,k=1\n2(Jxy,µ\nij,k− Jyx,µ\nij,k) )\nshow a rather slow decay with the distance rij. The\nisotropic SLC parameters Jiso,µ=x\nij,j, which have only a\nweak dependence on the SOC, are about one order\nof magnitude larger than the DSLC. All other SOC-\ndriven parameters shown in Fig. 11, characterizing the\ndisplacement-induced contributions to MCA, are much\nsmaller than the DSLC.17\nIII. SUMMARY\nTo summarize, we have considered a multi-level atom-\nistic approach commonly used to simulate finite temper-\natureand dynamical magneticpropertiesof solids, avoid-\ning in particular time-consuming TD-SDFT calculations.\nTheapproachisbasedonaphenomenologicalparameter-\nized spin Hamiltonian which allows to separate the spin\nand orbital degrees of freedom and that way to avoid the\ndemanding treatment of complex spin-dependent many-\nbody effects. As these parameters are fully determined\nby the electronic structure of a system, they can be de-\nduced from the information provided by relativistic band\nstructure calculations based on SDFT. 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By tuning the bias field, one can alter the resonant frequency at which the propagating acoustic\nwaves hybridize with the magnetic modes, and thereby control transmission and reflection coefficients of the acoustic\nwaves. The scattering coefficients exhibit Breit-Wigner/Fano resonant behaviour akin to inelastic scattering in atomic\nand nuclear physics. Employing oblique incidence geometry, one can effectively enhance the strength of magneto-\nelastic coupling, and thus countermand the magnetic losses due to the Gilbert damping. We apply our theory to discuss\npotential benefits and issues in realistic systems and suggest routes to enhance performance of the proposed devices.\nOptical and, more generally, wave-based computing\nparadigms gain momentum on a promise to replace and com-\nplement the traditional semiconductor-based technology.1The\nenergy savings inherent to non-volatile memory devices has\nspurred the rapid growth of research in magnonics,2,3in which\nspin waves4are exploited as a signal or data carrier. Yet, the\nprogress is hampered by the magnetic loss (damping).5,6In-\ndeed, the propagation distance of spin waves is rather short in\nferromagnetic metals while low-damping magnetic insulators\nare more difficult to structure into nanoscale devices. In con-\ntrast, the propagation distance of acoustic waves is typically\nmuch longer than that of spin waves at the same frequencies.7\nHence, their use as the signal or data carrier could reduce the\npropagation loss to a tolerable level. Notably, one could con-\ntrol the acoustic waves using a magnetic field by coupling\nthem to spin waves within magnetostrictive materials.8–10To\nminimize the magnetic loss, the size of such magneto-acoustic\nfunctional elements should be kept minimal. This implies\ncoupling propagating acoustic waves to confined spin wave\nmodes of finite-sized magnetic elements. As we show below\nthis design idea opens a route towards hybrid devices combin-\ning functional benefits of magnonics2,3with the energy effi-\nciency of phononics.7,11,12\nThe phenomena resulting from interaction between coher-\nent spin and acoustic waves have already been addressed\nin the research literature: the spin wave excitation of prop-\nagating acoustic waves7,13–15and vice versa,8,16–18acous-\ntic parametric pumping of spin waves,19–21magnon-phonon\ncoupling in cavities22–24and mode locking,25magnonic-\nphononic crystals,26,27Bragg scattering of spin waves from a\nsurface acoustic wave induced grating,28–30topological prop-\nerties of magneto-elastic excitations,15,31acoustically driven\nspin pumping and spin Seebeck effect,32,33and optical excita-\ntion and detection of magneto-acoustic waves.34–40However,\nstudies of the interaction between propagating acoustic waves\nand spin wave modes of finite-sized magnetic elements, which\nare the most promising for applications, have been relatively\nscarce to date.10,34,36,39\nHere, we explore theoretically the class of magneto-\na)Electronic mail: V .V .Kruglyak@exeter.ac.uk\nθHB\nM\nI\nTω\nRω\nM NM NM\nFIG. 1: The prototypical magneto-elastic resonator is a thin\nmagnetic slab (M) of width d, biased by an external field HB,\nand embedded into a non-magnetic (NM) matrix. The\nacoustic wave with amplitude Iincident at angle qinduces\nprecession of the magnetisation vector Mvia the\nmagneto-elastic coupling. As a result, the wave is partly\ntransmitted and reflected, with respective amplitudes Tw\nandRw.\nacoustic devices in which the signal is carried by acoustic\nwaves while the magnetic field controls its propagation via\nthe magnetoelastic interaction in thin isolated magnetic in-\nclusions as shown in Fig. 1. By changing the applied mag-\nnetic field, one can alter the frequency at which the incident\nacoustic waves hybridize with the magnetic modes of the in-\nclusions. Thereby, one can control the acoustic waves by the\nresonant behaviour of Breit-Wigner and Fano resonances in\nthe magnetic inclusion.41We find that the strength of the res-\nonances is suppressed by the ubiquitous magnetic damping\nin realistic materials, but this can be mitigated by employing\noblique incidence geometry. To compare magneto-acoustic\nmaterials for such devices, we introduce a figure of merit. The\nmagneto-elastic Fano resonance is identified as most promis-\ning in terms of frequency and field tuneability. To enhance res-\nonant behaviour, we explore the oblique incidence as a means\nby which to enhance the figure of merit.\nWe consider the simplest geometry in which magneto-arXiv:1906.07297v2  [physics.app-ph]  4 May 2020Controlling acoustic waves using magneto-elastic Fano resonances 2\nelastic coupling can affect sound propagation. A ferromag-\nnetic slab (\"magnetic inclusion\") of thickness d, of the or-\nder of 10 nm, is embedded within a non-magnetic medium\n(Fig.1). The slab is infinite in the Y\u0000Zplane, has satu-\nration magnetization Ms, and is biased by the applied field\nHB=HBˆz. Due to the magneto-elastic coupling, this equilib-\nrium configuration is perturbed by shear stresses in the xz- and\nyzplanes associated with the incident acoustic wave.\nTo derive the equations of motion, we represent the mag-\nnetic energy density Fof the magnetic material as a sum of the\nmagneto-elastic FMEand purely magnetic FMcontributions.42\nTaking into account the Zeeman and demagnetizing energies,\nwe write FM=\u0000m0HBM+m0\n2(NxM2\nx+NyM2\ny), where Nx(y)\nare the demagnetising coefficients, Nx+Ny=1,Mis the\nmagnetization and m0is the magnetic permeability. In a crys-\ntal of cubic symmetry, the magnetoelastic contribution takes\nthe form43\nFME=B\nM2så\ni6=jMiMjui j+B0\nM2så\niM2\niuii;i;j=x;y;z;(1)\nwhere B0and Bare the linear isotropic and anisotropic\nmagneto-elastic coupling constants, respectively.44The strain\ntensor is ujk=1\n2(¶jUk+¶kUj), where Ujare the displace-\nment vector components. To maximize the effect of the cou-\npling B, we consider a transverse acoustic plane wave incident\non the slab from the left and polarized along the bias field, so\nthatUx=Uy=0,Uz=U(x;y;t). The non-vanishing com-\nponents of the strain tensor are uxz=1\n2¶xUanduyz=1\n2¶yU,\nandFMEis linear in both MandU:\nFME=B\nMs(Mxuxz+Myuyz): (2)\nThe magnetization dynamics in the slab is due to the effec-\ntive magnetic field, m0Heff=\u0000dF=dM. We define mas\nthe small perturbation of the magnetic order, i.e. jmj\u001cMs.\nLinearizing the Landau-Lifshitz-Gilbert equation,4we write\n\u0000¶mx\n¶t=gm0(HB+NyMs)my+gB¶U\n¶y+a¶my\n¶t;(3)\n¶my\n¶t=gm0(HB+NxMs)mx+gB¶U\n¶x+a¶mx\n¶t;(4)\nwhere gis the gyromagnetic ratio and ais the Gilbert\ndamping constant. To describe the acoustic wave, we in-\nclude the magneto-elastic contribution to the stress, s(ME)\njk=\ndFME=dujk, into the momentum balance equation:\nr¶2U\n¶t2=¶\n¶x\u0012\nC¶U\n¶x+B\nMsmx\u0013\n+¶\n¶y\u0012\nC¶U\n¶y+B\nMsmy\u0013\n;\n(5)\nwhere C=c44is the shear modulus and ris the mass density.\nThe non-magnetic medium is described by Eq.(5) with B=0.\nSince the values of C,B, and Nx;yare constant within each\nindividual material, we shall seek solutions of the equations in\nthe form of plane waves U;mx(y)µexp[i(kw;xx+kw;yy\u0000wt)].\nFrom herein, we consider all variables in the Fourier domain.For the magnetization precession in the magnetic layer driven\nby the acoustic wave, we thus obtain\nmx=gB(wkw;y+iewykw;x)\nw2\u0000ewxewyU; (6)\nmy=igB(ewxkw;y+iwkw;x)\nw2\u0000ewxewyU; (7)\nwhere we have denoted wx(y)=gm0(HB+Nx(y)Ms)and\newx(y)=wx(y)\u0000iwa. The complex-valued wave number kw;x\nis given by the dispersion relation\nk2\nw;x=r\nCw2\u0000\nw2\u0000ewxewy\u0001\n\u0000k2\nw;y\u0010\nw2\u0000ewxewy+gB2\nMsCewx\u0011\nh\nw2\u0000ewxewy+gB2\nMsCewyi ;\n(8)\nwhere kw;yis equal to that of the incident wave, and the\nbranch with Im kw;x>0 describes a forward wave decaying\ninto the slab. Eq. (8) describes the hybridization between\nacoustic waves and magnetic precession at frequencies close\nto ferromagnetic resonance (FMR) at frequency wFMR, with\nlinewidth GFMR. The frequency at which the precession am-\nplitudes (Eqs. (6) and (7)) diverge is given by the condition\n(wFMR+iGFMR=2)2=ewxewy. In the limit of small a, this\nyields wFMR=wxwyandGFMR=a(wx+wy). Away from\nthe resonance, Eq. (8) gives the linear dispersion of acous-\ntic waves. In the non-magnetic medium ( B=0), one finds\nk2\n0=w2r0=C0. Here and below, the subscript ’0’ is used to\nmark quantities pertaining to the non-magnetic matrix.\nTo calculate the reflection and transmission coefficients, Rw\nandTw, for a magnetic inclusion, we introduce the mechanical\nimpedance as Z=isxz=wUw. Solution of the wave matching\nproblem can then be expressed via the ratio of load ( ZME) and\nsource ( Z0) impedances. For impedances in the forward (F)\nand backward (B) directions in the magnetic slab, we find\nZ(F=B)\nw;ME=Ckw;x\nw0\n@1+gB2\nCM sewy\u0007iwkw;y\nkw;x\nw2\u0000ewxewy1\nA: (9)\nHere, the ‘-’ and ‘+’ signs correspond to (F) and (B), re-\nspectively. For the non-magnetic material, Eq. (9) recov-\ners the usual acoustic impedance45Z0=cosqpr0C0. Due\nto magnon-phonon hybridization, Re Z(F=B)\nw;MEdiverges at wFMR\nand vanishes at a nearby frequency wME. For a=0, the latter\nis given by\nwME=s\nwxwy\u0000gB2\nMsCwy: (10)\nReflection Rwand transmission Twcoefficients are then\nfound via the well-known relations45as\nRw=(ehw+1)(1\u0000hw)sin(kw;xd)\n(ehwhw+1)sin(kw;xd)+i(hw+ehw)cos(kw;xd);\n(11)\nTw=i(hw+ehw)\n(ehwhw+1)sin(kw;xd)+i(hw+ehw)cos(kw;xd);\n(12)Controlling acoustic waves using magneto-elastic Fano resonances 3\nwhere dis the thickness of the magnetic inclusion, hw=\nZ(F)\nME=Z0andehw=Z(B)\nME=Z0.46In close proximity to the res-\nonance, the impedances changes rapidly. Expanding Eq. (11)\nnearwMEin the limit kwd\u001c1, we obtain\nRw=iGR=2\n(w\u0000wME)+iGR=2eif+R0; (13)\nf=\u00002 arctan\u0014C\nC0rwx\nwytanq\u0015\n;\nwhere R0represents a smooth non-resonant contribution due\nto elastic mismatch at the interfaces, while frepresents a res-\nonant phase, which is non-zero for finite qand approaches p\nrapidly. In a system with no magnetic damping, the hybridiza-\ntion yields a resonance of finite linewidth GR,\nGR=gB2\n2MsC2cosqp\nr0C0\u0012\nwycos2q+C2\nC2\n0wxsin2q\u0013\nd:\n(14)\nThe origin of this linewidth can be explained as follows. Due\nto the magneto-elastic coupling incident propagating acoustic\nmodes can be converted into localised magnon modes. These\nmodes in turn either decay due to the Gilbert damping or are\nre-emitted as phonons. The rates of these transitions are pro-\nportional to GFMR andGR, respectively, and the total decay\nrate is G=GR+GFMR. This is similar to resonant scattering\nin quantum theory47, such that GRandGFMRare analogous to\nthe the elastic (Ge)and inelastic (Gi)linewidths respectively.\nWhen a=0,GFMRvanishes, and G=GR.\nAcoustic waves in the geometry of Fig. 1 can be scattered\nvia several channels. E.g. in a non-magnetic system ( B=0),\nelastic mismatch can yield Fabry-Pérot resonance due to the\nquarter wavelength matching of dand the acoustic wave-\nlength. However, this occurs at very high frequencies, which\nwe do not consider here. To understand the resonant magneto-\nelastic response, it is instructive to consider first the case of\nnormal incidence ( q=0), when the demagnetising energy\ntakes a simplified form due to the lack of immediate inter-\nfaces to form surface poles in ythe direction, so that Nx=1\nandNy=0. Including magneto-elastic coupling ( B6=0), we\nplot the frequency dependence of RwandTwusing Eq. (11)\nand (12) in Fig.2. To gain a quantitative insight, we analysed\na magnetic inclusion made of cobalt ( r=8900kgm\u00003,B=\n10MPa, C=80GPa, g=176GHzT\u00001,M=1MAm\u00001), em-\nbedded into a non-magnetic matrix ( r0=3192kgm\u00003;C0=\n298GPa). To highlight the resonant behaviour, we first sup-\npress ato 10\u00004. The reflection coefficient exhibits an asym-\nmetric non-monotonic dependence, shown as a black curve in\nFig.2(a), characteristic of Fano resonance.27,41This line shape\ncan be attributed to coupling between the discrete FMR mode\nof the magnetic inclusion and the continuum of propagating\nacoustic modes in the surrounding non-magnetic material.41\nIf the two materials had matching elastic properties, Rwwould\nexhibit a symmetric Breit-Wigner lineshape.47The transmis-\nsion shown in Fig.2(b) exhibits an approximately symmetric\ndip near the resonance.48The absorbancejAwj2=1\u0000jRwj2\u0000\njTwj2, shown in Fig.2(c) exhibits a symmetric peak, since theacoustic waves are damped in our model only due to the cou-\npling with spin waves.\nTo consider how the magneto-elastic resonance is affected\nby the damping, we also plot the response for aof 10\u00003and\n10\u00002, red and blue curves in Fig.2, respectively. An increase\nofafrom 10\u00004to 10\u00003significantly suppresses and broad-\nens the resonant peak. For a more common, realistic value of\n10\u00002the resonance is quenched entirely. A stronger magne-\ntoelastic coupling (i.e. high values of B) could, in principle,\ncountermand this suppression. This, however, is also likely\nto enhance the phonon contribution to the magnetic damping,\nleading to a correlation between Bandaobserved in realistic\nmagnetic materials.49\nTo characterise the strength of the Fano resonance, we note\nthat the fate of the magnon excited by the incident acoustic\nwave is decided by the relation between the emission rate GR,\nsee Eq. (14), and absorption rate GFMR. Hence, we introduce\nthe respective figure of merit as ¡=GR=GFMR. This quan-\ntity depends upon the material parameters, device geometry,\nand bias field. As seen from the first terms on the l.h.s. of\nEqs. (6) and (7), the relation between the dynamic magnetisa-\ntion components mx;yare determined by the quantities wxand\nwy. Equating these terms, one finds mxµmyp\nwy=wx, i.e. the\nprecession of mis highly elliptical,50due to the demagnetis-\ning field along x. This negatively affects the phonon-magnon\ncoupling for normal incidence ( ky=0): the acoustic wave\ncouples only to mx, as given by the second term in Eqs. (6)\nand (7). One way to mitigate this is to increase HB, mov-\ning the ratio wy=wxcloser to 1 and thus improving the figure\nof merit. To compare different magneto-elastic materials, the\ndependence on the layer thickness dand elastic properties of\nthe non-magnetic matrix (i.e. r0andC0) can be eliminated by\ncalculating a ratio of the figures of merit for the compared ma-\nterials. The comparison can be performed either at the same\nvalue of the bias field, or at the same operating frequency. The\nlatter situation is more appropriate for a device application,\nbut to avoid unphysical parameters, we present our results for\nthe same m0HB. An example of such comparisons for yttrium\niron garnet (YIG), cobalt (Co) and permalloy (Py) is offered\nin Table I.\nAnother way to improve ¡is to employ the oblique inci-\ndence ( q6=0), in which the acoustic mode is also coupled to\nthe magnetisation component my. The latter is not suppressed\nby the demagnetisation effects if Ny\u001c1. The resulting en-\nhancement in ¡is reflected in the full equation by the inclu-\nsion of wxandwyfromGR,\n¡=GR\nGFMR=gdB2\n2p\nr0C0\u0010\nHBcos2q+C2\nC2\n0Mssin2q\u0011\naC2M2scosq;(15)\nwhere wx\u001dwyandHB\u001cMsis assumed. For small q, the\napproximation Nx'1 and Ny'0 still holds. As a result, non-\nzeroqincreases peak reflectivity, as seen in Fig.3. The evolu-\ntion of the curves in Fig.3 with qis explained by the variation\nof the phase fof the resonant scattering relative to that of the\nnon-resonant contribution R0. The latter changes its sign at in-\ncidence angle of about 30\u000e, which yields a nearly symmetric\ncurve (blue), and an inverted Fano resonance at larger anglesControlling acoustic waves using magneto-elastic Fano resonances 4\n7.10 7.12 7.14 7.16 7.180.000.050.100.150.200.25|R(f)|(a)Damping, α:\n10−4\n10−3\n10−2\n7.10 7.12 7.14 7.16 7.18\nFrequency, f (GHz)0.750.800.850.900.951.00|T(f)|(b)\n7.10 7.12 7.14 7.16 7.180.00.10.20.3|A(f)|2(c)\nFIG. 2: The frequency dependence of the absolute values of (a) reflection and (b) transmission coefficients and (c) absorbance\nis shown for a 20nm thick magnetic inclusion. The vertical dashed and solid black lines represent the ferromagnetic resonance\nfrequency wFMRand magneto-elastic resonance frequency wMErespectively. The non-magnetic and magnetic materials are\nassumed to be silicon nitride and cobalt, respectively, with parameters given in the text. The bias field is m0HB=50mT, which\nleads to fME\u00197:138 GHz.\n(green). Although larger incidence angles may be hard to im-\nplement in a practical device, the resonant scattering is still\nenhanced at smaller angles.\nAbove, we have focused on the simplest geometry that ad-\nmits full analytic treatment. To implement our idea exper-\nimentally, particular care should be taken about the acous-\ntic waves polarization and propagation direction relative to\nthe direction of the magnetization. Indeed, our choice max-\nimises magnetoelastic response. If however, the polariza-\ntion is orthogonal to the bias field HB, i.e. Uz=0, the cou-\npling would be second-order in magnetization components\nmx;y, and would not contribute to the linearized LLG equation.\nFurthermore, we have neglected the exchange and magneto-\ndipolar fields that could arise due to the non-uniformity of the\nmagnetization. To assess the accuracy of this approximation,\nwe note that the length scale of this non-uniformity is set by\nthe acoustic wavelength l, of about 420nm for our parame-\nters rather than by the magnetic slab thickness d. The asso-\n7.00 7.05 7.10 7.15 7.20 7.25\nFrequency, f (GHz)0.010.020.030.040.050.06|R(f)|0◦\n15◦\n30◦45◦\nFIG. 3: Peak R(f)is enhanced and slightly shifted to the left\nin the oblique incidence geometry ( q>0\u000e). Coloured curves\nrepresent specific incidence angles sweeping from 0\u000eto 45\u000e.\nModerate Gilbert damping of a=10\u00003is assumed. The\ndashed vertical line corresponds to the magnetoelastic\nresonance frequency.\n0 10 20 30 40\nAngle, θ (deg.)0.000.020.040.06Figure of Merit, Υ\n0.050.100.150.200.25\nΓR(10−1)/FMR(ns)−1ΓFMR\nΓRΥFIG. 4: Figure of merit ¡and radiative linewidth GRare both\nenhanced in the oblique incidence geometry ( q>0\u000e).\nFerromagnetic linewidth GFMRremains unchanged. Co is\nassumed with a=10\u00003:\nciated exchange field is m0Ms(klex)2'9mT. The k-dependent\ncontributions to the magneto-dipole field vanish at normal in-\nTABLE I: Comparison of the figure of merit ¡for different\nmaterials, assuming d=20nm, m0HB=50mT and\nC0=298GPa.\nParameters YIG Co Py\n¡(q=0\u000e) 4:3x10\u000021:7x10\u000032:7x10\u00004\nGR(ns\u00001) 1 :9x10\u000047:5x10\u000032:0x10\u00004\nGFMR (ns\u00001) 4 :4x10\u000034.3 0.74\n¡(q=30\u000e) 4:1x10\u000022:5x10\u000032:8x10\u00004\nGR(ns\u00001) 1 :8x10\u000041:1x10\u000022:1x10\u00004\nGFMR (ns\u00001) 4 :4x10\u000034.3 0.74\nfME=wME=2p(GHz) 2.97 7.14 6.26\nB(MJm\u00003) 0.55 10 -0.9\nC(GPa) 74 80 50\nr(kgm\u00003) 5170 8900 8720\na 0:9x10\u000041:8x10\u000024:0x10\u00003\nMs(kAm\u00001) 140 1000 760Controlling acoustic waves using magneto-elastic Fano resonances 5\ncidence but may become significant at oblique incidence, giv-\ningm0Mskyd'98mT at q=15\u000e. In principle, these could\nincrease the resonant frequency of the slab by a few GHz but\nwould complicate the theory significantly. 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Depassier\nInstituto de F\u0013 \u0010sica,\nPonti\fcia Universidad Cat\u0013 olica de Chile\nCasilla 306, Santiago 22, Chile\nAbstract\nNumerical simulations of domain wall propagation in thin nanotubes when an external magnetic\n\feld is applied along the nanotube axis have shown an unexpected behavior described as a transition\nfrom a linear to a magnonic regime. As the applied magnetic \feld increases, the initial regime of\nlinear growth of the speed with the \feld is followed by a sudden change in slope accompanied by\nthe emission of spin waves. In this work an analytical formula for the speed of the domain wall that\nexplains this behavior is derived by means of an asymptotic study of the Landau Lifshitz Gilbert\nequation for thin nanotubes. We show that the dynamics can be reduced to a one dimensional\nhyperbolic reaction di\u000busion equation, namely, the damped double Sine Gordon equation, which\nshows the transition to the magnonic regime as the domain wall speed approaches the speed of\nspin waves. This equation has been previously found to describe domain wall propagation in weak\nferromagnets with the mobility proportional to the Dzyaloshinskii-Moriya interaction constant, for\nPermalloy nanotubes the mobility is proportional to the nanotube radius.\nPACS numbers: 75.78.-n, 75.78.Fg\n1arXiv:1809.06278v3  [cond-mat.mes-hall]  17 Sep 2019I. INTRODUCTION\nMagnetic domain wall propagation is a subject of much current interest due to its possible\napplications in magnetic memory devices. Understanding and controlling the motion of\ndomain walls is essential for applications. In the micromagnetic approach, the magnetization\nis governed by the Landau Lifshitz Gilbert (LLG) equation [1, 2]\n@~ m\n@t=\u0000\r0~ m\u0002~He\u000b(~ m) +\u000b~ m\u0002@~ m\n@t(1)\nwhere~ mis the unit magnetization vector, that is, the magnetization ~M=Ms~ m, whereMsis\nthe constant saturation magnetization, a property of the material. The constant \r0=j\rj\u00160,\nwhere\ris the gyromagnetic ratio of the electron and \u00160is the magnetic permeability of\nvacuum. The parameter \u000b>0 is the dimensionless phenomenological Gilbert damping con-\nstant. The e\u000bective magnetic \feld ~He\u000bincludes the physical interactions and the external\napplied \feld ~Ha. The di\u000berent physical phenomena that must be included in the e\u000bective\n\feld and the geometry of the ferromagnetic material together with the intrinsic nonlinear-\nity of the problem imply that exact analytical solutions are generally nonexistent so that\nnumerical and approximate analytic methods have been developed to understand experi-\nmental results and predict new phenomena. The exact solution of Walker [3, 4] developed\nfor an in\fnite medium with an easy axis, a local approximation for the demagnetizing \feld,\nincluding exchange interaction and under the action of an external magnetic \feld along the\neasy axis, shows that when the applied \feld is small, the speed of the domain wall increases\nlinearly with the \feld. When the applied \feld reaches a critical value, the Walker \feld Hw;\nthe magnetization enters into a precessing motion. This behavior, which is encountered\neven when additional physical e\u000bects and di\u000berent geometries are studied, puts a limit to\nthe maximum speed that a domain wall can achieve.\nFor applications it is desirable to have stable domain walls and to reach high propagation\nvelocities. For such purpose di\u000berent physical e\u000bects and geometries have been considered.\nNumerical simulations for thin Permalloy nanotubes under the action of an external \feld\nalong the nanotube axis showed unexpected behavior [5, 6]. For small \felds the speed in-\ncreases linearly with the \feld, reaching a plateau at relatively low applied \feld and very\nhigh velocity. No instability nor Walker breakdown of the domain wall was observed for\nthis material in the parameter regime studied. This unexpected behavior occurs for a spe-\n2ci\fc chirality of the domain wall, namely right handed domain walls, for which the radial\ncomponent of the magnetization remains small throughout the motion [6].\nThe main result of this manuscript is the derivation of an analytical expression for the\nspeed of the domain wall which explains the linear increase at small \felds, the reaching of a\nplateau and the high values of the velocity. For Permalloy, a material of negligible uniaxial\nanisotropy, we \fnd that the speed is given by\nv=\r0RHap\n\u000b2+\u00160R2H2\na=(2A); (2)\nwhereAis the exchange constant [7] and R the thin nanotube radius. For small applied\n\feld we recover the linear regime [8],\nvL=\r0\n\u000bRHa; (3)\nwhereas for large applied \feld the speed tends to the constant value\nv1=\r0s\n2A\n\u00160= 1006 ms\u00001for Permalloy (4)\nwhich we identify with the minimal phase speed of spin waves. Following the notation used\nfor weak ferromagnets, notice that Eqn. (2) can be written as v=\u0016Ha=p\n1 +\u00162H2\na=v2\n1with\nmobility\u0016= (dv=dHa)jHa=0=\r0R=\u000b. The rise of the speed with the \feld is very fast; for a\nPermalloy nanotube of radius R= 55 nanometers, for an external \feld \feld B=\u00160Ha= 2\nmT, Eqn. (2) yields v= 893 m s\u00001.\nThe suppression of the Walker breakdown together with a slowdown of a domain wall\nas it approaches the phase speed of spin waves has been encountered previously in di\u000berent\nproblems. In antiferromagnets with Dzyaloshinskii-Moriya interaction (DMI) the mobility\nwas found to be proportional to the DMI constant [9{11]. See the recent review [12] for ad-\nditional references. Similar behavior was found in rough nanowires [13, 14] and in nanowires\nwith a strong hard axis perpendicular to the wire when an external \feld is applied along the\nwire [15{17]. A theoretical explanation for the e\u000bect of spin waves on Bloch walls was given\nin [18] where it was shown that the transition to the magnonic regime may occur before or\nafter the Walker breakdown depending on the parameters of the problem. See also [19, 20].\nIn all these works bulk matter or thin \flms were the subject of study. For Permalloy nan-\notubes the numerical simulations of [5, 6] show that the sudden change of slope in the rate\n3of increase of the speed of domain walls is accompanied by Cherenkov spin wave emission\nonce the DW speed exceeds the phase speed of the spin waves.\nIn the present work we study theoretically the DW propagation in Permalloy nanotubes\nand \fnd that curvature acts as an additional anisotropy and plays an equivalent role to the\nDMI in weak ferromagnets. The parallel between curvature of a nanotube and DMI has\nbeen observed in [21{23] among others. In [23] it is shown that the analytical expression of\nthe dispersion relation for spin waves in a nanotube has the same mathematical form as the\ndispersion relation for spin waves in thin \flms with DMI. Here we \fnd this mathematical\nanalogy in the mobility of the domain wall. See [24] for a recent comprehensive review on\nthe dynamics of magnetic nanotubes.\nAlthough simulations have been carried out for Permalloy, in the derivation below we will\nallow a material with non negligible uniaxial anisotropy for greater generality.\nII. STATEMENT OF THE PROBLEM\nConsider a thin nanotube with an easy direction along the nanotube axis which we\nchoose as the zaxis. The dynamic evolution of the magnetization is governed by the LLG\nequation (1). A right handed orthogonal cylindrical coordinate system ( \u001a;';z ) is introduced\nas shown in Fig.II in terms of which the unit magnetization vector is written as ~ m=\nm\u001a(\u001a;';z )^\u001a+m'(\u001a;';z ) ^'+mz(\u001a;';z )^z.\nFIG. 1: Cylindrical coordinate system in the nanotube.\nFor su\u000eciently thin tubes the demagnetizing \feld can be approximated by a local ex-\npression with the saturation magnetization acting as an e\u000bective radial hard axis anisotropy\n[8, 25, 26]. In this approximation and including exchange energy, uniaxial anisotropy energy,\n4demagnetization energy and Zeeman energy, the micromagnetic energy can be written as\n[7, 8]\nE=Z\n\nd3x(Ajr~ mj2+Ku(1\u0000m2\nz) +\u00160M2\ns\n2m2\n\u001a\u0000Hamz); (5)\nwhere \n is the material volume of the nanotube, Ais the exchange constant, Kuthe uniaxial\nanisotropy and an external \feld ~Ha=Ha^zhas been applied along the axis. The e\u000bective\n\feld is given by\n~He\u000b=\u00001\n\u00160Ms\u000eE\n\u000e~ m:\nIn a very thin nanotube variations of the magnetization with radius may be neglected so\nthat the unit magnetization depends only on the polar coordinate 'and the axial position\nz. With~ m=~ m(';z), the e\u000bective magnetic \feld can be written as [8]\n~He=2A\n\u00160Ms\u00141\nR2@2~ m\n@'2+@2~ m\n@z2\u0015\n+2Ku\n\u00160Msmz^z\u0000Msm\u001a^\u001a+Ha^z: (6)\nIntroducing Msas unit of magnetic \feld, and introducing the dimensionless space and time\nvariables\u0018=z=R and\u001c=\r0Mstwe rewrite equations (1) and (6) in dimensionless form as\nd~ m\nd\u001c=\u0000~ m\u0002~he\u000b+\u000b~ m\u0002d~ m\nd\u001c(7)\nwith\n~he\u000b=A0\u0014@2~ m\n@'2+@2~ m\n@\u00182\u0015\n+kumz^z\u0000m\u001a^\u001a+ha^z (8)\nwherehais the dimensionless applied \feld. The dimensionless numbers that have appeared\nareku= 2Ku=(\u00160M2\ns) andA0, the square of the ratio between the exchange length lex=\np\n2A=\u0016 0M2\nsand the radius, that is, A0= 2A=(\u00160M2\nsR2). Equations (7) and (8) describe\nthe dynamics of the problem.\nNumerical simulations [5, 6] have been performed for Permalloy for which the exchange\nconstantA= 1:3\u000210\u000011J m\u00001,Ms= 8\u0002105A m\u00001,Ku\u00190 and the external applied\n\feld does not exceed 10\u00002Ms. The nanotube used in simulations has inner radius R, and\nwidthwwithw << R: Here we neglect the variations with radius and consider the range\nR= 55\u0000100\u000210\u00009m. The vacuum permeability \u00160= 4\u0019\u000210\u00007N A\u00002so that\u00160Ms\u00191T.\nWe take the value \r0= 2:21\u0002105s\u00001T\u00001. For Permalloy the exchange length is lex= 5:68\nnm and for a radius of 80 nm lex=R= 0:071. The uniaxial anisotropy vanishes, ku= 0, and\nthe dimensionless applied \feld is in the range 0 <ha<10\u00002. The damping parameter \u000bin\nnumerical simulations ranges from 0 :01 to 0:03.\n5In the numerical simulations of Permalloy nanotubes [5, 6] it is observed that a right\nhanded vortex-type domain wall is stable against the Walker breakdown when driven with\na magnetic \feld. The handedness of the domain wall in [5, 6] is de\fned in relation to the\napplied \feld, a propagating domain wall is called right handed if ( ~ m\u0002~Ha)\u0001^\u001a > 0. Left\nhanded domain walls become unstable and convert into the other, stable chirality. In this\nwork we are interested in the speed of the stable DW, which will be selected through the\nscaling in the asymptotic solution.\nIII. ASYMPTOTIC SOLUTION\nIn this section we perform an asymptotic analysis of the LLG equation to \fnd a reduced\nmodel for the evolution of the domain wall as the applied \feld increases. The reduced model\nwill be valid for a restricted parameter range which is chosen based on the numerical results\ndescribed above for Permalloy. We are interested in right handed vortex walls for which\nthe radial component of the magnetization is small, m\u001a\u001c1 [5, 6]. Introducing a small\ndimensionless parameter \u000fwe write this condition as\nm\u001a=\u000f~m\u001a: (9)\nThe normalization condition ~ m2= 1 becomes\nm2\n'+m2\nz= 1\u0000\u000f2~m2\n\u001a: (10)\nWe will model a situation in which the ratio lex=Rand the Gilbert constant are of the same\norder in\u000fas the radial component of the magnetization. We assume that the applied \feld\nand uniaxial anisotropy are of an order smaller. Let then\nA0=\u000f2~A; k u=\u000f2~ku; ha=\u000f2~ha; \u000b =\u000f~\u000b: (11)\nIt is found that a consistent asymptotic approach can be obtained if a new time scale s=\u000f\u001c\nis introduced. With these scalings, the components of the e\u000bective magnetic \feld can be\nwritten as\n6(~he\u000b)\u001a=\u0000\u000f~m\u001a\u00002\u000f2~A@m'\n@'+\u000f3~A\u0000\nr2\ns~m\u001a\u0000~m\u001a\u0001\n=\u000fH0\n\u001a+\u000f2H1\n\u001a+\u000f3H2\n\u001a;(12a)\n(~he\u000b)'=\u000f2~A(r2\nsm'\u0000m') + 2\u000f3~A@~m\u001a\n@'\n=\u000f2H1\n'+\u000f3H2\n';(12b)\n(~he\u000b)z=\u000f2\u0010\n~ha+~Ar2\nsmz+~kumz\u0011\n=\u000f2H1\nz; (12c)\nwherer2\ns=@\u0018\u0018+@''and where we grouped terms according to the power of \u000fso that\nH0\n\u001a=\u0000~m\u001a,H1\n'=~A(r2\nsm'\u0000m') andH1\nz=~ha+~Ar2\nsmz+~kumz:In obtaining these\nexpressions for the e\u000bective \feld the property @^\u001a=@' = ^'; @^'=@' =\u0000^\u001ais used.\nIntroducing the scaling for \u000bandm\u001ain the LLG equation, we obtain at leading order in\n\u000f;\n_~m\u001a=\u0000(m'H1\nz\u0000mzH1\n') + ~\u000b(m'_mz\u0000mz_m'); (13a)\n_m'=\u0000mzH0\n\u001a; (13b)\n_mz=m'H0\n\u001a; (13c)\nwhere a dot represents a derivative with respect to the scaled time variable sand the\nsubindices represent the components of each vector.\nThe normalization condition (10) implies that, at leading order, we may write\nm'= sin\u0012(\u0018;';s ); m z= cos\u0012(\u0018;';s ): (14)\nIt follows then that equations (13b) and (13c) are equivalent and imply\n_\u0012=\u0000H0\n\u001a= ~m\u001a: (15)\nReplacing the value of ~ m\u001afrom (15) in (13a) together with the expressions for the e\u000bective\n\feldH1\n';H1\nz, the evolution equation for \u0012is found to be\n\u0012+ ~\u000b_\u0012=~A(\u0012\u0018\u0018+\u0012'')\u0000sin\u0012\u0010\n~ha+ (~A+~ku) cos\u0012\u0011\n; (16)\n7where the subscripts in \u0012denote derivatives with respect to \u0018and'respectively. Notice\nthat one may go back to the original unscaled variables and the small parameter \u000fcancels\nout.\nIn what follows we study cylindrically symmetric domain walls, for which \u0012'= 0 and\nidentify the evolution equation with the damped double Sine Gordon equation,\n@2\u0012\n@\u001c2+\u000b@\u0012\n@\u001c=A0\u0012\u0018\u0018\u0000sin\u0012(ha+ (A0+ku) cos\u0012); (17)\na particular case of hyperbolic reaction di\u000busion equation, for which the existence and\nstability of traveling waves have been studied rigorously in [27, 28].\nThis equation has been derived in the analysis of domain wall propagation in weak ferro-\nmagnets, [9{11, 29] and in systems with a strong easy plane [16, 30]. In [11] the dependence\nof mobility on the Dzyaloshinskii constant is derived with great detail. A common feature\nin these problems is the sudden decrease in the rate of increase of the speed with the applied\n\feld.\nThis equation has the same traveling wave solutions as the reaction di\u000busion equation\n\u000b_\u0012=A0\u0012\u0018\u0018\u0000sin\u0012(ha+ (A0+ku) cos\u0012) but with velocity c=cr=p\n1 +c2\nr=A0wherecris\nthe speed of fronts of the reaction di\u000busion equation [27]. We give the explicit expression\nfor the head to head (HH) domain wall, the tail to tail solution is similar. The HH solution\nis found to be the usual domain wall pro\fle,\n\u0012(\u0018;t) = 2 arctan\u0014\nexp\u0012\u0018\u0000c\u001c\n\u0001\u0013\u0015\n(18)\nwith the speed cand domain wall width \u0001 given by\nc=r\nA0\nA0+kuhap\n\u000b2+ (A0+ku)\u00001h2\na; \u0001 =\u000bc\nha: (19)\nThe leading order magnetization ~ m=m'^'+mz^zis given by\n~ m= sech\u0012\u0018\u0000c\u001c\n\u0001\u0013\n^'\u0000tanh\u0012\u0018\u0000c\u001c\n\u0001\u0013\n^z: (20)\nThe external \feld is applied along the zaxis so the magnetization is a right handed ( m'\u00150)\nhead to head domain wall as de\fned in [6]. The small radial component of the magnetization\nis calculated from (15).\nFor small applied \feld we recover the linear regime, that is, the speed increases linearly\nwith the \feld, and the domain wall width tends to a constant value, that is,\nlim\nha!0c=r\nA0\nA0+kuha\n\u000b;lim\nha!0\u0001 =r\nA0\nA0+ku: (21)\n8In this limit the dynamics is primarily governed by the reaction di\u000busion equation \u000b\u0012\u001c=\nA0\u0012\u0018\u0018\u0000sin\u0012(ha+ (A0+ku) cos\u0012) as already found in [8].\nIn terms of the physical parameters the dimensional domain wall width for small \feld\n\u000e= \u0001Rand speedvLcan be written as\n\u000e=s\nA\nA\nR2+Ku; vL=\r0Ha\n\u000b\u000e\nFor Permalloy, Ku= 0 and\u000e=Rin agreement with the results for a static domain\nwall in a thin nanotube [31]. The speed vLcoincides with the low \feld Walker solution\nvW=\r0Hap\nA=K=\u000b , withKan e\u000bective anisotropy A=R2. In the limit of large radius the\ndomain wall width for an in plane magnetized thin \flm,p\nA=Kuis recovered.\nFor large applied \feld the speed tends to a constant value and the domain wall width\ndecreases as the \feld increases,\nlim\nha!1c=p\nA0lim\nha!1\u0001 =\u000bpA0\nha: (22)\nThis limiting value for the speed corresponds to the minimal value of the phase velocity\nfor spin waves, vpmin. In e\u000bect, consider a material like Permalloy with vanishing uniaxial\nanisotropy for simplicity. The dispersion relation for the DSG equation (17), for vanishing\ndamping and vanishing applied \feld, is given by !DSG=pA0p\n1 +k2, so that the phase\nspeed is a decreasing function of kwhich tends asymptotically topA0askgrows. The full\ndispersion relation for spin waves in a thin nanotube, in the absence of damping and applied\n\feld, with vanishing uniaxial anisotropy, is given by [32]\n!=pA0\n2p\n(1 +k2) +A0(1 +k2)2 (23)\nin the units used in this work. We see that for small A0the full dispersion relation coincides,\nup to a constant, with !DSG.\nThe evolution equation (17) shows the transition from the low \feld regime where the\nspeed of the domain wall increases linearly with the \feld to the regime where the domain wall\nspeed approaches vpminand is slowed down by emitting spin waves. In order to capture the\nlarge applied \feld regime where the DW speed exceeds vpminand Cherenkov emission occurs\na di\u000berent scaling is needed. The DW width shrinks with increasing \feld, \u0001 \u0019\u000bpA0=ha,\nwhich indicates that at larger \felds a new scaling for the longitudinal coordinate \u0018is required.\n9v[m/s]\n<latexit sha1_base64=\"j9SroW/XuGmsdepu2GtGB3geGE0=\">AAACDHicbZC7SgNBFIbPejfe4qWzGWIEQYi7NlqKNpYKxgSyS5idnDWDsxdmzgbjklfwGWy1thNb38HSN3GSWGj0h4GP/z+HOfxhpqQh1/1wpqZnZufmFxZLS8srq2vl9Y1rk+ZaYF2kKtXNkBtUMsE6SVLYzDTyOFTYCG/Phnmjh9rINLmifoZBzG8SGUnByVrt8la1ynrMJ7yjgrXiAxMMqtV2ecetuSOxv+B9w85Jxd9/BICLdvnT76QijzEhobgxLc/NKCi4JikUDkp+bjDj4pbfYMtiwmM0QTG6fsB2rdNhUartS4iN3J8bBY+N6cehnYw5dc1kNjT/y1o5RcdBIZMsJ0zE+KMoV4xSNqyCdaRGQapvgQst7a1MdLnmgmxhJb+DkU+FT10kPijZTrzJBv7C9WHNc2vepS3nFMZagG2owB54cAQncA4XUAcB9/AIT/DsPDgvzqvzNh6dcr53NuGXnPcvE2+bZw==</latexit><latexit sha1_base64=\"Y2klHEeoJsELu4qSFIuhBBCz4zM=\">AAACDHicbZC7SgNBFIZnvbve4qWzGZIVBCHu2mgZtLFUMCpklzA7OZsMzl6YORuMS17Bygew1dpObH2HlL6Jk0uh0R8GPv7/HObwh5kUGl13YM3Mzs0vLC4t2yura+sbpc2ta53mikOdpzJVtyHTIEUCdRQo4TZTwOJQwk14dzbMb7qgtEiTK+xlEMSsnYhIcIbGapZ2HId2qY9wjwVtxIc66DtOs1Rxq+5I9C94E6jUyv7B06DWu2iWvvxWyvMYEuSSad3w3AyDgikUXELf9nMNGeN3rA0NgwmLQQfF6Po+3TNOi0apMi9BOnJ/bhQs1roXh2YyZtjR09nQ/C9r5BidBIVIshwh4eOPolxSTOmwCtoSCjjKngHGlTC3Ut5hinE0hdl+CyIfCx87gKxvm0686Qb+wvVR1XOr3qUp55SMtUR2SZnsE48ckxo5JxekTjh5IM/khbxaj9ab9W59jEdnrMnONvkl6/MbM7Oc7Q==</latexit><latexit sha1_base64=\"Y2klHEeoJsELu4qSFIuhBBCz4zM=\">AAACDHicbZC7SgNBFIZnvbve4qWzGZIVBCHu2mgZtLFUMCpklzA7OZsMzl6YORuMS17Bygew1dpObH2HlL6Jk0uh0R8GPv7/HObwh5kUGl13YM3Mzs0vLC4t2yura+sbpc2ta53mikOdpzJVtyHTIEUCdRQo4TZTwOJQwk14dzbMb7qgtEiTK+xlEMSsnYhIcIbGapZ2HId2qY9wjwVtxIc66DtOs1Rxq+5I9C94E6jUyv7B06DWu2iWvvxWyvMYEuSSad3w3AyDgikUXELf9nMNGeN3rA0NgwmLQQfF6Po+3TNOi0apMi9BOnJ/bhQs1roXh2YyZtjR09nQ/C9r5BidBIVIshwh4eOPolxSTOmwCtoSCjjKngHGlTC3Ut5hinE0hdl+CyIfCx87gKxvm0686Qb+wvVR1XOr3qUp55SMtUR2SZnsE48ckxo5JxekTjh5IM/khbxaj9ab9W59jEdnrMnONvkl6/MbM7Oc7Q==</latexit><latexit sha1_base64=\"/+j9p+rP1jjQly2r7+bfVNMThgg=\">AAACDHicbZC5TgMxFEU9YQthC0tHY5EgUYUZGigjaCiDRBYpM4o8zpvEimeR/SYijPILfAMt1HSIln+g5E9wlgISrmTp6N735KfrJ1JotO0vK7eyura+kd8sbG3v7O4V9w8aOk4VhzqPZaxaPtMgRQR1FCihlShgoS+h6Q9uJnlzCEqLOLrHUQJeyHqRCARnaKxO8ahcpkPqIjxgRtvhufbG5XKnWLIr9lR0GZw5lMhctU7x2+3GPA0hQi6Z1m3HTtDLmELBJYwLbqohYXzAetA2GLEQtJdNrx/TU+N0aRAr8yKkU/f3RsZCrUehbyZDhn29mE3M/7J2isGVl4koSREiPvsoSCXFmE6qoF2hgKMcGWBcCXMr5X2mGEdTWMHtQuBi5mIfkI0LphNnsYFlaFxUHLvi3Nml6vW8nTw5JifkjDjkklTJLamROuHkkTyTF/JqPVlv1rv1MRvNWfOdQ/JH1ucP6dqZ3g==</latexit>µ0Ha[mT]\n<latexit sha1_base64=\"x36GB0uNt1OORYSkO/833Q6joLA=\">AAACE3icbZA7SwQxFIXv+HZ9rVraBF1BEJYZGy1FG0sFV4WdYchk77jBZGZI7ojLsIU/wt7OVms7sfUHWPpPzO5a+DoQ+DjnXpKcpFDSku+/e2PjE5NT0zOztbn5hcWl+vLKmc1LI7AlcpWbi4RbVDLDFklSeFEY5DpReJ5cHQ7y82s0VubZKfUKjDS/zGQqBSdnxfW1RoOFuox9dhRzFhLeUMXa+jTqNxpxfcNv+kOxvxB8wcb+erh9DwDHcf0j7OSi1JiRUNzaduAXFFXckBQK+7WwtFhwccUvse0w4xptVA0/0WebzumwNDfuZMSG7veNimtrezpxk5pT1/7OBuZ/WbukdC+qZFaUhJkYXZSWilHOBo2wjjQoSPUccGGkeysTXW64INdbLexgGlIVUheJ92uuk+B3A3/hbKcZ+M3gxJVzACPNwBqswxYEsAv7cATH0AIBt/AAj/Dk3XnP3ov3Ohod8752VuGHvLdPusud3g==</latexit><latexit sha1_base64=\"49/6L556qISZOKAPMkLNKSKyNQ4=\">AAACE3icbZDNSgMxFIUz/lv/qi7dBFtBEMqMG10W3XSp0KrQGYZMeqcNJjNDckcsQxe+gz6DW127E7c+QJe+iWnrQlsPBD7OuZckJ8qkMOi6Q2dufmFxaXlltbS2vrG5Vd7euTJprjm0eCpTfRMxA1Ik0EKBEm4yDUxFEq6j2/NRfn0H2og0aWI/g0CxbiJiwRlaKyzvVavUV3no0kbIqI9wjwVtq2YwqFbDcsWtuWPRWfB+oFLf948eh/X+RVj+8jspzxUkyCUzpu25GQYF0yi4hEHJzw1kjN+yLrQtJkyBCYrxJwb0wDodGqfangTp2P29UTBlTF9FdlIx7JnpbGT+l7VzjE+DQiRZjpDwyUVxLimmdNQI7QgNHGXfAuNa2LdS3mOacbS9lfwOxD4WPvYA2aBkO/GmG5iFq+Oa59a8S1vOGZloheyRfXJIPHJC6qRBLkiLcPJAnskLeXWenDfn3fmYjM45Pzu75I+cz2/bD59k</latexit><latexit sha1_base64=\"49/6L556qISZOKAPMkLNKSKyNQ4=\">AAACE3icbZDNSgMxFIUz/lv/qi7dBFtBEMqMG10W3XSp0KrQGYZMeqcNJjNDckcsQxe+gz6DW127E7c+QJe+iWnrQlsPBD7OuZckJ8qkMOi6Q2dufmFxaXlltbS2vrG5Vd7euTJprjm0eCpTfRMxA1Ik0EKBEm4yDUxFEq6j2/NRfn0H2og0aWI/g0CxbiJiwRlaKyzvVavUV3no0kbIqI9wjwVtq2YwqFbDcsWtuWPRWfB+oFLf948eh/X+RVj+8jspzxUkyCUzpu25GQYF0yi4hEHJzw1kjN+yLrQtJkyBCYrxJwb0wDodGqfangTp2P29UTBlTF9FdlIx7JnpbGT+l7VzjE+DQiRZjpDwyUVxLimmdNQI7QgNHGXfAuNa2LdS3mOacbS9lfwOxD4WPvYA2aBkO/GmG5iFq+Oa59a8S1vOGZloheyRfXJIPHJC6qRBLkiLcPJAnskLeXWenDfn3fmYjM45Pzu75I+cz2/bD59k</latexit><latexit sha1_base64=\"xBgp5p2BvHq86gbNtgDnUBYYwik=\">AAACE3icbZC9TsMwFIWd8lfKX4Cxi0WLxFQlLDBWsDCC1BakJooc94ZatZPIvkFUUQcegmdghZkNsfIAjLwJbunA35EsfTrnXtk+cS6FQc97dyoLi0vLK9XV2tr6xuaWu73TM1mhOXR5JjN9FTMDUqTQRYESrnINTMUSLuPR6TS/vAFtRJZ2cJxDqNh1KhLBGVorcuvNJg1UEXn0LGI0QLjFkvZVJ5w0m5Hb8FreTPQv+HNokLnOI/cjGGS8UJAil8yYvu/lGJZMo+ASJrWgMJAzPmLX0LeYMgUmLGefmNB96wxokml7UqQz9/tGyZQxYxXbScVwaH5nU/O/rF9gchyWIs0LhJR/XZQUkmJGp43QgdDAUY4tMK6FfSvlQ6YZR9tbLRhAEmAZ4BCQTWq2E/93A3+hd9jyvZZ/4TXaJ/N2qqRO9sgB8ckRaZMzck66hJM78kAeyZNz7zw7L87r12jFme/skh9y3j4BkUWcVQ==</latexit>02468020040060080010001200FIG. 2: Speed of the domain wall versus applied \feld in millitesla for a thin Permalloy nanotube of\nradius R= 55 nanometers. At low \feld the speed increases linearly with the \feld, after a sudden\nchange in slope the speed tends to a constant value v1at large \felds. The dashed lines show the\nlimiting speeds vLandv1.\nThe damped DSG equation captures the emission of spin waves that occurs below but close\ntovpmin.\nA di\u000berent transition occurs at hKPP\na= 2A0when the speed of the reaction di\u000busion\nequationcrchanges from a pushed to a pulled or KPP front [33] and crbecomes proportional\nto the square root of the applied \feld.\nIn what follows consider Permalloy for which ku= 0. Going back to dimensional quanti-\nties, the speed of the domain wall for Permalloy is given by Eq. (2) with the limiting values\nat low and high \felds Eq. (3) and Eq. (4). In Fig. 2 the graph of the speed as a function\nof the applied \feld shows the gradual change from the linear to the magnonic regime. We\nhave used the values given above for Permalloy.\nAn approximate estimate of the \feld H\u0003\naat which this transition occurs is obtained by\nthe intersection vL(H\u0003\na) =v1which yields\nH\u0003\na=\u000b\nRs\n2A\n\u00160:\nFor Permalloy we obtain v1= 1006 m s\u00001, andB\u0003\na=\u00160H\u0003\na= 0:001 T. The transition to the\nKPP regime occurs at a much higher \feld, BKPP\na=\u00160HKPP\na= 0:021 T and is not associated\nto the transition from the linear to the magnonic regime. In this simple model the order of\nmagnitude of the speed and the value of the \feld at which the transition from the linear to\nthe magnonic regime occurs agrees with the order of magnitude of the numerical simulations\nof the LLG equation.\n10IV. SUMMARY\nWe studied the dynamics of a vortex domain wall in a thin nanotube by means of an\nasymptotic study of the Landau-Lifshitz Gilbert equation in a parameter regime based on\nexisting numerical simulations [5, 6]. The numerical simulations on Permalloy nanotubes in\na certain range of radii showed that when an external \feld is applied along the axis, domain\nwalls of one type of chirality, for which the radial magnetization remains small during the\nmotion, are stable and can reach high speeds. Initially the speed increases linearly with\nthe applied \feld, and at higher \felds the rate of increase is slowed down by the emission\nof spin waves. No Walker breakdown was observed in the parameter range considered in\nthe numerical studies. Domain walls of the opposite chirality are unstable and as the \feld\nincreases they convert into DW of stable chirality.\nThe purpose of this work was to understand analytically the behavior of the speed of\ndomain walls of stable chirality as a function of the applied \feld. Through an asymptotic\nanalysis the LLG was reduced to the damped double sine-Gordon equation from which an\nexplicit analytic formula for the speed as a function of the applied \feld was obtained together\nwith the leading order DW pro\fle. This model captures the initial regime of linear growth\nof the speed followed by a slowdown in the rate of increase through the emission of spin\nwaves before reaching the minimal phase speed of the spin waves, which is an upper bound\non the speed of the DW in this model. The order of magnitude of the speed and the value\nof the applied \feld where the transition from the linear to the magnonic regime occurs is in\nagreement with the numerical results of [5, 6]. In order to reach higher \felds and capture\nthe Cherenkov spin wave emission process a di\u000berent asymptotic regime is necessary.\nFor Permalloy, which has vanishing uniaxial anisotropy, the ratio of the exchange constant\nwith the square of the radius of the nanotube A=R2plays the role of an e\u000bective uniaxial\nanisotropy which leads to a mobility proportional to the nanotube radius. In constrast, for\nweak ferromagnets the mobility is proportional to de Dzyaloshinkii-Moriya constant. That\nthe e\u000bect of curvature acts as an equivalent e\u000bective anisotropy was already shown in [8, 22],\nand an analogy between the e\u000bect of DMI and curvature was found in the dispersion relation\nof spin waves in a nanotube [23]. The results in this manuscript show a similar e\u000bect when\nstudying the transition from the linear to the magnonic DW regime in nanotubes.\nAn analytical approach to the regime of higher \feld, where Cherenkov emission occurs,\n11will be the subject of future study.\nV. ACKNOWLEDGMENTS\nThis work was partially supported by Fondecyt (Chile) project 116{0856.\n[1] L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjet., 8, 153 (1935).\n[2] T. L. Gilbert, Ph.D. Thesis, Illinois Institute of Technology (1956), partially reprinted in IEEE\nTrans. Mag., 40, 3443 (2004).\n[3] L.R. Walker, Bell Telephone Laboratories Memorandum, 1956 (unpublished). An account of\nthis work is given in J.F. Dillon, Jr., Magnetism, Vol. III, edited by G.T. Rado and H. Suhl\n(Academic, New York, 1963).\n[4] N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406 (1974).\n[5] M. Yan, A. C. K\u0013 akay, F. Garc\u0013 \u0010a-Sanchez and R. Hertel, Appl. Phys. Lett. 99, 122505 (2011).\n[6] R Hertel, J. Phys.: Condens. Matter 28, 483002 (2016).\n[7] R. Hubert, A. Sch afer, Magnetic Domains (Springer-Verlag Berlin Heidelberg, 1998).\n[8] A. Goussev, J. M. Robbins and V. Slastikov, EPL 105, 67006 (2014).\n[9] E. M. Gyorgy, and F. B. Hagedorn, J. Appl. Phys. 39, 88 (1968).\n[10] A. K. Zvezdin, JETP Letters 29513 (1979).\n[11] N. Papanicolaou, Phys. Rev. B 55, 12290-12308 (1997).\n[12] E. G. Galkina and B. A. Ivanov, Low Temp. Phys. 44618 (2018).\n[13] Y. Nakatani, A. Thiaville and J. Miltat, Nature Materials 2, 521 (2003).\n[14] A. Thiaville and Y. Nakatani, Domain-Wall Dynamics in Nanowires and Nanostrips in: Hille-\nbrands B., Thiaville A. (Eds) Spin Dynamics in Con\fned Magnetic Structures III. Topics in\nApplied Physics, vol 101. Springer, Berlin, Heidelberg (2006).\n[15] R. Wieser, U. Nowak and K. D. Usadel, Phys. Rev. B 69, 064401 (2004).\n[16] R. Wieser, E. Y. Vedmedenko and R. Wiesendanger, Phys. Rev. B 81, 024405 (2010).\n[17] X. S. Wang and X. R. Wang, Phys, Rev. B 90184415 (2014).\n[18] D. Bouzidi and H. Suhl, Phys. Rev. Lett. 652587 (1990).\n[19] I. A. Akhiezer and A. E. Borovik, Soviet Phys. JETP 25885 (1967).\n12[20] V. G. Baryakhtar, B. A. Ivanov and M. V. Chetkin, Sov. Phys. Usp. 28, 563 (1986).\n[21] A. Goussev, J. M. Robbins, V. Slastikov and O. A. Tretiakov, Phys. Rev. B 93054418 (2016).\n[22] Y. Gaididei, V.P. Kravchuk and D. D. Sheka, Phys. Rev. Lett. 112257203 (2014).\n[23] J. A. Ot\u0013 alora, M. Yan, H. Schultheiss, R. Hertel and A. K\u0013 akay, Phys. Rev. Lett. 117, 227203\n(2016).\n[24] M. Sta\u0014 no and O. Fruchart, Ch. 3 of Handbook of Magnetic Materials vol. 27, North Holland\n(Elsevier), pp. 155-267 (2018).\n[25] G. Carbou, Math. Models Methods Appl. Sci., 11, 1529 (2001).\n[26] R. V. Kohn and V. Slastikov , Arch. Ration. Mech. Anal., 178, 227 (2005).\n[27] K. P. Hadeler, Proc. Edinburgh Math. Soc. 31, 89-97 (1988).\n[28] T, Gallay and R. Joly, Ann. Sci de l'Ecole Normale Sup\u0013 erieure, 42, 103-140 (2009).\n[29] H. J. Mikeska, J. Phys. C 13, 2913 (1980).\n[30] H. How, R. C. O'Handley and F. R. Morgenthaler, Phys. Rev. B 40, 4808 (1989).\n[31] P. Landeros and A. S. Nu~ nez, J. Appl. Phys. 108033917 (2010).\n[32] A. L. Gonz\u0013 alez, P. Landeros and A. S. N\u0013 u~ nez, J. Mag. Mag. Mater. 322, 530 (2010).\n[33] M. C. Depassier, EPL 108, 37008 (2014).\n13"    },    {        "title": "1711.07455v1.Spin_Pumping_in_Ion_beam_Sputtered_Co__2_FeAl_Mo_Bilayers_Interfacial_Gilbert_Damping.pdf",        "content": "Spin Pumping in Ion-beam Sputtered Co2FeAl/Mo Bilayer s: \nInterfacial Gilbert Damping  \nSajid Husain, Vineet Barwal, and Sujeet Chaudhary*  \nThin Film Laboratory, Department of Physics, Indian Institute of Technology Delhi, New Delhi 110016 (INDIA)  \nAnkit Kumar, Nilamani Behera, Serkan Akansel, and Peter Svedlindh  \nÅngström Laboratory, Department of Engineering Sciences, Box 534, SE -751 21 Uppsala, Sweden  \nAbstract  \nThe spin pumping mechanism and associated interfacial Gilbert damping are demonstrated  in \nion-beam sputtered Co2FeAl  (CFA)/Mo bilayer thin films employing ferromagnetic resonance  \nspectroscopy . The d ependence  of the net spin current transportation on Mo layer thickness, 0 to \n10 nm, and the enhancement of the net effective  Gilbert damping are reported . The experimental \ndata has been analyzed using spin pumping  theory  in terms of  spin current pumped through the \nferromagnet/nonmagnetic metal interface  to deduce the  effective  spin mixing conductance  and \nthe spin -diffusion length , which  are estimated to be 1.16(±0.19 )×1019 m−2 and 3.50±0.35nm, \nrespectively. The damping constant is found to be 8 .4(±0.3)×10-3 in the Mo(3.5nm) capped \nCFA(8nm) sample corresponding to a  ~42% enhancement of the original Gilbert damping \n(6.0(± 0.3)×10-3) in the uncapped CFA layer. This is  further confirm ed by insertin g a Cu dusting \nlayer which reduce s the spin transport across the CFA/Mo interface. The Mo layer thickness \ndependent net spin current density is found to lie in the ra nge of 1-3 MAm-2, which  also provides \nadditional quantitative evidence of spin pumping in this bilayer thin film system .  \n*Author for correspondence: sujeetc@physics.iitd.ac.in  \n \n  I. INTRODUCTION  \nMagnetic damping is an exceedingly importan t property for spintronic devices  due to its \ninfluence on  power  consumption  and information  writing in the spin-transfer torque random  \naccess memor ies ( STT-MRAMs)  [1][2]. It is therefore of high importance to study the  \ngeneration, manipulation , and detection of the flow of spin angular momentum to enable the \ndesign of efficient  spin-based magneti c memories and logic devices  [3]. The transfer of spin \nangular momentum known as  spin pumping in ferromag netic (FM)/ nonmagnetic (NM) bilayer s \nprovide s information of how the precession of the magnetization transfer s spin angular \nmomentum into the adjacent  nonmagnetic  metallic  layer  [4]. This transfer ( pumping ) of spin \nangular momentum slows down the precession and leads  to an enhance ment of  the effective \nGilbert damping constant  in FM/NM bilayers . This enhancement  has been  an area of intensive \nresearch since the novel mechanism (theory) of spin pumping was proposed by Arne Brataas et \nal.  [5] [6]. The amount of spin pumping  is quantified by the magnitude of the spin current \ndensity  at the FM/NM  interface  and theoretically  [7] described as \n4eff\nS effdgdt mJm\n  where  \nm is the magnetization unit vector, \neff\nSJ is the effective spin current density pumped into the NM \nlayer from the FM layer  (portrayed in Fig. 1), and \neffg  is the spin mixing conductance  which is \ndetermined  by the reflection  coefficient s of conductance channels at FM/NM interface  [5]. \nTo date, a number of  NM metals , such as  Pt, Au, [5], Pd  [8][9],-Ta [10] and Ru  [11], \netc. have been e xtensively investigated with regards to their performance as spin sink material  \nwhen in contact with a FM . It is to be noted here that none of the Pt, Pd, Ru, and Au is an earth \nabundant  material  [12]. Thus , there is a natural need  to search for new non-magnetic material s \nwhich could generate  large spin current at the FM/NM in terface . In this study , we have explored the potential of the transition metal molybdenum (Mo)  as a new candidate  material  for spin \npumping  owing to the fact that  Mo possesses a large spin-orbit coupling  [13]. To the best of our \nknowledge, Mo has not been used till date for the study  of spin pumping effect in a FM/NM  \nbilayer system .  \nIn a FM/NM  bilayer  and/or multilayer  system s, there are several mechanisms for \ndissipation of the spin angular momentum  which are categorized as intrinsic and extrinsic . In the \nintrinsic  category , the magnon -electron coupling , i.e., spin -orbit coupling (SOC)  contributes \nsignificant ly [14]. Among the extrinsic  category , the two-magnon scattering  (TMS ) mechanism \nis linked to the inhomogeneity  and interface/surface roughness  of the heterostructure , \netc.  [15] [16] [17]. For large SOC , interfacial d-d hybridization between the NM and FM layers \nis highly desirable  [16]. Thus, the FM -NM interfacial hybridization is expected to result in \nenhancement of the transfer of spin angular momentum from the FM to the NM layer , and hence \nthe NM layer can act as a spin reservoir  (sink)  [18]. But, the NM metallic layer  does not always \nact as a perfect spin reservoir  due to the spin accumulation effect which prevents transfer of \nangular momentum  to some extent and a s a result , a backflow of  spin-current towards the  \nFM  [6] is estabished . While the flow of spin angular momentum through the FM/NM  interface \nis determine d by the effective spin-mixing conductance  \n()effg  at the interface , the spin backflow  \nis governed  by the spin diffusion length \n()d . It is emphasized  here that t hese parameters  (\neffg  \nand \nd ) are primarily  tuned  by appropriate selection  of a suitable NM layer.  \nIn this work,  we have performed ferromagnetic resonance  (FMR) measurements to \nexplore the spin pumping phenomenon and associated interfacial Gilbert damping enhancement  \nin the Co2FeAl(8nm) /Mo(\nMot) bilayer system , \nMot is the thickness of Mo , which is varie d from 0 to 10  nm. The \nMot dependent net spin current transfer across the interface  and spin diffusion \nlength of Mo are estimated . The choice of employing the Heusler alloy CoFe 2Al (CFA) as a thin \nFM layer lies in its half metallic character anticipated at room temperature  [19] [20], a trait \nwhich is highly desirable in any spintronic device operating at room temperature.  \nII. EXPERIMENTAL DETAILS  \nThe CFA  thin films with fixed thickness of 8 nm were  grown  on naturally oxidized Si(100) \nsubstrate at 573K temperature using an ion-beam  sputtering deposition system ( NORDIKO -\n3450). The substrate temperature (573K) has been selected  following the growth optimization \nreported in our previous reports  [21] [20] [22]. On the top of the CFA layer , a Mo film with \nthickness \nMot  (\nMot=0, 0.5, 1.0, 1.5, 2.0, 3.0, 4, 5, 7, 8 and 10 nm) was deposited  in situ  at room \ntemperature . In addition , a trilayer structure of CFA(8)/Cu(1)/Mo(5) was also prepared to \nunderstand  and confirm  the effect of an additional interface on the Gilbert damping  (spin \npumping ). Numbers in parenthesis are film thicknesses in nm. All the samples were prepared  at a \nconstant working pressure  of ~8.5×10-5 Torr (base vacuum  ~ 1.010-7 Torr); Ar gas was directly \nfed at 4 sccm  into the rf-ion source  operated at 75W  with the deposition rate s of 0.03nm/s  and \n0.02nm/s for CFA and Mo, respectively . The deposition rate for Cu was 0.07nm/s at 80 W. The \nsamples were then cut to 1×4 mm2 to record the FMR spectra employing  a homebuilt  FMR set-\nup [21] [23]. The data was collected  in DC-magnetic field sweep mode by keeping the \nmicrowave frequency fixed . The saturation  magnetization was measured  using  the Quantum \nDesign make Physical Property Measurement System (Model PPMS Evercool -II) with the \nvibrating sample  magnetometer option (QD PPMS -VSM). The film density, thickness  and \ninterface /surface  roughness were estimated  by simulating the  specular X -ray reflectivity (XRR) \nspectra using the PANalytical X’Pert reflectivity software (Ver. 1.2 with segmented fit). To determine surface morphology /microstru cture (e.g., roughness) , topographical imaging  was \nperformed using the ‘Bruker dimension ICON  scan assist’  atomic fo rce microscope (AFM).  All \nmeasurements were performed at  room temperature.  \nIII. RESULTS AND DISCUSSIONS  \nA. X-ray Reflectivity and A tomic Force Microscopy : Interface/surface analysis  \nFigure  2 shows the specular XRR  spectra  recorded on all the CFA(8)/Mo(\nMot ) bilayer thin films . \nThe fitting parameters were accurately determined  by simu lating (red lines) the experimental  \ncurves  (filled circles)  and are presented  in Table -I. It is evident that for the smallest NM layer \nthickness , Mo(0.5nm) , the estimated value s of the roughness from XRR and AFM are slightly \nlarger in comparison to the thickness of the Mo layer which indicates that the surface coverage of \nMo layer is not enough to cover all of the  CFA  surface  in the CFA(8)/Mo(0.5) bilayer sample  \n(modeled in Fig. 3(a)). For \nMot  ≥ 1nm, the film roughness is smaller than the thickness \n(indicating that t he Mo layer coverage is uniform as modeled in Fig s. 3(b)-(c)). For the  thicker \nlayer s of Mo (\nMot≥ 5nm) the estimated values of the surface roughness as estimated from  both \nXRR and AFM are found to be similar ~0.6nm (c.f. the lowest right panel in Fig. 2).  \nB. Ferromagnetic Resonance  Study  \nThe FMR  spectra were  recorded  on al l sample s in 5 to 11 GHz range of microwave frequenc ies. \nFigure. 4(a) shows the FMR spectra  recorded on the CFA(8)/Mo( 5) bilayer thin film . The  FMR \nspectra \n()FMRI  were fitted with the derivative of symmetric and anti -symmetric Lorentzian \nfunction s to extract the line -shape parameters, i.e., resonant field \nrH  and linewidth\nH , given \nby [24] [21]: \n22\n2\n2222\n22()\n() ()22 2\n( ) ( )22FMR\ndc rdc r\ndcS ext A ext\next ext\nr dc rUIH\nHH HHH HH\nSA\nHF H F H\nHHSA\nHHHH\nH\n        \n                 \n          , (1) \nwhere \nS extFH  and \nA extFH  are the symmetric and anti -symmetric Lorentzian functions, \nrespectively, with S and A being  the corresponding coefficients.  Symbol ‘U’ refers to the raw \nsignal voltage from the VNA . The linewidth  \nH  is the full width at half maxim um (FWHM) , \nand \ndcH  is the applied DC-magnetic  field.  \nThe f vs. \n0 rH  plots are  shown  in Fig. 4(b). These are  fitted using t he Kittel’s formul a [25]: \n 0()2r K r K eff f H H H H M\n   \n ,  (2) \nwhere 𝛾 is the gyromagnetic ratio ; \n/Bg\n  (1.76×1011s-1T-1) with \ng being the Lande’s  \nsplitting factor ; taken as 2,  \n0 effM  is the effective saturation magnetization, and  \n0 KH  is the \nuniaxial anisotropy field. The value s of \n0 effM  are comparable to the values of \n0 SM  (obtained \nfrom VSM measurements) as is shown in Fig . 4(c). Figure 4(e) shows  the variation  of \n0 KH  \nwith \nMot from which the decrease in \n0 KH  with increas e in \nMot  is clearly evident . This observed \nreduction  in \n0 KH  could possibly stem from the spin a ccumulation increasing with increasing  \nMot\n [26]. The FMR spectra was also recorded on  CFA (8)/Cu(1)/Mo(5) trilayer thin film for the  \ncomparison with the results of CFA (8)/Mo(5) bilayer. The magnitudes of  \n0 effM  (\n0 kH ) for \nCFA (8)/Mo(5) and CFA (8)/Cu(1)/Mo(5) are found to be 1.33±0.08 T (0.55±0.15mT) and  1.30±0.04 T (3.21±0.13 mT) , respectivel y. Further, Fig. 4(d) shows the \n0 rH  vs. \nMot behavior at \ndifferent constant frequenc ies ranging between  5 to 11 GHz. The observed values of \n0 rH  are \nconstant for all the Mo capped layers which  clearly  indicate s that the dominant contribution to \nthe observed resonance spectra arises from the intrinsic effect, i.e., magnon -electron \nscattering  [27]. \nC. Mo t hickness -dependen t spin pumping  \nFigure 5(a) shows t he linewidth  \n0µH  vs. f (for clarity, the results are shown  only for a \nfew sel ected film samples ). The frequency dependent linewidth can mainly  have  two \ncontributions ; the intrinsic  magnon -electron scattering  contribution,  and the extrinsic  two-\nmagnon scattering  (TMS ) contribution. The extrinsic  TMS  contribution in linewidth has been \nanalysed  (not presented here) using the methods given by Arias and Mills  [28]. A similar \nanalysis was reported  in one of our previous studies on the CFA/Ta  system  [21]. For the present  \ncase, t he linewidth analysis shows that inclusion of the TMS part does not affect the Gilbert \ndamping , which means  the TMS  contribution  is negligible in our case.  Now , the effective  Gilbert \ndamping constant \neff can be estimated using,  \n0\n04efffHH\n  \n. (2) \nHere, \n0H  is the frequency independent contribution  from  sample  inhomogeneity , while the \nsecond term corresponds to the frequency dependent contribution  associated with the intrinsic \nGilbert relaxation . Here , \neff , defined as  \neff SP CFA   , is the effective  Gilbert damping which includes the intrinsic  value of CFA  \n()eff  and a spin pumping  contribution  (\nSP ) from the \nCFA/Mo  bilayer . \nThe extracted effective Gilbert damping constant values for different \nMot  are shown  in \nFig 5(b). An enhancement of the  Gilbert damping constant with the increase of the Mo layer \nthickness is clearly observed , which is anticipated  owing to the transfer  of spin angular momenta \nby spin pumping from CFA to the Mo layer at the CFA /Mo interface . The value  of \neff  is found \nto increase  up to  8.4(± 0.3)×10-3 with the increase in  \nMot (≥ 3.5nm) , which corresponds to ~42% \nenhancement of the damping constant due to spin pumpin g. It is remarkable that such a large \nchange in Gilbert damping is observed for the CFA/Mo  bilayer ; the change is comparabl e to \nthose reported when a high SOC NM such as  Pt [8], Pd [29] [9], Ru [11], and Ta [30] is \nemployed in FM/NM bilayer s. Here , we would like to mention that the enhancement of the \nGilbert dampin g can , in principle , also be explained  by extrinsic two-magnon scattering  (TMS ) \ncontribution s in CFA/Mo(\nMot ) bilaye rs by considering the variation  of \nrH  with NM \nthickness  [27]. In our case, the  \n0 rH  is constant  for all \nMot  (c.f. Fig . 4(d)).  Thus the extrinsic \ncontribution induced increase  in \neff  is negligibly small  and hence  the enhancement of the \ndamping is dominated by the  spin pumping  mechanism . The estimated values of  \n0 0µH  are \nfound to vary  from 0.6 to 2.5 mT in the CFA/Mo(\nMot ) thin films . The variation in \n0 0µH  is \nassigned  to the finite, but small, statistical variations in sputtering conditions between  samples \nwith different  \nMot. \nFurther, to affirm the spin pumping in the CFA/Mo bilayer system, a copper  (Cu) dusting \nlayer was inserted at the CFA/Mo interface. Fig ure 5(c) compares the linewidth  vs. f plot of the CFA (8)/Mo(5) and CFA (8)/Cu(1)/Mo(5) heterostructures. The Gilbert damping was found to \ndecrease from 8.4(± 0.3)×10-3 to 6.4(± 0.3)×10-3 after inserting the Cu (1) thin layer, which is \ncomparable to the value of the uncapped CFA (3.5) sample. It may be noted that Cu has a very \nlarge  spin diffusion length  (\nd~300nm)  but weak SOC  strength  [32]. Due to the weak SOC,  the \nasymmetry in the band structure at the FM/Cu interface would thus lead to a non -equilibrium \nspin accumulation  at the CFA/ Cu interface  [33]. This spin accumulation  opposes the transfer of \nangular momentum into the Mo layer  and hence the Gilbert damping value , after insertion of the \ndusting layer , is found very similar to that of the single layer  CFA film. It  is also known  that \nenhancement of damping in the FM layer (when coupled to the NM layer ) can occur due to  the \nmagnetic proximity effect  [34]. However, we did not find any evidence in favor of the proximity \neffect as the effective saturation magnetization  did not show any increase on the inserti on of the  \nultrathin  Cu dusting layer at CFA/Mo bilayer interface , which support s our claim of absence of  \nspin pumping in the CFA/Cu/Mo trilayer sample . \nThe flow of angular momentum across the FM/NM bilayer interface is determined  by the \neffective  complex spin-mixing conductance \ng Re(g ) Im(g )eff eff eff i   , defined as the flow of \nangular momentum per unit area through the FM/NM metal interface  created by the  precessing  \nmoment s in the FM layer . The term effective  spin-mixing conductance is being used  because it \ncontain s the forward and  backflow  of spin momentum at the FM/NM interface. The imaginary \npart of the spin-mixing conductance  is usually  assumed to be negligibly small   \nRe(g ) Im(g )eff eff \n as compared  to the real part  [35] [36], and therefore, to determine the real \npart of the spin-mixing conductance , the obtained  \nMot dependent G ilbert damping  is fit ted with \nthe relation  [29],  21Re(g ) 14Mo\ndt\nB\neff CFA eff\nS CFAgeMt \n   , (3) \nwhere \nCFAis the damping for a single layer CFA without Mo capping  layer, \nRe(g )eff  is given  \nin unit s of m-2, \nB  is the Bohr magneton,  and \nCFAt  is a CFA  layer thickness . The exponential \nterm describes the reflection of spin -current from Mo/air  interface . Figure 5(b) shows the  \nvariation of  the effective Gilbert damping constant with  \nMot and the fit using  Eqn. (3) (red line) . \nThe values of \nRe(g )eff  and \nd are found to be 1. 16(±0.19 )×1019 m-2 and 3.5±0.35 nm, \nrespectively. The  value of the spin-mixing conductance is comparable to those recent ly reported \nin FM/Pt (Pd) thin films  such as  Co/Pt ( 1-4 ×1019 m-2) [8] [33], YIG/Pt  (9.7 ×1018 m-2)  [37], \nFe/Pd (1×1020 m-2)  [9], and Py/Pd(Pt) (1.4(3.2) ×1018 m-2) [34].  \nWe now calculate the net intrinsic interfacial spin mixing conductance \nG  which \ndepends on the thickness and the nature of the NM layer  as per the relation   [9]  [38], \n11\n4( ) Re(g ) 1 tanh3Mo\nMo eff\ndtGt\n\n \n, (4) \nwhere \n24( / )Z e c\n  is a material dependent  param eter (Z is the atomic number of Mo i.e., 42 \nand c is the speed of light) whose value for Mo is 0.0088 . Using Eq.  (4), \n()Mo Gt values have \nbeen compu ted for variou s \nMot ; the results  are shown in Fig. 6(a). The \nMot  dependence of \nG  \nclearly suggest s that the spin mixing conductance  critically depend s on the NM layer properties . \nFor bilayers with \nMot  6 nm, \nG  attains its saturation value, which is quite  comparable with \nthose reported for Pd and Pt  [34]  [37]. Understandably, such a large value of the spin mixing  conductance  will yield a large spin current  into the adjacent NM layer  [6] [7] [37] [33]. In the \nnext section, we have estimated the spin current from the experimental FMR data and discuss the \nsame with regards to spin pumping in further detail . \nD. Spin current generation in Mo due to spin pumping  \nThe enhancement of the Gilbert damping  observed in the CFA (8)/Mo(\nMot) bilayers  (Fig. 5(b)) is \ngenerally  interpreted in terms of  the spin -current generated in Mo layer  by the spin pumping \nmechanism at the bilayer interface  (Fig. 1). The associated net effective  spin current density  in \nMo is described by  the relation  [38] [39]: \n \n00 02 22 2\n2\n0224 2( ) G ( )8 4eff eff rf eff\nS Mo Mo\neffeffMM h eJ t t\nM   \n\n\n, (5) \nwhere\n2f and \nrfhis the rf-field (26 A/m) in the strip -line of our co-planar waveguide.  \nG ( )Mot\n is the net intrinsic inte rfacial spin mixing conductance discussed  in the previous \nsection  (Fig.  6). The estimated values of  \n()eff\nS MoJt  for differen t microwave frequencies are shown  \nin Fig. 7. It is clearly  observed  that the spin current density increase s with the increase  in \nMot, the \nincrease becomes relatively less at higher  \nMot , which indicate s the progressive spin current \ngeneration in Mo . Such an appreciable change in current density  directly provide s evidence of \nthe interfacial enhancement of the Gilbert damping  in these CFA/Mo bilayers . \nFurther,  it would be interesting  to investigate the effect on the  spin current generation in \nMo layer if  an ultra thin dusting layer  of Cu is inserted at the CFA/Mo interface . In princip le, on \ninsertion of a thin Cu layer , the spin pumping should cease  because of the unmatched band \nstructure between the CFA /Cu and Cu/M o interfaces owing to the insignificant SOC in Cu. This is in consonance with the observed decrease in Gilbert damping back to  the value for the \nuncapped CFA layer  (c.f. Fig. 5(c) and associated discussion ). The spi n-mixing conductance of \nthe trilayer heterostructure can be evaluated by \n0 g/eff B eff S CFAg M t   [29], where \nsp eff CFA    \n is the spin-pump ing induced Gilbert damping contribution which for the \nCFA/Cu/Mo trilayer is quite small , i.e., 4.0(±0.3) ×10-4 after Cu insertion.  For the trilayer, \ngeff  is \nfound to be 1.49 (±0.12) ×1017 m-2 which is two order s of magnitude small er compared  to that of \nthe CFA/Mo bilayers. Furthermore, u sing the values of  \ngeff , \n0 effM  and \neff  for the \nCFA/Cu/Mo trilayer hetero structure in Eqn. (5) and for f = 9GHz, the spin current density is \nfound be 0.0278 (±0.001 3) MA/m2, which is two order  of magnitude smaller than that in the \nCFA/Mo bilayers. Thus, t he reduction in \neff  and \neff\nSJ  subsequent to Cu dusting is quite \ncomparable  to previously reported results  [33] [40]. \nIV. CONCLUSIONS  \nWe have systematically investigated the  changes in the spin dynamics in the ion-beam sputtered \nCo2FeAl ( CFA )/Mo(\nMot) bilayer s for various \nMot  at constant CFA thickness  of 8nm . Increasing \nthe Mo layer thickness to its spin diffusion length; CFA (8)/Mo(\nMot =\nd), the effective  Gilbert \ndamping constant increases to  8.4(± 0.3)×10-3 which  corresponds to about  ~42% enhancement  \nwith respect to  the \neff  value of 6.0(± 0.3)×10-3 for the uncapped CFA layer  (i.e., without the top \nMo layer ). We interpret our results based on the spin -pumping effect s where in the effective spin-\nmixing conductance , and spin -diffusion length are found to be 1.16(±0.19 )×1019 m−2 and \n3.50±0.35nm, respectively.  The spin pumping is further confirmed  by inserting  an ultrathin  Cu \nlayer at the CFA/Mo interface. The overall effect of the damping constant  enhancement observed when Mo is deposited  over CFA is remarkably  comparable to the far less -abundant  non-\nmagnetic  metals that are currently  being used  for spin pumping  applications . From this view  \npoint , the demonstration of the new material , i.e., Mo, as a suitable spin pumping medium is \nindispensable  for the development  of novel STT spintronic  devices . \nACKNOWLEDGMENT S \nOne of the authors SH acknowledge s the Department of Science and Technology , Govt. of  India \nfor providing the INSPIRE Fellowship. Authors thank the NRF  facilit ies of IIT Delhi for AFM \nimaging . This work was in part supported by Knut and Alice Wallenberg (KAW)  Foundation \nGrant No. KAW 2012.0031 . We also acknowledge the Ministry  of Information  Technology,  \nGovernment  of India  for providing the  financial  grant . \n  REFERENCES:  \n[1] W. Kang, Z. Wang, H. Zhang, S. Li, Y. Zhang, and W. Zhao, Glsvlsi 299 (2017).  \n[2] E. Eken, I. 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Here , \nFMt , \nMot, and σ refer to the density, thickness, and \ninterface width of the individual layers , respectively.  \n CFA  (Nominal thickness = 8 nm) Mo MoOx  \nS.No.  \n1 \n2 \n3 \n4 \n5 \n6 \n7 \n8 \n9 \n10 (gm/cc)±0.06  \n7.35 \n7.31 \n7.50 \n7.50 \n7.00 \n7.00 \n7.29 \n7.22 \n7.00 \n7.64 tFM(nm)±0.01  \n7.00 \n8.17 \n7.22 \n8.18 \n7.00 \n8.28 \n8.00 \n7.79 \n8.12 \n8.00 σ(nm) ±0.03  \n0.20 \n0.35 \n0.80 \n0.37 \n1.00 \n0.56 \n0.44 \n0.98 \n0.15 \n0.17 (gm/cc)±0.0 5 \n6.05 \n8.58 \n10.50  \n9.94 \n9.50 \n10.45  \n9.43 \n10.50  \n9.29 \n9.23 tMo(nm)±0.01  \n0.58 \n1.00 \n1.50 \n2.00 \n3.00 \n3.46 \n4.86 \n6.47 \n8.21 \n10.26  σ(nm) ±0.03  \n0.94 \n0.54 \n0.52 \n0.64 \n0.60 \n0.78 \n0.26 \n0.67 \n0.64 \n0.67 (gm/cc)±0.06  \n4.07 \n4.04 \n5.00 \n4.38 \n5.17 \n4.38 \n6.50 \n4.81 \n4.00 \n5.00 t(nm)±0.01  \n0.97 \n0.82 \n1.08 \n0.85 \n1.00 \n1.01 \n0.98 \n1.03 \n0.96 \n1.17 σ(nm) ±0.03  \n0.59 \n0.35 \n0.5 \n0.45 \n0.37 \n0.4 \n0.56 \n0.62 \n0.8 \n0.73 \n  Figure captions  \nFIG. 1. (color online) Schematic of the CFA/Mo bilayer structure  used in our work portrayed \nfor an example of spin current density \neff\nSJ  generated at the CFA/Mo interface  by spin pumping . \nFIG. 2 XRR spectra and the AFM topographical images of Si/CFA( 8)/Mo(\nMot ). In the \nrespective XRR spectra, circles represent the recorded experimental data points, and lines \nrepresent the simulated profiles. The estimated values of the surface roughness  in the entire \nsample series as obtained from XRR and AFM topographical measurements are compared in the \nlowest  right panel. The simulated parameters are presented  in the Table -I. All AFM images were \nrecorded  on a scan area of 10×10 m2. \nFIG.3 : The atomic representation  (model)  of the growth  of th e Mo layer (yellow sphere) on \ntop of the CFA (blue spheres) layer. The film changes from discontinuous to continuous as the \nthickness of the Mo layer is increased. Shown are the 3 different growth stages of the films: (a) \nleast coverage (b) partial coverage and (c) full coverage . \nFIG. 4: (a) Typical FMR spectra recorded  at various frequencies  (numbers in graph  are the \nmicrowave frequencies in GHz)  for the Si/SiO 2/CFA(8)/Mo(5)  bilayer sample (symbols \ncorrespond to experimental data and red  lines are fits  to the Eqn. (1)) Inset: FMR spectra of CFA \nsingle layer (filled circles) and CFA(8)/Mo(2) bilayer (open circles) samples measured at 5GHz \nshowing the increase in linewidth due to spin pumping. (b) The resonance field  \n0 rH  vs. f for all \nthe samples ( red lines are the fits to the Eqn. (2).  (c) Effective magnetization (scale on left) and \nsaturation magnetization (scale on right) vs.\nMot . The solid line represents  the bulk value of the \nsaturation magnetization  of Co 2FeAl . (d) The resonance field \n0 rH  vs. \nMot at different constant frequencies  for CFA(8)/Mo(\nMot ) bilayer thin films.  (e) Anisotropy field \n0 KH  vs. \nMot . (f) \nComparison  of \n0 rH  vs. f for the CFA(8)/Mo(5) and CFA(8)/Cu(1)/Mo(5) samples.  \nFIG. 5: (a) Linew idth vs. frequency for Si/SiO 2/CFA(8)/Mo(\nMot ) bilayer thin films. (b) \nEffective  Gilbert damping constant vs. Mo layer thicknesses. (c) \n0H  vs. f for CFA(8)/ Mo(5) \nand CFA(8)/Cu(1)/Mo(5) films.  \nFIG. 6 : Intrinsic s pin-mixing conductance vs. \nMot of the CFA (8)/Mo(\nMot) bilayers . \nFIG. 7. The effective spin current density (generated in Mo) vs. \nMot at different microwave \nfrequencies calculated using Eqn. (5)  \n   \nFIG. 1  \n  \n \nFIG. 2 \n  \n \nFIG. 3 \n  \n \nFIG. 4  \n  \n \n \nFIG. 5 \n  \n \nFIG. 6  \n  \n \n \nFIG. 7  \n \n"    },    {        "title": "2109.12605v1.Transition_state_dynamics_of_a_driven_magnetic_free_layer.pdf",        "content": "Transition state dynamics of a driven magnetic free layer\nJohannes M ¨ogerlea, Robin Schuldta, Johannes Rei \u000ba, J¨org Maina, Rigoberto Hernandezb,c,\u0003\naInstitut f¨ ur Theoretische Physik I, Universit¨ at Stuttgart, 70550 Stuttgart, Germany\nbDepartment of Chemistry, Johns Hopkins University, Baltimore, Maryland 21218, USA\ncDepartments of Chemical &Biomolecular Engineering, and Materials Science and Engineering, Johns Hopkins University, Baltimore, Maryland\n21218, USA\nAbstract\nMagnetization switching in ferromagnetic structures is an important process for technical applications such as data\nstorage in spintronics, and therefore the determination of the corresponding switching rates becomes essential. We\ninvestigate a free-layer system in an oscillating external magnetic field resulting in an additional torque on the spin.\nThe magnetization dynamics including inertial damping can be described by the phenomenological Gilbert equation.\nThe magnetization switching between the two stable orientations on the sphere then requires the crossing of a potential\nregion characterized by a moving rank-1 saddle. We adopt and apply recent extensions of transition state theory for\ndriven systems to compute both the time-dependent and average switching rates of the activated spin system in the\nsaddle region.\nKeywords: magnetization switching, ferromagnetic free-layer system, Landau–Lifshitz–Gilbert equation, transition\nstate theory, normally hyperbolic invariant manifold, stability analysis\n1. Introduction\nIn recent years, the promise of spintronics to emerging technological applications has attracted growing interest\nleading to extensive research e \u000borts in experimental [ 1–8] and theoretical physics [ 1,2,9–13]. The relative simplicity\nand accuracy of the single-domain models for ferromagnetic structures has proven to be a popular choice for charac-\nterizing such spintronics applications. Specifically, these models describe the macro spin-dynamics underlying the\nGilbert equation [ 14–16]. The landscape of the corresponding potential includes two minima at the stable spin up\nandspin down positions which are separated by a rank-1 saddle in certain configurations [ 17,18]. The typical goal\nin spintronics applications is to achieve and control the magnetization switching within a target timescale—viz., a\nspecified rate. This can be achieved, for example, through application of a spin torque [ 19]. An alternative approach\nis microwave-assisted magnetic recording—more specifically, microwave-assisted switching [ 20–23]—where a mi-\ncrowave field perpendicular to the easy axis is used in conjunction with a static external field along the easy axis in\norder to facilitate the magnetization switching. Multiple variations of this scheme have been proposed [24–27], some\nof which rely solely on rotating AC fields perpendicular to the easy axis [ 28,29]. In this paper, we focus on a single\nAC field along the easy axis without any static external fields.\nIn chemical reactions, the transition from reactants to products is typically marked by a barrier region with a\nrank-1 saddle that has exactly one unstable direction called the reaction coordinate, while the remaining orthogonal\nmodes are locally stable and are associated with other bound internal motions. The dynamical crossing of a rank-1\nsaddle in such chemical systems can be described by transition state theory ( TST) [30–36], which then allows for\nthe calculation of rate constants and the flux. However, TST is not restricted to chemical reactions as it has been\napplied in many other fields, including, e. g., atomic physics [ 37], solid state physics [ 38], cluster formation [ 39,40],\ndi\u000busion dynamics [ 41,42], and cosmology [ 43–45]. Notably, the theory has also been extended to time-dependent\n\u0003Corresponding author\nEmail address: r.hernandez@jhu.edu (Rigoberto Hernandez)\nPreprint submitted to Communications in Nonlinear Science and Numerical Simulation September 28, 2021arXiv:2109.12605v1  [cond-mat.mtrl-sci]  26 Sep 2021driven systems [ 46]. Although originally framed using perturbation theory [ 47–50], the requisite locally recrossing-free\ndividing surface ( DS) and instantaneous decay rates in TST can now be obtained with more generally-applicable\nmethods [51–55] as employed here.\nThus, the central result of this paper is the demonstration of the applicability of time-dependent TST to characterize\nthe dynamical crossing of a macrospin across a time-dependent rank-1 saddle using the recent advances cited above.\nIn the language of TST, the spin up andspin down regions can be interpreted as reactants andproducts , and the\nmagnetization switching corresponds to the “chemical” reaction. An important di \u000berence between the previous systems\nto which TST has so far been applied, and the ferromagnetic systems described by the Gilbert equation lies in the\ngeometry of the phase-space structure. Typically, a Hamiltonian system with ddegrees of freedom is described by\na (2d)-dimensional phase space with dcoordinates and dassociated velocities or momenta. The Gilbert equation,\nhowever, is a first-order di \u000berential equation for the dynamics of the magnetic moment on a sphere, i. e., there are\nno independent velocities or momenta. Therefore, the dynamics is e \u000bectively that of a one degree of freedom ( DoF)\nsystem [ 14,56–58]. Nevertheless, within this domain a DScan be associated with the neighborhood of the rank-1\nsaddle. In analogy to chemical reactions, we conjecture that the reactive flux across this DSis associated with the\ndecay rate of the spin flip. In this context, the reactive flux is that of all the trajectories that are reactants (viz., spin up )\nin the infinite past and products (viz., spin down ) in the infinite future. In transition state theory, the reactive flux is\napproximated by the sum of the positive velocities (headed in the direction of the product) over the surface, and it is\nexact if no trajectory recrosses the DS.\nWe show that recent extensions of TST for systems with time-dependent moving saddles [ 51–54] can indeed be\napplied to a ferromagnetic single-domain system with a two-dimensional phase space describing the orientation of the\nmagnetic moment on the sphere and the dynamics following the Gilbert equation. The system can even be driven by a\ntime-dependent external magnetic field. The free-layer system and the applied methods are introduced in Sec. 2. The\napplicability of TST relies on the fact that for any time t, the two-dimensional phase space exhibits a stable and unstable\nmanifold, which intersect in a point on the normally hyperbolic invariant manifold ( NHIM ). A locally recrossing-free\nDSseparating the spin down andspin up regions in phase space can be attached to this point. The time-dependent\nmoving points of the NHIM form the transition state ( TS) trajectory, which is a periodic orbit when the free-layer\nsystem is driven by an oscillating magnetic field.\nThe TStrajectories are the starting point for the calculation of rate constants, and the characterization of the\nmagnetization switching. Through application of the ensemble method and the local manifold analysis ( LMA )\ndeveloped in Ref. [ 54], we obtain the time-dependent instantaneous rates along TStrajectories at various amplitudes\nand frequencies of the driving external magnetic field. We also find in Sec. 3 that the time-averaged rates along the TS\ntrajectories depend significantly on the external driving.\n2. Theory and methods\nHere, we briefly discuss our model (cf. Fig. 1) motivated by a free-layer system [ 17] and present the equations,\nwhich describe the spin dynamics of this model including external driving. Then we introduce the basic ideas of TST\nand the methods, which will be applied for the computation of the instantaneous and average rates of the magnetization\nswitching.\n2.1. Spin dynamics in a driven free-layer model\nThe model addressed here is based on a magnetic single-domain layer with variable magnetization M, known as a\nfree layer . This layer is modeled in analogy to Stoner and Wohlfarth [ 59,60] including a demagnetization field for a\nthin film (shape anisotropy) [ 60,61]. A periodic external magnetic field is added to drive the magnetization. This field\nis intended as a generic placeholder for some externally applied torque—e. g., certain types of spin torque [ 62] such as\nthe one stated below—and must not necessarily be realized by a magnetic coil or antenna.\nFor a classical description of the spin system we start from the Gilbert equation [14, 57]\n˙M=\u0000\rM\u0002H+\u000b\nMSM\u0002˙M (1)\nto describe the motion of a magnetic moment M=\u0000\rS, with Sthe spin,\rthe gyromagnetic ratio, and MS=jMj\nthe saturation magnetization [cf. Fig. 1(b)]. The magnetic moment Mis damped by a strength proportional to the\n2HextM(t)\nfree layer(a)\nˆex\nˆeyˆezM(t)\nHdHan\nHextHeff(b)Figure 1: (a) Schematic of a magnetic single-domain layer with variable magnetization M(free layer) in an external magnetic field Hext. (b) Magnetic\nfield components governing the evolution of the free layer’s magnetization M. The e \u000bective field Hacting upon Mconsists of the demagnetization\nfield Hd=\u0000MSmxˆex, the magnetocrystalline anisotropy field Han=HKmzˆez, and the external driving Hext=Hext\nzsin(!t)ˆez. The material’s easy\naxis is aligned with the z-axis.\ncoe\u000ecient\u000band can be driven by the e \u000bective magnetic field H. Because the velocity ˙Mis orthogonal to M, the length\nof the magnetic moment is conserved and therefore we can write M=MSmwithjmj=1 and mbeing dimensionless.\nThe implicit di \u000berential equation (1)can be brought to an explicit form. Substituting ˙Mon the right-hand side\nof Eq. (1)with the equation itself and using the relation M\u0002(M\u0002˙M)=(M\u0001˙M)M\u0000M2˙M=\u0000M2\nS˙Mas well as\nM=MSmwe obtain the Landau–Lifshitz–Gilbert (LLG) equation [14, 16]\n˙m=\u0000\r\n1+\u000b2m\u0002[H+\u000b(m\u0002H)]. (2)\nHere, we investigate the motion of a magnetic moment in a free-layer model described by the potential [61]\nU=M2\nS\n2m2\nx\u0000MSHK\n2m2\nz\u0000MSHext\nzsin(!t)mz, (3)\nwhere HKis the anisotropy constant of the free layer. A magnetization switching induced by an additional torque\nmodifying the dynamics of Eq. (2), can in principle be achieved by various ways [ 20–29]. For the description of spin\ntorque in a pinned-layer system, Slonczewski introduced an additional term to the standard Gilbert equation, depending\non the polarization of the pinned layer [ 19]. In this model, the spin torque is proportional to the applied electron\ncurrent Iflowing trough the pinned layer and, thus, can in principle become oscillating if an AC-source is used [ 63,64].\nWhile this specific type of spin torque cannot be represented purely by an additional magnetic field term, others—e. g.,\nManchon and Zhang [ 62]—have suggested spin torques that can. Due to the fact that the influence of some spin torques\ncan be reformulated as an additional e \u000bective field acting on the spin dynamics [ 62,65], we directly add our applied\nfield expression into the e \u000bective field, leading to significant simplifications [ 16]. The last term in Eq. (3)describes\nsuch an oscillating external magnetic field in zdirection with amplitude Hext\nzand frequency !. The e \u000bective magnetic\nfield then reads\nH=\u00001\nMSrmU=0BBBBBBBB@\u0000MSmx\n0\nHKmz+Hext\nzsin(!t)1CCCCCCCCA. (4)\nFor the free-layer system with parameters based on Refs. [ 22,27,29] the saturation magnetization and the\ngyromagnetic ratio read\nMS=1\u0002106A m\u00001and\r=2:217\u0002105m A\u00001s\u00001, (5)\nrespectively. Using these values as units, we can set MS=1and\r=1for computations with dimensionless parameters.\nIn the following, we choose\nMS=1 ,\r=1 ,\u000b=0:01 , HK=0:5 , (6)\n3m\nx−1\n0\n1my−101mz\n−101(a)\n−π 0 +π\nϕ0π/2πθ(b)\n−0.20.00.20.4\nUFigure 2: The free-layer potential (3)(a) on the sphere and (b) in the ( ';\u0012) plane. The saddle points at \u0012=\u0019=2 and'=\u0006\u0019=2 mark the regions of the\nTS, which must be crossed for the magnetization switching. A typical trajectory with higher friction \u000b=0:1, propagated without external driving\nfrom a spin-down state to a spin-up state, is shown in cyan (or light gray in print) in both panels. Vertical markers highlight the part of the trajectory\nshown in (b).\nand an external magnetic field with amplitude and frequency\nHext\nz=0:15 and!=\u0019=8 (7)\nas reference parameters, if not stated otherwise. This corresponds to HK=5\u0002105A m\u00001,Hext\nz=1:5\u0002105A m\u00001,\nand!=2\u0019=13:86 GHz in the problem defined in Ref. [ 66] with the standard material parameters of permalloy. This\napplied field and frequency are well in the range of typical experimental conditions.\nTo take advantage of the symmetry of the system one can transform the LLG equation (2)in spherical angular\ncoordinates \u0012and', i. e.,\n˙\u0012=\r\n1+\u000b2\u0010\nH'+\u000bH\u0012\u0011\n, ˙'=\r\n1+\u000b21\nsin\u0012\u0010\n\u0000H\u0012+\u000bH'\u0011\n, (8)\nfor\u0012<f0;\u0019gand with the projections of the e \u000bective field to the basis vectors, H\u0012=H\u0001ˆe\u0012andH'=H\u0001ˆe'. The\npotential (3)on the sphere for the spin system without external driving is illustrated in Fig. 2(a). The reduced equations\nof motion (8)for the two spherical angular variables describe a two-dimensional problem with the potential U(\u0012;')\npresented in Fig. 2(b). Two energy minima are located at the poles with \u0012=0 and\u0012=\u0019, which make the potential (3)\na perfect candidate to observe magnetization switching. At the equator \u0012=\u0019=2 there are two local energy maxima for\n'=0 and'=\u0019and two rank-1 saddles for '=\u0000\u0019=2 and'= +\u0019=2.\nA typical trajectory without external driving illustrating the magnetization switching is drawn in both panels (a)\nand (b) of Fig. 2. It starts on the spin down side of the sphere ( \u0012 > \u0019= 2), crosses the saddle region of the potential\nnear\u0012=\u0019=2,'=\u0000\u0019=2 and approaches the spin up position (\u0012\u00190) on a spiral caused by the damping term in the\nGilbert equation (1). We are interested in spin-flip processes crossing the regions close to one of the rank-1 saddles,\nand investigate in the following, without loss of generality, spin flips crossing the rank-1 saddle near '= +\u0019=2.\n2.2. Transition state theory\nThe free-layer system, described by the potential (3), features a rank-1 saddle point at \u0012='=\u0019=2, as shown in\nFig. 2(b). This saddle can act as a bottleneck of the spin dynamics, which makes it a candidate for the application of\nTST models [ 30,31,33,36]. In typical scenarios for a chemical reaction, a one-dimensional reaction path—e. g., the\nminimum energy path [ 67]—characterizes the progress of the reaction. A rank-1 saddle point separates reactants from\nproducts along this unstable mode, and can be used to naively characterize the flux and associated reaction rate. In this\ncontext, it acts as a TS. In higher dimensions, the other degrees of freedom are stable and are referred to as orthogonal\nmodes. More generally, the TSmarks the transition between reactants and products through the location of a DS. Here,\nwe apply TST to a magnetization switching in the free-layer system—e. g., from the “reactant” state spin up to the\n“product” state spin down —caused by a time-dependent driving of the system via an external magnetic field. To achieve\nthis aim, we resort to recent extensions of TST to time-dependent driven systems [52–55].\n4π/4π/2 3π/4\nϕπ/4π/23π/4θ(a)\nNHIMsaddleI\nIIIII\nIVWuWs\nWs\nWuDS\n−1 0 1\n[ϕ−ϕ‡(t)]/10−4−202[θ−θ‡(t)]/10−4\n(b)\nDS\nWuWs∆ϕu ∆ϕs\n∆θu∆θs\nt0t0+ ∆tFigure 3: (a) Phase-space structure of the driven free-layer system introduced in Sec. 2.1. The stable and unstable manifolds, WsandWuseparate\nfour di \u000berent regions marked ( I)–(IV) (see text). The intersection of the manifold’s closures forms the NHIM . The DSattached to this point\nseparates the spin up andspin down regions in phase space. The external driving causes the NHIM to detach from the saddle point. (b) Schematic of\nthe geometric structure that underlies the rate constant expressions summarized in Sec. 2.2.2. Initially, an equidistant spin ensemble connecting Ws\non the reactant side with the DSparallel toWuis generated. Upon time propagation, parts of the ensemble undergo spin flips as they move through\nthe DS. The resulting ensemble is still equidistant, parallel to Wu, and connected to Ws.\n2.2.1. Phase-space structure and TS trajectory\nIn the free-layer system introduced above, the magnetization switching is related to a change in the \u0012coordinate—\ne. g., from\u0012&0 to\u0012.\u0019in an uptodown spin state. In applying TST to resolve the activated dynamics of a spin, it\nthus appears natural to take the angle \u0012as the reaction coordinate and 'as an orthogonal mode. However, an important\ndi\u000berence between the spin system described by the equations of motion in (8), and systems typically addressed by\nTST requires some considerations, discussed below, to make the analogy complete.\nIn a chemical or mechanical system with ddegrees of freedom the dynamics is typically described by dsecond-order\ndi\u000berential equations for the coordinates or, in the Hamilton formalism, by 2 dfirst-order di \u000berential equations for the\ncoordinates and canonical momenta in the 2 d-dimensional phase space. In the spin system, the LLG equation results\nin the first-order di\u000berential equations (8)for the two coordinates \u0012and', i. e., there are no canonical momenta p\u0012\nandp', which belong to these coordinates. Nevertheless, TST can be applied to this system. The crucial point is that\nthe two-dimensional phase space of the spin system consisting of the two coordinates \u0012and'is treated in formal\nmathematical analogy to the two-dimensional phase space of a one DoF Hamiltonian system with a reaction coordinate\nand the corresponding canonical momentum.\nThe phase-space structure of the driven spin system in the vicinity of the rank-1 saddle at a given time tis illustrated\nin Fig. 3(a). Note that the reaction coordinate \u0012is the ordinate and 'the abscissa, which di \u000bers from corresponding\npresentations in Refs. [ 52–55,68], where the reaction coordinate is chosen as the abscissa and the corresponding\nvelocity along the ordinate. The stable and unstable manifolds WsandWuseparate four di \u000berent regions, where\n(I) the spin stays down , (II) the spin stays up, (III) the spin switches from uptodown , and ( IV) the spin switches\nfrom down toup, when the system is propagated backwards and forwards in time. One subtlety regarding the time\npropagation of the spins should be noted: Due to the damping of the magnetic field by the term proportional to \u000bin\nEq.(1)the spin without external driving always moves towards a potential minimum, i. e., the spin up orspin down\nposition when propagated forwards in time. However, it moves towards one of the potential maxima located at \u0012=\u0019=2,\n'=0 or\u0012=\u0019=2,'=\u0019(see Fig. 2) when propagated backwards. Therefore, appropriate cuto \u000bs for the propagation of\ntrajectories must be introduced to obtain the correct classification to one of the regions ( I)–(IV) in Fig. 3(a). Failing to\ndo so can lead to visible artifacts, or it can cause the classification algorithm to not terminate. Similar problems in\ndissipative chemical systems have been discussed in Ref. [ 69]. In our case, we have found 0:1\u0019<'< 0:9\u0019to yield\nreliable results.\nThe intersection of the stable and unstable manifold is a point ( 'z;\u0012z) on the NHIM . Such points do not leave\nthe saddle region when propagated forwards or backwards in time. Therefore, these points describe spins that reside\npermanently in an unstable intermediate state roughly in xdirection that is neither spin up norspin down . Note that\nfor driven systems the points of the NHIM in general do not coincide with the time-dependent position of the saddle\n5marked by the black point in Fig. 3(a). The line with constant angle \u0012=\u0012zrepresents a recrossing-free DS, which\nseparates the “reactants” and “products” in TST, i. e., a spin with \u0012<\u0012zisspin up and a spin with \u0012>\u0012zisspin down .\nIn case of periodic driving of the spin system by a time-dependent external magnetic field, the points on the NHIM\nfollow a periodic orbit with the same period as the external driving. This orbit is called the TStrajectory, and is of\nfundamental importance for the computation of rate constants.\nFor the numerical construction of the NHIM , we resort to the binary contraction method ( BCM ) introduced in\nRef. [ 68]. For a given time t, the algorithm in the BCM is initialized by defining a quadrangle with each of its corners\nlying exclusively within one of the four regions in the ( ';\u0012) plane shown in Fig. 3(a). In each iterative step, we first\ndetermine an edge’s midpoint. Then, the adjacent corner corresponding to the same region as that midpoint is moved\nto the midpoint’s position. By repeating this interleaved bisection procedure in turn for all edges, the quadrangle\nsuccessively contracts and converges towards the intersection of the stable and unstable manifolds, i. e., a point on the\nNHIM. This method is numerically very e \u000bective and e \u000ecient for systems such as the one addressed here.\n2.2.2. Decay rates\nThree di \u000berent methods for calculating decay rates in driven systems have recently been introduced and applied in\nthe literature [ 54,55]. Here, we adopt these methods with appropriate modifications for the free-layer system. The\nresulting decay rates are a measure of the instability of specific trajectories near the saddle. They di \u000ber significantly\nfrom the Kramers rate [ 70] used in the theory of chemical reactions but nevertheless provide insight about the rate\nprocess.\nEnsemble method. The conceptually simplest method for calculating decay rates keis by means of propagation of\nan ensemble. In analogy to Ref. [ 54] we identify a line segment parallel to the unstable manifold that satisfies the\nproperty: it lies on the reactant side between the stable manifold and the DSat a distance that is small enough to allow\nfor linear response and large enough to suppress numerical instability. At t=t0, a spin ensemble is placed on this line\nas illustrated by blue dots in Fig. 3(b) and propagated in time to yield a time-dependent spin up population N\"(t). The\nensemble at time t=t0+ \u0001tis marked in Fig. 3(b) by red and orange dots. Spins, which have crossed the DS(red dots)\narespin down and thus cause a decrease of the population N\"(t) (see the orange dots) with increasing time. In principle,\none can now obtain a reaction rate constant keby fitting an exponential decay N\"(t)/exp[\u0000ke(t\u0000t0)]to the spin up\npopulation. This, however, is not possible in all systems because the decay in N\"(t) can be nonexponential. Instead, we\nuse the more general approach described in Ref. [54], which involves examining the instantaneous decays\nke(t)=\u0000˙N\"(t)\nN\"(t). (9)\nLocal manifold analysis. The ensemble method is computationally expensive because it requires the propagation\nof a large number of spins for su \u000eciently long time. An alternative method, called the LMA , can be used to obtain\ninstantaneous spin-flip rates purely from the geometry of the stable and unstable manifolds in phase space. The LMA\nis based on the observation that the equations of motion (8)can be linearized in the local vicinity of a trajectory mz(t)\non the NHIM with the Jacobian\nJ(t)=@(˙\u0012;˙')\n@(\u0012;')\f\f\f\f\f\fmz(t). (10)\nWith the (not necessarily normalized) directions of the stable and unstable manifolds WsandWuat time tgiven\nby (\u0001's;\u0001\u0012s) and ( \u0001'u;\u0001\u0012u) with \u0001\u0012s= \u0001\u0012u, as marked in Fig. 3(b), and using the linearization of the equations of\nmotion (8)with the Jacobian (10) for the propagation of the spin ensemble, we finally obtain an analytical expression\nfor the instantaneous rates of the magnetization switching\nkm(t)=\u0000˙N\"(t)\nN\"(t)=\u0000lim\n\u0001t!0N\"(t+ \u0001t)\u0000N\"(t)\n\u0001tN\"(t)=@˙\u0012\n@'\f\f\f\f\f\fmz(t) \u0001'u\n\u0001\u0012u\u0000\u0001's\n\u0001\u0012s!\n. (11)\nThese rates can be calculated independently at di \u000berent times t, which allows for computations in parallel. Note that\n\u0001's=\u0001\u0012sand\u0001'u=\u0001\u0012uare the inverse slopes of the stable and unstable manifolds WsandWuin Fig. 3(b), and thus\nthe instantaneous rate km(t) in Eq. (11) is mainly determined by the di \u000berence of these two inverse slopes. This di \u000bers\n61.4 1.6\nϕ1.251.501.752.00θ(a)\n0.00 0.25 0.50 0.75 1.00\nt/T1.341.361.381.401.42k(b) Hint\nz= 0.00 (static)\nHint\nz= 0.15,ω=π/8\nHint\nz= 0.30,ω=π/8\nHint\nz= 0.05,ω=π/8\nHint\nz= 0.15,ω=π/4\nHint\nz= 0.15,ω=π/16Figure 4: (a) A selection of TStrajectories of the free-layer system with the potential (3)described by the LLG equation (2). The static TStrajectory\nwithout external magnetic field ( Hext\nz=0) is marked by a black dot at \u0012='=\u0019=2. The TStrajectory with the reference parameters given in Eqs. (6)\nand(7)is shown as solid black line. TStrajectories with driving parameters deviating from Eq. (7)are drawn with colored dash or dash-dotted lines.\nThe elliptical shape and orientation of the TStrajectories depends strongly on the driving by the oscillating external magnetic field. (b) Instantaneous\nrates (dark lines) and mean rates (pale lines) for some of the periodic TS trajectories shown in (a).\nfrom, e. g., Ref. [ 54], where the instantaneous rate is related to the slopes of the stable and unstable manifolds; the\ninverse slopes in Eq. (11) occur because the reaction coordinate \u0012is not the abscissa but the ordinate in Figs. 3(a) and\n3(b). As discussed above, the angles \u0012and'are not canonical variables as is typical in applications of TST to systems\nwith Hamiltonian dynamics [ 45,54,55]. This manifests in a nontrivial and time-dependent prefactor (@˙\u0012.\n@')\f\f\fmz(t)\nin Eq. (11), which is an element of the Jacobian (10). In the limiting case of a Cartesian reaction coordinate xwith\ncanonical momentum p=m˙x(where mis the particle mass), the corresponding element of the Jacobian reduces to a\nconstant@˙x/@p=1=m[54].\nFloquet method. The average decay rates kFacross time-dependent barriers can also be obtained directly using a\nFloquet stability analysis [ 51,54]. While this method is computationally much cheaper than the ensemble method and\nthe LMA, it cannot yield instantaneous rates.\nTo obtain the time-independent rate constant kFfor a given TStrajectory on the NHIM , we linearize the equations\nof motion using the Jacobian (10). By integrating the di \u000berential equation\n˙\u001b(t)=J(t)\u001b(t) with \u001b(0)=1, (12)\nwe then obtain the system’s fundamental matrix \u001b(t). When considering trajectories with period T,M=\u001b(T) is called\nthe monodromy matrix. Its eigenvalues muandms, termed Floquet multipliers, can be used to determine the Floquet\nrate constant\nkF=1\nT(lnjmuj\u0000lnjmsj). (13)\nAs shown below, the Floquet rate constant kFagrees perfectly with the instantaneous rates ke(t) and km(t) when the\nlatter two are averaged over one period Tof the TS trajectory.\n3. Results and discussion\nWe now present and discuss the TStrajectories and the related instantaneous and averaged decay rates obtained for\nthe free-layer system with and without driving by an oscillating external magnetic field. TStrajectories in the ( ';\u0012)\nphase space at various amplitudes Hext\nzand frequencies !of the driving field relative to the reference system of Eq. (7)\nare shown in Fig. 4(a). The static TStrajectory without external driving is marked by a black dot at \u0012='=\u0019=2,\nwhich coincides with the position of the static saddle in Fig. 2. When driven by an oscillating external field, the TS\ntrajectories become periodic orbits with the same period as the driving. The elliptical shape and orientation of the orbits\nstrongly depend on the amplitude and frequency of the driving. The black solid line marks the TStrajectory with the\n7π/4 π/2\nω0.00.20.4Hint\nz\n1.201.251.301.351.40\n¯kFigure 5: Mean rates ¯kas function of the frequency !and amplitude Hext\nzof the external magnetic field. The diamonds mark the parameters used in\nFig. 4(b).\nreference parameters given in Eqs. (6)and(7). The dashed lines mark TStrajectories, where either the amplitude Hext\nz\n(blue lines) or the frequency !(red lines) deviates from these reference parameters. For the chosen sets of parameters\ninvestigated here, the driving frequency mostly a \u000bects the shape of the orbits, whereas the driving amplitude has a large\ninfluence on the orbit size while preserving the shape approximately.\nRate constants, which are related to the TStrajectories in Fig. 4(a), have been computed with the methods introduced\nin Sec. 2.2.2 and are shown in Fig. 4(b). The (dark lines mark the instantaneous rates obtained by the LMA as functions\noft=Twhere T=2\u0019=! is the period of the corresponding TStrajectory. As can be seen, the oscillation amplitude of\nthe instantaneous rates at high amplitude Hext\nz=0:3 of the driving field (dark blue line) is slightly higher than that of the\nsystem with Hext\nz=0:15 ( black line). This trend continues for Hext\nz=0:05, where the oscillation is almost unnoticeable.\nThe pale lines present the averaged rate constants. Here, the increase of the Hext\nzfrom 0:15(light gray line) to 0:3(pale\nblue line) causes a significant decrease in the averaged rate constant. The dark and pale red lines in Fig. 4(b) mark the\ninstantaneous and averaged decay rate of the system at lower frequency !=\u0019=16 and higher frequency !=\u0019=4 of the\noscillating magnetic field. The instantaneous rate fluctuates much stronger around the mean value. The alternation in\nthe strength of these fluctuations is strong evidence for a sign change in the modulation amplitude around the reference\nfrequency. As mentioned above, the rate constants obtained as time averages of the instantaneous rates over one period\nof the TS trajectory agree perfectly with the rate constants computed using the Floquet method.\nFinally, the dependence of the averaged decay rate ¯kon the amplitude Hext\nzand frequency !of the magnetic field is\nreported in Fig. 5. The diamonds mark the parameters of the TStrajectories shown in Fig. 4(b). A minimum in the\nrates lies near the corner with low frequencies !and high amplitudes Hext\nz.\n4. Conclusion and outlook\nWe have investigated magnetization switching in a ferromagnetic free-layer system. The dynamics of the magnetic\nmoment is described by the Gilbert equation (1). We have shown that TST can be applied to its two-dimensional phase\nspace even though the Gilbert equation does not have the expected structure of a Hamiltonian system with coordinates\nand canonical momenta. We obtained the periodic TStrajectories of the free-layer system driven by an additional\noscillating external magnetic field. In turn, these form the basis for the calculation of the instantaneous and averaged\ndecay rates. The rates significantly depend on the time-dependent driving, i. e., the amplitude and frequency of the\nexternal magnetic field. The magnetization switching can thus be controlled by the external driving.\nIn this paper, we have assumed that the time derivative ˙mof the magnetic moment follows the magnetic field\nwithout relaxation, as described by the Gilbert equation (1). In future work, the model for the free-layer system could\nbe extended by taking into account relaxation of the spins [ 71], which requires one to enlarge the phase space from two\nto four dimensions. TST will then allow us to study the influence of the relaxation on the decay rates.\nPerhaps surprisingly, an increase in the field Hext\nzmostly leads to a decrease in the mean rate in Fig. 5. This\nis perhaps a consequence of the intermediate friction regime wherein the population of activated spins—i. e., those\n8that would go over the barrier—are dampened by the dissipation. Moreover, as the driving frequency increases, the\nmoving trajectory explores a wider oscillation potentially averaging—and suppressing—the di \u000berence in the curvatures\nassociated with the stable and unstable directions that contributes to the rate. Resolution of this phenomenon remains a\nchallenge for future work.\nIn summary, this work suggests that the application of recent advances in locally nonrecrossing TST to magnetization\nswitching could be helpful in future work addressing dynamics in spintronics.\nDeclaration of competing interest\nThe authors declare that they have no known competing financial interests or personal relationships that could have\nappeared to influence the work reported in this paper.\nCRediT authorship contribution statement\nJohannes M ¨ogerle: Methodology, Software, Formal analysis, Investigation, Writing – Original Draft. Robin\nSchuldt: Methodology, Formal analysis, Investigation, Writing – Original Draft. Johannes Rei \u000b:Methodology,\nSoftware, Validation, Resources, Data Curation, Writing – Review & Editing, Visualization. J¨org Main: Conceptual-\nization, Methodology, Formal analysis, Resources, Writing – Original Draft, Writing – Review & Editing, Supervision,\nProject administration, Funding acquisition. Rigoberto Hernandez: Conceptualization, Writing – Review & Editing,\nProject administration, Funding acquisition.\nAcknowledgments\nFruitful discussions with Robin Bardakcioglu, Matthias Feldmaier, and Andrej Junginger are gratefully acknowl-\nedged. The German portion of this collaborative work was supported by Deutsche Forschungsgemeinschaft (DFG)\nthrough Grant No. MA1639 /14-1. RH’s contribution to this work was supported by the National Science Foundation\n(NSF) through Grant No. CHE-1700749. This collaboration has also benefited from support by the European Union’s\nHorizon 2020 Research and Innovation Program under the Marie Skłodowska-Curie Grant Agreement No. 734557.\nReferences\n[1]C. M. Schneider, B. Zhao, R. Kozhuharova, S. Groudeva-Zotova, T. M ¨uhl, M. Ritschel, I. M ¨onch, H. 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Lattery1†, Delin Zhang2†, Jie Zhu1, Jian-Ping Wang2*, and Xiao jia Wang1* \n1Department of Mechanical  Engineering, University of Minnesota, Minneapolis, MN 55455, USA  \n2Department of Electrical and Computer Engineering , University of Minnesota, Minneapolis, MN \n55455, USA   \n†These authors contributed equally to this work.  \n*Corres ponding a uthor s: wang4940@umn.edu  & jpwang@umn.edu   \n \nAbstract:  Perpendicular magnetic materials with low damping constant and high thermal stability \nhave great potential for realizing high -density, non -volat ile, and low -power consumption \nspintronic devices, which can  sustain operation reliability for high processing temperatures. In this \nwork, we study the Gilbert damping constant ( α) of perpendicularly magnetized W/CoFeB/MgO \nfilms with a high perpendicular magnetic anisotropy (PMA) and superb thermal stability. The α of \nthese PMA films annealed at different temperatures ( Tann) is determined via an all -optical T ime-\nResolved Magneto -Optical Kerr Effect method. We find that α of these W/CoFeB/MgO PMA \nfilms decreases with increasing Tann, reaches a minimum of α = 0.016 at Tann = 350 °C, and then \nincreases to 0.024 after post -annealing at 400 °C. The minimum α observed at 350 °C is \nrationalized by two competing effects  as Tann becomes higher : the enhanced crystallization of \nCoFeB and dead -layer growth occurring at the two interfaces of the CoFeB layer. We further \ndemonstrate that α of the 400 °C -annealed W/CoFeB/MgO film is comparable to that of a reference \nTa/CoFeB/MgO PMA fil m annealed at 300 °C , justif ying the enhanced thermal stability of the W -\nseeded CoFeB films.  2 \n I. INTRODUCTION  \nSince the first demonstration of perpendicular magnetic tunnel junctions with \nperpendicular magnetic anisotropy (PMA) Ta/CoFeB/MgO stacks  [1], interfacial PMA materials \nhave been extensively studied as promising candidates for ultra -high-density and low -power \nconsumption spintronic devices, including spin -transfer -torque magnetic rando m access memory \n(STT -MRAM)  [2,3] , electrical -field induced magnetization switching  [4-6], and spin -orbit torque \n(SOT) devices  [7-9]. An interfacial PMA stack typically consists of a thin ferromagnetic layer \n(e.g., CoFeB) sandwiched between a heavy metal layer ( e.g., Ta) and an oxide layer ( e.g., MgO). \nThe heavy metal layer interface  with the ferromagnetic layer  is responsible for the spin Hall effect , \nwhich is  favorable  for SOT and skyrmion devices  [10,11] . The critical switching current ( Jc0) \nshould be minimized to decrease the power consumption  of perpendicular STT -MRAM and SOT \ndevices . Reducing  Jc0 requires  the exploration of  new materials with low Gilbert damping  constant \n(α), large spin Hall angle ( θSHE), and large effective anisotropy ( Keff) [12,13] .  \nIn addition, spintronic devices need to sustain operation reliability for processing \ntemperatures as high as 400 °C  for their integration with existing CMOS fabrication technologies,  \nproviding the standard  back -end-of-line process compatibility  [14]. Based on this requirement, the \nmagnetic properties of a PMA material shou ld be thermally stable at annealing temperature s (Tann) \nup to  400 °C. Unfortunately, Ta/CoFeB/MgO  PMA films commonly used in spintronic devices \ncannot survive with Tann higher than 350 °C, due to Ta diffusion or CoFeB oxidation at the \ninterfaces  [15,16] . The diffusion of Ta atoms can act as scattering  sites to increase the spin -flip \nprobability  [17] and lead to a higher Gilbert Damping constant ( α), a measure of the energy \ndissipation from the magnetic precession into phonons or magnons  [18].  3 \n Modifying the composition of thin-film stack s can prevent heavy metal diffusion , which  \nis beneficial to both lowering  α and improving thermal stability  [19]. Along this line , new  \ninterfacial PMA stacks have been developed, such as  Mo/CoFeB/MgO, to circumvent the \nlimitation on device processing temperatures  [20,21] . While Mo/CoFeB/MgO films can indeed \nexhibit PMA at temperatures higher than 400 °C , they cannot be used for SOT devices due to the \nweak spin Hall effect of the Mo layer  [20,21] . Recently, W/CoFeB/MgO PMA thin films have \nbeen proposed because of their PMA property at high post -annealing temperature  [22], and  the \nlarge spin Hall angle of the W laye r (θSHE  ≈ 0.30)  [23], which is twice that of a Ta layer \n(θSHE  ≈ 0.12 ~ 0.15)  [9]. While  there have been a few scattered studies demonstrating the promise \nof fabricating SOT devices using the W/CoFeB/MgO stacks, special attention  has been given to \ntheir PMA properties and functionalities as SOT devic es [24]. A systematic investigation is lacking \non the effect of Tann on α of W/CoFeB/MgO PMA thin films.  \n \nII. SAMPLE PREPERATION AND MAGNETIC CHARACTERIZATION  \nIn this work, we grow a series of W(7)/Co 20Fe60B20(1.2)/MgO(2)/Ta(3) thin films on \nSi/SiO 2(300) substrates (thickness in nanometers) with a magnetron sputtering system \n(<5×1 08 Torr). These films are then post -annealed at varying temperatures (Tann = 250 ~ 400  °C) \nwithin a high -vacuum furnace (<1×106 Torr)  and their magnetic properties and damping constants \nas a function of Tann are systematically investigated . For comparison, a reference sample of \nTa(7)/Co 20Fe60B20(1.2)/MgO(2)/Ta(3) is also prepared to examine the effect of seeding layer to \nthe damping constant of these PMA films. The saturation magnetization ( Ms) and anisotropy  of \nthese films are measured with the Vibrating Sample Magnetometer (VSM)  module of a Physical 4 \n Property Measurement System . Figure  1 plots the magnetic hysteresis  loops and  associated  \nmagnetic properties extracted from VSM measurements.  \n \n \nFigure 1. Room temperature magnetic hysteresis loops of W/CoFeB/MgO PMA thin films post -\nannealed at ( a) 250 °C, (b) 300 °C, (c) 350 °C, and ( d) 400 °C. Black and red curves  denote external \nmagnetic field ( Hext) applied along and perpendicular to the film plane, res pectively . (e-g) Plots of \nthe saturation magnetization ( Ms), effective interfacial anisotropy ( Keff × t), and interfacial \nanisotropy ( Ki) as functions of Tann. \n \nWith the increase of Tann, Ms for the W/ CoFeB /MgO films decreases from ~780  to \n~630  emu/cm3 [Fig. 1 (e)]. The effective interfacial anisotropy  [(Keff × t) depicted in Fig. 1 (f)] \nshows a n increasing trend  with Tann (from ~0.18  to ~0.34 erg/cm2 when Tann increases from 250 to \n350 °C) and saturates at Tann = 350 °C. The positive values of Keff × t suggest that these \nW/CoFeB /MgO films maintain high PMA properties at elevated temperatures including 400  °C, \ndemonstrating their enhanced thermal stability compared to Ta/ CoFeB /MgO films that can only \nsustain PMA up to 350 ° C. Removing the influence  of the demagnetization energy from Keff × t 5 \n results in the  interfacial anisotropy ( Ki), which changes from 0.6 to 0.7 erg/cm2 with the increase \nof Tann up to 350  °C and then decreases to ~0.6 erg/cm2 at Tann = 400  °C [Fig. 1(f)]. Details about \nthe determination of Ki and Keff are provided in Section S1 of the Supp lemental  Material  (SM).  \n \n \nFigure 2 . The dead -layer extraction results. ( a), (b), (c), and ( d) represent the series of samples \nannealed at 250, 300, 350, and 400 C respectively. The tdead value is the extrapolated x-axis \nintercept from the linear fitting of the thickness -dependent saturation magnetization area product \n(Ms×t). \n \nWe attribute the decrease of Ki at high Tann to the growth of a dead layer at the CoFeB  \ninterfaces, which becomes prominent at higher Tann. To quantitatively determine the thickness of \nthe dead  layer as Tann increases, we prepare four sets of PMA stacks of \nW(7)/CoFeB( t)/MgO(2)/Ta( 3). One set contains five stacks with varying thicknesses for the \nCoFeB layer ( t = 1.2, 1.5, 1.8, 2.2, and 2.5 nm) and is post -annealed at a fixed Tann. Four Tann of 6 \n 250, 300, 350, and 400 °C  are used for four sets of the PMA stacks, respectively. The anne aling \nconditions are the same as those for the W(7)/CoFeB(1.2)/MgO(2)/Ta(3) samples for discussed \npreviously. We measure the magnetic hysteresis loops  of these samples using VSM and plot their \nsaturation magnetization area product (\nsMt ) as a function of film thickness ( t) in Fig. 2. Linear \nextrapolati on of the \nsMt  data provides the dead -layer thickness, at which the magnetization \nreduces to zero  as illustrate d by the x-axis intercept in Fig. 2.  \n \nIII. TR-MOKE MEASUREM ENTS  \nThe magnetization dynamics of these PMA thin films are determined using the all-optical  \nTime-Resolved Magneto -Optical Kerr Effect (TR -MOKE) method  [25-29]. This  pump -probe  \nmethod utilizes ultra -short laser pulses to thermally demagnetize the sample  and probe  the \nresulting Kerr rotation angle ( θK). In the polar -MOKE configuration,  θK is proportional to the \nchange of the out-of-plane  component of magnetization [Mz in Fig. 3(a)] [30]. Details  of the TR -\nMOKE setup are provided  in Section S2 of the SM.  \nThe TR-MOKE signal is fitted to the equation \n//\nK sin 2t C tA Be D ft e      , \nwhere A, B, and C are the offset, amplitude, and exponential decaying constant of the thermal \nbackground , respectively . D denotes the amplitude  of oscillations , f is the resonance  frequency , φ \nis a phase shift (related to the demagnetization process), and  is the relaxation time of \nmagnetization precession.  Directly from TR -MOKE measurements, an effective damping constant \n(αeff) can be extracted based on the relationship αeff = 1/(2πf). However, αeff is not an intrinsic \nmaterial property; rather, it depends on measurement conditions, such as the applied field direction \n[θH in Fig. 3(a)], the magnitude of the applied field ( Hext), and inhomogeneities of the sample ( e.g. \nlocal variation in the magnetic properties of the sample)  [31,32] .  7 \n  \n \nFigure  3. (a) Definition of the parameters and angles used in TR -MOKE experiments. The red \ncircle indicates the magnetization precession. θ is the equilibrium direction of the magnetization . \nθK is measured by the probe beam at a given time delay (Δ t). (b) The TR -MOKE data (open \nsymbols) and model fitting of  θK (black curves) for the 400 °C sample at 76° , for varying Hext from \n2.0 to 20 kOe.  \n \n \nTo obtain the Gilbert damping constant, the inhomogene ous contribution needs to be \nremoved from αeff, such that the remaining value of damping is a n intrinsic  material property  and \nindependent of the measurement conditions. To determine the inhomogeneous broadening in the 8 \n sample, the effective anisotropy field (\nk,eff eff s 2/ H K M ) needs to be pre -determined from either \n(1) the magnetic hysteresis loops; or (2) the fitting results of  f vs. Hext obtained from TR -MOKE.  \nThe resonance frequency, f, can be related to Hext through the Smit -Suhl approach  by identifying \nthe second derivatives of the total magnetic free energy, which combines a Zeeman energy, an \nanisotropy energy, and a demagnetization energy [33-35]. For a perpen dicularly magnetized thin \nfilm,  f is defined by Eqs. ( 1-4) [35]. \n12 f H H\n\n,      (1) \n 2\n1 ext H k,eff cos cos H H H      \n,     (2) \n  2 ext H k,eff cos cos 2 H H H      \n,     (3) \n  ext H k,eff2 sin sin 2HH  \n.             (4) \nThis set of equations permits calculat ion of  f with the material gyromagnetic ratio ( ), Hext, \nθH, Hk,eff, and the angle between the equilibrium magnetization di rection and the surface normal \n[θ, determined by Eq. ( 4)]. The measured values of f as a function of Hext can be fitted to Eq. (1) \nby treating and Hk,eff as fitting parameters. To minimize the fitting errors resulting from the \ninhomogeneous broadening effect that is pronounced at the low fields, we use measured \nfrequencies  at high fields ( Hext > 10 kOe) to determine Hk,eff. \nWith a known value of Hk,eff , the Gilbert damping constant of the sample can be determined \nthrough a fitting of the inverse relaxation time (1/ ) to Eq. (5). The two terms of Eq. (5) take into \naccount , respectively,  contributions from the intrinsic Gilbert damping of the materials (first term) \nand inhomogeneous broadening (second term)  [31]:  \n 1 2 k,eff\nk,eff1 1 1\n22dH H HdH   \n,     (5) 9 \n where H1 and H2 are related to the curvature of the magnetic free energy surface as defined by \nEqs. (2) and (3) [35,36] . The second term on the right side of Eq. (5) capture s the inhomogeneous \neffect by attributing it to a spatial variation in the magnetic properties (Δ Hk,eff), analogous  to the \nlinewidth broadening effect in F erromagnetic Resonance  measurements  [37]. The magnitude of \nk,eff/d dH\n can be calculated once the relationship of ω vs. Hext is determined with a numerical \nmethod. Both α and Δ Hk,eff (the inhomogeneous term related to the amount of spatial variation in \nHk,eff) are determined via the fitting of the measured 1/  based on Eq. ( 5). In this way, we can \nuniquely extract the field -independent  α, as an intrinsic material property, from the ef fective \ndamping ( αeff), which is directly obtained from TR -MOKE and dependent on Hext. \nIt should be noted here that the inhomogeneous broadening of the magnetization precession \nis presumably due to the multi -domain structure of the materials, which becomes  negligible in the \nhigh-field regime ( Hext >> Hk,eff) as the magnetization direction  of multiple magnetic domains  \nbecomes uniform. This is also reflected by the fact that the derivative in the second term of Eq.  (5) \napproaches zero for the high -field regim e [38]. \n \nIV. RESULTS AND DISCUSSION  \nThe measurement method is validated  by measuring  the Tann = 400 C at multiple angles \n(θH) of the external magnetic field direction.  By repeating this meas urement at varying  θH, we can \nshow that α is an intrinsic material property , independent of θH. Figure 4(a) plots the resonance \nfrequenc ies derived from TR -MOKE and model fittings for the 400 °C sample at two field \ndirections ( θH = 76° and 89° ).  For the data acquired at θH = 89°, a minimum f occurs at Hext ≈ Hk,eff. \nThis corresponds to the smallest amplitude of magnetization precession, when the equilibrium \ndirection of the magnetization is aligned with the applied field direction at the magnitud e of Hk,eff 10 \n [35]. The dip at this local minimum  diminishes when θH decreases, as reflected by the comparison \nbetween the red ( θH = 89°) and blue ( θH = 76°) lines in Fig. 4(a). With the Hk,eff extracted from the \nfitting of frequency data with θH = 89°, we generate the plot of theoretically predicted f vs. Hext \n[θH = 76° theory, blue line in Fig. 4(a)], which agrees well with experimental data [open square s \nin Fig. 4(a)].  \n \n \nFigure 4. (a) Measured f vs. Hext results for the 400 C sample at θH = 89° (open circles) and \nθH = 76° (open squares) and corresponding modeling at θH = 89° (red line) and θH = 76° (blue line) . \n(b) The measured inverse of relaxation time  (1/) at θH = 89° (open symbols) and the fitting of 1/  \nbased on Eq. (5) (dotted line). For reference, the first term of 1/  in Eq. (5) is also plotted (solid \nline), which accounts for the contribution from the Gilbert damping  only. (c) αeff as a function of \nHext for θH = 89° (red circles). Black circles are the extracted Gilb ert damping, which is \nindependent of Hext. The black dotted line shows the average of this extracted damping; ( d) and ( e) \ndepict similar plots of 1/  and damping constants for θH = 76°. Error bars in ( b) through ( e) come \nfrom the uncertainty in the mathematical fitting.  11 \n  \nThe inverse relaxation time (1/ should also have a minimum  value near Hk,eff for θH = 89° \nif the damping was purely from  Gilbert damping [as shown by the solid lines in Figs.  4(b) and \n4(d)]; however, the measured data do  not follow this trend. Adding the inhomogeneous term \n[dotted lines in Figs. 4(b) and 4(d)] more accurately describes the field  dependen ce of the measured \n1/[open symbols  in Figs. 4(b) and 4(d)] It should be noted that the dip of the predicted 1/ occurs \nwhen the frequency derivative term in Eq.  (5) approaches  zero; however,  this is not captured by \nthe measurement due to the finite interval over which we vary  Hext. Figures 4(c) and 4(e) depict  \nthe field -dependent effective damping ( αeff) and the Gilbert damping ( α) as the intrinsic material’s \nproperty obtained from fitting the measured 1/.  \nWith the knowledge that the value of α extracted with this method is the intrinsic material \nproperty, we repeat this data reduction technique for the annealed W/CoFeB /MgO samples \ndiscussed  in Fig. 1. The symbols in Fig. 5 represent  the resonance frequency and damping \nconstants (both effective damping and Gilbert damping) for all samples measured at θH ≈ 90°. The \nfittings for the resonance frequency [red lines , from Eq.  (1)] are also shown to demonstrate  the \ngood agreement between our TR -MOKE measurement and theoretical prediction. The \nuncertainties of f, , and Hk,eff are calculated from the least-square s fitting uncertainty and the \nuncertainty of measuring Hext with the Hall sensor.  12 \n \n \nFigure 5 . Results for f (a-d) and αeff (e-h, on a log scale) for individual samples. For comparison, \nthe Gilbert damping constant α is also plotted by subtracting the inhomogeneous terms from αeff. \nThe dashed line in (e -h) indicat es the average α. All samples are measured at θH = 90° except for \nthe 400 C sample ( θH = 89°).  \n \nThe summary of the anisotropy and damping measured via TR -MOKE is shown in Fig. 6. \nFigure 6(a) plots Hk,eff obtained from VSM (black open circle s) and TR -MOKE  (blue open \nsquares) , both of which exhibit a monotonic increasing trend as Tann becomes higher.  \nDiscrepancies in Hk,eff from these two methods can be attributed to the difference in the size of the \nprobing region , which is highly localized in TR -MOKE but sample -averaged in VSM. Since Hk,eff \ndetermined  from  TR-MOKE is obtained from fitting the measured frequency for a localized region, \nwe expect these values  more consistently  describ e the magnetization precession th an those \nobtained from VSM. The increase in Hk,eff  with Tann can be partially attributed to the crystallization 13 \n of the CoFeB layer  [32]. For temperatures higher than 350 °C, this increasing trend  of Hk,eff begins \nto lessen, presumably  due to the diffusion of W atoms into the CoFeB layer , which is more \npronounced at higher Tann. The W diffusion process  is also responsible for the decrease in Ms of \nthe CoFeB layer as Tann increases  [Fig. 1 (e)]. Subsequently, the  decrease in Ms leads to a further -\nreduced demagnetizing energy  and thus a  larger Hk,eff.  \nSimilar observation of Ms has been reported in literature for Ta/CoFeB/MgO PMA \nstructures and attribute d to the growth of a dead layer  at the heavy metal/CoFeB inter face [1]. \nFigure 6(b) summarizes tdead as a function of  Tann with tdead increas ing from 0.17 to 0.53 nm as Tann \nchanges fr om 250 to 400  C, as discussed in Section II.  \n \n \nFigure 6. Summary of the magnetic properties of W -seeded CoFeB as a function of Tann. (a) The \ndependence of Hk,eff on Tann obtained from both the VSM (black open circles) and TR-MOKE \nfitting (blue  open squares ). (b) The dependence of dead -layer thickness on Tann. (c) Damping \nconstants as a function of Tann. The minim um damping constant of α = 0.016 occurs at 350°C. The \nvalues for the all samples are obtained from measurements at θH = ~90°. For comp arison, α of the  \nreference Ta/CoFeB/MgO PMA sample annealed at 300 °C is also shown as a red triangle in ( c). 14 \n  \nFigure  6(c) depicts the dependence of α on Tann, which first decreases with Tann, reaches a \nminimum of 0.016 at 350 °C, and then increases as Tann rises to 400 °C. Similar trends have been \nobserved for Ta/CoFeB/MgO previously (minimum α at Tann = 300 °C) [32]. We speculate that \nthis dependence of damping on Tann is due to two competing effects: (1) the inc rease in \ncrystallization in the CoFeB layer with Tann which reduces the damping, and (2) the growth of a \ndead layer, which results from the diffusion of W and B atoms and is prominent at higher Tann. At \nTann = 400 °C, the dead -layer formation leads to a la rger damping presumably due to an increase \nin scattering sites (diffused atoms) that contribute to spin -flip events, as described by the Elliot -\nYafet relaxation mechanisms [17]. The observation that our W -seeded samples still sustain \nexcellent PMA properties at Tann = 400 °C confirms  their enhanced thermal stability, compared \nwith Ta/CoFeB/MgO stacks which fail at Tann = 350 °C or higher.  \nThe damping constants are comparable for the W/CoFeB/MgO and Ta/CoFeB/MgO films \nannealed at 300 °C, both of which are higher than that of the W/ CoFeB /MgO PMA with the \noptimal Tann of 350 °C.  Nevertheless,  our work focuses on the enhanced thermal stability of W -\nseeded CoFeB PMA films while still maintaining a relatively low damping constant. Such an \nadvantage enables W -seeded CoFeB layers to be viable and promising alternatives to \nTa/CoFeB/MgO , which is currently widely used in spintronic devices.  \n \nV. CONCLUSION  \nIn summary, we deposit a series of W -seeded CoFeB  PMA films with varying annealing \ntemperatures up to 400  °C and conduct ultrafast all -optical TR -MOKE measurements to study \ntheir magnetization precession dynamics. The Gilbert damping, as a n intrinsic  material property, \nis proven to be independent of meas urement conditions, such as the amplitudes and directions of 15 \n the applied field. The damping constant varies with Tann, first decreasing and then increasing, \nleading to a minimum of α = 0.016 for the sample anneale d at 350  °C. Due to the  dead -layer \ngrowth , the damping constant slightly increases to α = 0.024 at Tann = 400  °C, comparable to the \nreference Ta/ CoFeB /MgO PMA film annealed at 300°C, which demonstrates the improved \nenhanced thermal stability of W/ CoFeB /MgO over the Ta/ CoFeB /MgO structures. This strongly \nsuggests the great potential of W/ CoFeB /MgO PMA material systems for future spintronic device \nintegration that requires materials to sustain a processing temperature as high as 400 °C.  \n  16 \n Acknowledgements  \nThis work is supported by C -SPIN  (award #: 2013 -MA-2381) , one of six centers of STARnet, a \nSemiconductor Research Corporation program, sponsored by MARCO and DARPA. The authors \nwould like to thank Prof.  Paul Crowell and Dr. Changjiang Liu for valuable discussions.  \n \nSupplemental  Materials  Available:  Complete description of the TR -MOKE mea surement \nmethod, the angular dependent result summary , and the  description for determining the  interface \nanisotropy.  \n \n  17 \n References  \n[1] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo, S. Kanai, J. \nHayakawa, F. Matsukura, and H. Ohno, A perpendicular -anisotropy CoFeB -MgO magnetic tunnel \njunction, Nat. Mater. 9, 721 (2010).  \n[2] H. Sato, E. C. I. Enobio, M. 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B 92, 224402, 224402 (2015).  \n "    },    {        "title": "2106.08528v2.Spin_Torque_driven_Terahertz_Auto_Oscillations_in_Non_Collinear_Coplanar_Antiferromagnets.pdf",        "content": "Spin-Torque-driven Terahertz Auto Oscillations in Non-Collinear Coplanar\nAntiferromagnets\nAnkit Shukla\u0003and Shaloo Rakhejay\nHolonyak Micro and Nanotechnology Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL 61801\n(Dated: January 19, 2022)\nWe theoretically and numerically study the terahertz auto oscillations, or self oscillations, in thin-\n\flm metallic non-collinear coplanar antiferromagnets (AFMs), such as Mn 3Sn and Mn 3Ir, under the\ne\u000bect of antidamping spin torque with spin polarization perpendicular to the plane of the \flm. To\nobtain the order parameter dynamics in these AFMs, we solve three Landau-Lifshitz-Gilbert equa-\ntions coupled by exchange interactions assuming both single- and multi-domain (micromagnetics)\ndynamical processes. In the limit of a strong exchange interaction, the oscillatory dynamics of the\norder parameter in these AFMs, which have opposite chiralities, could be mapped to that of two\ndamped-driven pendulums with signi\fcant di\u000berences in the magnitude of the threshold currents\nand the range of frequency of operation. The theoretical framework allows us to identify the in-\nput current requirements as a function of the material and geometry parameters for exciting an\noscillatory response. We also obtain a closed-form approximate solution of the oscillation frequency\nfor large input currents in case of both Mn 3Ir and Mn 3Sn. Our analytical predictions of threshold\ncurrent and oscillation frequency agree well with the numerical results and thus can be used as com-\npact models to design and optimize the auto oscillator. Employing a circuit model, based on the\nprinciple of tunnel anisotropy magnetoresistance, we present detailed models of the output power\nand e\u000eciency versus oscillation frequency of the auto oscillator. Finally, we explore the spiking\ndynamics of two unidirectional as well as bidirectional coupled AFM oscillators using non-linear\ndamped-driven pendulum equations. Our results could be a starting point for building experimen-\ntal setups to demonstrate auto oscillations in metallic AFMs, which have potential applications in\nterahertz sensing, imaging, and neuromorphic computing based on oscillatory or spiking neurons.\nI. INTRODUCTION\nTerahertz (THz) radiation, spanning from 100 Giga-\nhertz (GHz) to 10 THz, are non-ionizing, have short\nwavelength, o\u000ber large bandwidth, scatter less, and are\nabsorbed or re\rected di\u000berently by di\u000berent materials.\nAs a result, THz electronics can be employed for safe\nbiomedical applications, sensing, imaging, security, qual-\nity monitoring, spectroscopy, as well as for high-speed\nand energy-e\u000ecient non-von Neumann computing (e.g.,\nneuromorphic computing). THz electronics also has po-\ntential applications in beyond-5G communication sys-\ntems and Internet of Things. Particularly, the size of\nthe antennae for transmitting the electromagnetic signal\ncould be signi\fcantly miniaturized in THz communica-\ntion networks [1{8]. These aforementioned advantages\nand applications have led to an intense research and de-\nvelopment in the \feld of THz technology with an aim\nto generate, manipulate, transmit, and detect THz sig-\nnals [3, 9]. Therefore, the development of e\u000ecient and\nlow power signal sources and sensitive detectors that op-\nerate in the THz regime is an important goal [3].\nMost coherent THz signal sources can be categorized\ninto three types | particle accelerator based sources,\nsolid state electronics based sources, and photonics based\nsources [3, 9]. Particle accelerator based signal genera-\ntors include free electron lasers [10], synchrotrons [11],\n\u0003ankits4@illinois.edu\nyrakheja@illinois.eduand gyrotrons [12]. While particle accelerator sources\nhave the highest power output, they require a large\nand complex set-up [13]. Solid state generators include\ndiodes [14{16], transistors [17, 18], frequency multipli-\ners [19], and Josephson junctions [20], whereas photonics\nbased signal sources include quantum cascade lasers [21],\ngas lasers [22], and semiconductor lasers [23]. Solid-state\ngenerators are e\u000ecient at microwave frequencies whereas\ntheir output power and e\u000eciency drop signi\fcantly above\n100 GHz [13]. THz lasers, on the other hand, provide\nhigher output power for frequencies above 30 THz [24],\nhowever, their performance for lower THz frequencies\nis plagued by noise and poor e\u000eciency [13]. Here, we\npresent the physics, operation, and performance bench-\nmarks of a new type of nanoscale THz generator based\non the ultra-fast dynamics of the order parameter of an-\ntiferromagnets (AFMs) when driven by spin torque.\nSpin-transfer torque (STT) [25, 26] and spin-orbit\ntorque (SOT) [27] enable electrical manipulation of ferro-\nmagnetic order in emerging low-power spintronic radio-\nfrequency nano-oscillators [28]. When a spin current\ngreater than a certain threshold (typically around 108\u0000\n109A=cm2[28, 29]) is injected into a ferromagnet (FM)\nat equilibrium, the resulting torque due to this current\npumps in energy which competes against the intrinsic\nGilbert damping of the material. When the spin torque\nbalances the Gilbert damping, the FM magnetization un-\ndergoes a constant-energy steady-state oscillation around\nthe spin polarization of the injected spin current. Such\noscillators are nonlinear, current tunable with frequencies\nin the range of hundreds of MHz to a few GHz with out-\nput power in the range of nano-Watt (nW). They are alsoarXiv:2106.08528v2  [cond-mat.mes-hall]  15 Jan 20222\ncompatible with the CMOS technology [28]; however, the\ngeneration of the THz signal using FMs would require\nprohibitively large amount of current, which would lead\nto Joule heating and degrade the reliability of the elec-\ntronics. It would also lead to electromigration and hence\nirreversible damage to the device set-up [30].\nAFM materials, which are typically used to exchange\nbias [31] an adjacent FM layer in spin-valves or magnetic\ntunnel junctions for FM memories and oscillators, have\nresonant frequencies in the THz regime [32{35] due to\ntheir strong exchange interactions. It was suggested that\nSTT could, in principle, be used to manipulate the mag-\nnetic order in conducting AFMs [36], leading to either\nstable precessions for their use as high-frequency oscilla-\ntors [37, 38] or switching of the AFM order [39] for their\nuse as magnetic memories in spin-valve structures. The\nSOT-based spin Hall e\u000bect (SHE), on the other hand,\ncould enable the use of both conducting [40{44] and in-\nsulating [29, 43, 45, 46] AFMs in a bilayer comprising an\nAFM and a non-magnetic (NM) layer, for its use as a\nhigh frequency auto-oscillator [47].\nTable I lists the salient results from some of the re-\ncently proposed AFM oscillators. These results, how-\never, are reported mainly for collinear AFMs, while de-\ntailed analyses of the dynamics of the order parameter in\nthe case of non-collinear AFMs is lacking. In this paper,\n\frstly, we \fll this existing knowledge gap in the model-\ning of auto oscillations in thin-\flm non-collinear copla-\nnar AFMs like Mn 3Ir;Mn3Sn, or Mn 3GaN under the ac-\ntion of a dc spin current. Secondly, we compare their\nperformance (generation and detection) against that of\ncollinear AFMs such as NiO for use as a THz signal\nsource. In the case of NiO, inverse spin Hall e\u000bect (iSHE)\nis employed for signal detection, whereas in this work we\nutilize the large magnetoresistance of metallic AFMs. Fi-\nnally, we investigate these auto oscillators as possible can-\ndidates for neuron emulators. Considering that the spin\npolarization is perpendicular to the plane of the AFM\nthin-\flm, three possible device geometries are identi\fed\nand presented in Fig. 1 for the generation and detection\nof auto oscillations in metallic AFMs.\nFigure 1(a) is based on the phenomena of spin injec-\ntion and accumulation in a local lateral spin valve struc-\nture. Charge current, I write, injected into the structure\nis spin-polarized along the magnetization of FM 1(FM 2)\nand gets accumulated in the NM. It then tunnels into the\nAFM with the required perpendicular spin-polarization\n(adapted from Ref. [50]). The bottom MgO layer used\nhere would reduce the leakage of charge current into\nthe metallic AFMs considered in this work. This would\nreduce chances of Joule heating in the AFM thin-\flm\nlayer. On the other hand, in Fig. 1(b), spin \flter-\ning [51] technique is adopted wherein, a conducting AFM\nis sandwiched between two conducting FMs. Di\u000ber-\nent scattering rates of the up-spins and down-spins of\nthe injected electron ensemble at the two FM interfaces\nresults in a perpendicularly polarized spin current as\nshown. The structure in Fig. 1(c) generates spin polar-\nFM1\nIwrite\nFM2NMJs\nnpAFMMgOPtLoadIdeal Bias Tee\nFM1FM2\nIwriteAFM\nMgOIdeal Bias Tee\nLoad\nJsnpIread\nFM1\nNMLoadIdeal Bias Tee(a)\n(b) (c)xyzIread\nIwriteIread\nPt\nMgO\nJsnpAFMMgOLbtCbt\nCbt Lbt LbtCbtFIG. 1. Device geometries to inject perpendicularly polar-\nized spin current in thin-\flm metallic AFMs. In all the cases,\nIwrite is the charge current injected to generate spin current,\nwhereas, I readis the charge current injected to extract the\noscillations as a transduced voltage signal using the princi-\nples of tunnel anisotropy magnetoresistance (TAMR). (a) Lat-\neral spin valve structure leads to spin accumulation in NM\nfollowed by injection into the AFM. (b) Perpendicular spin\nvalve structure spin \flters the injected charge current. (c)\nFM/NM/AFM trilayer structure generates spin current due\nto interfacial spin-orbit torque.\nization perpendicular to the interface due to the interfa-\ncial SOTs generated at the FM/NM interface (adapted\nfrom Ref. [52, 53]). In this case, the spin current injected\ninto the AFM has polarization along both yandzdirec-\ntion; however, the interface properties could be tailored\nto suppress the spin polarization along y[52]. In order\nto extract the THz oscillations of the order parameter as\na measurable voltage signal, the tunnel anisotropy mag-\nnetoresistance (TAMR) measurements are utilized [54].\nIn this work, we establish the micromagnetic model\nfor non-collinear coplanar AFMs with three sublattices\nalong with the boundary conditions in terms of both the\nsublattice magnetizations (Section II A), as well as the\nN\u0013 eel order parameter (Section II B). We show that in the\nmacrospin limit the oscillation dynamics correspond to\nthat of a damped-driven pendulum (Section III A and\nSection III B). The oscillation dynamics of AFM mate-\nrials with two di\u000berent chiralities in then compared in\nSection III C. We use the TAMR detection scheme to\nextract the oscillations as a voltage signal and present\nmodels of the output power and e\u000eciency as a function\nof the oscillator's frequency (Section IV). This is followed\nby a brief investigation of the e\u000bect of inhomogeneity due\nto the exchange interaction on the dynamics of the AFM\norder (Section V). Finally, we discuss the implication of\nour work towards building coherent THz sources in Sec-3\nTABLE I. Recent numerical studies on electrically controlled AFM THz oscillators. The investigated AFM materials, the\ndirection of their uniaxial anisotropy axis ueand that of the spin polarization of the injected spin current npare listed. Salient\nresults along with the schemes to extract the oscillation as a voltage signal are also brie\ry stated. Ref. [44] does not provide\nthe name of a speci\fc AFM, however, an AFM with uniaxial anisotropy is considered.\nRef. AFM material ue np Salient Features Detection Schemes\n[45] NiO x x a) THz oscillations for current above a threshold iSHE\nb) Feedback in AFM/Pt bilayer sustains oscillation\n[29] NiO x -z a) Hysteretic THz oscillation in a biaxial AFM iSHE\nb) Threshold current dependence on uniaxial anisotropy\n[40]\u000b\u0000Fe2O3 a)z y a) Monodomain analysis of current driven oscillations in -\nAFM insulators with DMI\nb)y y Similar to [45] -\n[41]\u000b\u0000Fe2O3 x y a) Canted net magnetization due to DMI Dipolar radiation\nb) Small uniaxial anisotropy leads to low power THz frequency\n[42] CuMnAs ;Mn2Auy z a) Low dc current THz signal generation due to N\u0013 eel SOT -\nb) Phase locked detector for external THz signal\n[43] NiO x Varied a) Comparison of analytical solutions to micromagnetic results AMR/SMR\nb) E\u000bect of DMI on hysteretic nature of dynamics\n[48] Mn 2RuxGa z y a) Generation of spin current in single AFM layer AMR\nb) Oscillation dependence on reactive and dissipative torques\n[44] Uniaxial Ani. z Varied a) Non-monotonic threshold current variation with np -\nb) E\u000bects of anisotropy and exchange imperfections\n[49] Mn 3Sn x-y plane z a) E\u000bective pendulum model based on multipole theory AHE\n[46] NiO;Cr2O3 x Varied a) General e\u000bective equation of a damped-driven pendulum iSHE\nb) Analytic expression of threshold current and frequency\nOur Mn 3Sn;Mn3Ir x-y plane z a) Di\u000berent numerical and analytic models TAMR\nWork b) Inclusion of generation current for TAMR e\u000eciency\nc) Non-linear dynamics of bidirectional coupled oscillators\ntion VI, and towards hardware neuron emulators for neu-\nromorphic computing architecture in Section VII. Some\nof the salient results from this work are listed in Table I.\nII. THEORY\nA. Magnetization Dynamics\nWe consider a micromagnetic formalism in the con-\ntinuum domain [43, 55] under which a planar non-\ncollinear AFM is considered to be composed of three\nequivalent interpenetrating sublattices, each with a con-\nstant saturation magnetization Ms[56]. Each sublattice,\ni(= 1, 2 or 3), is represented as a vector \feld mi(r;t)\nsuch that for an arbitrary r=r0,kmi(r0;t)k= 1. The\ndynamics of the AFM under the in\ruence of magnetic\n\felds, damping, and spin torque is assumed to be gov-\nerned by three Landau-Lifshitz-Gilbert (LLG) equations\ncoupled by exchange interactions. For sublattice i, the\nLLG is given as [57]\n@mi\n@t=\u0000\r\u00160\u0000\nmi\u0002He\u000b\ni\u0001\n+\u000bi\u0012\nmi\u0002@mi\n@t\u0013\n\u0000!smi\u0002(mi\u0002np)\u0000\f!s(mi\u0002np);(1)\nwheretis time in seconds, He\u000b\niis the position dependent\ne\u000bective magnetic \feld on i,\u000biis the Gilbert dampingparameter for i, and\n!s=~\n2e\rJs\nMsda(2)\nis the frequency associated with the input spin current\ndensity,Js, with spin polarization along np. Here,dais\nthe thickness of the AFM layer, ~is the reduced Planck's\nconstant,\u00160is the permeability of free space, eis the el-\nementary charge, and \r= 17:6\u00021010T\u00001s\u00001is the gyro-\nmagnetic ratio. For all sublattices, the spin polarization,\nnp, is assumed to be along the zaxis. Finally, \fis a mea-\nsure of the strength of the \feld-like torque as compared\nto the antidamping-like torque. The e\u000bect of \feld-like\ntorque on the sublattice vectors here is the same as that\nof an externally applied magnetic \feld|canting towards\nthe spin polarization direction. Results presented in the\nmain part of this work do not include the e\u000bect of the\n\feld-like torque; however, a small discussion on the same\nis presented in the supplementary material [58].\nThe e\u000bective magnetic \feld, He\u000b\niat each sublattice,\nincludes contributions from internal \felds as well as ex-\nternally applied magnetic \felds and is obtained as\nHe\u000b\ni(r;t) =\u00001\n\u00160Ms\u000eF\n\u000emi(r;t); (3)\nwhere\u000e\n\u000emi=@\n@mi\u0000r\u0001@\n@(rmi), andFis the energy\ndensity of the AFM, considered in our work. It is given\nas4\nF=X\nhi;ji\u0000\nJmi\u0001mj+Aijrmi\u0001rmj\u0001\n+Aii3X\ni=1(rmi)2+3X\ni=1Khm2\ni;z\u0000Ke(mi\u0001ue;i)2\n+DX\nhi;jiz\u0001(mi\u0002mj) +Dii3X\ni=1(mi;zr\u0001mi\n\u0000(mi\u0001r)mi;z) +DijX\nhi;ji((mi;zr\u0001mj\n\u0000(mi\u0001r)mj;z)\u0000(mj;zr\u0001mi\u0000(mj\u0001r)mi;z))\n\u00003X\ni=1\u00160MsHa\u0001mi;(4)\nwherehi;jirepresents the sublattice ordered pairs (1 ;2),\n(2;3) and (3;1).\nThe \frst three terms in Eq. (4) represent exchange en-\nergies. HereJ(>0) is the homogeneous inter-sublattice\nexchange energy density whereas Aii(>0) andAij(<\n0) are the isotropic inhomogeneous intra- and inter-\nsublattice exchange spring constants, respectively. The\nnext two terms in Eq. (4) represent magnetocrystalline\nanisotropy energy for biaxial symmetery upto the low-\nest order withKe(>0) andKh(>0) being the easy and\nhard axes anisotropy constants, respectively. We assume\nthat the easy axes of sublattices 1, 2 and 3 are along\nue;1=\u0000(1=2)x+ (p\n3=2)y,ue;2=\u0000(1=2)x\u0000(p\n3=2)y\nandue;3=x, respectively, and an equivalent out of plane\nhard axis exists along the zaxis. The next three terms\nrepresent the structural symmetry breaking interfacial\nDzyaloshinskii-Moriya Interaction (iDMI) energy density\nin the continuum domain. Its origin lies in the interaction\nof the antiferromagnetic spins with an adjacent heavy\nmetal with a large spin-orbit coupling [59, 60]. Here, we\nassume the AFM crystal to have Cnvsymmetry [61] such\nthat the thin-\flm AFM is isotropic in its plane, and D,\nDii, andDijrepresent the e\u000bective strength of homoge-\nneous and inhomogeneous iDMI, respectively, along the\nzdirection. Finally, the last term in Eq. (4) represents\nthe Zeeman energy due to an externally applied mag-\nnetic \feld Ha. Now, using Eq. (4) in Eq. (3) we get the\ne\u000bective \feld for sublattice ias\nHe\u000b\ni=X\nj\nj6=i\u0012\n\u0000J\n\u00160Msmj+Aij\n\u00160Msr2mj\u0013\n+2Aii\n\u00160Msr2mi\n\u00002Kh\n\u00160Msmi;zz+2Ke\n\u00160Ms(mi\u0001ue;i)ue;i\n+Dz\u0002(mj\u0000mk)\n\u00160Ms\u00002Dii\n\u00160Ms((r\u0001mi)z\u0000rmi;z)\n\u0000Dij\n\u00160Ms((r\u0001(mj\u0000mk))z\u0000r(mj;z\u0000mk;z))\n+Ha;\n(5)where (i;j;k ) = (1;2;3);(2;3;1);or (3;1;2), respectively.\nIn order to explore the dynamics of the AFM, we adopt\na \fnite di\u000berence discretization scheme and discretize the\nthin-\flm of dimension L\u0002W\u0002dainto smaller cells of\nsizesL\u0002sW\u0002sd. Each of these cells is centered around\nposition rsuch that mi(r;t) denotes the average mag-\nnetization of the spins within that particular cell [43].\nFinally, we substitute Eq. (5) in Eq. (1) and use fourth-\norder Runge-Kutta rule along with the following bound-\nary conditions for sublattice iof the thin-\flm considered\n(see supplementary material [58]):\n2Aii@mi\n@\u0011\u0011\u0011+AijX\nj\nj6=imi\u0002\u0012@mj\n@\u0011\u0011\u0011\u0002mi\u0013\n+Diimi\u0002(\u0011\u0011\u0011\u0002z)\n+Dijmi\u0002(mi\u0002((\u0011\u0011\u0011\u0002z)\u0002(mk\u0000mj))) = 0;(6)\nwhere\u0011\u0011\u0011is the normal vector perpendicular to a surface\nparallel to xory. The above equation ensures that the\nnet torque due to the internal \felds on the boundary\nmagnetizations of each sublattice is zero in equilibrium\nas well as under current injection [62]. For the energy\ndensity presented in Eq. (4), the \felds at the boundary\nare non-zero only for inhomogeneous inter- and intra-\nsublattice exchange, and Dzyaloshinskii-Moriya interac-\ntions. Finally, in the absence of DMI, we have the Neu-\nmann boundary condition@mi\n@\u0011\u0011\u0011= 0, which implies that\nthe boundary magnetization does not change along the\nsurface normal \u0011\u0011\u0011.\nFor all the numerical results presented in this work\nwe solve the system of Eqs. (1), (5), and (6) with the\nequilibrium state as the starting point. The equilibrium\nsolution in each case was arrived at by solving these three\nequations for zero external \feld and zero current with a\nlarge Gilbert damping of 0 :5.\nB. N\u0013 eel Order Dynamics\nThe aforementioned micromagnetic modeling ap-\nproach assuming three sublattices is extremely useful in\nexploring the physics of the considered AFM systems. It\nis, however, highly desirable to study an e\u000bective dynam-\nics of the AFMs under the e\u000bect of internal and external\nstimuli in order to gain fundamental insight. Therefore,\nwe consider an average magnetization vector mand two\nstaggered order parameters n1andn2to represent an\nequivalent picture of the considered AFMs. These vec-\ntors are de\fned as [37, 56, 63]\nm\u00111\n3(m1+m2+m3); (7a)\nn1\u00111\n3p\n2(m1+m2\u00002m3); (7b)\nn2\u00111p\n6(\u0000m1+m2); (7c)5\nsuch thatkmk2+kn1k2+kn2k2= 1. The energy land-\nscape (Eq. (4)) can then be represented as\nF\n3=3J\n2m2+Am(rm)2+An\n2\u0010\n(rn1)2+ (rn2)2\u0011\n+Kh\u0000\nm2\nz+n2\n1;z+n2\n2;z\u0001\n\u0000Ke\n2\u00123\n2(n1;x\u0000n2;y)2\n+1\n2(n1;y+n2;x)2+mx\u0010\nmx\u0000p\n2(n1;x\u0000n2;y)\u0011\n+my\u0010\nmy+p\n2(n1;y+n2;x)\u0011\n+ 4n1;xn2;y\u0011\n+p\n3Dz\u0001(n1\u0002n2) +Dii(mzr\u0001m\u0000(m\u0001r)mz\n+n1;zr\u0001n1\u0000(n1\u0001r)n1;z+n2;zr\u0001n2\n\u0000(n2\u0001r)n2;z) +p\n3Dij(n1;zr\u0001n2\u0000(n1\u0001r)n2;z\n\u0000n2;zr\u0001n1+ (n2\u0001r)n1;z)\u0000\u00160MsHa\u0001m;\n(8)\nwhereAm=\u0000\nAii+Aij\u0001\nandAn=\u0000\n2Aii\u0000Aij\u0001\n.\nAn equation of motion involving the staggered order\nparameters can be obtained by substituting Eq. (1) in the\n\frst-order time derivatives of Eq. (7) and evaluating each\nterm carefully (see supplementary material [58]). How-\never, an analytical study of such an equation of motion\nthat consists of contributions from all the energy terms\nof Eqs. (4) or (8) would be as intractable as the dynam-\nics of individual sublattices itself. Therefore, we consider\nthe case of AFMs with strong inter-sublattice exchange\ninteraction such that J \u001djDj\u001dK e. This corresponds\nto systems with ground state con\fned to the easy-plane\n(x\u0000yplane) and those that host kmk\u001c1 (weak ferro-\nmagnetism), n1?n2, andkn1k\u0019kn2k\u00191=p\n2 [56, 63].\nHowever, when an input current is injected in the system,\nthe sublattice vectors cant towards the spin polarization\ndirection leading to an increase in the magnitude of m\nwhile decreasing that of n1andn2. Spin polarization\nalong the zdirection and an equal spin torque on each\nsublattice vector ensures that n1andn2have negligible\nzcomponents at all times (Eqs. (7b), (7c)). Therefore,\nwe consider\nn1(r;t) =\u00150\n@p\n1\u0000n2\n1zcos'(r;t)p\n1\u0000n2\n1zsin'(r;t)\nn1z(r;t)1\nA; (9a)\nn2(r;t) =\u00150\n@p\n1\u0000n2\n2zcos('(r;t)\u0006\u0019=2)p\n1\u0000n2\n2zsin('(r;t)\u0006\u0019=2)\nn2z(r;t)1\nA;(9b)\nwhere'is the azimuthal angle from the xaxis and\njn1zj;jn2zj\u001c1.\nThe two choices for n2correspond to two di\u000berent\nclasses of materials|one with a positive (+ \u0019=2) chiral-\nity and the other with a negative ( \u0000\u0019=2) chirality [56].\nMaterials that have a negative (positive) value of Dcor-\nrespond to + \u0019=2(\u0000\u0019=2) chirality because the respective\ncon\fguration reduces the overall energy of the system.\nL12phase of AFMs like Mn 3Ir;Mn3Rh, or Mn 3Pt is ex-\npected to host + \u0019=2 chirality whereas the hexagonalphase of AFMs like Mn 3Sn;Mn3Ge, or Mn 3Ga is ex-\npected to host\u0000\u0019=2 chirality [56, 64].\nIII. SINGLE DOMAIN ANALYSIS\nA. Positive Chirality\nDe\fning n3= (n1\u0002n2)=\u0015, and considering the case\nof +\u0019=2 chirality, it can be shown that mis just a de-\npendent variable of the N\u0013 eel order dynamics. To a \frst\norder, mcould be expressed as [56, 63]\nm=\u00001\n!E(n1\u0002_ n1+n2\u0002_ n2+n3\u0002_ n3\u0000\r\u00160Ha\n\u0000\f!snp);\n(10)\nwhere!E= 3\rJ=Ms. One can then arrive at the equa-\ntion of motion for the N\u0013 eel vectors as\nn1\u0002\u0002\n n1\u0000c2r2n1\u0000!E!Kn1+!E!Kh(n1\u0001z)z\n+!E!D(n2\u0002z) +!E!ii\nDn1+!E!ij\nDn2+\u000b!E_n1\n\u0000!E!s(n1\u0002np)] +n2\u0002\u0002\n n2\u0000c2r2n2\u0000!E!Kn2\n+!E!Kh(n2\u0001z)z\u0000!E!D(n1\u0002z) +!E!ii\nDn2\n\u0000!E!ij\nDn1+\u000b!E_n2\u0000!E!s(n2\u0002np)i\n+n3\u0002n3\n+\r\u00160(n1\u0002_n1+n2\u0002_n2+n3\u0002_n3)\u0002Ha\n\u0000\r\u00160_Ha= 0;\n(11)\nwherec=p\n!E\rAn=Ms,!Kh= 2\rKh=Ms,!K;n1=\n!K\n4\u0000\u0000\nn1;y+n2;x+p\n2my\u0001\n^ x+ (n1;x+ 3n2;y\n+p\n2mx\u0001\n^ y\u0001\n,!D=p\n3\rD\nMs,!K;n2=!K\n4((n1;y+n2;x\n+p\n2my\u0001\n^ x+\u0000\nn1;x+ 3n2;y+p\n2mx\u0001\n^ y\u0001\n,\n!ii\nD;ni=2\rDii\nMs((r\u0001ni)z\u0000rni;z),\nand!ij\nD;ni=p\n3\rDij\nMs((r\u0001ni)z\u0000rni;z).\nThe equations of motion (Eqs. (10) and (11)) derived\nhere are useful in the numerical study of textures like\ndomain walls, skyrmions, and spin-waves in AFMs with\nbiaxial anisotropy under the e\u000bect of external magnetic\n\feld and spin current. However, here we are interested\nin analytically studying oscillatory dynamics of the order\nparameter in thin-\flm AFMs, therefore, we neglect in-\nhomogeneous interactions compared to the homogeneous\n\felds. Using Eq. (9) in Eq. (11) and neglecting the time\nderivative of n1zandn2z, we have\n'+\u000b!E_'+!E!K\n2sin 2'+!E!s= 0; (12)\nwhere!K= 2\rKe=Ms. This indicates that in the limit\nof strong exchange interaction, the dynamics of the stag-\ngered order parameters is identical to that of a damped-\ndriven non-linear pendulum [65]. This equation is iden-\ntical to the case of collinear AFMs such as NiO when the6\n(c) (d)\n(a) (b)Negative Chirality Positive Chirality\nFIG. 2. Stationary solution for AFMs with positive (a, b) and negative chirality (c, d). (a) Sublattice magnetization for currents\nbelow the threshold current Jth1\ns. WhenJs= 0 (equilibrium state), the sublattice vectors micoincide with the easy axes ue;i,\nwhereas for a non-zero current smaller than Jth1\ns, the macrospins have stationary solutions other than the equilibrium solution,\nas depicted by dashed and dotted line. The zcomponent of these vectors is zero. (b) An equivalent representation of (a) through\nthe staggered order parameters n1andn2. They are perpendicular to each other and have zero out-of-plane component. The\nthinner dash-dotted gold arrows through the thicker arrows of n1represent the analytic expression of the stationary solution\n'=\u00001\n2sin\u00001\u0010\n2!s\n!K\u0011\n. The average magnetization mis vanishingly small, as can also be noticed from Eq. (10). (c) Sublattice\nmagnetization at equilibrium ( Js= 0). Thezcomponent of these vectors is zero. Here, only the sublattice vector m3coincides\nwith its corresponding easy axis. On the other hand, m1andm2are oriented such that the energy due to DMI is dominant\nover anisotropy. (d) An equivalent representation of (c) through the staggered order parameters n1andn2, and average\nmagnetization m.n1andn2are almost perpendicular to each other with a negative chirality, as assumed in Eq.(9). A small\nin-plane net magnetization (shown by the magni\fed green arrow) also exists in this case [56].\n(a) (b) (c) (d)\n(e) (g) (f) (h)Positive Chirality Negative Chirality\nα = 0.01α = 0.01 α = 0.01\nα = 0.01da = 4 nm da = 4 nm\nda = 4 nm da = 4 nm\nFIG. 3. Upper panel (a-d) shows the time averaged frequency as a function of input spin current, whereas the lower panel (e-h)\nshows the FFT of the oscillations corresponding to the cases marked by the dashed red boxes above. In the time averaged\nfrequency response in the upper panel, the dashed black lines denote the analytic expression of frequency (Eq. (17)). (a), (c)\nFrequency response for di\u000berent \flm thicknesses for \u000b= 0:01. (e), (g) FFT of the signal corresponding to Js=Jth2\ns. (b),\n(d) Frequency response for di\u000berent damping constants for da= 4 nm. (f), (h) FFT of the signal corresponding to \u000b= 0:1\nandJth1\ns, respectively. Positive chirality: The numerical values of the average frequency match very well against the analytic\nexpression for lower damping and large current. On the other hand, non-linearity and, hence, higher harmonics are observed\nfor small current and large damping. Negative chirality: The numerical values of the average frequency exactly match against\nthe analytic expression for all values of damping and input current considered here.\ndirection of spin polarization is perpendicular to the easy-\nplane [29, 44, 66]. However, the dynamics of the non-\ncollinear coplanar AFMs discussed here is signi\fcantlydi\u000berent in the direction of the spin torques, magnitude\nof threshold currents as well as the range of possible fre-\nquencies. Here, the sin 2 'dependence signi\fes a two-fold7\nanisotropy symmetric system.\nB. Negative Chirality\nFor the case of\u0000\u0019=2 chirality, it can be shown that m\nis a dependent variable of the N\u0013 eel order; however, in this\ncase there are additional in-plane terms that arise due to\na competition between the DMI, exchange coupling and\nmagnetocrystalline anisotropy. To a \frst order, mis ex-\npressed as [56, 67] (also see supplementary material [58])\nm=\u00001\n!E(n1\u0002_ n1+n2\u0002_ n2+n3\u0002_ n3\u0000\r\u00160Ha\n\u0000\f!snp)\u0000!K\n2!E(cos'x\u0000sin'y);\n(13)\nwhich is used to arrive at the equation of motion for the\nN\u0013 eel vectors as\nn1\u0002\u0002\n n1\u0000c2r2n1\u0000!E!Kn1+!E!Kh(^ n1\u0001z)z\n+!E!D(n2\u0002z) +!E!ii\nDn1+!E!ij\nDn2+\u000b!E_n1\n\u0000!E!s(n1\u0002np)] +n2\u0002\u0002\n n2\u0000c2r2n2\u0000!E!Kn2\n+!E!Kh(^ n2\u0001z)z\u0000!E!D(n1\u0002z) +!E!ii\nDn2\n\u0000!E!ij\nDn1+\u000b!E_n2\u0000!E!s(n2\u0002np)i\n+n3\u0002n3\n+\r\u00160(n1\u0002_n1+n2\u0002_n2+n3\u0002_n3)\u0002Ha\u0000\r\u00160_Ha\n\u0000!K\n2(sin'x+ cos'y) _'\u0000\r\u00160!K\n2(Ha;zsin'x\n+Ha;zcos'y\u0000(Ha;xsin'+Ha;ycos')z) = 0:\n(14)\nSimilar to the previous case, we are interested in a the-\noretical analysis of the oscillation dynamics in thin \flm\nAFMs with negative chirality. Therefore, we use Eq. (9)\nin Eq. (14) and neglect all the inhomegeneous interac-\ntions to arrive at a damped-driven linear pendulum equa-\ntion given as\n'+\u000b!E_'+!E!s= 0: (15)\nHere the dependence of the dynamics on anisotropy is\nnot zero but very small, and it scales proportional to\n!3\nK\n!2\nEcos 6'[67]. However, for a \frst-order approxima-\ntion in mand dynamics in the THz regime, it can be\nsafely ignored. The cos 6 'dependence implies that these\nmaterials host a six-fold anisotropic symmetry. Though\nthis equation is similar to that obtained for the case of\na collinear AFM with spin polarization along the easy\naxis [44], the dynamics is signi\fcantly di\u000berent from that\nof the collinear AFM.\nC. Comparison of Dynamics for Positive and\nNegative Chiralities\nHere, we contrast the dynamics of AFM order param-\neter for positive and negative chiralities. The numericalresults presented in this section are obtained in the single-\ndomain limit assuming thickness da= 4 nm,\u000b= 0:01,\nMs= 1:63 T,Ke= 3 MJ=m3,J= 2:4\u0002108J=m3,\nD=\u000020 MJ=m3for positive chirality or 20 MJ =m3for\nnegative chirality [56], unless speci\fed otherwise.\nFigure 2 shows the stationary solutions of the thin-\n\flm AFM system with di\u000berent chiralities. For the case\nof positive chirality, it can be observed from Fig. 2(a)\nthat in equilibrium the sublattice vectors micoincide\nwith the easy axes ue;i. When a non-zero spin current is\napplied, the equilibrium state is disturbed; however, be-\nlow a certain threshold, Jth1\ns, the system dynamics con-\nverge to a stationary solution in the easy-plane of the\nAFM, indicated by dashed blue, and dotted red set of\narrows. An equivalent representation of the stationary\nsolutions in terms of the staggered order parameters is\npresented in Fig. 2(b). n1andn2are perpendicular to\neach other with zero out-of-plane component for all val-\nues of the input currents. The gold dash-dotted arrows\npassing through n1correspond to the stationary solu-\ntions given as '=\u00001\n2sin\u00001\u0010\n2!s\n!K\u0011\n, obtained analytically\nby setting both _ 'and 'as zero in Eq. (12). In positive\nchirality material, the average magnetization mis van-\nishingly small in the stationary state. This can also be\nperceived from Eq. (10) as we do not consider any exter-\nnal \feld. Since these materials have a two-fold symmetry,\nthey also host '=\u0019\u00001\n2sin\u00001\u0010\n2!s\n!K\u0011\nstationary states.\nFor the case of negative chirality, it can be observed\nfrom Fig. 2(c) that in equilibrium only the sublattice\nvector m3coincides with the its corresponding easy axis,\nwhereas both m1andm2are oriented such that the en-\nergy due to DMI is dominant over anisotropy, which in\nturn lowers the overall energy of the system. It can be\nobserved from Fig. 2(d) that n1andn2are almost per-\npendicular to each other. A small in-plane net magneti-\nzation exists in this case and is shown here as a zoomed\nin value (zoom factor = 100x) for the sake of compari-\nson to staggered order parameters. Due to the six-fold\nanisotropy dependence other equilibrium states, wherein\neither of m1orm2coincide with their easy axis while the\nother two sublattice vectors do not, also exist. Finally,\ndue to the small anisotropy dependence, non-equilibrium\nstationary states exist for much lower currents [68] than\nthose considered here, and therefore are not shown.\nFor materials with positive chirality, the system be-\ncomes unstable when the input spin current exceeds the\nthreshold,Jth1\ns. The resultant spin torque pushes the\nN\u0013 eel vectors out of the easy-plane, and they oscillate\naround the spin polarization axis, np=z;with THz fre-\nquency due to strong exchange. This threshold current\nis given as [29, 44, 46]\nJth1\ns=da2e\n~Ms\n\r!K\n2=da2e\n~Ke; (16)\nwhile the frequency of oscillation in the limit of large\ninput current (neglecting the sin 2 'term) from Eq. (12)8\n(a) (b)\n(c) (d)Positive Chirality Negative Chirality\nFIG. 4. The out-of-plane (z) component of the average magnetization, m, and n3for di\u000berent values of input currents for both\npositive and negative chirality. Positive chirality: (a) When current is increased from zero but to a value below the threshold\n(0:95Jth1\ns),mzis zero. However, it increases to a large value when Js=Jth1\ns.mzdecreases again to a smaller value when\nthe current is decreased to Js= 0:86Jth1\ns. Finally, when the current is further reduced below the lower threshold to 0 :9Jth2\ns,\nmzbecomes zero again. (b) n3is initially equal to 1 =p\n2, but decreases in magnitude during the AFM dynamics since the\nmagnitude of mincreases when the sublattice vectors move out of the plane. As soon as the current is lowered below Jth2\ns,\nthe system goes to a stationary state and n3= 1=p\n2. Negative chirality: (c) Since the threshold current in this case is small,\nnon-zeromzis observed for all values of current considered here. (d) n3decreases in magnitude when current increases but\napproaches\u00001=p\n2 for lower values of current. Here \u000b= 0:01, andda= 4 nm.\nis given as[44, 46]\nf=1\n2\u0019!s\n\u000b=1\n2\u0019~\n2e\rJs\nMsda1\n\u000b: (17)\nAdditionally, for small damping, there exists a lower\nthreshold,Jth2\ns< Jth1\ns, which is equal to the current\nthat pumps in the same amount of energy that is lost in\none time period due to damping [29, 44, 46]. The lower\nthreshold current is given as\nJth2\ns=da2e\n~Ms\n\r2\u000b\n\u0019p!E!K=da2e\n~2\u000b\n\u0019p\n6JKe:(18)\nThe presence of two threshold currents enables energy-\ne\u000ecient operation of the THz oscillator in the hysteretic\nregion [29].\nThe average frequency response as a function of the\ninput spin current and the Fourier transform of the os-\ncillation dynamics is plotted in the left panel of Fig. 3\nfor materials with positive chirality. It can be observed\nfrom Fig. 3(a) that the fundamental frequency for dif-\nferent \flm thicknesses scales as predicted by Eq. (17)\nexcept for low currents near Jth2\nswhere non-linearity in\nthe form of higher harmonics appears as seen from the\nFFT response in Fig. 3(e). Next, Figs. 3(b), (f) show\nthat the non-linearity in the frequency response for low\ninput current increases as the value of the damping co-\ne\u000ecient increases. This is expected as the contribution\nfrom the uniaxial anisotropy (sin 2 'term) becomes sig-\nni\fcant owing to large damping and low current making\nthe motion non-uniform (  '6= 0) [29].\nIn the case of materials with negative chirality, small\nequivalent anisotropy suggests that the threshold current\nfor the onset of oscillations is very small, while the fre-\nquency of oscillations increases linearly with the spin cur-\nrent considered here and is given by Eq. (17). Indeed the\nsame can be observed from Figs. 3(c), (d) where the re-\nsults of numerical simulations exactly match the analyticexpression. The FFT signal in Figs. 3(g), (h) contains\nonly one frequency corresponding to uniform rotation of\nthe order ( '= 0). This coherent rotation of the order\nparameter with such small threshold current [68] in AFM\nmaterials with negative chirality opens up the possibil-\nity of operating such AFM oscillators at very low en-\nergy for frequencies ranging from MHz-THz. It can also\nbe observed from Fig. 3(b), (d) that for lower values of\ndamping, such as \u000b= 0:005, the frequency of oscillations\nsaturates for input current slightly above Jth1\ns. This is\nbecause the energy pumped into the system is larger than\nthat dissipated by damping. As a result the sublattice\nvectors move out of the easy-plane and get oriented along\nthe spin polarization direction (slip-\rop). For larger val-\nues of damping, the same would be observed for larger\nvalues of current. Finally, we would like to point out that\nthe values of both Jth1\nsandJth2\nsobserved from numerical\nsimulations were slightly di\u000berent from their analytical\nvalues for di\u000berent damping constants, similar to that\nreported in Ref. [43] for collinear AFMs.\nFigure 4 shows the out-of-plane (z) components of m\nandn3for non-zero currents for AFMs with di\u000berent chi-\nralities. For negative chirality materials, the steady-state\nzcomponent of both mandn3does not oscillate with\ntime ( '\u00190), whereas, for the case of positive chirality,\nthe steady-state zcomponents of both mandn3show\nsmall oscillations with time (  '6= 0) similar to the case\nof NiO with spin polarization along the hard axis [29].\nIt can be observed from Fig. 4(a) that for positive chi-\nrality, as current increases from below the upper thresh-\nold current (0.95 Jth1\ns) toJth1\ns, the out-of-plane compo-\nnent of magnetization vectors and hence the average mag-\nnetization mincreases from zero to a larger value. Due\nto the hysteretic nature of the AFM oscillator, the mag-\nnitude of mreduces when current is lowered but is non-\nzero as long as the input current is above Jth2\ns. Similarly,\nit can be observed from Fig. 4(b) that the out-of-plane9\ncomponent of n3, which was initially 1 =p\n2, decreases\nas the current increases above Jth1\ns. When the current\nis lowered to a value below Jth1\ns(0:86Jth1\nshere) , the\nmagnitude of the out-of-plane component of n3increases\nagain and eventually saturates to 1 =p\n2 when the current\nis lowered further below Jth2\ns(0:9Jth2\ns, here).\nIt can also be observed from Fig. 4(c) that for negative\nchirality AFMs, the out-of-plane component of the av-\nerage magnetization although small is non-zero even for\nsmall currents due to the lower value of threshold current.\nOn the other hand, nz\n3in Fig. 4(d) decreases in magnitude\nfrom an initial value of \u0015= 1=p\n2 to\u0015<1=p\n2 as current\nincreases sincekmkincreases. The values of current are\nassumed to be the same for both positive and negative\nchirality AFMs for the sake of comparison. Next, using\n(a)\n(b)\n(c)\n(d)\nFIG. 5.mzfor four di\u000berent values of input current Js.mz\nincreases with current and so does the frequency of oscillation.\nHere (a)Js= 1:6Jth2\ns, (b)Js= 0:9Jth1\ns. They are both inside\nthe hysteretic region bounded by Jth2\nsandJth1\ns. (c)Js=Jth1\ns,\nand (d)Js= 1:5Jth1\nslies outside the hysteretic region. These\nresults correspond to \u000b= 0:01, andda= 4 nm.\nEq. (9) in Eq. (10), it can be shown that _ 'is directly pro-\nportional to mz[56]. Therefore, to present the features\nof angular velocity with input current, we show mzfor\nfour di\u000berent values of input current Jsfor positive chi-\nrality material in Fig. 5 . Here, Figs. 5(a)-(c) correspond\nto the hysteretic region, whereas Fig. 5(d) is for current\noutside the hysteretic region. As mentioned previously,\nan increase in current increases the spin torque on the\nsublattice vectors which leads to an increase in mzand\nhence _'.\nIV. SIGNAL EXTRACTION\nAn important requirement for the realization of an\nAFM-based auto-oscillator is the extraction of the gen-\nerated THz oscillations as measurable electrical quanti-ties viz. voltage and current. It is expected for the ex-\ntracted voltage signal to oscillate at the same frequency\nas that of the N\u0013 eel vector and contain substantial out-\nput power ( >1\u0016W) [69]. In this regard, the landmark\ntheoretical work on NiO based oscillator [29] suggested\nthe measurement of spin pumped [70] time varying in-\nverse spin Hall voltage [71] across the heavy metal (Pt)\nof a NiO=Pt heterostructure. However, the time varying\nvoltage at THz frequency requires an AFM with signif-\nicant in-plane biaxial anisotropy [29, 69], thus limiting\nthe applicability of this scheme to only select AFM ma-\nterials. In addition, the output power of the generated\nsignal is sizeable (above 1 \u0016W) only for frequencies below\n0:5 THz [69]. A potential route to overcoming the afore-\nmentioned limitations is coupling the AFM signal genera-\ntor to a high-Q dielectric resonator, which would enhance\nthe output power even for frequencies above 0 :5 THz [41].\nThis method, however, requires devices with sizes in the\n10's micrometers range for frequencies above 2 THz and\nfor the AFMs to possess a tilted net magnetization in\ntheir ground state [69]. A more recent theoretical work\non collinear AFM THz oscillators [43] suggested employ-\ning Anisotropy Magnetoresistance (AMR) or Spin Mag-\nnetoresistance (SMR) measurements in a four terminal\nAFM/HM spin Hall heterostructure. This would enable\nthe extraction of the THz oscillations as longitudinal or\ntransverse voltage signals. However, the reported values\nof both AMR and SMR at room temperatures in most\nAFMs is low and would, in general, require modulating\nthe band structure for higher values [72].\nIdeal Bias \nRLC\nLR(t)\nA TJ\nCbtLbt\nPac(a)\nRLPac (b)\nZth\nUthIread\nFIG. 6. (a) An equivalent circuit representation of Fig. 1\n(adapted from [69]). The generation (write) current is not\nshown in the circuit, although its e\u000bect is included as a varia-\ntion in the resistance R(t) through its frequency dependence.\n(b) Thevenin equivalent of (a).\nA recent theoretical work [69] proposed employing a\nfour terminal AFM tunnel junction (ATJ) in a spin Hall\nbilayer structure with a conducting AFM to e\u000bectively\ngenerate and detect THz frequency oscillations as vari-\nations in the tunnel anisotropy magnetoresistance [54].\nA DC current passed perpendicularly to the plane of\nthe ATJ generates an AC voltage, which is measured\nacross an externally connected load. It was shown that\nboth the output power and its e\u000eciency decrease as fre-\nquency increases, nevertheless, it was suggested that this\nscheme could be used for signal extraction in the fre-\nquency range of 0 :1\u000010 THz, although the lateral size10\nof the tunnel barrier required for an optimal performance\ndepends on the frequency of oscillations (size decreases as\nthe frequency increases) [69]. The analysis presented in\nRef. [69], however, neglects the generation current com-\npared to the read current while evaluating the e\u000eciency\nof power extraction. But it can be observed from the\nresults in Section III C that the threshold current, and,\ntherefore, the generation current, depend on AFM ma-\nterial properties, such as damping, anisotropy, and ex-\nchange constants, and could be quite large. Therefore, in\nour work we include the e\u000bect of the generation current\nto accurately model the power e\u000eciency of the TAMR\nscheme.\n(a) (b)\nFIG. 7. (a) Output power and (b) e\u000eciency dependence on\nthe area of cross-section of the tunnel barrier for di\u000berent fre-\nquencies. The thickness of the barrier is \fxed to db= 1 nm.\nThe e\u000bect of write current and the input power associated\nwith it is not considered here, therefore, these results are in-\ndependent of the choice of the AFM material.\nIn order to evaluate the performance of the TAMR\nscheme, an equivalent circuit representation (adapted\nfrom Ref. [69]) of the device setup of Fig. 1 is shown\nin Fig. 6(a), while its Thevenin equivalent representation\nis shown in Fig. 6(b). The circuits in Fig. 6 only repre-\nsent the read component, while the THz generation com-\nponent is omitted for the sake of clarity. In Fig. 6(a),\nthe dashed red box encloses a circuit representation of\nthe ATJ, comprising a series combination of an oscillat-\ning resistance R(t) =R0+ \u0001Rcos!tand inductance\nL=\u00160db, connected in parallel to a junction capacitor\nC=Ac\u000f\u000f0=db(assumed parallel plate). The constant\ncomponent, R0, in the oscillating resistance, R(t), is the\nequilibrium resistance of the MgO barrier and is given\nasR0=RA(0) exp(\u0014db)\nAc. Here,RA(0) is the resistance-\narea product of a zero-thickness tunnel barrier, \u0014is the\ntunneling parameter, dbis the barrier thickness, and Ac\nis the cross-sectional area. The pre-factor, \u0001 R, of the\ntime varying component of R(t) is the resistance variation\ndue to the oscillation of the magnetization vectors with\nrespect to the polarization axis. \u0001 R= (\u0011=(2 +\u0011))R0,\nwhere\u0011is the TAMR ratio of the barrier and depends\non the temperature and material properties.\nDue to the \row of the DC current, Iread, an alter-\nnating voltage develops across the ATJ, which is mea-\nsured across an externally connected load RL, separated\nfrom the ATJ via an ideal bias tee (enclosed in the green\ndashed box). The bias-tee, characterized by an induc-\ntanceLbtand a capacitance Cbt, and assumed to have noTABLE II. List of common antiferromagnetic materials Mn 3X\nand their associated parameters. Here Msis in Tesla,Keis\nin kJ=m3, andJis in MJ=m3. Sign ofDwhich decides the\nchirality is also mentioned.\nX\u000b M sKeJ D Ref.\nIr 0:01 1:63 3000 240 - [56]\nPt 0:013 1:37 10 280 - [73]\nRh 0:013 2:00 10 230 - [73{75]\nGa 0:008 0:54 100 110 + [75{78]\nSn 0:003 0:50 110 59 + [67, 75, 77, 79]\nGe 0:0009 0:28 1320 77 + [75, 77, 78, 80]\nGaN 0:1 0:69 10 280 - [81, 82]\nNiN 0:1 1:54 10 177 - [81, 82]\nvoltage drop across it, blocks any DC current from \row-\ning into the external load. Therefore, the AC voltage of\nthe ATJ is divided only into its impedance (a combina-\ntion ofR0,L, andC) and that of the load RL[69].\nNext, we simplify the ATJ circuit into a Thevenin\nimpedance Zthand voltage Uthas shown in Fig. 6(b).\nThey are evaluated as\nZth=R0+j!L\n(1\u0000\u0018) +j\f; (19)\nand\nUth=Uac\n(1\u0000\u0018) +j\f; (20)\nwherej=p\u00001,!= 2\u0019f,\u0018=!2LC,\f=!R0C, and\nUac=Iread\u0001R. The output voltage and average power\nacross the load can then be obtained as\nUL=UthRL\nZth+RL=Uacr\n1 +jp+r(1\u0000\u0018+j\f);(21)\nand\nPL=1\n2jULj2\nRL=U2\nac\n2RLr2\n1 +qr2+ 2r+p2; (22)\nwherer=RL=R0,q= (1\u0000\u0018)2+\f2, andp=!L\nR0. Finally,\nthe e\u000eciency of the power extraction can be obtained as\n\u0010=PL\nPin\n=0:5r\n1 +qr2+ 2r+p21\nI2\nwriteRGenR0=U2ac+ 1;(23)\nwhereRGenis the resistance faced by the generation cur-\nrent. It can be observed from Eqs. (21)-(23) that the\noutput voltage, output power, and the e\u000eciency of power\nextraction decrease with an increase in frequency since \u0018,\n\f,q, andpincrease with ![69, 83].\nConsidering that the load impedance is \fxed to 50 \nby the external circuit, one can only optimize the source\nimpedance to achieve PL>1\u0016W andUL>1 mV. In\nthis regard, the resistance of the source tunnel barrier can11\nTABLE III. Material Parameters of the NM, and at the\nNM/AFM interface.\nParameters Values Ref.\ngM 3:8\u00021010S/m2[50]\ngm 3:8\u0002109S/m2[50]\n\u001aCu 6\u000210\u00009\n m2[50]\ntCu 5 nm [50]\nbe altered by either varying the thickness of the tunnel\nbarrier,db, or its cross-sectional area, Ac. However, the\noptimum values of dbandAcfor the desired output sig-\nnals is frequency dependent, and, therefore, tunnel barri-\ners of di\u000berent sizes would be required for di\u000berent oper-\nating frequencies [69, 83]. For all estimates, we consider\ndb= 1 nm,\u0011= 1:3,\u0014= 5:6 nm\u00001,RA(0) = 0:14 \n\u0016m2,\nand\u000f= 9:8 [69]. For reliable operation of the tunnel bar-\nrier, we consider the electric \feld across the barrier to be\nE= 0:3 V=nm [69], which is below the barrier break-\ndown \feld. Ignoring the e\u000bect of the generation current\nin Eq. (23), as suggested in Ref. [69], we deduce from\nFig. 7 that the optimal cross-sectional area Ac\u00190:36\n\u0016m2forf= 0:1 THz,Ac\u00190:25\u0016m2forf= 1 THz,\nAc\u00190:16\u0016m2forf= 10 THz.\nIrSnPtRhGeGaGaNNiN\nX10-710-510-3ζ(%)\nf=2.0THz\nFIG. 8. Power e\u000eciency for di\u000berent materials Mn 3X listed in\nTable II. The dashed horizontal line shows the expected e\u000e-\nciency of\u0011= 0:011% for the optimized geometry ( db= 1 nm,\nandAc= 0:24\u0016m2) if write current is neglected. The e\u000e-\nciency, however, decreases signi\fcantly due to the inclusion\nof write current. Here the AFM thin-\flm thickness dais as-\nsumed to be 4 nm.\nTable II lists the material properties of various con-\nducting AFMs. Depending on the sign of their DMI\nconstant, these AFMs could host moments with either\na positive or a negative chirality. The closed-form model\npresented in Eq. (17) can be used to evaluate the re-\nquired spin current for frequency f= 2 THz, regardless\nof the chirality since frequency scales linearly with theinput current in this region (see Fig. 3). For a given spin\ncurrent density ( Js), the charge current density ( Jwrite)\nfor the lateral spin-valve structure of Fig. 1(a) is given as\nJwrite =gM+gm\ngM\u0000gmJs: (24)\nwheregMandgmare the conductance of the majority-\nand minority-spin electrons at the NM (Cu)/FM inter-\nface. The input power required to start the oscilla-\ntions is given as ( JwriteAc)2RCu, whereRGen=RCu=\n\u001aCuLCu\nACu=\u001aCupAc\ntCupAcis the resistance of the copper\n(NM) underneath the bottom MgO. In order to evaluate\nthe resistance of the copper layer, we have assumed its\nlength and width to be the same as MgO and the AFM\nthin-\flm.\nThe e\u000eciency of power extraction for the listed AFM\nmaterials is presented in Fig. 8. The dashed horizon-\ntal line denotes the expected e\u000eciency of \u0011= 0:011% if\nthe e\u000bect of generation current is neglected and the area\nof cross-section of MgO is optimized for f= 2:0 THz.\nHowever, it can be observed that the e\u000eciency decreases\nsigni\fcantly i.e. by a few orders when the input power\ndue to the generation current in included in the analy-\nsis. For materials with large damping and large uniaxial\nanisotropy constants, the required generation current is\nhigher leading to lower e\u000eciency. This result shows that\nfurther optimization of the device geometry for di\u000berent\nmaterials is required to increase the e\u000eciency.\nThis method of power extraction could be more suit-\nable for materials with negative chirality. We can observe\nfrom Fig. 9 that the output power as well as the e\u000eciency\nfor both Mn 3Sn and Mn 3Ge for frequencies between 0.1\nTHz and 2.0 THz are signi\fcant. The required genera-\ntion current for Mn 3Ge is smaller than that for Mn 3Sn,\ntherefore, the e\u000eciency is higher for the former. Also,\nthe e\u000eciency of power extraction increases with decrease\nin area of cross-section in both the cases but this is ac-\ncompanied by a decrease in output power.\nIt might be possible to increase the output power and\noverall e\u000eciency of the system if the material properties\nof the tunnel barrier such as \u0011;\u0014, andRA(0) could be\naltered. Large room temperature tunneling magnetore-\nsistance in an ATJ is feasible either by using a tunnel\nbarrier other than MgO [84] or inserting a di\u000busion bar-\nrier to enhance magneto-transport [85]. Here we adopted\nthe TAMR extraction scheme because we have consid-\nered metallic AFMs so a DC current through the ATJ\nstructure can be easily applied. In addition, the three-\nor four-terminal compact ATJ structure along with its\nsmall lateral size enables dense packing of several such\nTHz oscillators on a chip accompanied with a net in-\ncrease in the output power and e\u000eciency of the oscillator\narray [72]. For example, with an array of 10 \u000210 such\nAFM oscillators excited in parallel, the output power and\ne\u000eciency could be scaled up by 100 \u0002compared to the\nresults presented in Figs. 7 and 8.12\n(a) (b)\n(c) (d)\nFIG. 9. Upper panel: (a) Output power and (b) e\u000eciency for\nMn3Sn. Lower panel: (c) Output power and (d) e\u000eciency for\nMn3Ge. Two di\u000berent cross-section size of the MgO barrier is\nconsidered. The output power depends only on the frequency\nof oscillation and therefore is same for both the materials.\nThe e\u000eciency of power extraction depends on the generation\ncurrent, which is lower for Mn 3Ge, leading to a higher value\nof e\u000eciency in that case.\nV. EFFECTS OF INHOMOGENEITY DUE TO\nEXCHANGE INTERACTION\nThe results presented in Section III C correspond to\nthe case of a single-domain AFM particle and are, there-\nfore, independent of the lateral dimensions of the thin-\n\flm. This can also be deduced from the equations of the\nthreshold current and the average oscillation frequency.\nHowever, when the lateral dimensions of the AFM thin-\n\flm exceed several 10's of nm, micromagnetic analysis\nmust be carried out. In this section, we analyze the dy-\nnamics in thin-\flm AFMs of varying dimensions within a\nmicromagnetic simulation framework. We consider AFM\nthin-\flms of dimensions 50 nm \u000250 nm and investigate\nthe e\u000bect of the inhomogeneity due to exchange interac-\ntions. In each case, the thin-\flm was divided into smaller\ncubes, each of size 1 nm \u00021 nm\u0002danm, since the do-\nmain wall width \u0001 0=p\n(2Aii\u0000Aij)=(2Ke)>1 nm for\nKecorresponding to Mn 3Ir as listed in Table II. It can\nbe observed from Fig. 10 that for materials with positive\nchirality the e\u000bects of inhomogeneity becomes important\nfor low currents. On the other hand, for materials with\nnegative chirality, inhomogeneities do not appear to have\nany e\u000bect. For positive chirality materials, the numerical\nvalues of frequency for di\u000berent spring constants deviates\nsigni\fcantly from that obtained from the single domain\nsolution, as well as analytic results. In this case, the hys-\nteretic region reduces in size since the lower threshold\ncurrent increases in magnitude as compared to the the-\noretical prediction as can be observed from Fig. 10(a).\nPositive Chirality Negative Chirality\n(a) (b)\n(c)                                               (d)FIG. 10. Frequency vs. input current for di\u000berent values\nof inhomogeneous exchange constants (intra-sublattice (a, c),\ninter-sublattice (b, d)) for both positive and negative chirality.\nIn all cases \u000b= 0:01, andda= 4 nm. Other parameters\ncorrespond to those of Mn 3Ir as listed in Table II for both\npositive and negative chirality materials with the exception\nof the sign ofDfor the latter.\nWhile we have not included the e\u000bect of inhomogeneous\nDMI in our work, we expect such interactions to lead to\nthe formation of domain walls in the thin-\flm similar to\nthe case of collinear AFMs [43]. A more detailed analy-\nsis of the dynamics of the positive chirality materials due\nto variation in exchange interaction as well as inhomoge-\nneous DMI would be carried out in a future publication.\nVI. DISCUSSION\nWe focused on the dynamics of the order parameters\nin exchange dominant non-collinear coplanar AFMs with\nboth positive (+ \u0019=2) and negative (\u0000\u0019=2) chiralities as-\nsociated to the orientation of equilibrium magnetization\nvectors. In both these classes of AFMs, the exchange en-\nergy is minimized for a 2 \u0019=3 relative orientation between\nthe sublattice vectors. Next, the negative (positive) sign\nof the iDMI coe\u000ecient minimizes the system energy for\ncounterclockwise (clockwise) ordering of m1;m2, andm3\nin the x\u0000yplane leading to positive (negative) chiral-\nity. Finally, all the sublattice vectors coincide with their\nrespective easy axis only in the case of the positive chiral-\nity materials due to the relative anticlockwise orientation\nof the easy axes. On the other hand, the negative chi-\nrality materials have a six-fold symmetry wherein only\none of the sublattice vectors can coincide with its respec-\ntive easy axis. As a result, these AFM materials with\ndi\u000berent chiralities have signi\fcantly distinct dynamics\nin the presence of an input spin current. For AFM ma-\nterials with + \u0019=2 chirality, oscillatory dynamics are ex-13\ncited only when the injected spin current overcomes the\nanisotropy, thus indicating the presence of a larger cur-\nrent threshold. Moreover, the dynamics in such AFMs is\nhysteretic in nature. Therefore, it is possible to sustain\noscillations by lowering the current below that required\nto initiate the dynamics as long the energy pumped in by\nthe current overcomes that dissipated by damping. On\nthe other hand, in the case of \u0000\u0019=2 chirality AFMs, os-\ncillations can be excited when signi\fcantly smaller spin\ncurrent with appropriate spin polarization is injected into\nthe AFM. Hence, \u0000\u0019=2 chirality AFMs may be more\namenable to tuning the frequency response over a broad\nfrequency range, from the MHz to the THz range [68].\nThe oscillation of the AFM N\u0013 eel vectors can be mea-\nsured as a coherent AC voltage with THz frequencies\nacross an externally connected resistive load through the\ntunnel anisotropic magnetoresistance measurements for\nboth +\u0019=2 and\u0000\u0019=2 chirality materials. In general, as\nthe frequency increases, the magnitude of both the out-\nput power and the e\u000eciency of power extraction decrease,\nhowever, it is possible to enhance both these quantities\nby optimizing the cross-sectional area of the tunnel junc-\ntion. This, however, is limited due to larger threshold\ncurrent requirement for materials with large damping.\nTherefore, a hybrid scheme of electrically synchronized\nAFM oscillators on a chip could be used to further en-\nhance the power and e\u000eciency [86, 87].\n(a) (b)\nFIG. 11.mzfor larger damping, \u000b= 0:1, andda= 4 nm. (a)\nNon-coherent (spike-like) signals near the threshold current\nJs= 1:1Jth1\ns. (b) Coherent signal for larger current Js=\n1:5Jth1\ns. The angular frequency is directly proportional to\nmz, and therefore it would show the exact same features (in\nthe absence of any external \feld) for the chosen values of\ncurrent.\nMetallic AFMs such as Mn 3Ir and Mn 3Sn could be\nconsidered as examples of + \u0019=2 and\u0000\u0019=2 chiralities,\nrespectively. Recently, thin-\flms with di\u000berent thickness\nranging from 1 nm to 5 nm of both these materials have\nbeen grown using UHV magnetron sputtering [88{91].\nIn addition, di\u000berent values of damping constants have\nbeen reported for Mn 3Sn [56, 77]. Therefore, we expect\nthe results presented in Sections III C, IV and V to be\nuseful for benchmarking THz dynamics in experimental\nset-ups with such thin \flms metallic antiferromagnets.\n(a) (b)\n(c) (d)FIG. 12. Time dynamics (single and train of spikes) of a single\n\\neuron\" for di\u000berent input currents and frequencies. The net\ninput current should be greater than the threshold current\n(\u0015 > 0:2) for a non-zero dynamics. For an input current\nabove the threshold, as the external frequency increases the\ndynamics changes from (a) bursts of spikes to (c) single spikes\nto (d) no spikes ( \u0015= 0:3). As the input current increases to\n\u0015= 0:4 the range of external frequency where the spiking\nbehaviour is observed increases.\nVII. POTENTIAL APPLICATIONS\nNeurons in the human brain could be thought of as a\nnetwork of coupled non-linear oscillators, while the stim-\nuli to excite neuronal dynamics is derived from the neigh-\nboring neurons in the network [8, 92{94]. For materials\nwith +\u0019=2 chirality, a non-linear behaviour was observed\nfor large damping, and input currents near the threshold\ncurrent,Jth1\ns, in Fig. 3(b), (f). This non-linearity cor-\nresponds to Dirac-comb-like magnetization dynamics, as\nshown in Fig. 11(a), and is similar to the dynamics of\nbiological neurons in their spiking behaviour as well as\na dependence on the input threshold. However, unlike a\nbiological neuron which shows various dynamical modes\nsuch as spiking, bursting, and chattering [95], the dy-\nnamics here shows only spikes and does not show any\nrefractory (\\resting\") period. Recent works [96, 97] have\nshown that it is possible to generate single spiking as\nwell as bursting behaviours using NiO-based AFM oscil-\nlators by considering an input DC current below Jth1\ns,\nand superimposing it with an AC current. As the AC\ncurrent changes with time, the total current could either\ngo above the threshold, thereby triggering a non-linear\nresponse, or below the threshold current resulting in a\n\\resting\" period. Here we explore the possibility of spik-\ning behaviours in + \u0019=2 chirality materials such as Mn 3Ir\nunder the e\u000bect of an input spin current. We use the non-\nlinear pendulum model of Eq. (12) and study the possible\ndynamics in case of a single oscillator, two unidirectional\ncoupled oscillators, and two bidirectional coupled oscil-14\nlators.\nA. Ultra-fast Hardware Emulator of Neurons\nWe consider a large damping of \u000b= 0:1 while the other\nmaterial parameters correspond to that of Mn 3Ir as listed\nin Table II. Next we choose an input current Js(t) =\nJdc\ns+Jac\ns(t), whereJdc\ns= 0:8Jth1\nsis the dc component of\nthe input current, superimposed with a smaller ac signal\nJac\ns(t) =\u0015Jth1\nscos(2\u0019fact). The time dynamics of this\nnon-linear oscillator is governed by\n'+\u000b!E_'+!E!K\n2sin 2'+!E!s(t) = 0; (25)\nwhere!s(t)(/Js(t)) is the time varying input current.\n(a) (b)\nFIG. 13. The dynamics of two neuron system with unidirec-\ntional coupling at fac= 60 GHz, and \u0015= 0:3. The dotted\nblue curve corresponds to the \frst neuron. (a) Second neuron\nshows no spike for \u0014= 0:028 but a single spike for \u0014= 0:032.\n(b) The single spiking behaviour changes to bursts with three\nspikes as\u0014increases and coupling strengthens.\nFigure 12 presents the dynamics of Eq. (25) for dif-\nferent input current and frequencies. Firstly, it can be\nobserved that the input current must be greater than\nthe threshold current to excite any dynamics viz. \u0015\nmust be greater than 0.2 (dotted line corresponding to\n\u0015= 0:2 shows no spikes for any value of external fre-\nquency). Secondly, for input currents above the thresh-\nold viz.\u0015=f0:3;0:4g, a train of spikes is observed for\nlower frequency of 20 GHz in Fig. 12(a). However, as\nfrequency of the input excitation increases the number of\nobserved spikes decreases for both values of current con-\nsidered here (Fig. 12(b, c)). Finally, it can be observed\nfrom Fig. 12(d) that for very large frequency the spiking\nbehaviour vanishes for lower current ( \u0015= 0:3) but per-\nsists for higher current ( \u0015= 0:4). For higher values of\ncurrent, the cut-o\u000b frequency is higher. This observed\nspiking behaviour is indeed similar to that of biological\nneurons [95]. Here, however, the observed dynamics is\nvery fast in the THz regime and thus the AFM oscilla-\ntors could be used as the building blocks of an ultra-high\nthroughput brain-inspired computing architecture.\n(a)\n(b)\n(c)\n(d)FIG. 14. The dynamics of two neuron system with unidirec-\ntional coupling at fac= 180 GHz. The dashed blue curve\ncorresponds to the \frst neuron. \u0015= 0:3: (a) Single spike\nfor\u0014= 0:04 but not for \u0014= 0:036. (b) The single spik-\ning behaviour become prominent as the coupling strengthens.\n\u0015= 0:4: (c) Single spike for \u0014= 0:032 in response to a double\nspiking behaviour of the \frst neuron. (d) For larger \u0014second\nneuron shows bursting dynamics with two spikes.\nB. Two unidirectional coupled arti\fcial neurons\nA network composed of interacting oscillators forms\nthe basis of the oscillatory neurocomputing model pro-\nposed by Hoppensteadt and Izhikevich [98]. In such a\nnetwork, the dynamics of an oscillating neuron (or a\n\\node\") is controlled by the incoming input signal as\nwell as its coupling to neighboring neurons. To inves-\ntigate this coupling behaviour we consider a system of\ntwo unidirectional coupled neurons. The \frst neuron is\ndriven by an external signal and its dynamics is governed\nby Eq. (25). The dynamics of the second neuron, on the\nother hand, depends on the output signal of the \frst neu-\nron as well as the coupling between the two neurons. It\nis governed by\n'j+\u000b!E_'j+!E!K\n2sin 2'j+!E!s\n\u0000\u0014ij!E_'isgn(!s) = 0;\n(26)\nwhere\u0014ij=\u0014is the unidirectional coupling coe\u000ecient\nfrom neuron i= 1 toj= 2. There is no feedback from\nthe second neuron to the \frst and therefore \u0014ji= 0. In\naddition to the input from the \frst neuron, the second\nneuron is also driven by a constant DC current Jdc\ns2(/!s\nin Eq. (26)). We choose this DC current to be the same\nas that for the \frst neuron viz. Jdc\ns2= 0:8Jth1\ns. The\ndynamics of the second neuron for two di\u000berent external\ninput currents ( \u0015=f0:3;0:4g) and frequencies ( fac=\nf60;180gGHz) is presented in Figs. 13 and 14.15\nFirstly, it can be observed that in all the cases the sec-\nond neuron shows a spiking behaviour only for \u0014above\na certain value. Secondly, for \u0015= 0:3 andfac= 60 GHz,\nwherein the \frst neuron shows bursting behaviour con-\nsisting of three spikes, the second neuron shows a single\nspike (Fig. 13(a)) for lower value of \u0014, and three spikes\nfor stronger coupling (Fig. 13(b)). This behaviour is due\nto the threshold dependence of the second neuron as well\nas due to its inertial dynamics. Similar behaviour is also\nobserved for \u0015= 0:4, andfac= 180 GHz in Figs. 14(c),\n(d). Thirdly, for \u0015= 0:3 andfac= 180 GHz, wherein the\n\frst neuron shows a single spike, Fig. 14(a) shows that\ncompared to the case of fac= 60 GHz a slightly higher\nvalue of\u0014is now required to excite the second neuron.\nThe single spiking behaviour of the second neuron be-\ncomes more prominent as the coupling strength increases\nbecause of a stronger input as shown in Fig. 14(b). Re-\ncently, it was suggested that this coupled behaviour of\nTHz arti\fcial neurons could be used to build ultra-fast\nmulti-input AND, OR, and majority logic gates [96].\n(a)\n(b)\n(c)\n(d)\nFIG. 15. The dynamics of two neuron system with bidirec-\ntional coupling at fac= 180 GHz, and \u0015= 0:3. First neu-\nron shows bursting behaviour in this system while the second\nneuron follows the \frst neuron for all values of \u0014. As the cou-\npling between the two neurons increase the number of spikes\nfor both the neurons increases.\nC. Two bidirectional coupled arti\fcial neurons\nIn some circuits it is possible that the coupling be-\ntween any two neurons is bidirectional. In such cases, in\naddition to a forward coupling from the \frst neuron to\nthe second, a feedback exists from the second neuron to\nthe \frst. The dynamics of each neuron of this coupled\nsystem is governed by Eq. (26), however, !s=!s(t) for\nthe \frst neuron, as discussed previously. We consider\n\u001412=\u001421=\u0014. Figures 15 and 16 show the dynamics of\nthe two neurons of this coupled system with the coupling\u0014atfac= 180 GHz for \u0015= 0:3 and 0.4, respectively.\n(a) (b)\n(c) (d)\nFIG. 16. The dynamics of two neuron system with bidirec-\ntional coupling at fac= 180 GHz, and \u0015= 0:4. Second\nneuron \fres when the coupling is above a certain threshold\nwhich in turn leads to another spike for the \frst neuron. As\nthe coupling between the two neurons increase the number of\nspikes for both the neurons increases.\nFirstly, Fig. 15(a) shows that for \u0014= 0:04 the dynam-\nics of both _'1and _'2are almost similar to that presented\nin Fig. 14(a), viz. the e\u000bects of coupling is very small.\nHowever, as the coupling between the two neurons in-\ncreases (Fig. 15(b)-(d)), a positive feedback is established\nbetween the two neuron leading to dynamics with two or\nmore spikes, in general. This is observed after the sec-\nond neuron has \fred, at least once, because the positive\nfeedback leads to a net input greater than the threshold\ncurrent to the \frst neuron, even though the external in-\nput has reduced below the threshold. Similar behavior is\nalso observed in the case of \u0015= 0:4, although at lower\nvalues of coupling, as presented in Fig. 16. The results\nallude to the threshold behaviour of the neurons, inertial\nnature of the dynamics, and a dependence of the dynam-\nics on the phase di\u000berence between the two neurons. The\ndynamics of two bidirectional coupled arti\fcial neurons\npresented here could be the \frst step towards building\nAFM-based recurrent neural networks or reservoir com-\nputing [99], instead of the slower FM-based coupled os-\ncillator systems [100, 101].\nVIII. CONCLUSION\nIn this work, we numerically and theoretically explore\nthe THz dynamics of thin-\flm metallic non-collinear\ncoplanar AFMs such as Mn 3Ir and Mn 3Sn, under the\naction of an injected spin current with spin polarization\nperpendicular to the plane of the \flm. Physically, these\ntwo AFM materials di\u000ber in their spin con\fguration viz.\npositive chirality for Mn 3Ir, and negative chirality for16\nMn3Sn. In order to explore the dynamics numerically,\nwe solve three LLG equations coupled to each other via\ninter-sublattice exchange interactions. We also analyze\nthe dynamics theoretically in the limit of strong exchange\nand show that it can be mapped to that of a damped-\ndriven pendulum if the e\u000bects of inhomogeneity in the\nmaterial are ignored. We \fnd that the dynamics of Mn 3Ir\nis best described by a non-linear pendulum equation and\nhas a hysteretic behaviour, while that of Mn 3Sn in the\nTHz regime is best described by a linear pendulum equa-\ntion and has a signi\fcantly small threshold for oscillation.\nThe hysteretic dynamics in the case of Mn 3Ir allows for\npossibility of energy e\u000ecient THz coherent sources. On\nthe other hand, a small threshold current requirement\nin the case of Mn 3Sn indicates the possibility of e\u000e-\ncient coherent signal sources from MHz to THz regime.\nWe employ the TAMR detection scheme to extract the\nTHz oscillations as time-varying voltage signals across\nan external resistive load. Including inhomogeneous ef-\nfects leads to a variation in the dynamics | the lowerthreshold current for sustaining the dynamics increases,\nthe hysteretic region reduces, and the frequency of oscil-\nlation decreases for lower current levels. Finally, we also\nshow that the non-linear behaviour of positive chirality\nmaterials with large damping could be used to emulate\narti\fcial neurons. An interacting network of such oscil-\nlators could enable the development of neurocomputing\ncircuits for various cognitive tasks. 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Based on a quantized spin+environment Hamiltonian, we here\nderive a general spin operator equation of motion that describes three-dimensional precession and damp-\ning and consistently accounts for e\u000bects arising from memory, coloured noise and quantum statistics.\nThe LLG equation is recovered as its classical, Ohmic approximation. We further introduce resonant\nLorentzian system{reservoir couplings that allow a systematic comparison of dynamics between Ohmic\nand non{Ohmic regimes. Finally, we simulate the full non-Markovian dynamics of a spin in the semi{\nclassical limit. At low temperatures, our numerical results demonstrate a characteristic reduction and\n\rattening of the steady state spin alignment with an external \feld, caused by the quantum statistics\nof the environment. The results provide a powerful framework to explore general three-dimensional\ndissipation in quantum thermodynamics.\nThe continued miniaturisation of critical components\nin consumer electronics and neighbouring technologies\nwill require a deeper understanding of thermal noise\nand general thermodynamic principles beyond the clas-\nsical macroscopic world. Quantum thermodynamics [1{\n3] has emerged as a \feld addressing the conceptual\nchallenges related to the exchange of energy and in-\nformation at the nanoscale. Recent advances include\nstudies of heat transport in quantum systems [4{11],\nthe characterisation of memory e\u000bects in their dynam-\nics [12{19], and clari\fcation of the impact of quan-\ntum coherence and correlation on thermodynamic pro-\ncesses [20{24]. The establishment of a generalised\nthermodynamic framework, valid for nanoscale systems\nthat strongly couple to environmental modes, is well\nunder way [25{33], and for magnetic molecules an\nenvironment-induced renormalisation of the anisotropy\nhas been predicted [34]. Two open quantum sys-\ntems models have served as the workhorse for many\nof these conceptual studies; the Caldeira-Leggett model\nfor quantum Brownian motion [8, 35, 36] and the spin-\nboson model of a spin (or many spins) coupled to a\none-dimensional harmonic bath [5, 6, 33, 37{39]. These\ndescribe a very wide range of physical situations extend-\ning to studies of quantum e\u000bects in bio-chemical reac-\ntions [40], where they are used to model exciton-phonon\ninteractions [41].\nUntil now few nanoscale technologies have required\nthe use of advanced open quantum systems techniques.\nBut advances in engineering magnetic materials for\nmagnetic hard drives at unprecedented length and time-\nscales [42] are likely to require a more detailed picture\nof spin dynamics including memory and quantum sig-\nnatures. Here we introduce a three-dimensional open\nquantum system model to characterise the quantum\nBrownian motion of spins in magnetic materials.\nMagnetic behaviour has been studied extensively\nbased on the classical phenomenological Landau{\nLifshitz{Gilbert (LLG) equation [43{47]\n@M\n@t=\rM\u0002\u0014\nBe\u000b\u0000\u0011G@M\n@t\u0015\n; (1)\n\u0003janet@qipc.orgwhich is routinely solved with micromagnetic and atom-\nistic simulations. Here Mis the magnetic moment, \r\nis the gyromagnetic ratio and Be\u000bis an e\u000bective mag-\nnetic \feld which includes the external \feld [106], ex-\nchange and anisotropy e\u000bects, as well as stochastic mag-\nnetic noiseb/p\nTstemming from an environment at\ntemperature Tthat was added by Brown [107] in 1963\n[48]. The \fnal term on the right of (1) is the so{called\n\\Gilbert damping\" term and the positive constant \u0011Gis\nthe damping parameter [108], which is often rewritten\nas\u0011G=\u0011=jMjj\rjwith a unit-free \u0011.\nGilbert damping is not derived from microscopic prin-\nciples, but chosen as the simplest term that could\nserve to align the magnetic moment with the applied\n\feld [43]. As we will see, it contains no memory which\nis increasingly seen as a limitation [49, 50]. Advances\nin engineering magnetic materials at the nanoscale\nand manipulating them on ultrafast timescales indicate\nthat a theory beyond the classical LLG equation is re-\nquired [47]. Early attempts have pursued a path integral\nderivation of a quantum spin dynamics equation [51], as\nwell as other conceptually related classical and quantum\nderivations [52, 53]. These derivations were not directly\napplied to the calculation of magnetization dynamics\nor steady states, nor have they been connected to re-\ncent generalizations of Gilbert damping that include in-\nertial terms [49, 50, 54{59] or provided an assessment\nof quantum e\u000bects.\nHere we go further and develop a comprehensive and\nquantum-thermodynamically consistent theory suitable\nto describe the quantum dynamics of spins in magnetic\nmaterials including non{Markovian damping, coloured\nnoise and quantum zero-point \ructuations. Unlike\nthe conceptually pioneering Caldeira-Leggett model that\nhas few experimental realisations, the developed three-\ndimensional quantum spin model is directly applicable\nfor atomistic spin dynamics simulations [47, 60], ultra-\nfast magnetism experiments [61], and systems exhibiting\nanisotropic damping [62].\nThe paper is organised as follows: In section I the\ngeneral quantum spin dynamics equation for spin oper-\nator precession in three dimensions is derived. For the\nsimplest, Ohmic, coupling this equation is found to re-\nduce to the memory-free LLG equation. In section II we\nintroduce Lorentzian couplings as a systematic method\nfor exploring non-Markovian dynamical regimes in gen-arXiv:2009.00600v2  [quant-ph]  7 Jul 20212\neral open quantum systems, including spins. Finally,\nin section III we detail a numerical method to simulate\nnon-Markovian dynamics, and present results for a sin-\ngle classical spin that illustrate the di\u000berences between\nspin dynamics and steady states arising with non-trivial\nmemory, coloured noise, and quantum bath statistics in\ncomparison to those obtained with the memory-free LLG\nequation. Conclusions and open questions are discussed\nin section IV.\nI. Quantum spin dynamics equation\nA. System+environment Hamiltonian\nWe begin by introducing the quantized Hamiltonian\ndescribing the di\u000berent contributions to the total energy\nof the system, consisting of spins as well as environmen-\ntal degrees of freedom (e.g. electrons and phonons),\ngiven by\n^H=^HS+^HR+^Vint; (2)\nwhere ^HSis the bare spin Hamiltonian operator which\ncaptures the spin energy in external \felds and interac-\ntions between spins, ^HRis the environmental or reser-\nvoir Hamiltonian, and ^Vintis the interaction between\nthe spins and the reservoir.\nWe choose ^HSas the sum of the interaction with\na homogeneous external \feld Bext[109] and the ex-\nchange interaction between three-dimensional spin vec-\ntor operators ^S(n)= (^S(n)\n1;^S(n)\n2;^S(n)\n3) at sitesnof a\nlattice [110],\n^HS=\u0000\rX\nn^S(n)\u0001Bext\u00001\n2X\nn;m6=n^S(n)\u0001J(nm)^S(m):(3)\nHereJ(nm)is the exchange tensor for spin pairs (n;m)\n[111], which can include the Dzyaloshinskii-Moriya in-\nteraction [47]. It is straightforward to include additional\nenergetic terms in the bare spin Hamiltonian, such as en-\nergies associated with magnetic anisotropy. Instead of\nthe magnetic moment Mused in Eq. (1), we will here\nwork with the spin angular momentum Sproportional\ntoM,M=\rS, where\ris the gyromagnetic ratio.\nIn the following we will assume the gyromagnetic ratio\n\r=\u0000ge\u0016B=~=\u00001:76\u00011011s\u00001T\u00001for an electron.\nThe reservoir Hamiltonian is commonly modelled as\na set of harmonic oscillators [35, 64], and we here fol-\nlow the continuous reservoir approach by Huttner and\nBarnett [64], taking the reservoir Hamiltonian as\n^HR=1\n2X\nnZ1\n0d!\u0014\u0010\n^\u0005(n)\n!\u00112\n+!2\u0010\n^X(n)\n!\u00112\u0015\n:(4)\nIt describes a continuous frequency reservoir at each lat-\ntice siten, where ^\u0005(n)\n!and^X(n)\n!are (three-dimensional)\nmomentum and position operators of the reservoir os-\ncillator with frequency !. The position operators ^X(n)\n!\nphysically represent variations in the environment to\nwhich the spin at site nresponds, see illustration Fig. 1,\nas for example, in magnon{phonon mediated loss [65].\nUnlike most system+environment Hamiltonians which\nassume one-dimensional coupling, we here take the spin-\nreservoir interaction to be of the three-dimensional form\n^Vint=\u0000\rX\nn^S(n)\u0001Z1\n0d!C(n)\n!^X(n)\n!: (5)This coupling allows angular momentum transfer as well\nas energy transfer between the spins and the environ-\nment. HereC(n)\n!is a three-dimensional coupling tensor\nand a function of frequency !. At each!, the coupling\ntensor determines the strength of the coupling of each\nspin to its reservoir oscillators at frequency !, thus act-\ning as a frequency \flter. As we shall see, the choice of\nthe couplingC(n)\n!will determine the damping of the spin\ndynamics as well as the stochastic noise experienced by\nthe spins.\nFor readers concerned about time-reversal symmetry\nof^Vintin (5), we note that ^X(n)\n!should be interpreted\nas an e\u000bective magnetic \feld seen by the spins due to\ntheir interaction with the environment, which has the\nsame time symmetry as ^S(n)[112]. Indeed, the the-\nory of magnetic materials developed here on the basis\nof the system+environment Hamiltonian ^His analo-\ngous to macroscopic QED, an e\u000bective medium theory\nwhich successfully describes quantum electromagnetism\nin dielectric materials [64, 66, 67]. Instead of trying to\ngive a fully microscopic description that accounts for\nevery light-matter interaction in the material, macro-\nscopic QED characterizes electromagnetic materials in\nterms of two measured frequency dependent suscepti-\nbilities. The quantum Hamiltonian is then written in\nterms of these susceptibilities, and can be used in ap-\nplications from predicting the Lamb shift to the Casimir\ne\u000bect [66, 68].\nB. Equations of motion\nHaving set up the full Hamiltonian (2) of the spins\nand environment degrees of freedom allows the study of\nthe spins' reduced state dynamics ^\u001aS(t) = trR[^\u001aSR(t)]\nof the total state\n^\u001aSR(t) =ei^Ht=~(^\u001aS(0)\n^\u001aR)e\u0000i^Ht=~; (6)\nwhere the reservoir state ^\u001aR=e\u0000\f^HR=tr[e\u0000\f^HR]is\nthermal at some inverse temperature \f= 1=kBT. In\nwhat follows it will be more convenient to work instead\nin the Heisenberg picture where the state is station-\nary,^\u001aSR(0), while the time dependence of an opera-\ntor^O(t)is governed by the commutator d^O(t)=dt=\n(i=~)[^H;^O(t)]. Expectation values at time tcan then\nbe obtained as\ntr[^\u001aSR(t)^O(0)] = tr[^\u001aSR(0)^O(t)]: (7)\nFIG. 1. Illustration of Hamiltonian model. In addition to\nprecessing in an external \feld Bext, and coupling to its spin\nneighboursmwith strengthJ(nm), each spin ^S(n)couples\nto its environmental mode (phonons, electrons) at frequency\n!with a coupling function C(n)\n!. All environmental modes\nare assumed to be thermal at the same temperature T.3\nUsing the standard commutation relations for the spin\noperators (and orbital angular momentum operators\nin general), [^S(n)\nj;^S(m)\nk] = i ~\u000emnP\nk\u000fjkl^S(n)\nl, and\nfor the position/momentum operators, [^X(n)\n!;j;^\u0005(m)\n!0;k] =\ni~\u000enm\u000ejk\u000e(!\u0000!0), we obtain the following equations\nof motion for the spin operators ^S(n)(t)\nd^S(n)\ndt=^S(n)\u00022\n4\r\u0010\nBext+^B(n)\nenv\u0011\n+X\nm6=n\u0016J(nm)^S(m)3\n5;\n(8)\nwhere \u0016J(nm)= (1=2)[J(nm)+ (J(mn))T]is the sym-\nmetrized exchange tensor and ^B(n)\nenv=R1\n0d!C(n)\n!^X(n)\n!\nis a magnetic \feld operator generated by the reservoir\noscillator positions ^X(n)\n!at siten.\nIn turn, the equations of motion for these operators\nare\nd2^X(n)\n!\ndt2+!2^X(n)\n!=\rC(n) T\n!^S(n); (9)\ni.e. the reservoir oscillators are driven by the motion\nof the spins, with the (transposed) coupling tensors\nC(n) T\n! governing the degree of driving for each of the\ncontinuum of oscillators. We assume retarded bound-\nary conditions so that the reservoir responds only to\nthe past behaviour of the spins. The retarded Green\nfunction,G!(t\u0000t0) = \u0002(t\u0000t0) sin(!(t\u0000t0))=!obeys\n(@2\nt+!2)G!=\u000e(t\u0000t0), and Eq. (9) can then be solved\nexactly by\n^X(n)\n!(t) =r\n~\n2!\u0010\n^a(n)\n!e\u0000i!t+^a(n)y\n!e+i!t\u0011\n+\rZ1\nt0dt0G!(t\u0000t0)C(n) T\n!^S(n)(t0):(10)\nHere, ^a(n)\n!and^a(n)y\n! are (vectors of) bosonic ladder op-\nerators with their components obeying [^a(n)\n!;j;^a(m)y\n!0;k] =\n\u000enm\u000ejk\u000e(!\u0000!0). Classically these correspond to the\ntwo integration constants for the di\u000berential equation\n(9) which set the initial amplitude and velocity of the\noscillator.\nSubstituting the reservoir solutions (10) into the\nequations of motion for the spins (8), we obtain the \frst\nresult: The Heisenberg{Langevin equation that governs\nthree-dimensional quantum spin dynamics under the in-\n\ruence of memory and coloured quantum noise is\nd^S(n)(t)\ndt=^S(n)(t)\u0002\u0014\n\rBext+X\nm6=n\u0016J(nm)^S(m)(t)\n+\r^b(n)(t) +\r2Zt\nt0dt0K(n)(t\u0000t0)^S(n)(t0)\u0015\n:(11)\nThe term ^b(n)(t)is a Hermitian magnetic noise operator\nfor siten,\n^b(n)(t) =Z1\n0d!r\n~\n2!C(n)\n!\u0012\n^a(n)\n!e\u0000i!t+h:c:\u0013\n;(12)which plays the role of the stochastic noise \frst de-\nscribed by Brown [48]. Here it arises from the spin's\ninteraction with its reservoir. As we will see below, the\nbath noise can be coloured and contain quantum zero-\npoint \ructuations. In addition to the coloured noise\n^b(n), a kernel tensorK(n)(t\u0000t0)appears in Eq. (11),\nwhich captures the damping of the spins. It arises from\nthe coupling tensor C!and is given by\nK(n)(\u001c) = \u0002(\u001c)Z1\n0d!C(n)\n!C(n) T\n!\n!sin (!\u001c);(13)\nwhere\u001c=t\u0000t0. Here \u0002is the Heaviside function\nwhich makes the spin's dynamics at time t, see (11), a\nfunction of the spin's state at previous times t0\u0014t0<t\n[113].\nThe three-dimensional spin dynamics equation (11)\ndescribes the evolution of spin operators and ex-\nplicitly includes memory of the past dynamics (non-\nMarkovianity). This contrasts with previous derivations\nof quantum spin dynamics in the form of a master equa-\ntion [69] which can be solved numerically. But to obtain\nthe master equation a range of simplifying assumptions\nwhere made, including as weak spin-environment cou-\npling, as well as the Markov and secular approximations.\nThese approximations are quite strong and may not al-\nways be justi\fed for a given spin system. We further\nremark that a di\u000berent method of including (classical)\ncoloured noise is based on the Miyazaki-Seki approach\n[70]. Similar to (8), this model assumes that the equa-\ntion of motion of the spins is coupled to a stochastic\nequation of motion for the lattice enabling transfer of\nenergy and angular momentum between the lattice and\nthe spin systems. Di\u000berent coupling potentials, such as\nharmonic and Morse potentials, have been considered\nand the spin-lattice coupling impact on the magnetisa-\ntion has been characterised [71].\nThe above spin+environment Hamiltonian and its\nresulting equations of motion share many similarities\nwith the well-known Caldeira-Leggett model for har-\nmonic quantum Brownian motion [26, 35] and the spin-\nboson model [33, 37, 39]. A key di\u000berence is the three-\ndimensional nature of the reservoir interaction in (5)\nwhich leads to the cross product in (11). The spin-boson\nmodel is recovered as a special case for spin 1=2oper-\nators and rank-1 coupling tensors, see Appendix A1.\nC. Fluctuation-Dissipation Relation and ^S2\nThe presence of the kernel in (11) gives rise to spin\ndamping that can signi\fcantly di\u000ber from Gilbert damp-\ning, as explored further in section II. Previous generali-\nsations to Gilbert damping have considered the spins' in-\nteractions with the lattice and electron motion [54, 71{\n73], and included inertial terms [49] and further memory\nterms within a damping kernel [55{58]. First measure-\nments have recently con\frmed the presence of inertial\ncorrections [50, 59]. But these studies have not con-\nsidered how the noise may have to change from the\nstandard white magnetic noise that is included in the\ne\u000bective magnetic \feld in the LLG equation (1).\nA separate strand of reasoning has argued that to ex-\ntend the LLG equation to the quantum setting requires\nthat the stochastic noise included in the e\u000bective \feld4\nBe\u000bshould have a quantum distribution rather than\nthe classical Boltzmann one [60, 74{77]. These studies\npartially reproduce experimentally measured magnetiza-\ntion over temperature curves. But the noise considered\nvanishes at low temperatures, whereas the full Bose-\nEinstein distribution famously has non-zero quantum\n\ructuations even at T = 0 K. Other pioneering inves-\ntigations have discussed the e\u000bect of coloured noise and\nnon-Markovian e\u000bects on the ultrafast demagnetization\nrate [70], while not requiring the \ructuation-dissipation\ntheorem (FDT) to hold and not including quantum ef-\nfects.\nThe quantum spin dynamics equation (11) brings\nthese pieces, the damping kernel, quantum statistics and\ncoloured noise, together in a quantum thermodynami-\ncally consistent framework. To see this we now prove\nthat the reservoir's two signatures - the stochastic \feld\n^b(n)and memory kernel K(n)- always ful\fl the (quan-\ntum) \ructuation-dissipation theorem, for any choice of\ncoupling tensorC(n)\n!.\nThe reservoir power spectrum ~P(!)is de\fned as the\n(quantum symmetrised [114]) expectation value of the\nautocorrelation function of the magnetic noise ^b(n)(t)\nin the thermal reservoir state ^\u001aR=e\u0000\f^HR=tr[e\u0000\f^HR],\nand then taking the Fourier transform from the time to\nthe frequency domain [115], i.e.\n~P(nm)\njk(!) =Z1\n\u00001d\u001cei!\u001cDn\n^b(n)\nj(t);^b(m)y\nk(t\u0000\u001c)oE\n\f\n2;(14)\nwheref:;:gis the anti-commutator. Inserting (12),\none \fnds that the only non-trivial contributions to\nthis expression come from the bosonic ladder operator\nexpectation values hf^a(n)\n!;l;^a(m)y\n!0;l0gi\f=\u000enm\u000ell0\u000e(!\u0000\n!0) coth(\f~!=2). Meanwhile by Eq. (13) the square of\nthe coupling tensor, C(n)\n!C(n) T\n! , is related to the imagi-\nnary part of the Fourier transform of the damping kernel\nK(n)(\u001c), i.e.\nC(n)\n!C(n) T\n! =2!\n\u0019Im[~K(n)(!)]: (15)\nHence one obtains for allchoices of the interaction ten-\nsorC(n)\n!in Eq. (5) the quantum \ructuation-dissipation\ntheorem (FDT)\n~P(nn)\nqu(!) =~Im[~K(n)(!)] coth~!\n2kBT; (16a)\n~P(nn)\ncl(!) =2kBT\n!Im[~K(n)(!)]forkBT\u001d~!\n2;(16b)\nwhere the last line is the well-known classical high-\ntemperature (or low frequency) approximation, and the\n\frst line is the low-temperature limit where the Bose-\nEinstein distribution of the reservoir oscillator modes\nbecomes important [116].\nFinally, the spin dynamics equation (11) leaves the\nsquare of the spin operators a constant of motion, since\nd\f\f^S(n)\f\f2\ndt=i\r~\n2X\nl[^S(n)\nl;B(n)\ne\u000b;l] = 0; (17)see Appendix A3. Here ^B(n)\ne\u000b=Bext+^B(n)\nenv+\n1\n\rP\nm6=n\u0016J(nm)^S(m)is an e\u000bective magnetic \feld at\nsitensee Eq. (8), which commutes with all compo-\nnents of ^S(n). This con\frms that the spin length j^S\f\fis\na constant in time, e.g. for a spin-1/2 ^S=~\n2^\u001bwith ^\u001bj\nthe Pauli matrices, one has a constant ^S2=~2\n4^\u001b2=\n3~2\n412.\nII. Coupling functions\nThe general quantum spin dynamics equation (11) is\nspeci\fed completely by the coupling tensor C!in (5)\nthat sets the interaction between spins and reservoirs.\nHere we will show that the standard LLG equation (1)\narises as a special case of (11), for a particular choice\nof couplingC!. We will further introduce a class of\nLorentzian coupling functions which allow a systematic\nexploration of spin dynamics behaviours beyond the LLG\nequation.\nFor simplicity from here on we will drop the lattice\nsite superscripts and consider isotropic coupling tensors\nC!=C! 13withC!a scalar function, and similarly for\nthe corresponding kernels K(\u001c) =K!(\u001c) 13and power\nspectra ~P(!) =~P(!) 13. These choices are appropriate\nfor 3D materials in which all spins couple to the environ-\nment in the same manner in all spatial directions. For\nother materials, such as 2D layers, non-isotropic cou-\npling tensors can be considered.\nA. Ohmic coupling and recovery of LLG equation\nIn the open quantum systems literature, coupling\nfunctions that are linear in frequency are referred to as\n\\Ohmic\", while those proportional to higher and lower\npowers of!are called super- and sub-Ohmic, respec-\ntively. For the magnetic system considered here, Ohmic\ncoupling means\nCOhm\n!=r\n2\u0011Gcos(!\u000f+)\n\u0019!; (18)\nwhere\u0011Gis a positive constant with units of kg =A2m2s,\nas in Eq. (1), and \u000f+is an in\fnitesimal positive constant\nwhich we take to zero at the end of the calculation. The\ncorresponding kernel (13) is close to instantaneous,\nKOhm(\u001c) =\u0011G\n\u0019\u0002(\u001c)Z1\n0d!![sin(!(\u001c+\u000f+))\n+ sin(!(\u001c\u0000\u000f+))]\n=\u0000\u0011G\u0002(\u001c)d\nd\u001c\u000e(\u001c\u0000\u000f+); (19)\nfrom which we can see that a positive value of \u000f+en-\nsures the response kernel KOhmis causal. For Ohmic\ncoupling (18), the damping term in (11) reduces to\n\r2^S(t)\u0002Zt\nt0dt0KOhm(t\u0000t0)^S(t0)\n=\u0000\r2\u0011G^S(t)\u0002d^S(t\u0000\u000f+)\ndt;(20)\ni.e. Ohmic coupling results in a damping term propor-\ntional to a \frst order time derivative of the system vari-\nable. In the magnetic case this is ^S, while for quantum5\nBrownian motion it would be the oscillator position [35].\nInserting (20) into the general quantum spin dynam-\nics equation (11), we see that it recovers (the quan-\ntum version of) the LLG equation (1) with \u0011Gbeing\nidenti\fed as the damping parameter. This becomes the\nwell{known classical equation in the limit of large spins.\nConsequently we will also refer to the Ohmic coupling\nfunction (18) as \\LLG coupling\" [117].\nGiven a coupling function it is straightforward to \fnd\nthe corresponding power spectrum (16a) governing the\ntime correlations of the quantum noise. For COhm\n! the\nquantum power spectrum is\n~POhm\nqu(!) =\u0011G~!coth\u0012~!\n2kBT\u0013\n; (21)\nwhere we have taken the limit \u000f+!0. The Ohmic\nquantum power spectrum (21) is proportional to the\ndamping parameter \u0011G, and tends to a linear (diverg-\ning) function of frequency in the low temperature limit\n~!\u001dkBT, see Fig. 2g+h. However, power spectra\ndiverging with frequency are unphysical. Coupling any\nsystem to environmental modes must go to zero at large\nenough frequencies, such as the interaction of spins with\nlattice phonons or the e\u000bect of conduction electrons\nscattering o\u000b the spins. A common way of integrat-\ning this physical information into Ohmic coupling is to\nintroduce a cut-o\u000b, i.e. an upper frequency to which the\ncoupling grows linearly, and after which it is set to 0 or\ndecays to 0 in either algebraic or exponential form [38].\nA di\u000berent approach was taken in [60, 74] for the mod-\nelling of spin dynamics, where a semi{quantum Ohmic\npower spectrum similar to (21) was chosen, but with the\nzero{point noise subtracted ensuring that at T= 0K it\nvanishes for all frequencies.\nIn the high temperature limit kBT\u001d~!, Eq. (21)\nreduces to the classical power spectrum\n~POhm\ncl(!) = 2\u0011GkBT; (22)\nwhich is frequency independent (white) noise. Thus\nOhmic coupling plus the high temperature approxima-\ntion to the power spectrum recovers the LLG equation\n(1) commonly used to simulate magnetic materials over\na wide range of temperatures.\nAs a \frst step to unravel how the dynamics predicted\nby the quantum spin equation (11) can deviate from\nthat predicted by the LLG equation (1), one can expand\nthe spin vector (operator) at time t0in Eq. (11) around\nthe end point of integration to arbitrary order,\n\r2^S(t)\u0002Zt\nt0dt0K(t\u0000t0)^S(t0)\n=\r21X\nm=1\u0014m^S(t)\u0002@m\nt^S(t):(23)\nThe0-th order coe\u000ecient, \u00140does not appear in the ex-\npansion of the damping operator (23), as ^S(t)\u0002^S(t) =\n0. For the quantum operators ^S(t)this is ensured by\nthe angular momentum commutation relations, which\nmake this cross product anti{Hermitian. As outlined in\nAppendix A2, when using any truncated form of ex-\npansion (23) one must employ an explicitly Hermitianform of Eq. (11), and hence any anti-Hermitian con-\ntributions drop out. The higher expansion coe\u000ecients\n\u0014m>0are proportional to the mthone{sided moment of\nthe memory kernel\n\u0014m=(\u00001)m\nm!Z1\n0d\u001c\u001cmK(\u001c); (24)\nwhere we have assumed that the dynamics has been\nrunning for some time longer than the kernel decay time,\nfor which one can replace the initial time t0by\u00001.\nFor the Ohmic kernel only the \frst moment is non-\nzero and corresponds to the (negative) damping param-\neter,\n\u0014Ohm\n1=\u0011GZ1\n0d\u001c\u001cd\nd\u001c\u000e(\u001c\u0000\u000f+) =\u0000\u0011G;(25)\nwhile\u0014Ohm\nm>1= 0 . This complete lack of higher mo-\nments, which would maintain a certain degree of mem-\nory in the dynamics, shows that Ohmic coupling dynam-\nics can only be an approximation to any real dynamics.\nFor example within magnetism, the memory-free form\nof damping (20) is known as Gilbert damping [43], and\nis almost universally used to describe magnetization dy-\nnamics through the LLG equation (1). However, \\iner-\ntial\" corrections to such dynamics have been proposed\n[49] and their presence was recently con\frmed experi-\nmentally [50].\nB. Lorentzian coupling\nHere we provide a tool to systematically study dy-\nnamics beyond the Ohmic case, allowing one to include\nmemory and coloured noise e\u000bects in a manner consis-\ntent with the quantum \ructuation dissipation theorem\n(16a). We consider the class of Lorentzian coupling\nfunctions\nCLor\n!=s\n2A\u0000\n\u0019!2\n(!2\n0\u0000!2)2+!2\u00002; (26)\nwhereAis a coupling amplitude, with the following\nproperties: i) for small !,CLor\n!grows linearly with !\nand can be approximated by an Ohmic coupling func-\ntion, ii) at large !,CLor\n!smoothly decays to zero, and\niii) at some intermediate frequency !0,CLor\n!has a reso-\nnant peak with some width \u0000. This peak characterises\nthe con\fned range and relative strength of system-\nenvironment interaction with two parameters. Alter-\nnative \\peaks\" such as Gaussians or top hat functions\ncould be considered, but here we chose the Lorentzian\nshape due to the fact that many expressions can be\nsolved analytically and, as we will demonstrate in section\nIII, Lorentzian couplings allow us to e\u000eciently simulate\nnon-Markovian dynamics.\nWe call the above functions \\Lorentzian coupling\"\nsince the corresponding damping kernel in the frequency\ndomain is the widely studied Lorentzian response\nKLor(!) =A\n!2\n0\u0000!2\u0000i!\u0000; (27)\nwhere the imaginary part is obtained from (15) and the\nreal part is determined using the usual Kramers{Kronig\nrelations. In the time-domain the Lorentzian memory\nkernel is\nKLor(\u001c) = \u0002(\u001c)Ae\u0000\u0000\u001c\n2sin(!1\u001c)\n!1; (28)6\nwhere!1=q\n!2\n0\u0000\u00002\n4and\u0000=2can now be interpreted\nas the kernel decay rate. For the coupling function\n(26), the collective response of the environment is thus\nequivalent to a single harmonic oscillator of resonant\nfrequency!0and damping rate \u0000=2[78]. From the\nquantum FDT (16a) it follows that the corresponding\npower spectrum is\n~PLor\nqu(!) =A\u0000~!\n(!2\n0\u0000!2)2+!2\u00002coth\u0012~!\n2kBT\u0013\n;(29)\nwhich takes its largest values at frequencies close to\n!0and tends to zero as !\u00003at large!. This power\nspectrum di\u000bers from classical Ohmic noise (22) in two\nimportant respects. Firstly, the quantum mechanical\ntreatment means that the low temperature noise is not\nproportional to temperature, and does not vanish at\nzero temperature. Secondly, even in the high temper-\nature limit, the noise spectrum is frequency dependent\n(coloured), unlike the white noise of Eq. (22). The pre-\nsented theory thus captures both of these aspects in a\nconsistent quantum thermodynamic framework.\nUnlike Ohmic coupling (18) which depends on a single\nparameter\u0011G, the Lorentzian coupling function (26),\nand hence its kernel and spectrum, depends on three\nparameters. These allow a systematic study of di\u000berent\nregimes of the environment response. Speci\fcally, the\nmemory time of the environment can be continuously\nvaried by changing !0and\u0000, which can lead to very\ndi\u000berent magnetic behaviour. Beyond the spin dynam-\nics explored here, the proposed Lorentzian coupling may\nalso be a useful tool for the characterisation of quan-\ntum Brownian motion of a variety of systems, including\noscillators and free particles.\nC. Two coupling regimes\nTo better understand the relation between the Ohmic\nand Lorentzian coupling functions, one may consider\ntheir kernel moments \u0014min expansion (23). In con-\ntrast to the Ohmic case, for the Lorentzian kernel (28)\nall\u0014mare non-zero and given by\n\u0014Lor\nm=(\u00001)mA\n!1!2(m+1)\n0Im\"\u0012\u0000\n2+ i!1\u0013m+1#\n:(30)\nThe \frst relevant two moments are\n\u0014Lor\n1=\u0000A\u0000\n!4\n0and\u0014Lor\n2=A\n!6\n0(\u00002\u0000!2\n0);(31)\nand when comparing to the Ohmic case, one \fnds that\nthe \frst Lorentzian moment can be identi\fed with minus\nthe damping parameter \u0011G, i.e.\u0014Lor\n1=\u0000A\u0000\n!4\n0=\u0000\u0011G.\nFor a material with a given damping parameter \u0011Gthis\n\fxes one of the Lorentzian parameters, i.e.\nCLor\n!=s\n2\u0011G!2\n\u0019!4\n0\n(!2\n0\u0000!2)2+!2\u00002; (32)\nwhich now only depends on the two parameters !0and\n\u0000. For a speci\fc material these may be approximately\ndetermined through information contained in the den-\nsity of states of the environment to which the spins\ncouple [79].Inserting expansion (23) with Lorentzian moments\n(30) in the quantum spin dynamics equation (11) one\ncan distinguish two di\u000berent dynamical situations.\nOhmic regime: When the resonant frequency !0and\nthe damping rate \u0000of the reservoir coupling is much\nlarger than the spin operators' typical frequency of mo-\ntion,!0\u001d!, each successive term in expansion (23) is\nsmaller by an extra factor of (!=! 0). In the limit of in\f-\nnite!0but \fnite damping parameter \u0011G, the Lorentzian\ndamping term in (11) thus tends to the Ohmic one (20),\n\u0000\r2\u0011G^S(t)\u0002@t^S(t).\nNon-Ohmic regime: When the environmental fre-\nquency!0is comparable to typical spin motion frequen-\ncies,!0\u0019!the Ohmic approximation to the Lorentzian\nkernel begins to fail. The \frst deviation in (11) is a new\nterm proportional to \u0014Lor\n2. I.e. in addition to the Gilbert\ndamping term, one adds the term \r2\u0014Lor\n2^S(t)\u0002@2\nt^S(t)\ncontaining a second time derivative of the spin operator\n^S. By analogy with the classical equation of motion for\na massive body, this is known as an \\inertial\" modi\fca-\ntion to the spin dynamics [49]. As the ratio \u00142=\u00141has\nthe dimensions of time, one may introduce an \\inertial\ntimescale\"\u001cin[49], which for the Lorentzian is\n\u001cin=\u0014Lor\n2\n\u0014Lor\n1=!2\n0\u0000\u00002\n!2\n0\u0000: (33)\nA large inertial time \u001cinindicates the presence of non-\nMarkovian dynamics, i.e. dynamics that has a certain\ndegree of memory. For a high quality factor resonance\n!0\u001d\u0000the inertial time is (half) the kernel's decay\ntime,\u001cd= 2=\u0000. For increasing resonance width \u0000, the\ninertial time \u001cindecreases and memory e\u000bects become\nless important. In magnetic systems this timescale de-\ntermines the time over which nutation oscillations are\nobserved in the precession of the spin. Such inertial cor-\nrections to standard magnetism have very recently been\nobserved for the \frst time in ultrafast experiments on\nthin \flms [50]. Curiously, the inertial timescale becomes\nnegative when \u0000>! 0, a fact that may be the subject\nof future investigation.\nSimilarly to the kernel expansion (23), one can also\nexpand the Lorentzian power spectrum in frequency, see\nAppendix A4. Generally, only the odd moments \u00142m+1\nappear in the power spectrum. This implies that if a ker-\nnel only has non-trivial \frst ( \u00141) and second ( \u00142) mo-\nments, while higher moments vanish ( \u0014m>2= 0) then\nthe power spectrum will still be given by the (quantum)\nOhmic one (21), with \u0011G=\u0000\u00141. When third or higher\nmoments are non-zero, then the power spectrum of the\nnoise will deviate from the Ohmic case at all tempera-\ntures.\nTo summarise, here we have demonstrated that\nLorentzian coupling functions, kernels and power spec-\ntra provide a systematic framework to explore sys-\ntem dynamics that arises from inertial terms and other\nmemory e\u000bects, while recovering the standard Ohmic\nlimit whenever the Lorentzian resonance frequency !0\nis much larger than the typical system frequencies.\nD. Unit-free variables and Lorentzian parameters\nIn section III we perform semi{classical simulations\nof the dynamics of Eq. (11), for the Lorentzian kernel\nof Sec. II B. For this purpose we re{write expressions in7\n01 4 7 ω/ωL00.511.5cωa)\nLorentzian with\nω0=7ωL\nt/prime= t - 15ω−1\nLt00.30.6k(t−t/prime)c)\n01 4 7 ω/ωL03.06.0˜p(ω)\ne)\n0.6LLG+cl.\nLLG+qu.\nLor+qu. \n01 4 7 ω/ωL00.150.3˜p(ω)\ng)\n0.003\n01 4 7 ω/ωL00.511.5cωb)\nLorentzian with\nω0=1.4ωL\nt/prime= t - 15 ω−1\nLt00.30.6k(t−t/prime)d)\n01 4 7 ω/ωL03.06.0˜p(ω)\nf)T=200 K\n0.6LLG+cl.\nLLG+qu.\nLor+qu.\n01 4 7 ω/ωL00.150.3˜p(ω)\nh)T=1 K\n0.003\nFIG. 2. Comparison of coupling functions, memory kernels and power spectra: Top panels show (a) coupling function c!,\n(c) time dependent damping kernel k(t\u0000t0), and magnetic noise power spectrum ~p(!)at (e)T= 200 K, and (g)T= 1 K, for\nLorentzian coupling with parameter Set 1 (34a) (solid blue). Typical spin dynamics frequencies !2[0;2:5!L]are highlighted\n(yellow shading) in the frequency domain. Bottom panels shows the same quantities for Lorentzian coupling with parameter\nSet 2 (34b) (solid red). In (a{b) the Lorentzian coupling functions are compared to the LLG (Ohmic) approximation (magenta\ndash-dotted). In (e{h) the Lorentzian power spectrum is compared to the quantum LLG approximation (cyan squares), and\nits high temperature white noise limit (magenta dash-dotted).\nthe operator equation (11) in terms of a unit-free set of\nquantities. The time coordinate tis replaced with the\nunit-free coordinate !Lt, where!L=j\rjjBextjis the\nLarmor frequency. In addition the spin operator ^Swith\nlargest eigenvalue S0is re{written in terms of a unit-free\noperator ^swith largest eigenvalue 1,^S= sign(\r)S0^s.\nThe sign of the gyromagnetic ratio is included in the\nde\fnition of ^sso that ^saligns with the magnetic \feld,\nwhatever the sign of \r.\nFrom Eq. (11) we can see that the damping ker-\nnel has dimensions of magnetic \feld squared divided\nby angular momentum, which leads us to identify a\nunit-free damping kernel k(t\u0000t0)throughK(t\u0000t0) =\njBextj2S\u00001\n0k(t\u0000t0). Similarly, the unit-free coupling\nfunctionc!is de\fned through C!=jBextjS\u00001=2\n0c!\nand the unit-free spectral functions ~pthrough ~P=\n~jBextj2S\u00001\n0!\u00001\nL~p.\nLooking at the Lorentzian kernel K(t\u0000t0)in (27),\nthe pulling of dimensions can be achieved by setting\nthe kernel amplitude to A=jBextj2S\u00001\n0\u000bwhere\u000b\nnow is a frequency. For the simulations we choose the\nLorentzian parameters !0;\u0000and\u000bto be independent\nof spin length S0. Through (5) this implies an inter-\naction energy scaling of ^Vint/S0p\nA/pS0which\nsets the scaling of the interaction versus self-energy to\n^Vint=^HS/1=pS0. Apart from it being implied by\ndimensional analysis, such scaling is heuristically plau-\nsible in many physical contexts. E.g. it is similar to\nthe increasing ratio of the surface (where reservoir in-\nteraction occurs) to volume (self-energy) for decreasing\nsystem sizes. For microscopic systems for which ^Vintis\nno longer small in comparison to ^HSa thermodynamic\ntreatment beyond the weak coupling limit [26, 27, 81], a\nlimit tacitly assumed in standard thermodynamics, may\nbe required.\nSimilarly for Ohmic coupling leading to the LLG equa-tion (1), the above scaling choice amounts to choos-\ning\u0011G/A/1=S0implying that \u0011=\u0011G\r2S0=\n\u000b\u0000!2\nL=!4\n0is assumed to be independent of S0. In phys-\nical situations where this assumption is not justi\fed,\none may instead choose \u0011and\u000bto depend on the spin\nlengthS0.\nFor a spin in an external \feld Bextthe typical fre-\nquency of the dynamics is set by the Larmor frequency,\n!L. In the simulations discussed in section III we will\nuse the following two sets of Lorentzian parameters, all\nexpressed in terms of !L,\nSet 1):!0= 7:0!L\u000b= 10:0!L\u0000 = 5:0!L\n\u0011= 0:02\u001cin= 0:1!\u00001\nL\u001cd= 0:4!\u00001\nL(34a)\nSet 2):!0= 1:4!L\u000b= 0:16!L\u0000 = 0:5!L\n\u0011= 0:02\u001cin= 1:7!\u00001\nL\u001cd= 4!\u00001\nL:(34b)\nIn the second rows we have also listed the equivalent\nunit-free Gilbert damping \u0011, the inertial timescale \u001cin,\nand the memory kernel decay time \u001cdfor the Lorentzian\nkernel. Figs. 2-5, show plots obtained with Lorentzian\ncoupling functions (26) with parameter Sets 1 and 2,\nwhich are shown in blue and red, respectively. The\nOhmic LLG approximation is shown in magenta when\nthe classical reservoir (22) is considered, and in cyan\nwhen the quantum reservoir (21) is considered. The\nexternal \feld is set to Bext= 10 Tezwithezthe unit-\nvector inz-direction throughout.\nParameter Set 1 has been chosen to have a reso-\nnant frequency !0much larger than the characteristic\nspin precession frequency !L(Ohmic regime). Conse-\nquently we can truncate the series given in Eq. (23) to\nleading order and recover the Ohmic form (20) typically\nconsidered in magnetism theory. The validity of this ap-\nproximation is demonstrated in the top row of Fig. 2.\nFig. 2a) shows that the Lorentzian coupling function cLor\n!\nis well approximated by LLG (Ohmic) coupling for the8\nrelevant frequency range, while Fig. 2c) shows that the\nkernel is approximately instantaneous on the timescale\n!\u00001\nL, in line with Ohmic damping for which \u0014Ohm\nm>1= 0.\nFig. 2e) shows that at high temperature ( T= 200 K)\nand for relevant frequencies !\u0019!L, the power spec-\ntrum ~pLoris well approximated by the quantum Ohmic\n(LLG) power spectrum (21), and its classical limit (22).\nFig. 2g) shows that at lower temperatures ( T= 1 K)\nquantum noise becomes important. Here the Ohmic ap-\nproximation (21) remains valid, while its classical limit\n(22) is invalid.\nParameter Set 2 is chosen such that it has a resonant\nfrequency!0comparable to the precession frequency !L\n(non-Ohmic regime). Here it is inaccurate to truncate\nthe series (23) and the damping will be fundamentally\nnon{Ohmic. To directly compare with Ohmic dynam-\nics generated by Lorentzian coupling with Set 1, both\nparameter sets have been chosen to correspond to the\nsame unit-free Gilbert damping parameter, \u0011= 0:02.\nThe failure of the Ohmic approximation is demonstrated\nin the bottom row of Fig. 2. Fig. 2b) shows that the\nlinear approximation to the coupling function fails in\nthe relevant frequency range. For this set of parame-\nters the damping kernel now exhibits signi\fcant mem-\nory and Fig. 2d) shows that the response persists over\na timescale of several !\u00001\nL. This memory kernel im-\nplies, through the FDT (16), a coloured quantum noise\npower spectrum ~pLor\nqu. As shown in Fig. 2h), this coloured\nLorentzian ~pLor\nqu(red) di\u000bers from the LLG counterpart,\n~pOhm\nqu (cyan). Furthermore, Fig. 2f) shows that in the\nhigh temperatures ~pLor\nqu(red) also di\u000bers very signi\f-\ncantly from the LLG power spectrum ~pOhm\ncl (magenta).\nThe presence of both memory and coloured quantum\nnoise for the Lorentzian with parameter Set 2 are both\nsignatures of a thermostat that substantially deviates\nfrom the classical Ohmic assumptions and, as we will\nsee in the next section, leads to markedly di\u000berent short\ntime dynamics and steady state of sz.\nIII. Semi-classical spin dynamics simula-\ntions\nThe general spin dynamics equation (11) is an oper-\nator equation for quantum spins in a lattice, each inter-\nacting with neighbouring spins and with a bosonic reser-\nvoir. Solving the quantum dynamics using, for example,\nLorentzian coupling, kernel and spectrum, is rather di\u000e-\ncult without approximations, even numerically, and such\nexploration is left for future work.\nTo make progress here, we will numerically solve\nthe full non-Markovian for a semi-classical version of\nEq. (11), while including coloured quantum noise and\nmemory e\u000bects arising from the coupling to the envi-\nronment. It replaces the quantum spin operator ^Swith\na classical spin vector S, and the quantum stochastic\nnoise \feld vector ^bwith a stochastic classical \feld vec-\ntorbwith statistics that obey the quantum \ructuation{\ndissipation theorem (16a). This semi{classical approach\nis currently used in the theory of molecular and ionic\ndynamics [82, 83], and was perhaps \frst applied by\nKoch [84] to include the e\u000bects of quantum \ructuations\nin Josephson junctions. It has been justi\fed through an\nexpansion of a forward{backward path integral [85, 86](note the remark of Caldeira and Leggett on pg. 589\nof [35]), and is valid when the potential energy can be\nexpanded in the path integral to \frst order in the devi-\nations from the average path. The validity of applying\nthis approach to the decay of metastable states was in-\nvestigated in detail in [87].\nHere we simulate a single spin allowing us to illustrate\nthe impact of memory e\u000bects and the reservoir's quan-\ntum statistics on the spin dynamics and steady state.\nThe simulation details presented below can readily be\nextended to multiple interacting spins and could be in-\ntegrated in sophisticated atomistic spin dynamics simu-\nlations such as those used in [47, 60].\nA. How to simulate coloured noise and memory\nkernel\nHere we detail how to e\u000eciently simulate non-\nMarkovian dynamics that arises as a result of Lorentzian\ncoupling (26) for the example of spins vectors. Numer-\nical implementation of (11) requires both - the integra-\ntion of the kernel with the spin state of previous time\nsteps and the inclusion of coloured noise as follows.\nDropping the spin index and for simplicity assuming\nany isotropic kernel K(\u001c) =13K(\u001c), the three vector\ncomponents bj(t)forj= 1;2;3of the magnetic noise\n(12) are implemented as [88]\nbj(t) =Z1\n\u00001dt0F(t\u0000t0)\u0018j(t0); (35)\nwhere\u0018j(t0)is standard white Gaussian noise for the j-\nth component, which is delta correlated h\u0018j(t)\u0018k(t0)i=\n\u000ejk\u000e(t\u0000t0). The \\coloured noise\" comes from choosing\nF(t\u0000t0)as the Fourier transform of the square root of\nthe power spectrum associated with the kernel through\n(16), i.e.\nF(t\u0000t0) =Z1\n\u00001d!\n2\u0019e\u0000i!(t\u0000t0)q\n~P(!); (36)\nwhich can be implemented using a fast Fourier trans-\nform. To simulate the e\u000bect of a Lorentzian damping\nkernel (27) we numerically integrate [89] the following\nset of \frst order coupled di\u000berential equations for the\nspin vectorSand two dummy vectors VandW:\ndS(t)\ndt=\rS(t)\u0002(Bext+b(t) +V(t));\ndV(t)\ndt=W(t); (37)\ndW(t)\ndt=\u0000!2\n0V(t)\u0000\u0000W(t) +A\rS(t):\nThe integrated values of the dummy vectors and the\nspin are separated by the time step dt. Solving these\nequations is equivalent to solving the integro{di\u000berential\nequation (11) for a Lorentzian kernel, see Appendix A5,\nbut is numerically more straightforward to implement.\nB. Single trajectories for di\u000berent couplings and\nnoises.\nWe wish to illustrate on a single trajectory level, the\ndi\u000berences between the dynamics predicted by (11) with\neither an approximately Ohmic (Set 1) or non-Ohmic\n(Set 2) Lorentzian coupling function, as well as the dy-\nnamics predicted by the standard LLG equation. At9\n0 6π 12π 18π-1-0.500.51sz,sx,|s|a)T=1 K S=1/2,\n0 6π 12π 18π-1-0.500.51sz,sx,|s|c)T=200 K S=200/2,\nsz for LLG+cl. +0.1\nsz for LLG+qu. +0.05\n|s| for Lor(Set1)+qu.\nsx for Lor(Set1)+qu.\nsz for Lor(Set1)+qu.\n0 6π 12π 18π\nωLt-1-0.500.51sz,sx,|s|b)T=1 K S=1/2,\n0 6π 12π 18π\nωLt-1-0.500.51sz,sx,|s|d)T=200 K S=200/2,\nsz for LLG+cl. +0.1\nsz for LLG+qu. +0.05\n|s| for Lor(Set2)+qu.\nsx for Lor(Set2)+qu.\nsz for Lor(Set2)+qu.\nFIG. 3. Sample of stochastic short-time spin dynamics for di\u000berent couplings and noises. Stochastic short-time dynamics of\nsz(blue in top row & red in bottom row), sx(green) andjsj(black dashed) according to Eq. (11) with Lorentzian coupling\nfunctionCLor\n!, for a classical spin initially in state s= (\u00001;0;0). All traces in the four panels are generated from the same\nsample of Gaussian noise, enabling a direct comparison. Shown are the dynamics for Set 1 (top row) and Set 2 (bottom row),\nand two spin+temperature pairs: S0= 1~=2andT= 1 K (left column), and S0= 200 ~=2andT= 200 K (right column).\nAlso shown in all four panels are the sz-dynamics according to the LLG equation Eq. (1) with damping parameter \u0011\u00190:02\nfor two types of noise: the classical \rat white noise power spectrum Eq. (22) (magenta) and the quantum noise power\nspectrum Eq. (21) (cyan). All cyan and magenta plots are o\u000b-set by +0.05 and +0.1, respectively, to avoid overlapping.\nThe external magnetic \feld is set to Bext= (0;0;10T)setting the timescale to !\u00001\nL\u00190:57\u000110\u000012s, and the simulation\ntime interval is dt= 0:15!\u00001\nL.\n\frst, because the dynamics is intrinsically stochastic,\ntrajectories will naturally di\u000ber and cannot readily be\ncompared. However, looking at the noise generation in\nEqs. (35) and (36), one can see that the same white\nnoise\u0018j(t)forj= 1;2;3may be used as a seed to cre-\nate comparable \\stochastic\" noise for di\u000berent power\nspectra ~P(!).\nFig. 3 shows the stochastic short time dynamics of\na single classical spin for two pairs of spin length and\ntemperature, S0= 1~=2atT= 1 K (left panel) for a\nsingle electron, and S0= 200 ~=2atT= 200 K (right\npanel) for a mesoscopic cluster of spins with a combined\nlarger e\u000bective spin.\nThe dynamics is obtained according to Eq. (11)\nfor Lorentzian coupling (26) with S0-scalingA=\njBextj2S\u00001\n0\u000b, for parameter sets Set 1 (top panel, blue)\nand Set 2 (bottom panel, red), and with the quan-\ntum coloured noise given by (29). For comparison we\nalso show the short time dynamics according to the\nLLG equation (1) with S0-scaling\u0011G=A\u0000=!4\n0=\njBextj2S\u00001\n0!\u00002\nL\u0011with the Gilbert damping parameter\n\u0011= 0:02common to both Lorentzian parameter sets,\nsee (34). That implies that the top and bottom LLG\nplots are identical. For the standard LLG equation two\ntypes of noise are considered - high-temperature classi-\ncal noise (magenta) see Eq. (22), and quantum noise\n(cyan) see Eq. (21). Since the same white noise time\nseries is used as a seed for producing the stochastic noise\nfor all traces, we can compare them directly. We willhere focus on sz, the component of s=sign(\r)S=S0\naligned with the external \feld Bext.\nThree features stand out in Fig. 3: i) as expected\nfrom section II, the dynamics generated with Eq. (11)\nwith Lorentzian Set 1 (top, blue) closely matches the dy-\nnamics obtained with the LLG equation (1) with quan-\ntum noise (cyan) for both spin-temperature pairs, ii)\nthe quantum statistics of the reservoir (cyan) at low\ntemperatures (left) introduces di\u000berences to the LLG\ndynamics compared to the LLG dynamics obtained with\nclassical noise (magenta), and iii) memory e\u000bects that\nare present for Lorentzian Set 2 (bottom, red) result in\nsigni\fcantly di\u000berent dynamics from that arising with\nthe memory-free Lorentzian Set 1 (top, blue).\nWe remark that due to the spin/temperature ratio\nbeing the same for the two spin-temperature pairs, the\nLLG equation with classical noise (magenta) integrates\nto exactly the same dynamics in left and right panel, see\nAppendix A8. This scaling relation ceases to be true for\nthe LLG equation with quantum noise (cyan). Another\ndi\u000berence to note is that in Fig. 3a{d) the dynamics for\nLorentzian parameter Set 1 (blue) varies more rapidly\nin time than for Lorentzian parameter Set 2 (red). This\nis due to the high frequency content of Set 1's power\nspectrum, see Fig. 2e+g).\nFinally, the spin component sx(green) and the spin-\nvector lengthjsj(black) are shown for the Lorentzian\ncoupling with Set 1 and Set 2 in the top and bottom\npanels of Fig. 3, respectively. The plots of jsjshow that10\n012π 36π 60π 84π\nωLt00.250.50.751/angbracketleftbig\nsz/angbracketrightbig\n a)\n0.84\n0.28\n0.27\n0.29S0=1/2,\nT=1 K\n012π 36π 60π 84π\nωLt00.250.50.751/angbracketleftbig\nsz/angbracketrightbigb)\n0.84\n0.84\n0.85\n0.85S0=200/2,\nT=200 K\n/angbracketleftbig\nsz/angbracketrightbig\n for LLG+cl./angbracketleftbig\nsz/angbracketrightbig\n for LLG+qu./angbracketleftbig\nsz/angbracketrightbig\n for Lor(Set1)+qu./angbracketleftbig\n|s|/angbracketrightbig\n for Lor(Set1)+qu./angbracketleftbig\nsz/angbracketrightbig\n for Lor(Set2)+qu.\nFIG. 4. Ensemble averaged spin relaxation dynamics. Averaged dynamics hszi, averaged over 500 stochastic traces up to\ntimetmax= 2\u0019\u000248!\u00001\nL. Shown are the dynamics according to Eq. (11) with Lorentzian coupling functions for parameter\nSet 1 (blue) and Set 2 (red). Also shown is the ensemble averaged dynamics according to the LLG equation with damping\nparameter\u0011\u00190:02for two types of noise: classical white noise (magenta) and full quantum noise (cyan). The two panels\nshow two spin+temperature pairs: S0= 1~=2andT= 1 K (left), and S0= 200 ~=2andT= 200 K (right). Note that\nblue, cyan and magenta curves lie on top of each other in b). As discussed in section II D, the plots assume that both Aand\n\u0011Gscale as/S\u00001\n0with spin size S0, making\u000band\u0011the same for the two spin sizes. The external magnetic \feld is set to\nBext= (0;0;10T)and the simulation time interval is dt= 0:15!\u00001\nL.\nthe numerical integration of Eq. (11) indeed leads to a\nconstant spin-vector length jsj= 1, i.e. no renormali-\nsation is required.\nC. Ensemble-averaged hszitrajectories.\nFig. 4 shows the ensemble averaged hsziover time,\naveraged over 500 stochastic trajectories. We now high-\nlight two important features in Fig 4. Firstly, at low\ntemperatures (left) the quantum statistics of the reser-\nvoir (blue, red, cyan) results in a much depleted value of\nhszi, roughly at around 0:28, in comparison to that ob-\ntained with the LLG equation with classical noise (ma-\ngenta), ca 0:85. This indicates that for this particu-\nlar choice of spin length and temperature the quantum\ncharacter of the reservoir strongly a\u000bects the value of\nhszi, as further discussed below, and the classical high-\ntemperature limit taken in (16b) would not be appropri-\nate. For the high temperature T= 200 K + larger spin\npairS0= 200 ~=2(right), the di\u000berence between clas-\nsical and quantum statistics of the reservoir can be ne-\nglected andhszisettles at 0:85independent of whether\nthe spin dynamics integration was done for Eq. (11) with\neither Set 1 (blue) or Set 2 (red), or for Eq. (1) with\neither classical (magenta) or quantum noise (cyan).\nSecondly, for the large spin-temperature pair (right),\nthere is clear evidence of a much quicker relaxation\nto steady state (by a factor of a third) for Lorentzian\nSet 2 (red) compared to the other plots (blue, cyan,\nmagenta). This is a non-Markovian e\u000bect that arises\nbecause the memory kernel for Set 2 has an appreciable\nmemory over time, see Fig. 2d), while the other memory\nkernels are (close to) instantaneous. This quicker equi-\nlibration occurs because the non-Markovian kernel leads\nto a smoother dynamics which in turn is more quickly\nsampled by the dynamical system.\nD. Steady state hszias a function of tempera-\nture.\nFig. 5 shows the average steady state spin value hszi\nas a function of temperature T, found by time-averaging\na single trajectory szover late times, from 0:75tmax to\ntmax= 2\u0019\u00027200!\u00001\nL. There are two key observationsto make in Fig. 5. Firstly, for both the small spin (left)\nand the large spin (right) the steady state hsziobtained\nwith the LLG equation and classical noise (22) matches\nthe standard statistical physics prediction hszistat phys =\ncoth\u0010\nS0!L\nkBT\u0011\n\u0000kBT\nS0!L, see Appendix A6.\nSecondly, for simulations that include the full quan-\ntum noise (cyan, blue, red) in the dynamics of the small\nspin at low temperatures (left), we observe reduced hszi\nvalues in the range 0.2-0.4 at T= 0 K, i.e., well be-\nlow the classical value of 1. This arises because the\npower spectrum ~PLor\nqu, given through the quantum FDT\n(16a), includes quantum \ructuations which remain even\nforT!0K. The steady state curves with quantum\nnoise also show a characteristic \\\rattening\" compared\nto the steep decay of the steady state with tempera-\nture for classical noise (magenta). Qualitatively speak-\ning, this quantum zero point noise, when compared to\nclassical noise, is as if thermal noise is \\on\" even at\n0K. I.e. taking the Larmor frequency as the relevant\nfrequency, and setting ~POhm\nqu(0K) =~POhm\ncl(Tcl)for the\nOhmic coupling, for example, de\fnes a classical temper-\nature ofTcl= 6:7K \\equivalent\" to the quantum zero-\ntemperature case. For S0= 1~=2the classical statisti-\ncal physics steady state value at Tclishszistat phys\u00190:3.\nThis indeed is of comparable size to the hszivalues ob-\ntained with quantum noise at T= 0K. The correspond-\ning steady state value for the large spin ( S0= 200 ~=2)\nishszistat phys\u00190:995\u00191, see Fig. 5c).\nGenerally, for the classical temperature Tcl=~!L\n2kBwhich is \\equivalent\" to the quantum zero-point noise,\none obtainshszistat phys = coth\u00002S0\n~\u0001\n\u0000~\n2S0, which only\ndepends on the spin length S0while being independent\nof the \feld strength jBextj. With increasing S0this\nfunction rises very sharply from \u00190:3to1. For ex-\nample for spin length S0= 5~=2, the quantum zero\ntemperature value is hszistat phys\u00190:8and its decay\nwith increasing temperate is shown in Fig. 6b) in Ap-\npendix A7.\nThe middle panel, Fig. 5b), gives an alternative illus-\ntration of the steady state value for the small spin as a\nfunction of environment temperature T. It shows the11\n01 4 8 12 16 20\nT / K00.250.50.751/angbracketleftbig\nsz/angbracketrightbig\na)\nS0=1/2\nstat. phys.\nLLG+cl.\nLLG+qu.\nLor(Set1)+qu.\nLor(Set2)+qu.\n01 4 8 12 16 20\nT / K00.250.50.751m(T)\nb)\nS0=1/2\n0 800 1600 2400 3200 4000\nT / K00.250.50.751/angbracketleftbig\nsz/angbracketrightbig\nc)\nS0=200/2\nstat phys\nLLG+cl.\nLLG+qu.\nLor(Set1)+qu.\nLor(Set2)+qu.\nFIG. 5. Steady statehsziof a spin interacting with a (quantum) reservoir at temperature T.For a spin in an external\nmagnetic \feld Bext= (0;0;10T)and interacting with a thermal reservoir, the time averaged value of hsziis obtained\nby integrating Eq. (11) for Lorentzian parameter Set 1 (blue crosses) and Set 2 (red dots), as well as by integrating the\nLLG equation with damping parameter \u0011= 0:02for two types of noise: classical white noise (magenta dash-dotted) and\nfull quantum noise (cyan dashed). The spin lengths are S0= 1~=2(panel a and b), and S0= 200 ~=2(panel c). For\nthe small spin in panel a), the three curves with quantum noise (cyan, blue, red) start at hszi-values in the range 0.2-0.4\natT= 0 K. For both S0values, the steady state for the LLG equation with classical noise (magneta) coincide well with\nhszistat phys (black), derived from classical statistical mechanics for a thermal distribution, see Appendix A6. Note that all\ncurves lie on top of each other in panel c). Panel b) shows the same plot as panel a) - but with the y-axis \\rescaled\"\nasm(T) =hsz(T)i=hsz(0)iso that all plots start at 1 at T= 0 K. While the magenta curve remains the same as in\na), the rescaled blue, red and cyan curves now show a \rattened decay behaviour that somewhat resembles the corrected\nmagnetization curves for real materials analysed in [90]. Error bars for the four simulations (blue, red, cyan, magenta) are\nindicated at a) T= 1 K and c)T= 200 K. The simulation time interval is dt= 0:15!\u00001\nL.\nsame plot as panel a), but with the y-axis rescaled as\nm(T) =hsz(T)i=hsz(0)i. All plots now start at 1 at\nT= 0 K, independent of whether the dynamics was in-\ntegrated with quantum or classical noise. Interestingly,\nthe overarching behaviour of the resulting curves bears\nsome resemblance with heuristically rescaled magneti-\nzation curves that match experimental data [90, 91].\nRunning high-end atomistic simulations of Eq. (11),\ninstead of (1), for multiple interacting classical spins\nwould answer if the quantum power spectrum's impact\non their low temperature magnetisation behaviour as\nwell as their Curie temperature is the reason for the\napparent rescaling.\nWe note that larger numerical uncertainties arise for\nthe quantum noise because an additional scale is present\nin comparison to classical noise, see Appendix A8. Er-\nror bars obtained from an ensemble of simulations are\nindicated at one temperature value in a) and c) for all\nfourhszicurves in Fig. 5.\nIV. Conclusions and open questions\nWe have derived a general quantum spin dynam-\nics equation, Eq. (11), capable of describing three-\ndimensional precession and damping. The terms aris-\ning from the reservoir interaction are treated in a quan-\ntum thermodynamically consistent manner, by tracing\nthe origin of both the memory kernel, K(\u001c), and the\nstochastic noise, ^b(t), to a single coupling function,\nC!. Secondly, Lorentzian coupling functions were pro-\nposed and shown to provide a systematic means to in-\nvestigate di\u000berent dynamical regimes - from Ohmic to\nnon-Ohmic dynamics which is subject to memory and\ncoloured noise. We showed that only in the Ohmic\nregime, the standard LLG equation with Gilbert damp-\ning, widely used in magnetism, is recovered. Finally, we\nprovided details of how to include Lorentzian memory\nand coloured noise in numerical simulations of open sys-tem dynamics. For the example of a single spin vector,\nwe illustrated that a non-Ohmic Lorentzian kernel leads\nto a faster equilibration time of hsziin comparison to\nthe Ohmic (LLG) regime. We also discussed the steady\nstate di\u000berences that arise when the full quantum ther-\nmostat with quantum zero-point noise is employed, in\ncontrast to classical white noise.\nThe above three ingredients provide a complete\nframework for the simulation of damped three-\ndimensional precession including memory and coloured\nnoise. It can readily be adapted in atomistic spin dy-\nnamics simulations [47, 60] that solve the dynamics of\nmillions of interacting spins.\nThe theory presented here will be a useful tool for\ninvestigating non{Markovian behaviour, opening up a\nnumber of avenues for future research at the intersec-\ntion of quantum thermodynamics, magnetic materials\nand beyond. For example, it is an open question to\nclarify the connection between the three-dimensional\nprecession described by the spin equation (11) and ro-\ntational Brownian motion. The orientation of a non-\nsymmetric rotating body behaves analogously to the\nthree-dimensional spin vector, and the motion and vis-\ncosity of a gas surrounding a rotating body simulta-\nneously act on its motion while obeying the FDT as\ndiscussed in recent work [92, 93].\nWithin magnetism, for particular materials of inter-\nest, detailed models of the coupling functions C!can\nbe developed that are based on an understanding of\nthe interactions between spins, phonons, and electrons\nin the material [79]. Coupling to optical modes may\nfurther be included to describe, for example, whispering\ngallery photon-magnon coupling which leads to an e\u000bec-\ntive Gilbert damping term that can take either sign [94].\nA direct experimental characterisation of a material's\ndamping kernelKthat determines memory and noise\nin (11) may be attempted, for example with high \feld\nexperiments such as those recently reported in [50]. Of12\nparticular interest are dynamical features beyond the in-\nertial kernel approximation, which will also modify the\nnoise spectrum at larger temperatures.\nWhile we here discussed scalar couplings to the en-\nvironment in depth, Eq. (11) does hold for any real\n3x3 matrixC!describing the spin-environment interac-\ntion in three dimensions. Anisotropic coupling tensors\ncan be implemented, suitable for describing magnetiza-\ntion dynamics within thin \flms [62], where one direction\nis coupled di\u000berently to environmental modes than the\nother two. One simpli\fcation of our three-dimensional\nmodel is to choose a coupling tensor such that spins in-\nteract with only one-dimensional environmental modes.\nThis reduces the theory to the spin-boson model, see\nAppendix A1, whose quantum thermodynamic proper-\nties have been discussed very extensively, recently for\nexample in [33].\nMicroscopic heat transport in spin systems can also be\nanalysed by allowing non{equilibrium situations where\nindividual reservoir modes at frequencies !and for spins\nnare thermal - but at di\u000berent temperatures. This will\nresult in spin dynamics that shu\u000fes energy from one\nreservoir mode to another, and could result in two- and\nmore-temperature models. For example, the possibility\nof di\u000berent phonon modes, each with their own tem-\nperature, to couple with di\u000berent strengths to electrons\nhas recently been analysed in [95] for a magnetic system\nexcited by an ultra-short laser pulse. Furthermore, in de-\nriving the FDT we have assumed bosonic environmental\nmodes but it would be insightful to identify changes to\nthe properties of equation (11) that arise when the spins\ncouple directly to electrons, or fermionic modes in gen-\neral [14, 15, 96].\nBeyond the quantum character of the reservoir, it\nwill be important to numerically solve the full quan-\ntum dynamics according to Eq. (11), including spin\noperators interacting with neighbouring spin operators.\nAdvanced quantum numerical methods such as Hier-\narchical Equations Of Motion (HEOM) [97], and the\nrecently proposed time-evolving matrix product oper-\nators (TEMPO) method [98] will be required to e\u000e-\nciently describe the time evolution of even just a sin-gle quantum spin coupled to a non-Markovian environ-\nment. For multiple interacting spins at low tempera-\ntures one can expect entanglement between the spins\nbeing present during the short-time dynamics, and even\nin steady state [99{101]. Unfortunately, evaluating such\nproperties will very quickly become a numerically hard\nproblem, requiring advanced numerical techniques such\nas density-matrix renormalisation group (DMRG) [102]\nto \fnd realistic approximate solutions. Vice versa, solv-\ning (11) within the classical spin vector approximation\nwhile including a full quantum power spectrum for the\nenvironmental modes, may prove insightful and numer-\nically tractable in the context of \fnding suitable mod-\nels for noise in quantum computing hardware, such\nas superconducting qubits that are held in the mK\nrange [103]. The results may also inform implementa-\ntions of Young's double slit experiment with a levitated\nsingle magnetic domain nanoparticle using the Einstein-\nde Haas e\u000bect [104, 105].\nAcknowledgments\nWe thank Karen Livesey, Richard Evans, Marco Berritta,\nStefano Scali, Federico Cerisola, Luis Correa, James\nCresser, Claudia Clarke, Ian Ford and Rob Hicken\nfor inspiring discussions, Carsten Henkel and Richard\nEvans for comments on a draft of this manuscript,\nand Somayyeh Nemati for iron's Lorentzian parame-\nters mentioned in [79]. SARH thanks Paul Kinsler\nfor pointing out the stupidity of numerically solving\nan integro{di\u000berential equation when an ODE will do.\nSARH also acknowledges funding from the Royal So-\nciety and TATA (RPG-2016-186). CRJS and JA ac-\nknowledge support and funding from the EPSRC Centre\nfor Doctoral Training in Electromagnetic Metamaterials\nEP/L015331/1. 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Schollwoeck, \\The density-matrix renormalization\ngroup\", Rev. Mod. Phys. 77, 259 (2005).\n[103] A.D. King, J. Carrasquilla, et al. , \\Observation of\ntopological phenomena in a programmable lattice of\n1,800 qubits\", Nature 560, 456 (2018)\n[104] C.C. Rusconi, V. P ochhacker, K. Kustura, J.I. Cirac,\nand O. Romero-Isart, \\Quantum Spin Stabilized Mag-\nnetic Levitation\", Phys. Rev. Lett. 119, 167202 (2017)\n[105] H. Pino, J. Prat-Camps, K. Sinha, B. Prasanna\nVenkatesh, O. Romero-Isart, \\On-chip quantum inter-\nference of a superconducting microsphere\", Quantum\nSci. Technol. 3, 025001 (2018).\n[106] Note that Eq. (1) is expressed in SI rather than Gaus-\nsian units (as in e.g. [45]).\n[107] Not the Brownian motion Brown!\n[108] Using vector identities the time derivative on the\nright hand side of Eq. (1) can be eliminated and the\nequation becomes @M=@t=\r0M\u0002Be\u000b\u0000\u0015M\u0002\n(M\u0002Be\u000b)with\r0and\u0015functions of \r;\u0011GandjMj.\n[109] We use SI rather than Gaussian units, so we have B\n(units mass=(charge\u0002time) ) rather than an H-\feld\n(units charge=(length\u0002time) ).\n[110] Here the spins are discrete and positioned on a lattice,\nbut one could also use a continuum description, as in\nmicromagnetics [44].\n[111] Note that tensors and vectors are set in calligraphic\nand bold font, respectively, and that scalar products\nbetween vectors are indicated with \u0001, while a tensor\nfollowed by a tensor or a vector is to be understood as\nmatrix multiplication.\n[112] Note that in contrast to what is typically done in\nCaldeira{Leggett type models[35] no counter term has\nbeen included here. In any case, coupling to a spin\nwould result in a term proportional to S2/ 1, which\nwill incur an o\u000bset in the overall Hamiltonian that does\nnot a\u000bect the dynamics.[113] The Fourier transform ~K(n)(!)of the kernelK(n)(\u001c)\nautomatically satis\fes the Kramers{Kronig rela-\ntions [63], connecting the dissipative and reactive parts\nof the response kernel, as is required for any causal re-\nsponse.\n[114] The autocorrelation function of two reservoir operators\nAandByin the thermal reservoir state ^\u001aRis de\fned\nas the expectation value of the Hermitian operator,\nh\b\nA(t);By(t\u0000\u001c)\t\ni\f=2. In the classical case Aand\nBycommute at all times, removing the need for this\ndistinction.\n[115] The Fourier transform is here de\fned as\n~f(!) =R1\n\u00001d\u001ce+i!\u001cf(\u001c), with the inverse\nf(\u001c) =R1\n\u00001d!\n2\u0019e\u0000i!\u001c~f(!).\n[116] The power spectrum given in (16) is the correct gen-\neral version for any kernel K(n)(t), ful\flling the full\nquantum FDT [63]. For a Gilbert damping kernel a\npower spectrum proportional to ~!=(exp ( ~!=kBT)\u0000\n1)was given in [60, 74{76]. This is missing the quan-\ntum ground state contribution of ~!=2, which acts\nas stochastic noise on the spin system even at zero\ntemperature.\n[117] Note that although (20) is not an explicitly Hermitian\noperator, Appendix A2 shows that Eq. (11) is never-\ntheless equivalent to a Hermitian equation of motion.16\nAppendix\nA1. Recovery of the spin-boson model\nThe spin-boson model is recovered as a special case\nof the three-dimensional Hamiltonian (2), and hence its\ndynamics is also given by Eq. (11). To see this one may\ndrop the site index n, and choose the external \feld as\nBext=B0(cos(\u0012)ez\u0000sin(\u0012)ex); (38)\nfor some angle \u0012. Taking spin 1=2operators ^S=\n(~=2)\u001band the coupling tensor as (C!)jk=C!\u000ejk\u000ej1\nwith a scalar coupling function C!, one recovers from\n(2) the one-dimensional spin-boson Hamiltonian\n^H=\u0000\rB0(cos(\u0012)^Sz\u0000sin(\u0012)^Sx)\n\u0000\r^SxZ1\n0d! C!^x! (39)\n+1\n2Z1\n0d!h\n(^px;!)2+!2(^x!)2i\n+^H2D\nR;\nwhere ^H2D\nRis a decoupled two-dimensional reservoir\nthat can be dropped from the dynamics.\nA2. Hermiticity of the quantum spin dynamics\nequation\nThe quantum spin dynamics equation (11) is not writ-\nten in an explicitly Hermitian form. The integral term\ncontaining the damping kernel K(n)includes an operator\nproduct that does not equal its conjugate transpose\n^S(n)(t)\u0002^S(n)(t0)6=\u0000^S(n)(t0)\u0002^S(n)(t): (40)\nNevertheless equation (11) is Hermitian, as it is the time\nintegral that commutes with ^S(n)(t)\n\u0014Zt\nt0K(n)(t\u0000t0)^S(n)(t0);^S(n)(t)\u0015\n= 0 (41)\nThis can be veri\fed from an observation that Eq. (11)\nis simply a re{written form of the explicitly Hermitian\nequation (8).\nAny confusion can be avoided through re{writing\nEq. (11) in an equivalent but explicitly Hermitian form\nd^S(n)(t)\ndt=\r\n2[^S(n)(t)\u0002^B(n)\ne\u000b(t)\n\u0000^B(n)\ne\u000b(t)\u0002^S(n)(t)](42)\nwhere the e\u000bective magnetic \feld operator at time tand\nsitenis given by\n^B(n)\ne\u000b(t) =Bext+1\n\rX\nm6=n\u0016J(nm)^S(m)(t) +^b(n)(t)\n+\rZt\nt0dt0K(n)(t\u0000t0)^S(n)(t0):(43)\nIn the Hermitian form (42) it is clear, for example\nthat any term proportional to ^S(n)(t)appearing in ^B(n)\ne\u000b\ndoes not a\u000bect the evolution of the spin operator, even\nthough the operator cross product\n^S(n)(t)\u0002^S(n)(t) = i~^S(n)(t) (44)is non{zero. A consequence of this result is that the\nzeroth order term in the expansion of the damping op-\nerator (23) does not contribute to the evolution of the\nspin operator.\nWe note that Eq. (41) implies that only the sum of\nallthe terms in the damping kernel expansion (23) com-\nmutes with the spin operator. When using a truncated\nform of the expansion (23) we must therefore use the\nexplicitly Hermitian equation of motion (42).\nA3. ^S2is a constant of motion of Eq. (11)\nTo evaluate the derivative of (^S(n)(t))2we \frst ex-\npress Eq. (11) in explicitely Hermitian form (42). Drop-\nping site index and time for simplicity, we \fnd\ndj^S(t)j2\ndt=X\nj \n^Sjd^Sj\ndt+d^Sj\ndt^Sj!\n=\r\n2X\njkl\u000fjkl\u0010\n^Sj^Sk^Bl+^Bl^Sk^Sj\u0011\n=i~\r\n2X\nl[^Sl;^Bl]\n= 0; (45)\nwhere we have applied the angular momentum com-\nmutation relations, interchanged indices, and used the\nanti-symmetric property of \u000fjkm. The \fnal line follows\nfrom the fact that the spin and the e\u000bective magnetic\n\feld commute.\nA4. Lorentzian power spectrum expansion\nSimilar to the damping kernel term expansion (23),\nin moments (30) and time-derivatives, the Lorentzian\npower spectrum (29) can be expanded in powers of fre-\nquency!, as\n~PLor\nqu(!) =1X\nm=0(\u00001)m+1!2m+1\u0014Lor\n2m+1coth\u0012~!\n2kBT\u0013\n;\n(46)\nwhere we have kept the quantum coth unexpanded. The\n\u0014Lor\nmare the same coe\u000ecients as those given in (30).\nFor small frequencies !the \frst term in the series (46)\ndominates and the power spectrum takes the (quantum)\nOhmic form\n~PLor\nqu(!)\u0019\u0000!\u0014Lor\n1coth\u0012~!\n2kBT\u0013\n; (47)\nwhere comparison with (21) again shows that \u0000\u0014Lor\n1is\nthe e\u000bective Gilbert damping constant.\nBeyond the Ohmic regime, one can see in (46) that\nonly the odd moments \u0014Lor\n2m+1contribute. Therefore\nthe inertial term \u0014Lor\n2, which is the \frst deviation of the\ndamping kernel from Ohmic behaviour, does not change\nthe quantum \ructuations in (16). Only when the third\norder time derivative of the spin operator contributes\nsigni\fcantly to equation (11), will memory e\u000bects begin\nto colour the spectrum away from the (quantum) Ohmic\nform (21).\nA5. Set of equations for kernel simulation\nHere we show that the simulation of the kernel in\nEq. (11) can be achieved by numerically integrating a set17\n01 4 8 12 16 20\nT / K00.250.50.751/angbracketleftbig\nsz/angbracketrightbig\na)\nS0=1/2\n0520 40 60 80 100\nT / K00.250.50.751/angbracketleftbig\nsz/angbracketrightbig\nb)\nS0=5/2\n0 800 1600 2400 3200 4000\nT / K00.250.50.751/angbracketleftbig\nsz/angbracketrightbig\nc)\nS0=200/2\nstat phys\nLLG+cl.\nLLG+qu.\nLor(Set1)+qu.\nLor(Set2)+qu.\nFIG. 6. Same notation and Lorentzian parameters as in Fig. 5 but here also showing spin length S0= 5~=2in panel b).\nPanels a) and c) show hsziforS0= 1~=2andS0= 200 ~=2, respectively, as in main text. Each curve is obtained by, for each\ntemperature T,time-averaging over the late times of a single stochastic trajectory, from 0:75tmax totmax= 2\u0019\u00028000!\u00001\nL.\nof \frst order coupled di\u000berential equations. We assume\na single spin and rewrite Eq. (11) as\ndS(t)\ndt=\rS(t)\u0002\u0014\nBext+b(t) +V(t)\u0015\n; (48)\nwhere we have de\fned V(t) =\rRt\nt0dt0K(t\u0000t0)S(t0).\nFurthermore de\fning W(t) =dV(t)\ndt, now leads to a\ndi\u000berential equation for W(t):\ndW(t)\ndt=\rZt\nt0dt0d2K(t\u0000t0)\ndt2S(t0); (49)\nwhere we have assumed K(0) = 0 and _K(0) = 0 . Ex-\npressingK(t\u0000t0)through its Fourier transform ~K(!),\nchoosing a Lorentzian kernel (27) and considering the\nexpressionZ(t) :=dW(t)\ndt+\u0000W(t)+!2\n0V(t), we obtain\nZ(t)=A\rZt\nt0dt0\u000e(t\u0000t0)S(t0): (50)\nRearranging gives\ndW(t)\ndt=\u0000\u0000W(t)\u0000!2\n0V(t) +A\rS(t); (51)\nas stated in the main text. (Note that the assumption\nK(0) = 0 and _K(0) = 0 is ful\flled for the Lorentzian\nkernel, (28), since the Heaviside function \u0002(\u001c) = 1 for\n\u001c >0, and zero elsewhere.)\nA6. Statistical physics prediction for hszias func-\ntion of temperature\nFor a classical spin Sof lengthS0=n~=2in an\nexternal \feld Bext=Bezthe thermal average hszi\nis determined by the Boltzmann distribution for the\nHamiltonian H=\u0000\rS\u0001Bextat inverse temperature\n\f= 1=kBT,\nhSzistat phys =Z+S0\n\u0000S0dSzSze\u0000\f(\u0000\rSzB)\nZa=@alnZa;(52)\nwithZa=R+S0\n\u0000S0dSzeaSzwherea=\f\rB . This gives\nZa=2 sinh(aS0)\na; (53)\nand hence\nhSzistat phys\nS0= coth (\f\rBS 0)\u00001\n\f\rBS 0;(54)\nhszistat phys = coth\u0012n~!L\n2kBT\u0013\n\u00002kBT\nn~!L; (55)where!L=j\rBjandsz=sign(\r)Sz\nS0. In the mag-\nnetism literature, sometimes a reduced temperature ex-\nperienced by a spin with n6= 1 is de\fned as Tred=T=n ,\ni.e. the temperature is e\u000bectively reduced in comparison\nto the temperature experienced by a spin with n= 1.\nA7. Steady state hsziplot for spin S0= 5~=2\nAs discussed in the main text, the impact of the quan-\ntum zero-point noise on the steady state hszivalue at\nT= 0K is very highly spin length dependent. For some\nmaterials a fundamental spin value of S0= 1~=2will\nnot be appropriate. For example Iron (III) has 5 elec-\ntrons in the outer dshell, and then from Hund's rules\nthe spin is maximized to S= 5=2, and the orbital an-\ngular momentum is zero, L= 0. Therefore J=S,\nLand\u0013 e g{factor equals 2, and the gyromagnetic ratio re-\nmains the electron gyromagnetic ratio. Fig. 6b) shows\nthe steady statehsziplot as a function of temperature\nfor spinS0= 5~=2, next to those for spin S0= 1~=2\n(a) andS0= 200 ~=2(c). Thehszivalue is below 1, at\n\u00190:8, but the reduction is far less severe than for the\nspin-1/2.\nA8. Scales in classical and for quantum ther-\nmostats\nHere we establish the set of scales determining the\ndynamics described by Eqs. (1) and (11) with either\nclassical orquantum power spectra.\nFor a single spin, i.e. ignoring exchange terms etc.,\none may rescale the LLG equation (1) using M=\rS=\nj\rjS0swith spin lengthjSj=S0. One obtains\nds\ndt=\rs\u0002\u0014\n^Bcl\ne\u000b(t)\u0000j\rjS0\u0011Gds\ndt\u0015\n; (56)\nwhere the spin length S0and\u0011Gappear together, set-\nting the \frst scale. Furthermore the e\u000bective \feld in-\ncluding classical stochastic noise with power spectrum\n(22), is given through (35) and (36) by\n^Bcl\ne\u000b(t) =Bext+r\n2S0\u0011GkBT\nS0\u0018(t): (57)\nHere we have introduced an S0so that\u0011Gappears to-\ngether with it, and we \fnd the second scale to be given\nbyT=S 0. The third scale is clearly set by the strength\nof the external \feld, Bext. We note that if one chooses\nthe sameS0\u0011Gvalue for di\u000berent spin lengths, i.e. as-\nsumes\u0011Gscales as 1=S0, then only two scales are left,\nBextandT=S 0.18\nHowever, for the quantum Ohmic power spectrum the\ncomponents of the stochastic noise can be written as\nbj=Z1\n\u00001dt0Z!c\n\u00001d!\n2\u0019e\u0000i!(t\u0000t0)\n\u0002s\nS0\u0011G~!\nS0coth\u0012~!\n2kBT\u0013\n\u0018j(t0); (58)\nClearly, in the quantum case the temperature Tnow\nappears separately from spin length S0, thus introduc-\ning an additional scale in comparison to the classical\ncase. Moreover the fact that the frequency integration\nfor the stochastic \feld does not simplify as in (57) means\nthat relaxation to the steady state at low temperatures(where the cothxcannot be approximated as 1=x) will\nbe much more noisy than in the high temperature case.\nThus in our simulations, this additional scale leads to\nlarger uncertainties in the steady state results, as seen\nin Fig. 5a).\nFinally, we note that for the integration of the quan-\ntum Ohmic power spectrum in (58) we have introduced\na frequency cut-o\u000b !cby hand, which is necessary at\nlow temperatures to avoid the integral diverging. At low\ntemperatures, this cut-o\u000b will set an additional, some-\nwhat arti\fcial, scale of the problem. Importantly, such\ncut-o\u000b is not required for the Lorentzian coupling since\nthe power spectrum (29) decays at high frequencies,\neven at low T."    },    {        "title": "2006.03400v2.Controlling_the_nonlinear_relaxation_of_quantized_propagating_magnons_in_nanodevices.pdf",        "content": "Controlling the nonlinear relaxation of quantized p ropagating magnons in nanodevices \nM. Mohseni,  1, *  Q. Wang,  2 B. Heinz, 1,3  M. Kewenig,  1 M. Schneider,  1 F. Kohl,  1 B.  Lägel,  4 C. Dubs,  5 A. V. \nChumak,  2 and P. Pirro 1 \n \n1 Fachbereich Physik and Landesforschungszentrum OPT IMAS, Technische Universität Kaiserslautern, 67663 \nKaiserslautern, Germany \n2 Faculty of Physics, University of Vienna, Boltzman ngasse 5, A-1090 Vienna, Austria \n \n3 Graduate School Materials Science in Mainz, Staudi ngerweg 9, 55128 Mainz, Germany \n \n4 Nano Structuring Center, Technische Universität Kai serslautern, 67663 Kaiserslautern, Germany \n \n5 INNOVENT e.V., Technologieentwicklung, Prüssingstr aße 27B, 07745 Jena, Germany \n \nRelaxation of linear magnetization dynamics is well  described by the viscous Gilbert damping pro- \ncesses. However, for strong excitations, nonlinear damping processes such as the decay via magnon-mag-\nnon interactions emerge and trigger additional rela xation channels. Here, we use space- and time-resol ved \nmicro-focused Brillouin light scattering spectrosco py and micromagnetic simulations to investigate the  \nnonlinear relaxation of strongly driven propagating  spin-waves in yttrium iron garnet nanoconduits. We  \nshow that the nonlinear magnon relaxation in this h ighly quantized system possesses intermodal feature s, \ni.e. magnons scatter to higher-order quantized mode s through a cascade of scattering events. We furthe r \nshow how to control such intermodal dissipation pro cesses by quantization of the magnon band in single -\nmode devices, where this phenomenon approaches its fundamental limit. Our study extends the knowledge \nabout nonlinear propagating spin-waves in nanostruc tures which is essential for the construction of ad - \nvanced spin-wave elements as well as the realizatio n of Bose-Einstein condensates in scaled systems.  \n \nRelaxation of magnons, the quanta of spin waves \n(SWs), due to magnetic damping is a complicated \nprocess and involves different (non)linear contribu - \ntions. Relaxation mechanisms which can be de- \nscribed by the phenomenological Gilbert damping  \ndrive the magnetization towards its equilibrium sta te \nby e.g. dissipating the energy to the lattice. It i s one \nof the key elements of performance in many practica l \ndevices and fundamental phenomena  [1–10]. \nDissipation of the energy can be more intricate \nfor strongly driven excitations, where nonlinear re - \nlaxation mechanisms via magnon-magnon interac- \ntions open up additional dissipation channels  [11–\n17]. Unlike the Gilbert damping, these types of in-\ntrinsic dissipation processes can redistribute the \nmagnon energy within the magnon spectrum [18–\n27]. \n       The classical works of Suhl predicted that l arge \namplitude uniform magnetization oscillations lead t o the onset of instability processes, allowing the no n- \nlinear relaxation of strongly driven magnons by a d e- \ncay into secondary magnon modes  [25]. In particu- \nlar, the common second-order Suhl instability pro- \ncess can be: ( i) a disadvantage since it comes along \nwith detrimental influence on the magnon transport \nand decay characteristics, potentially dominating t he \ncompeting linear damping  [17,22,28] , or, ( ii ) an ad- \nvantage by providing additional degrees of freedom \nof magnon transport for device architectures and \nquantum computing concepts  [23,29,30]. So far, \nmost of such investigations in scaled systems, whic h \nare of large interest for applications, have been c ar- \nried out for standing SW modes with vanishing mo- \nmentum ( k = 0), e.g. the Ferromagnetic resonance \n(FMR) mode. However, SWs carrying a momentum \nare not only essential for applications, but they p os- \nsess an enriched physics behind their nonlinear ins ta- \nbilities due to the increased amount of potential s cat- \ntering channels. Nevertheless, little investigation s \nhave been carried out in this direction yet. Recent development of ultra-low damping na- \nnoscale systems based on YIG, the most promising \nhosts for SWs, provides access to quasi-1D systems \nwith highly quantized magnon spectra  [31,32] . By \nimposing limitations on the available relaxation \nchannels due to the strong quantization of the mag-\nnon band, and a drastically modified SW character- \nistics including the SW dispersion relation, mode \nprofile and their ellipticity, nonlinear SW dynamic s \nin such devices can be different compared to contin - \nuous films and quasi-2D systems  [33-34]. Further- \nmore, recent experimental and theoretical studies o f \nSW dynamics and magnon condensates in nano- \nscopic systems  [32, 35,36] enforce us to better un - \nderstand nonlinear SW dynamics and magnon ther- \nmalization processes in nano-scaled 1D systems.  \n \nHere, we use space- and time-resolved mi- \ncro-focused Brillouin light scattering (µBLS) to un - \ncover the mechanism of nonlinear relaxation of \nstrongly driven propagating magnons via the second-\norder Suhl instability in YIG nanoconduits. We \ndemonstrate how magnons nonlinearly relax to other \nquantized modes via four-magnon scattering pro- \ncesses, and such nonlinear processes can be con- \ntrolled using quantization of the magnon band.  \n \nTo demonstrate the effect of quantization on \nthe nonlinear dynamics, we use two exemplary mag- \nnonic nanoconduits structured from a Liquid Phase \nEpitaxial (LPE) YIG film grown on top of a Gado- \nlinium Gallium Garnet (GGG) substrate  [37]. The \nmulti-mode nanoconduit with a lateral width of w = \n400 nm (Fig. 1a) and a thickness of d = 85 nm was \nfabricated using a hard mask and ion beam milling \nprocess  [31]. A comparative single-mode conduit \nwith a smaller width of w = 100 nm and d= 44 nm \nwas fabricated using a similar method (Fig. 1b). SW s \nin both devices are excited by a microwave antenna \nwhich is placed on top of the nanoconduits by elec-\ntron beam lithography and a lift-off process [31]. Ap- \nplying a microwave rf  current to the antenna gener- \nates a dynamic Oersted field which in return excite s \nSWs resonantly, see supplemental materials SM \n[40]. The detection of the generated SWs has been carried out using space- and time-resolved \nµBLS  [38]. An incident laser light with an effecti ve \nspot size of 300 nm (focused by a ×100 microscope \nobjective with a numerical aperture NA=0.85) is \nused to probe the SWs through the GGG substrate \nunder the antenna. The inelastically scattered ligh t \nwas analyzed using a tandem Fabry-Perot interfer- \nometer to obtain the frequency and intensity of the  \nmagnons.   \n \nFIG. 1. (a)-(b) SEM images of the w = 400 nm (multi-\nmode) and w =100 nm (single-mode) wide conduits \n(shaded in orange), respectively. (c)-(d) Magnon ba nd \nstructures of the multi-mode and single-mode condui ts, \nrespectively. Color plots are obtained by micromagn etic \nsimulations and dashed lines from analytical calcul ations. \nNote the different scales of the frequencies. (e)-( f) Meas- \nured spin-wave spectra of the multi-mode and single -\nmode conduits in the presence of different powers, respec- \ntively. The excited modes are represented by the ye llow \ndots in (c) and (d). \nA static external field (µ 0He = 60 mT) saturates \nthe nanoconduits along their length. Thus, the wave  \nvector of the propagating SWs is parallel to the ma g- \nnetization vector, /g2193 ‖  /g2169 , and waveguide (WG) \nmodes appears [32].  The width of the multi-mode \nwaveguide is large enough to ensure dipolar pinning  \nof the spins at the edges, while spins at the edges  of \nthe single-mode conduit are fully unpinned [32]. \nMoreover, due to the interplay between the contribu - \ntions of the dipolar and exchange energy to the SW \ndispersion, the different WG modes are well quan- \ntized on the frequency axis. The dispersion relatio n \nof the fundamental mode and the first two WG \nmodes are shown in Fig 1c-d, in which the dashed \nlines are analytical results based on method discus ses \nin Ref  [32], and the color plot is obtained by mic ro- \nmagnetic simulations using the MuMax 3.0 pack- \nage  [39, 40]. The fundamental mode and higher or- \nder WG modes are labeled as n = 0 and n = 1, 2 re- \nspectively. Please note that the spectrum is much \nmore dilute in the 100 nm wide conduit due to the \nhigher contribution of the exchange energy to the \nmagnon band structure, which leads to a strong quan - \ntization and the absence of degenerate states among  \nmodes (single-mode system for wave vectors below \napprox. 40 rad/µm).  \nWe first set the rf  frequency to f = 3.85 GHz \nwhere dipolar SWs having a wave vector of kx = 1.5 \nrad/µm are excited in the multi-mode device [40]. T o \ncharacterize the linear SW dynamics, we set the rf  \npower to P = 10 dBm and measure the intensity of \nthe generated magnons as displayed in Fig 1e (black  \ncircles). Up to  P = 18 dBm, only the frequency of the \nresonantly driven SW mode is observed (red and \ngreen triangles). A further increase in the rf  power up \nto P = 20 dBm (blue curve) leads to the appearance \nof two additional peaks in the SW frequency spec- \ntrum labeled as f – and f + in Fig. 1e. We refer to these \nmagnons as secondary magnons which are modes \npopulated by nonlinear scattering processes. They \nhave the lowest threshold for the observed instabil ity \nprocess and can fulfill the fundamental conservatio n \nlaws to permit the scattering process [22]. The en-\nergy and momentum conservation laws of these pro- \ncesses generally  read  [18,20,22,28], \n \n/g1858/g2869+ /g1858/g2870= /g1858/g2871+ /g1858/g2872,   /g2193/g2869+ /g2193/g2870= /g2193/g2871+ /g2193/g2872          (1) \n \nwhere two magnons with the frequencies f 1 & f 2 and \nmomenta k 1 & k 2 scatter to two magnons with the \nfrequencies f 3 & f 4 and momenta k 3 & k 4. Note that \nthe lateral component of the k vector is symmetric, \nand the out of plane component is zero in this fre-quency range due to the small thickness.   In our ex- \nperiments, two magnons with a frequency of f = 3.85 \nGHz scatter finally to two magnons with the frequen - \ncies of f 3 = f - = 3.25 GHz and  f 4 =  f + = 4.45 GHz. \nWe note that this process is not a special peculiar ity \nof the chosen spectral position, see SM [40]. \nFor comparison, we now investigate the \nsame nonlinear process in the comparative single-\nmode waveguide. We set the f = 3.71 GHz and meas- \nure the intensity of the driven mode as shown in Fi g. \n1f.  Clearly, even in the presence of high powers l ike \nP = 20 dBm, side peaks cannot be observed, evidenc- \ning the absence of a similar nonlinear dissipation \nprocesses. Here, only the µBLS intensity drops at \nhigh powers which is caused by the nonlinear fre- \nquency shift of the dispersion relation and possibl e \nimpacts of the higher temperature [31, 41]. In prin ci- \nple, the absence of side peaks demonstrates that su ch \nscattering processes can be efficiently suppressed in \nnarrower conduits where the magnon band structure \nis highly quantized and therefore, the fundamental \nconservation laws required for the scattering pro- \ncesses cannot be fulfilled. \nTo understand the fundamental differences \nbetween the two waveguide types, let us investigate  \nthe observed nonlinear dynamics in the multi-mode \nconduit in more detail. A nonlinear scattering inst a- \nbility is characterized by a clear threshold of the  ini- \ntial magnon intensity which is required for its on-\nset [18,22,24,42]. Neglecting SW radiation losses, \nthe threshold magnon amplitude is defined by the ef - \nfective relaxation frequency of the secondary mag- \nnons divided by the four-magnon coupling \nstrength  [16,22]. To investigate the threshold beh av- \nior in the multi-mode conduit in which the scatteri ng \nis observed, we sweep the rf  power for a fixed fre- \nquency f = 3.85 GHz as shown in Fig. 2. Once the \ninstability threshold is reached at P = 18 dBm (indi- \ncated by the black arrow), the growth rate of the d i- \nrectly excited magnon intensity as a function of mi - \ncrowave power drops. Increasing the power to P = \n19 dBm leads to an abrupt increase of the intensity  \nof the secondary magnons  labeled as f + and f – (indi- \ncated by the gray arrow). From this power ( P = 19 \ndBm) on, the intensity growth rate of the directly ex- \ncited mode with respect to the power is decreased, evidencing that the energy transfers to the seconda ry \nmagnon modes. \nFIG. 2. Spin-wave amplitude in the multi-mode  conduit as \na function of microwave excitation power when f = 3.85 \nGHz. The secondary magnons created by the second or der \nSuhl instability are denoted as f + and f -. The back and \ngray arrows indicate the onset of instability and t he rise of \nthe secondary magnons, respectively.  \nA closer look on Fig. 2 near the instability \nthreshold opens the question what happens when the \ninstability threshold is approached at P = 18 dBm \nand the µBLS intensity of the initially excited mod e \ndrops, while the amplitudes of the secondary mag- \nnons  at f+ and f- are still at the thermal level, implying \nthe absence of magnon scattering to these modes. \nWe perform micromagnetic simulations to \nuncover the wave vector of the scattered magnons \nand address the discussed question. Figure 3a shows  \nthe frequency spectrum of the simulated multi-mode \nconduit ( f = 3.85 GHz) in which different amplitude \nof the rf  currents are used to drive the system. For a \nsmall rf  current equal to irf  = 4 mA, only the reso- \nnantly excited SWs can be observed in the frequency  \nspectrum (black curve). The corresponding popula- \ntion of the magnon band is depicted in Fig. 3b show - \ning the wave vector of kx = 1.5 rad/µm of the directly \nexcited mode. Increasing the rf  current to a higher \nvalue of irf  = 8 mA increases the amplitude and the \nlinewidth of the resonant SWs (red curve in Fig \n3.a) [28]. As shown in Fig. 3c, this is related to the \nonset of a first-level four magnon scattering proce ss \nin which the frequency of the magnons is conserved.  \nSuch a process cannot be observed in the measured frequency spectrum of the conduit, but it can mani-\nfest itself in the observed drop of the directly ex cited \nmode intensity with power.  As evidenced by the sim- \nulations, two incoming magnons from the resonantly \ndriven mode with opposite momenta scatter to two \noutgoing magnons at the same frequency, but with \ndifferent momenta. The scattered magnons populate \nthe fundamental mode ( n = 0) at a higher wave num- \nber of /g1863/g3051= 30 /g1870/g1853/g1856/µ/g1865 , and two spectral position \nat the first WG mode ( n = 1). These frequency-con- \nserving  scattering processes which are similar to \nplane films  [25] are indicated by the pink arrows in \nFig. 3c, and can also be observed in the single-mod e \nconduit, see SM [40]. \nA further increase of the rf  current to irf  = 13 \nmA leads to the onset of the sideband peaks in the \nfrequency spectrum (blue curve in Fig 3.a), similar  \nto the experiments. As evidenced from the simulated  \nband structure (Fig. 3d), this is due to the second  \nlevel of the magnon scattering cascade. Once the \nmagnons scattered by the first level process to the  \nn=1 WG mode reach a critical amplitude, they un- \ndergo themselves another second order instability. In \nthis process, two magnons with the frequency of f = \n3.85 GHz  and identical momentum of  /g1863/g3051=\n10.7 /g1870/g1853/g1856/µ/g1865  at the first WG mode ( n = 1), scatter \nto two outgoing magnons with the frequencies of f - \n= 3.46 GHz and f + = 4.24 GHz at the fundamental \nmode ( n = 0) and the second WG mode ( n = 2), re- \nspectively. The simulated values are in very good \nagreement with the experimentally obtained frequen-\ncies. \nIn Fig. 3d, this type of frequency-noncon- \nserving  scattering is represented by the red arrows. \nThe scattered magnons feature /g1863/g3051/g2878= 14.3 /g1870/g1853/g1856/µ/g1865  \nand /g1863/g3051/g2879= 7.1 /g1870/g1853/g1856/µ/g1865 , assuring momentum conser- \nvation laws given by 2/g1863/g3051= /g1863 /g3051/g2878+ /g1863 /g3051/g2879. We note that \nthe second scattering step clearly shows that the f i- \nnite momentum of the ingoing magnon opens the op- \nportunity to scatter to two new, different frequenc ies \nand thus, to redistribute the magnon energy towards  \nthe bottom of the spectrum and to higher frequencie s \n(modes). Unlike the first level process, it involve s \nonly magnons of a single propagation direction (+ k \nor –k) and can only occur for propagating waves. \nThis is evidenced by the momentum and energy con- \nservation laws which require a finite sum of the mo - \nmenta of the two incoming magnons to allow for a \nfrequency non-degenerated splitting. This is a sign if- \nicant difference to the nonlinear instabilities of the \nFMR mode without momentum ( kx = 0) in which \nmagnon instabilities are always degenerated [43-44] . \nThus, if the FMR undergoes a second-order instabil-\nity, this process never leads to a redistribution o f the \nmagnon energy across the spectrum.  \n \n \nFIG. 3. Results of the micromagnetic simulations in  the \nmulti-mode conduit. (a) spin-wave frequency spectra  \nwhen the microwave current varies. (b-d) Magnon ban d \nstructures (linear scale) of the driven system corr espond \nto the black, red and blue curves in (a), respectiv ely. The \nscaling of b-d is independent from each other.  \nThe properties of the cascade-like magnon \nscattering events coupling different waveguide \nmodes in the multi-mode waveguide and the absence \nof this effect in the single mode waveguide also im - \nplies that thermalization of magnons is significant ly \nchanged in systems with strongly diluted spectra \ncompared to earlier investigations in systems which  \nquasi-continuous spectra. \nThe simulations also explain the observed \npeculiarity in the threshold curve of the experimen ts \nas were discussed in the context of Fig. 2. Indeed,  the \nmagnons scattered to higher wave numbers via the \nfirst level frequency-conserving scattering process  (Fig. 3c) cannot be detected experimentally due to \nthe maximum detectable momentum using µBLS \nspectroscopy, which is approximately kx ~ 21 rad/µm \nin our experiments [38]. This explains at least par - \ntially the decrease of the measured magnon intensit y \nat the driving frequency. Since the different level s of \nthe cascade process have different threshold powers , \nthe nonlinear scattering to the secondary magnon \nmodes at different frequencies is observed at a \nslightly higher power than the start of the drop of  the \nintensity at the directly excited frequency. In add i- \ntion, the limited wave vector sensitivity of the BL S \ncan pose inconsistency for the SW amplitude ob- \nserved in the simulations compared to the experi- \nments. \n \nFIG. 4. Time-resolved spin-wave amplitude measured by \nµBLS spectroscopy. (a) Beginning of the pulse. (b) End \nof the pulse. Black arrows indicate the onset and t he decay \nof the instability, respectively. Note that the dec ay rates \ncorrespond to the intensity of the magnons.  \nTo further characterize the impact of the \nnonlinear relaxation on the total relaxation of the  sys- \ntem  [16], we perform time-resolved µBLS measure- \nments in the multi-mode conduit. The measured in- \ntensity of the driven and secondary magnons at the \nbeginning and the end of a 1µs long microwave rf  \npulse ( f = 3.85 GHz and P = 24 dBm) at the meas- \nurement position are shown in Fig. 4a-b. Figure 4a \nillustrates that the resonantly driven SW mode (blu e \ncurve) undergoes the second-level four-magnon \nscattering after t ~ 4 ns, evidenced by the rise of the \nsecondary magnons (yellow and red curves). This is \nindicated by the black arrow in Fig. 4a. Note that the \ngrowth rate of the driven mode drops immediately \nwhen the rise of the secondary magnons sets in, evi - \ndencing the conservation of the energy in the nonli n- \near redistribution process.  \nThe decay of the magnons at the end of pulse \nis presented in Fig. 4b. In particular, the decay o f the \nsecondary magnons begins once the intensity of the \ndriven SW mode is decayed enough after t ~ 4 ns \n(indicated by the black arrow). More interestingly,  \nthe decay of the magnons at the resonantly driven \nfrequency to the thermal level includes two steps \nmanifesting the high nonlinearity of the dynamics. \nFirst, it decays with an exponential decay time of t1,d \n= 19 ns, which is accompanied by the decay of the \nsecondary magnons at f+ and f-. Afterwards, it de- \ncays with a longer exponential decay time of t2,d = 24 \nns suggesting a transition from a nonlinear relaxat ion \nto a linear relaxation with a lower decay rate. In other \nwords, the first decay includes an energy flow to t he \nsecondary magnons which acts as an additional dis- \nsipation channel for the driven magnons. After the \nsecondary magnons decayed to the thermal level, thi s \nadditional dissipation channel is switched off, whi ch \nleads to a slower decay time of the driven SWs. \n In summary, we explored the nonlinear re- \nlaxation of strongly driven propagating spin waves \nin nanodevices. The finite momentum of the mag- \nnons investigated in our study provides an addition al \nplayground for the nonlinear magnon instability pro - \ncesses. Furthermore, it was shown that such inter- \nmodal dissipation process is strongly suppressed in  \nsystems with a strongly quantized magnon band (sin-\ngle-mode systems), suggesting the fundamental lim- \nitation of this process in nanodevices.  This can o pen \na new avenue for coherent nonlinear nano-mag- \nnonics. The nonlinear dynamics studied in this lett er \nare general and thus, can be applied to devices bas ed \non other deposition techniques as well. Our study c an \nbe used for several device architectures, namely, f re- \nquency mixers  [45], squeezed states [46], signal a nd \ndata processing units [29, 47-50], and quantum com-\nputing concepts [23], and further open doors to eng i- \nneered dissipation of magnons in nanodevices.  Acknowledgments \n The authors thank Burkard Hillebrands for \nsupport and valuable discussions. This project is \nfunded by the Deutsche Forschungsgemeinschaft \n(DFG, German Research Foundation) - TRR 173 - \n268565370 (“Spin+X”, Project B01) and by the pro- \nject - 271741898, the European Research Council \nwithin the Starting Grant No. 678309 “MagnonCir- \ncuits” and by the Austrian Science Fund (FWF) \nthrough the project I 4696-N. 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Morton,1, 6Goki\nEda,7, 2, 8and Hidekazu Kurebayashi1, 6, 9\n1)London Centre for Nanotechnology, University College London, 17-19 Gordon Street, London, WCH1 0AH,\nUK\n2)Department of Physics, Faculty of Science, National University of Singapore, 2 Science Drive 3, Singapore 117542,\nSingapore\n3)National Physical Laboratory, Hampton Road, Teddington TW11 0LW, UK\n4)Institute for Condensed Matter Physics and Complex Systems, School of Physics and Astronomy, The University of Edinburgh,\nEdinburgh EH9 3FD, UK\n5)Higgs Centre for Theoretical Physics, The University of Edinburgh, Edinburgh EH9 3FD,\nUK\n6)Department of Electronic & Electrical Engineering, UCL, London WC1E 7JE, United Kingdom\n7)Centre for Advanced 2D Materials, National University of Singapore, 6 Science Drive 2, Singapore 117546,\nSingapore\n8)Department of Chemistry, Faculty of Science, National University of Singapore, 3 Science Drive 3, Singapore 117543,\nSingapore\n9)WPI Advanced Institute for Materials Research, Tohoku University, 2-1-1, Katahira, Sendai, 980- 8577,\nJapan\n(Dated: 1 May 2023)\nLayered van der Waals (vdW) magnets can maintain a\nmagnetic order even down to the single-layer regime and\nhold promise for integrated spintronic devices. While the\nmagnetic ground state of vdW magnets was extensively\nstudied, key parameters of spin dynamics, like the Gilbert\ndamping, crucial for designing ultra-fast spintronic de-\nvices, remains largely unexplored. Despite recent studies\nby optical excitation and detection, achieving spin wave\ncontrol with microwaves is highly desirable, as modern in-\ntegrated information technologies predominantly are op-\nerated with these. The intrinsically small numbers of\nspins, however, poses a major challenge to this. Here, we\npresent a hybrid approach to detect spin dynamics medi-\nated by photon-magnon coupling between high-Q super-\nconducting resonators and ultra-thin flakes of Cr 2Ge2Te6\n(CGT) as thin as 11 nm. We test and benchmark our tech-\nnique with 23 individual CGT flakes and extract an upper\nlimit for the Gilbert damping parameter. These results are\ncrucial in designing on-chip integrated circuits using vdW\nmagnets and offer prospects for probing spin dynamics of\nmonolayer vdW magnets.\nINTRODUCTION\nvan der Waals (vdW) materials1–3consist of individual\natomic layers bonded by vdW forces and can host different\ntypes of collective excitations such as plasmons, phonons and\nmagnons. Strong coupling between these excitation modes\nand electromagnetic waves (i.e. photonic modes) creates con-\nfined light-matter hybrid modes, termed polaritons. Polaritons\na)Electronic mail: c.zollitsch@ucl.ac.ukin vdW materials are an ideal model system to explore a va-\nriety of polaritonic states5,6, e.g. surface plasmon polaritons\nin graphene7,8and exciton polaritons in a monolayer MoS 2\nembedded inside a dielectric microcavity9. These states can\nbe further modified by electrostatic gating16, as well as by\nhetero-structuring with dissimilar vdW layers1.\nNumerous studies on magnon polaritons (MPs)11,12have\nbeen using macroscopic yttrium iron garnet (YIG) cou-\npled to either three-dimensional cavities13or to on-chip\nresonators14,15, with potential applications in ultra-fast infor-\nmation processing, non-reciprocity or microwave to optical\ntransduction. By reducing the number of excitations, MPs\nfind application in the quantum regime e.g., magnon number\ncounting via an electromagnetically coupled superconducting\nqubit16,17or as a building block for Bell state generation18.\nThe rapidly developing research around polaritons and\nspecifically MPs has so far, been little studied in magnetic\nvdW materials due to the relatively recent discoveries of long-\nrange magnetic order in vdW systems at the few monolayer\nregime9,20,21, in addition to its technically challenging real-\nization. Stable MP states are formed by strongly coupling the\nmagnetic field oscillation of a resonant photon to the collec-\ntive magnetization oscillation in a magnetic material. This\nstrong coupling is achieved when the collective coupling rate\ngeffis larger than the average of both system loss rates. In a\nsimplified picture, geffscales linearly with the strength of the\noscillating magnetic field of a resonator and the square root\nnumber of spins14. For studies involving bulk magnetic mate-\nrials and low quality and large microwave resonators, strong\ncoupling is achieved when geff=2pis in the MHz range, which\nis accomplished with relative ease due to the abundance of\nspins in bulk magnetic materials. A reduction of the bulk di-\nmensions down from mm to mm and nm scales, the typical\nlateral dimensions and thickness of vdW material monolay-arXiv:2206.02460v2  [cond-mat.mtrl-sci]  28 Apr 20232\ners, results in a decrease of the coupling strength by at least 6\norders of magnitude. Commonly used microwave resonators\nare not able to produce strong enough oscillating magnetic\nfields to compensate for such a reduction in absolute number\nof spins. Only by advanced resonator design and engineering\nthe regime of strongly coupled MPs in monolayer vdW mag-\nnetic materials can be accomplished, granting access to spin\ndynamic physics at a true 2d monolayer limit and research on\nMPs in nano-scale devices where the whole range of on-chip\ntuning and engineering tools, such as electric fields or device\ndesign, are available.\nMagnons or magnon polaritons have been observed in mag-\nnetic vdW materials, but it had been restricted to either to the\noptical frequency range22,23or a large thickness limit24,25, re-\nspectively. Here, we present our attempt of detecting spin\ndynamics in ultra-thin vdW magnetic materials and the cre-\nation of MPs by magnon-photon coupling in the microwave\nfrequency range, using superconducting resonators optimized\nfor increased magnon-photon coupling. By using microwave\nresonators with a small mode volume, we not only increased\nits oscillating magnetic field strength but also matched it more\nefficiently to the size of nanoscale vdW flakes. Our work\npresents a fundamental cornerstone for a general blueprint\nfor designing and developing magnon-photon hybrids for any\ntype of ultra-thin or monolayer vdW magnetic material, en-\nabling research on on-chip microwave applications for (quan-\ntum) information processing.\nRESULTS\nIn this article, we report on the observation of spin dynam-\nics and the creation of MPs at the onset of the high cooper-\nativity regime with the vdW ferromagnet CGT of nm scale\nthickness, demonstrating a pathway towards stable magnon-\nphoton polariton creation. We combine a precise transfer\nprocess of exfoliated CGT flakes and high sensitivity su-\nperconducting resonators, to access and study the dynami-\ncal response of coupled photon-magnon states in a small-\nvolume (nm-thick and \u0016m-sized) CGT flake (illustrated in\nFig. 1 (a)). High-quality-factor superconducting lumped el-\nement resonators are chosen to be the counterpart due to\ntheir extremely small mode volume ( \u00196000\u0016m3) and con-\nsequently strong oscillating magnetic fields ( B1\u001925nT, see\nSI for resonator quality-factors and B1-field distributions), re-\nsulting in high spin sensitivities4,26. At cryogenic temper-\natures, we perform low-power microwave spectroscopy on\nmultiple resonator-vdW-flake hybrids, covering a frequency\nrange from 12GHz to 18GHz for a variety of thickness. Sam-\nples consist of up to 12 resonators on a single chip, all capac-\nitively coupled to a common microwave transmission line for\nread-out (see SI for details). Multiple peaks of spin-wave res-\nonances are observed for each CGT flake measured. The spin-\nwave modes are closely spaced in frequency and show a large\noverlap. We employ a semi-optimized fitting model to pro-\nduce a good estimate for the collective coupling strength and\nmagnetic linewidth. By taking the resonance value of the most\nprominent peak of each spectrum, we find that all measuredpoints can be fitted very well by a single curve calculated by\nthe Kittel formula with bulk CGT parameters. Furthermore,\nwe extracted the linewidth for the thinnest CGT flake inves-\ntigated, 11nm or 15 monolayers (ML), the only device ex-\nhibiting well separated spin-wave modes. This allowed a fully\nquantitative analysis and we determined an upper limit of the\nGilbert damping parameter of 0 :02. This value is comparable\nto the damping reported for 3d transition metal ferromagnets,\nsuggesting that magnetic vdW flakes have the potential for the\nfabrication of functional spintronic devices.\nWe investigate the dynamics of nm-thick CGT flakes, us-\ning superconducting lumped element resonators made of NbN\n(see methods for fabrication details and SI and Ref. [28] for\nmore performance details). The advantages of a lumped ele-\nment design are the spatial separation of the oscillating mag-\nnetic field B1and electric field E1and the concentration of\nB1within a narrow wire section of the resonators, as indi-\ncated in Fig. 1 (a). Additionally, the B1field distribution is\nhomogeneous along the length of the narrow wire section (see\nfinite element simulations in SI). This magnetic-field concen-\ntration is our primary reason to use this type of resonator in\norder to reduce the photon mode volume as well as achieve\na considerable mode overlap between the resonator photon\nmode and CGT magnon mode, and consequently, a large cou-\npling strength. We therefore transfer CGT flakes onto these\n5 μm\nB0CGTB1a\nbcE1\n0 4 8\nx (μm)y (nm)\n102030\n1240\nMCGT\nCrGeTe\nB1,extent  ≈ 2 μm\nFIG. 1. Magnon-photon coupling between thin CGT and a super-\nconducting resonator. a Schematic of a resonator shows the design\nin detail, indicating the areas of high E1-field (yellow) and B1-field\n(green) intensities, as well as the orientation of the externally applied\nfield B0. Finally, a schematic zoom in of the section loaded with a\nCGT flake is shown. The collective coupling between a microwave\nphoton and the magnetization of the CGT is illustrated, as well as the\napproximate extent of the microwave B1-field. bMicrograph image\nof a CGT flake transferred onto the narrow section of a resonator. c\nAFM image of the CGT flake together with a height profile along the\nblue solid line in the AFM image. The red solid line is a fit to the\nflake thickness. The results of this resonator are presented in Fig. 2.3\n12.8112.8212.83 ω/2π (GHz)\n560 580 600 620\nMagnetic Field (mT)640 66012.841.0 0.9 0.8|S21|20.7\n|S21|21.0\n0.9\n0.8a b\nc\n580 600\nMagnetic Field (mT)620234\nκeff/2π (MHz)0510 ωres/2π (MHz)\nd\n640\n+ 12820 MHz\n0.7550 mT\n598 mT\n614 mT\n670 mT\n12.81 12.82 12.83\nω/2π (GHz)12.84\nFIG. 2. Magnon-photon coupling observed in resonator microwave transmission. a jS21j2as a function static magnetic field B0and\nfrequency, with the microwave transmission encoded in the color. The results are obtained from the resonator shown in Fig 1 (b) and (c),\nfeaturing a loaded quality factor of QL=4600. bjS21j2as a function of frequency at fixed magnetic fields, indicated in aby dashed vertical\nlines. canddResonance frequency wresand effective loss rate keffas a function of magnetic field. Note the multiple resonance peaks,\nindicating multiple CGT FMRs. The dashed orange lines are results from the semi-optimized fit. dexemplary includes the individual peaks of\nwhich the orange dashed lines consists. The green bar in canddhighlights the main mode.\nnarrow sections (Fig. 1 (b)). Details of CGT flake transfers\nare described in the methods section. Optical imaging and\natomic force microscopy (AFM) measurements are used to\ncharacterise the size and thickness of the CGT flakes (see\nFig. 1 (c)). Measured thicknesses range from 153 \u000623nm\ndown to 11\u00061:8nm (15 ML), enabling a thickness dependent\nstudy of CGT flakes and their coupling to the resonators.\nWe measured the microwave transmission jS21j2as a func-\ntion of frequency and externally applied magnetic field B0for\neach resonator at a temperature of 1 :8K, using a microwave\npower of approximately \u000080dBm at the resonator chip. Fig-\nure 2 (a) shows the resulting 2D plot of jS21j2for a resonator\nloaded with a 17nm \u00060:8nm thick CGT flake (see Fig. 1 (b)\nand (c) for the respective micrograph and AFM images). A\nresonator peak can be clearly observed for each magnetic\nfield, with its resonance frequency wresdecreasing with in-\ncreasing magnetic field. The reduction of the frequency is\na result of a slow degradation of the superconductivity by\nB0, which in general exhibits a parabolic dependence29. For\n580mT\u0014B0\u0014630mT the resonator prominence is reduced,\nhighlighted byjS21j2as a function of frequency for four con-\nstant B0values in Fig. 2 (b). Within this field range, the mode\nresonance has been modified due to its hybridization with the\nmagnetic modes of the CGT flake. To further quantify the in-\nteraction, we fit each jS21j2profile by a Fano resonance line-\nshape (solid orange lines in Fig. 2 (b)) to account for an asym-\nmetric resonance peak due to additional microwave interfer-\nence in the circuitry30,31,\njS21j2=S0+A(qkeff=2+w\u0000wres)2\n(keff=2)2+ (w\u0000wres)2: (1)\nHere, S0is the microwave transmission baseline, Athe peak\namplitude, qdescribes the asymmetry of the lineshape and\nkeffrepresents the effective loss rate of the hybrid system (seeSI for resonator parameters before and after CGT transfer for\nall resonators). Figure 2 (c) shows wresof the hybrid system\nas a function of B0.wresexperiences a dispersive shift when\nthe photon mode and the magnon mode hybridize, indicating\nan onset of a strong interaction between the two individual\nsystems14,17,32–34. We observe multiple shifts in wres, suggest-\ning an interaction of several magnon modes with the resonator\nin our experiment.\nSignatures of the resonator–CGT-flake coupling are also\ncharacterised by keffof the hybrid system (Fig. 2 (d)). keffis\nenhanced from the value of the resonator loss rate k0due to\nan additional loss introduced by the magnon system charac-\nterized by the loss rate g14,32,35. Consistent with the B0de-\npendence of wres,keffshows a rich structure, having its main\npeak at 598mT, together with less prominent peaks distributed\naround it. Based on a formalism for coupled-harmonic-\noscillator systems in the high cooperativity regime32–34, we\nuse the following to analyse our experimental results with\nmultiple peaks:\nwres=wres;0+mB2\n0++n\nå\nk=\u0000ng2\neff;kDk\nD2\nk+g2; (2)\nkeff=k0++n\nå\nk=\u0000ng2\neff;kg\nD2\nk+g2: (3)\nwith the detuning factor for each resonance as Dk=\ngCGTmB\n¯h\u0000\nB0\u0000BFMR ;k\u0001\n. Here, wres;0is the resonator resonance\nfrequency at B0=0T and mrepresents the curvature of the\nresonance frequency decrease due to the applied magnetic\nfield. BFMR ;kis the CGT FMR field, gCGT the g-factor of\nCGT and geff;kgives the collective coupling strength between\nphoton and magnon mode. The summation is over all reso-\nnance modes kpresent on the low or high field (frequency)4\nside of the main resonance mode, where ngives the number\nof modes on one side. For simplicity, we assume a symmet-\nric distribution of modes about the main mode. The large\nnumber of multiple modes and their strong overlap prevent\na reliable application of a fully optimized fit to the data, due\nto the large number of free parameters required. In an ef-\nfort to gain a good estimate of the model parameters we ap-\nply the model functions Eq. (2) and (3) in a two-step semi-\noptimized fashion (see SI for details). With this approach, we\narrive at a model in good agreement with wresandkeff(see\norange dashed lines in Fig. 2 (c), (d), exemplary showing the\nindividual peaks of the orange dashed line in Fig. 2 (d) and\nthe SI for additional results and data). We can reproduce the\ndata using g=2p=94:03\u00065:95MHz and a collective cou-\npling strength of the main mode of 13 :25\u00061MHz. Together\nwith k0=2p=1:4\u00060:02MHz the system resides at the onset\nof the high cooperativity regime, classified by the cooperativ-\nityC=g2\neff=k0g=1:3>113,32. In this regime, magnon polari-\ntons are created and coherently exchange excitations between\nmagnons and resonator photons on a rate given by geff. The\ncreated MPs are, however, short lived and the excitations pre-\n100 200 500 700\nResonance Field BFMR (mT)300 400 600 0051015ωFMR/2π (GHz)500\nResonance Field BFMR (mT)600 7001518ωFMR/2π (GHz)\n12a\nb\n11 31 51 71 91 111 131151Flake Thickness (nm)\nFIG. 3. Summary of CGT-FMR conditions. a Extracted CGT res-\nonance fields and frequencies from the set of resonators loaded with\nCGT flakes of different thickness. Resonance values are taken from\nthe most prominent peaks in keff. The solid curve is calculated us-\ning the Kittel formalism presented in10, using same parameters, with\ngCGT =2:18,m0Ms=211:4mT and Ku=3:84\u0002104J=m3.bWider\nmagnetic field range of awhere the CGT flake thickness for the dif-\nferent symbols is indicated by the color gradient given in a.dominately dissipate in the magnonic system, as geff\u001cg.\nOur analysis suggests that the separation of the different\nFMR modes is of the same order of magnitude as the loss rate\n(see SI for additional data). We consider that these are from\nstanding spin wave resonances, commonly observed for thin\nmagnetic films12and with one reported observation in bulk of\nthe vdW material CrI 338. In thin-film magnets under a static\nmagnetic field applied in-plane, the magnetic-dipole interac-\ntion generates two prominent spin wave branches for an in-\nplane momentum, the backward volume spin wave (BVMSW)\nand magnetostatic surface spin wave (MSSW) modes39,40.\nThese spin wave modes have different dispersion relations,\nhaving higher (MSSWs) and lower (BVMSWs) resonance\nfrequencies with respect to that of the uniform FMR mode.\nWe calculate the distance of these standing spin-wave modes\nbased on magnetic parameters of bulk CGT as well as the lat-\neral dimensions of the flakes (see SI for more details). We\ncan find spin waves having a frequency separation within\n100MHz and 200MHz (3 :3mT to 6 :6mT in magnetic field\nunits), which are consistent with our experimental observa-\ntion in terms of its mode separation. However, the irregular\nshape of the CGT flakes renders exact calculations of spin\nwave mode frequencies very challenging. We also consid-\nered a possibility that each layer of CGT might have different\nmagnetic parameters (e.g. chemical inhomogeneity), and thus\nproducing different individual resonance modes. Our numer-\nical simulations based on atomistic spin dynamics14,15rule\nout this possibility, as resonance modes from individual lay-\ners average to a single mode as soon as a fraction of 10% of\ninter-layer exchange coupling is introduced (see SI for more\ndetails). Therefore, we speculate that the multiple mode na-\nture we observe in our experiments is likely originating from\nintrinsic properties of the CGT flakes.\nFigure 3 shows the extracted wFRM as a function of BFMR\nfor each resonator–CGT-flake hybrid. The experimental val-\nues are in excellent agreement with a curve calculated by the\nKittel equation with magnetic parameters for bulk CGT10,\nfrom which the data exhibits a standard deviation of less than\n5%. This agreement, achieved by independent characteri-\nzations of 23 CGT flakes measured by superconducting res-\nonators, is experimental evidence that the magnetic parame-\nters that determine the dispersion of wFRM (BFMR), i.e. the\nCGT g-factor gCGT, saturation magnetization Msand uniaxial\nanisotropy Ku, exhibit little thickness dependence in exfoli-\nated CGT flakes, and are not disturbed by the transfer onto\nthe resonator structure. We note, that this demonstrates that\nvdW magnetic materials are particularly attractive for device\napplications, as they are less prone for contamination from\nexfoliation.\nFinally, we present our analysis of kefffor a resonator with\na 11\u00061:8nm CGT flake in Fig.4. With the thickness of a\nsingle layer of CGT being 0 :7nm9, this flake consists of 15\nmonolayers and is the thinnest in our series. Figure 4 (a) and\n(b) show wresandkeffas a function of B0, respectively. While\nthe response of the CGT flake shows a prominent signature\ninkeff, the CGT FMR is considerably more subtle in wres.\nThis highlights the excellent sensitivity of the high-Q super-\nconducting resonators in our study. kefffeatures five well-5\nseparated peaks with the main peak at B0=547mT, which\nenables us to perform a single-peak fully optimized analy-\nsis for each, in contrast to our multi-step analysis for the re-\nmainder of the devices. We assume the additional peaks are\nBVMSW modes, as discussed in the previous section. How-\never, the splitting is about four times larger than compared to\nall other investigated devices, which would result in a signifi-\ncantly shorter wavelength. Thickness steps can lead to a wave-\nlength down-conversion13, however, due to the irregular shape\nandB1inhomogeneities it is difficult to exactly calculate the\nspin wave frequencies (see SI for further details). From the\nmain peak profile, we extract geff=2p=3:61\u00060:09MHz,\ng=2p=126:26\u00068:5MHz and k0=2p=0:92\u00060:05MHz. We\ncompare the experimental value of geffwith a numerically cal-\nculated geff;simu, using the dimensions of the CGT flake de-\ntermined by AFM measurements (see SI for details). The\ncalculation yields geff;simu=2p=8:94MHz, lying within the\nsame order of magnitude. The overestimation is likely due\nto in-perfect experimental conditions, like non-optimal place-\nment of the flake, uncertainties in the thickness and dimen-\nsion determination as well as excluding the additional modes\nin the calculation (see SI). With g\u001dgeffandC=0:11, the\nhybrid system is in the weak coupling regime13, but due to\nthe highly sensitive resonator with its small k0the response\nfrom the magnon system can still be detected. With the ex-\ntracted g=2pwe can give an upper limit of the Gilbert damp-\ning in CGT, by calculating aupper =g=wFMR. We find aupper as\n0:021\u00060:002, which is comparable to other transition metal\nmagnetic materials44, and is in very good agreement with a\npreviously reported effective Gilbert damping parameter de-\ntermined by laser induced magnetization dynamics45. Here,\nwe emphasise that the actual Gilbert damping value is lower\ndue to a finite, extrinsic inhomogeneous broadening contribu-\ntion.\nWe further use these results to benchmark the sensitivity\nof our measurement techniques. The detection limit is given\nby comparing the main peak height characterised by g2\neff=g\nand the median noise amplitude which is 18kHz in Fig. 4 (b)\nwhere g2\neff=2pg= 103 kHz. By assuming the same lateral di-\nmensions and scale the thickness down to a single monolayer,\nwhile keeping gconstant, we calculate the expected signal re-\nduction numerically by geff;simu;1ML=geff;simu;15ML to 0.26. We\nobtain (0:26geff)2=2pg=7kHz for the monolayer limit. Al-\nthough this suggests the noise amplitude is greater than the\nexpected peak amplitude, we can overturn this condition by\nimproving the coupling strength by optimising the resonator\ndesign, enhancing the exfoliation and flake transfer as well as\nby reducing the noise level by averaging a number of mul-\ntiple scans. Superconducting resonators with mode volumes\nof about 10 \u0016m3have been realised46, a reduction of 2 orders\nof magnitude compared to our current design. This would\ntranslate to an order of magnitude improvement in geff. Fur-\nthermore, this flake covers about 4% of the resonator. By\nassuming maximised coverage a 5 times enhancement of geff\ncan be achieved. Both approaches would make the detection\nof monolayer flakes possible.\nIn summary, we provide the first demonstration of photon-\nmagnon coupling between a superconducting resonator and\n520 540 600 6400.900.951.001.05\nκeff/2π (MHz)\nMagnetic Field (mT)560 580 620\nωres/2π (MHz)\n122801229012300a\nbFIG. 4. Magnon-photon coupling for the thinnest CGT flake. a\nResonance frequency wresandbeffective loss rate keffas a func-\ntion of magnetic field of a resonator loaded with the thinnest CGT,\nconsisting of 15 ML. The resonator’s loaded quality factor is 6938.\nThe solid orange lines are results a fit to Eq.(2) and (3), respectively.\nThe errorbars in brepresent the standard deviation from the Fano\nresonance lineshape fit to the resonator transmission.\nnm-thick vdW flakes of CGT, using a total of 23 devices\nwith different CGT flakes of thickness from 153nm down to\n11nm. By employing a coupled-harmonic-oscillator model,\nwe extract the coupling strength, magnetic resonance field\nand relaxation rates for both photon and magnon modes in\nour devices. From our semi-broadband experiments, we find\nthat the magnetic properties of exfoliated CGT flakes are ro-\nbust against the transfer process, with a standard deviation of\nless than 5% to expected resonance values from bulk param-\neters. Notably, this suggests that vdW magnetic materials can\nbe pre-screened at bulk to identify the most promising mate-\nrial for few layer device fabrication. The upper limit of the\nGilbert damping in the 15 ML thick CGT flake is determined\nto be 0 :021, which is comparable to commonly used ferro-\nmagnetic thin-films such as NiFe and CoFeB and thus mak-\ning CGT attractive for similar device applications. We high-\nlight that the damping parameter is key in precessional mag-\nnetisation switching47,48, auto-oscillations by dc currents49,50,\nand comprehensive spin-orbit transport in vdW magnetic sys-\ntems51. The presented techniques are readily transferable\nto other vdW magnetic systems to study spin dynamics in\natomically-thin crystalline materials. While creating stable\nmagnon polaritons is still an open challenge due to the large\nloss rate gof the CGT magnon system, this work offers an\nimportant approach towards its achievement. There are still\npotential improvements to the measurement sensitivity such\nas resonator mode volume reduction by introducing nm scale\nconstrictions52,53and use of exfoliation/transfer techniques to\nproduce larger flakes to enhance the mode overlap (hence cou-6\npling strength)54,55. With concerted efforts, the formation of\nmagnon polaritons in few layers vdW materials will become\nfeasible.\nMETHODS\nSuperconducting Resonators: The resonators were fab-\nricated by direct laser writing and a metal lift-off process.\nThe individual 5mm \u00025mm chips are scribed from an in-\ntrinsic, high resistivity ( r>5000Wcm) n-type silicon wafer\nof 250 \u0016m thickness. For a well defined lift-off, we use a\ndouble photoresist layer of LOR and SR1805. The resonator\nstructures are transferred into the resist by a Heidelberg Di-\nrect Writer system. After development, \u001850nm NbN are de-\nposited by magnetron sputtering in a SVS6000 chamber, at\na base pressure of 7 \u000210\u00007mbar, using a sputter power of\n200W in an 50:50 Ar/N atmosphere held at 5 \u000210\u00003mbar,\nwith both gas flows set to 50 SCCM28. Finally, the lift-off is\ndone in a 1165 solvent to release the resonator structures.\nCGT Crystal Growth: CGT crystals used in this study\nwere grown via chemical vapour transport. To this end, high-\npurity elemental precursors of Cr (chips, \u001599:995%), Ge\n(powder,\u001599:999%), and Te (shots, 99 :999%) were mixed\nin the molar weight ratio Cr:Ge:Te = 10:13.5:76.5, loaded into\na thick-wall quartz ampule and sealed under the vacuum of\n\u001810\u00005mbar. Then, the ampule was loaded into a two-zone\nfurnace, heated up and kept at 950\u000eC for 1 week to homog-\nenize the precursors. To ensure high-quality growth, the am-\npule was slowly cooled (0 :4\u000eC=h) maintaining a small tem-\nperature gradient between the opposite ends of the ampule.\nOnce the ampule reached 500\u000eC, the furnace was turned off\nallowing the ampule to cool down to room temperature nat-\nurally. The large ( \u00181cm) single-crystalline flakes were ex-\ntracted from the excess tellurium and stored in the inert envi-\nronment.\nCGT Flake Transfer: Devices for this study were made\nvia transfer of single-crystalline thin flakes on top of the super-\nconducting resonators. The flakes were first exfoliated from\nbulk crystals on the clean surface of a home-cured PDMS\n(polydimethylsiloxane, Sylgard 184) substrate. The thickness\nof the CGT flakes on PDMS was estimated through the con-\ntrast variation with transmission optical microscopy. Then,\nthe selected flake was transferred to a resonator. The trans-\nfer was performed in air at room temperature. To minimize\nthe air exposure, the entire process of exfoliation, inspection\nand transfer was reduced to 10-15 min per resonator. For\nthe flakes thicker than 50nm, the strong optical absorption of\nCGT prevented the accurate thickness estimation with optical\ncontrast. 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E.J.G.S. acknowledges computational resources\nthrough CIRRUS Tier-2 HPC Service (ec131 Cirrus Project)\nat EPCC (http://www.cirrus.ac.uk) funded by the University\nof Edinburgh and EPSRC (EP/P020267/1); ARCHER UK Na-\ntional Supercomputing Service (http://www.archer.ac.uk) via\nProject d429. E.J.G.S acknowledges the Spanish Ministry\nof Science’s grant program “Europa-Excelencia” under grant\nnumber EUR2020-112238, the EPSRC Early Career Fellow-\nship (EP/T021578/1), and the University of Edinburgh for\nfunding support. D.S. acknowledges EPSRC funding through\nthe Centre for Doctoral Training in Advanced Characteri-\nsation of Materials (EP/L015277/1) and European Union’s\nHorizon 2020 Research and Innovation program under grantagreement GrapheneCore3, number 881603 and the Depart-\nment for Business, Energy and Industrial Strategy through the\nNPL Quantum Program.\nAUTHOR CONTRIBUTION\nC.W.Z, S.K. and H.K. conceived the experimental project.\nResonator design and optimization was done by J.O’S.,\nO.W.K, C.W.Z and supervised by J.J.L.M. Resonator fabri-\ncation and characterization was done by C.W.Z. CGT crystals\nwere grown by I.A.V . and exfoliated and transferred by I.A.V .\nand N.V .T.T. and supervised by E.G.. D.S. measured AFM on\nthe CGT flakes on the resonators. C.W.Z. performed the ex-\nperiments and the data analysis with input from S.K. and H.K.\nAtomistic spin dynamics simulations were carried out by M.S.\nsupervised by E.J.G.S.. C.W.Z., M.S., I.A.V . and H.K. wrote\nthe manuscript with input from all authors.\nCOMPETING INTERESTS\nThe Authors declare no conflict of interests.\nSupplemental Material - Probing spin dynamics of ultra-thin van der Waals magnets via\nphoton-magnon coupling\nI. MICROWAVE SETUP AND MEASUREMENT\nVNA\nMW out MW in\n-20\ndB+32\ndB\nDUT\nB0Cryostat\nFIG. S1. Microwave delivery and detection setup. Schematic of the microwave delivery and detection circuit. The image shows the coplanar\nwaveguide transmission line. A resonator chip is placed on top of the transmission line for read out. On the right, a schematic layout of the\nresonators on a single chip is shown.2\nFigure S1 shows a schematic of the used microwave measurement setup. We are using a Keysight E5071C vector network\nanalyzer (VNA) to deliver and detect microwaves. The VNA is connected to a low temperature probe, fitted into a closed\ncycle helium cryostat and cooled to a base temperature of about 1 :8K. The microwave signal is transmitted into the cryostat\nand is attenuated by \u000020dB. The attenuator is positioned just before the sample box and provides a thermal anchoring for\nthe center conductor of the coaxial cable to minimize the thermal load onto the sample. The output line is equipped with a\nLow Noise Factory LNC6_20C cryogenic amplifier, operating between 6 \u000020GHz with an average amplification of +32dB.\nThe transmitted and amplified signal is finally detected by the VNA. Figure S1 also shows an image of the coplanar waveguide\ntransmission line PCB, loaded with a resonator ship, of which a schematic shows the resonator layout on a single chip. The\nresonators on the chip are capacitively coupled to the transmission line PCB. Upon resonance the transmission through the\nPCB is reduced, indicating the resonator resonance. The cryostat is equipped with a mechanical rotation stage and prior to the\nmeasurements the superconducting resonators are carefully aligned to the externally applied static magnetic field B0, such that\nthe field is in the plane of the superconductor and along the narrow section of the resonators.\nFigure S2 shows the raw uncalibrated microwave transmission, ranging from 10GHz to 18GHz. The transmission is domi-\nnated by imperfections in our microwave circuitry, masking the small signals from the superconducting resonators. Thus, we\nperformed a simple thru calibration of the microwave transmission to remove contributions from the setup, prior each magnetic\nfield dependent measurement. Here, we exploit the magnetic field tunability of our superconducting resonators. Before calibra-\ntion, we set the frequency range of the measurement. We change the applied magnetic field such that the resonator’s resonance\nfrequency is tuned out of the set frequency range. With a frequency window just showing the transmission of the setup we\nperform the thru calibration. After calibration we set the magnetic field back to its starting value, resulting in a background\ncorrected spectrum with just the resonator feature on it.\n10 11 12 13 14 15 16 17 18-1010\n0\nRaw Transmission |S21| (dB)\nFrequency (GHz)\nFIG. S2. Raw broadband microwave transmission signal. Logarithmic microwave transmission jS21jas a function of frequency between\n10GHz and 18GHz at a temperature of 1 :8K.\nII. RESONATOR CHARACTERIZATION\nIn this study, we fabricate twelve superconducting lumped element resonators on each of three resonator chips were fabricated\nusing the same design (see schematic Fig. 1 (a) in the main text). Prior to transfer of the CGT flakes, we characterized the res-\nonators at a temperature of 1 :8K and zero applied magnetic field, using microwave powers of about \u000080dBm at the resonators,\nwhich is well below the bifurcation limit starting above \u000060dBm. Due to finite fabrication tolerances the resonator parameters\nhave some variation, while some didn’t work at all. However, the targeted resonance frequencies are well reproducible and very\nsimilar for the 3 different chips. We compare the resonator parameters before and after transfer of the CGT flakes and collate the\nparameters in Tab. I. Note, the resonator parameters with the CGT flakes on were obtained with a static magnetic field applied\nin the plane of the superconductor, but far detuned from the CGT FMR. In addition, we add the respective thickness of the flake\non each resonator, acquired from AFM measurements. Here, we give the values of the thickest region of a given flake on a\nresonator, as the thickest region will dominate the FMR signal. Due to the arbitrary shape of exfoliated flakes, some exhibit\nregions of different thickness, as seen e.g. in Fig. S5 (h) and (i).3\nTABLE I. Resonator Parameters\nChip Number wres;before (MHz) QL;before wres;after(MHz) QL;after CGT Thickness (nm)\n1 12165 1978 12063 5733 16.2\u00061.3\n1 13303 7357 13177 4950 -\n1 13968 5575 13860 4679 49.4\u00063.5\n1 14184 6492 14048 5627 153.1\u000623.3\n1 16648 6606 16470 5021 23.5\u00062.5\n1 17431 3215 17237 6826 23.8\u00066.4\n1 17959 7595 17790 3963 26.2\u00064.1\n2 12285 360 12153 7135 49.1\u00069.1\n2 12669 3600 12548 6693 102.8\u00065.6\n2 12782 3448 12648 6557 105.9\u00063.9\n2 13393 4643 13244 4501 34.4\u00064.1\n2 13760 6858 13620 5488 95.9\u00065.9\n2 14395 9048 14201 4139 36.7\u00064.3\n2 16075 7283 - - -\n2 17048 6541 16811 4241 75.5\u00065.4\n3 12043 6114 11899 6044 59.7\u000632.8\n3 12456 2716 12314 6938 11.4\u00061.8\n3 12996 5828 12848 4600 17\u00060.8\n3 13422 6517 13272 5461 89.8\u00067.5\n3 13719 6800 13582 6608 -\n3 14238 9184 14064 5420 73.5\u00068.4\n3 15390 8680 15219 6030 30.5\u00064.2\n3 15821 2386 15604 4769 33.1\u00069.9\n3 16430 7518 16193 5780 30.1\u000638.1\n3 17308 6521 17054 5569 137.9\u00063.4\n3 18111 3542 17870 4643 50.2\u00066.9\nIII. RESONATOR AND COUPLING SIMULATION\nWe use finite element and numerical simulations to optimize our resonator design. Key requirements of our resonators are\na strong resilience to externally applied static magnetic fields and a small mode volume. To achieve a large field resilience we\nreduced the area of the resonator to minimize effects of the magnetic field on the superconducting film. Further, we designed the\nresonators such that they act as lumped element resonators. Here, the resonance frequency is given by the total capacitance and\ninductance of the structure, with wres=1=p\nLC, analogues to a parallel LC circuit. This allows us to locally separate oscillating\nelectric and magnetic fields and also to concentrate the magnetic fields in more confined regions, resulting in very small mode\nvolumes. To verify the lumped element nature of our resonators we performed finite element simulations, using CST Microwave\nStudio. Figure S3 shows the resulting magnitude of the E-field (left side) and H-field (right side) distribution along the resonator\nstructure for the resonator design producing the results shown in Fig. 2 in the main text. The E-field is concentrated along the\nparallel running wire sections, with its strength approaching zero along the narrow wire section. The opposite is the case for the\nH-field, where it is zero along the parallel wire sections and strongly concentrated along the narrow wire section. Note, that the\nH-field magnitude is homogeneous along the whole of the narrow wire section.\nThe CST Microwave Studio at hand allowed us a simulation with perfect electric conductors. This is sufficient to model\nthe general electric and magnetic energy distributions and resonance frequencies, however, not to simulate the corresponding\noscillating magnetic field distribution, created by a superconducting rectangular wire. To this end, we numerically solve the\nBiot-Savart law for a rectangular wire cross-sectionS1, assuming a superconducting current distribution Jx;zS2,\nB1;x;z=m0\n2pZw=2\n\u0000w=2Zd=2\n\u0000d=2J\u0002r\n(x\u0000x0)2+ (z\u0000z0)2dx0dz0; (S1)\nwith the vectors as J= (0;J(x;z);0)Tandr= (x\u0000x0;0;z\u0000z0)Tandm0being the magnetic constant. The integration is performed\nover the cross-section of the wire, of width wand thickness d. We define the wire cross-section in the x-z-plane, with win x-\ndirection and din z-direction. The length of the wire is along the y-direction. For a superconducting wire, the current is\nnot homogeneously distributed over the cross-section of the wire. Current is only flowing on the surface and is exponentially\ndecaying towards the center of the wire. The characteristic length scale is given by the London penetration depth lL. We use the4\nFIG. S3. Finite element simulations of resonator. CST Microwave Studio simulation of the distribution of E-fields and H-fields across the\nresonator structure. The color encoded fields represent the magnitude values.\nfollowing expression for the current distributionS2\nJ(x;z) =J1 \ncoshz0=lL\ncoshd=lL\"\nCcoshx0=l1\ncoshw=l1+1\u0000cosh x0=l2=coshw=l2p\n1\u0000(x0=w)2#\n+J2\nJ1coshx0=lL\ncoshw=lL!\n; (S2)\nwhere\nJ2\nJ1=1:008\ncoshd=lLs\nw=l?\n4\u0003l?=lL\u00000:08301lL=l?;\nC=\u0010\n0:506p\nw=2l?\u00110:75\n;\nl1=lLp\n2lL=l?;\nl2=0:774 l2\nL=l?+0:5152l?;\nl?=lL=2d:\nThe prefactors J1andJ2define the amplitude of the current density and hence the absolute value of the oscillating magnetic field\nB1. We define J1by normalizing the vacuum B1field to the energy density stored in the resonatorS3,S4\n1\n2¯hwres\n2=1\n2m0Z\nB2\n1dV=1\n2m0B2\n1Vm; (S3)\nwith Vmrepresenting the resonator mode volume. The additional factor of1=2on the left hand side of S3 takes into account that\nonly half of the total energy is stored in the magnetic fieldS5. As our resonator design is a quasi 1-dimensional structure we have\nto define boundaries for the mode volume in the x- and z-direction. A common assumption is to use the width of the conductor\nwire wS6. For simplicity, we approximate the x-z-area of the mode distribution with the area of an ellipse. For the last dimension\nwe use the length of the narrow wire section, supported by the CST Microwave Studio simulations (see Fig. S3). In total we find5\nthe mode volume to be Vm= ((p3:0\u0016m\u00022:025\u0016m)\u0000w\u0002d)\u0002300\u0016m=5696\u0016m3. Figure S4 shows the resulting distribution\nof the oscillating magnetic field for the cross-section of the rectangular wire of width w=2\u0016m and thickness d=50nm. The\nmagnitudejB1;x;zjis encoded in the color and the arrows indicate the B1;xandB1;zcomponents of the oscillating field.\n-2000200z (nm)400\n-400\nx (μm)0 -1 1 2 3 -2 -320\n15\n10\n5\n|B| (nT)\n30\n25\nFIG. S4. Cross-section of resonator magnetic field distribution. Calculated magnitude of the magnetic field distribution around the cross-\nsection of a rectangular superconducting wire. The wire cross-section lies in the center, indicated by the grey rectangular. The red arrows show\nthe direction of the magnetic field.\nWith the simulated B1field distribution we can calculate the position dependent single photon - single spin coupling strength\ng0(r)S3,S4for each magnetic moment per unit cell of CGT (ab-plane 0 :68nmS7,S8, along the c-axis 0 :7nmS9). Summation over\nall CGT unit cells Nwithin the mode volume of the resonator results in the collective coupling strength\ngeff=s\nN\nå\ni=1jg0(ri)j2=gCGTmB\n2¯hs\nN\nå\ni=1jB1(ri)j2=gCGTmB\n2¯hNys\nN\nå\ni=1h\n(B2\nx;i+B2\nz;i)i\n: (S4)\nHere, mBis the Bohr magneton, Nyis the number of unit cells along the y-direction and gCGTis the g-factor for CGT for which\na value of 2 :18S10is used. Note, we give the collective coupling strength for spin1=2and for linear polarized microwavesS3. For\nthe calculation of gefffor the resonator loaded with 15 monolayers of CGT we extracted its lateral dimension from the AFM\nmeasurements (see Fig. S4 (g)) to 2 \u0016m along the x-direction and 12 \u0016m along the y-direction. The flake is assumed to lie directly\non top of the superconducting wire without any gap in between. For these values the simulation yields geff=2p=8:94MHz,\nwhich is about a factor 2 :5 larger than the experimentally determined value of 3 :61MHz. The overestimation of the simulation\nmost likely results from non-ideal conditions in the experiment. The corresponding flake lies at the top end of the resonators\nnarrow wire section (see Fig. S5 (g)), where B1is concentrated. The finite element simulations show that in this area the field\nstrength is already declining, resulting in a reduced coupling strength. Further, AFM can overestimate the thickness of a flake\nslightly for when there is a gap between resonator surface and flakeS9. The calculation also not includes the multiple peaks\nobserved in the experiment, which - depending on their real nature - can distribute the magnon density over all resonant peaks.\nNevertheless, we can use the simulation to estimate the signal reduction by scaling down the thickness of the flake to a single\nmonolayer. Reducing the simulation to a single monolayer, while keeping the lateral dimensions, results in geff=2p=2:33MHz,\na reduction by a factor of 0 :26.\nIV. AFM MEASUREMENTS ON CGT FLAKES\nAfter the transfer of the CGT flakes onto the individual resonators and after measuring FMR, we characterized the thickness\nof the flakes by AFM. Figure S5 shows a selection of height profile maps from the three resonator chips, including a height\nprofile along the inductor wire of the resonator (blue line in the AFM profile images in Fig. S5). To extract the thickness we fit\nthe steps in the height profile (red or green lines in the height profiles in Fig. S5). Note, the height values are relative values with\nan arbitrary offset. Figure S5 (g) shows the thinnest flake of this study, where the processed FMR data is shown in Fig. 4 in the\nmain text.6\n10 μm\n5 μm0 5\nx (μm)y (nm)40506070\n5 μmy (nm)\n02040\n0 4\nx (μm)2 6\n5 μmy (nm)\n506070\n0 4\nx (μm)2 680\n8\n5 μm\ny (nm)\n02040\n0 10\nx (μm)5 1560\n2080\ny (nm)050100\n0 10\nx (μm)5\n5 μm\n5 μm\ny (nm)\n02040\n0 10\nx (μm)5 15\n5 μm 10 μmy (nm)101520\n0 6\nx (μm)325\ny (nm)\n03060\n0 20\nx (μm)1090\ny (nm)50100\n0 6\nx (μm)3150 120\n30(c) (b) (a)\n(f) (e) (d)\n(i) (h) (g)9\nFIG. S5. AFM measurements. AFM profile images with respective height profile (above) along the resonator inductor wire (blue and\npurple lines in profile images, with the arrow indicating scan direction). a-cfigures for resonator chip 1 (refer to Tab. I), having resonance\nfrequencies with CGT of 17237MHz, 17790MHz and 16470MHz, respectively. d-ffigures for resonator chip 2 (refer to Tab. I), having\nresonance frequencies with CGT of 13244MHz, 12063MHz and 13620MHz, respectively. g-iimages for resonator chip 3 (refer to Tab. I),\nhaving resonance frequencies with CGT of 12314MHz, 13272MHz and 17054 ;MHz, respectively. The red and green solid lines are fits to the\nheight profiles.\nV. ANALYSIS AND ADDITIONAL FMR DATA\nWe analyze our experimental data, using the model functions (2) and (3) from the main text in a two-step semi-optimized\nfashion. The main intention for this approach is to minimize the number of free parameters in our model functions. In a first7\ncoarse step, we match the collective coupling strength geff;kto fit the experimental data, assume a constant separation between\nthe individual magnon modes at BFMR ;kand the same magnon loss rate gfor all modes and determine the resonator loss rate\nk0from the resonator transmission far detuned from the FMR with the CGT flakes. This results in 3 free parameters for the\nfirst stage of our analysis, the magnon loss rate g,BFMR of the main mode and the constant separation between the BFMR ;k.\nAfter this first step we arrive at a best fit to the envelope of the experimental data, however with not matching amplitudes. In a\nconsecutive second step, we manually optimize the geff;kto arrive at a model in good agreement with wresandkeff(see dashed\nlines in Fig. S6).\nFig. S6 shows additional results from the corresponding FMR measurements performed on the in Fig. S5 showed resonators.\nAs described in the main text, the measurements were performed at a temperature of 1 :8K and recording the microwave trans-\nmissionjS21j2as a function of the static magnetic field. Analyzing the microwave transmission by fitting a Fano resonance\nlineshape to it we extract the effective loss rate of the resonator, interacting with the CGT keff. Figure S6 shows the resulting\nkeffas a function of the magnetic field. In general, the response of the CGT FMR is complex and varies for the different res-\nonators. The resonance lineshape is not well described by just a single Lorentzian and requires multiple peaks to produce a\ngood agreement. For some resonators, keffexhibits obvious peaks, residing on a broader spectrum (see Fig. S6 (c), (f) and (i)).\nTogether with the observation of well and clearly separated peaks for the resonator loaded with the thinnest CGT flake of 11nm,\nwe motivating the multiple peak analysis as presented in the main text. However, as the individual peaks are overlapping for the\nremainder of the resonators we only applied a qualitative analysis.\nκeff/2π (MHz)580 600\nMagnetic Field (mT)62036912\n525 550\nMagnetic Field (mT)5751.61.82.0\n600\n625\n675 700\nMagnetic Field (mT)725246\n7508\n10\n500 525\nMagnetic Field (mT)5501.251.301.35\n5751.40\n520 560\nMagnetic Field (mT)6000.900.951.00\n6401.05\n650 675\nMagnetic Field (mT)7004812\n72516\n675 700\nMagnetic Field (mT)7251.41.61.8\nκeff/2π (MHz)\n750 650 700\nMagnetic Field (mT)750369\n800\n12\n500 520\nMagnetic Field (mT)540123\n560\n(c) (b) (a)\n(f) (e) (d)\n(i) (h) (g)\nκeff/2π (MHz)\nκeff/2π (MHz)\nκeff/2π (MHz)\nκeff/2π (MHz)\nκeff/2π (MHz)\nκeff/2π (MHz)\nκeff/2π (MHz)\nFIG. S6. Additional data on magnon-photon coupling of CGT-resonator devices. Results from FMR measurements with effective loss rate\nkeff=2pas a function of the static magnetic field. a-cresults for resonator chip 1 (refer to Tab. I), having resonance frequencies with CGT\nof 17237MHz, 17790MHz and 16470MHz, respectively. d-fresults for resonator chip 2 (refer to Tab. I), having resonance frequencies with\nCGT of 13244MHz, 12063MHz and 13620MHz, respectively. g-iresults for resonator chip 3 (refer to Tab. I), having resonance frequencies\nwith CGT of 12314MHz, 13272MHz and 11899MHz, respectively. The orange solid lines are semi-optimized fits, as described in the main\ntext. The errorbars in the figures represent the standard deviation from the Fano resonance lineshape fit to the respective resonator transmission.\nFigure S7 shows the extracted collective coupling strength geffas a function of the square root of the FMR active volume. We\ndefine the active volume as the overlap of the oscillating magnetic field B1and the CGT flake lying on the resonator. The B1\nfield distribution, discussed in Sec. III, is used to estimate the extend of the B1and is taken as 2 \u0016m. From AFM measurements\nand microscope images we extract the thickness and lateral dimensions of the flakes to calculate the final active volume. As\nthe collective coupling is proportional to the square root of the number of magnetic momentsS3, which are interacting with the\nresonator field, it follows that geffscales linearly with the square root of the active volume. This linear trend is highlighted by\nthe orange solid line in Fig. S7. The majority of the extracted data follows this linear trend very well, corroborating our analysis.\nOnly 3 data points deviate strongly from the rest of the data, which we attribute to significant inhomogeneities in the CGT-flakes,\nmaking the volume estimation inaccurate. These data points are highlighted in red in Fig. S7.8\n0 1 2 3 4 51030\n20Collective Coupling geff/2π (MHz)\nSquare Root of Active FMR Volume ( μm3/2)\nFIG. S7. Scaling of the collective coupling. Collective coupling strength geffas a function of the FMR active CGT-flake volume. The orange\nline highlights the linear trend of geffwith increasing volume. The red symbols are regarded as outliers, as these flakes show inhomogeneities,\nleading to inaccurate volume estimations. The star symbol represents data from the thinnest flake (see data in Fig. 4 in the main text) and the\npentagon symbol data from the 17nm flake (see data in Fig. 2 in the main text) The errorbars give confidence values for the extracted values.\nVI. MAGNETO-STATIC SPIN-WAVE DISPERSION IN THIN-FILM MAGNETS WITH PERPENDICULAR ANISOTROPY\nHere we describe the spin-wave mode frequency in a thin-film magnet with perpendicular anisotropy along the film normal.\nWe consider this at the magnetic-dipole limit where the wavelength is relatively large and the exchange interaction contribution\nto the spin-wave dispersion is neglected. Furthermore, standing spin-wave modes along the thickness direction are also ruled out\nsince these modes only appear at much higher frequencies than the main mode, where we consistently observe additional peaks\nat both higher and lower frequencies from the main mode. The mode (angular) frequency ( w) for wavevector k=0 when we\napply a magnetic field Balong one of the film plane directions can be given by Eq. 3d in Ref.S11as:\n\u0012w\ng\u00132\n=B\u0012\nB+m0Ms\u00002Ku\nMs\u0013\n: (S5)\nHere, g,MsandKuare the gyromagnetic ratio, saturation magnetization and the perpendicular anisotropy energy density, respec-\ntively. Note, that the total field within m0Ms\u00002Ku\nMsis negative for perpendicularly-magnetized films which we consider in this\nsection. Within the magnetic-dipole limit, the demagnetization term m0Msis modified for spin-waves with finite k, depending\non the relative orientation between the Msandkdirections. Here we follow the expression given in Serga et al.S12. For pure\nbackward volume magnetostatic modes where kkMs(illustrated in Fig. S8), the mode frequency becomes:\n\u0012wBVMSW\ng\u00132\n=B\u0012\nB+m0Ms\u00121\u0000e\u0000kt\nkt\u0013\n\u00002Ku\nMs\u0013\n; (S6)\nwhere tis the thickness of the magnet. Note, that this expression is only valid for the case where Msis colinear to B, meaning that\njBj>jm0Ms\u00002Ku\nMsj. To the limit of k!0, the term (1\u0000e\u0000kt)=ktis reduced to unity, consistent to Eq. (S5). When kis nonzero,\nwe can observe that wBVMSW becomes smaller than that for k= 0, exhibiting a negative group velocity for this spin-wave mode.\nAs the opposite extreme where k?Ms(illustrated in Fig. S8), the resonance frequency becomes larger than that for k= 0 and is\ncalled magneto-static surface spin-wave mode. The mode frequency expression for this mode is given by:\n\u0012wMSSW\ng\u00132\n=B\u0012\nB+m0Ms\u00002Ku\nMs\u0013\n+m2\n0M2\ns\u0010\n1\u0000e\u00002kt\u0011\n: (S7)\nHere, m2\n0M2\ns\u0000\n1\u0000e\u00002kt\u0001\nis the spin-wave correction term which goes to zero for k!0 (hence consistent to Eq. (S5)) and\nbecomes positive for k>0, meaning that wMSSW becomes larger as soon as spin-waves gain momentum along this direction.\nWe use these two expressions in an effort to explain the origin of the multiple peaks in our experiments. Figure S8 plots the\ncalculated wBVMSW =2pandwMSSW =2pas a function of wavevector k. The range of wavevector is chosen such that the resulting\nresonance frequencies are within the same order of magnitude as the observed mode splittings in the experiment ( µ100MHz).9\nWavevector (μm-1)6 4 8 10 12 2 012.712.812.9Resonance Frequency (GHz)13.0\n12.6\n12.5\n12.4Kittel\nBVMSW\nMSSWB0B0100 MHz\nFIG. S8. Spin-wave dispersion. Spin-wave resonance frequency for BVMSW (green solid line) and MSSW (yellow solid line) as a function\nof wavevector. The dashed blue line is the resonance frequency of the k=0 main mode. The parameters used are B0=598mT, gCGT =2:18,\nm0Ms=194:3mT and Ku=3:84\u0002104J=m3and a thickness of 17nm. The grey area highlights a 100MHz margin relative to the main mode,\nindicating the order of magnitude of the mode splitting observed in the experiment. The arrows on the right hand side illustrate the relative\nwavevector orientations of the BVMSW and MSSW spin-wave modes with respect to the static magnetic field.\nThe corresponding wavelength to a 100MHz resonance offset to the main mode are about 2 :2\u0016m and 620nm for wBVMSW\nandwMSSW , respectively. These values are within a reasonable scale for our different lateral CGT flake dimensions under\ninvestigation. This suggests that spin-wave modes are likely the origin of the multiple resonance peaks observed.\nThe thinnest CGT flake shows, however, a deviation from this behaviour. We only observe modes at lower frequencies, which\nwould indicate to BVMSW modes. Calculating the respective shortest wavelength results in 225nm, which is significantly\nshorter than for the other devices. We assume that the placement and irregular shape are likely to cause this difference. First,\nthis flake is placed at the very edge of the inductor wire, where the B1field strength is declining (see Fig. S3), reducing the FMR\nactive area. Thickness steps can lead to a wavelength down-conversionS13, however, with the overall irregular shape of the flake\nit is difficult to define a length scale for a standing spin wave mode.\nVII. ATOMISTIC SPIN DYNAMICS SIMULATIONS OF FMR\nTo study the ferromagnetic resonance in CGT we perform atomistic spin dynamics simulationsS14,S15. The magnetic Hamil-\ntonian employed in the simulations is given by:\nH=\u00001\n2å\ni;jSiJi jSj\u0000å\niDi(Si·e)2\u0000å\nimiSi·(B0+B1) (S8)\nwhere i,jrepresent the atoms index, Ji jrepresents the exchange interaction tensor, Dithe uniaxial anisotropy, which for\nCGT is orientated out of plane ( e= (0;0;1)) and B0the external static magnetic field applied in-plane during the ferromagnetic\nresonance simulations and B1=B1sin(2pnt)the oscillating field applied perpendicular with respect to B0. The CGT system\nhas been parameterized from first principle methodsS9, up to the third nearest neighbor intralayer and interlayer exchange. The\nexchange values have also been re-scaled by Gong et al.S9with a 0.72 factor to obtain the experimental TCand multiplied by\nS2to match the magnetic Hamiltonian. The magnetic moment or Cr is considered 3.26 mBS16and the uniaxial anisotropy has\na value of 0 :05 meV as extracted from first principle methodsS9. The parameters used in the simulations are given in Table II.\nFMR calculations have previously been employed for atomistic models, and can reproduced well the variation of linewidth with\ntemperature, for example, in recording media systemsS17. Hence, in the current simulations we use the same setup of frequency\nswept FMRS17and we obtain the spectra by performing a Fourier transform of the magnetisation component parallel to the\noscillating field. Since these calculations are done close to 0K, no averaging is require to reduce the thermal noise. To excite the\nFMR mode, we apply a DC field in-plane of 0.9 T on x-direction and an AC field perpendicular to the DC field, on y-direction.\nThe Fourier transform has been performed for the y-component of magnetisation for 5ns after an initial 1ns equilibration time.\nA thermal bath coupling has been chosen in agreement with the upper limit of the Gilbert damping observed in experiments.\nThe system size we performed FMR on is a 4-layer CGT system, with lateral size of 6 :91nm\u000211:97nm, periodic boundary\nconditions in xy and total of 1600 atoms. The small system size has been used to reduce the computational cost associated10\nQuantity Symbol quantity units\nTimestep ts 0.1 fs\nThermal bath coupling a 0.02\nGyromagnetic ratio ge 1.760859\u00021011rad s\u00001T\u00001\nMagnetic moment mB 3.26S16mB\nUniaxial anisotropy Di 0.05S9meV/link\nSimulation temperature T 0.001 K\nStatic magnetic field B0 0.9, 0.7 T\nOscillating magnetic field amplitude B0 0.001 T\nFMR frequency n varied GHz\nIntralayer exchange, NN J1 2.71S9meV/link\nIntralayer exchange, 2NN J2 - 0.058S9meV/link\nIntralayer exchange, 3NN J3 0.115S9meV/link\nInterlayer exchange, NN Jz\n1-0.036S9meV/link\nInterlayer exchange, 2NN Jz\n20.086S9meV/link\nInterlayer exchange, 3NN Jz\n30.27S9meV/link\nTABLE II. Simulation parameters for FMR on CGT system\n.\nwith FMR simulations. Experiments have showed modified g-factors due to photon-magnon coupling hence hereby we propose\na simple model where the properties of the individual layers have been modified to include different gyromagnetic ratio, as\nillustrated in Fig. S9 a.\nWe can define the resonance frequencies for each magnetic layer using the Kittel equation in the case of in-plane applied field\nwith perpendicular anisotropy B?u:\nw=gp\nB0(B0\u0000B?u) (S9)\nWe next investigate the FMR signal for a few cases assuming the CGT monolayers at low or strong interlayer exchange cou-\nplings J0\nz=0;0:1%;10%;100% Jz, where Jzcorresponds to the pristine interlayer exchange (Fig. S9 b-c). In the low interlayer\nexchange regime ( J0\nz=0;0:1%Jz), the CGT presents multiple peaks with each frequency corresponding to the layer dependent\ngyromagnetic ratio, g-n(g1) =16:81GHz, n(g2) =25:22GHz, n(g3) =33:62GHz. At J0\nz=0:1%J0\nz(Fig. S9 b) we can still\nobserve resonance peaks corresponding to each individual layer. However by increasing the exchange coupling to 10% J0\nzor\nhigher (Fig. S9 c) there is a single FMR peak indicating that the system behave coherently with all layers having the same FMR\nfrequency. The single FMR frequency corresponds to the average magnetic properties of the CGT layers. Small variations of\nthe resonance frequency as function of the inter-layer exchange coupling can be observed which these being correlated to the\ntransition of the system from the multi-peaks regime to a coherent excitation. By calculating the damping of the highest reso-\nnance peaks from a Lorenzian fit, we reobtain the damping corresponding to the input thermal bath coupling, 0 :02 with a relative\ntinny error\u00185%. Overall, the interlayer exchange coupling locks the dynamics of individual layers coherently together without\nallowing multiple frequencies at the FMR signalS18.\n[S1]A. E. Primenko, M. A. Osipov, and I. A. Rudnev, Technical Physics 62, 1346 (2017).\n[S2]L. H. Lee, T. P. 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Cao, W. Bao, C. Wang, Y . Wang, Z. Q. Qiu, R. J. Cava, S. G. Louie, J. Xia, and X. Zhang, Nature 546,\n265 (2017).\n[S10]S. Khan, C. W. Zollitsch, D. M. Arroo, H. Cheng, I. Verzhbitskiy, A. Sud, Y . P. Feng, G. Eda, and H. Kurebayashi, Physical Review B 100, 134437 (2019).\n[S11]M. Farle, Reports on Progress in Physics 61, 755 (1998).\n[S12]A. A. Serga, A. V . Chumak, and B. Hillebrands, Journal of Physics D: Applied Physics 43, 264002 (2010).\n[S13]J. Stigloher, T. Taniguchi, M. Madami, M. Decker, H. S. Körner, T. Moriyama, G. Gubbiotti, T. Ono, and C. H. Back, Applied Physics Express 11, 053002\n(2018).\n[S14]D. A. Wahab, M. Augustin, S. M. Valero, W. Kuang, S. Jenkins, E. Coronado, I. V . Grigorieva, I. J. Vera-Marun, E. Navarro-Moratalla, R. F. Evans, et al. ,\nAdvanced Materials 33, 2004138 (2021).\n[S15]A. Kartsev, M. Augustin, R. F. Evans, K. S. Novoselov, and E. J. G. Santos, npj Computational Materials 6, 1 (2020).\n[S16]I. A. Verzhbitskiy, H. Kurebayashi, H. Cheng, J. Zhou, S. Khan, Y . P. Feng, and G. Eda, Nature Electronics 3, 460 (2020).11\nFIG. S9. Atomistic simulations. a, Schematic of the crystal structure of CGT with atoms defined by different colours. b,FMR spectra of 4\nlayer CGT where the layers are low interayer exchange coupled (0 ;0:1%J0z, where J0zis the pristine CGT interlayer exchange). c,Similar as b,\nbut with the layers at a strong exchange coupling (10% ;100% J0z). The solid lines in b-crepresent a Lorenzian fit to the numerical data.\n[S17]M. Strungaru, S. Ruta, R. F. Evans, and R. W. Chantrell, Physical Review Applied 14, 014077 (2020).\n[S18]Data inputs/plots utilised for Supplementary Figure S7 (atomistic simulations) can be found at the following GitHub repository."    },    {        "title": "1701.08076v2.Structural_scale__q__derivative_and_the_LLG_Equation_in_a_scenario_with_fractionality.pdf",        "content": "Structural scale q\u0000derivative and the LLG-Equation in a scenario with\nfractionality\nJ.Weberszpil\u0003\nUniversidade Federal Rural do Rio de Janeiro, UFRRJ-IM/DTL and\nAv. Governador Roberto Silveira s/n- Nova Iguaçú, Rio de Janeiro, Brasil, 695014.\nJ. A. Helayël-Netoy\nCentro Brasileiro de Pesquisas Físicas-CBPF-Rua Dr Xavier Sigaud 150, and\n290-180, Rio de Janeiro RJ Brasil.\n(Dated: November 5, 2018)\nIn the present contribution, we study the Landau-Lifshitz-Gilbert equation\nwith two versions of structural derivatives recently proposed: the scale\u0000\nq\u0000derivative in the non-extensive statistical mechanics and the axiomatic met-\nric derivative, which presents Mittag-Leffler functions as eigenfunctions. The\nuse of structural derivatives aims to take into account long-range forces, possi-\nble non-manifest or hidden interactions and the dimensionality of space. Hav-\ning this purpose in mind, we build up an evolution operator and a deformed\nversion of the LLG equation. Damping in the oscillations naturally show up\nwithout an explicit Gilbert damping term.\nKeywords: Structural Derivatives, Deformed Heisenberg Equation, LLG Equation, Non-extensive\nStatistics, Axiomatic Deformed Derivative\nI. INTRODUCTION\nIn recent works, we have developed connections and a variational formalism to treat deformed or metric\nderivatives, considering the relevant space-time/ phase space as fractal or multifractal [1] and presented a\n\u0003Electronic address: josewebe@gmail.com\nyElectronic address: helayel@cbpf.br arXiv:1701.08076v2  [math-ph]  28 Feb 20172\nvariational approach to dissipative systems, contemplating also cases of a time-dependent mass [2].\nThe use of deformed-operators was justified based on our proposition that there exists an intimate\nrelationship between dissipation, coarse-grained media and a limit energy scale for the interactions. Con-\ncepts and connections like open systems, quasi-particles, energy scale and the change in the geometry\nof space–time at its topological level, nonconservative systems, noninteger dimensions of space–time con-\nnected to a coarse-grained medium, have been discussed. With this perspective, we argued that deformed\nor, we should say, Metric or Structural Derivatives, similarly to the Fractional Calculus (FC), could allows\nus to describe and emulate certain dynamics without explicit many-body, dissipation or geometrical terms\nin the dynamical governing equations. Also, we emphasized that the paradigm we adopt was different\nfrom the standard approach in the generalized statistical mechanics context [3–5], where the modification\nof entropy definition leads to the modification of the algebra and, consequently, the concept of a derivative\n[1, 2]. This was set up by mapping into a continuous fractal space [6–8] which naturally yields the need\nof modifications in the derivatives, that we named deformed or, better, metric derivatives [1, 2]. The\nmodifications of the derivatives, accordingly with the metric, brings to a change in the algebra involved,\nwhich, in turn, may lead to a generalized statistical mechanics with some adequate definition of entropy.\nThe Landau-Lifshitz-Gilbert (LLG) equation sets out as a fundamental approach to describe physics\nin the field of Applied Magnetism. It exhibits a wide spectrum of effects stemming from its non-linear\nstructure, and its mathematical and physical consequences open up a rich field of study. We pursue the\ninvestigation of the LLG equation in a scenario where complexity may play a role. The connection between\nLLG and fractionality, represented by an \u000b\u0000deformation parameter in the deformed differential equations,\nhas not been exploited with due attention. Here, the use of metric derivatives aims to take into account\nlong-range forces, possible non-manifest or hidden interactions and/or the dimensionality of space.\nIn this contribution, considering intrinsically the presence of complexity and possible dissipative effects,\nand aiming to tackle these issues, we apply our approach to study the LLG equation with two metric\nor structural derivatives, the recently proposed scale \u0000q\u0000derivative [2] in the nonextensive statistical\nmechanics and, as an alternative, the axiomatic metric derivative (AMD) that has the Mittag-Leffler\nfunction as eigenfunction and where deformed Leibniz and chain rule hold - similarly to the standard\ncalculus - but in the regime of low-level of fractionality. The deformed operators here are local. We3\nactually focus our attention to understand whether the damping in the LLG equation can be connected\nto some entropic index, the fractionality or even dimensionality of space; in a further step, we go over\ninto anisotropic Heisenberg spin systems in (1+1) dimensions with the purpose of modeling the weak\nanisotropy effects by means of some representative parameter, that depends on the dimension of space or\nthe strength of the interactions with the medium. Some considerations about an apparent paradox in the\nmagnetization or angular damping is given.\nOur paper is outlined as follows: In Section 2, we briefly present the scale \u0000q\u0000derivative in a nonex-\ntensive context, building up the q\u0000deformed Heisenberg equation and applying to tackle the problem of\nthe LLG equation; in Section 3, we apply the axiomatic derivative to build up the \u000b\u0000deformed Heisen-\nberg equation and to tackle again the problem of LLG equation. We finally present our Conclusions and\nOutlook in Section 4.\nII. APPLYING SCALE \u0000q\u0000DERIVATIVE IN A NONEXTENSIVE CONTEXT\nHere, in this Section, we provide some brief information to recall the main forms of scale \u0000q\u0000derivative.\nThe readers may see ref. [1, 2, 6] for more details.\nSome initial claims here coincide with our work of Refs. [1, 2] and the approaches here are in fact based\non local operators [1].\nThe local differential equation,\ndy\ndx=yq; (1)\nwith convenient initial condition, yields the solution given by the q-exponential, y=eq(x)[3–5].\nThe key of our work here is the Scale \u0000q\u0000derivative (Sq-D) that we have recently defined as\nD\u0015\n(q)f(\u0015x)\u0011[1 + (1\u0000q)\u0015x]df(x)\ndx: (2)\nThe eigenvalue equation holds for this derivative operator, as the reader can verify:\nD\u0015\n(q)f(\u0015x) =\u0015f(\u0015x): (3)4\nA.q\u0000deformed Heisenberg Equation in the Nonextensive Statistics Context\nWith the aim to obtain a scale \u0000q\u0000deformed Heisenberg equation, we now consider the scale\u0000q\u0000\nderivative [2]\ndq\ndtq= (1 + (1\u0000q)\u0015xd\ndx(4)\nand the Scale - q\u0000Deformed Schrödinger Equation [2],\ni~D\u0015\nq;t =\u0000~2\n2mr2 \u0000V =H ; (5)\nthat, as we have shown in [2], is related to the nonlinear Schrödinger equation referred to in Refs. [10]\nas NRT-like Schrödinger equation (with q=q0\u00002compared to the q\u0000index of the reference) and can be\nthought as resulting from a time\u0000scale\u0000q\u0000deformed-derivative applied to the wave function  .\nConsidering in eq.(5),  (~r;t) =Uq(t;t0) (~r;t0), theq\u0000evolution operator naturally emerges if we take\ninto account a time\u0000scale\u0000q\u0000deformed-derivative (do not confuse with formalism of discrete scale time\nderivative):\nUq(t;t0) =e(\u0000i\n~MqHqt)\nq: (6)\nHere,Mqis a constant for dimensional regularization reasons. Note that the q-deformed evolution\noperator is neither Hermitian nor unitary, the possibility of a q\u0000unitary asUy\nq(t;t0)\nqUq(t;t0) =1could\nbe thought to come over these facts. In this work, we assume the case where the commutativity of Uqand\nHholds, but the q\u0000unitarity is also a possibility.\nNow, we follow similar reasonings that can be found in Ref.[12] and considering the Sq-D.\nSo, with these considerations, we can now write a nonlinear Scale\u0000q\u0000deformed Heisenberg Equation\nas\nD\u0015\nt;q^A(t) =\u0000i\n~Mq[^A;H]; (7)\nwhere we supposed that UqandHcommute and Mqis some factor only for dimensional equilibrium.5\nB.q\u0000deformed LLG Equation\nTo build up the scale \u0000q\u0000deformed Landau-Lifshitz-Gilbert Equation, we consider eq.(7), with ^A(t) =\n^Sq\nD\u0015\nt;q^Sq(t) =\u0000i\n~Mq[^Sq;H]; (8)\nwhere we supposed that UqandHcommute.\nH=\u0000gq\u0016B\n~Mq^Sq\u000e~Heff: (9)\nHere,~Heffis some effective Hamiltonian whose form that we shall clearly write down in the sequel.\nThe scale\u0000q\u0000deformed momentum operator is here defined as bp\u0015\nq0=\u0000i~Mq0[1 +\u0015(1\u0000q0)x]@q\n@xq:\nConsidering this operator, we obtain a deformed algebra, here in terms of commutation relation between\ncoordinate and momentum\n\u0002\n^xq\ni;^pq\nj\u0003\n={[1 +\u0015(1\u0000q0)x]~Mq0\u000e{jI (10)\nand, for angular momentum components, as\nh\n^Lq\ni;^Lq\nji\n={[1 +\u0015(1\u0000q0)x]~Mq^Lq\nk: (11)\nTheq0factor in ^xq0\n{;^pq0\nj;^Lq0\ni;^Lq0\nj;Mq0is only an index and qis not necessarily equal to q0.\nThe resulting scale \u0000q\u0000deformed LLG equation can now be written as\nD\u0015\nt;q^Sq(t) =\u0000[1 +\u0015(1\u0000q0)x]gq\u0016B\n~Mq^Sq\u0002~Heff: (12)\nTake ^mq\u0011\rq^Sq; \rq0\u0011[1+\u0015(1\u0000q0)x]gq\u0016B\n~Mq.\nIf we consider that the spin algebra is nor affected by any emergent effects, we can take q0= 1.\nConsidering the eq.(7) with ^A(t) = ^Sqand ^mq=j\rqj^Sqandq0= 1; we obtain the q\u0000time deformed\nLLG dynamical equation for magnetization as\nD\u0015\nt;q^mq(t) =\u0000j\rj^mq\u0002~Heff: (13)6\nConsidering ~Heff=H0^k;we have the solution:\nmx;q=\u001acosq(\u00120) cosq(\rH0t) +\u001asinq(\u00120) sinq(\rH0t): (14)\nIn the figure, \u00120= 0:\nFigure 1: Increase/Damping- cosq(x)\n.\nIII. APPLYING AXIOMATIC DERIVATIVE AND THE \u000b\u0000DEFORMED HEISENBERG\nEQUATION\nNow, to compare results with two different local operators, we apply the axiomatic metric derivative.\nFollowing the steps on [12] and considering the axiomatic MD [13], there holds the eigenvalue equation\nD\u000b\nxE\u000b(\u0015x\u000b) =\u0015E\u000b(\u0015x\u000b);whereE\u000b(\u0015x\u000b)is the Mittag-Leffler function that is of crucial importance to\ndescribe the dynamics of complex systems. It involves a generalization of the exponential function and\nseveral trigonometric and hyperbolic functions. The eigenvalue equation above is only valid if we consider\n\u000bvery close to 1:This is what we call low-level fractionality [13]. Our proposal is to allow the use o Leibniz\nrule, even if it would result in an approximation. So, we can build up an evolution operator:\nU\u000b(t;t0) =E\u000b(\u0000i\n~\u000bHt\u000b); (15)\nand for the deformed Heisenberg Equation\nD\u000b\ntAH\n\u000b(t) =\u0000i\n~\u000b[AH\n\u000b;H]; (16)7\nwhere we supposed that U\u000bandHcommute.\nTo build up the deformed Landau-Lifshitz-Gilbert Equation, we use the eq. (16), and considering and\nspin operator ^S\u000b(t), in such a way that we can write the a deformed Heisenberg equation as\nD\u000b\nt^S\u000b(t) =\u0000i\n~\u000b[^S\u000b;H]; (17)\nwhith\nH=\u0000g\u000b\u0016B\n~\u000b^S\u000b\u000e~Heff: (18)\nHere,~Heffis some effective Hamiltonian whose form that we will turn out clear forward.\nNow, consider the deformed momentum operator as [9, 11, 12]\nbp\u000b=\u0000i(~)\u000bMx;\u000b@\u000b\n@x\u000b: (19)\nTakingthisoperator, weobtainadeformedalgebra, hereintermsofcommutationrelationforcoordinate\nand momentum\n\u0002\n^x\u000b\ni;^p\u000b\nj\u0003\n={\u0000(\u000b+ 1)~\u000bM\u000b\u000e{jI (20)\nand for angular momentum components as\nh\n^L\u000b\ni;^L\u000b\nji\n={\u0000(\u000b+ 1)~\u000bM\u000b^L\u000b\nk: (21)\nThe resulting the \u000b\u0000deformed LLG equation can now be written as\nJ\n0D\u000b\nt^S\u000b(t) =\u0000M\u000b\u0000(\u000b+ 1)g\u000b\u0016B\n~\u000b^S\u000b\u0002~Heff: (22)\nIf we take ^m\u000b\u0011\r\u000b^S\u000b,\r\u000b\u0011M\u000b\u0000(\u000b+1)g\u000b\u0016B\n~\u000b, we can re-write the equation as the \u000b\u0000deformed LLG\nJ\n0D\u000b\nt^m\u000b(t) =\u0000j\r\u000bj^m\u000b\u0002~Heff; (23)\nwith~Heff=H0^k. We have the Solution of eq.(23):\nm\u000bx=Acos\u00120E2\u000b(\u0000!2\n0t2\u000b) +Asin\u00120:x:E 2\u000b;1+\u000b(\u0000!2\n0t2\u000b): (24)8\nIn the figure below, the reader may notice the behavior of the magnetization, considering \u00120= 0.\nFigure 2: a) Damping of oscillations. In the figure \f= 1. b) Increase of oscillations\n.\nFor\u000b= 1;the solution reduces to mx=Acos(!0t+\u00120), the standard Simple Harmonic Oscillator\nsolution for the precession of magnetization.\nThe presence of complex interactions and dissipative effects that are not explicitly included into the\nHamiltonian can be seen with the use of deformed metric derivatives. Without explicitly adding up\nthe Gilbert damping term, the damping in the oscillations could reproduce the damping described by\nthe Gilbert term or could it disclose some new extra damping effect. Also, depending on the relevant\nparameter, the q\u0000entropic parameter or for \u000b, the increasing oscillations can signally that it is sensible\nto expect fractionality to interfere on the effects of polarized currents as the Slonczewski term describes.\nWe point out that there are qualitative similarities in both cases, as the damping or the increasing of the\noscillations, depending on the relevant control parameters. Despite that, there are also some interesting\ndifferences, as the change in phase for axiomatic derivative application case.\nHere, we cast some comments about an apparent paradox: If we make, as usually done in the literature\nfor LLG, the scalar product in eq. (13) with, ^mq;we obtain an apparent paradox that the modulus of\n^mqdoes not change. On the other hand, if instead of ^m\u000b;we proceed now with a scalar product with ~Heff\nand we obtain thereby the indications that the angle between ^m\u000band~Heffdoes not change. So, how to\nexplain the damping in osculations for ^mq?This question can be explained by the the following arguments.\nEven the usual LLG equation, with the term of Gilbert, can be rewritten in a form similar to eq. LLG9\nwithout term of Gilbert. See eq. (2.7) in the Ref. [14]. The effective ~Hefffield now stores information\nabout the interactions that cause damping. In our case, when carrying out the simulations, we have taken\n~Heffas a constant effective field. Here, we can argue that the damping term, eq. (2.8) in Ref. [14] being\nsmall, this would cause the effective field ~Heff=\u0000 !H(t) +\u0000 !k(\u0000 !S\u0002\u0000 !H)to be approximately\u0000 !H(t). In this\nway, the scalar product would make dominate over the term of explicit dissipation. This could, therefore,\nexplain the possible inconsistency.\nIV. CONCLUSIONS AND OUTLOOK\nIn short:\nHere, we tackle the problem of LLG equations considering the presence of complexity and dissipation\nor other interactions that give rise to the term proposed by Gilbert or the one by Slonczewski.\nWith this aim, we have applied scale - q\u0000derivative and the axiomatic metric derivative to build up\ndeformed Heisenberg equations. The evolution operator naturally emerges with the use of each case of the\nstructural derivatives. The deformed LLG equations are solved for a simple case, with both structural or\nmetric derivatives.\nAlso, in connection with the LLG equation, we can cast some final considerations for future investiga-\ntions:\nDoes fractionality simply reproduce the damping described by the Gilbert term or could it disclose\nsome new effect extra damping?\nIs it sensible to expect fractionality to interfere on the effects of polarized currents as the Slonczewski\nterm describes?\nThese two points are relevant in connection with fractionality and the recent high precision measure-\nments in magnetic systems may open up a new venue to strengthen the relationship between the fractional\nproperties of space-time and Condensed Matter systems.\n[1] J. Weberszpil, Matheus Jatkoske Lazo and J.A. Helayël-Neto, Physica A 436, (2015) 399–404.10\n[2] Weberszpil, J.; Helayël-Neto, J.A., Physica. A (Print), v. 450, (2016) 217-227; arXiv:1511.02835 [math-ph].\n[3] C. Tsallis, J. Stat. Phys. 52, (1988) 479-487.\n[4] C. Tsallis, Brazilian Journal of Physics, 39, 2A, (2009) 337-356.\n[5] C. Tsallis, Introduction to Nonextensive Statistical Mechanics - Approaching a Complex World (Springer,\nNew York, 2009).\n[6] Alexander S. Balankin and Benjamin Espinoza Elizarraraz, Phys. Rev. E 85, (2012) 056314.\n[7] A. S. Balankin and B. Espinoza, Phys. Rev. E 85, (2012) 025302(R).\n[8] Alexander Balankin, Juan Bory-Reyes and Michael Shapiro, Phys A, in press, (2015)\ndoi:10.1016/j.physa.2015.10.035.\n[9] Weberszpil, J. ; Helayël-Neto, J. A., Advances in High Energy Physics, (2014), p. 1-12.\n[10] F. D. Nobre, M. A. Rego-Monteiro, and C. Tsallis, Phys. Rev. Lett. 106, (2011) 140601.\n[11] J.Weberszpil, C.F.L.Godinho, A.ChermanandJ.A.Helayël-Neto, In: 7thConferenceMathematicalMethods\nin Physics - ICMP 2012, 2012, Rio de Janeiro. Proceedings of Science (PoS). Trieste, Italia: SISSA. Trieste,\nItalia: Published by Proceedings of Science (PoS), 2012. p. 1-19.\n[12] J. Weberszpil and J. A. Helayël-Neto, J. Adv. Phys. 7, 2 (2015) 1440-1447, ISSN 2347-3487.\n[13] J. Weberszpil, J. A. Helayël-Neto, arXiv:1605.08097 [math-ph]\n[14] M. Lakshmanan, Phil. Trans. R. Soc. A (2011) 369, 1280–1300 doi:10.1098/rsta.2010.0319"    },    {        "title": "0709.2937v2.Theory_of_current_driven_magnetization_dynamics_in_inhomogeneous_ferromagnets.pdf",        "content": "arXiv:0709.2937v2  [cond-mat.mes-hall]  23 Jan 2008Theory of current-driven magnetization dynamics in\ninhomogeneous ferromagnets\nYaroslav Tserkovnyak,1Arne Brataas,2and Gerrit E. W. Bauer3\n1Department of Physics and Astronomy,\nUniversity of California, Los Angeles, California 90095, U SA\n2Department of Physics, Norwegian University of\nScience and Technology, NO-7491 Trondheim, Norway\n3Kavli Institute of NanoScience, Delft University\nof Technology, 2628 CJ Delft, The Netherlands\n(Dated: November 8, 2018)\nAbstract\nWe give a brief account of recent developments in the theoret ical understanding of the interac-\ntion between electric currents and inhomogeneous ferromag netic order parameters. We start by\ndiscussingthephysicaloriginofthespintorquesresponsi bleforthisinteraction andconstructaphe-\nnomenological description. We then consider the electric c urrent-induced ferromagnetic instability\nand domain-wall motion. Finally, we present a microscopic j ustification of the phenomenological\ndescription of current-driven magnetization dynamics, wi th particular emphasis on the dissipative\nterms, theso-called Gilbertdamping αandtheβcomponent oftheadiabatic current-driventorque.\n1I. INTRODUCTION\nFerromagnetism is a correlated state in which, at sufficiently low temp eratures, the elec-\ntrons align their spins in order to reduce the exchange energy. Belo w the Curie temperature\nTc, the free energy F[M] then attains its minimum at a finite magnetization M∝negationslash= 0 with\nan arbitrary direction, thus spontaneously breaking the spin-rot ational symmetry. Crystal\nanisotropies that are caused by the spin-orbit interaction and sha pe anisotropies governed\nby the magnetostatic dipolar interaction pin the equilibrium magnetiza tion direction to a\ncertain plane or axis. At low temperatures, T≪Tc, fluctuations of the magnitude of the\nmagnetization around the saturation value Ms(the so-called Stoner excitations) become\nenergetically unfavorable. The remaining low-energy long-waveleng th excitations are spin\nwaves (or magnons, which, in technical terms, are viewed as Goldst one modes that restore\nthe broken symmetry). These are slowly varying modulations of the magnetization direction\nin space and time.\nA phenomenological description of the slow collective magnetization d ynamics without\ndissipation proceeds from the free energy F[M(r)] as a functional of the inhomogeneous\n(and instantaneous) magnetic configuration M(r) [1]. The equation of motion\n∂M(r,t)\n∂t=γM(r,t)×δF[M]\nδM(1)\npreserves the total free energy of the system, since the rate o f change of the magneti-\nzation is perpendicular to the “gradient” of the free energy. The f unctional derivative\nof the free energy with respect to the local magnetization is called t heeffective field:\nHeff(r,t) =−δF[M]/δM. [In the following, we use the abbreviations ∂t=∂/∂tfor the\npartial derivative in time and ∂M=δ/δMfor the functional derivative with respect to M.]\nIn the presence of only an externally applied magnetic field H,F[M] =−/integraltext\nd3rM(r)·H(r),\nsoγis identified as the effective gyromagnetic ratio. In general, Heffincludes the crys-\ntal anisotropy due to spin-orbit interactions, modulation of the ex change energy due to\nmagnetization gradients, and demagnetization fields due to dipole-d ipole interactions. The\nLandau-Lifshitz equation (1) qualitatively describes many ferroma gnetic resonance (FMR)\n[2, 3] and Brillouin light scattering [4] experiments. With M=Msm, wheremis the\nmagnetic direction unit vector, we may rewrite Eq. (1) as\n∂tm(r,t) =−γm(r,t)×Heff(r,t). (2)\n2By definition, the effective field on the right-hand side of Eq. (2) is de termined by the\ninstantaneous magnetic configuration. This can be true only if the m otion is so slow that\nall relevant microscopic degrees of freedom manage to immediately r eadjust themselves to\nthe varying magnetization. If this is not the case, the effective field acquires a finite time\nlag that to lowest order in frequency can be schematically expanded as\n˜Heff→ −∂MF[M(r,t−τ)]≈Heff−τ(∂tM·∂M)Heff, (3)\nwhereτisa characteristic delay time. This dynamic correction tothe instant aneous effective\nfield,δHeff, leads to a new term in the equation of motion ∝m×δHeff. Although in general\nnonlocal and anisotropic [5], it makes sense to identify first the simple st, i.e., local and\nisotropic contribution. We can then construct two new terms out o f the vectors mand\n∂tm. The first one, ∝m×∂tm, is dissipative, meaning that it is odd under time reversal\n(i.e., under the transformation t→ −t,H→ −H, andm→ −m), and thus violates the\ntime-reversal symmetry of the Landau-Lifshitz equation (2). Th is argument leads to the\nLandau-Lifshitz-Gilbert (LLG) equation [6, 7]:\n∂tm(r,t) =−γm(r,t)×Heff(r,t)+αm(r,t)×∂tm(r,t), (4)\nintroducing the phenomenological Gilbert damping constant α. The second local and\nisotropic term linear in ∂tmand perpendicular to m, that can be composed out of m\nand∂tmis proportional to ∂tmand can be combined with the left-hand side of the LLG\nequation. In principle, any physical process that contributes to t he Gilbert damping can\nthus also renormalize the gyromagnetic ratio γ. The latter should therefore be interpreted\nas an effective parameter in the equation of motion (4). The second law of thermodynamics\nrequires that αγ≥0, which guarantees that the dissipation of energy P∝Heff·∂tm≥0\n(assuming the magnetization dynamics are slow and isolated from any external sinks of\nentropy). Since the implications of small modifications of the gyroma gnetic ratio are mi-\nnor, we will be mainly concerned with the Gilbert damping constant α. Furthermore, we\nwill focus on dissipative effects due to spin dephasing by magnetic or s pin-orbit impurities\n[8, 9, 10, 11, 12, 13], noting that many other Gilbert damping mechan isms have been pro-\nposed in the past [14, 15, 16, 17, 18, 19, 20, 21]. According to the fl uctuation-dissipation\ntheorem, the dissipation, whatever its microscopic origin is, must be accompanied by a\nstochastic contribution h(r,t) to the effective field. Assuming Gaussian statistics with a\n3white noise correlator [22], which is valid in the classical limit with charac teristic frequen-\ncies that are sufficiently small compared to thermal energies:\n∝angb∇acketlefthi(r,t)hj(r′,t′)∝angb∇acket∇ight= 2kBTα\nγMsδijδ(r−r′)δ(t−t′). (5)\nEqs. (1)-(5) form the standard phenomenological basis for unde rstanding dynamics of fer-\nromagnets [2, 3, 4], in the absence of an applied current or voltage b ias.\nII. CURRENT-DRIVEN MAGNETIZATION DYNAMICS\nIn order to understand recent experiments on current-biased m agnetic multilayers [23,\n24, 25, 26, 27, 28, 29, 30, 31] and nanowires [32, 33, 34, 35, 36, 3 7, 38, 39], Eq. (4) has to be\nmodified [40, 41, 42]. The leading correction has to take into account the finite divergence\nof the spin-current density in conducting ferromagnets, with mag netization texture that has\nto be brought into compliance with the conservation of angular mome ntum. We have to\nintroduce a new term\ns0∂tmi|torque=∇·ji (6)\nin the presence of a current density jifor spin-icomponent, where s0is the total equilibrium\nspin density along −m(the minus sign takes into account that electron spin and magnetic\nmoment point in opposite directions). By adding this term to the right -hand side of Eq. (4)\nas a contribution to ∂tmi, we assume that the angular momentum lost in the spin current\nis fully added to the magnetization. This is called the spin-transfer to rque [42, 43, 44, 45].\nThesimplest approximationforthespin-current density inthebulko fisotropicferromagnets\n[43, 45, 46] is ji=Pjmi, wherePis a material-dependent constant that converts charge-\ncurrent density jinto spin-current density. The underlying assumption here is that s pins are\ncarried by the electric current such that the spin-polarization axis adiabatically follows the\nlocal magnetization direction. This is the case for a large exchange fi eld that varies slowly\nin space. This condition is fulfilled very well in transition-metal ferrom agnets. The spin\nconversion factor\nP=/planckover2pi1\n2eσ↑−σ↓\nσ↑+σ↓(7)\ncharacterizes the polarization of the spin-dependent conductivit yσs(s=↑ors=↓) with↑\nchosen along −m. Hence\n∂tm=−γm×Heff+P\ns0(j·∇)m, (8)\n4where we took into account local charge neutrality by ∇·j= 0.\nThe phenomenological Eq. (8) is “derived” without taking into accou nt spin relaxation\nprocesses. Its inclusion requires some care since both Gilbert damp ing and spin-transfer\ntorque are nontrivially affected [47, 48, 49, 50, 51, 52]. Spin relaxat ion is generated by\nimpurities with potentials that do not commute with the spin density op erator, such as\na quenched random magnetic field or spin-orbit interaction associat ed with randomly dis-\ntributed non-magnetic impurities [13, 47, 49, 50, 51]. In the absenc e of an applied current\nj, imperfections with potentials that mix the spin channels contribute to the Gilbert con-\nstantα[8, 9, 10, 13, 50]. It is instructive to interpret the right-hand side o f the equation\nof motion (8) as an analytic expansion of driving and damping torques in∇and∂t. The\nLLG equation (4) corresponds to the most general (local and isot ropic) expression for the\ndamping to the zeroth order in ∇and first order in ∂t. We will not be concerned with\nhigher order terms in ∂t, since the characteristic frequencies of magnetization dynamics a re\ntypically small on the scale of the relevant microscopic energies, at le ast in metallic systems.\nThe contribution to the effective field due to a finite magnetic stiffnes s [1] is proportional\nto∇2. In the presence of inversion symmetry, the terms proportional to∇cannot appear\nwithout applied electric currents. In the following, we focus on the c urrent-driven terms lin-\near in∇, assuming that the spatial variations in the magnetization direction are sufficiently\nsmooth to rule out higher-order contributions. The dynamics of iso tropic spin-rotationally\ninvariant ferromagnets can then in general be described by the ph enomenological equation\nof motion [47, 49, 53]\n∂tm=−γm×Heff+αm×∂tm+P\ns0(1−βm×)(j·∇)m, (9)\nin which αandβcharacterize those terms that break time-reversal symmetry. Both arise\nnaturally in the presence of spin-dependent impurities [47, 49, 50]. E ven though in practice\nβ≪1,itgivesanimportantcorrectiontothecurrent-driven spin-tra nsfertorque[47, 49, 53],\nasdiscussedinmoredetailbelow. Forthespecialcaseof α=β: Eq.(9)canthenberewritten\n(after multiplying it by 1+ αm×on the left) as\n∂tm=−γ∗m×Heff+αγ∗m×Heff×m+P\ns0(j·∇)m, (10)\nwhereγ∗=γ/(1 +α2). The dissipative term proportional to m×Heff×mis called the\nLandau-Lifshitz damping. Eq. (9) cannot be transformed into equ ation (10) if α∝negationslash=β[in\n5which case Eq. (10) necessarily retains a βterm]. A special feature of Eq. (10) appears when\nHeff(m) is time independent and translationally invariant. A general solution m(r,t) in the\nabsence of an electric current j= 0 (such as a static domain wall or a spin wave) can be\nused to construct a solution\n˜m(r,t) =m(r+Pjt/s0,t) (11)\nof Eq. (10) for an arbitrary uniform j. This unique feature of the solutions of Eq. (9) only\narises for α=β.\nInterestingly, the argument above has been turned around by Ba rnes and Maekawa [54],\nwho find that Galilean invariance of a system would dictate α=β. Galilean invariance\nrequires the existence of solutions of the form ˜m(r−vt), where ˜m(r) is an arbitrary static\nsolution (say, a domain wall) and vis an arbitrary velocity. As explained above, this is\nonly possible when α=β. However, the general validity of the Galilean invariance assump-\ntion for the current-carrying state needs to be discussed in more detail from a microscopic\npoint of view. The Galilean invariance argument [54] implies that the bias -induced electron\ndrift exactly corresponds to the domain-wall velocity, since other wise electron motion would\npersist in the frame that moves with the domain wall. Referring to Eq. (11), we must,\ntherefore, identify v=−Pj/s0with the average electron drift velocity in the presence of the\ncurrentj. We argue in the following that this is indeed true in certain special limits , but\nis not generic, however. In the itinerant Stoner model for ferrom agnets, the spin-dependent\nDrude conductivity reads σs∝nsτs, wherensandτsare the densities and scattering times\nof spins, respectively. When there is no asymmetry between the scatterin g times,τ↑=τ↓,\nvindeed equals the electron drift velocity and Galilean invariance is effec tively fulfilled. In\ngeneral, however, since the spin dependence of wave functions an d densities of states for\nthe electrons at the Fermi energy lead to different scattering cro ss sections in conducting\nferromagnets, the equality between −Pj/s0and the average drift velocity disappears. In\nthe simplest model of perturbative white-noise impurity potentials, for example, 1 /τs∝νs,\nwhereνsis the spin-dependent density of states. Assuming parabolic free- electron bands\nand weak ferromagnets, in which the ferromagnetic exchange split ting is much less than the\nFermi energy, the domain-wall velocity v=−Pj/s0actually becomes 2 /3 of the average\ndrift velocity ¯v,\nv=−Pj\ns0=n↑τ↑−n↓τ↓\nn↑τ↑+n↓τ↓n↑v↑+n↓v↓\nn↑−n↓≈n↑/ν↑−n↓/ν↓\n(n↑−n↓)(ν↑+ν↓)/2¯v≈2\n3¯v.(12)\n6Here,vsis the spin- selectron drift velocity, and we used the relation νs∝n1/3\ns, which\nis valid in three dimensions. Clearly, the potential disorder breaks Ga lilean invariance.\nAn identity of αandβcan therefore not be deduced from general symmetry principles.\nFurthermore, spin-orbit interaction or magnetic disorder that st rongly affect the values of α\nandβ(see below) also break Galilean invariance at the level of the microsco pic Hamiltonian.\nNevertheless, for itinerant ferromagnets we show below that α∼β(where by ∼we mean\n“of the order”), with α≈βin the simplest model of weak and isotropic spin-dephasing\nimpurities [49], which implies that deviations from translational invarian ce are not very\nimportant in metallic ferromagnets, such as transition metals and th eir alloys, in which\nthe Stoner model is applicable. Very recently, two independent gro ups measured α≈βin\npermalloy nanowires [36, 37].\nLet us also consider the s−dmodel of ferromagnetism. When, as is usually done, the d-\norbital lattice is assumed spatially locked, Galilean invariance is broken even in the absence\nof disorder. In this case, the ratio α/βdeviates strongly from unity, although it remains to\nbe relatively insensitive to the strength of spin-dependent impuritie s. In other words, αand\nβscale similarly with the strength of spin-dephasing processes, and t heir ratio appears to\nbe determined mainly by band-structure effects and the nature (r ather than the strength)\nof the disorder [49, 50]. A predictive material-dependent theory of magnetization damping\nand current-induced domain wall motion that transcends the toy m odels mentioned above\nis beyond the scope of our paper.\nThe form of the equation of motion (10) for the special case α=βhas also triggered\nthe suggestion [55] that the Landau-Lifshitz form of damping, ∝m×Heff×m, is more\nnatural than the Gilbert form, ∝m×∂tm. In our opinion, however, such a distinction\nis purely semantic. Both forms are odd under time reversal, and one can easily imagine\nsimple models in which either form arises more naturally than the other : For example, a\nBloch-like T2relaxation added to the Stoner model naturally leads to a Landau-L ifshitz\nform of damping [49], whereas the dynamic interface spin pumping [56, 57] very generally\nobeys the Gilbert damping form. Moreover, mathematically, both eq uations are identical\n(in the absence of any additional torques), since we have shown ab ove that the Landau-\nLifshitz form of damping follows from the Gilbert one simply by multiplying both sides of\nthe LLG form by 1+ αm×from the left (and vice versa by 1 −αm×). Only at the special\npointα=β, the Landau-Lifshitz form (10) does not involve the “ βterm.” In that limit,\n7it may be a more transparent expression for the equation of motion . On the other hand,\nwe noted above that in general α∝negationslash=βand the ratio α/βdepends on material and sample.\nThe current-driven dynamics of domain walls and other spatially nonu niform magnetization\ndistributions turn out to be very sensitive to small deviations of α/βfrom unity, which\nstrongly reduces any advantage a Landau-Lifshitz damping formu lation might have over the\nGilbert phenomenology. In general, we therefore prefer to use th e Gilbert phenomenology.\nUnder time reversal, the electric current as well as the magnetizat ion vector change sign\nand the adiabatic current-induced torque is symmetric, thus nond issipative. The Ohmic\ndissipation generated by this current does not depend on the magn etization texture in this\nlimit and is intentionally disregarded. Saslow [58] prefers to discuss a t orque driven by\nvoltage rather current, which, after inserting Ohm’s law for the cu rrent, becomes odd under\ntime reversal and thus appears dissipative (see Ref. [59] for anot her discussion of this point).\nObviously, the βcorrection torque is odd for the current-biased and even for the voltage-\nbiased configurations. The current-bias picture appears to be mo re natural, since it reflects\nthe absence of additional dissipation by the magnetization texture in the adiabatic limit as\nwell as the close relation between the βcorrection and the Gilbert dissipation.\nIII. CURRENT-DRIVEN INSTABILITY OF FERROMAGNETISM\nLet us now pursue some special aspects of the solutions of the phe nomenological Eq. (9),\nhighlighting the role of various parameters, before we discuss the m icroscopic derivation\nof the magnetization dynamics in Sec. V. It is interesting, for examp le, to investigate\nthe possibility to destabilize a single-domain ferromagnet by sufficient ly large spin torques\n[43, 46]. We consider a homogeneous ferromagnet with an easy-axis anisotropy along the x\naxis, characterized by the anisotropy constant K, and an easy-plane anisotropy in the xy\nplane, with the anisotropy constant K⊥, see Fig. 1. Typically, the anisotropies originate\nfrom the demagnetization fields: For a ferromagnetic wire, for exa mple, the magnetostatic\nenergy is lowest when the magnetization is in the wire direction, so tha t there are no stray\nfield lines outside the ferromagnet. The effective field governing mag netization dynamics is\nthen given by\nHeff= (H+Kmx)x−K⊥mzz+A∇2m, (13)\n8where we also included an applied field Halong the xaxis and the exchange coupling\nparametrized by the stiffness constant A.A,H,K,K ⊥≥0. We then look for spin-wave\nsolutions of the form\nm=x+uei(q·r−ωt), (14)\nplugging it into Eq. (9) with the effective field (13) in the presence of a constant current\ndensityj, and linearizing it with respect to small deviations u. Whenα= 0 and j= 0, we\nrecover the usual spin-wave dispersion (Kittel formula):\nω0(q) =γ/radicalbig\n(H+K+Aq2)(H+K+K⊥+Aq2). (15)\nAfiniteαγ >0results ina negative Im ω(q), asrequired by thestability oftheferromagnetic\nstate. Asufficientlylargeelectriccurrentmay, however, reverse thesignofIm ω(q)forcertain\nwave vectors q, signaling the onset of an instability. The critical value of the curren t for the\ninstability corresponds to the condition Im ω(q) = 0. Straightforward manipulations based\non Eqs. (9) and (13) show that this condition is satisfied when\nP\ns0/parenleftbigg\n1−β\nα/parenrightbigg\n(q·jc) =±ω0(q). (16)\nwhich leads to a critical current density\njc=jc0\n|1−β/α|. (17)\njc0is the lowest current satisfying equation ( P/s0)(q·jc0) =ω0(q) for some q, where the\nleft-hand side can be loosely interpreted as the current-induced D oppler shift to the nat-\nural frequency given by the right-hand side [46]. According to Eq. ( 17), a current-driven\ninstability is absent when α=β. This conclusion is in line with the arguments leading to\nEq. (11): For the special case of α=β, a spin-wave solution in the presence of a finite\ncurrent density jwould acquire a frequency boost proportional to q·j, but with a stable\namplitude. Note that in general the onset of the current-driven f erromagnetic instability\nis significantly modified by the existence of βeven with β≪1, provided that the ratio\nβ/αis appreciable. In fact, αis typically measured to be ∼0.001−0.01, and the existing\nmicroscopic theories [49, 50] predict βto be not too different from α.\n9FIG. 1: Transverse head-to-head (N´ eel) domain wall parall el to the yaxis in the easy xyplane.\nThe uniform magnetization has two stable solutions m=±xalong the easy axis x, which is\ncharacterized by the anisotropy constant K. These are approached far away from the domain wall:\nm→ ±xatx=∓∞, respectively. In equilibrium, the magnetization directi onmis forced into the\nxyplane by the easy-plane anisotropy parametrized by K⊥. A weak magnetic field Hor electric\ncurrentjapplied along the xaxis can induce a slow domain-wall drift along the xaxis, during\nwhich the magnetization close to the domain wall is tilted sl ightly out of the xyplane. At larger H\norj(above the so-called Walker threshold), the magnetization is significantly pushed out of the xy\nplane and undergoes precessional motion during the drift. I n the moving frame, the magnetization\nprofile may remain still close to the equilibrium one.\nIV. CURRENT-DRIVEN DOMAIN-WALL MOTION\nEven more interesting phenomena are associated with the effect of the applied electric\ncurrent on a stationary domain-wall. In particular, we wish to discus s how the spin torques\nmove and distort a domain wall. These questions date back about thr ee decades [40],\nalthough only relatively recently they sparked an intense activity by several groups [49,\n50, 51, 53, 54, 60]. This is motivated by the growing number of intrigu ing experiments\n[32, 33, 34, 35, 36, 38, 39] as well as the promise of practical pote ntial, such as in the so-\ncalled racetrack memory [61] or magnetic logics [62]. Current-induce d domain-wall motion\nis a central topic of the present review.\nFor not too strong driving currents and in the absence of any signifi cant transverse dy-\nnamics, one can make progress analytically by using the one-dimensio nal Walker ansatz,\nwhich was first employed in studies of magnetic-field driven domain-wa ll dynamics [63].\n10This approach has proven useful in the present context as well [6 0, 64, 65]. The key idea\nis to approximately capture the potentially complex domain-wall motio n by few parameters\ndescribing the displacement of its center and a net distortion of the domain-wall structure.\nIn a quasi-one-dimensional set-up, such as a narrow magnetic wire , the domain wall is con-\nstrained to move along a certain axis, whereas the transverse dyn amics are suppressed. This\nregime is relevant for a number of existing experiments, although it s hould be pointed out\nthat the common vortex-type domain walls do not necessarily fall int o this category. Let us\nconsider an idealized situation with an effective field (13) and an equilibr ium domain wall\nmagnetization in the xyplane. The magnetization prefers to be collinear with the xaxis\ndue to the easy-axis anisotropy K. A transverse head-to-head domain wall parallel to the y\naxis corresponds to a magnetization direction that smoothly rotat es in the xyplane between\nxatx→ −∞and−xatx→ ∞, as sketched in Fig. 1.\nThe collective domain-wall dynamics can be described by the center p ositionX(t) and\nan out-of-plane tilting angle Φ( t). [For a more technically-interested reader, we note that\nin the effective treatment of Ref. [60], these variables are canonica lly conjugate.] There is\nalso a width distortion, but that is usually considered less important. WhenH < K, the\ntwo uniform stable states are m=±x. When H= 0, a static transverse head-to-head\ndomain-wall solution centered at x= 0 is given by\nϕ(x)≡0,lntanθ(x)\n2=x\nW, (18)\nwhere position-dependent angles ϕandθparametrize the magnetic configuration:\nm= (mx,my,mz) = (cosθ,sinθcosϕ,sinθsinϕ). (19)\nW=/radicalbig\nA/Kis the wall width, which is governed by the interplay between the stiffn essA\nthat tends to smooth the wall extent and the easy-axis anisotrop yKthat tends to sharpen\nthe wall.\nThe external magnetic field Hor the current density jalong the xaxis disturb the static\nsolution (18), distorting the domain-wall structure and displacing it s position. At weak field\nand current biases, magnetic dynamics can be captured by the Walk er ansatz [63, 64]:\nϕ(r,t)≡Φ(t),lntanθ(r,t)\n2≡x−X(t)\n˜W(t). (20)\nHere, it is assumed that the driving perturbations ( Handj) are not too strong, such that\nthe wall preserves its shape, except for a small change of its width ˜W(t) and a uniform\n11out-of-plane tilt angle Φ( t).X(t) parametrizes the net displacement of the wall along the\nxaxis. Note that although ϕis assumed to be spatially uniform, it has an effect on the\nmagnetization direction only when m∝negationslash=±x, i.e., only near the wall center. A more detailed\ndiscussion concerning the range of validity of this approximation can be found in Ref. [63].\nInserting the ansatz (20) into the equation of motion (9) with j=jx(since the other current\ndirections do not couple to the wall), and using Eq. (13) for the effec tive field, one finds\n[53, 64]\n˙Φ+α˙X\n˜W=γH−βPj\ns0˜W,\n˙X\n˜W−α˙Φ =γK⊥sin2Φ\n2−Pj\ns0˜W,\n˜W=/radicalBigg\nA\nK+K⊥sin2Φ. (21)\nIt iseasyto verify thatthestaticsolution(18)is consistent withth ese equations when H= 0\nandj= 0. Two different dynamic regimes can be distinguished based on Eqs. (21): When\nthe driving forces are weak, a slightly distorted wall moves at a cons tant speed, ˙X= const,\nand constant tilt angle, ˙Φ = 0 (assuming constant Handj). The corresponding Walker\nansatz (20) then actually provides the exact solution, which is appr oached at long times\nafter the constant driving field and/or current are switched on [6 3, 64]. Beyond certain\ncritical values of Horj, called Walker thresholds, however, no solution with constant angle\nΦ and constant velocity ˙Xexist. Both undergo periodic oscillations in time, albeit with a\nfinite average drift velocity ∝angb∇acketleft˙X∝angb∇acket∇ight ∝negationslash= 0. In the spacial case of α=β, Eqs. (21) are exact at\narbitrary dc currents when H= 0: According to Eq. (11), the static domain-wall solution\n(of an arbitrary domain-wall shape) then simply moves with velocity −Pj/s0without any\ndistortions. When β∝negationslash=α, the Walker threshold current diverges when βapproaches α,\nreminiscent of the critical current (17) discussed in the previous s ection.\nFor subthreshold fields and currents with Φ( t)→const as t→ ∞, the steady state\nterminal velocity is given by [47]\nv=˙X(t→ ∞) =γH˜W−βPj/s0\nα. (22)\nIn particular, when j= 0, the wall depicted in Fig. 1 moves along the direction of the\napplied magnetic field Hin order to decrease the free energy [63]. Let us in the following\n12focus on the current-driven dynamics with H= 0. At a finite but small j, the wall is slightly\ncompressed according to\n1−˜W\nW≈(Pj/s0)2\n2γ2AK⊥/parenleftbigg\n1−β\nα/parenrightbigg2\n, (23)\nwhereW=/radicalbig\nA/Kis the equilibrium width. When α=β, the domain-wall velocity\nv→ −Pj/s0. In this case, if we consider the electron spins following the magnetiz ation\ndirection from ±mto∓mon traversing the domain wall with current density j, the entire\nangular momentum change is transferred to the domain-wall displac ement. In this sense,\nthe ratio β/αcan be loosely interpreted as a spin-transfer efficiency from the cu rrent density\nto the domain-wall motion.\nOnly when α=β, the rigidly moving domain-wall solution is exact at arbitrary current\ndensities, leading to an infinite Walker threshold current. The latter becomes finite and\ndecreases with β < α, approaching a finite value jt0atβ= 0 [60], see Fig. 2. In the absence\nof a strong disorder pinning centers, as assumed so far, jt0∝K⊥(which is also the case\nwith the Walker threshold fieldin the absence of an applied current [63]), with an average\nvelocity that slightly above the threshold reads\n∝angb∇acketleft˙X∝angb∇acket∇ight ∝/radicalBig\nj2−j2\nt0. (24)\nSee theβ= 0 curve in Fig.2. At finite β, thedepinning current is determined by the pinning\nfields, which should be included into the effective field (13). The domain -wall velocity at\ncurrents slightly above the depinning current is predicted in Ref. [5 4] to grow linearly with\nj.\nSo far in our discussion, we have completely disregarded the random noise contribution\nto the magnetization dynamics. As noted above, see Eq. (5), ther mal fluctuations are\nubiquitous in dissipative systems. Below the (zero-temperature) d epinning currents, applied\ncurrentscandrivethedomainwallwithfiniteaveragevelocity ∝angb∇acketleftv∝angb∇acket∇ightonlybythermalactivation.\nThe question how ln ∝angb∇acketleftv∝angb∇acket∇ightscales with the current at low temperatures and weak currents is\nof fundamental interest beyond the field of magnetism. Experimen ts on thermally-activated\ndomain-wall motion in magnetic semiconductors [33, 39] reveal a “cr eep” regime [66], in\nwhich the effective thermal-activation barrier diverges at low curre nt density j, so that\nln∝angb∇acketleftv∝angb∇acket∇ightscales as const −j−µ, with an exponent µ∼1/3. This is inconsistent with the\ntheory based on the Walker ansatz for rigid domain-wall motion [67], w hich yields a simple\nlinear scaling of the effective activation barrier and ln ∝angb∇acketleftv∝angb∇acket∇ight ∝const +j. A refinement of the\n13FIG. 2: Average current-driven domain-wall velocity vnumerically calculated using the Walker\nansatz [Eqs. (21)] in Ref. [53]. Here, the domain-wall width has been approximated by its equilib-\nrium value, ˜W≈W, assuming K⊥≪K. The curves are very similar to the full micromagnetic\nsimulations [53]. u=−Pj/s0has the units of velocity (proportional to electron drift ve locity) and\nvw=γK⊥ζ/2 is its value for j=jt0. The length ζ≈W, if we assume K⊥≪K(as was done\nin this calculation), while ζ≈/radicalbig\n2A/K⊥in the opposite limit, K⊥≫K, which is relevant for a\nthin-film with large demagnetization anisotropy K⊥= 4πMs(in which case ζis called exchange\nlength) [64]. α= 0.02, and we refer to Ref. [53] for the remaining details.\nWalker-ansatz treatment [68] cannot explain the experiments eith er. A scaling theory of\ncreep motion close to the critical temperature [39] does offer a qua litative agreement with\nmeasurements by Yamanouchi et al.[39]. However, the intrinsic spin-orbit coupling in p-\ndoped (Ga,Mn)As leads to current-driven effects beyond the stan dard spin-transfer theories,\nsee, e.g., Refs. [69, 70], which needs to be understood better in the present context.\nEven at zero temperature, there are stochastic spin-torque so urces in the presence of\nan applied current, which stem from the discreteness of the angula r momentum carried by\nelectron spins, in analogy with the telegraph-like shot noise of electr ic current carried by\ndiscrete particles. A theoretical study of the combined thermal a nd shot-noise contributions\n14to the stochastic torques for inhomogeneous magnetic configura tions [71] did not yet explore\nconsequences forthedomain-wall dynamics. Forexample, itisnotk nown whether shot noise\nassists thecurrent-driven domain-wall depinning atlowtemperatu res. Questions alongthese\nlines pose challenging problems for future research.\nEffects beyond the theory discussed above are generated by non adiabatic spin torques,\nwhich lead to higher-order in ∂tand∇terms in the equation of motion (9). It is in principle\npossible to extend linear-response diagrammatic Green’s function c alculation [13, 49, 50]\nby systematically calculating higher-order terms as an expansion in t he small parameters,\ni.e., spin-wave frequency and momentum [72]. A dynamic correction to the spin torque in\nEq. (9) has been found in Refs. [49, 72], which comes down to replacin gβ→β+n(/planckover2pi1/∆xc)∂t,\nwhere ∆ xcis the ferromagnetic exchange splitting and n= 1(2) for the Stoner ( s−d)\nmodel [49, 72]. Since this term scales like ∂t∇, it is symmetric under time reversal and\ntherefore nondissipative. Although this dynamic correction is rath er small at the typical\nFMR frequencies /planckover2pi1ω≪∆xc, it can cause significant effects at large currents [72]. Starting\nfromaninhomogeneousequilibriumconfiguration[72, 73,74,75], suc hasamagneticspiralor\na domain wall, one can capture nonadiabatic terms in the equation of m otion that vanish in\nlinear response withrespect totheuniformmagnetizationconsider ed inRefs. [13, 49, 50, 51].\nFor strongly-inhomogeneous magnetic structures, perturbativ e expansions around a uni-\nform magnetic state fail. For example, for sharp domain walls, the eff ective equations (21)\ndescribing wall dynamics and displacement acquire a new term, which c an be understood\nas a force transferred by electrons reflected at the potential b arrier caused by the domain\nwall [60, 76]. Electron reflection at a domain wall increases the resist ance. Adiabaticity\nimplies a vanishing intrinsic domain-wall resistance (see, however, Re f. [69] for a model with\nstrong intrinsic spin-orbit coupling). The force term, becomes impo rtant only for abrupt\nwalls with width W∼λxc≡/planckover2pi1vF/∆xc. Such nonadiabatic effects are not expected to be\nstrong in metallic ferromagnets, where typically W≫λxc∼λF(the Fermi wavelength).\nDilute magnetic semiconductors [such as (Ga,Mn)As] are a different c lass of materials with\nlongerλxcand a strong spin-orbit coupling [70].\nIn metallic systems, effects of the spin-torque in the most relevant regime of slow dy-\nnamics,/planckover2pi1ω≪∆xc, with smooth walls, W≫λxc, and at moderate applied currents is in\nour opinion captured by the adiabatic terms linear in ∂tand∇. We will now discuss the\nmicroscopic basis for Eq. (9) containing such terms.\n15V. MICROSCOPIC THEORY OF MAGNETIZATION DYNAMICS\nOnce the phenomenological equation for current-driven magnetiz ation dynamics is re-\nduced to the form (9), which requires smooth magnetization variat ion, slow dynamics, and\nisotropic ferromagnetism, the remaining key questions concern th e magnitude and relation\nbetween the two dimensionless parameters αandβ. The size of the Gilbert damping con-\nstantαis a long-standing open question in solid-state physics, and even a br ief review of the\nrelevant ideas and literature is beyond the scope of this paper. A re cent model calculation\nhighlighting the multitude of relevant energy scales that control ma gnetic damping can be\nfound in Ref. [12]. Here, we discuss only the ratio β/α, since it is of central importance\nfor macroscopic current-driven phenomena. As noted above, th e ratioβ/αdetermines, for\nexample, the onset of the ferromagnetic current-driven instabilit y [see Eq. (17)] as well as\nthe Walker threshold current (both diverging when β/α→1). The subthreshold current-\ndriven domain-wall velocity is proportional to β/α[see Eq. (22)], while β/α= 1 is a special\npoint, at which the effect of a uniform current density jon the magnetization dynamics is\neliminated in the frame of reference that moves with velocity v=−Pj/s0[see Eq. (11)].\nAlthough the exact ratio β/αis a system-dependent quantity, some qualitative aspects not\ntoo sensitive to the microscopic origin of these parameters have re cently been discussed\n[13, 49, 50].\nIn Ref. [49], we developed a self-consistent mean-field approach, in which itinerant elec-\ntrons are described by a time-dependent single-particle Hamiltonian\nˆH= [H0+U(r,t)]ˆ1+γ/planckover2pi1\n2ˆσ·(H+Hxc)(r,t)+ˆHσ, (25)\nwhere the unit matrix ˆ1 and the vector of the Pauli matrices ˆσ= (ˆσx,ˆσy,ˆσz) form a basis\nfor the Hamiltonian in spin space. H0is the crystal Hamiltonian including kinetic and\npotential energy. Uis the scalar potential consisting of disorder and applied electric-fie ld\ncontributions. The total magnetic field consists of the applied, H, and exchange, Hxc, fields.\nFinally, thelasttermintheHamiltonian, ˆHσ, accountsforspin-dephasing processes, e.g, due\nto quenched magnetic disorder or spin-orbit scattering associate d with impurity potentials.\nThis last term is responsible for low-frequency dissipative processe s affecting αandβin the\ncollective equation of motion (9).\nIn time-dependent spin-density-functional theory [44, 77, 78] o f itinerant ferromagnetism,\n16the exchange field Hxcis a functional of the time-dependent spin-density matrix\nραβ(r,t) =∝angb∇acketleftΨ†\nβ(r)Ψα(r)∝angb∇acket∇ightt (26)\nthat should be computed self-consistently from the Schr¨ odinger equation corresponding to\nˆH. The spin density of conducting electrons is given by\ns(r) =/planckover2pi1\n2Tr[ˆσˆρ(r)]. (27)\nFocusing on low-energy magnetic fluctuations that are long range a nd transverse, we restrict\nour attention to a single parabolic band. Consideration of realistic ba nd structures is pos-\nsible from this starting point. We adopt the adiabatic local-density ap proximation (ALDA,\nessentially the Stoner model) for the exchange field:\nγ/planckover2pi1Hxc[ˆρ](r,t)≈∆xcm(r,t), (28)\nwith direction m=−s/slocked to the time-dependent spin density (27) (assuming γ >0).\nIn another simple model of ferromagnetism, the so-called s-dmodel, conducting selec-\ntrons interact with the exchange field of the delectrons which are assumed to be localized to\nthe crystal lattice sites. The d-orbital electron spins are supposed to account for most of the\nmagnetic moment. Because d-electron shells have large net spins and strong ferromagnetic\ncorrelations, they are usually treated classically. In a mean-field s-ddescription, therefore,\nconducting sorbitals are described by the same Hamiltonian (25) with an exchange field\n(28). The differences between the Stoner and s-dmodels for the magnetization dynamics\nare rather minor and subtle. In the ALDA/Stoner model, the excha nge potential is (on the\nscale of the magnetization dynamics) instantaneously aligned with th e total magnetization.\nIn contrast, the direction unit vector min thes-dmodel corresponds to the dmagnetization,\nwhich is allowed to be misaligned with the smagnetization, transferring torque between the\nsanddmagnetic moments. Since most of the magnetization is carried by the latter, the\nexternal field Hcouples mainly to the dspins, while the sspins respond to and follow the\ntime-dependent exchange field (28). As ∆ xcis usually much larger than the external (includ-\ning demagnetization and anisotropy) fields that drive collective magn etization dynamics, the\ntotal magnetic moment will always be very close to m. A more important difference of the\nphilosophy behind the two models is the presumed shielding of the dorbitals from exter-\nnal disorder. The reduced coupling with dissipative degrees of free dom would imply that\n17their dynamics are much less damped. (Whether this is actually the ca se in real systems\nremains to be proven, however.) Consequently, the magnetization damping has to come\nfrom the disorder experienced by the itinerant selectrons. As in the case of the itinerant\nferromagnets, the susceptibility has to be calculated self-consist ently with the magnetization\ndynamics parametrized by m. For more details on this model, we refer to Refs. [10, 49].\nWith the above differences in mind, the following discussion is applicable t o both models. In\norder to avoid confusion, we remark that the equilibrium spin density s0introduced earlier\nrefers to the total spin density, i.e., d- pluss-electron spin density, while Eq. (27) refers\nonly to the latter. The Stoner model is more appropriate for trans ition-metal ferromagnets\nbecause of the strong hybridization between dands,pelectrons. Magnetic semiconductors\nare characterized by deep magnetic impurity states for which the s-dmodel may be a better\nchoice.\nThe single-particle itinerant electron response to electric and magn etic fields in Hamil-\ntonian (25) is all that is needed to compute the magnetization dynam ics microscopically.\nAs mentioned above, the distinction between the Stoner and s-dmodels will appear only\nat the end of the day, when we self-consistently relate m(r,t) to the itinerant electron spin\nresponse. Before proceeding, we observe that since the consta ntsαandβwhich parametrize\nthe magnetic equation of motion (9) affect the linear response to a s mall transverse applied\nfield with respect to a uniform magnetization, we can obtain them by a linear-response\ncalculation for the single-domain bulk ferromagnet. The large-scale magnetization texture\nassociated with a domain wall does not affect the value of these para meters, in the consid-\nered limit. The linear response to a small magnetic field is complicated by the presence of\nan electrically-driven applied current, however. Since the Kubo for mula based on two-point\nequilibrium Green’s functions is insufficient to calculate the response t o simultaneous mag-\nnetic and electric fields, we chose to pursue a nonequilibrium (Keldysh ) Green’s function\nformalism in Refs. [13, 49]. A technically impressive equilibrium Green’s fu nction calcula-\ntion has been carried out in Ref. [50], which to a large extent confirme d our results, but also\ncontributed some important additions that will be discussed below.\nThe central quantity in the kinetic equation approach [13, 49] is the nonequilibrium\ncomponent of the 2 ×2 distribution function ˆfk(r,t). In the quasiparticle approximation,\nvalidwhen∆ xc≪EF[49], thekineticequationcanbereducedtoasemiclassical Boltzmann -\nlike equation that accounts for electron drift in response to the ele ctric field as well as the\n18spin precession in the magnetic field. The nonequilibrium component of the spin density\nreadss′= (/planckover2pi1/2)/integraltext\nd3kfk/(2π)3, wherefk= Tr[ˆfkˆσ]:\n∂ts′−∆xc\n/planckover2pi1z×s′−∆xcs\n/planckover2pi1z×u=−/planckover2pi1\n2/integraldisplayd3k\n(2π)3(vk·∂r)fk−s′+su\nτσ. (29)\nshere is the equilibrium spin density of itinerant electrons, vk=∂kεks//planckover2pi1is the momentum-\ndependent group velocity, and the magnetization direction m=z+uis assumed to undergo\na small precession urelative to the uniform equilibrium direction z. The first term on the\nright-hand side is the spin-current divergence and the last term is t he spin-dephasing term\nintroduced phenomenologically in Ref. [49] and studied microscopically in Ref. [13]. As\ndetailed in Ref. [49], the spin currents have to be calculated from the full kinetic equation\nand then inserted in Eq. (29). The final result (for the Stoner mod el) is given by Eq. (10)\nor, equivalently, Eq. (9), with α=β. The latter is proportional to the spin-dephasing rate:\nβ=/planckover2pi1\nτσ∆xc. (30)\nThe derivation assumes ω,τ−1\nσ≪∆xc//planckover2pi1, which is typically the case in real materials suffi-\nciently below the Curie temperature. The s-dmodel yields the same result for β, but\nα=ηβ (31)\nis reduced by the η=s/s0ratio, i.e., the fraction of the itinerant to the total angular\nmomentum. [Note that Eq. (31) is also valid for the Stoner model sinc e thens0=s.] For\nthes-dmodel, the equation of motion (9) clearly cannot be reduced to Eq. ( 10), since α∝negationslash=β.\nThe steady-state current-driven velocity (22) for both mean-fi eld models becomes\nv=−βPj\nαs0=−Pj\ns, (32)\nwheresis the itinerant electron spin density. Interestingly, the velocity (3 2) is completely\ndetermined by properties of the conducting electrons, even for t hes−dmodel. In the Drude\nmodel,\nv∝Eτ\nm∗, (33)\nwhereEis the applied electric field, τis the characteristic momentum scattering time, and\nm∗is the effective mass of the itinerant bands at the Fermi energy. We expect the velocity\n(33), which is essentially the conducting electron drift velocity, to b e suppressed for the\n19s-dmodel if the dorbitals are coupled to their own dissipative bath, which has not been\nincluded in the above treatment.\nRef. [50] refines these results by relaxing the assumption that ∆ xc≪EFand by consid-\nering also anisotropic spin-dephasing impurities, which results in α∝negationslash=βfor both Stoner and\ns-dmodels. Ref.[51]laterofferedaKeldysh functional-integral appro achleading tothesame\nresults. (These authors also found stochastic torques express ed in terms of thermal fluctu-\nations (5) in the weak current limit; see, however, Ref. [71] for add itional current-induced\nstochastic terms present in the case of an inhomogeneous magnet ization.) Consider, for\nexample, weak magnetic disorder described by the potential ˆHσ=h(r)·ˆσwith Gaus-\nsian white-noise correlations ∝angb∇acketleftha(r)hb(r′)∝angb∇acket∇ight ∝Uaδabδ(r−r′), where Ua=U⊥(U/bardbl) whenais\nperpendicular (parallel) to the equilibrium magnetization direction. (S pin-orbit interaction\nassociated with scalar disorder gives similar results.) For isotropic dis order,U⊥=U/bardbl, and\n∆xc≪EF,α/β≈ηwithη=s/s0, as was already discussed (reducing to η= 1 for the\nStoner model). Even for larger exchange, the correction to this α/βratio turns out to be\nrather small: For parabolic bands, for example, α/β≈[1−(∆xc/EF)2/48]η. This ratio is\nmore sensitive to anisotropies U/bardbl∝negationslash=U⊥, however, so that in general α/β∝negationslash=ηeven in the\nlimit ∆ xc/EF→0 [50].\nVI. SUMMARY AND OUTLOOK\nOur microscopic understanding is based on a mean-field approximatio n, in which itiner-\nant electrons interact self-consistently with a space- and time-de pendent exchange field. We\npresented results for the local-spin-density approximation and th e mean-field s-dmodel. We\nidentified a relation between dissipative terms parametrized by αandβand spin-dephasing\nscattering potentials. The central result for the collective low-fr equency long-wavelength\ncurrent-driven magnetization dynamics can be formulated as a gen eralization of the phe-\nnomenological Landau-Lifshitz-Gilbert equation, accounting for t he current-driven torques.\nOne should in general also include stochastic terms due to thermal fl uctuations as well as\nnonequilibrium shot-noise contribution in the presence of applied cur rentj[71]. Despite\nsome recent efforts, stochastic effects remain to be relatively une xplored both theoretically\nand experimentally, however.\nThe most important parameter that determines the effect of an ele ctric current on the\n20collective magnetization dynamics in extended systems is the ratio β/α. We find that\nthis ratio is not universal and in general depends on details of the ba nd structure and\nspin-dephasing processes. Nevertheless, simple models give α∼βwith the special limit\nα≈βfor the Stoner model with weak and isotropic spin-dephasing disord er. Solving the\nmagnetization equation of motion for a domain wall is rather straight forward at low dc\ncurrents, when the wall is only slightly compressed. The domain-wall motion can then be\nmodeled within the Walker ansatz, based on parametrizing the magne tic dynamics in terms\nof wall position and spin distortion. Two regimes can then be distinguis hed: At the lowest\ncurrents, the wall moves steadily in the presence of a constant un iform current, while above\nthe so-called Walker threshold, the magnetization close to the wall c enter starts oscillating,\nresulting in a singular dependence of the average velocity on the app lied current.\nThe values of the αandβparameters are not affected by the magnetization textures.\nMicromagnetic simulations can provide better understanding of exp erimental results in the\nregimeswheredomainwallsarenotwell describedbyaone-dimensiona l model. Experiments\ncan contribute to the understanding by studying ferromagnets w ith systematic variations\nof impurity types and concentration, for Py and other different ma terials. Experimental\ninvestigation of creep in metallic ferromagnets at temperatures fa r below the critical ones,\nas compared to studies [33, 39] on magnetic semiconductors close t o the Curie transition,\nare highly desirable in order to advance our understanding.\nBesides realistic microscopic evaluations of the key parameters αandβ, the collective\ncurrent-driven magnetization dynamics pose many theoretical ch allenges, in the spirit of\nclassical nonlinear dynamical systems. Current-driven magnetism displays a rich behavior\nwell beyond what can be achieved by applied magnetic fields only. At su percritical currents,\nferromagnetism becomes unstable, possibly leading to chaotic dyna mics [79], although al-\nternative scenarios have been also suggested [80]. Domain-wall dyn amics in a medium with\ndisordered pinning potentials pose an interesting yet, at weak applie d currents, tractable\nproblem. Spin torques and dynamics in sharp walls and the role of stro ng intrinsic spin-orbit\ncoupling (relevant for dilute magnetic semiconductors) are not yet completely understood.\nOscillatory domain-wall motion under ac currents and in curved geom etries is also starting\nto attract attention both experimentally and theoretically [35, 81]. Another direction of\nrecent activities concern the backaction of a moving domain wall on t he charge degrees of\nfreedom [82, 83, 84, 85, 86].\n21With the exciting recent and forthcoming experimental developmen ts, the questions con-\ncerning interactions of the collective ferromagnetic order with elec tric currents will certainly\nchallenge theoreticians for many years to come. The prospects of using purely electric means\nto efficiently manipulate magnetic dynamics are also promising for prac tical applications.\nVII. ACKNOWLEDGMENTS\nWewould like tothanktheEditors forcarefullyreading themanuscrip t andmaking many\nuseful comments. 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Orenstein1,2\n1Materials Science Division, Lawrence Berkeley National La boratory, Berkeley, California 94720, USA\n2Department of Physics, University of California, Berkeley , California 94720, USA and\n3Physik Department E21, Technische Universit¨ at M¨ unchen, D-85748 Garching, Germany\n(Dated: DatedAugust 30, 2018)\nWe study the magnetic excitations of itinerant helimagnets by applying time-resolved optical spectroscopy\nto Fe0.8Co0.2Si. Optically excited oscillations of the magnetization in the helical state are found to disperse\nto lower frequency as the applied magnetic field is increased ; the fingerprint of collective modes unique to\nhelimagnets,knownashelimagnons. Theuseoftime-resolve dspectroscopyallowsustoaddressthefundamen-\ntal magnetic relaxation processes by directly measuring th e Gilbert damping, revealing the versatility of spin\ndynamics inchiralmagnets.\nTheconceptofchiralitypervadesallofscience,havingpro -\nfound implications in physics, chemistry and biology alike .\nIn solids, relativistic spin-orbit coupling can give rise t o the\nDzyaloshinskii-Moriya (DM) interaction,2,3imparting a ten-\ndency for the electron spins to form helical textures with a\nwell-definedhandednessincrystalslackinginversionsymm e-\ntry. Helical spinorderisespeciallyinterestingwhenthem ag-\nnetismarisesfromthesameelectronsresponsibleforcondu c-\ntion as is the case in doped FeSi which displays unconven-\ntional magnetoresitence,4,5helimagnetism,6and the recently\ndiscovered Skyrmion lattice.7,8The excitations of helimag-\nnets have been studied over the past 30 years, culminating\nrecently in a comprehensive theory of spin excitations call ed\nhelimagnons.9,10Signatures of helimagnons have been ob-\nserved in neutron scattering11and microwave absorption,12\nyet little is known about their magnetodynamics and relax-\nation phenomenaon the sub-picosecondtimescales on which\nmagnetic interactions occur. Understanding the dynamics,\nhowever,isofgreatimportanceregardingspintransfertor que\neffects in chiral magnets, and related proposed spintronic s\napplications.13–15\nIn this work we study the dynamics of collective spin ex-\ncitationsin the itineranthelimagnetFe 0.8Co0.2Si. Ouroptical\npump-probemeasurementsidentifyanomalousmodesatzero\nwavevector ( q=0) which we identify unmistakably as heli-\nmagnons. These helimagnons manifest as strongly damped\nmagnetization oscillations that follow a characteristic s caling\nrelation with respect to temperature and magnetic field. The\nsub-picosecond time resolution of our technique enables de -\nterminationof the intrinsic Gilbert dampingparameterwhi ch\nis foundto be oneorderof magnitudelargerthan in localized\nsystems, revealing the versatility of the spin-lattice int erac-\ntions available in the emergent class of DM-driven helimag-\nnets.\nDespite being a non-magnetic insulator, FeSi is trans-\nformed into an itinerant magnet upon doping with cobalt.4,16\nWe have chosen Fe 0.8Co0.2Si for our study because it can\neasily be prepared in high quality single crystals17with a\nreasonably high magnetic ordering temperature TN, and its\nexotic equilibrium properties are well characterized, ope n-\ning the door for non-equilibrium dynamical studies. Small-\nangle neutron scattering8was used to determine the phase\ndiagram and has revealed helimagnetic spin textures belowTN=30 K that emerge from the interplay between the fer-\nromagnetic exchange and DM interactions. In zero magnetic\nfield the spins form a proper helix with a spatial period of\n≈350˚A,18whereasfinitefieldscantthespinsalongthehelix\nwavevector, kh, (see Fig. 1(c))inducinga conicalstate witha\nnet magnetization. Sufficiently high fields, H≥Hc, suppress\nthe conical order in favor of field alignment of all spins. In\ntheexperimentsreportedhere,femtosecondpulsesoflinea rly\npolarized 1.5 eV photons from a Ti:Sapphire oscillator were\nused to excite a (100) oriented single crystal of Fe 0.8Co0.2Si\nat near normal incidence. The changes induced in the sam-\nple by the pump pulse were probed by monitoringthe reflec-\ntion and Kerr rotation of time-delayed probe pulses from the\nsame laser. In order to minimize laser heating of the sam-\nple the laser repetition rate was reduced to 20 MHz with an\nelectro-optic pulse picker, ensuring that thermal equilib rium\nwasreachedbetweensuccessivepumppulses. Signaltonoise\nwas improved by modulation of the pump beam at 100 KHz\nandsynchronouslock-indetectionofthereflectedprobe. Ke rr\nrotation was measured using a Wollaston prism and balanced\nphotodiode. All temperature and field scans presented in thi s\nwork were performed from low to high TandH||(100)after\nzero-fieldcooling.\nFig. 1 shows the transient reflectivity, ΔR/R, as a function\nof temperature and magnetic field. At high temperature we\nobserve a typical bolometric response from transient heati ng\nof the sample by the pump pulse (Fig 1 (a)).19This is char-\nacterized by a rapid increase in reflectivity, followed by tw o-\ncomponent decay on the fs and ps timescales, corresponding\nto the thermalizationtimes between differentdegreesof fr ee-\ndom (electron, spin, lattice, etc.).20As the sample is cooled\nbelowTN, the small thermal signal is beset by a much larger\nnegative reflectivity transient (Fig. 1 (b)) with a decay tim e\nof roughly τR≈175 ps at low temperature (Fig. 3 (b)). A\nnatural explanation for this is that the pump pulse weakens\nthe magnetic order below TN, which in turn causes a change\nin reflectivity via the resulting shift of spectral weight to low\nenergy.21Thetemperaturedependenceof the peak ΔR/Rval-\nues is plotted in Fig. 1 (c) for several applied fields, showin g\nonlyweakfielddependence.\nToaccessthemagnetizationdynamicsmoredirectlywean-\nalyze the polarizationstate of the probe pulses, which rota tes\nby an angle θKupon reflection from the sample surface, in2\nFIG.1: Timedependence ofthepump-inducedtransientreflec tivityΔR/Rinthe(a)paramagneticand(b)helimagneticstates. Thetem perature\ndependence of the maximum ΔR/Ris plottedin(c)for several applied magnetic fields.\nproportion to the component of the magnetization along the\nlight trajectory. The change in Kerr rotation induced by the\noptical pump, ΔθK, is shown in Fig. 2 as a function of tem-\nperature and field. The upper panels show temperature scans\nat fixedmagneticfield,while afieldscan atfixedtemperature\nisshowninpanel(d). Weobservethat ΔθKchangessignas H\nisreversed(notshown),andgoestozeroas Hgoesto zeroor\nas temperatureis raised above TN. Oscillationsof the magne-\ntization are clearly visible in the raw data below 25 K in the\nhelimagneticphase.\nIn order to analyze the magnetization dynamics, we use a\nsimple phenomenological function that separates the oscil la-\ntory and non-oscillatorycomponentsseen in the data. It con -\nsists ofadecayingsinusoidaloscillation,\nΔθK=e−t\nτK[A+Bsin(ωt)] (1)\nwitha timedependentfrequency,\nω(t)=2πf0/bracketleftBig\n1+0.8/parenleftBig\ne−t\nτK/parenrightBig/bracketrightBig\n(2)\nwhich decays to a final value ω0. We emphasize that there is\nonly a single decay time τKdescribing the magneto dynam-\nics, and it is directly related to the Gilbert damping parame -\nterα=(2πf0τK)−1. This function produces excellent fits to\nthe data as illustrated in Fig. 3 (a), allowing accurate extr ac-\ntion of the oscillation frequencies and decay times shown in\nFigs. 3 (b)-(d). The oscillation frequency is reduced as ei-\nther field or temperature is increased, while the decay time\nτKis roughly constant and equal to τRbelow 20 K. As the\ntemperature is raised towards the phase transition, the rel ax-\nation time τKdiverges, which can be understood in terms of\na diverging magnetic correlation length due to the presence\nof a critical point. The similarity between the decay times τR\nandτKwithin the ordered phase reflects strongly correlated\ncharge and spin degrees of freedom, and supports the notion\nthatΔR/Risdeterminedbythemagneticorder.\nThemagneticoscillationfrequencyreaches f0≈4.8GHzat\nlowtemperature,whichcorrespondstoaLarmorprecessiono f\nspinssubjectedtoafieldof170mT,whichisroughlythecrit-\nical field Hcrequiredto destroy the spin helix. This, togetherwith the fact that the oscillation frequencyis nonzero only in\nthehelicalstate,suggeststhattheoscillationsarecomin gfrom\nexcitations unique to the helical structure. It is well know n\nthat magnetization oscillations can be optically induced b y\nultrafast generation of coherent magnons,24–26however, or-\ndinary magnons cannot explain our data as their frequency\nwouldincreasewith H,oppositetowhatisseenin Fig. 3(c).\nBased on these observations, we propose the following in-\nterpretation of our results: In the helical magnetic phase, the\npump photons weaken the magnetic order through the ultra-\nfast demagnetization process.27As described above, this re-\nduction in magnetic order gives rise to a decrease in the re-\nflectivity at 1.5 eV which is nearly field independent. As a\nmagnetic field is applied the spins become canted along the\nhelix wavevector,giving rise to a macroscopic magnetizati on\nwhichweobserveinKerrrotationviaitscomponentalongthe\nprobelight trajectory. The demagnetizationfromthe pumpi s\nresponsible for the initial peak seen in the ΔθKtime traces,\nand is captured by the exponential component of our fitting\nfunction (green curve in Fig 3 (a)). The pump photons also\nlaunch a coherentspin wave, giving rise to the oscillations in\nΔθK(red curve in Fig. 3 (a)). The form of the oscillatory\ncomponent goes like sin (ωt)rather than [1−cos(ωt)], sug-\ngesting impulsive stimulated Raman scattering as the mecha -\nnism of excitation.25The anomalousfield dependenceshown\nin Fig. 3 (c) leads to the unambiguous conclusion that the\noptically excited spin waves are the fundamental modes of\nhelimagnetstermedhelimagnons.10Specifically,theoptically\naccessible helimagnon mode consists of the constituents of\nthe spin helix precessing in-phase about their local effect ive\nfield. Since this local effective field is reduced during the u l-\ntrafast demagnetizationprocess, the oscillation frequen cyde-\ncreases as a function of time delay as the field recovers, ne-\ncessitatingthetimedependentfrequencyinEq. 1. Theabili ty\nto resolve helimagnons with femtosecond time resolution at\nq=0isuniquetoouropticalprobe,andcomplimentsneutron\nscatteringwhichisrestrictedtomappinghelimagnonbands at\nhigherq. This regionof reciprocalspace is particularlyinter-\nestinginthecaseofhelimagnetsastheperiodicityintrodu ced\nbythehelicalspintexturegeneratesbandsthatarecentere dat\nq=±khand therefore have finite frequency modes at q=03\nFIG. 2: (a),(b),(c) Time dependence of the pump-induced cha nge in Kerr rotation, ΔθK, as a function of temperature for several applied\nmagneticfields. (d) ΔθKasafunctionofmagneticfieldforseveraltemperatures. Cur vesareoffsetforclarity. Alsoshownisaschematicphase\ndiagram, adapted from Reference 8,withredarrows illustra tingthe temperature and fieldscans usedin(a)-(d).\neven in the absence of a gap. This is in contrast to ordinary\nmagnonsinwhichthebandsaregenerallycenteredat q=0so\nthattheassociatedmodehaszerofrequency. Wenotethatour\nobservationsare in agreementwith previouswork on the col-\nlectivemodesofskyrmions28whichcoexistwithhelimagnons\nin theA-Phase(see Fig. 3).12Theappearanceofthese modes\nis not expected in our data as their corresponding oscillati on\nperiodsexceedtheobserveddampingtimein Fe 0.8Co0.2Si.\nInordertoquantitativelytestthehelimagnoninterpretat ion\nwetaketheexpressionforthe q=0helimagnonfrequencyin\nanexternalmagneticfield,\nf0=gµBHc/radicalbig\n1+cos2θ (3)\nwheregistheeffectiveelectron g-factor,µBistheBohrmag-\nneton,andπ\n2−θistheconicalanglei.e. theamountthespins\nare canted away from khby the applied field H. Ignoringde-\nmagnetization effects of the spin waves themselves, we can\nwrite sinθ=H\nHc, whereHcis the critical field at which the\nspinsall alignwiththe field andthe helimagnonceasesto ex-\nist asa well-definedmode. Thenweobtain,9\nf0=gµBHc/radicalBigg\n1−1\n2/parenleftbiggH\nHc/parenrightbigg2\n(4)\nwhich expresses the magnon frequency as a function of ap-\nplied field. This expression fits the data remarkably well as\nshown in Fig. 3 (c), capturingthe decrease in frequencywith\nincreasing Hwhich is unique to helimagnons. However, due\ntothefactthattheoscillationperiodexceedsthedampingt imeforfieldsabove75mT,itisnotpossibletoextractthevalueo f\nthecritical field Hcinthis system. The solidline in Fig. 3(d)\nisafittotheform f0∝/radicalBig\n1−T\nTNwhichgives TNasafunction\nofHin reasonableagreementwithpublisheddata.8\nThe Gilbert damping parameter can be directly obtained\nfrom the measured decay times through the relation α=\n(2πf0τK)−1, which gives a value of α≈0.4 for the heli-\nmagnetic phase of Fe 0.8Co0.2Si. This is an order of magni-\ntude larger than what was seen in insulating Cu 2OSeO3,12\nwhere helimagnetism arises from localized rather than itin -\nerant spins. The contrast in dynamics between these systems\nis critical in the context of potential spintronic applicat ions\nbasedonhelimagnetismwherethereisatradeoffbetweenfas t\nswitching which requires large damping, and stability whic h\nreliesonlowdamping.\nIn summary, this work demonstrates ultrafast coherent op-\ntical excitation of spin waves in an itinerant DM-driven spi n\nsystem and reveals the underlying spin dynamics. We iden-\ntifytheseexcitationsashelimagnonsthroughtheiranomal ous\nfield dependence and explain our observations with a com-\nprehensive model. Our experiments directly yield the intri n-\nsic Gilbertdampingparameter,revealingastrikingdiffer ence\nin spin relaxationphenomenabetweenitinerant andlocaliz ed\nhelimagnets. The results elucidate the dynamicsof collect ive\nmodes common to the actively studied B20 transition metal\ncompounds that codetermine their performance in potential\nspinbasedapplications.\nAcknowledgments: The work in Berkeley was supported\nby the Director, Office of Science, Office of Basic Energy4\nFIG.3: (a) Exemplary ΔθKoscillation data (blue circles) and fit (black line) using th e model described inthe text. The fitis decomposed into\nan exponential term (green curve) and an oscillatory term (r ed curve). The fitting function uses a single time constant τKfor all terms which\nis plotted in panel (b) as a function of temperature and field. For comparison we also plot the decay time of the reflectivity ,τR, averaged over\nall fields. The solid lines are guides to the eye. Panels (c) an d (d) show the reduced magnetization oscillation frequency for field scans and\ntemperature scans respectively, andsolidlines are fitstot he data as described inthe maintext.\nSciences,MaterialsSciencesandEngineeringDivision,of the\nU.S. Department of Energy under Contract No. DE-AC02-\n05CH11231. C.P. and A.B. acknowledge support through\nDFGTRR80(FromElectronicCorrelationstoFunctionality) ,\nDFG FOR960 (Quantum Phase Transitions), and ERC AdG\n(291079, TOPFIT). A.B. acknowledges financial support\nthrough the TUM graduate school. D.M. acknowledges sup-portfromtheAlexandervonHumboldtfoundationandS.A.P.\nacknowledgessupportfrom the SimonsFoundation. C.P. and\nA.B. also thank S. Mayr, W. M¨ unzer, and A. Neubauer for\nassistance.\n∗Theseauthorscontributedequallytothiswork.\n†Email address: jdkoralek@lbl.gov and meier@berkeley.edu\n2I. E.Dzyaloshinskii, Sov. Phys.JETP 5, 1259 (1957).\n3T. Moriya, Phys. Rev. 120, 91(1960).\n4N. Manyala etal., Nature404, 581 (2000).\n5N. 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Kapeliovich and T. L. Perelman, Sov. Phy s.\nJETP39, 375 (1975).\n21F.P.Mena etal.,Phys. Rev. B 73, 085205 (2006).\n22S.A.Brazovskii,S.G.Dmitriev,Sov.Phys.JETP 42,497(1976).\n23M. Janoschek, M. Garst, A. Bauer, P. Krautscheid, R. Georgii ,\nP.B¨ oni, andC.Pfleiderer,arXiv:1205.4780v1(2012).\n24M. van Kampen etal.,Phys. Rev. Lett. 88, 227201 (2002).\n25A. M.Kalashnikova et al.,Phys.Rev. B 78, 104301 (2008).\n26D. Talbayev et al.,Phys. Rev. Lett. 101, 097603 (2008).\n27A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod. Phys. 82,\n2731 (2010).\n28M. Mochizuk, Phys. Rev. Lett. 108, 017601 (2012)."    },    {        "title": "1507.06748v1.Boosting_Domain_Wall_Propagation_by_Notches.pdf",        "content": "arXiv:1507.06748v1  [cond-mat.mes-hall]  24 Jul 2015Boosting Domain Wall Propagation by Notches\nH. Y. Yuan and X. R. Wang1,2,∗\n1Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong\n2HKUST Shenzhen Research Institute, Shenzhen 518057, China\nWereportacounter-intuitivefindingthatnotchesinanothe rwise homogeneousmagnetic nanowire\ncan boost current-induced domain wall (DW) propagation. DW motion in notch-modulated wires\ncan be classified into three phases: 1) A DW is pinned around a n otch when the current density\nis below the depinning current density. 2) DW propagation ve locity is boosted by notches above\nthe depinning current density and when non-adiabatic spin- transfer torque strength βis smaller\nthan the Gilbert damping constant α. The boost can be manyfold. 3) DW propagation velocity is\nhindered when β > α. The results are explained by using the Thiele equation.\nPACS numbers: 75.60.Ch, 75.78.-n, 85.70.Ay, 85.70.Kh\nI. INTRODUCTION\nMagnetic domain wall (DW) motion along a nanowire\nunderpins many proposals of spintronic devices1,2. High\nDW propagation velocity is obviously important because\nit determines the device speed. In current-driven DW\npropagation,many efforts havebeen devoted to high DW\nvelocity and low current density in order to optimize de-\nvice performance. The issue of whether notches can en-\nhance current-induced DW propagation is investigated\nhere.\nTraditionally, notches are used to locate DW\npositions1–4. Common wisdom expects notches to\nstrengthen DW pinning and to hinder DW motion. In-\ndeed, in the field-driven DW propagation, intentionally\ncreated roughness slows down DW propagation although\nthey can increase the Walker breakdown field5. Unlike\nthe energy-dissipation mechanism of field-induced DW\nmotion6, spin-transfer torque (STT)7–10is the driven\nforce behind the current-driven DW motion. The torque\nconsists of an adiabatic STT and a much smaller non-\nadiabatic STT9,10. In the absence of the non-adiabatic\nSTT, there exists an intrinsic pinning even in a homoge-\nneous wire, below which a sustainable DW motion is not\npossible11,12. Interestingly, there are indications13that\nthe depinningcurrentdensityofaDWtrappedinanotch\nis smaller than the intrinsic threshold current density in\nthe absence of the non-adiabatic STT. Although there is\nno intrinsic pinning1,10in the presence ofa non-adiabatic\nSTT, It is interesting to ask whether notches can boost\nDW propagation in the presence of both adiabatic STT\nand non-adiabatic STT.\nIn this paper, we numerically study how DW propa-\ngates along notch-modulated nanowires. Three phases\nare identified: pinning phase when current density is be-\nlow depinning current density ud; boosting phase and\nhindering phase when the current density is above ud\nandthe non-adiabaticSTT strength βissmallerorlarger\nthan the Gilbert damping constant α, respectively. The\naverage DW velocity in boosting and hindering phases\nis respectively higher and lower than that in the wire\nwithout notches. It is found that DW depinning is facili-tated by antivortex nucleation. In the case of β < α, the\nantivortexgenerationis responsiblefor velocityboost be-\ncause vortices move faster than transverse walls. In the\nother case of β > α, the longitudinal velocity of a vor-\ntex/antivortex is slower than that of a transverse wall in\nahomogeneouswallandnotcheshinderDWpropagation.\nII. MODEL AND METHOD\nWe consider sufficient long wires (with at least 8\nnotches) of various thickness and width. It is well known\nthat14narrowwiresfavoronlytransversewallswhilewide\nwires prefer vortex walls. Transverse walls are the main\nsubjects of this study. A series of identical triangular\nnotches of depth dand width ware placed evenly and\nalternately on the two sides of the nanowires as shown in\nFig. 1a with a typical clockwise transverse wall pinned\nat the center of the first notch. The x−,y−andz−axis\nare along length, width, and thickness directions, respec-\ntively. The magnetization dynamics of the wire is gov-\nerned by the Landau-Lifshitz-Gilbert (LLG) equation\n∂m\n∂t=−γm×Heff+αm×∂m\n∂t−(u·∇)m+βm×(u·∇)m,\nwherem,γ,Heff, andαare respectively the unit vec-\ntor of local magnetization, the gyromagnetic ratio, the\neffective field including exchange and anisotropy fields,\nand the Gilbert damping constant. The third and fourth\nterms on the right hand side are the adiabatic STT and\nnon-adiabatic STT10. The vector uis along the electron\nflow direction and its magnitude is u=jPµB/(eMs),\nwherej,P,µB,e, andMsare current density, current\npolarization,the Bohrmagneton, the electronchargeand\nthe saturation magnetization, respectively. For permal-\nloy ofMs= 8×105A/m,u= 100 m/s corresponds\ntoj= 1.4×1012A/m2. In this study, uis lim-\nited to be smaller than both 850 m/s (corresponding to\nj≃1.2×1013A/m2!) and the Walkerbreakdowncurrent\ndensity because current density above the values gener-\natesintensivespinwavesaroundDWsandnotches,which\nmakes DW motion too complicated to be even described.2\nxy\nz\n(b) (a) \nL\nw\ndj\nFIG. 1. (color online) (a) A notch-modulated nanowire. L\nis the separation between two adjacent notches. The color\ncodes the y−component of mwith red for my= 1, blue for\nmy=−1 and green for my= 0. The white arrows denote\nmagnetization direction. (b) The phase diagram in β−u\nplane. A is the pinning phase; B is the boosting phase; and C\nis the hindering phase. Vortices are (are not) generated nea r\nnotches by a propagating DW in C1 (C2). Inset: The notch\ndepth dependence of depinning current udwhen notch width\nis fixed at w= 48 nm.\nDimensionless quantity βmeasures the strength of non-\nadiabatic STT and whether βis larger or smaller than α\nis still in debate10,15,16. The LLGequation isnumerically\nsolved by both OOMMF17andMUMAX18packages19. The\nelectric current density is modulated according to wire\ncross section area while the possible change of current\ndirection around notch is neglected. The material pa-\nrameters are chosen to mimic permalloy with exchange\nstiffness A= 1.3×10−11J/m,α= 0.02 andβvarying\nfrom 0.002 to 0.04. The mesh size is 4 ×4×4 nm3.\nIII. RESULTS\nA. Transverse walls in wide wires: boosting and\nhindering\nThis is the focus of this work. Our simulations on\nwires of 4 nm thick and width ranging from 32 nm to\n128 nm and notches of d= 16 nm and wvarying from\n16 nm to 128 nm show similar behaviors. Domain walls\nin these wires are transverse. Results presented below\nare on a wire of 64 nm wide and notches of w= 48 nm.\nThree phases can be identified. A DW is pinned at a\nnotch when uis below a depinning current density ud.\nThis pinning phase is denoted as A (green region) in Fig.\n1b. Surprisingly, udincreases slightly with β, indicatingthat the β-term actually hinders DW depinning out of\na notch although it is responsible for the absence of the\nintrinsic pinning in a uniform wire (see discussion below\nfor possible cause). When uis above ud, a DW starts to\npropagate and it can either be faster or slower than the\nDW velocity in the corresponding uniform wire, depend-\ning on relative values of βandα.\nWhenβ < α, DW velocity is boosted through antivor-\ntexgenerationat notches. Thisphaseisdenoted asphase\nB. When β > α, the boosting of DW propagation is sup-\npressed no matter vortices are generated (phase C1) or\nnot (phase C2). The upper bound of the phase plane\nis determined by the Walker breakdown current density\nandu= 850 m/s. If the current density is larger than\nthe upper bound, spin waves emission from DW20and\nnotches are so strong that new DWs may be created.\nAlso, the Walker breakdown is smaller than the depin-\nning value udforβ >0.04. Thus the phase plane in Fig.\n1b is bounded by β= 0.04. Although the general phase\ndiagram does not change, the phase boundaries depend\non the wire and notch specificities. The inset is notch\ndepth dependence of the depinning current when w= 48\nnm andβ= 0.0121.\nBoosting phase: The boost of DW propagation for β < α\ncan be clearly seen in Fig. 2. Figure 2a is the average\nDW velocity ¯ vas a function of notch separation Lfor\nu= 600 m/s > ud. ¯vis maximal around an optimal\nnotch separation Lp, which is close to the longitudinal\ndistance that an antivortex travels in its lifetime. Lp\nincreases with βand it is respectively about 1.5 µm, 2\nµm, and 4 µm forβ= 0.005 (squares), 0.01 (circles) and\n0.015 (up-triangles). This result suggests that the an-\ntivortex generation and vortex dynamics are responsible\nfor the DW propagation boost. Filled symbols in Fig. 2b\nare ¯vfor various current density when Lpis used. For a\ncomparison, DW velocities in the corresponding homoge-\nneous wires are also plotted as open symbols which agree\nperfectlywith ¯ v=βu/αdiscussedbelow. Take β= 0.005\nas an example, ¯ vis zero below ud= 550 m/s and jumps\nto an average velocity ¯ v≃550 m/s at ud, which is about\nfour times of the DW velocity in the homogeneous wire.\nAs the current density further increases, the average ve-\nlocityalsoincreasesandisapproximatelyequalto u. The\ninset of Fig. 2b shows the instantaneous DW velocity for\nβ= 0.005 and u= 600 m/s. Blue dots denote the mo-\nments at which the DW is at notches. Right after the\ncurrent is turned on at t= 0 ns, the instantaneous DW\nvelocity is very low until an antivortex of winding num-\nberq=−123,24is generatednear the notch edge at 0.5ns\n(see discussion and Fig. 9 below). The motion of the an-\ntivortex core drags the whole DW to propagate forward\nat a velocity around 600 m/s. The antivortex core anni-\nhilatesitselfatthebottomedgeofthewireaftertraveling\nabout 1.5 µm and the initial transverse wall reverses its\nchirality at the same time24. Surprisingly, the reversal of\nDW chiralityleads to a significantincreasesofDW veloc-\nity as shown by the peaks of the instantaneous velocity\nat about 2.0ns in the inset. Another antivortex of wind-3\n(a) \n(b) \nFIG. 2. (color online) (a) L−dependence of average DW ve-\nlocity ¯vforu= 600 m/s, α= 0.02, andβ= 0.005 (squares),\n0.01 (circles), 0.015 (up-triangles). The dash lines are βu/α.\n(b)u−dependence of ¯ vforβ= 0.005 (squares), 0.01 (circles),\n0.015 (up-triangles). Open symbols are DW velocity in the\ncorresponding homogeneous wires. Straight lines are βu/α. ¯v\nis above βu/αwhenu > u d. Inset: instantaneous DW speed\nforu= 600 m/s, β= 0.005, and L= 1.5µm. The blue dots\nindicate the moments when the DW is at notches.\ning number q=−1 is generated at the second notch and\nDW propagation speeds up again. Once the antivortex\ncore forms, it pulls the DW out of notch. This process\nthen repeats itself and the DW propagates at an average\nlongitudinal velocity of about 600 m/s. A supplemental\nmovie corresponding to the inset is attached25.\nHindering phase: Things are quite different for β > α.\nFigure 3a shows that ¯ vincreases monotonically with L\nforβ= 0.025, 0.03 and 0.035, which are all larger than\nα. In order to make a fair comparison with the results of\nβ < α, Fig. 3b is the current density dependence of ¯ vfor\nL= 2µm andβ= 0.025 (filled squares), 0.03 (filled cir-\ncles)and0.035(filledup-triangles). Again,DWvelocities\nin the corresponding homogeneouswires are presented as\nopen symbols. Take β= 0.025 as an example, although\nthe average velocity jumps at the depinning current den-\nsity 565 m/s, it’s still well below the DW velocity in the\ncorresponding uniform wire. The inset of Fig. 3b shows(b) (a) \nFIG. 3. (color online) (a) L−dependence of ¯ vforu= 600\nm/s and β= 0.025 (squares), 0.03 (circles), and 0.035 (up-\ntriangles), all larger than α= 0.02. The dash lines are βu/α.\n(b)u−dependence of ¯ vforL= 2µm. Fill symbols (squares\nforβ= 0.025, circles for β= 0.03, and up-triangles for\nβ= 0.035) are numerical data in notched wire of w= 48\nnm andd= 16 nm. Open circles are DW velocity of the cor-\nrespondinghomogeneous wire. Straight lines are βu/α. Inset:\ninstantaneous DW velocity for u= 600 m/s and β= 0.025.\nThebluedotsdenotethemomentswhentheDWisatnotches.\nthe instantaneous DW velocity for u= 600 m/s. An\nantivortex is generated at the first notch. In contrast\nto the case of β < α, the antivortex slows down DW\npropagation velocity below the value in the correspond-\ning uniform wire. Moreover, the transverse wall keeps its\noriginal chirality unchanged when the antivortex is anni-\nhilated at wire edge, and no vortex/antivortex is gener-\nated at the second notch. However, another antivortex\nis generated at the third notch. This is the typical cycle\nof phase C1. As uincreases above 640 m/s, phase C1\ndisappears and the DW passes all the notches without\ngenerating any vortices. This motion is termed as phase\nC2. For β >0.025, only phase C2 is observed. In C2,\nDW profile is not altered, and the average DW velocity\nis slightly below that in a uniform wire.4\n(b) (a) \nFIG. 4. (color online) (a) u−dependence of ¯ vforβ=\n0.01 (filled circles) and 0.015 (filled up-triangles). (b)\nu−dependence of ¯ vforβ= 0.03 (filled squares) and 0.035\n(filled up-triangles). Open symbols are DW velocity in the\ncorresponding homogeneous wires. Straight lines are βu/α. ¯v\nis below βu/αwhenu > u d. The nanowire is 8 nm wide and\n1 nm thick while the notch size is 10 nm wide and 2 nm deep\nfor (a) and 50 nm wide and 2 nm deep for (b). The separation\nof adjacent notches is 100 nm.\nB. Transverse walls in very narrow wires\nOne interesting question is whether notches can boost\nDW propagation in very narrow wires such that the nu-\ncleation of a vortex/antivortex is highly unfavorable. To\naddress this issue, Fig. 4a are u−dependence of the av-\nerage DW velocity on a 8 nm wide wire for β < α(circles\nforβ= 0.01 and up-triangles for β= 0.015) with (filled\nsymbols) and without (open symbols) notches. When\nnotches are placed, notch depth is 2 nm, L= 100 nm,\nw= 10 nm. DW velocity in the corresponding ho-\nmogeneous wire (open symbols) follows perfectly with\n¯v=βu/α(straight lines). It is clear that averaged DW\nvelocity in the notched wire (filled symbols) is below the\nvalues of the DW velocity in the corresponding homo-\ngeneous wire. Take β= 0.015 as an example, ¯ vis zero\nbelowud= 310 m/s and jumps to an average velocity\n¯v≃168 m/s at ud, which is below the DW velocity in\nthe corresponding uniform wire.\nThings are similar for β > α. Figure 4b is the cur-\nrent density dependence of ¯ vforβ= 0.03 (filled squares)\nand 0.035 (filled up-triangles). Again, DW velocities in\nhomogeneous wire are presented as open symbols for a\ncomparison. The averaged DW velocity in the notched\nwire (filled symbols) is below the values of the DW ve-\nlocity in the corresponding homogeneous wire.\nC. Vortex walls in very wide wires\nAlthough our main focus is on transverse walls, it\nshould be interesting to ask whether DW propagation\nboost can occur for vortex walls. It is well-known that\na vortex/antivortex wall is more stable for a much wider\nwire in the absence of a field and a current14. One\nmay expect that DW propagation boost would not oc-\ncur in such a wire because the boost comes from vor-\ntex/antivortex generation near notches and a such vor-\ntex/antivortex exists already in a wider wire even in the\nmy\n+1 -1 0200 nm 1.5 ns 13.5 ns \n18.0 ns 26.0 ns 0 ns \n47.5 ns 14.5 ns 0 ns (c) \n(d) (a) (b) \nFIG. 5. (color online) (a) u−dependence of ¯ vforβ=\n0.01 (filled circles) and 0.015 (filled up-triangles). (b)\nu−dependence of ¯ vforβ= 0.025 (filled squares) and 0.03\n(filled circles). Open symbols are DW velocity in the corre-\nsponding homogeneous wires. Straight lines are βu/α. ¯vis\nabove (below) βu/αwhenu > u dandβ < α(β > α). The\nnanowire is 520 nm wide and 10 nm thick while the rectan-\ngular notch is 160 nm wide and 60 nm deep. The separation\nof adjacent notches is 8 µm. (c) and (d) The spin configu-\nrations in a uniform wire (a) and in a notched wire (b) at\nvarious moments for β= 0.01 andu= 650 m/s. The time\nis indicated on the bottom-right corner of each configuratio n.\nThe color codes the value of myand color bar is shown in the\nbottom-right corner.\nabsence of a current. However, DW propagation boost\nwas still observed as shown in Fig. 5 for a wire of 520\nnm wide and 10 nm thick. Rectangular notches of 60\nnm deep and 160 nm wide are separated by L= 8µm.\nWhenβ < α(Fig. 5a: circles for β= 0.01 and up-\ntriangles for β= 0.015), the average DW propagation\nvelocities in the notched wire (filled symbols) is higher\nthan the DW velocity in the corresponding homogeneous\nwire (open symbols) when u > ud. Figure 5b shows that\nthe average DW propagation velocities in a notched wire\n(filled symbols) is lower than that in the corresponding\nhomogeneous wire (open symbols) for β > α(squares for\nβ= 0.025 and circles for β= 0.03). Figure 5c shows the\nspin configurations of the DW in the homogeneous wire\nofβ= 0.01before a current is applied (the left configura-\ntion) and during the current-driven propagation (middle\nand right configurations). When a current u= 650 m/s\nis applied at 0 ns, a vortex wall moves downward. The\nvortex was annihilated at wire edge, and the vortex wall\ntransformintoatransversewall. TheDWkeepsitstrans-\nverse wall profile and propagates with velocity of βu/α\n(solid lines in Fig. 5a and 5b). The middle and right\nconfigurations are two snapshots at 14.5 ns and 47.5 ns.\nTime is indicated in the bottom-right corner. Figure 5d\nare snapshots of DW spin configurations in the notched\nwire ofβ= 0.01 when a current u= 650 m/s is applied5\nu\n    m\nxxy\nzu\n    m\nx(a) (b)\nFIG. 6. (color online) Directions of vortex core magnetizat ion\n(red symbols) and non-adiabatic torque (blue symbols) for a\nclockwise transverse wall (a) and a counterclockwise trans -\nverse wall (b). The dots (crosses) represent ±z-direction.\natt= 0 ns. At t= 0 ns, a vortex wall is pinned near\nthe first notch. Right after the current is turned on, the\nvortex wall starts to depin and complicated structures\nmay appear during the depinning process as shown by\nthe snapshot at t= 1.5 ns. At t= 13.5 ns, the DW\ntransforms to a transverse wall and propagates forward.\nWhen the transverse wall reaches the second notch at\naboutt= 18.0 ns, new vortex core nucleates near the\nnotch and drags the whole DW to propagate forward. In\ncontrast to the case of homogeneous wire where a prop-\nagating DW prefers a transverse wall profile, DW with\nmore than one vortices can appear as shown by the snap-\nshot att= 26.0 ns. The vortex core in this structure\nboosts DW velocity above the average DW velocity of a\nuniform wire. This finding may also explain a surprising\nobservation in an early experiment4that depinning cur-\nrent does not depend on DW types. A vortex wall under\na current transforms into a transverse wall before depin-\nning from a notch. Thus both vortex wall and transverse\nwall have the same depinning current.\nIV. DISCUSSION\nA. Depinning process analysis\nEmpirically, we found that vortex/antivortex polarity\nis uniquely determined by the types of transverse wall\nand current direction. This result is based on more than\ntwenty simulations that we have done by varying var-\nious parameters like notch geometry, wire width, mag-\nnetic anisotropy, damping etc. Within the picture that\nDW depinning starts from vortex/antivortex nucleation,\ntheβ−dependence of depinning current density udcan\nbe understood as follows. For a clockwise (counter-\nclockwise) transverse wall and current in −xdirection,\np= +1 (p=−1), as shown in Fig. 6. If one as-\nsumes that vortex/antivortex formation starts from the\nvortex/antivortex core, it means that the core spin ro-\ntates into + z-direction for a vortex of p= 1. For a clock-\nwise wall, β-torque ( βm×∂m\n∂x) tends to rotate core spin\nin−z-direction, as shown in Fig. 6a, so the presence of\na smallβ-torque tries to prevent the nucleation of vor-\ntices. Thus, the larger βis, the higher udwill be. This\nmay be the reason why the depinning current density ud\nincreases as βincreases.(a) (b)\nFIG. 7. (a) Depinning current density as a function of an\nexternal field. A 0.4 ns field pulse in the x-direction is turned\non simultaneously with the current. The shape of a pulse of\nH= 100 Oe is shown in the inset. Since the depinning field\nof the wire (64 nm wide and 4 nm thick) is 150 Oe, the field\namplitude is limited to slightly below 150 Oe in the curve. (b )\nDepinningcurrentdensityas afunctionof nanowire thickne ss.\nOur simulations suggest that DW depinning starts\nfrom vortex/antivortexnucleation. Adiabatic spin trans-\nfertorquetendstorotatethespinsattheedgedefectnear\na notch out of plane and to form a vortex/antivortex\ncore. Thus, any mechanisms that help (hinder) the\ncreation of a vortex/antivortex core shall decrease (in-\ncrease) the depinning current density ud. To test this\nhypothesis, we use a magnetic field pulse of 0.4 ns along\n±x−direction (shown in the inset of Fig. 7a) such that\nthe field torque rotates spins out of plane. Figure 7a\nis the numerical results of the magnetic field depen-\ndence of the depinning current density for a 64 nm wide\nwire with triangular notches of 48 nm wide and 16 nm\ndeep. The non-adiabatic coefficient is β= 0.01. As\nexpected, uddecreases (increases) with field when it is\nalong -x−direction (+ x−direction) so that spins rotate\ninto +z−direction (- z−direction). All other parameters\nare the same as those for Fig. 2.\nIf the picture is correct, one should also expect the\ndepinning current density depends on the wire thick-\nness. The shape anisotropy impedes vortex core for-\nmation because it does not favor a spin aligning in the\nz−direction. The shape anisotropy decreases as the\nthickness increases. Thus, one should expect the depin-\nning current density decreases with the increase of wire\nthickness. Indeed, numerical results shown in Fig. 7b\nverifiestheconjecture. All otherparametersarethe same\nas that in Fig. 7a ( H= 0).\nB. Width effects on the depinning current density\nThe DW propagating boost shown above is from the\nwire in which the notch depth (16 nm) is relatively big in\ncomparisonwith wire width (64 nm). Naturally, one may\nask whether the DW propagation boost exists also in a\nwire when the notch depth is much smaller than the wire\nwidth. To address the issue, we fix the notch geometry\nand vary the wire width. Figure 8 is the nanowire width\ndependence of depinning current density when the notch6\n(a) (b) \n(c) (d) 50 nm 50 nm \nFIG. 8. (color online) (a) and (b) are nanowire width de-\npendence of depinning current density for β= 0.005 (a) and\nβ= 0.01 (b), respectively. The wire thickness is 4 nm and\nnotch size is fixed at 48 ×16 nm2. (c) and (d) are the real\nconfigurations of initial domain walls pinned at the notch fo r\n64 nm and 160 nm wide wires, respectively. The color coding\nis the same as that of Fig. 5. The blue jagged lines indicate\nthe profiles of triangular notches.\nsize is fixed at 48 ×16 nm2. Figures 8a and 8b show the\nphase boundary between vortex-assisted boosting phase\nand the pinning phase. DW propagation boost exists\nwhen nanowire width is one order of magnitude larger\nthan the notch depth. The top view of the wire and spin\nconfigurations for 64 nm wide and 160 nm wide wires are\nshown in Fig. 8c and Fig. 8d, respectively.\nC. DW Propagation and vortex dynamics\nDW propagation boost and slow-down by vortices can\nbe understood from the Thiele equation10,26,27,\nF+G×(v−u)+D·(αv−βu) = 0,(1)\nwhereFis the external force related to magnetic field\nthat is zero in our case, Gis gyrovector that is zero for a\ntransverse wall and G=−2πqplM s/γˆ zfor a 2D vortex\nwall, where qis the winding number (+1 for a vortex and\n-1foranantivortex), pisvortexpolarity( ±1forcorespin\nin±zdirection) and lis the thickness of the nanowire. D\nis dissipation dyadic, whose none zero elements for a vor-\ntex/antivortex wall are Dxx=Dyy=−2MsWl/(γ∆)27,\nwhereWis nanowire width and ∆ is the Thiele DW\nwidth26.vis the DW velocity.\nFor a transverse wall, v=βu/α(solid lines) agrees\nperfectly with numerical results (open symbols) in ho-\nmogeneous wires as shown in Figs. 2b and 3b without\nany fitting parameters. For a vortex wall, the DW veloc-\nity is\nvy=1\n1+α2W2/(π2∆2)W\nπqp∆(α−β)u,(2)\nvx=u\n1+α2W2/(π2∆2)/parenleftbigg\n1−β\nα/parenrightbigg\n+βu\nα.(3)vydepends on DW width, αas well as β/α. For a given\nvortexwall, vyhas opposite sign for β < αandβ > α. In\nterms of topological classification of defects23, the edge\ndefect of the transverse DW at the first notch (Fig. 1a)\nhas winding number q=−1/2, and this edge defect can\nonlygivebirthtoanantivortexof q=−1andp= 1while\nitself changes to an edge defect of q= 1/2 as shown in\nFig. 9a. Empirically, we found that antivortexpolarityis\nuniquely determined by the types of transverse wall and\ncurrent direction. A movie visualizing the DW propa-\ngation in boosting phase is shown in the Supplemental\nMovie25. All the parameters are the same as the inset of\nFig. 2b. The three segments of identical length 1200 nm\nare connected in series to form a long wire. When β < α,\nthe antivortex moves downward ( vy<0) to the lower\nedge defect of winding number of q= 1/2. The lower\nedge defect changes its winding number to q=−1/2\nand the transverse DW reverses its chirality24when the\nvortex merges with the edge defect. Then another an-\ntivortex of winding number q=−1 andp=−1 is gen-\nerated at the second notch on the lower wire edge and\nit moves upward ( vy>0). The DW reverses its chiral-\nity again at upper wire edge when the antivortex dies.\nThen this cycle repeats itself. The spin configurations\ncorresponding to various stages are shown in the lower\npanels of Fig. 9a. When β > α, as shown in Fig. 9b,\nthe antivortex of q=−1 andp= +1 moves upward\nsincevy>0. The chirality of the original transverse\nwall shall not change when the antivortex is annihilated\nat the upper edge defect because of winding number con-\nservation. No antivortex is generated at the even number\nnotches and same type of the antivortex is generated at\nodd number notches, hence the transverse wall preserves\nits chirality throughout propagation. The corresponding\nspin configurations are shown in the lower panels of Fig.\n9b.\nThe second term in Eq. (3) (for vx) isβu/α, the same\nas the transverse DW velocity in a homogeneous wire\n(straight lines in Figs. 2b and 3b). The first term de-\npends on DW properties as well as βandα. It changes\nsign atβ=α.vxis larger than βu/αin the presence of\nvortices if β < α. Therefore, in this case vortex genera-\ntions and vortex dynamics boost DW propagation. For\nsmallαand to the leading order correction in αandβ,\nEq. (3) becomes vx=u−(α2−αβ)uW2/(π2∆2). Thus,\nthe longitudinal velocity equals approximately uand de-\npends very weakly on β. This is what was observed in\nFig. 2b. vx=ucorresponds to the complete conversion\nof itinerant electron spins into local magnetic moments.\nAlthough the Thiele equation cannot explain why a DW\ngenerates vortices around notches in phase B, it explains\nwell DW propagation boost for β < α. This result is in\ncontrast to the field-driven DW propagation where vor-\ntex/antivortexgenerationreduces the Walker breakdown\nfield and inevitably slows down DW motion5,24.\nBefore conclusion, we would also like to point out that\nit is possible to realize both β < α(boosting phase) and\nβ > α(hindering phase) experimentally in magnetic ma-7\n(a) \n(b) +1/2 +1/2 +1/2 +1/2 \n+1/2 +1/2 \n+1/2 +1/2 \n+1/2 +1/2 \n-1/2 -1/2 -1/2 \n+1/2 +1/2 +1/2 +1/2 +1/2 +1/2 +1/2 +1/2 -1/2 -1/2 +1/2 -1/2 -1/2 -1/2 \n-1 \n-1 -1 \n-1 -1 \n-1/2 \n(s1) (s2) (s3) (s4) (s5) \ns1 \ns1 (s1) (s2) (s3) (s4) (s5) s2 \ns2 s3 \ns3 s4 \ns4 s5 \ns5 (s6) \n(s6) 50 nm \ns6 \ns6 \nFIG. 9. (color online) (a) Illustrations of changes of topol og-\nical defects (transverse DW edge defects and vortices) duri ng\nthebirthanddeathofvortices inPhase Bas aDWpropagates\nfrom the left to the right and the corresponding spin config-\nurations at various moments. Lines represent DWs. Big blue\ndots for vortices and open circles for edge defects of wind-\ning number −1/2 and filled black circles for edge defects of\nwinding number 1 /2. The color coding is the same as that of\nFig. 5. The blue jagged lines indicate the profiles of trian-\ngular notches. The nanowire is 64 nm wide and 4 nm thick.\nThe notch dimensions are 48 ×16 nm3. The interval between\nadjacent notches is L= 1500 nm. u= 600 m/s, β= 0.005.\n(b) Illustrations of changes of topological defects in Phas e C1\nand the the corresponding spin configurations at various mo-\nments. The nanowire is 64 nm wide and 4 nm thick. The\nnotch dimensions are 48 ×16 nm2. The interval between ad-\njacent notches is L= 2000 nm. u= 600 m/s, β= 0.025.terials like permalloy with damping coefficient engineer-\ning. A recent study28demonstrated that αof permalloy\ncan increaseby four times througha dilute impurity dop-\ning of lanthanides (Sm, Dy, and Ho).\nV. CONCLUSIONS\nIn conclusion, notches can boost DW propagation\nwhenβ < α. The boost is facilitated by antivortex\ngeneration and motion, and boosting effect is optimal\nwhen two neighboring notches is separated by the dis-\ntance that an antivortex travels in its lifetime. In the\nboosting phase, DW can propagate at velocity uthat\ncorresponds to a complete conversion of itinerant elec-\ntron spins into local magnetic moments. When β > α,\nthe notches always hinder DW propagation. According\nto Thiele’s theory, the generation of vortices increases\nDW velocity for β < αand decreases DW velocity when\nβ > α. This explains the origin of boosting phase and\nhindering phase. Furthermore, it is found that a vortex\nwall favored in a very wide wire tends to transform to a\ntransverse wall under a current. This may explain exper-\nimental observation that the depinning current density is\nnot sensitive to DW types.\nVI. ACKNOWLEDGMENTS\nWe thank Gerrit Bauer for useful comments. HYY ac-\nknowledges the support of Hong Kong PhD Fellowship.\nThis work was supported by NSFC of China (11374249)\nas well as Hong Kong RGC Grants (163011151 and\n605413).\n∗Corresponding author: phxwan@ust.hk\n1S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320,\n190 (2008).\n2D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson, D.\nPetit, and R. P. Cowburn, Science 309, 1688 (2005).\n3M. Kl¨ aui, C. A. F. Vaz, J. A. C. Bland, W. Wernsdorfer,\nG. Faini, E. Cambril, L. J. Heyderman, F. Nolting, and U.\nR¨ udiger, Phys. Rev. Lett. 94, 106601 (2005).\n4M. Hayashi, L. Thomas, C. Rettner, R. Moriya, X. Jiang,\nand S. S. P. Parkin, Phys. Rev. Lett. 97, 207205 (2006).\n5Y. Nakatani, A. Thiaville, and J. Miltat, Nat. Mater. 2,\n521 (2003).\n6X. R. Wang, P. Yan, J. Lu and C. He, Ann. Phys. (N.Y.)\n324, 1815 (2009); X. R. Wang, P. Yan, and J. Lu, Euro-\nphys. Lett. 86, 67001 (2009).\n7L. Berger, J. Appl. Phys. 55, 1954 (1984).\n8J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n9S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).10A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Euro-\nphys. Lett. 69, 990 (2005).\n11Z. Li and S. Zhang, Phys. Rev. B 70, 024417 (2004).\n12G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601\n(2004).\n13H. Y. Yuan and X. R. Wang, European Physical Journal\nB (in press); arXiv:1407.4559 [cond-mat.mes-hall]\n14R. D. McMichael and M. J. Donahue, IEEE Trans. Magn.\n33, 4167 (1997).\n15G. S. D. Beach, C. Knutson, C. Nistor, M. Tsoi, and J. L.\nErskine, Phys. Rev. Lett. 97, 057203 (2006).\n16L. Thomas, R. Moriya, C. Rettner, and S. S. P. Parkin,\nScience330, 1810 (2010).\n17http://math.nist.gov/oommf.\n18A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F.\nGarcia-Sanchez, F. B. V. Waeyenberge, AIP Adbances 4,\n107133 (2014).8\n19OOMMF package was used in the early stage of this re-\nsearch. In order to simulate a long and wide wire, we\nswitched to MUMAX package. Two packages give almost\nidentical results on shorter wires, and the results present ed\nhere were generated from MUMAX.\n20B. Hu and X. R. Wang, Phys. Rev. Lett. 111, 027205\n(2013); X. S. Wang, P. Yan, Y. H. Shen, G. E.W. Bauer,\nand X. R. Wang, Phys. Rev. Lett. 109, 167209 (2012).\n21Notch geometry affects depinning current because of\nthe change of current density and perpendicular shape\nanisotropy (see Ref. 22) in notch area. Both effects help to\ngenerate vortices and thus reduce the depinning current.This may explain the result.\n22A. Aharoni, J. Appl. Phys. 83, 3432 (1998).\n23O. Tchernyshyov and G. -W. Chern, Phys. Rev. Lett. 95,\n197204 (2005).\n24H. Y. Yuan and X. R. Wang, J. Magn. Magn. Mater. 368,\n70 (2014).\n25See Supplemental Material at [URL] for DW propagation\nin boosting phase.\n26A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973).\n27D. L. Huber, Phys. Rev. B 26, 3758 (1982).\n28S. G. Reidy, L. Cheng, and W. E. Bailey, Appl. Phys. Lett.\n82, 1254 (2003)."    },    {        "title": "0809.2910v1.Spin_transfer_torque_induced_reversal_in_magnetic_domains.pdf",        "content": "arXiv:0809.2910v1  [cond-mat.other]  17 Sep 2008Spin-transfer torque induced reversal in\nmagnetic domains\nS. MurugeshaM. Lakshmananb\naDepartment of Physics & Meteorology, IIT-Kharagpur, Kharagpu r 721 302, India\nbCentre for Nonlinear Dynamics, School of Physics, Bharathid asan University,\nTiruchirappalli 620024, India\nAbstract\nUsing the complex stereographic variable representation f or the macrospin, from\na study of the nonlinear dynamics underlying the generalize d Landau-Lifshitz(LL)\nequation with Gilbert damping, we show that the spin-transf er torque is effectively\nequivalent to an applied magnetic field. We study the macrosp in switching on a\nStoner particle due to spin-transfer torque on application of a spin polarized cur-\nrent. We find that the switching due to spin-transfer torque i s a more effective\nalternative to switching by an applied external field in the p resence of damping. We\ndemonstrate numerically that a spin-polarized current in t he form of a short pulse\ncan be effectively employed to achieve the desired macro-spin switching.\nKey words: Nonlinear spin dynamics, Landau-Lifshitz equation, Spin- transfer\ntorque, Magnetization reversal\nPACS:75.10.Hk, 67.57.Lm, 75.60.Jk, 72.25.Ba\n1 Introduction\nIn recent times the phenomenon of spin-transfer torque has gained much at-\ntention in nanoscale ferromagnets[1,2,3]. Electromigration refers to the recoil\nlinearmomentumimpartedontheatomsofametalorsemiconductor asalarge\ncurrent is conducted across. Analogously, if the current is spin-p olarized, the\ntransfer of a strong current across results in a transfer of spin angular momen-\ntum to the atoms. This has lead to the possibility of current induced s witch-\ning of magnetization in nanoscale ferromagnets. With the success o f GMR,\n∗Corresponding author. Tel: +91 431 2407093, Fax:+91 431 240 7093\nEmail address: lakshman@cnld.bdu.ac.in (M. Lakshmanan).\nPreprint submitted to Chaos, Solitons and Fractals 7 Septem ber 2021this has immense application potential in magnetic recording devices s uch as\nMRAMs[3,4,5,6]. The phenomenon has been studied in several nanomag netic\npile geometries. The typical set up consists of a nanowire[3,7,8,9,10,11 ], or a\nspin-valve pillar, consisting of two ferromagnetic layers, one a long f erromag-\nneticpinnedlayer, and another small ferromagnetic layer or film, separated\nby a spacer conductor layer (see Figure 1). The pinned layer acts a s a reser-\nvoir for spin polarized current which on passing through the conduc tor and\non to the thin ferromagnetic layer induces an effective torque on th e spin\nmagnetization in the thin film ferromagnet. A number of experiments have\nbeen conducted on this geometry and the phenomenon has been co nvincingly\nconfirmed [12,13,14,15]. Although the microscopic quantum theory of the phe-\nnomenon is fairly well understood, interestingly the behavior of the average\nspin magnetization vector can be described at the semi-classical lev el by the\nLL equation with an additional term[16].\nj\nxyz\nPinned layer Conductor Thin film ConductorS S p\nFig. 1. A schematic diagram of the spin-valve pillar. A thin fi lm ferromagnetic layer\nwith magnetization Sis separated from long ferromagnetic layer by a conductor. ˆSp\nis the direction of magnetization in the pinned region, whic h also acts as a reservoir\nfor spin polarized current.\nFrom a different point of view, several studies have focused on mag netic pulse\ninduced switching of the macro-magnetization vector in a thin nanod ot un-\nder different circumstances [17,18,19,20]. Several experimental st udies have\nalso focussed on spin-current induced switching in the presence of a magnetic\nfield, switching behavior for different choices of the angle of the app lied field,\nvariation in the switching time, etc., [12,21,22,23,24,25]. A numerical stu dy\non the switching phenomenon induced by a spin current in the presen ce of\na magnetic field pulse has also been investigated very recently in [26]. A s an\nextension to two dimensional spin configurations, the switching beh avior on a\nvortex has been studied in [27].\nIn this article, by investigating the nonlinear dynamics underlying the gener-\nalized Landau-Lifshitz equation with Gilbert damping, we look at the ex citing\npossibility of designing solid state memory devices at the nanoscale, w herein\nmemory switching is induced using a spin polarized current alone, witho ut\nthe reliance on an external magnetic field. We compare earlier studie d switch-\ning behavior for the macro-magnetization vector in a Stoner partic le [17] in\nthe presence of an external magnetic field, and the analogous cas e wherein\nthe applied field is now replaced by a spin polarized current induced spin -\n2transfer torque, i.e., with the thin film in the first case replaced by a s pin\nvalve pillar. It will be shown that a pulse of spin polarized current is mor e\neffective in producing a switching compared to an applied field. In doing so we\nrewrite the system in terms of a complex stereographic variable inst ead of the\nmacro-magnetization vector. This brings a significant clarity in unde rstanding\nthe nonlinear dynamics underlying the macrospin system. Namely, it w ill be\nshown that, in the complex system, the spin-transfer torque is eff ectively an\nimaginary applied magnetic field. Thus the spin-transfer term can ac complish\nthe dual task of precession of the magnetization vector and dissip ation.\nThe paper is organized as follows: In Section 2 we discuss briefly the m odel\nsystem and the associated extended LL equation. In Section 3 we in troduce\nthe stereographic mapping of the constant spin magnetization vec tor to a\ncomplex variable, and show that the spin-transfer torque is effect ively an\nimaginary applied magnetic field. In Section 4 we present results from our\nnumerical study on spin-transfer torque induced switching pheno menon of the\nmacro-magnetization vector, for a Stoner particle. In particular , we study two\ndifferent geometries for the free layer, namely, (a) an isotropic sp here and (b)\nan infinite thin film. In applications to magnetic recording devices, the typ-\nical read/write time period is of the order of a few nano seconds. We show\nthat, in order to achieve complete switching in these scales, the spin -transfer\ntorque induced by a short pulse of sufficient magnitude can be affirma tively\nemployed. We conclude in Section 5 with a discussion of the results and their\npractical importance.\n2 The extended LL equation\nThe typical set up of the spin-valve pillar consists of a long ferromag netic\nelement, or wire, with magnetization vector pinned in a direction indica ted by\nˆSp, as shown in Figure 1. It also refers to the direction of spin polarizat ion of\nthe spin current. A free conduction layer separates the pinned ele ment from\nthe thin ferromagnetic film, or nanodot, whose average spin magne tization\nvectorS(t) (of constant magnitude S0) is the dynamical quantity of interest.\nThe cross sectional dimension of the layers range around 70 −100nm, while\nthe thickness of the conduction layer is roughly 2 −7nm[3,20]. The free layer\nthus acts as the memory unit, separated from the pinned layer cum reservoir\nby the thin conduction layer. It is well established that the dynamics of the\nmagnetization vector Sin the film in the semiclassical limit is efficiently de-\nscribed by an extended LL equation[16]. If ˆ m(={m1,m2,m3}=S/S0) is the\nunit vector in the direction of S, then\ndˆ m\ndt=−γˆ m×/vectorHeff+λˆ m×dˆ m\ndt−γag(P,ˆ m·ˆSp)ˆ m×(ˆ m׈Sp),(1)\n3a≡ℏAj\n2S0Ve. (2)\nHere,γis the gyromagnetic ratio (= 0 .0176Oe−1ns−1) andS0is the satura-\ntion magnetization (Henceforth we shall assume 4 πS0= 8400, the saturation\nmagnetization value for permalloy). The second term in (1) is the phe nomeno-\nlogical dissipation term due toGilbert[28] with damping coefficient λ. The last\nterm is the extension to the LL equation effecting the spin-transfe r torque,\nwhereAis the area of cross section, jis the current density, and Vis the\nvolume of the pinned layer.′a′, as defined in (2), has the dimension of Oe, and\nis proportional to the current density j.g(P,ˆ m·ˆSP) is given by\ng(P,ˆ m·ˆSp) =1\nf(P)(3+ˆ m·ˆSp)−4;f(P) =(1+P)3\n(4P3/2),(3)\nwheref(P)isthepolarizationfactorintroducedbySlonczewski [1],and P(0≤\nP≤1) is the degree of polarization of the pinned ferromagnetic layer. F or\nsimplicity, we take this factor gto be a constant throughout, and equal to 1.\n/vectorHeffisthe effective fieldacting onthespin vector due toexchange intera ction,\nanisotropy, demagnetization and applied fields:\n/vectorHeff=/vectorHexchange+/vectorHanisotropy+/vectorHdemagnetization +/vectorHapplied,(4)\nwhere\n/vectorHexchange=D∇2ˆ m, (5)\n/vectorHanisotropy=κ(ˆ m·ˆ e/bardbl)ˆ e/bardbl, (6)\n∇·/vectorHdemagnetization =−4πS0∇·ˆ m. (7)\nHere,κis the strength of the anisotropy field. ˆ e/bardblrefers to the direction of\n(uniaxial) anisotropy, In what follows we shall only consider homogen eous\nspin states on the ferromagnetic film. This leaves the exchange inte raction\nterm in (4) redundant, or D= 0, while (7) for /vectorHdemagnetization is readily solved\nto give\n/vectorHdemagnetization =−4πS0(N1m1ˆ x+N2m2ˆ y+N3m3ˆ z), (8)\nwhereNi,i= 1,2,3 are constants with N1+N2+N3= 1, and {ˆ x,ˆ y,ˆ z}are the\northonormal unit vectors. Equation (1) now reduces to a dynamic al equation\nfor a representative macro-magnetization vector ˆ m.\nIn this article we shall be concerned with switching behavior in the film p urely\ninduced by the spin-transfer torque term, and compare the resu lts with earlier\nstudies on switching due to an applied field [17] in the presence of dissip ation.\nConsequently, it will be assumed that /vectorHapplied= 0 in our analysis.\n43 Complex representation using stereographic variable\nIt proves illuminating to rewrite (1) using the complex stereographic variable\nΩ defined as[29,30]\nΩ≡m1+im2\n1+m3, (9)\nso that\nm1=Ω+¯Ω\n1+|Ω|2;m2=−i(Ω−¯Ω)\n1+|Ω|2;m3=1−|Ω|2\n1+|Ω|2.(10)\nFor the spin valve system, the direction of polarization of the spin-p olarized\ncurrentˆSpremains a constant. Without loss of generality, we chose this to be\nthe direction ˆ zin the internal spin space, i.e., ˆSp=ˆ z. As mentioned in Sec.\n2, we disregard the exchange term. However, for the purpose of illustration,\nwe choose /vectorHapplied={0,0,ha3}for the moment but take ha3= 0 in the later\nsections. Defining\nˆ e/bardbl={sinθ/bardblcosφ/bardbl,sinθ/bardblsinφ/bardbl,cosθ/bardbl} (11)\nand upon using (9) in (1), we get\n(1−iλ)˙Ω =−γ(a−iha3)Ω+im/bardblκγ/bracketleftBig\ncosθ/bardblΩ−1\n2sinθ/bardbl(eiφ/bardbl−\nΩ2e−iφ/bardbl)/bracketrightBig\n−iγ4π S0\n(1+|Ω|2)/bracketleftBig\nN3(1−|Ω|2)Ω−N1\n2(1−Ω2−|Ω|2)Ω\n−N2\n2(1+Ω2−|Ω|2)Ω−(N1−N2)\n2¯Ω/bracketrightBig\n,(12)\nwherem/bardbl=ˆ m·ˆ e/bardbl. Using (10) and (11), m/bardbl, and thus (12), can be written\nentirely in terms of Ω.\nIt is interesting to note that in this representation the spin-trans fer torque\n(proportional to the parameter a) appears only in the first term in the right\nhand side of (12) as an addition to the applied magnetic field ha3but with a\nprefactor −i. Thus the spin polarization term can be considered as an effective\napplied magnetic field. Letting κ= 0, and N1=N2=N3in (12), we have\n(1−iλ)˙Ω =−γ(a−iha3)Ω, (13)\nwhich on integration leads to the solution\nΩ(t) = Ω(0) exp( −(a−iha3)γt/(1−iλ))\n= Ω(0) exp( −a+λha3\n1+|λ|2γt) exp(−iaλ−ha3\n1+|λ|2γt). (14)\n5The first exponent in (14) describes relaxation, or switching, while t he second\nterm describes precession. From the first exponent in (14), we no te that the\ntime scale ofswitching is given by 1 /(a+λha3).λbeing small, thisimplies that\nthe spin-torque term is more effective in switching the magnetization vector.\nFurther, letting ha3= 0, we note that in the presence of the damping term\nthe spin transfer produces the dual effect of precession and diss ipation.\nTo start with we shall analyze the fixed points of the system for the two cases\nwhich we shall be concerned with in this article: (i) the isotropic spher e char-\nacterized by N1=N2=N3= 1/3, and (ii) an infinite thin film characterized\nbyN1= 0 =N3,N2= 1.\n(i) First we consider the case when the anisotropy field is absent, or κ= 0.\nFrom (12) we have\n(1−iλ)˙Ω =−aγΩ−iγ4πS0\n1+|Ω|2/bracketleftBig\nN3(1−|Ω|2)Ω−N1\n2(1−Ω2−|Ω|2)Ω\n−N2\n2(1+Ω2−|Ω|2)Ω−(N1−N2)\n2¯Ω/bracketrightBig\n.(15)\nIn the absence of anisotropy ( κ= 0), we see from (15) that the only fixed\npoint is Ω 0= 0.To investigate the stability of this fixed point we expand (15)\nup to a linear order in perturbation δΩ around Ω 0. This gives\n(1−iλ)δ˙Ω =−aγδΩ−iγ4πS0[N3−1\n2(N1+N2)]δΩ+iγ2πS0(N1−N2)δ¯Ω.(16)\nFor the isotropic sphere, N1=N2=N3= 1/3, (16) reduces to\n(1−iλ)δ˙Ω =−aγδΩ. (17)\nWe find the fixed point is stable since a >0. For the thin film, N1= 0 =\nN3,N2= 1. (16) reduces to\n(1−iλ)δ˙Ω =−aγδΩ+iγ2πS0δΩ−iγπS0δ¯Ω. (18)\nThis may be written as a matrix equation for Ψ ≡(δΩ,δ¯Ω)T,\n˙Ψ =MΨ, (19)\nwhereMis a matrix obtained from (18) and its complex conjugate, whose\ndeterminant and trace are\n|M|=(a2+3π2S2\n0)γ2\n1+λ2;Tr(M) =(−2a−4πS0λ)γ\n1+λ2.(20)\nSince|M|is positive, the fixed point Ω 0= 0 is stable if Tr|M|<0, or,\n(a+2πS0λ)>0.\n6The equilibrium point (a), Ω 0= 0, corresponds to ˆ m=ˆ z. Indeed this holds\ntrue even in the presence of an applied field, though we have little to d iscuss\non that scenario here.\n(ii) Next we consider the system with a nonzero anisotropy field in the ˆ z\ndirection. (12) reduces to\n(1−iλ)˙Ω =−aγΩ+iκγ(1−|Ω|2)\n(1+|Ω|2)Ω−iγ4πS0\n(1+|Ω|2)/bracketleftBig\nN3(1−|Ω|2)Ω\n−N1\n2(1−Ω2−|Ω|2)Ω−N2\n2(1+Ω2−|Ω|2)Ω−(N1−N2)\n2¯Ω/bracketrightBig\n.(21)\nHere again the only fixed point is Ω 0= 0.As in (i), the stability of the\nfixed point is studied by expanding (21) about Ω 0to linear order. Following\nthe same methodology in (i) we find the criteria for stability of the fixe d\npoint for the isotropic sphere is ( a+λκ)>0, while for the thin film it is\n(a+λ(κ+2πS0))>0.\n(iii) With nonzero κin an arbitrary direction the fixed point in general moves\naway from ˆ z.\nFinally, it is also of interest to note that a sufficiently large current lea ds to\nspin wave instabilities induced through spin-transfer torque [31,32]. In the\npresent investigation, however, we have assumed homogeneous m agnetization\nover the free layer, thus ruling out such spin wave instabilities. Rece ntly we\nhaveinvestigated spinwave instabilitiesoftheSuhltypeinduced bya napplied\nalternating field in thin film geometries using stereographic represen tation[30].\nIt will be interesting to investigate the role of a spin-torque on such instabil-\nities in the spin valve geometry using this formulation. This will be pursu ed\nseparately.\n4 Spin-transfer torque induced switching\nWenowlookattheinterestingpossibilityofeffectingcompleteswitchin g ofthe\nmagnetization using spin-transfer torque induced by a spin curren t. Numerical\nstudies on switching effected on a Stoner particle by an applied magne tic field,\nor in the presence of both a spin-current and applied field, in the pre sence of\ndissipation and axial anisotropy have been carried out recently and switching\nhas been demonstrated [17,26]. However, the intention here is to ind uce the\nsame using currents rather than the applied external fields. Also, achieving\nsuch localized magnetic fields has its technological challenges. Spin-t ransfer\ntorque proves to be an ideal alternative to accomplish this task sinc e, as we\nhave pointed out above, it can be considered as an effective (albeit c omplex)\n7magnetic field. In analogy with ref. [17], where switching behavior due to an\napplied magnetic field has been studied we investigate here switching b ehav-\nior purely due to spin-transfer torque, on a Stoner particle. Nume rical results\nin what follows have been obtained by directly simulating (12) and makin g\nuse of the relations in (10), for appropriate choice of parameters . It should\nbe remembered that (12) is equivalent to (1), and so the numerical results\nhave been further confirmed by directly numerically integrating (1) also for\nthe corresponding parameter values. We consider below two sample s differing\nin their shape anisotropies, reflected in the values of ( N1,N2,N3) in the de-\nmagnetization field: a) isotropic sphere, N1=N2=N3= 1/3 and b) a thin\nfilmN1= 0 =N3,N2= 1. The spin polarization ˆSpof the current is taken\nto be in the ˆ zdirection. The initial orientation of ˆ mis taken to be close to\n−ˆ z. In what follows this is taken as 170◦fromˆ zin the (z−x) plane. The\norientation of uniaxial anisotropy ˆ e/bardblis also taken to be the initial direction\nofˆ m. With these specified directions for ˆSpandˆ e/bardblthe stable fixed point is\nslightly away from ˆ z, the direction where the magnetization ˆ mis expected to\nswitch in time. A small damping is assumed, with λ= 0.008. The magnitude\nof anisotropy κis taken to be 45 Oe. As stated earlier, for simplicity we have\nconsidered the magnetization to be homogeneous.\n4.1 Isotropic sphere\nIt is instructive to start by investigating the isotropic sphere, whic h is char-\nacterized by the demagnetization field with N1=N2=N3= 1/3. With these\nvalues for ( N1,N2,N3), (12) reduces to\n(1−iλ)˙Ω =−aγΩ+im/bardblκγ/bracketleftBig\ncosθ/bardblΩ−1\n2sinθ/bardbl(eiφ/bardbl−Ω2e−iφ/bardbl)/bracketrightBig\n(22)\nA constant current of a= 10Oeis assumed. Using (2), for typical dimen-\nsions, this equals a current density of the order 108A/cm2. We notice that\nfor the isotropic sample the demagnetization field does not play any r ole in\nthe dynamics of the magnetization vector. In the absence of aniso tropy and\ndamping the spin-transfer torque term leads to a rapid switching of Sto the\nˆ zdirection. This is evident from (22), which becomes\n˙Ω =−aγΩ, (23)\nwith the solution Ω = Ω 0e−aγt, and the time scale for switching is given by\n1/aγ. Figure 2.a shows the trajectory traced out by the magnetization vector\nS, for 5 ns, initially close to the −ˆ zdirection, switching to the ˆ zdirection.\nFigure 2.b depicts the dynamics with anisotropy but no damping, all ot her\nparameters remaining same. While the same switching is achieved, this is\n8more smoother due to the accompanying precessional motion. Not e that with\nnonzero anisotropy, ˆ zis not the fixed point any more. The dynamics with\ndamping but no anisotropy (Figure 2.c) resembles Figure 2.a, while Figu re\n2.d shows the dynamics with both anisotropy and damping.\n-1\n-0.5\n 0\n 0.5\n 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(d)\nxyz\n-1\n-0.5\n 0\n 0.5\n 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(c)\nxyz-1\n-0.5\n 0\n 0.5\n 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(b)\nxyz\n-1\n-0.5\n 0\n 0.5\n 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(a)\nxyz\nFig. 2. Trajectory of the magnetization vector m, obtained by simulating (12) for\nthe isotropic sphere ( N1=N2=N3= 1/3), and using the relations in (10), for\na= 10Oe(a)withoutanisotropyanddamping,(b)withanisotropybut nodamping,\n(c) without anisotropy but nonzero damping and (d) with both anisotropy and\ndamping nonzero. The results have also been confirmed by nume rically integrating\n(1). The arrows point in the initial orientation (close to −ˆ z) and the direction of the\nspincurrent ˆ z. Evolution shownis for aperiodof 5 ns.Note that thefinal orientation\nis not exactly ˆ zin the case of nonzero anisotropy ((b) and (d)).\nIt may be noticed that Figures 2.c and 2.d resemble qualitatively Figure s\n2.a and 2.b, respectively, while differing mainly in the time taken for the\nswitching. It is also noticed that switching in the absence of anisotro py is\nfaster. Precession assisted switching has been the favored reco rding process in\nmagnetic memory devices, as it helps in keeping the exchange interac tion at\na minimum[18,19]. The sudden switching noticed in the absence of anisot ropy\nessentially refers to a momentary collapse of order in the magnetic m edia.\n9This can possibly lead to strong exchange energy and a breakdown o f our\nassumption regarding homogeneity of the magnetization field. Howe ver, such\nrapid quenching assisted by short high intensity magnetic pulses has in fact\nbeen achieved experimentally [33].\nA comparison with reference [17] is in order. There it was noted that with an\napplied magnetic field, instead of a spin torque, a precession assiste d switch-\ning was possible only in the presence of a damping term. In Section 3 we\npointed out how the spin transfer torque achieves both precessio n and damp-\ning. Consequently, all four scenarios depicted in Figure 2 show switc hing of\nthe magnetization vector without any applied magnetic field.\n4.2 Infinite thin film\nNext we consider an infinite thin film, whose demagnetization field is give n by\nN1= 0 =N3andN2= 1. With these values (12) becomes\n(1−iλ)˙Ω =−aγΩ+im/bardblκγ/bracketleftBig\ncosθ/bardblΩ−1\n2sinθ/bardbl(eiφ/bardbl−Ω2e−iφ/bardbl)/bracketrightBig\n−iγ4πS0/parenleftbigg1−|Ω|2\n1+|Ω|2/parenrightbigg\nΩ.(24)\nHere againin the absence of anisotropy Ω = 0 is the only fixed point. Th us the\nspin vector switches to ˆ zin the absence of damping and anisotropy (Figure\n3a). In order to achieve this in a time scale of 5 ns, we find that the value of a\nhas to be of order 50 Oe. Again the behavior is in stark contrast to the case\ninduced purely by an applied field[17], wherein the spin vector traces o ut a\ndistorted precessional trajectory. As in Sec. 4.1, the trajecto ry traced out in\nthe presence of damping is similar to that without damping (Figure 3c) . The\ncorresponding trajectories traced out in the presence of anisot ropy are shown\nin Figures 3b and 3d.\n4.3 Switching of magnetization under a pulsed spin-polariz ed current\nWe noticed that in the absence of uniaxial anisotropy, the constan t spin polar-\nized current can effect the desired switching to the orientation of ˆSp(Figure\n2). This is indeed the fixed point for the system (with no anisotropy) . Fig-\nure 2 traces the dynamics of the magnetization vector in a period of 5 ns, in\nthe presence of a constant spin-polarized current. However, fo r applications in\nmagnetic media we choose a spin-polarized current pulseof the form shown\nin Figure 4. It may be recalled here that, as was observed in 4.2, with a spin\n10-1\n-0.5\n 0\n 0.5\n 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(d)\nxyz\n-1\n-0.5\n 0\n 0.5\n 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(c)\nxyz-1\n-0.5\n 0\n 0.5\n 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(b)\nxyz\n-1\n-0.5\n 0\n 0.5\n 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(a)\nxyz\nFig. 3. Trajectory of the magnetization vector Sin a period of 5 ns, obtained as\nearlier by numerically simulating (12), and also confirming with (1), with demagne-\ntization field with by N1= 0 =N3,N2= 1 and a= 50Oe, and all the parameter\nvalues are as earlier. As in Figure 1, the ˆSand initial orientation are indicated\nby arrows. (a) Without anisotropy or damping, (b) nonzero an isotropy but zero\ndamping, (c) without anisotropy but nonzero damping and (d) both anisotropy and\ndamping nonzero. As earlier, in the presence of nonvanishin g anisotropy, the fixed\npoint is not the ˆ zaxis.\npolarized current of sufficient magnitude, the switching time can inde ed be\nreduced. We choose a pulse, polarized as earlier along the ˆ zdirection, with\nrise time and fall time of 1 .5ns, and a pulse width, defined as the time interval\nbetween half maximum, of 4 ns. We assume the rise and fall phase of the pulse\nto be of a sinusoidal form, though, except for the smoothness, t he switching\nphenomenon is independent of the exact form of the rise or fall pha se.\nIn Figures 5 and 6, we show trajectories of the spin vector for a pe riod of\n25ns, for the two different geometries, the isotropic sphere and a thin fi lm.\nThe action of the spin torque pulse, as in Figure 4, is confined to the fi rst\n5ns. We notice that, with the chosen value of a, this time period is enough\n11Rise time Fall time\nPulse width\n 0 20 40 60 80 100 120 140 160\n 0  1  2  3  4  5a (Oe)\ntime (ns)\nFig. 4. Pulse form showing the magnitude of a, or effectively the spin-polarized\ncurrent. The rise and fall phase are assumed to be of a sinusoi dal form. The rise\nand fall time are taken as 1 .5ns, and pulse width 4 ns. The maximum magnitude\nofais 150Oe.\nto effect the switching. In the absence of anisotropy, the directio n ofˆSpis\nthe fixed point. Thus a pulse of sufficient magnitude can effect a switc hing in\nthe desired time scale of 5 ns. From our numerical study we find that in order\nfor this to happen, the value of ahas to be of order 150 Oe, or, from (2),\na current density of order 109A/cm2, a magnitude achievable experimentally\n(see for example [34]). Comparing with sections 4.1 and 4.2, we note th at the\nextra oneorder ofmagnitude in current density isrequired due to t he duration\nof the rise and fall phases of the pulse in Figure(4). Here again we co ntrast\nthe trajectories with those induced by an applied magnetic field [17], w here\nthe switching could be achieved only in the presence of a uniaxial aniso tropy.\nIn Figure 5b for the isotropic sphere with nonzero crystal field anis otropy,\nwe notice that the spin vector switches to the fixed point near ˆ zaxis in the\nfirst 5ns. However the magnetization vector precesses around ˆ zafter the\npulse has been turned off. This is because in the absence of the spin- torque\nterm, the fixed point is along ˆ e/bardbl, the direction of uniaxial anisotropy. Due to\nthe nonzero damping term, the spin vector relaxes to the direction ofˆ e/bardblas\ntime progresses. The same behavior is noticed in Figure 6b for the th in film,\nalthough the precessional trajectory is a highly distorted one due to the shape\nanisotropy.\n12-1\n-0.5\n 0\n 0.5\n 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(b)\nxyz\n-1\n-0.5\n 0\n 0.5\n 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(a)\nxyz\nFig. 5. Evolution of the magnetization vector Sin a period of 25 nsinduced by the\nspin-polarized current pulse in Figure 4, (a) with and (b) wi thout anisotropy for\nthe isotropic sample, with N1=N2=N3= 1/3 all other parameters remaining\nsame. A nonzero damping is assumed in both cases. The current pulse acts on the\nmagnetization vector for the first 5 ns. In both cases switching happens in the first\n5ns. In the presence of nonzero anisotropy field, (b), the magnet ization vector\nprecesses to the fixed point near ˆ z.\n-1\n-0.5\n 0\n 0.5\n 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(b)\nxyz\n-1\n-0.5\n 0\n 0.5\n 1-1-0.5 0 0.5 1-1-0.5 0 0.5 1z(a)\nxyz\nFig. 6. Evolution of the magnetization vector Sin a period of 25 nsinduced by\nthe spin-polarized current pulse in Figure 4, for a infinite t hin film sample, with\nN1= 0 =N3, andN2= 1, all other parameters remaining same, along with a\nnonzero damping. (a) Without anisotropy and (b) with anisot ropy. As in Figure 5,\nswitching happens in the first 5 ns.\n5 Discussion and conclusion\nWe have shown using analytical study and numerical analysis of the n onlinear\ndynamics underlying the magnetization behavior in spin-valve pillars th at a\nvery effective switching of macro-magnetization vector can be ach ieved by a\nspin transfer-torque, modeled using an extended LL equation. Re writing the\n13extended LL equation using the complex stereographic variable, we find the\nspin-transfer torque term indeed acts as an imaginary applied field t erm, and\ncan lead to both precession and dissipation. It has also been pointed out why\nthe spin-torque term is more effective in switching the magnetization vector\ncompared to the applied field. On application of a spin-polarized curre nt the\naverage magnetization vector in the free layer was shown to switch to the\ndirection of polarization of the spin polarized current. For a consta nt current,\nthe required current density was found to be of the order of 108A/cm2. For\nrecording in magnetic media, switching is achieved using a stronger po larized\ncurrent pulse of order 109A/cm2. Currents of these magnitudes have been\nachieved experimentally.\nAcknowledgements\nThe work forms part of a research project sponsored by the Dep artment of\nScience andTechnology, Government ofIndia anda DSTRamannaFe llowship\nto M. L.\nReferences\n[1] J. C. Slonczewski. Current-driven excitation of magnet ic multilayers J. Mag.\nMag. Mat. 1996; 159: L1-L7.\n[2] L. Berger. Emission of spin waves by a magnetic multilaye r traversed by a\ncurrent Phys. Rev. B 1996; 54: 9353-58.\n[3] M. D. Stiles and J. Miltat. Spin-Transfer Torque and Dyna mics Topics Appl.\nPhy. 2006; 101: 225-308.\n[4] S. A. Wolf, A. Y. Chtchelkanova and D. M. Treger. Spintron ics A retrospective\nand perspective IBM J. Res. Dev. 2006; 50: 101-110.\n[5] R. K. Nesbet. 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Rev. Lett 2004; 92: 02 6602.\n[32] S. Adam, M. L. Polianski and P. W. Brouwer. Current-indu ced transverse spin-\nwave instability in thin ferromagnets: Beyond linear stabi lity analysis Phys.\nRev. B 2006; 73: 024425.\n[33] I. Tudosa, C. Stamm, A. B. Kashuba, F. King, H. C. Siegman n, J. St¨ ohr, G.\nJu, B. Lu and D. Weller. The ultimate speed of magnetic switch ing in granular\nrecording media Nature 2004; 428: 831-33.\n[34] M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang, M. Seck, M . Tsoi and\nP. Wyder. Excitation of a Magnetic Multilayer by an Electric Current Phys.\nRev. Lett. 1998; 80: 4281-84.\n16"    },    {        "title": "2107.00982v3.Anomalous_Gilbert_Damping_and_Duffing_Features_of_the_SFS___boldmath___varphi_0___Josephson_Junction.pdf",        "content": "arXiv:2107.00982v3  [cond-mat.supr-con]  25 Aug 2021Anomalous Gilbert Damping and Duffing Features of the SFS ϕ0Josephson Junction\nYu. M. Shukrinov1,2, I. R. Rahmonov1,3, A. Janalizadeh4, and M. R. Kolahchi4\n1BLTP, JINR, Dubna, Moscow Region, 141980, Russia\n2Dubna State University, Dubna, 141980, Russia\n3Umarov Physical Technical Institute, TAS, Dushanbe 734063 , Tajikistan\n4Department of Physics, Institute for Advanced Studies in Ba sic Sciences, P.O. Box 45137-66731, Zanjan, Iran\n(Dated: August 26, 2021)\nWe demonstrate unusual features of phase dynamics, IV-char acteristics and magnetization dy-\nnamics of the ϕ0Josephson junction at small values of spin-orbit interacti on, ratio of Josephson to\nmagnetic energy and Gilbert damping. In particular, an anom alous shift of the ferromagnetic reso-\nnance frequency with an increase of Gilbert damping is found . The ferromagnetic resonance curves\nshow the Duffing oscillator behaviour, reflecting the nonline ar nature of Landau-Lifshitz-Gilbert\n(LLG) equation. Based on the numerical analysis of each term in LLG equation we obtained an\napproximated equation demonstrated both damping effect and Duffing oscillator features. The re-\nsulting Duffing equation incorporates the Gilbert damping in a special way across the dissipative\nterm and the restoring force. A resonance method for the dete rmination of spin-orbit interaction in\nnoncentrosymmetric materials which play the role of barrie r inϕ0junctions is proposed.\nIntroduction. The Josephson junctions (JJ) with the\ncurrent-phaserelation I=Icsin(ϕ−ϕ0), wherethephase\nshiftϕ0is proportional to the magnetic moment of ferro-\nmagneticlayerdetermined bythe parameterofspin-orbit\ninteraction, demonstratea number ofunique featuresim-\nportant for superconducting spintronics, and modern in-\nformation technology [1–6]. The phase shift allows one\nto manipulate the internal magnetic moment using the\nJosephson current, and the reverse phenomenon which\nleads to the appearance of the DC component in the su-\nperconducting current [7–9].\nInteractive fields can bring nonlinear phenomena of\nboth classical, and quantum nature. A basic example\nis the magnons strongly interacting with microwave pho-\ntons [10]. As a result we could name Bose-Einstein con-\ndensation of such quasiparticles, i.e. magnons [11, 12],\nand synchronization of spin torque nano-oscillators as\nthey coherently emit microwave signals in response to\nd.c. current [13]. It is interesting that (semi)classical an-\nharmonic effects in the magnetodynamics described by\nthe Landau-Lifshitz-Gilbert (LLG) model in thin films or\nheterostructures [14, 15], and the quantum anharmonic-\nity in the cavity mangnonics [16] can well be modeled\nby so simple a nonlinear oscillator as Duffing. The cor-\nresponding Duffing equation contains a cubic term and\ndescribesthe oscillationsofthe variousnonlinearsystems\n[17].\nDespite the fact that nonlinear features of LLG are\nstudied often during a long time and in different systems,\nmanifestation of the Duffing oscillator behavior in the\nframeworkofthisequationisstill notcompletelystudied.\nCloser to our present investigation, in the study of the\ndynamics of antiferromagnetic bimeron under an alter-\nnatingcurrent,Duffingequationformsagoodmodel, and\nthis has applications in weak signal detection [14, 18, 19].\nAs another application with Duffing oscillator at work,\nwe can mention the ultra thin Co 20Fe60B20layer, andits largeangle magnetizationprecessionunder microwave\nvoltage. There are also ‘foldover’ features, characteris-\ntic of the Duffing spring, in the magnetization dynamics\nof the Co/Ni multilayer excited by a microwave current\n[15, 20, 21]. But nonlinear features of ϕ0Josephson junc-\ntions have not been carefully studied yet. In this Letter,\nwe show that the Duffing oscillator helps in the under-\nstanding of the nonlinear features of ϕ0Josephson junc-\ntions at small values of system parameters.\nCoupling of superconducting current and magnetiza-\ntion and its manifestation in the IV-characteristics and\nmagnetizationdynamicsopensthedoorfortheresonance\nmethod determination of spin-orbit intensity in noncen-\ntrosymmetric materials playing the role of barrier in ϕ0\njunctions. As it is well known, the spin-orbit interaction\nplays an important role in modern physics, so any novel\nmethod for its determination in real materials would be\nvery important. There are a series of recent experiments\ndemonstrating the modification of Gilbert damping by\nthe superconducting correlations (see Ref.[22] and cita-\ntionstherein). Inparticular, the pronouncedpeaksin the\ntemperature dependence of Gilbert damping have been\nobserved for the ferromagnetic insulator/superconductor\nmultilayers [23] which might be explained by the pres-\nence of spin relaxation mechanisms like the spin-orbit\nscattering [22]. Here, we use the noncentrosymmetric\nferromagnetic material as a weak link in ϕ0junctions.\nThe suitable candidates may be MnSi or FeGe, where\nthe lack of inversion center comes from the crystalline\nstructure [8].\nThe Gilbert damping determines the magnetization\ndynamics in ferromagnetic materials but its origin is not\nwell understood yet. Effect of nonlinearity on damp-\ning in the system is very important for application of\nthese materials in fast switching spintronics devices. Our\nstudy clarifies such effects. In Ref.[24] the authors dis-\ncuss the experimental study of temperature-dependent2\nGilbert damping in permalloy (Py) thin films of varying\nthicknesses by ferromagnetic resonance, and provide an\nimportant insight into the physical origin of the Gilbert\ndamping in ultrathin magnetic films.\nIn this Letter we demonstrate an anomalous depen-\ndence of the ferromagnetic resonance frequency with an\nincrease of the Gilbert damping. We find that the reso-\nnance curves demonstrate features of Duffing oscillator,\nreflecting the nonlinear nature of LLG equation. The\ndamped precession of the magnetic moment is dynami-\ncally driven by the Josephson supercurrent, and the res-\nonance behavior is given by the dynamics of the Duffing\nspring. The resonance methods for the determination of\nspin-orbit interaction in the ϕ0junction are proposed.\nModel and Methods. In the considered SFS ϕ0junc-\ntion (see Fig.1) the superconducting phase difference ϕ\nand magnetization Mof the F layer are two coupled dy-\nnamical variables. Based on the LLG equation for the\nFigure 1: Schematic view of SFS ϕ0Josephson junction. The\nexternal current applied along xdirection, ferromagnetic easy\naxis is along zdirection.\nmagnetic moment Mwith effective magnetic field Heff,\nresistively capacitively shunted junction (RCSJ) model,\nand Josephson relation for the phase difference ϕ, we de-\nscribe dynamics of the SFS ϕ0junction by the system of\nequations in normalized variables\ndm\ndt=ωFheff×m+α/parenleftbigg\nm×dm\ndt/parenrightbigg\n,\nheff=Grsin(ϕ−rmy)/hatwidey+mz/hatwidez, (1)\ndV\ndt=1\nβc[I−V+rdmy\ndt−sin(ϕ−rmy)],\ndϕ\ndt=V,\nwheremis vector of magnetization with components\nmx,y,z, normalized to the M0=/bardblM/bardbland and satisfy-\ning the constraint/summationtext\ni=x,y,zm2\ni(t) = 1,ωF= ΩF/ωc,\nΩF=γK/νis ferromagnetic resonance frequency, γis\nthe gyromagnetic ratio, Kis an anisotropic constant, ν\nis the volume of the ferromagnetic F layer, αis the phe-\nnomenologicaldamping constant(Gilbert damping), heff\nis the vector of effective magnetic field, normalized to\ntheK/M0(heff=HeffM0/K),G=EJ/(Kν) relation\nof Josephson energy to magnetic one, ris a parameter\nof spin-orbit coupling, ϕis phase difference of JJ, Vis\nvoltage normalized to the Vc=IcR,Iccritical current\nof JJ,Rresistance of JJ, βc= 2eIcCR2//planckover2pi1is McCumberparameter, Cis capacitance of JJ, Iis bias current nor-\nmalized to the Ic. In this system of equation time tis\nnormalized to the ω−1\nc, whereωc= 2eIcR//planckover2pi1is character-\nistic frequency. In the chosen normalization, the average\nvoltage corresponds to the Josephson frequency ωJ.\nFerromagnetic resonance in ϕ0junction. The ferro-\nmagnetic resonance features are demonstrated by aver-\nage voltage dependence of the maximal amplitude of the\nmycomponent ( mmax\ny), taken at each value of bias cur-\nrent. To stress novelty and importance of our finding,\nwe first present the analytical results for average volt-\nage dependence of mmax\nyalong IV-characteristics in the\nferromagnetic resonance region. As it was discussed in\nRefs.[8, 25, 26], in case Gr≪1,mz≈1, and neglecting\nquadratic terms mxandmy, we get\n/braceleftBigg\n˙mx=ξ[−my+GrsinωJt−αmx]\n˙my=ξ[mx−αmy],(2)\nwhereξ=ωF/(1 +α2). This system of equations can\nbe written as the second order differential equation with\nrespect to the my\n¨my=−2αξ˙my−ξ2(1+α2)my+ξ2GrsinωJt.(3)\nCorresponding solution for myhas the form\nmy(t) =ω+−ω−\nrsinωJt−α++α−\nrcosωJt,(4)\nwhere\nω±=Gr2ωF\n2ωJ±ωF\n((ωJ±ωF)2+(αωJ)2),(5)\nand\nα±=Gr2ωF\n2αωJ\n((ωJ±ωF)2+(αωJ)2).(6)\nSo,mydemonstrates resonance with dissipation when\nJosephson frequency is approaching the ferromagnetic\none (ωJ→ωF). The maximal amplitude mmax\nyas a\nfunction of voltage (i.e., Josephson frequency ωJ) at dif-\nferentα, calculated using (4), is presented in Fig.2 (a).\nWe see the usual characteristicvariation of the resonance\ncurve with an increase in dissipation parameter when the\nmaximal amplitude and position of resonance pick cor-\nresponds to the damped resonance. We note that the\nanalytical result (4) were obtained in the case Gr≪1.\nPresented in Fig.2(b) results of numerical simulations\nmmax\ny(V) dependence at different values of dissipation\nparameter αdemonstrate the essential differences with\nthe results followedfrom the analytical consideration(4).\nWe note also that the strong coupling of the supercon-\nducting phase difference ϕand magnetization Mof the\nF layermanifests itself by appearanceof subharmonics of\nthe resonance at ω= 1/2,1/3,1/4 demonstrated in the\ninset to Fig.2(b).3\nFigure 2: (a) Analytical results for maximal amplitude mmax\ny\nin the ferromagnetic resonance region for different α; (b)\nNumerical results for maximal amplitude of magnetization\nmy−component at each values of bias current and voltage\nalong IV-characteristics of the ϕ0junction in the ferromag-\nnetic resonance region for various α. Inset shows the man-\nifestation of the resonance subharmonics. Parameters are:\nβc= 25, G=0.05, r=0.05, ωF= 0.5.\nWe stress two important features followed from the\npresented results. First, the ferromagnetic resonance\ncurves show the foldover effect, i.e., the features of Duff-\ning oscillator. Different from a linear oscillator, the non-\nlinear Duffing demonstrates a bistability under external\nperiodic force [27]. Second, the ferromagnetic resonance\ncurves demonstrate an unusual dependence of the reso-\nnance frequency as a function of Gilbert damping α. As\nshown in Fig. 3(a), an increase in damping leads to a\nnonuniform change in the resonant frequency, i.e., with\nan increase in damping the resonance maximum shifts\ntoωFat small α, but then moves to the opposite side,\ndemonstrating the usual damped resonance. So, with\nan increase in α, unusual dependence of the resonance\nvoltage transforms to the usual one. For the parameters\nchosen, the critical value of this transformation is around\nα= 0.02−0.03. We call this unusual behaviour of the\nresonance maximum of mmax\nyas an “α-effect”. Both the\nα−effect and Duffing features in our system appear due\nto the nonlinear features of the system dynamics at small\nFigure 3: (a) α-dependence of the resonance curve mmax\ny(V)\npeak presented in Fig.2 in the damping parameter interval\n[0.006 – 0.2]. Dashed line indicates ferromagnetic resonan ce\nposition; (b) Comparison of the resonance curves mmax\ny(V)\ncalculated by full LLG equation (1) and the approximate\nequation (8).\nG,r,α≪1. To prove it, we have carried out the nu-\nmerical analysis of each term of LLG full equation (first\ntwo equations in (1)) for the set of model parameters\nG= 0.05,r= 0.05α= 0.005. After neglecting the\nterms of order 10−6, we have\n˙mx\nξ=−mymz+Grmzsin(ϕ−rmy)−αmxm2\nz,\n˙my\nξ=mxmz−αmym2\nz, (7)\n˙mz\nξ=−Grmxsin(ϕ−rmy)+αmz(m2\nx+m2\ny),\nInthisapproximationweobserveboththe“ α–effect”and\nDuffing oscillator features. Neglecting here the last term\nαmz(m2\nx+m2\ny) in third equationfor ˙ mz, which is orderof\n10−4, leadstothe losingoftheDuffing oscillatorfeatures,\nbut still keeps alpha-effect. We note that equation (7)\nkeeps the time invariance of the magnetic moment, so\nthat term plays an important role for manifestation of\nDuffing oscillator features by LLG equation.\nThe generalized Duffing equation for ϕ0junction.\nThe LLG is a nonlinear equation and in case of simple\neffective field it can be transformed to the Duffing equa-\ntion [14, 17]. Such transformation was used in Ref.[17]\nto demonstrate the nonlinear dynamics of the magnetic\nvortex state in a circular nanodisk under a perpendicular\nalternating magnetic field that excites the radial modes\nof the magnetic resonance. They showed Duffing-type\nnonlinear resonance and built a theoretical model corre-\nsponding tothe Duffing oscillatorfromthe LLG equation\nto explore the physics of the magnetic vortex core polar-\nity switching for magnetic storage devices.\nThe approximated LLG system of equations (7)\ndemonstrates both α-effect and features of Duffing os-\ncillator. As demonstrated in the Supplemental Materials\n[28], the generalizedDuffing equation forthe ϕ0junction,\n¨my+2ξα˙my+ξ2(1+α2)my\n−ξ2(1+α2)m3\ny=ξ2GrsinωJt.(8)4\ncan be obtained directly from the LLG system of equa-\ntions.\nAs we see, for small enough Gandr, it is only the\ndimensionless damping parameter αin LLG that plays a\nrole in the dynamics of the system. We can think of a\nharmonic spring with a constant that is hardened or soft-\nened by the nonlinear term. For a usual Duffing spring,\nwith independent coefficients of the various terms, the\nresonancepeak relative to the harmonic (linear) resonant\nfrequency folds over to the smaller (softening) or larger\n(hardening) frequencies. In the frequency response, the\ninterplay of the specific dependence of each coefficient on\nαplays an important role and as Fig.3(a) shows, there is\na particular αthat brings the resonant frequency closest\nto ferromagnetic resonance.\nSimulations of the mydynamics in the framework of\nDuffing equation can explain observed foldover effect in\nthe frequency dependence of mmax\ny. Comparison the re-\nsults followed from analytical approximate equation (8)\nand results of full equation (1) for maximal amplitude of\nmmax\nyin the ferromagnetic resonance region is presented\nin Fig.3(b). So, the magnetization dynamics in the SFS\nϕ0-junction due to the voltage oscillations can effectively\nbe described by a scalar Duffing oscillator, synchronizing\nthe precession of the magnetic moment with the Joseph-\nson oscillations.\nEffect of spin-orbit interactions. As we mentioned\nabove, the spin-orbit interaction plays an important role\nin different fields of modern physics. Here we have sug-\ngested a novel method for its determination in real non-\ncentrosymmetric ferromagnetic materials like MnSi or\nFeGe, where the lack of inversion center comes from\nthe crystalline structure Ref.[8] and which play role a\nweak link in ϕ0junctions. Based on the obtained re-\nsults, presented in Fig.4, we propose different versions of\nthe resonance method for the determination of spin-orbit\ninteraction in these materials. Particularly, in Fig.4(a)\nwe present the simulation results of maximal amplitude\nmmax\nybased on (1) at G= 0.05,α= 0.01 at different\nvalues of spin-orbit parameter rin the ferromagnetic res-\nonance region. This case corresponds to the nonlinear\napproximation leading to the Duffing equation (8). The\nsame characteristics calculated by equation (1) for larger\nvalueα= 0.1, i.e. corresponding to the linear approxi-\nmation (3) are presented in Fig.4(b). As it was expected,\nin caseα= 0.01 the foldover effect is more distinct.\nIn Fig.4(c) the r-dependence of the resonancepeak po-\nsition, obtained from the simulation results of full equa-\ntion atα= 0.01 andα= 0.1 for the same set of model\nand simulation parameters is demonstrated. We stress\nhere that nonlinear features of LLG equation leading to\nthe Duffing’s shift of the mmax\nypeak of main harmonic\nwith r presented in Fig.4(c) show the manifestation of\nnonlinearity.\nDespite the noted differences between results for α=\n0.01 andα= 0.1 , we see in both cases a monotonic\nFigure 4: (a) Voltage dependence of mmax\nyin the ferromag-\nnetic resonance region at different values of spin-orbit int er-\naction based on (1) at G= 0.05,α= 0.01. Inset enlarges\nthe main harmonic; (b) The same as in (a) for α= 0.1; (c)\nShift ofmmax\nypeak as a function of spin-orbit interaction at\ntwo values of Gilbert damping; (d) r-dependence of the main\nharmonic and subharmonics peaks in case (a); (e) The same\nas in (d) for the case (b).\nlinear increase of mmax\nypeak of main harmonic and sub-\nharmonics with rdemonstrated in Fig.4(d) and Fig.4(e).\nSuch lineardependence canbe noted fromEq. (6) ofRef.\n[14], but the authors did not discuss it. This dependence\nmight serve as a calibrated curve for spin-orbit interac-\ntion intensity, thus creating the resonance methods for r\ndetermination.\nConclusions. Based on the reported features of the\nϕ0Josephson junction at small values of spin-orbit in-\nteraction, ratio of Josephson to magnetic energy and\nGilbert damping, we have demonstrated that the cou-\npled superconducting current and the magnetic moments\nin theϕ0-junction result in the current phase relation in-5\ntertwining with the ferromagnetic LLG dynamics. The\nferromagnetic resonance clearly shows this interplay. In\nparticular, an anomalous shift of the ferromagnetic res-\nonance frequency with an increase of Gilbert damping\nis found. The ferromagnetic resonance curves demon-\nstrate features of Duffing oscillator, reflecting the nonlin-\near nature of LLG equation. The obtained approximated\nequation demonstrates both damping effect and Duffing\noscillator features. We have shown that due to the non-\nlinearity, asmodeledbythe generalizedDuffing equation,\nthe parameters of the system can compensate each other\nresulting in unusual response. The position of the maxi-\nmum can shift towards and then away from the expected\nresonant frequency, as the damping is decreased. There\nare also foldover effects that was explained by the pro-\nposed model. A resonance method for the determination\nof spin-orbit interaction in noncentrosymmetric materi-\nals which play the role of barrier in ϕ0junctions was\nproposed.\nThe experimental testing of our results would in-\nvolve SFS structures with ferromagnetic material having\nenough small value of Gilbert damping. Potential candi-\ndate for experimental realization could be ferromagnetic\nmetals or insulators which have small values of damping\nparameter ( α∼10−3−10−4). In Ref.[29] the authors\nreport on a binary alloy of cobalt and iron that exhibits\na damping parameterapproaching10−4, which is compa-\nrable to values reported only for ferrimagnetic insulators\n[30, 31]. Using superconductor-ferromagnetic insulator-\nsuperconductor on a 3D topological insulator might be\na way to have strong spin-orbit coupling needed for ϕ0\nJJ and small Gilbert dissipation for α-effect [5]. We note\nin this connection that the yttrium iron garnet YIG is\nespecially interesting because of its small Gilbert damp-\ning (α∼10−5). The interaction between the Joseph-\nson current and magnetization is determined by the ra-\ntio of the Josephson to the magnetic anisotropy energy\nG=EJ/(Kν) and spin-orbit interaction r. The value of\nthe Rashba-type parameter rin a permalloy doped with\nPt[32] and in the ferromagnets without inversion sym-\nmetry, like MnSi or FeGe, is usually estimated to be in\nthe range 0 .1−1. The value of the product Grin the ma-\nterialwith weakmagneticanisotropy K∼4×10−5KA−3\n[33], and a junction with a relatively high critical current\ndensity of (3 ×105−5×106)A/cm2[34] is in the range\n1−100. It givesthe set offerromagneticlayerparameters\nand junction geometry that make it possible to reach the\nvalues used in our numerical calculations for the possible\nexperimental observation of the predicted effect.\nNumerical simulations were funded by the project 18-\n71-10095oftheRussianScientificFund. A.J.andM.R.K.\nare grateful to IASBS for financial support.[1] Jacob Linder and W. A. Jason Robinson, Nature Physics\n11, 307 (2015).\n[2] Yu. M. Shukrinov, Accepted for UFN.\nDOI:https://doi.org/10.3367/UFNe.2020.11.038894\n[3] A.A. Mazanik, I.R. Rahmonov, A.E. Botha, and Yu.M.\nShukrinov, Phys. Rev. Applied 14, 014003 (2020).\n[4] M. Nashaat and Yu. M. Shukrinov, Physics of Particles\nand Nuclei Letters, 17, 79. (2020).\n[5] I. V. Bobkova , A. M. Bobkov, I. R. Rahmonov, A. 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Xie, J. S. Moodera, and W. Han,\nPhys. Rev. B 97, 224414 (2018).\n[24] Yuelei Zhao, Qi Song, See-Hun Yang, Tang Su1, Wei\nYuan, Stuart S. P. Parkin, Jing Shi and Wei Han, Sci.\nRep.6, 22890 (2016).\n[25] Shukrinov, Y.M., Rahmonov, I.R., Phys. Part. Nuclei 51,\n816 (2020).\n[26] Yu. M. Shukrinov, I. R. Rahmonov, and A. E. Botha.,\nLow Temp. Phys. 46, 932 (2020).\n[27] IvanaKovacic, Michael JBrennan. The DuffingEquation\n: Nonlinear Oscillators and their Behaviour. — John Wi-6\nley and Sons, 2011.\n[28] See Supplemental Material for details of our procedure\nto get the generalized Duffing equation from Landau-\nLifshitz-Gilbert system of equations.\n[29] M.A.W.Schoen, D.Thonig, M.L.Schneider, T. J.Silva,\nH. T. Nembach, O. Eriksson, O. Karis and J. M. Shaw,\nNature Physics 12, 842 (2016).\n[30] O. A. Kelly, A. Anane, R. Bernard, J. B. Youssef, C.\nHahn, A. H. Molpeceres, C. Carr´ et´ ero, E. Jacquet, C.\nDeranlot, P. Bortolotti, R. Lebourgeois, J.-C. Mage, G.\nde Loubens, O. Klein, V. Cros, and A. Fert, Appl. Phys.\nLett.103, 082408 (2013).[31] M. C. Onbasli, A. Kehlberger, D. H. Kim, G. Jakob, M.\nKl¨ aui, A. V. Chumak, B. Hillebrands, and C. A. Ross,\nAPL Mater. 2, 106102 (2014).\n[32] A. Hrabec, F. J. T. Goncalves, C. S. Spencer, E. Aren-\nholz, A. T. N’Diaye, R. L. Stamps, and C. H. Marrows,\nPhys. Rev. B 93, 014432 (2016).\n[33] A. Yu. Rusanov, M. Hesselberth, J. Aarts, and A. I.\nBuzdin, Phys. Rev. Lett. 93, 057002 (2004).\n[34] J. W. A. Robinson, F. Chiodi, M. Egilmez, G. B. Hal´ asz\nand M. G. Blamire, Scientific Report 2, 699 (2012).arXiv:2107.00982v3  [cond-mat.supr-con]  25 Aug 2021Supplemental Material to “Anomalous Gilbert Damping and Du ffing Features of the\nSFSϕ0Josephson Junction”\nYu. M. Shukrinov1,2, I. R. Rahmonov1,3, A. Janalizadeh4, and M. R. Kolahchi4\n1BLTP, JINR, Dubna, Moscow Region, 141980, Russia\n2Dubna State University, Dubna, 141980, Russia\n3Umarov Physical Technical Institute, TAS, Dushanbe 734063 , Tajikistan\n4Department of Physics, Institute for Advanced Studies in Ba sic Sciences, P.O. Box 45137-66731, Zanjan, Iran\n(Dated: August 26, 2021)\nHere, we demonstrate by numerical methods that a generalize d Duffing equation can be obtained\ndirectly from LLG system of equations, for small system para meters of S/F/S junction.\nBoth the α−effect and Duffing features obtained by\nLLG system of equations appear due to the nonlinear\nfeatures of its dynamics at small G,r,α≪1. To proveit,\nwe have carried out the numerical analysis of each term\nof LLG full equation (first two equations in the equation\n(1) of the main text) for the set of model parameters\nG= 0.05,r= 0.05α= 0.005. After neglecting the\nterms of order 10−6, we have\n˙mx\nξ=−mymz+Grmzsin(ϕ−rmy)−αmxm2\nz,\n˙my\nξ=mxmz−αmym2\nz, (1)\n˙mz\nξ=−Grmxsin(ϕ−rmy)+αmz(m2\nx+m2\ny),\nThe procedure is as follows. Expanding mn\nzin a series\nwith the degree of ( mz−1) we can find\nmn\nz=nmz−(n−1). (2)\nFrom expression m2\nx+m2\ny+m2\nz= 1 and (2), we obtain\nmz=2−m2\ny\n2. (3)\nUsing approximation sin( ϕ−rmy) = sin(ωJt) in (1),\ndifferentiatingsecondequationofthe system(1) andsub-\nstituting ˙ mx,mxand ˙mzfrom first second and third\nequations of the system (1), respectively and using the\nexpression (2), (3) and assuming mz= 1 only in denom-\ninators, we come to a second order differential equation\nwith respect to my\n¨my=a1˙m3\ny+a2my˙m2\ny+a3m4\ny˙my+a4m2\ny˙my+a5˙my\n+a6m5\ny+a7m3\ny+a8my−c1˙m2\nysinωJt (4)\n+c2m4\nysinωJt+c3m2\nysinωJt+AsinωJt.The numerical calculation for the used set of model\nparameters allows us to estimate each of the terms in the\nequation, as presented in Table I.\nNow, if we neglect those terms smaller than 10−4, the\nequation (4) takes on the form of Duffing equation with\nTable I: Numerical analysis of equation (4) terms.\na1α\nξa1˙m3\ny∼1.76×10−5\na2 α2a2my˙m2\ny∼3.4×10−8\na3ξα3a3m4\ny˙my∼7.7×10−12\na4ξ(3α−α3)a4m2\ny˙my∼2×10−5\na52ξα a5˙my∼6×10−4\na6ξ2(α2+2α4)a6m5\ny∼5.56×10−9\na7ξ2(1+α2−α4)a7m3\ny∼3.7×10−3\na8ξ2(1+α2)a8my∼6.1×10−2\nc1 Gr c1˙m2\nysinϕ∼3.6×10−5\nc22ξ2α2Grc2m4\nysinϕ∼5.3×10−11\nc3ξ2Gr(α2−2)c3m2\nysinϕ∼4.5×10−5\nAξ2Gr AsinωJt∼6.25×10−4\ndamping dependent coefficients, i.e., we have a general-\nization of the Duffing equation\n¨my+2ξα˙my+ξ2(1+α2)my\n−ξ2(1+α2)m3\ny=ξ2GrsinωJt.(5)"    },    {        "title": "2106.14858v3.Stability_of_a_Magnetically_Levitated_Nanomagnet_in_Vacuum__Effects_of_Gas_and_Magnetization_Damping.pdf",        "content": "Stability of a Magnetically Levitated Nanomagnet in Vacuum: E\u000bects of Gas and\nMagnetization Damping\nKatja Kustura,1, 2Vanessa Wachter,3, 4Adri\u0013 an E. Rubio L\u0013 opez,1, 2and Cosimo C. Rusconi5, 6\n1Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria.\n2Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria.\n3Max Planck Institute for the Science of Light, Staudtstra\u0019e 2, 91058 Erlangen, Germany\n4Department of Physics, University of Erlangen-N urnberg, Staudtstra\u0019e 7, 91058 Erlangen, Germany\n5Max-Planck-Institut f ur Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany.\n6Munich Center for Quantum Science and Technology,\nSchellingstrasse 4, D-80799 M unchen, Germany.\n(Dated: June 1, 2022)\nIn the absence of dissipation a non-rotating magnetic nanoparticle can be stably levitated in a\nstatic magnetic \feld as a consequence of the spin origin of its magnetization. Here we study the\ne\u000bects of dissipation on the stability of the system, considering the interaction with the background\ngas and the intrinsic Gilbert damping of magnetization dynamics. At large applied magnetic \felds\nwe identify magnetization switching induced by Gilbert damping as the key limiting factor for\nstable levitation. At low applied magnetic \felds and for small particle dimensions magnetization\nswitching is prevented due to the strong coupling of rotation and magnetization dynamics, and\nthe stability is mainly limited by the gas-induced dissipation. In the latter case, high vacuum\nshould be su\u000ecient to extend stable levitation over experimentally relevant timescales. Our results\ndemonstrate the possibility to experimentally observe the phenomenon of quantum spin stabilized\nmagnetic levitation.\nI. INTRODUCTION\nThe Einstein{de Haas [1, 2] and Barnett e\u000bects [3] are\nmacroscopic manifestations of the internal angular mo-\nmentum origin of magnetization: a change in the mag-\nnetization causes a change in the mechanical rotation\nand conversely. Because of the reduced moment of in-\nertia of levitated nano- to microscale particles, these ef-\nfects play a dominant role in the dynamics of such sys-\ntems [4{10]. This o\u000bers the possibility to harness these\ne\u000bects for a variety of applications such as precise magne-\ntometry [11{16], inertial sensing [17, 18], coherent spin-\nmechanical control [19, 20], and spin-mechanical cool-\ning [21, 22] among others. Notable in this context is\nthe possibility to stably levitate a ferromagnetic parti-\ncle in a static magnetic \feld in vacuum [23, 24]. Stable\nlevitation is enabled by the internal angular momentum\norigin of the magnetization which, even in the absence of\nmechanical rotation, provides the required angular mo-\nmentum to gyroscopically stabilize the system. Such a\nphenomenon, which we refer to as quantum spin stabi-\nlized levitation to distinguish it from the rotational stabi-\nlization of magnetic tops [25{27], relies on the conserva-\ntive interchange between internal and mechanical angular\nmomentum. Omnipresent dissipation, however, exerts\nadditional non-conservative torques on the system which\nmight alter the delicate gyroscopic stability [26, 28]. It\nthus remains to be determined if stable levitation can\nbe observed under realistic conditions, where dissipative\ne\u000bects cannot be neglected.\nIn this article, we address this question. Speci\f-\ncally, we consider the dynamics of a levitated magnetic\nnanoparticle (nanomagnet hereafter) in a static magnetic\n\feld in the presence of dissipation originating both fromthe collisions with the background gas and from the\nintrinsic damping of magnetization dynamics (Gilbert\ndamping) [29, 30], which are generally considered to be\nthe dominant sources of dissipation for levitated nano-\nmagnets [8, 13, 31{33]. Con\fned dynamics can be ob-\nserved only when the time over which the nanomagnet is\nlevitated is longer than the period of center-of-mass os-\ncillations in the magnetic trap. When this is the case, we\nde\fne the system to be metastable . We demonstrate that\nthe system can be metastable in experimentally feasible\nconditions, with the levitation time and the mechanism\nbehind the instability depending on the parameter regime\nof the system. In particular, we show that at weak ap-\nplied magnetic \felds and for small particle dimensions\n(to be precisely de\fned below) levitation time can be\nsigni\fcantly extended in high vacuum (i.e. pressures be-\nlow 10\u00003mbar). Our results evidence the potential of\nunambiguous experimental observation of quantum spin\nstabilized magnetic levitation.\nWe emphasize that our analysis is particularly timely.\nPresently there is a growing interest in levitating and con-\ntrolling magnetic systems in vacuum [9, 34, 35]. Current\nexperimental e\u000borts focus on levitation of charged para-\nmagnetic ensembles in a Paul trap [19, 36, 37], diamag-\nnetic particles in magneto-gravitational traps [38{40], or\nferromagnets above a superconductor [14, 20, 41]. Lev-\nitating ferromagnetic particles in a static magnetic trap\no\u000bers a viable alternative, with the possibility of reaching\nlarger mechanical trapping frequencies.\nThe article is organized as follows. In Sec. II we in-\ntroduce the model of the nanomagnet, and we de\fne\ntwo relevant regimes for metastability, namely the atom\nphase and the Einstein{de Haas phase. In Sec. III and\nIV we analyze the dynamics in the atom phase and thearXiv:2106.14858v3  [cond-mat.mes-hall]  31 May 20222\nFigure 1. (a) Illustration of a spheroidal nanomagnet levi-\ntated in an external \feld B(r) and surrounded by a gas at the\ntemperature Tand the pressure P. (b) Linear stability dia-\ngram of a non-rotating nanomagnet in the absence of dissipa-\ntion, assuming a= 2b. Blue and red regions denote the stable\natom and Einstein{de Haas phase, respectively; hatched area\nis the unstable region. Dashed lines show the critical values\nof the bias \feld which de\fne the two phases. In particular,\nBEdH,1\u00115\u0016=[4\r2\n0(a2+b2)M],BEdH,2\u00113 [\u0016B02=(4\r0M)]1=3,\nandBatom = 2kaV=\u0016. Numerical values of physical parame-\nters used to generate panel (b) are given in Table I.\nEinstein{de Haas phase, respectively. We discuss our re-\nsults in Sec. V. Conclusions and outlook are provided in\nSec. VI. Our work is complemented by three appendices\nwhere we de\fne the transformation between the body-\n\fxed and laboratory reference frames (App. A), analyze\nthe e\u000bect of thermal \ructuations (App. B), and provide\nadditional \fgures (App. C).\nII. DESCRIPTION OF THE SYSTEM\nWe consider a single domain nanomagnet levitated in\na static1magnetic \feld B(r) as shown schematically in\nFig. 1(a). We model the nanomagnet as a spheroidal\nrigid body of mass density \u001aMand semi-axes lengths a;b\n(a > b ), having uniaxial magnetocrystalline anisotropy,\nwith the anisotropy axis assumed to be along the major\nsemi-axisa[42]. Additionally, we assume that the mag-\nnetic response of the nanomagnet is approximated by a\npoint dipole with magnetic moment \u0016of constant mag-\nnitude\u0016\u0011j\u0016j, as it is often justi\fed for single domain\nparticles [42, 43]. Let us remark that such a simpli\fed\nmodel has been considered before to study the classical\ndynamics of nanomagnets in a viscous medium [31, 44{\n49], as well as to study the quantum dynamics of mag-\nnetic nanoparticles in vacuum [5, 13, 50, 51]. Since the\nmodel has been successful in describing the dynamics of\nsingle-domain nanomagnets, we adopt it here to inves-\ntigate the stability in a magnetic trap. In particular,\nour study has three main di\u000berences as compared with\n1We denote a \feld static if it does not have explicit time depen-\ndence, namely if @B(r)=@t= 0.Table I. Physical parameters of the model and the values used\nthroughout the article. We calculate the magnitude of the\nmagnetic moment as \u0016=\u001a\u0016V, where\u001a\u0016=\u001aM\u0016B=(50amu),\nwith\u0016Bthe Bohr magneton and amu the atomic mass unit.\nParameter Description Value [units]\n\u001aM mass density 104[kg m\u00003]\na;b semi-axes see main text [m]\n\u001a\u0016 magnetization 2 :2\u0002106[J T\u00001m\u00003]\nka anisotropy constant 105[J m\u00003]\n\r0 gyromagnetic ratio 1 :76\u00021011[rad s\u00001T\u00001]\nB0 \feld bias see main text [T]\nB0\feld gradient 104[T m\u00001]\nB00\feld curvature 106[T m\u00002]\n\u0011 Gilbert damping 10\u00002[n. u.]\nT temperature 10\u00001[K]\nP pressure 10\u00002[mbar]\nM molar mass 29 [g mol\u00001]\n\u000bc re\rection coe\u000ecient 1 [n. u.]\nprevious work. (i) We consider a particle levitated in\nhigh vacuum, where the mean free path of the gas parti-\ncles is larger than the nanomagnet dimensions (Knudsen\nregime [52]). This leads to gas damping which is gen-\nerally di\u000berent from the case of dense viscous medium\nmostly considered in the literature. (ii) We consider\ncenter-of-mass motion and its coupling to the rotational\nand magnetic degrees of freedom, while previous work\nmostly focuses on coupling between rotation and mag-\nnetization only (with the notable exception of [48]). (iii)\nWe are primarily interested in the center-of-mass con\fne-\nment of the particle, and not in its magnetic response.\nWithin this model the relevant degrees of freedom of\nthe system are the center-of-mass position r, the linear\nmomentum p, the mechanical angular momentum L, the\norientation of the nanomagnet in space \n, and the mag-\nnetic moment \u0016. The orientation of the nanomagnet\nis speci\fed by the body-\fxed reference frame Oe1e2e3,\nwhich is obtained from the laboratory frame Oexeyez\naccording to ( e1;e2;e3)T=R(\n)(ex;ey;ez)T, where\n\n= (\u000b;\f;\r )Tare the Euler angles and R(\n) is the\nrotational matrix. We provide the expression for R(\n)\nin App. A. The body-\fxed reference frame is chosen such\nthate3coincides with the anisotropy axis. The magnetic\nmoment\u0016is related to the internal angular momentum F\naccording to the gyromagnetic relation \u0016=\r0F, where\n\r0is the gyromagnetic ratio of the material2.\n2The total internal angular momentum Fis a sum of the individ-\nual atomic angular momenta (spin and orbital), which contribute\nto the atomic magnetic moment. For a single domain magnetic\nparticle, it is customary to assume that Fcan be described as\na vector of constant magnitude, jFj=\u0016=\r0(macrospin approxi-\nmation) [43].3\nA. Equations of Motion\nWe describe the dynamics of the nanomagnet in the\nmagnetic trap with a set of stochastic di\u000berential equa-\ntions which model both the deterministic dissipative evo-\nlution of the system and the random \ructuations due to\nthe environment. In the following it is convenient to de-\n\fne dimensionless variables: the center-of-mass variables\n~r\u0011r=a,~p\u0011\r0ap=\u0016, the mechanical angular momen-\ntum`\u0011\r0L=\u0016, the magnetic moment m\u0011\u0016=\u0016, and\nthe magnetic \feld b(~r)\u0011B(a~r)=B0, whereB0denotes\nthe minimum of the \feld intensity in a magnetic trap,\nwhich we hereafter refer to as the bias \feld. Note that\nwe choose to normalize the position r, the magnetic mo-\nment\u0016and the magnetic \feld B(r) with respect to the\nparticle size a, the magnetic moment magnitude \u0016, and\nthe bias \feld B0, respectively. The scaling factor for an-\ngular momentum, \u0016=\r0, and linear momentum, \u0016=(a\r0),\nfollow as a consequence of the gyromagnetic relation.\nThe dynamics of the nanomagnet in the laboratory\nframe are given by the equations of motion\n_~r=!I~p; (1)\n_e3=!\u0002e3; (2)\n_~p=!Lr~r[m\u0001b(~r)]\u0000\u0000cm~p+\u001ep(t); (3)\n_`=!Lm\u0002b(~r)\u0000_m\u0000\u0000rot`+\u0018l(t); (4)\n_m=m\n1 +\u00112\u0002[!e\u000b\u0000\u0011m\u0002(!+!e\u000b+\u0011!\u0002m)\n+\u0010b(t)]: (5)\nHere we de\fne the relevant system frequencies: !I\u0011\n\u0016=(\r0Ma2) is the Einstein{de Haas frequency, with M\nthe mass of the nanomagnet, !L\u0011\r0B0is the Larmor\nfrequency,!A\u0011kaV\r0=\u0016is the anisotropy frequency,\nwithVthe volume of the nanomagnet and kathe ma-\nterial dependent anisotropy constant [43], !\u0011I\u00001L\nis the angular velocity, with Ithe tensor of inertia,\nand!e\u000b\u00112!A(m\u0001e3)e3+!Lb(~r). Dissipation is\nparametrized by the dimensionless Gilbert damping pa-\nrameter\u0011[29, 53], and the center-of-mass and rotational\nfriction tensors \u0000 cmand \u0000 rot, respectively [32]. The e\u000bect\nof stochastic thermal \ructuations is represented by the\nrandom variables \u001ep(t) and\u0018l(t) which describe, respec-\ntively, the \ructuating force and torque exerted by the\nsurrounding gas, and by \u0010b(t) which describes the ran-\ndom magnetic \feld accounting for thermal \ructuations\nin magnetization dynamics [54]. We assume Gaussian\nwhite noise, namely, for X(t)\u0011(\u001ep(t);\u0018l(t);\u0010b(t))Twe\nhavehXi(t)i= 0 andhXi(t)Xj(t0)i\u0018\u000eij\u000e(t\u0000t0).\nEquations (1-4) describe the center-of-mass and rota-\ntional dynamics of a rigid body in the presence of dis-\nsipation and noise induced by the background gas [32].\nThe expressions for \u0000 cmand \u0000 rotdepend on the parti-\ncle shape { here we take the expressions derived in [32]for a cylindrical particle3{, and on the ratio of the sur-\nface and the bulk temperature of the particle, which\nwe assume to be equal to the gas temperature T. Fur-\nthermore, they account for two di\u000berent scattering pro-\ncesses, namely the specular and the di\u000busive re\rection\nof the gas from the particle, which is described by a\nphenomenological interpolation coe\u000ecient \u000bc. The or-\nder of magnitude of the di\u000berent components of \u0000 cmand\n\u0000rotis generally well approximated by the dissipation\nrate \u0000\u0011(2Pab=M )[2\u0019M=(NAkBT)]1=2, wherePand\nMare, respectively, the gas pressure and molar mass,\nkBis the Boltzmann constant and NAis the Avogadro\nnumber. The magnetization dynamics given by Eq. (5)\nis the Landau-Lifshitz-Gilbert equation in the laboratory\nframe [8, 57], with the e\u000bective magnetic \feld !e\u000b=\r0.\nWe remark that Eqs. (1-5) describe the classical dynam-\nics of a levitated nanomagnet where the e\u000bect of the\nquantum spin origin of magnetization, namely the gy-\nromagnetic relation, is taken into account phenomeno-\nlogically by Eq. (5). This is equivalent to the equations\nof motion obtained from a quantum Hamiltonian in the\nmean-\feld approximation [24].\nLet us discuss the e\u000bect of thermal \ructuations on\nthe dynamics of the nanomagnet at subkelvin temper-\natures and in high vacuum. These conditions are com-\nmon in recent experiments with levitated particles [58{\n60]. The thermal \ructuations of magnetization dy-\nnamics, captured by the last term in Eq. (5), lead\nto thermally activated transition of the magnetic mo-\nment between the two stable orientations along the\nanisotropy axis [54, 61]. Such process can be quan-\nti\fed by the N\u0013 eel relaxation time, which is given by\n\u001cN\u0019(\u0019=! A)p\nkBT=(kaV)ekaV=(kBT). Thermal acti-\nvation can be neglected when \u001cNis larger than other\ntimescales of magnetization dynamics, namely the pre-\ncession timescale given by \u001cL\u00111=j!e\u000bj, and the Gilbert\ndamping timescale given by \u001cG\u00111=(\u0011j!e\u000bj). Con-\nsidering for simplicity j!e\u000bj\u00182!A, for a particle size\na= 2b= 1 nm and temperature T= 1 K, and the\nvalues of the remaining parameters as in Table I, the ra-\ntio of the timescales is of the order \u001cN=\u001cL\u0018103, and\nit is signi\fcantly increased for larger particle sizes and\nat smaller temperatures. We remark that, for the val-\nues considered in this article, \u001cNis much larger than the\nlongest dynamical timescale in Eqs. (1-5) which is associ-\nated with the motion along ex. Thermal activation of the\nmagnetic moment can therefore be safely neglected. The\nstochastic e\u000bects ascribed to the background gas, cap-\ntured by the last terms in Eqs. (3-4), are expected to be\nimportant at high temperatures (namely, a regime where\nMkBT\r2\n0a2=\u00162&1 [32]). At subkelvin temperatures and\nin high vacuum these \ructuations are weak and, con-\nsequently, they do not destroy the deterministic e\u000bects\n3The expressions for \u0000 cmand \u0000 rotfor a cylindrical particle\ncapture the order of magnitude of the dissipation rates for a\nspheroidal particle [55, 56].4\ncaptured by the remaining terms in Eqs. (1-5) [33]. In-\ndeed, for the values of parameters given in Table I and\nfora= 2b,MkBT\r2\n0a2=\u00162\u00190:8T=(a[nm]). For sub-\nkelvin temperatures and particle sizes a>1 nm, thermal\n\ructuations due to the background gas can therefore be\nsafely neglected.\nIn the following we thus neglect stochastic e\u000bects by\nsetting\u001ep=\u0018l=\u0010b= 0, and we consider only the de-\nterministic part of Eqs. (1-5) as an appropriate model\nfor the dynamics [8, 33, 54]. In App. B we carry out\nthe analysis of the dynamics including the e\u000bects of gas\n\ructuations in equations (1-5), and we show that the\nresults presented in the main text remain qualitatively\nvalid even in the presence of thermal noise. For the mag-\nnetic \feld B(r) we hereafter consider a Io\u000be-Pritchard\nmagnetic trap, given by\nB(r) =ex\u0014\nB0+B00\n2\u0012\nx2\u0000y2+z2\n2\u0013\u0015\n\u0000ey\u0012\nB0y+B00\n2xy\u0013\n+ez\u0012\nB0z\u0000B00\n2xz\u0013\n;(6)\nwhereB0;B0andB00are, respectively, the \feld bias, gra-\ndient and curvature [62]. We remark that this is not a\nfundamental choice, and di\u000berent magnetic traps, pro-\nvided they have a non-zero bias \feld, should result in\nsimilar qualitative behavior.\nB. Initial conditions\nThe initial conditions for the dynamics in Eqs. (1-5),\nnamely at time t= 0, depend on the initial state of the\nsystem, which is determined by the preparation of the\nnanomagnet in the magnetic trap. In our analysis, we\nconsider the nanomagnet to be prepared in the thermal\nstate of an auxiliary loading potential at the temperature\nT. Subsequently, we assume to switch o\u000b the loading\npotential at t= 0, while at the same time switching\non the Io\u000be-Pritchard magnetic trap. The choice of the\nauxiliary potential is determined by two features: (i) it\nallows us to simply parametrize the initial conditions by a\nsingle parameter, namely the temperature T, and (ii) it is\nan adequate approximation of general trapping schemes\nused to trap magnetic particles.\nRegarding point (i), we assume that the particle is lev-\nitated in a harmonic trap, in the presence of an external\nmagnetic \feld applied along ex. This loading scheme\nprovides, on the one hand, trapping of the center-of-mass\ndegrees of freedom, with trapping frequencies denoted by\n!i(i=x;y;z ). On the other hand, the magnetic moment\nin this case is polarized along ex. The Hamiltonian of the\nsystem in such a con\fguration reads Haux=p2=(2M) +P\ni=x;y;zM!2\nir2\ni=2+LI\u00001L=2\u0000kaVe2\n3;x\u0000\u0016xBaux, where\nBauxdenotes the magnitude of the external magnetic\n\feld, which we for simplicity set to Baux=B0in all our\nsimulations. At t= 0 the particle is released in the mag-\nnetic trap given by Eq. (6). For the degrees of freedomx\u0011(~r;~p;`;mx)T, we take as the initial displacement\nfrom the equilibrium the corresponding standard devia-\ntion in a thermal state of Haux. More precisely, xi(0) =\nxi;e+ (hx2\nii\u0000hxii2)1=2, wherexi;edenotes the equilib-\nrium value, andhxk\nii\u0011Z\u00001R\ndxxk\niexp[\u0000Haux=(kBT)],\nwithk= 1;2 and the partition function Z. For the Eu-\nler angles \nwe use \n 1(0)\u0011cos\u00001[\u0000p\nhcos2\n1i] and\n\ni(0)\u0011cos\u00001[p\nhcos2\nii] (i= 2;3). The initial condi-\ntions for e3follow from \nusing the transformation given\nin App. A.\nRegarding point (ii), the initial conditions obtained in\nthis way describe a trapped particle prepared in a ther-\nmal equilibrium in the presence of an external loading\npotential where the center of mass is decoupled from the\nmagnetization and the rotational dynamics. It is outside\nthe scope of this article to study in detail a particular\nloading scheme. However, we point out that an auxil-\niary potential given by Hauxcan be obtained, for exam-\nple, by trapping the nanomagnet using a Paul trap as\ndemonstrated in recent experiments [19, 21, 37, 63{70].\nIn particular, trapping of a ferromagnetic particle has\nbeen demonstrated in a Paul trap at P= 10\u00002mbar,\nwith center-of-mass trapping frequency of up to 1 MHz,\nand alignment of the particle along the direction of an\napplied \feld [19]. We note that particles are shown to\nremain trapped even when the magnetic \feld is varied\nover many orders of magnitudes or switched o\u000b. We re-\nmark further that alignment of elongated particles can\nbe achieved using a quadrupole Paul trap even in the\nabsence of magnetic \feld [55, 71].\nC. Linear stability\nIn the absence of thermal \ructuations, an equilibrium\nsolution of Eqs. (1-5) is given by ~re=~pe=`e= 0 and\ne3;e=me=\u0000ex. This corresponds to the con\fguration\nin which the nanomagnet is \fxed at the trap center, with\nthe magnetic moment along the anisotropy axis and anti-\naligned to the bias \feld B0. Linear stability analysis of\nEqs. (1-5) shows that the system is unstable, as expected\nfor a gyroscopic system in the presence of dissipation [28].\nHowever, when the nanomagnet is metastable, it is still\npossible for it to levitate for an extended time before\nbeing eventually lost from the trap, as in the case of a\nclassical magnetic top [25{27]. As we show in the fol-\nlowing sections, the dynamics of the system, and thus its\nmetastability, strongly depend on the applied bias \feld\nB0. We identify two relevant regimes: (i) strong-\feld\nregime, de\fned by bias \feld values B0> B atom, and\n(ii) weak-\feld regime, de\fned by B0< B atom, where\nBatom\u00112kaV=\u0016. This di\u000berence is reminiscent of the\ntwo di\u000berent stable regions which arise as a function of\nB0in the linear stability diagram in the absence of dis-\nsipation [see Fig. 1(b)] [23, 24]. In Sec. III and Sec. IV\nwe investigate the possibility of metastable levitation by\nsolving numerically Eqs. (1-5) in the strong-\feld and\nweak-\feld regime, respectively.5\nIII. DYNAMICS IN THE STRONG-FIELD\nREGIME: ATOM PHASE\nThe strong-\feld regime, according to the de\fnition\ngiven in Sec. II C, corresponds to the blue region in the\nlinear stability diagram in the absence of dissipation,\nshown in Fig. 1(b). This region is named atom phase\nin [23, 24], and we hereafter refer to the strong-\feld\nregime as the atom phase. This parameter regime corre-\nsponds to the condition !L\u001d!A;!I. In this regime, the\ncoupling of the magnetic moment \u0016and the anisotropy\naxise3is negligible, and, to \frst approximation, the\nnanomagnet undergoes a free Larmor precession about\nthe local magnetic \feld. In the absence of dissipation,\nthis stabilizes the system in full analogy to magnetic trap-\nping of neutral atoms [72, 73].\nIn Fig. 2(a-c) we show the numerical solution of\nEqs. (1-5) for nanomagnet dimensions a= 2b= 20\nnm and the bias \feld B0= 200 mT. As evidenced by\nFig. 2(a), the magnetization mxof the particle changes\ndirection. During this change, the mechanical angu-\nlar momentum lxchanges accordingly in the manifesta-\ntion of the Einstein{de Haas e\u000bect, such that the to-\ntal angular momentum m+`is conserved4. The dy-\nnamics observed in Fig. 2(a) is indicative of Gilbert-\ndamping-induced magnetization switching, a well-known\nphenomenon in which the projection of the magnetic mo-\nment along the e\u000bective magnetic \feld !e\u000b=\r0changes\nsign [30]. This is expected to happen when the applied\nbias \feldB0is larger than the e\u000bective magnetic \feld\nassociated with the anisotropy, given by \u0018!A=\r0. Mag-\nnetization switching displaces the system from its equi-\nlibrium position on a timescale which is much shorter\nthan the period of center-of-mass oscillations, estimated\nfrom [24] to be \u001ccm\u00181\u0016s. The nanomagnet thus shows\nno signature of con\fnement [see Fig. 2(b)].\nThe timescale of levitation in the atom phase is given\nby the timescale of magnetization switching, which we\nestimate as follows. As evidenced by Fig. 2(a-b), the\ndynamics of the center of mass and the anisotropy axis\nare approximately constant during switching, such that\n!e\u000b\u0019!e\u000b(t= 0). Under this approximation and as-\nsuming\u0011\u001c1, the magnetic moment projection mk\u0011\n!e\u000b\u0001m=j!e\u000bjevolves as\n_mk\u0019\u0011[!L+ 2!Amk](1\u0000m2\nk): (7)\nAccording to Eq. (7) the component mkexhibits switch-\ning ifmk(t= 0)&\u00001 and!L=2!A>1 [30], both of\nwhich are ful\flled in the atom phase. Integrating Eq. (7)\nwe obtain the switching time \u001c[de\fned as mk(\u001c)\u00110],\nwhich can be well approximated by\n\u001c\u0019ln\u0000\n1 +jmk(t= 0)j\u0001\n2\u0011(!L+ 2!A)\u0000ln\u0000\n1\u0000jmk(t= 0)j\u0001\n2\u0011(!L\u00002!A):(8)\n4We always \fnd the transfer of angular momentum to the center\nof mass angular momentum r\u0002pto be negligible.\nFigure 2. Dynamics in the atom phase. (a) Dynamics of\nthe magnetic moment component mx, the mechanical angular\nmomentum component lx, and the anisotropy axis component\ne3;xfor nanomagnet dimensions a= 2b= 20 nm and the bias\n\feldB0= 200 mT. For the initial conditions we consider\ntrapping frequencies !x= 2\u0019\u00022 kHz and!y=!z= 2\u0019\u000250\nkHz. Unless otherwise stated, for the remaining parameters\nthe numerical values are given in Table I. (b) Center-of-mass\ndynamics for the same case considered in (a). (c) Dynamics of\nthe magnetic moment component mk. Line denoted by circle\ncorresponds to the case considered in (a). Each remaining\nline di\u000bers by a single parameter, as denoted by the legend.\nDotted vertical lines show Eq. (8). (d) Switching time given\nby Eq. (8) as a function of the bias \feld B0and the major\nsemi-axisa. In the region left of the thick dashed line the\ndeviation from the exact value is more than 5%. Hatched\narea is the unstable region in the linear stability diagram in\nFig. 1.(b).\nThe estimation Eq. (8) is in excellent agreement with\nthe numerical results for di\u000berent parameter values [see\nFig. 2(c)].\nMagnetization switching characterizes the dynamics of\nthe system in the entire atom phase. In particular, in\nFig. 2(d) we analyze the validity of Eq. (8) for di\u000berent\nvalues of the bias \feld B0and the major semi-axis a, as-\nsumingb=a=2. The thick dashed line shows the region\nwhere Eq. (8) di\u000bers from the exact switching time, as\nestimated from the full dynamics of the system, by 5%;\nleft of this line the deviation becomes increasingly more\nsigni\fcant, with Eq. (8) predicting up to 20% larger val-\nues close to the stability border (namely, for bias \feld\nclose toBatom = 90 mT). We believe that the signi\f-\ncant deviation close to the border of the atom phase is\ndue to the non-negligible coupling to the anisotropy axis,6\nFigure 3. Dynamics in the Einstein{de Haas phase. (a) Motion of the system in the ey-ezplane until time t= 5\u0016s for\nnanomagnet dimensions a= 2b= 2 nm and the bias \feld B0= 0:5 mT. For the initial conditions we consider trapping\nfrequencies !x= 2\u0019\u00022 kHz and !y=!z= 2\u0019\u00021 MHz. For the remaining parameters the numerical values are given\nin Table I. (b) Dynamics of the projection mkand (c) dynamics of the anisotropy axis component e3;x, for the same case\nconsidered in (a). (d) Dynamics of the center-of-mass component ryand (e) dynamics of the magnetic moment component\nmxon a longer timescale, for the same values of parameters as in (a). (f) Escape time t?as a function of gas pressure P, for\ndi\u000berent con\fgurations in the Einstein{de Haas phase. Circles correspond to the case considered in (a). Each remaining case\ndi\u000bers by parameters indicated by the legend. (g) Escape time t?as a function of the major semi-axis a, with the values of\nthe remaining parameters as in (a). Dashed vertical line denotes the upper limit of the Einstein{de Haas phase, given by the\ncritical \feld BEdH,1 [see Fig. 1(b)].\nwhich results in additional mechanisms not captured by\nthe simple model Eq. (7). In fact, it is known that cou-\npling between magnetization and mechanical degrees of\nfreedom might have an impact on the switching dynam-\nics [74]. As demonstrated by Fig. 2(d), the switching\ntime is always shorter than the center-of-mass oscillation\nperiod\u001ccm, and thus no metastability can be observed in\nthe atom phase.\nLet us note that the conclusions we draw in Fig. 2\nremain valid if one varies the anisotropy constant ka,\nGilbert damping parameter \u0011, and the temperature T,\nas we show in App. C. Finally, we note that the dis-\nsipation due to the background gas has negligible ef-\nfects. In particular, for the values assumed in Fig. 2(a-b)\nthe timescale of the gas-induced dissipation is given by\n1=\u0000 = 440\u0016s.\nIV. DYNAMICS IN THE WEAK-FIELD\nREGIME: EINSTEIN{DE HAAS PHASE\nWe now focus on the regime of weak bias \feld, cor-\nresponding to the condition !L\u001c!A. In this regime\nmagnetization switching does not occur, and the dynam-\nics critically depend on the particle size. In the follow-\ning we focus on the regime of small particle dimensions,i.e.!L\u001c!I, which, as we will show, is bene\fcial for\nmetastability. In the absence of dissipation, this regime\ncorresponds to the Einstein{de Haas phase [red region\nin Fig. 1(b)] [23, 24]. The hierarchy of energy scales in\nthe Einstein{de Haas phase (namely, !L\u001c!A;!I) man-\nifests in two ways: (i) the anisotropy is strong enough to\ne\u000bectively \\lock\" the direction of the magnetic moment \u0016\nalong the anisotropy axis e3(!A\u001d!L), and (ii) accord-\ning to the Einstein{de Haas e\u000bect, the frequency at which\nthe nanomagnet would rotate if \u0016switched direction is\nsigni\fcantly increased at small dimensions ( !I\u001d!L),\nsuch that switching can be prevented due to energy con-\nservation [4]. In the absence of dissipation, the combina-\ntion of these two e\u000bects stabilizes the system.\nIn Fig. 3(a-c) we show the numerical solution of\nEqs. (1-5) for nanomagnet dimensions a= 2b= 2 nm\nand the bias \feld B0= 0:5 mT. The nanomagnet is\nmetastable, as evidenced by the con\fned center-of-mass\nmotion shown in Fig. 3(a). In Fig. 3(b-c) we show the\ndynamics of the magnetic moment component mkand\nthe anisotropy axis component e3;x, respectively, which\nindicates that no magnetization switching occurs in this\nregime. We remark that the absence of switching can-\nnot be simply explained on the basis of Eqs. (7-8). In\nfact, the simple model of magnetization switching, given7\nby Eq. (7), assumes that the dynamics of the rotation\nand the center-of-mass motion happen on a much longer\ntimescale than the timescale of magnetization dynam-\nics. However, in this case rotation and magnetization\ndynamics occur on a comparable timescale, as evidenced\nby Fig. 3(b-c). The weak-\feld condition alone ( !L\u001c!A)\nis thus not su\u000ecient to correctly explain the absence of\nswitching, and the role of particle size ( !L\u001c!I) needs\nto be considered.\nLet us analyze the role of Gilbert damping in this case.\nSince in the Einstein{de Haas phase mk\u00181, we de\fne\nm\u0011e3+\u000em, where\u000emrepresents the deviation of m\nfrom the anisotropy axis e3, and we assumej\u000emj\u001cje3j\n[see Fig. 3(b)]. This allows us to simplify Eq. (5) as\n\u000e_m\u0019!e\u000b\u0002\u000em\u0000\u0011[2!A+!3e3\u0001(m+`)]\u000em;(9)\nwhere!3\u0011\u0016=(\r0I3), withI3the principal moment of\ninertia along e3. As evidenced by Eq. (9), the only e\u000bect\nof Gilbert damping is to align mande3on a timescale\ngiven by\u001c0\u00111=[\u0011(2!A+!3)], irrespective of the dy-\nnamics of e3. For the values of parameters considered in\nFig. 3(a-c), \u001c0= 5 ns, and it is much shorter than the\ntimescale of center-of-mass dynamics, given by \u001ccm\u00181\n\u0016s. For all practical purposes, the magnetization in the\nEinstein{de Haas phase can be considered frozen along\nthe anisotropy axis. The nanomagnet in the presence of\nGilbert damping is therefore equivalent to a hard magnet\n(i. e.ka!1 ) [24].\nThe main mechanism behind the instability in the\nEinstein{de Haas phase is thus gas-induced dissipation.\nIn Fig. 3(d-e) we plot the dynamics of the center-of-\nmass component ryand the magnetic moment compo-\nnentmxon a longer timescale, for two di\u000berent values of\nthe pressure P. The e\u000bect of gas-induced dissipation is\nto dampen the center-of-mass motion to the equilibrium\nposition, while the magnetic moment moves away from\nthe equilibrium. Both processes happen on a timescale\ngiven by the dissipation rate \u0000. When ex=mx\u00190, the\nsystem becomes unstable and ultimately leaves the trap\n[see arrow in Fig. 3(d)]. We de\fne the escape time t?as\nthe time at which the particle position is y(t?)\u00115y(0),\nand we show it in Fig. 3(f) as a function of pressure Pfor\ndi\u000berent con\fgurations in the Einstein{de Haas phase,\nand forb=a=2. Fig. 3(f) con\frms that the dissipation\na\u000bects the system on a timescale which scales as \u00181=P.\nThe metastability of the nanomagnet in the Einstein{de\nHaas phase is therefore limited solely by the gas-induced\ndissipation, which can be signi\fcantly reduced in high\nvacuum. Finally, in Fig. 3(g) we analyze the e\u000bect of\nparticle size on metastability. Speci\fcally, we show the\nescape time t?as a function of the major semi-axis aat\nthe bias \feld B0= 0:5 mT, forb=a=2. The escape time\nis signi\fcantly reduced at increased particle sizes. This\ncon\frms the advantage of the Einstein{de Haas phase to\nobserve metastability, even in the presence of dissipation.V. DISCUSSION\nIn deriving the results discussed in the preceding sec-\ntions, we assumed (i) a single-magnetic-domain nanopar-\nticle with uniaxial anisotropy and constant magnetiza-\ntion, with the values of the physical parameters summa-\nrized in Table I, (ii) deterministic dynamics, i. e. the\nabsence of thermal \ructuations, (iii) that gravity can be\nneglected, and (iv) a non-rotating nanomagnet. Let us\njustify the validity of these assumptions.\nWe \frst discuss the values of the parameters given in\nTable I, which are used in our analysis. The material pa-\nrameters, such as \u001aM,\u001a\u0016,kaand\u0011, are consistent with,\nfor example, cobalt [75{78]. We remark that the uniax-\nial anisotropy considered in our model represents a good\ndescription even for materials which do not have an in-\ntrinsic magnetocrystalline uniaxial anisotropy, provided\nthat they have a dominant contribution from the uniaxial\nshape anisotropy. This is the case, for example, for fer-\nromagnetic particles with a prolate shape [75]. We point\nout that the values used here do not correspond to a spe-\nci\fc material, but instead they describe a general order\nof magnitude corresponding to common magnetic materi-\nals. Indeed, our results are general and can be particular-\nized to speci\fc materials by replacing the above generic\nvalues with exact numbers. As we show in App. C, the re-\nsults and conclusions presented here remain unchanged\neven when di\u000berent values of the parameters are con-\nsidered. The values used for the \feld gradient B0and\nthe curvature B00have been obtained in magnetic mi-\ncrotraps [62, 79{82]. The values of the gas pressure P\nand the temperature Tare experimentally feasible, with\nnumerous recent experiments reaching pressure values as\nlow asP= 10\u00006mbar [58, 68, 70, 83{85]. All the values\nassumed in our analysis are therefore consistent with cur-\nrently available technologies in levitated optomechanics.\nThermal \ructuations can be neglected at cryogenic\nconditions (as we argue in Sec. II A), as their e\u000bect is\nweak enough not to destroy the deterministic e\u000bects cap-\ntured by Eqs. (1-5). In particular, thermal activation of\nthe magnetization, as quanti\fed by the N\u0013 eel relaxation\ntime, can be safely neglected due to the large value of\nthe uniaxial anisotropy even for the smallest particles\nconsidered. As for the mechanical thermal \ructuations,\nwe con\frm that they do not modify the deterministic\ndynamics in App. B, where we simulate the associated\nstochastic dynamics.\nGravity, assumed to be along ex, can be safely ne-\nglected, since the gravity-induced displacement of the\ntrap center from the origin is much smaller than the\nlength scale over which the Io\u000be-Pritchard \feld signi\f-\ncantly changes [24]. Speci\fcally, the gravitational poten-\ntialMgx shifts the trap center from the origin r= 0\nalong exby an amount rg\u0011Mg= (\u0016B00), wheregis\nthe gravitational acceleration. On the other hand, the\ncharacteristic length scales of the Io\u000be-Pritchard \feld\nare given by \u0001 r0\u0011p\nB0=B00for the variation along\nex, and \u0001r0\u0011B0=B00for the variation o\u000b-axis. When-8\neverrg\u001c\u0001r0;\u0001r0, gravity has a negligible role in the\nmetastable dynamics of the system. In the parameter\nregime considered in this article, this is always the case.\nWe note that the condition to neglect gravity is the same\nas for a magnetically trapped atom, since both Mand\u0016\nscale with the volume.\nFinally, we remark that the analysis presented here\nis carried out for the case of a non-rotating nanomag-\nnet5. The same qualitative behavior is obtained even in\nthe presence of mechanical rotation (namely, considering\na more general equilibrium con\fguration with `e6= 0).\nThe analysis of dynamics in the presence of rotation is\nprovided in App. C. In particular, the dynamics in the\nEinstein{de Haas phase remains largely una\u000bected, pro-\nvided that the total angular momentum of the system is\nnot zero. In the atom phase, mechanical rotation leads to\ndi\u000berences in the switching time \u001c, as generally expected\nin the presence of magneto-mechanical coupling [74, 88].\nVI. CONCLUSION\nIn conclusion, we analyzed how the stability of a nano-\nmagnet levitated in a static magnetic \feld is a\u000bected by\nthe most relevant sources of dissipation. We \fnd that in\nthe strong-\feld regime (atom phase) the system is un-\nstable due to the Gilbert-damping-induced magnetiza-\ntion switching, which occurs on a much faster timescale\nthan the center-of-mass oscillations, thereby preventing\nthe observation of levitation. On the other hand, the sys-\ntem is metastable in the weak-\feld regime and for small\nparticle dimensions (Einstein{de Haas phase). In this\nregime, the con\fnement of the nanomagnet in a mag-\nnetic trap is limited only by the gas-induced dissipation.\nOur results suggest that the timescale of stable levitation\ncan reach and even exceed several hundreds of periods of\ncenter-of-mass oscillations in high vacuum. These \fnd-\nings indicate the possibility of observing the phenomenon\nof quantum spin stabilized magnetic levitation, which we\nhope will encourage further experimental research.\nThe analysis presented in this article is relevant for\nthe community of levitated magnetic systems. Speci\f-\ncally, we give precise conditions for the observation of\nthe phenomenon of quantum spin stabilized levitation\nunder experimentally feasible conditions. Levitating a\nmagnet in a time-independent gradient trap represents a\nnew direction in the currently growing \feld of magnetic\nlevitation of micro- and nanoparticles, which is interest-\ning for two reasons. First, the experimental observation\nof stable magnetic levitation of a non-rotating nanomag-\nnet would represent a direct observation of the quantum\nnature of magnetization. Second, the observation of such\n5Rotational cooling might be needed to unambiguously identify\nthe internal spin as the source of stabilization. Subkelvin cooling\nof a nanorotor has been recently achieved [86, 87], and cooling\nto\u0016K temperatures should be possible [56].phenomenon would be a step towards controlling and us-\ning the rich physics of magnetically levitated nanomag-\nnets, with applications in magnetometry and in tests of\nfundamental forces [9, 11, 34, 35].\nACKNOWLEDGMENTS\nWe thank G. E. W. Bauer, J. J. Garc\u0013 \u0010a-Ripoll, O.\nRomero-Isart, and B. A. Stickler for helpful discussions.\nWe are grateful to O. Romero-Isart, B. A. Stickler and\nS. Viola Kusminskiy for comments on an early ver-\nsion of the manuscript. C.C.R. acknowledges funding\nfrom ERC Advanced Grant QENOCOBA under the EU\nHorizon 2020 program (Grant Agreement No. 742102).\nV.W. acknowledges funding from the Max Planck So-\nciety and from the Deutsche Forschungsgemeinschaft\n(DFG, German Research Foundation) through Project-\nID 429529648-TRR 306 QuCoLiMa (\"Quantum Cooper-\nativity of Light and Matter\"). A.E.R.L. thanks the AMS\nfor the \fnancial support.\nAppendix A: Rotation to the body frame\nIn this appendix we de\fne the transformation ma-\ntrix between the body-\fxed and the laboratory reference\nframes according to the ZYZ Euler angle convention,\nwith the Euler angles denoted as \n= (\u000b;\f;\r )T. We\nde\fne the transformation between the laboratory frame\nOexeyezand the body frame Oe1e2e3as follows,\n0\n@e1\ne2\ne31\nA=R(\n)0\n@ex\ney\nez1\nA; (A1)\nwhere\nR(\n)\u0011Rz(\u000b)Ry(\f)Rz(\r) =0\n@cos\rsin\r0\n\u0000sin\rcos\r0\n0 0 11\nA\n0\n@cos\f0\u0000sin\f\n0 1 0\n\u0000sin\f0 cos\f1\nA0\n@cos\u000bsin\u000b0\n\u0000sin\u000bcos\u000b0\n0 0 11\nA:(A2)\nAccordingly, the components vj(j= 1;2;3) of a vector\nvin the body frame Oe1e2e3and the components v\u0017\n(\u0017=x;y;z ) of the same vector in the laboratory frame\nOexeyezare related as\n0\n@v1\nv2\nv31\nA=RT(\n)0\n@vx\nvy\nvz1\nA: (A3)\nThe angular velocity of a rotating particle !can be writ-\nten in terms of the Euler angles as != _\u000bez+_\fe0\ny+ _\re3,\nwhere ( e0\nx;e0\ny;e0\nz)T=Rz(\u000b)(ex;ey;ez)Tdenotes the\nframeOe0\nxe0\nye0\nzobtained after the \frst rotation of the9\nlaboratory frame Oexeyezin the ZYZ convention. By\nusing (A1) and (A2), we can rewrite angular velocity in\nterms of the body frame coordinates,\n!= _\u000b2\n4R(\n)\u000010\n@e1\ne2\ne31\nA3\n5\n3+_\f2\n4R(\r)\u000010\n@e1\ne2\ne31\nA3\n5\n2+ _\re3;\n(A4)\nwhich is compactly written as ( !1;!2;!3)T=A(\n)_\n,\nwith\nA(\n) =0\n@\u0000cos\rsin\fsin\r0\nsin\fsin\rcos\r0\ncos\f 0 11\nA: (A5)\nAppendix B: Dynamics in the presence of thermal\n\ructuations\nIn this appendix we consider the dynamics of a lev-\nitated nanomagnet in the presence of stochastic forces\nand torques induced by the surrounding gas. The dy-\nnamics of the system are described by the following set\nof stochastic di\u000berential equations (SDE),\nd~r=!I~pdt; (B1)\nde3=!\u0002e3dt; (B2)\nd~p= [!Lr~r[m\u0001b(~r)]\u0000\u0000cm~p] dt+p\nDcmdWp;(B3)\nd`= [!Lm\u0002b(~r)\u0000_m\u0000\u0000rot`] dt+p\nDrotdWl;\n(B4)\ndm=m\n1 +\u00112\u0002[!e\u000b\u0000\u0011m\u0002(!+!e\u000b+\u0011!\u0002m)]dt;\n(B5)\nwhere we model the thermal \ructuations as uncorrelated\nGaussian noise represented by a six-dimensional vector\nof independent Wiener increments (d Wp;dWl)T. The\ncorresponding di\u000busion rate is described by the tensors\nDcmandDrotwhich, in agreement with the \ructuation-\ndissipation theorem, are related to the corresponding dis-\nsipation tensors \u0000 cmand \u0000 rotasDcm\u00112\u0000cm\u001f;D rot\u0011\n2\u0000rot\u001f, where\u001f\u0011MkBT\r2\n0a2=\u00162.\nIn the following we numerically integrate Eqs. (B1-B5)\nusing the stochastic Euler method implemented in the\nstochastic di\u000berential equations package in MATLAB. As\nthe e\u000bect of thermal noise is more prominent for small\nparticles at weak \felds, we focus on the Einstein-de Haas\nregime considered in Sec. IV. We show that even in this\ncase the e\u000bect of thermal \ructuations leads to dynamics\nwhich are qualitatively very close to the results obtained\nin Sec. IV. In Fig. 4 we present the results of the stochas-\ntic integrator by averaging the solution of 100 di\u000berent\ntrajectories calculated using the same parameters consid-\nered in Fig. 3(a-c). The resulting average dynamics agree\nqualitatively with the results obtained by integrating the\ncorresponding set of deterministic equations Eqs. (1-5)\nFigure 4. Stochastic dynamics of a nanomagnet for the same\nparameter regime as considered in Fig. 3. (a) Average motion\nof the system in the y-zplane until time t= 5\u0016s. (b) Dy-\nnamics of center of mass along the ey(top) and ez(bottom)\ndirections. (c) Dynamics of the anisotropy axis component\ne3;x. (d) Numerical error as function of time. The simulations\nshow the results of the average of 100 di\u000berent realizations of\nthe system dynamics. In panels (b-d) the solid dark lines are\nthe average trajectories, while the shaded area represents the\nstandard deviation.\n[cfr. Fig. 3(a-c)]. The main e\u000bect of thermal excitations\nis to shift the center of oscillations of the particle's de-\ngrees of freedom around the value given by the thermal\n\ructuations. This is more evident for the dynamics of\ne3[cfr. Fig. 4(c) and Fig. 3(c)]. We thus conclude that\nthe deterministic equations Eqs. (1-5) considered in the\nmain text correctly capture the metastable behavior of\nthe system. We emphasize that the results presented in\nthis section include only the noise due to the surround-\ning gas. Should one be interested in simulating the ef-\nfect of the \ructuations of the magnetic moment, the Eu-\nler method used here is not appropriate, and the Heun\nmethod should be used instead [89].\nLet us conclude with a technical note on the numerical\nsimulations. In the presence of dissipation and thermal\n\ructuations the only conserved quantity of the system is\nthe magnitude of the magnetic moment ( jmj= 1). We\nthus use the deviation 1 \u0000jmj2as a measure of the numer-\nical error in both the stochastic and deterministic sim-\nulations presented in this article. For the deterministic\nsimulations the error stays much smaller than any other\nphysical degree of freedom of the system during the whole\nsimulation time. The simulation of the stochastic dynam-\nics shows a larger numerical error [see Fig. 4(d)], which\ncan be partially reduced by taking a smaller time-step\nsize. We note that, for the value of magnetic anisotropy\ngiven in Table I, the system of SDE is sti\u000b. This, together\nwith the requirement imposed on the time-step size by10\nthe numerical error, ultimately limits the maximum time\nwe can simulate to a few microseconds. However, this is\nsu\u000ecient to validate the agreement between the SDE and\nthe deterministic simulations presented in the article.\nAppendix C: Additional \fgures\nIn this appendix we provide additional \fgures.\n1. Dynamics in the atom phase\nIn Fig. 5 we analyze magnetization dynamics in the\natom phase as a function of di\u000berent system parame-\nters. In Fig. 5(a) we show how magnetization switching\nchanges as the anisotropy constant kais varied. We con-\nsider the bias \feld B0= 1100 mT, which is larger than\nthe value considered in the main text. This is done to en-\nsure thatB0>B atom for all anisotropy values. Fig. 5(a)\ndemonstrates that the switching time \u001c, given by Eq. (8),\nis an excellent approximation for the dynamics across a\nwide range of values for the anisotropy constant ka. The\nlarger discrepancy between Eq. (8) and the line showing\nthe case with ka= 106J/m3is explained by the prox-\nimity of this point to the unstable region (in this case\ngiven by the critical \feld Batom = 900 mT), and better\nagreement is recovered at larger bias \feld values.\nIn Fig. 5(b) we analyze the validity of Eq. (8) for dif-\nferent values of the Gilbert damping parameter \u0011and the\ntemperature T. The thick dashed line shows the region\nwhere Eq. (8) di\u000bers from the exact switching time by\n5%; below this line the deviation becomes increasingly\nmore signi\fcant. As evidenced by Fig. 5(b), \u001cshows lit-\ntle dependence on T; its order of magnitude remains con-\nstant over a wide range of cryogenic temperatures. On\nthe other hand, the dependence on \u0011is more pronounced.\nIn fact, reducing the Gilbert parameter signi\fcantly de-\nlays the switching time, leading to levitation times as\nlong as\u00181\u0016s.\nAdditionally, we point out that \u001cdepends on the \feldgradientB0and curvature B00through the initial con-\nditionmk(t= 0). In particular, magnetization switch-\ning can be delayed by decreasing B0, as this reduces\nthe initial misalignment of the magnetization and the\nanisotropy axis (i. e. jmk(t= 0)j!1).\n2. Dynamics in the presence of rotation\nIn Fig. 6 we consider a more general equilibrium con-\n\fguration, namely a nanomagnet initially rotating such\nthat in the equilibrium point Le=\u0000I3!Sex, with!S>0\ndenoting the rotation in the clockwise direction. This\nequilibrium point is linearly stable in the absence of dis-\nsipation [23, 24], with additional stability of the system\nprovided by the mechanical rotation, analogously to the\nclassical magnetic top [25{27].\nIn Fig. 6(a) we analyze how magnetization switching\nin the atom phase changes in the presence of rotation for\ndi\u000berent values of parameters. The rotation has a slight\ne\u000bect on the switching time \u001c, shifting it forwards (back-\nwards) in case of a clockwise (counterclockwise) rotation.\nThis is generally expected in the presence of magneto-\nmechanical coupling [74, 88].\nIn Fig. 6(b) we show the motion in the y-zplane in\nthe Einstein{de Haas phase for both directions of rota-\ntion. This can be compared with Fig. 3(a). The rotation\ndoes not qualitatively a\u000bect the dynamics of the system.\nThe di\u000berence in the two trajectories can be explained\nby a di\u000berent total angular momentum in the two cases,\nas in the case of a clockwise (counterclockwise) rotation\nthe mechanical and the internal angular momentum are\nparallel (anti-parallel), such that the total angular mo-\nmentum is increased (decreased) compared to the non-\nrotating case. This asymmetry arises from the linear\nstability of a rotating nanomagnet, and it is not a conse-\nquence of dissipation. In fact, we con\frm by numerical\nsimulations that the escape time t?as a function of the\npressurePshows no dependence on the mechanical ro-\ntation!S. Namely, even in the presence of mechanical\nrotation one recovers the same plot as shown in Fig. 3(f).\n[1] A. Einstein and W. J. de Haas, Experimental proof of\nthe existence of Amp\u0012 ere's molecular currents, Proc. K.\nNed. Akad. Wet. 18, 696 (1915).\n[2] O. W. Richardson, A mechanical e\u000bect accompanying\nmagnetization, Phys. Rev. 26, 248 (1908).\n[3] S. J. Barnett, Magnetization by rotation, Phys. Rev. 6,\n239 (1915).\n[4] E. M. Chudnovsky, Conservation of angular momentum\nin the problem of tunneling of the magnetic moment,\nPhys. Rev. Lett. 72, 3433 (1994).\n[5] C. C. Rusconi and O. Romero-Isart, Magnetic rigid rotor\nin the quantum regime: Theoretical toolbox, Phys. Rev.\nB93, 054427 (2016).\n[6] M. Ganzhorn, S. Klyatskaya, M. Ruben, and W. Werns-dorfer, Quantum Einstein-de Haas e\u000bect, Nat. Commun.\n7, 11443 (2016).\n[7] S. Viola Kusminskiy, H. X. Tang, and F. Marquardt,\nCoupled spin-light dynamics in cavity optomagnonics,\nPhys. Rev. A 94, 033821 (2016).\n[8] H. Keshtgar, S. Streib, A. Kamra, Y. M. Blanter, and\nG. E. W. Bauer, Magnetomechanical coupling and fer-\nromagnetic resonance in magnetic nanoparticles, Phys.\nRev. B 95, 134447 (2017).\n[9] B. A. Stickler, K. Hornberger, and M. S. 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In the region\nbelow the thick dashed line the deviation from the exact\nvalue is more than 5%.\nFigure 6. Dynamics of a nanomagnet initially rotat-\ning around the axis exwith frequencyj!Sj=(2\u0019) = 100\nMHz. (a) Magnetization switching in the atom phase.\nLine denoted by circle corresponds to the same set of pa-\nrameters as in Fig. 2(a). Each remaining line di\u000bers by a\nsingle parameter, as denoted by the legend. Dotted verti-\ncal lines show Eq. (8). (b) Motion in the y-zplane in the\nEinstein{de Haas phase, using the same numerical values\nof the parameters as in Fig. 3. Left panel: Clockwise\nrotation. Right panel: counterclockwise rotation.\n[11] D. F. Jackson Kimball, A. O. Sushkov, and D. Budker,\nPrecessing ferromagnetic needle magnetometer, Phys.\nRev. Lett. 116, 190801 (2016).\n[12] P. Kumar and M. Bhattacharya, Magnetometry via spin-\nmechanical coupling in levitated optomechanics, Opt.\nExpress 25, 19568 (2017).\n[13] Y. B. Band, Y. Avishai, and A. 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Mech. , P09008\n(2014)."    },    {        "title": "2101.02794v2.Mechanisms_behind_large_Gilbert_damping_anisotropies.pdf",        "content": "Mechanisms behind large Gilbert damping anisotropies\nI. P. Miranda1, A. B. Klautau2,∗A. Bergman3, D. Thonig3,4, H. M. Petrilli1, and O. Eriksson3,4\n1Universidade de São Paulo, Instituto de Física,\nRua do Matão, 1371, 05508-090, São Paulo, SP, Brazil\n2Faculdade de Física, Universidade Federal do Pará, Belém, PA, Brazil\n3Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden and\n4School of Science and Technology, Örebro University, Fakultetsgatan 1, SE-701 82 Örebro, Sweden\n(Dated: November 22, 2021)\nA method with which to calculate the Gilbert damping parameter from a real-space electronic\nstructure method is reported here. The anisotropy of the Gilbert damping with respect to the\nmagnetic moment direction and local chemical environment is calculated for bulk and surfaces\nof Fe 50Co50alloys from first principles electronic structure in a real space formulation. The size\nof the damping anisotropy for Fe 50Co50alloys is demonstrated to be significant. Depending on\ndetails of the simulations, it reaches a maximum-minimum damping ratio as high as 200%. Several\nmicroscopic origins of the strongly enhanced Gilbert damping anisotropy have been examined, where\nin particular interface/surface effects stand out, as do local distortions of the crystal structure.\nAlthough theory does not reproduce the experimentally reported high ratio of 400% [Phys. Rev.\nLett. 122, 117203 (2019)], it nevertheless identifies microscopic mechanisms that can lead to huge\ndamping anisotropies.\nIntroduction: Magnetic damping has a critical impor-\ntanceindeterminingthelifetime,diffusion,transportand\nstability of domain walls, magnetic vortices, skyrmions,\nand any nano-scale complex magnetic configurations [1].\nGiven its high scientific interest, a possibility to obtain\nthis quantity by means of first-principles theory [2] opens\nnew perspectives of finding and optimizing materials for\nspintronic and magnonic devices [3–8]. Among the more\npromising ferromagnets to be used in spintronics devices,\ncobalt-iron alloys demonstrate high potentials due to the\ncombination of ultralow damping with metallic conduc-\ntivity [4, 9].\nRecently, Li et al.[10] reported an observed, gi-\nant anisotropy of the Gilbert damping ( α) in epitaxial\nFe50Co50thin films (with thickness 10 −20nm) reach-\ning maximum-minimum damping ratio values as high as\n400%. TheauthorsofRef. [10]claimedthattheobserved\neffect is likely due to changes in the spin-orbit coupling\n(SOC) influence for different crystalline directions caused\nby short-range orderings that lead to local structural dis-\ntortions. This behaviour differs distinctly from, for ex-\nample, pure bcc Fe [11]. In order to quantitatively pre-\ndict the Gilbert damping, Kambersky’s breathing Fermi\nsurface (BFS) [12] and torque-correlation (TC) [13] mod-\nels are frequently used. These methods have been ex-\nplored for elements and alloys, in bulk form or at sur-\nfaces, mostly via reciprocal-space ab-initio approaches,\nin a collinear or (more recently) in a noncollinear con-\nfiguration [14]. However, considering heterogeneous ma-\nterials, such as alloys with short-range order, and the\npossibility to investigate element specific, non-local con-\ntributions to the damping parameter, there are, to the\nbest of our knowledge, no reports in the literature that\nrely on a real space method.\nIn this Letter, we report on an implementation of ab\ninitiodamping calculations in a real-space linear muffin-tin orbital method, within the atomic sphere approxi-\nmation (RS-LMTO-ASA) [15, 16], with the local spin\ndensity approximation (LSDA) [17] for the exchange-\ncorrelation energy. The implementation is based on the\nBFSandTCmodels, andthemethod(SupplementalMa-\nterial - SM, for details) is applied to investigate the re-\nported, huge damping anisotropy of Fe50Co50(100)/MgO\nfilms [10]. A main result here is the identification of\na microscopic origin of the enhanced Gilbert damping\nanisotropy of Fe50Co50(100) films, and the intrinsic rela-\ntionships to the local geometry of the alloy. Most signifi-\ncantly, wedemonstratethatasurfaceproducesextremely\nlarge damping anisotropies that can be orders of magni-\ntude larger than that of the bulk. We call the attention\nto the fact that this is the first time, as far as we know,\nthat damping values are theoretically obtained in such a\nlocal way.\nResults: We calculated: i)ordered Fe50Co50in theB2\nstructure (hereafter refereed to as B2-FeCo) ii)random\nFe50Co50alloysinbccorbctstructures, wherethevirtual\ncrystal approximation (VCA) was applied; iii)Fe50Co50\nalloys simulated as embedded clusters in a VCA matrix\n(host). In all cases VCA was simulated with an elec-\ntronic concentration corresponding to Fe50Co50. The ii)\nandiii)alloys were considered as in bulk as well as in\nthe (001) surface, with bcc and bct structures (here-\nafter correspondingly refereed as VCA Fe50Co50bcc,\nVCA Fe50Co50bct, VCA Fe50Co50(001) bcc and VCA\nFe50Co50(001) bct). The effect of local tetragonal distor-\ntions was considered with a localc\na= 1.09ratio (SM for\ndetails). All data for cluster based results, were obtained\nfrom an average of several different configurations. The\ntotal damping for a given site iin real-space ( αt, Eqs. S6\nand S7 from SM) can be decomposed in non-local, αij\n(i/negationslash=j), and local (onsite), αonsite(orαii,i=j) contri-\nbutions, each of them described by the tensor elements2\nανµ\nij=g\nmiπ/integraldisplay\nη(/epsilon1)Tr/parenleftBig\nˆTν\niImˆGijˆTµ\njImˆGji/parenrightBig\nd/epsilon1,(1)\nwheremiis the total magnetic moment localized in the\nreference atomic site i,µ,ν={x,y,z},ˆTis the torque\noperator, and η(/epsilon1) =∂f(/epsilon1)\n∂/epsilon1the derivative of the Fermi\ndistribution. The scalar αijparameter is defined in the\ncollinear regime as αij=1\n2(αxx\nij+αyy\nij).\nTo validate our methodology, the here obtained total\ndamping for several systems (such as bcc Fe, fcc Ni, hcp\nand fcc Co and B2-FeCo) were compared with estab-\nlished values available in the literature (Table S1, SM),\nwhere an overall good agreement can be seen.\nFig. 1 shows the non-local contributions to the damp-\ning for bcc Fe and B2-FeCo. Although the onsite contri-\nbutions are around one order of magnitude larger than\nthe non-local, there are many αijto be added and total\nnet values can become comparable. Bcc Fe and B2-FeCo\nhave very different non-local damping contributions. El-\nement resolved αij, reveal that the summed Fe-Fe in-\nteractions dominate over Co-Co, for distances until 2a\ninB2-FeCo. We observe that αijis quite extended in\nspace for both bcc Fe and B2-FeCo. The different con-\ntributions to the non-local damping, from atoms at equal\ndistance arises from the reduced number of operations in\nthe crystal point group due to the inclusion of SOC in\ncombination with time-reversal symmetry breaking. The\nB2-FeCo arises from replacing every second Fe atom in\nthe bcc structure by a Co atom. It is interesting that this\nreplacement (i.e. the presence of Co in the environment)\nsignificantly changes the non-local contributions for Fe-\nFe pairs , what can more clearly be seen from the Insetin\nFig. 1, where the non-local damping summed over atoms\nat the same relative distance for Fe-Fe pairs in bcc Fe\nandB2-FeCo are shown; the non-local damping of Fe-Fe\npairs are distinctly different for short ranges, while long\nranged (further than ∼2.25 Å) contributions are smaller\nand more isotropic.\nThe damping anisotropy, i.e. the damping change,\nwhen the magnetization is changed from the easy axis\nto a new direction is1\n∆αt=/parenleftBigg\nα[110]\nt\nα[010]\nt−1/parenrightBigg\n×100%, (2)\nwhereα[110]\ntandα[010]\ntare the total damping obtained\nfor magnetization directions along [110]and [010], re-\nspectively. Analogousdefinitionalsoappliesfor ∆αonsite.\n1We note that this definition is different to the maximum-\nminimum damping ratio, defined asα[110]\nt\nα[010]\nt×100%, from Ref.\n[10].We investigated this anisotropy in surfaces and in bulk\nsystems with (and without) tetragonal structural distor-\ntions. Our calculations for VCA Fe50Co50bcc show a\ndamping increase of ∼13%, when changing the magne-\ntization direction from [010]to[110](Table S2 in the\nSM). The smallest damping is found for the easy magne-\ntization axis, [010], which holds the largest orbital mo-\nment (morb) [18]. For VCA Fe50Co50bcc we obtained\na small variation of ∼2%for the onsite contribution\n(α[010]\nonsite = 8.94×10−4andα[110]\nonsite = 8.76×10−4),\nwhat implies that the anisotropy comes mostly from\nthe non-local contributions, particularly from the next-\nnearest neighbours. For comparison, ∆αt∼3%(with\n∆αonsite∼0.4%) in the case of bcc Fe, what corrobo-\nrates the reported [11] small bcc Fe anisotropy at room\ntemperature, andwiththebulkdampinganisotropyrates\n[19].\nWe also inspected the chemical inhomogeneity influ-\nence on the anisotropy, considering the B2-FeCo alloy,\nwhere the weighted average damping (Eq. S7 of SM)\nwas used instead. The B2-FeCo bcc (∼7%) and VCA\nFe50Co50bcc (∼13%) anisotropies are of similar magni-\ntudes. Both B2structure and VCA calculations lead to\ndamping anisotropies which are significantly lower than\nwhatwasobservedintheexperiments, anditseemslikely\nthatthepresenceofdisorderincompositionand/orstruc-\ntural properties of the Fe/Co alloy would be important\nto produce large anisotropy effects on the damping.\n\nα\nij\n\t\n×\n\t\n10\n-4\n−3\n−2\n−1\n0\n1\n2\n3\n4\n\nNormalized\n\t\ndistance\n1.0\n1.5\n2.0\n2.5\n3.0\n\n\t\nB2\n\t\nFe-Co\n\t\nB2\n\t\nFe-Fe\n\t\nB2\n\t\nCo-Co\n\t\nbcc\n\t\nFe-Fe\n\t\n−10\n0\n10\n20\n\n\t\n1.0\n1.5\n2.0\n2.5\n3.0\nFigure 1. (Color online) Non-local damping contributions,\nαij, in (Fe-centered) bulk B2-FeCo and bcc Fe, as a function\nof the normalized distance in lattice constant units a.Inset:\nNon-local contributions from only Fe-Fe pairs summed, for\neach distance, in bcc Fe bulk (empty blue dots) and in the\nB2-FeCo (full red dots). The onsite damping for Fe (Co) in\nB2-FeCo isαFe\nonsite = 1.1×10−3(αCo\nonsite = 0.8×10−3) and for\nbcc Fe it is αFe\nonsite = 1.6×10−3. The magnetization direction\nisz([001]). Lines are guides for the eyes.\nWeanalyzedtheroleoflocaldistortionsbyconsidering3\na hypothetical case of a large, 15%(c\na= 1.15), distortion\non thez-axis of ordered B2-FeCo. We found the largest\ndamping anisotropy ( ∼24%) when comparing the results\nwith magnetization in the [001](α[001]\nt= 10.21×10−3)\nand in the [010](α[010]\nt= 7.76×10−3) directions. This\nconfirms that, indeed, bct-like distortions act in favour of\nthe∆αtenhancement (and therefore, of the maximum-\nminimum damping ratio), but the theoretical data are\nnot large enough to explain the giant value reported ex-\nperimentally [10].\nNevertheless, in the case of an alloy, the local lattice\ndistortions suggested in Ref. [10] are most to likely occur\nin an heterogeneous way [20], with different distortions\nfor different local environments. To inspect this type\nof influence on the theoretical results, we investigated\n(Table S3, SM) clusters containing different atomic con-\nfigurations embedded in a VCA Fe50Co50matrix (with\nFe bulk lattice parameter); distortions were also consid-\nered such that, locally in the clusters,c\na= 1.15(Ta-\nble S4, in the SM). Moreover, in both cases, two types\nof clusters have to be considered: Co-centered and Fe-\ncentered. The αtwas then computed as the sum of the\nlocal and non-local contributions for clusters with a spe-\ncific central (Fe or Co) atom, and the average of Fe-\nand Co-centered clusters was taken. Fe-centered clus-\ntershaveshownlargeranisotropies, onaverage ∼33%for\nthe undistorted (∼74%for the distorted) compared with\n∼8%fortheundistortedCo-centeredclusters( ∼36%for\nthe distorted). Although these results demonstrate the\nimportance of both, local distortions as well as non-local\ncontributions to the damping anisotropy, they are not\nstill able to reproduce the huge observed [10] maximum-\nminimum damping ratio.\nWe further proceed our search for ingredients that\ncould lead to a huge ∆αtby inspecting interface effects,\nwhich are present in thin films, grain boundaries, stack-\ning faults and materials in general. Such interfaces may\ninfluence observed properties, and in order to examine\nif they are relevant also for the reported alloys of Ref.\n[10], we considered these effects explicitly in the calcu-\nlations. As a model interface, we considered a surface,\nwhat is, possibly, the most extreme case. Hence, we per-\nformed a set of αtcalculations for the Fe50Co50(001),\nfirst on the VCA level. Analogous to the respective bulk\nsystems, we found that the onsite contributions to the\ndamping anisotropy are distinct, but they are not the\nmain cause ( ∆αonsite∼18%). However, the lack of in-\nversion symmetry in this case gives a surprisingly large\nenhancement of ∆αt, thus having its major contribution\ncoming from the non-local damping terms, in particular\nfrom the next-nearest neighbours. Interestingly, negative\nnon-local contributions appear when αtis calculated in\nthe[010]direction. These diminish the total damping\n(the onsite contribution being always positive) and gives\nrise to a larger anisotropy, as can be seen by comparisonof the results shown in Table I and Table S5 (in the Sup-\nplemental Material). In this case, the total anisotropy\nwas found to be more than ∼100%(corresponding to a\nmaximum-minimum damping ratio larger than 200%).\nA compilation of the most relevant theoretical results\nobtained here is shown in Fig. 2, together with the ex-\nperimental data and the local density of states (LDOS)\natEFfor each magnetization direction of a typical atom\nin the outermost layer (data shown in yellow). As shown\nin Fig. 2, the angular variation of αthas a fourfold ( C4v)\nsymmetry, with the smallest Gilbert damping occurring\nat 90◦from the reference axis ( [100],θH= 0◦), for both\nsurface and bulk calculations. This pattern, also found\nexperimentally in [10], matches the in-plane bcc crys-\ntallographic symmetry and coincides with other mani-\nfestations of SOC, such as the anisotropic magnetoresis-\ntance [10, 21]. Following the simplified Kambersky’s for-\nmula [13, 22], in which (see SM) α∝n(EF)and, there-\nfore, ∆α∝∆n(EF), we can ascribe part of the large\nanisotropy of the FeCo alloys to the enhanced LDOS dif-\nferences at the Fermi level, evidenced by the close corre-\nlation between ∆n(EF)and∆αtdemonstrated in Fig. 2.\nThus, as a manifestation of interfacial SOC (the so-called\nproximity effect [23]), the existence of ∆αtcan be under-\nstoodintermsofRashba-likeSOC,whichhasbeenshown\nto play an important role on damping anisotropy [24, 25].\nAnalogous to the bulk case, the higher morboccurs where\nthe system presents the smallest αt, and the orbital\nmoment anisotropy matches the ∆αtfourfold symme-\ntry with a 90◦rotation phase (see Fig. S3, SM). Note\nthat a lower damping anisotropy than Co50Fe50(001) is\nfound for a pure Fe(001) bcc surface, where it is ∼49%\n(Table S2, SM), in accordance with Refs. [7, 26], with\na dominant contribution from the onsite damping val-\nues (conductivity-like character on the reciprocal-space\n[19, 27]).\nThe VCA surface calculations on real-space allows to\ninvestigate the layer-by-layer contributions (intra-layer\ndamping calculation), as shown in Table I. We find that\nthemajorcontributiontothedampingsurfaceanisotropy\ncomes from the outermost layer, mainly from the differ-\nence in the minority 3dstates around EF. The deeper\nlayers exhibit an almost oscillatory ∆αtbehavior, simi-\nlar to the oscillation mentioned in Ref. [28] and to the\nFriedel oscillations obtained for magnetic moments. The\ndamping contributions from deeper layers are much less\ninfluenced by the inversion symmetry breaking (at the\nsurface), as expected, and eventually approaches the typ-\nicalbulklimit. Therefore, changesintheelectronicstruc-\ntureconsiderednotonlytheLDOSoftheoutermostlayer\nbut a summation of the LDOS of all layers (including the\ndeeper ones), which produces an almost vanishing differ-\nence between θH= 0◦andθH= 45◦(also approaching\nthe bulk limit). The damping anisotropy arising as a sur-\nface effect agrees with what was observed in the case of\nFe [7] and CoFeB [29] on GaAs(001), where the damping4\n0o45o90o\n135o\n180o\n225o\n270o315o[100][110][010]\n[−110]\n0.10.20.3\nΔn(EF)\n(st./Ry−at.)\n0.0050.0100.015\nαt\nθH\nFigure 2. (Color online) Total damping and LDOS difference\natEF,∆n(EF), as a function of θH, the angle between the\nmagnetizationdirectionandthe [100]-axis. Squares: (redfull)\nVCAFe 50Co50(001)bcc. Triagles: (greenfull)averageover32\nclusters (16 Fe-centered and 16 Co-centered), with bcc struc-\nture at the surface layers (SM) embedded in a VCA medium;\n(gray open) similar calculations, but with a local lattice dis-\ntortion. Circles: (yellow open) ∆n(EF)betweenθH= 0◦and\nthe current angle for a typical atom in the outermost layer\nof VCA Fe 50Co50(001) bcc; (blue full) experimental data [10]\nfor a 10-nm Fe 50Co50/Pt thin film; (purple full) average bulk\nVCA Fe 50Co50bcc; and (brown full) the B2-FeCo bulk. Lines\nare guides for the eyes.\nanisotropy diminishes as the film thickness increases.\nTable I. Total intra-layer damping ( αt×10−3) and anisotropy,\n∆αt(Eq. 2), of a typical (VCA) atom in each Fe 50Co50(001)\nbcc surface layer for magnetization along [010]and[110]di-\nrections. In each line, the sum of all αijin the same layer is\nconsidered. Outermost (layer 1) and deeper layers (2-5).\nLayerαt[010]αt[110] ∆αt\n1 7.00 14.17 +102.4%\n2 1.28 1.16 −9.4%\n3 2.83 3.30 +16.6%\n4 2.18 1.99 −8.7%\n5 2.54 2.53 −0.4%\nWe also studied the impact of bct-like distortions in\nthesurface, initiallybyconsideringtheVCAmodel. Sim-\nilartothebulkcase,tetragonaldistortionsmaybeimpor-\ntant for the damping anisotropy at the surface, e.g. when\nlocal structural defects are present. Therefore, localized\nbct-like distortions of the VCA medium in the surface,\nparticularly involving the most external layer were inves-\ntigated. The structural model was similar to what was\nused for the Fe50Co50bulk, consideringc\na= 1.09(see\nSM). Our calculations show that tetragonal relaxations\naround a typical site in the surface induce a ∆αt∼75%,\nfromα[010]\nt= 8.94×10−3toα[110]\nt= 15.68×10−3. Themain effect of these distortions is an enhancement of the\nabsolute damping values in each direction with respect to\nthe pristine (bcc) system. This is due to an increase on\nαonsite, fromα[010]\nonsite = 7.4×10−3toα[010]\nonsite = 9.5×10−3,\nand fromα[110]\nonsite = 8.7×10−3toα[110]\nonsite = 11.7×10−3;\nthe resulting non-local contributions remains similar to\ntheundistortedcase. Theinfluenceofbct-likedistortions\non the large damping value in the Fe50Co50surface is in\nline with results of Mandal et al.[30], and is related to\nthe transition of minority spin electrons around EF.\nWe then considered explicit 10-atom Fe50Co50clusters\nembedded in a VCA FeCo surface matrix. The results\nfrom these calculations were obtained as an average over\n16 Fe-centered and 16 Co-centered clusters. We con-\nsidered clusters with undistorted bcc crystal structure\n(Fig. 2, yellow open circles) as well as clusters with lo-\ncal tetragonal distortions (Fig. 2, black open circles). As\nshown in Fig. 2 the explicit local tetragonal distortion\ninfluences the damping values ( α[010]\nt= 10.03×10−3and\nα[110]\nt= 14.86×10−3)andtheanisotropy, butnotenough\ntoreproducethehugevaluesreportedintheexperiments.\nA summary of the results obtained for each undis-\ntorted FeCo cluster at the surface is shown in Fig. 3:\nCo-centered clusters in Fig. 3(a) and Fe-centered clusters\nin Fig. 3(b). A large variation of αtvalues is seen from\nclustertocluster, dependingonthespatialdistributionof\natomic species. It is clear that, αtis larger when there is\na larger number of Fe atoms in the surface layer that sur-\nroundsthecentral, referenceclustersite. Thiscorrelation\ncan be seen by the numbers in parenthesis on top of the\nblue symbols (total damping for each of the 16 clusters\nthat were considered) in Fig. 3. We also notice from the\nfigure that the damping in Fe-centered clusters are lower\nthan in Co-centered, and that the [010]magnetization di-\nrection exhibit always lower values. In the Insetof Fig. 3\nthe onsite contributions to the damping, αonsite, and the\nLDOS atEFin the central site of each cluster are shown:\na correlation, where both trends are the same, can be ob-\nserved. The results in Fig. 3 shows that the neighbour-\nhood influences not only the local electronic structure at\nthe reference site (changing n(EF)andαonsite), but also\nmodifies the non-local damping αij, leading to the cal-\nculatedαt. In other words, the local spatial distribution\naffects how the total damping is manifested, something\nwhich is expressed differently among different clusters.\nThis may open up for materials engineering of local and\nnon-local contributions to the damping.\nConclusions: We demonstrate here that real-space\nelectronic structure, based on density functional theory,\nyield a large Gilbert damping anisotropy in Fe50Co50al-\nloys. Theory leads to a large damping anisotropy, when\nthe magnetization changes from the [010]to the [110]di-\nrection, which can be as high as ∼100%(or200%in the\nminimum-maximum damping ratio) when surface calcu-\nlations are considered. This is in particular found for5\n\u0001\u0001\u0002\u0003\u0004\u0005\u0006\u0007\n\u0005\u0004\u0005\b\u0005\u0007\u0005\t\u0005\n\u0001\n\u0001\u0001\n\u000b\f\u0001\r\u000e\u0006\u000f\u0010\u0011\u0002\u0010\u0012\u0010\u0013\u0001\u0002\u0001\u0014\u0005\u0004\u0005\u0015\u0001\u0002\u0001\u0014\u0004\u0004\u0005\u0015\u0001\u0001\u0016\u0016\u0003\u0004\u0005\u0006\u0007\u0005\u0004\u0005\b\u0005\u0007\u0005\n\u0001\u0002\u0003\u0004\u0005\b\u0005\u0007\u0005\t\u0005\n\u0001\b\t\u0017\u0018\u0004\u0005\u0004\b\u0004\t\u0004\u0017\u0001\u0006\u0007\u0007\b\t\n\u000b\n\f\b\u0001\u0002\u0003\u0004\u0005\b\t\n\u000b\n\f\u0006\u0007\u0007\b\t\u000b\u000b\n\f\b\u0001\u0002\u0003\u0004\u0005\b\t\u000b\u000b\n\f\n\u0001\u0001\u0002\u0003\u0001\u0004\u0005\u0006\u0007\n\u0005\b\u0004\u0005\u0004\b\t\u0005\n\u0001\n\u000b\f\r\u0002\u000e\u000f\u0001\u0010\f\u0011\u0012\u000e\u000f\t\u0013\u0014\u0015\u0004\u0005\u0004\t\u0004\u0013\u0004\u0014\u0001\u0016\u0012\u0017\u0001\u0018\u000e\u0006\u0019\u000e\u0010\u0002\u000e\u000f\u000e\u001a\u0001\u0001\u001b\u001b\u0003\u0001\u0004\u0005\u0006\u0007\u0005\b\u0004\u0005\u0004\b\t\u0005\n\u0001\u0002\u0003\u0004\u0005\u0004\u0014\u0004\u0015\t\u0005\t\t\n\u0001\t\u0013\u0014\u0015\u0004\u0005\u0004\t\u0004\u0013\u0004\u0014\u0001(0)(1)(2)(2)(2)(2)(2)(2)(2)(2)(2)(2)(3)(3)(4)(4)\n(1)(1)(2)(2)(3)(3)(3)(3)(3)(3)(3)(3)(3)(3)(4)(5)\nFigure 3. Damping for the [010](open circles) and [110](full\ncircles) magnetization directions for distinct types of 10-atom\nFe50Co50bcc clusters, embedded in VCA Fe 50Co50(001) bcc\nand without any distortion around the reference atom (for\nwhichαtandαonsiteare shown). (a) Co-centered and (b)\nFe-centered clusters. The quantity of Fe atoms in the surface\nlayers (near vacuum) are indicated by the numbers in paren-\nthesis and the results have been ordered such that larger val-\nues are to the left in the plots. Insets:αonsitefor the [010]\n(red open circles) and [110](blue filled circles) magnetization\ndirections, and corresponding local density of states, n(EF),\nat the Fermi level (green filled and unfilled triangles) at the\ncentral atom (placed in the outermost layer) for both types\nof clusters. Lines are guides for eyes.\ncontributions from surface atoms in the outermost layer.\nHence the results presented here represents one more ex-\nample, in addition to the well known enhanced surface\norbital moment [31], of the so-called interfacial spin-orbit\ncoupling. This damping anisotropy, which holds a bcc-\nlike fourfold ( C4v) symmetry, has a close relation to the\nLDOS difference of the most external layer at EF(ma-\njorly contributed by the minority dstates), as well as\nto the orbital moment anisotropy with a 90◦phase. As\na distinct example of an interface, we consider explicitly\nthe Fe50Co50cluster description of the alloy. In this case,\nbesides an onsite contribution, we find that the damp-\ning anisotropy is mostly influenced by non-local next-\nnearest-neighbours interactions.\nSeveral Gilbert damping anisotropy origins are also\ndemonstrated here, primarily related to the presence of\ninterfaces, alloy composition and local structural distor-\ntions (as summarized in Table S6, in the SM [32]). Pri-\nmarily we find that: ( i) the presence of Co introduces anenhanced spin-orbit interaction and can locally modify\nthe non-local damping terms; ( ii) the randomness of Co\nin the material, can modestly increase ∆αtas a total ef-\nfect by creating Co-concentrated clusters with enhanced\ndamping; ( iii) at the surface, the spatial distribution of\nFe/Co, increases the damping when more Fe atoms are\npresent in the outermost layer; and ( iv) the existence\nof local, tetragonal distortions, which act in favour (via\nSOC) of the absolute damping enhancement, by modify-\ning theαonsiteof the reference atom, and could locally\nchange the spin relaxation time. Furthermore, in rela-\ntionship to the work in Ref. [10], we show here that bulk\nlike tetragonal distortions, that in Ref. [10] were sug-\ngested to be the key reason behind the observed huge\nanisotropy of the damping, can in fact not explain the\nexperimental data. Such distortions were explicitly con-\nsidered here, using state-of-the-art theory, and we clearly\ndemonstrate that this alone can not account for the ob-\nservations.\nAlthough having a similar trend as the experimen-\ntal results of Ref. [10], we do not reproduce the most\nextreme maximum-minimum ratio reported in the ex-\nperiment,∼400%(or∆αt∼300%). The measured\ndamping does however include effects beyond the intrin-\nsic damping that is calculated from our electronic struc-\nturemethodology. Other mechanismsare knownto influ-\nence the damping parameter, such as contributions from\neddy currents, spin-pumping, and magnon scattering, to\nname a few. Thus it is possible that a significant part\nof the measured anisotropy is caused by other, extrin-\nsic, mechanisms. Despite reasons for differences between\nobservation and experiment on films of Fe50Co50alloys,\nthe advancements presented here provide new insights on\nthe intrinsic damping anisotropy mechanisms, something\nwhich is relevant for the design of new magnetic devices.\nAcknowledgements: H.M.P. and A.B.K. acknowledge\nfinancial support from CAPES, CNPq and FAPESP,\nBrazil. The calculations were performed at the computa-\ntional facilities of the HPC-USP/CENAPAD-UNICAMP\n(Brazil), at the National Laboratory for Scientific Com-\nputing (LNCC/MCTI, Brazil), and at the Swedish Na-\ntional Infrastructure for Computing (SNIC). I.M. ac-\nknowledge financial support from CAPES, Finance Code\n001, process n◦88882.332894/2018-01, and in the Insti-\ntutional Program of Overseas Sandwich Doctorate, pro-\ncess n◦88881.187258/2018-01. 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Bergman3, D. Thonig3,4, H. M. Petrilli1, and O. Eriksson3,4\n1Universidade de São Paulo, Instituto de Física,\nRua do Matão, 1371, 05508-090, São Paulo, SP, Brazil\n2Faculdade de Física, Universidade Federal do Pará, Belém, PA, Brazil\n3Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden and\n4School of Science and Technology, Örebro University, Fakultetsgatan 1, SE-701 82 Örebro, Sweden\n(Dated: November 22, 2021)\nI. Theory\nThe torque-correlation model, first introduced by\nKamberský [1], and later elaborated by Gilmore et al.\n[2], can be considered as both a generalization and an\nextended version of the breathing Fermi surface model,\nwhich relates the damping of the electronic spin orienta-\ntion, with the variation in the Fermi surface when the\nlocal magnetic moment is changed. In this scenario,\nand considering the collinear limit of the magnetic or-\ndering, due to the spin-orbit coupling (SOC), the tilting\nin magnetization ˆmby a small change δˆmgenerates a\nnon-equilibrium population state which relaxes within a\ntimeτtowards the equilibrium. We use an angle θ, to\nrepresent the rotation of the magnetization direction δˆm.\nIftheBlochstatesofthesystemsarecharacterizedbythe\ngenericbandindex natwavevector k(withenergies /epsilon1k,n),\nit is possible to define a tensorfor the damping, that has\nmatrix elements (adopting the isotropic relaxation time\napproximation)\nανµ=gπ\nm/summationdisplay\nn,mdk\n(2π)3η(/epsilon1k,n)/parenleftbigg∂/epsilon1k,n\n∂θ/parenrightbigg\nν/parenleftbigg∂/epsilon1k,m\n∂θ/parenrightbigg\nµτ\n~\n(S1)\nwhich accounts for both intraband ( n=m, conductivity-\nlike) and interband ( n/negationslash=m, resistivity-like) contribu-\ntions [2]. Here µ,νare Cartesian coordinate indices,\nthat will be described in more detail in the discussion\nbelow, while η(/epsilon1k,n) =∂f(/epsilon1)\n∂/epsilon1/vextendsingle/vextendsingle/vextendsingle\n/epsilon1k,nis the derivative of\nthe Fermi distribution, f, with respect to the energy\n/epsilon1, andn,mare band indices. Therefore, the torque-\ncorrelation model correlates the spin damping to vari-\nations of the energy of single-particle states with respect\nto the variation of the spin direction θ, i.e.∂/epsilon1k,n\n∂θ. Us-\ning the Hellmann-Feynmann theorem, which states that\n∂/epsilon1k,n\n∂θ=/angbracketleftψk,n|∂H\n∂θ|ψk,n/angbracketright, and the fact that only the spin-\norbit Hamiltonian Hsochanges with the magnetization\ndirection, the spin-orbit energy variation is given by\n∂/epsilon1k,n(θ)\n∂θ=/angbracketleftψk,n|∂\n∂θ/parenleftbig\neiσ·ˆnθHsoe−iσ·ˆnθ/parenrightbig\n|ψk,n/angbracketright(S2)\nin which σrepresents the Pauli matrices vector, and\nˆnis the direction around which the local moment hasbeen rotated. The expression in Eq. S2 can be eas-\nily transformed into∂/epsilon1k,n(θ)\n∂θ=i/angbracketleftψk,n|[σ·ˆn,Hso]|ψk,n/angbracketright\nand we call ˆT= [σ·ˆn,Hso]thetorqueoperator. In\nview of this, it is straightforward that, in the collinear\ncase in which all spins are aligned to the zdirection,\nσ·ˆn=σµ(µ=x,y,z), originating the simplest {x,y,z}-\ndependent torque operator ˆTµ. Putting together the in-\nformation on Eqs. S1 and S2, and using the fact that\nthe imaginary part of the Greens’ functions can be ex-\npressed, in Lehmann representation, as Im ˆG(/epsilon1±iΛ) =\n−1\nπ/summationtext\nnΛ\n(/epsilon1−/epsilon1n)2+Λ2|n/angbracketright/angbracketleftn|, then it is possible to write in\nreciprocal-space [3]:\nανµ=g\nmπ/integraldisplay /integraldisplay\nη(/epsilon1)Tr/parenleftBig\nˆTνImˆGˆTµImˆG/parenrightBig\nd/epsilon1dk\n(2π)3.(S3)\nIn a real-space formalism, the Fourier transformation\nof the Green’s function is used to find a very similar ex-\npression emerges for the damping element ανµ\nijrelative to\ntwo atomic sites iandj(at positions riandrj, respec-\ntively) in the material:\nανµ\nij=g\nmiπ/integraldisplay\nη(/epsilon1)Tr/parenleftBig\nˆTν\niImˆGijˆTµ\njImˆGji/parenrightBig\nd/epsilon1,(S4)\nwhere we defined mi= (morb+mspin)as the total mag-\nnetic moment localized in the reference atomic site iin\nthe pair{i,j}. The electron temperature that enters into\nη(ε)is zero and, consequently, the energy integral is per-\nformed only at the Fermi energy. In this formalism, then,\nthe intraband and interband terms are replaced by onsite\n(i=j) and non-local ( i/negationslash=j) terms. After calculation of\nall components of Eq. S4 in a collinear magnetic back-\nground, we get a tensor of the form\nαij=\nαxxαxyαxz\nαyxαyyαyz\nαzxαzyαzz\n, (S5)\nwhich can be used in the generalized atomistic Landau-\nLifshitz-Gilbert (LLG) equation for the spin-dynamics\nof magnetic moment on site i[4]:∂mi\n∂t=mi×/parenleftBig\n−γBeff\ni+/summationtext\njαij\nmj·∂mj\n∂t/parenrightBig\n. Supposing that all spins are\nparallel to the local zdirection, we can define the scalar2\nαvalue as the average between components αxxandαyy,\nthat is:α=1\n2(αxx+αyy).\nOnce one has calculated the onsite ( αonsite) and the\nnon-local (αij) damping parameters with respect to the\nsite of interest i, the total value, αt, can be defined as\nthe sum of all these α’s:\nαt=/summationdisplay\n{i,j}αij. (S6)\nIn order to obtain the total damping in an heteroge-\nneous atomic system (more than one element type), such\nas Fe50Co50(with explicit Fe/Co atoms), we consider the\nweighted average between the different total local damp-\ning values ( αi\nt), namely:\nαt=1\nMeff/summationdisplay\nimiαi\nt, (S7)\nwheremiis the local magnetic moment at site i, and\nMeff=/summationtext\nimiis the summed total effective magnetiza-\ntion. This equation is based on the fact that, in FMR ex-\nperiments, the magnetic moments are excited in a zone-\ncentered, collective mode (Kittel mode). In the results\npresented here, Eq. S7 was used to calculate αtofB2-\nFeCo, both in bcc and bct structures.\nII. Details of calculations\nThe real-space linear muffin-tin orbital on the atomic-\nsphere approximation (RS-LMTO-ASA) [5] is a well-\nestablished method in the framework of the DFT to de-\nscribetheelectronicstructureofmetallicbulks[6,7], sur-\nfaces [8, 9] and particularly embedded [10] or absorbed\n[11–14] finite cluster systems. The RS-LMTO-ASA is\nbased on the LMTO-ASA formalism [15], and uses the\nrecursion method [16] to solve the eigenvalue problem\ndirectly in real-space. This feature makes the method\nsuitable for the calculation of local properties, since it\ndoes not depend on translational symmetry.\nThe calculations performed here are fully self-\nconsistent, and the spin densities were treated within\nthe local spin-density approximation (LSDA) [17]. In all\ncases, we considered the spin-orbit coupling as a l·sterm\nincluded in each variational step [18–20]. The spin-orbit\nis strictly necessary for the damping calculations due to\nits strong dependence on the torque operators, ˆT. In\nthe recursion method, the continued fractions have been\ntruncated with the Beer-Pettifor terminator [21] after 22\nrecursion levels ( LL= 22). The imaginary part that\ncomes from the terminator was considered as a natural\nchoice for the broadening Λto build the Green’s func-\ntions ˆG(/epsilon1+iΛ), which led to reliable αparameters in\ncomparison with previous results (see Table S1).To account for the Co randomness in the experimental\nFe50Co50films [22], some systems were modeled in terms\nof the virtual crystal approximation (VCA) medium of\nFe50Co50, considering the bcc (or the bct) matrix to have\nthe same number of valence electrons as Fe50Co50(8.5\ne−). However, we also investigated the role of the Co\npresence, as well as the influence of its randomness (or\nordering), by simulating the B2(CsCl) FeCo structure\n(a=aFe). The VCA Fe50Co50andB2-FeCo bulks were\nsimulated by a large matrix containing 8393 atoms in\nreal-space, the first generated by using the Fe bcc lat-\ntice parameter ( aFe= 2.87Å) and the latter using the\noptimized lattice parameter ( a= 2.84Å). Thisachoice\nin VCA Fe50Co50was based on the fact that it is eas-\nier to compare damping results for Fe50Co50alloy and\npure Fe bcc bulk if the lattice parameters are the same,\nand the use of the aFehas shown to produce trustwor-\nthyαtvalues. On the other hand, bct bulk structures\nwithc\na= 1.15(B2-FeCo bct and VCA Fe50Co50bct) are\nbased on even larger matrices containing 49412 atoms.\nThe respective surfaces were simulated by semi-matrices\nof the same kind (4488 and 19700 atoms, respectively),\nconsidering one layer of empty spheres above the outer-\nmost Fe50Co50(or pure Fe) layer, in order to provide a\nbasis for the wave functions in the vacuum and to treat\nthe charge transfers correctly.\nWe notice that the investigations presented here are\nbased on a (001)-oriented Fe50Co50film, in which only a\nsmall lattice relaxation normal to the surface is expected\nto occur (∼0.1%[23]).\nDamping parameters of Fe-centered and Co-centered\nclusters, embedded in an Fe50Co50VCA medium, have\nbeen calculated (explicitly) site by site. In all cases,\nthese defects are treated self-consistently, and the po-\ntential parameters of the remaining sites were fixed at\nbulk/pristine VCA surface values, according to its envi-\nronment. When inside the bulk, we placed the central\n(reference) atom of the cell in a typical site far away\nfrom the faces of the real-space matrix, avoiding any un-\nwanted surface effects. We considered as impurities the\nnearest 14 atoms (first and second nearest neighbours,\nup to 1a) from the central atom, treating also this sites\nself-consistently, in a total of 15 atoms. We calculated\n10 cases with Fe and Co atoms randomly positioned: 5\nwith Fe as the central atom (Fe-centered) and 5 with Co\nas the central atom (Co-centered). An example (namely\ncluster #1 of Tables S3 and S4), of one of these clusters\nembedded in bulk, is represented in Fig. S1(a). As the\nself-consistent clusters have always a total of 15 atoms,\nthe Fe (Co) concentration is about 47% (53%) or vice-\nversa. On the other hand, when inside the surface, we\nplaced the central (reference) atom of the cluster in a\ntypical site of the most external layer (near vacuum),\nsince this has shown to be the layer where the damping\nanisotropy is larger. Therefore, we considered as impu-\nrities the reference atom itself and the nearest 9 atoms3\n(up to 1a), in a total of 10 atoms (and giving a perfect\n50% (50%) concentration). An example of one of these\nclusters embedded in a surface is shown in Fig. S1(b).\n(a)\n(b)\nFigure S1. (Color online) Schematic representation of an ex-\nampleof: (a)Fe-centered15-atomclusterembeddedinaVCA\nFe50Co50bcc bulk medium; (b) Co-centered 10-atom cluster\nembedded in a VCA Fe 50Co50(001) bcc surface medium. Yel-\nlow and blue spheres represent Fe and Co atoms, respectively,\nwhile gray atoms represent the VCA Fe 50Co50sites (8.5 va-\nlence e−). The Fe(Co) concentration in the clusters are: (a)\n53% (47%) and (b) 50% (50%) . The total number of atoms\nincluding the surrounding VCA sites are: (a) 339 and (b) 293.\nThey were all accounted in the sum to obtain αtat the central\n(reference) Fe (a) and Co (b) site.\nTo simulate a bct-like bulk distortion, the 8 first neigh-\nbours of the central atom were stretched in the cdi-\nrection, resulting in ac\na= 1.15ratio. On the other\nhand, when embedded on the Fe50Co50(001) bcc surface,\nthe central (reference) atom is placed in the outermost\nlayer (near vacuum), and we simulate a bct distortion\nby stretching the 4 nearest-neighbours (on the second\nlayer) to reproduce ac\na= 1.09ratio (the maximum per-\ncentage that the atoms, in these conditions, could be\nmoved to form a bct-like defect). In this case, a total\nof 10 atoms (the nearest 9 atoms from the central one\n– up to 1a– and the reference atom) were treated self-\nconsistently, analogous to as shown in Fig. S1(b). As in\nthe case of the pristine bcc Fe50Co50clusters embedded\nin the VCA surface, we considered a total of 32 10-atom\nclusters with different Fe/Co spatial distributions, being\n16 Fe-centered, and 16 Co-centered.III. Comparison with previous results\nTheab-initio calculation of the Gilbert damping, in\nthe collinear limit, is not a new feature in the literature.\nMainly, the reported theoretical damping results are for\nbulk systems [2, 4, 24–28], but, some of them even stud-\nied free surfaces [29]. Therefore, in order to demonstrate\nthe reliability of the on-site and total damping calcula-\ntions implemented here in real-space, a comparison of\nthe presently obtained with previous (experimental and\ntheoretical) results, are shown in Table S1. As can be\nseen, our results show a good agreement with previously\nobtainedαvalues, including some important trends al-\nreadypredictedbefore. Forexample, thereducedGilbert\ndamping of Co hcp with respect to the Co fcc due to\nthe reduction of the density of states at the Fermi level\n[24, 28], (∼10.92states/Ry-atom in the hcp case and\n∼16.14states/Ry-atom in the fcc case).\nIV. Details of the calculated damping values\nThe damping values obtained for the systems studied\nhere are shown in Tables (S2-S5). These data can be use-\nful for the full understanding of the results presented in\nthe main text. For easy reference, in Table S2 the αtof a\ntypical atom in each system (bulk or surface) for different\nspin quantization axes are shown. These data are plot-\nted in Fig. 2 of the main text. The obtained values show\nthat, indeed, for bulk systems the damping anisotropies\nare not so pronounced as in the case of Fe50Co50(001)\nbcc surface.\nAs observed in Table S2, the increase in αtwhen\nchanging from the bcc Fe50Co50(c\na= 1) to the bct\nFe50Co50bulk structure (c\na= 1.15) is qualitatively con-\nsistent to what was obtained by Mandal et al.[33] (from\nαt= 6.6×10−3in the bcc to αt= 17.8×10−3in the\nbct, withc\na= 1.33[33]).\nTables S3 and S4 refer to the damping anisotropies\n(∆αt) for all Fe-centered and Co-centered clusters stud-\nied here, with different approaches: ( i) bcc clusters em-\nbedded in the VCA medium (Table S3) and ( ii) bct-like\nclusters embedded in the VCA medium (Table S4).\nIn comparison with bct-like clusters, we found larger\nabsoluteαtvalues but lower damping anisotropies. In\nall cases, Fe-centered clusters present higher ∆αtper-\ncentages.\nIn Table S5 the onsite damping anisotropies ( ∆αonsite)\nfor each layer of the Fe50Co50(001) bcc surface (\"1\" repre-\nsents the layer closest to vacuum) are shown. In compar-\nison with the total damping anisotropies (Table I of the\nmain text), much lower percentages are found, demon-\nstrating that the damping anisotropy effect comes ma-\njorly from the non-local damping contributions.\nThe most important results concerning the largest\ndampinganisotropiesaresummarizedinTableS6, below.4\nTable S1. Total damping values ( ×10−3) calculated for some bulk and surface systems, and the comparison with previous\nliterature results. The onsite contributions are indicated between parentheses, while the total damping, αt, are indicated\nwithout any symbols. All values were obtained considering the [001]magnetization axis. The VCA was adopted for alloys,\nexcept for the Fe 50Co50bcc in theB2structure (see Eq. S7). Also shown the broadening Λvalue considered in the calculations.\nBulks a(Å) This work Theoretical Experimental Λ(eV)\nFe bcc 2.87 4.2(1.6) 1 .3[2]a/(3.6)[4] 1.9[30]/2.2[31]\nFe70Co30bcc 2.87 2.5(0.7) − 3−5[32]d\nFe50Co50bcc 2.87 3.7(1.0)[VCA]/ 2.3(1.0)[B2]1.0[25]c[VCA]/ 6.6[33] [B2] 2.3[27]\nNi fcc 3.52 27.8(57.7) 23 .7[34]/( 21.6[4])b26.0[31]/24.0[35]\nNi80Fe20(Py) fcc 3.52 9.8(12.1) 3 .9[25]c8.0[30]/5.0[35]\nCo fcc 3.61 [3] 3.2(5.3) 5 .7[28]/(3.9[4])b11.0[30]∼5×10−2\nCo hcp 2.48/4.04 [28] 2.1(6.2) 3 .0[28] 3.7[31]\nCo85Mn15bcc [36] 2.87 [28] 6.2(4.2) 6 .6[28] −\nCo90Fe10fcc 3.56 [37] 3.6(4.2) − 3.0[35]/4.8[37]\nSurfaces a(Å) This work Theoretical Experimental\nFe(001) bcc [110] 2.87 5.8(5.4)e− 7.2[38]h/6.5[39]i\nFe(001) bcc [100] 2.87 3.9(4.4)f∼4[29]g4.2[40]j\nNi(001) fcc 3.52 80.0(129.6)∼10[29]g/12.7[41]m22.1[42]l\nPdFe/Ir(111) [43] fcc 3.84 3.9(2.7)n− −\nPdCo/Ir(111) [44] fcc 3.84 3.2(14.7)o− −\naWith Λ∼2×10−2eV.\nbWith Λ = 5×10−3eV.\ncWith Λ∼1.4×10−4eV.\ndFor a 28%Co concentration, but the results do not significantly change for a 30%Co concentration. Range including results before and\nafter annealing.\neOf a typical atom in the more external surface layer (in contact with vacuum), in the [110] magnetization direction.\nfOf a typical atom in the more external surface layer (in contact with vacuum), in the [100] magnetization direction.\ngFor a (001) bcc surface with thickness of N= 8ML (the same number of slabs as in our calculations), and Λ = 10−2eV.\nhAnisotropic damping obtained for a 0.9 nm Fe/GaAs(001) thin film (sample S2 in Ref. [38]) in the [110] magnetization direction.\niAnisotropic damping obtained for a 1.14 nm Fe/InAs(001) thin film in the in-plane [110] hard magnetization axis.\njFor a 25-nm-thick Fe films grown on MgO(001).\nkFor epitaxial Fe(001) films grown on GaAs(001) and covered by Au, Pd, and Cr capping layers.\nlIntrinsic Gilbert damping for a free 4×[Co(0.2 nm)/Ni(0.6 nm)](111) multilayer. Not the same system as Ni(001), but the nearest system\nfound in literature.\nmFor a Co | Ni multilayer with Ni thickness of 4 ML (fcc stacking).\nnOf a typical atom in the Fe layer.\noOf a typical atom in the Co layer.\nThe alloys with short-range orders (SRO) are described\nas FeCo clusters (with explicit Fe and Co atoms) embed-\nded in the Fe50Co50VCA medium – with and without\nthe bct-like distortion. In this case, the damping is cal-\nculated as a weighted average (Eq. S7). As discussed in\nthe main text, it can be seen from Table S6 that distor-\ntions and disorder can increase the anisotropy but the\nmajor effect comes from the surface. We notice that the\nnumber of clusters considered is limited in the statistical\naverage.\nIV. Kambersky’s simplified formula\nInordertoconnecttheanisotropyoftheGilbertdamp-\ning to features in the electronic structure, we consider in\nthe following Kambersky’s simplified formula for Gilbert\ndamping [47, 48]α=1\nγMs/parenleftBigγ\n2/parenrightBig2\nn(EF)ξ2(g−2)2\nτ.(S8)\nHere,γis the gyromagnetic ratio, n(EF)represents the\nLDOS at the Fermi level, ξis the SOC strength, τis the\nelectron scattering time, Msis the spin magnetic mo-\nment, andgis the spectroscopic g-factor [35, 49]. Note\nthat Eq. S8 demonstrates the direct relation between\nαandn(EF), often discussed in the literature, e.g., in\nRef. [27]. Our first principles calculations have shown\nno significant change in ξ, upon variation of the mag-\nnetization axis, for the FeCo systems ( ξCo= 71.02meV\nandξFe= 53.47meV). Hence, we can soundly relate the\ndamping anisotropy ∆αtto∆n(EF).\nFigure S2 shows how the LDOS difference (per atom)\n∆n(E)between the [010]and [110]magnetization di-\nrections is developed in pure Fe(001) bcc and in VCA\nFe50Co50(001) bcc surfaces, respectively. In both cases,5\nTableS2. Totaldamping( αt×10−3)ofatypicalatomin\neach system for the spin quantization axes [010](θH=\n90◦) and [110](θH= 45◦); also shown for the [001]and\n[111]. Bulk and surface bct systems are simulated with\nc\na= 1.15.\nBulks\nBulk αt[010]αt[110] ∆αt\nFe bcc 4.18 4.31 +3.1%a\nB2-FeCo bcc 2.28 2.44 +7.2%\nB2-FeCo bct 7.76 8.85 +12.4%\nVCA Fe 50Co50bcc 3.70 4.18 +13.0%\nVCA Fe 50Co50bct 4.69 5.10 +8.7%\nαt[010]αt[001] ∆αt\nB2-FeCo bct 7.76 10.21 +24.1%\nVCA Fe 50Co50bct 4.69 5.75 +22.6%\nαt[010]αt[111] ∆αt\nFe bcc 4.18 4.56 +9.1%b\nSurfaces\nSurface αt[010]αt[110] ∆αt\nFe(001) bcc 3.85 5.75 +49.4%\nFe/GaAs(001) bcc [38] 4.7(7) 7.2(7) +53(27)%c\nFe/MgO(001) bcc [45] 3.20(25) 6.15(20) +92(14)%d\nVCA Fe 50Co50(001) bcc 7.00 14.17 +102.4%\nVCA Fe 50Co50(001) bct 15.20 14.80−2.6%\nαt[010]αt[001] ∆αt\nVCA Fe 50Co50(001) bct 15.20 15.56 +2.4%\nVCA Fe 50Co50(001) bcc 7.00 9.85 +40.7%\naMankovsky et al.[24] find a damping anisotropy of ∼12%\nfor bulk Fe bcc at low temperatures ( ∼50K) between\n[010] and [011] magnetization directions. For this result,\nthe definition α=1\n2(αxx+αyy)was used.\nbThis result agrees with Gilmore et al.[46], which find\nthat the total damping of pure Fe bcc presents its higher\nvalue in the [111] crystallographic orientation and the\nlower value in the [001] direction, except for high scatter-\ning rates. Also agrees with Mankovsky et al.[24] results.\ncAnisotropic damping obtained for a 0.9 nm Fe/GaAs(001)\nthin film (sample S2 in Ref. [38]) in the [010] and [110]\nmagnetization directions.\ndFor a Fe(15 nm)/MgO(001) film at T= 4.5K in the high-\nest applied magnetic field, in which only intrinsic contri-\nbutions to the anisotropic damping are left.\nthe chosen layer,denoted as first, is the most external\none (near vacuum). the VCA Fe50Co50(001) bcc we also\ncalculated ∆n(E)for all layers summed (total DOS dif-\nference).\nAs can be seen, although in all cases the quantity\n∆n(E)exhibits some oscillations, differently from what\nwe observe forthe pureFe(001) surface case, at the Fermi\nenergy, there is a non-negligible difference in the minor-\nity spin channel ( 3dstates) for the VCA Fe50Co50(001).\nConsidering the results presented in Table I (main text)\nthe larger contribution to the damping anisotropy comes\nfrom the most external layer. The results by Li et al.\n[22] indicate a small difference (for two magnetization\ndirections) of the total density of states at the FermiTable S3. Total damping anisotropy ( ×10−3) of all stud-\nied Co-centered and Fe-centered bcc clusters for the spin-\nquantization axis [010]and [110], considering the 15-atom\nFeCo cluster together with the VCA medium in the summa-\ntion for total damping.\nCo-centered\nCluster # αt[010]αt[110] ∆αt\n1 10.11 9.65 4.8%\n2 8.09 6.96 16.2%\n3 7.81 7.02 11.3%\n4 7.11 7.02 1.3%\n5 7.48 6.88 8.7%\nAverage 8.12 7.51 8.1%\nFe-centered\nCluster # αt[010]αt[110] ∆αt\n1 2.68 2.03 32.0%\n2 2.49 2.05 21.5%\n3 2.56 1.86 37.6%\n4 2.45 1.79 36.9%\n5 2.76 2.01 37.3%\nAverage 2.59 1.95 32.8%\nTable S4. Total damping anisotropy ( ×10−3) of all stud-\nied Co-centered and Fe-centered bcc clusters for the spin\nquantization axis [010]and [110], with bct-like distortions/parenleftbigc\na= 1.15/parenrightbig\n, considering the 15-atom FeCo cluster together\nwith the VCA medium in the summation for total damping.\nCo-centered\nCluster # αt[010]αt[110] ∆αt\n1 5.85 4.37 33.9%\n2 5.95 4.21 41.3%\n3 5.88 4.35 35.2%\n4 5.90 4.41 33.8%\n5 5.86 4.34 35.0%\nAverage 5.89 4.34 35.7%\nFe-centered\nCluster # αt[010]αt[110] ∆αt\n1 2.36 1.39 69.8%\n2 2.27 1.32 72.0%\n3 2.22 1.26 76.2%\n4 2.25 1.26 78.6%\n5 2.42 1.38 75.4%\nAverage 2.30 1.32 74.2%\nTable S5. Onsite damping ( αonsite×10−3) of a typical atom\nin each layer of the VCA Fe 50Co50(001) bcc for the spin quan-\ntization axis [010]and[110].\nLayerαonsite[010]αonsite[110] ∆αonsite\n1 7.36 8.70 +18.2%\n2 0.63 0.69 +9.5%\n3 1.41 1.44 +2.1%\n4 0.87 0.86−1.1%\n5 0.99 0.97−2.0%6\nTableS6. SummaryofthemainFe 50Co50dampinganisotropy\nresults for: pure ordered ( B2) alloy; pure random (VCA) bulk\nalloy; bcc bulk together with short-range order (SRO) clus-\nters (see Table S3); bulk together with bct-like distorted clus-\nters inside (see Table S4); surface calculations, in the pristine\nmode and with explicit bct-like clusters embedded (surface +\ndistortion). The maximum-minimum ratio according to Ref.\n[22] isα[110]\nt\nα[010]\nt×100%.\nStructure ∆αtMax-min ratio\nOrdered alloy bcc 7.2% 107.2%\nOrdered alloy bct 24.1% 124.1%\nRandom alloy bcc 13% 113%\nRandom alloy bct 22.6% 122.6%\nRandom alloy + SRO 14.9% 114.9%\nRandom alloy + SRO + Distortion 47.2% 147.2%\nSurface (external layer) 102.4% 202.4%\nSurface (ext. layer) + Distortion 75.4% 175.4%\n10-nm Co 50Fe50/Pt [22] (exp.) 281.3% 381.3%\n−2−1 0 1 2\n−0.02 −0.01  0  0.01  0.02Δn(E)[010]−[110]  (states/Ry−atom)\nEnergy (E−EF) (Ry)Fe(001) bcc (first)\nVCA Fe50Co50(001) bcc (first)\nVCA Fe50Co50(001) bcc (all)\nFigure S2. LDOS difference (per atom), ∆n(E), between the\n[010]and[110]magnetization directions, for both spin chan-\nnels (full lines for majority spin and dashed lines for minority\nspin states), in the outermost layer in pure Fe(001) bcc (in\nblack); outermost layer in VCA Fe 50Co50(001) bcc (in blue);\nand all layers summed in VCA Fe 50Co50(001) bcc (in red).\nlevel,N(EF), what the authors claim that could not ex-\nplain the giant maximum-minimum damping ratio ob-\nserved. So, in order to clarify this effect in the VCA\nFe50Co50(001) bcc, ∆n(E)was also calculated for the all\nlayers summed, what is shown in Fig. S2 (in red). This\ndifference is in fact smaller if we consider the DOS of\nthe whole system, with all layers summed. However, if\nwe consider only the most external layer, then the LDOS\nvariation is enhanced. This is consistent with our theo-\nretical conclusions. As we mention in the main text, this\ndo not rule out a role also played by local (tetragonal-\nlike) distortions and other bulk-like factors in the damp-\ning anisotropy.For the outermost layer of Fe(001) bcc, the calculated\nLDOSatEFis∼20.42states/Ry-atominthe[110]direc-\ntion and∼20.48states/Ry-atom in the [010] direction,\nwhich represents a difference of ∼0.3%and agrees with\nthe calculations performed by Chen et al.[38].\nV. Correlation with anisotropic orbital moment\nBesides the close relation exhibited between ∆αt\nand∆n(EF), we also demonstrate the existence of an\nanisotropic orbital moment in the outermost layer, in\nwhich the fourfold symmetry ( C4v) matches the damp-\ning anisotropy with a 90◦phase. Fig. S3 shows this\ncorrelation between ∆αtand∆morbfor two situations:\n(i) for a typical atom in the outermost layer of VCA\nFe50Co50(001) bcc (blue open dots); and ( ii) for a typi-\ncal atom in the VCA Fe50Co50bcc bulk, considering the\nsame ∆morbscale. For case ( i) we find orbital moments\ndifferencesmorethanoneorderofmagnitudehigherthan\ncase (ii).\n0o45o90o\n135o\n180o\n225o\n270o315o[100][110][010]\n[−110]\n0.51.52.5\nΔmorb\n(µB/atom × 10−3)\n0.0050.0100.015\nαt\nθH\nFigure S3. (Color online) Total damping and orbital moment\ndifference, ∆morbas a function of θH, the angle between the\nmagnetizationdirectionandthe [100]-axis. Squares: (redfull)\nVCA Fe 50Co50(001) bcc. Circles: (blue open) morbdifference\nbetweenθH= 90◦and the current angle for a typical atom in\nthe outermost layer of VCA Fe 50Co50(001) bcc; and (yellow\nfull) samemorbdifference but for a typical atom in the VCA\nFe50Co50bcc bulk (in the same scale). Lines are guides for\nthe eyes.\nVI. Contribution from next-nearest-neighbours\nFinally, we show in Fig. S4 the summation of all non-\nlocal damping contributions, αij, for a given normalized\ndistance in the outermost layer of VCA Fe50Co50(001)7\nbcc. As we can see, the next-nearest-neighbours from a\nreference site (normalized distanced\na= 1) have very dis-\ntinctαijcontributions to αtfor the two different mag-\nnetization directions ( [010]and[110]), playing an impor-\ntant role on the final damping anisotropy. We must note,\nhowever, that these neighbours in a (001)-oriented bcc\nsurface are localized in the same layer as the reference\nsite, most affected by the interfacial SOC. Same trend is\nobserved ford\na= 2, however less intense. This is con-\nsistent with our conclusions, about the relevance of the\noutermost layer on ∆αt.\n−3−2−1 0 1 2\n 0.5  1  1.5  2  2.5  3  3.5  4∑αij × 10−3\nNormalized distance[110]\n[010]\nFigure S4. 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Rev. 76, 743 (1949)."    },    {        "title": "1909.05315v2.Chaos_in_nanomagnet_via_feedback_current.pdf",        "content": "arXiv:1909.05315v2  [cond-mat.mes-hall]  23 Nov 2019Chaos in nanomagnet via feedback current\nTomohiro Taniguchi1, Nozomi Akashi2, Hirofumi Notsu3,4,\nMasato Kimura3, Hiroshi Tsukahara5, and Kohei Nakajima2\n1National Institute of Advanced Industrial Science and Tech nology (AIST),\nSpintronics Research Center, Tsukuba, Ibaraki 305-8568, J apan,\n2Graduate School of Information Science and Technology,\nThe University of Tokyo, Bunkyo-ku, 113-8656 Tokyo, Japan,\n3Faculty of Mathematics and Physics, Kanazawa University, K anazawa, Ishikawa 920-1164, Japan,\n4JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Ja pan,\n5High Energy Accelerator Research Organization (KEK), Tsuk uba, Ibaraki 305-0801, Japan\n(Dated: November 26, 2019)\nNonlinear magnetization dynamics excited by spin-transfe r effect with feedback current is studied\nbothnumericallyandanalytically. Thenumericalsimulati onoftheLandau-Lifshitz-Gilbertequation\nindicates the positive Lyapunov exponent for a certain rang e of the feedback rate, which identifies\nthe existence of chaos in a nanostructured ferromagnet. Tra nsient behavior from chaotic to steady\noscillation is also observed in another range of the feedbac k parameter. An analytical theory is\nalso developed, which indicates the appearance of multiple attractors in a phase space due to the\nfeedback current. An instantaneous imbalance between the s pin-transfer torque and damping torque\ncauses a transition between the attractors, and results in t he complex dynamics.\nPACS numbers:\nI. INTRODUCTION\nNonlinear dynamics can be found in a wide variety of\nphysical, chemical, and biological systems from small to\nlarge scale [1,2]. Recent observations of rich magnetiza-\ntion dynamics, such as switching, auto-oscillation (limit\ncycle), and synchronization, excited in a nanostructured\nferromagnet have also proved the applicability of nonlin-\near science to a fine structure [3–12]. These dynamics are\ndriven by spin current carried by, for example, conduct-\ning electrons in metals [13–15]. Since the spin current\nin metals can survive only within nanometer scale [16],\nthese magnetization dynamics had not been observedun-\ntil the development of fabrication technology of nanos-\ntructure was achieved. A new direction investigating the\napplicability of such nonlinear magnetization dynamics\nto non-von Neumann computing scheme, inspired by hu-\nman brain, has been growing very recently [17–20].\nAn attractive and intriguing phenomenon in nonlinear\nscience is chaos [21,22]. It should be noticed here that\nthe previous works in magnetism and spintronics have\nclarified that the magnetization dynamics in a nanos-\ntructured ferromagnet is sufficiently sufficiently well de-\nscribed by two dynamical variables [23–28]. For exam-\nple, the macrospin model has two dynamical variables\ndescribing the zenith and azimuth angles of the mag-\nnetization. The Thiele equation depicting the magnetic\nvortex or skyrmion dynamics includes two variables cor-\nresponding to the radius and phase of the core, whereas\nthe domain wall motion is represented by the center\nof the wall position and the tilted angle of the magne-\ntization at the center. On the other hand, according\nto the Poincar´ e-Bendixson theorem, chaos is prohibited\nin a two-dimensional system [21]. Therefore, an addi-\ntional degree of freedom is necessary to induce chaosin ferromagnets. In previous works, chaos has been\nstudied for systems with alternative current [29,30] or\nmagnetic and/or electric interaction between two ferro-\nmagnets [31,32]. The former makes the system nonau-\ntonomous, whereas the latter utilizes many-body system.\nAnother possible sourcecausing highly nonlinear dynam-\nics is feedback force with delay because the presence of\nthe delay makes the dimension of the system infinite [33].\nRecently, the modulation of the threshold current by the\nself-injection of the feedback current into the vortex fer-\nromagnet was theoretically predicted [34] and was ex-\nperimentally confirmed [35]. Complex dynamics in an\nin-plane magnetized ferromagnet with feedback current\nwas also found by numerical simulation [36]. However,\nit should be emphasized that the existence of the feed-\nback effect does not necessarily guarantee chaos. There-\nfore, a careful analysis is necessary for the magnetization\ndynamics in the presence of feedback effect in order to\nidentify chaos.\nThe purpose of this work is to develop a theoretical\nanalysis of the nonlinear magnetization dynamics in a\nnanostructured ferromagnet in the presence of feedback\ncurrent. We perform the numerical simulation of the\nLandau-Lifshitz-Gilbert (LLG) equation in spin torque\noscillator (STO), and find that the feedback current\ncauses highly nonlinear dynamics of the magnetization.\nThis work identifies chaos by the positive Lyapunov ex-\nponent, which is found in a certain range of the feedback\nrate, whereas transient behavior is also observed in an-\nother range of the feedback rate. We also develop an\nanalytical theory to reveal the origin of such complex\ndynamics. The bifurcation analysis indicates that the\nfeedback current results in the appearance ofmultiple at-\ntractors in the phase space. An instantaneous imbalance\nbetween the spin-transfer torque and damping torque al-\nlows a transition between these attractors, and induces2\npm\nIz\nxχIm.p\ntime (ns)0 20 40 60 80 1001.0\n0.5\n-0.5\n-1.00mx, mz\nmxmz\n1ns01\n-1(a) (b)\nFIG. 1: (a) Schematic view of the system. The direct current\nIis injected from the reference layer to the positive layer,\nwhereas the current, χIm·p, outputted from the STO is\ninjectedintotheSTOwithtimedelay τ. Thefeedbackcurrent\noscillates when the magnetization min the free layer is in a\ndynamical state. (b) Typical magnetization dynamics in the\nabsence of the feedback current. The inset shows an auto-\noscillation in a steady state.\nthe complex dynamics found in the numerical analysis.\nThe paper is organized as follow. In Sec. II, we de-\nscribe the structure of the STO and show the LLG equa-\ntion including feedback current. In Sec. III, the results\nof the numerical simulation of the LLG equation are pre-\nsented. The Lyapunov exponents and bifurcation dia-\ngrams as functions of the feedback rate and delay time\nare also presented. In Sec. IV, a theoretical analysis on a\nmultiple attractor is discussed. Section IVA summarizes\nthis work.\nII. SYSTEM DESCRIPTION\nIn this section, we describe the system under consider-\nation, and provide the comment on the numerical meth-\nods. The details of the algorithms are also given in the\nSupplemental Material [37] (which includes Ref. [38]).\nA. LLG equation\nThesystemunderconsiderationisschematicallyshown\nin Fig. 1(a). The unit vectors pointing in the magnetiza-\ntion directions in free and reference layers are denoted as\nmandp, respectively. Direct current, I, is injected from\nthe reference to free layer, and excites an auto-oscillation\nof the magnetization mvia spin-transfer effect [13,14].\nHere, we focus on the STO consisting of a perpendicu-\nlarly magnetized free layer and an in-plane magnetized\nreference layer because this type of STO can emit large\nemissionpowerwith narrowlinewidth [10], andtherefore,\nis of great interest from viewpoints of both fundamental\nand applied physics. The magnetization pin the refer-\nence layer points to the positive xdirection, whereas the\nzaxis is perpendicular to the film-plane. The magne-\ntization dynamics in the free layer is described by the\nLandau-Lifshitz-Gilbert (LLG) equation given by\ndm\ndt=−γm×H−γHsm×(p×m)+αm×dm\ndt(1)whereγandαare the gyromagnetic ratio and the\nGilbert damping constant respectively. The magnetic\nfieldH= [Happl+ (HK−4πM)mz]ezconsists of an\napplied field Happl, interfacial magnetic anisotropy field\nHK[39–41], and demagnetization field −4πM. The spin-\ntransfer torque strength, Hsis given by\nHs=/planckover2pi1ηI[1+χm(t−τ)·p]\n2e(1+λm·p)MV, (2)\nwhereMandVarethesaturationmagnetizationandthe\nvolume of the free layer, respectively. The spin-transfer\ntorque strength is characterized by the spin polarization\nηand spin-transfer torque asymmetry λ. The values of\nthe parameters used in this work are derived from the\nexperiment [10], as well as a theoretical analysis [42] as\nM= 1448.3 emu/c.c., HK= 18.616 kOe, Happl= 2.0\nkOe,V=π×602×2 nm3,η= 0.537,λ= 0.288,\nγ= 1.764×107rad/(Oe s), and α= 0.005. The cur-\nrent ofI= 1.0 mA corresponds to the current density of\n8.8 MA/cm2. An auto-oscillation in the absence of the\nfeedback is excited in this type of STO when the cur-\nrent magnitude becomes larger than a threshold value\n[42] (see also Appendix A for derivation),\nIc=4αeMV\n/planckover2pi1ηλ(Happl+HK−4πM),(3)\nwhich is about 1 .6 mA for the present parameters. Fig-\nure 1(b) shows a typical magnetization dynamics in the\nabsence of the feedback current, where the direct current\nisI= 2.5 mA. As shown, an auto-oscillation having a\nperiod of 0 .16 ns is excited after a relaxation time on the\norderof10ns. The inset ofFig. 1(b) showsthe dynamics\nofmx(red) and mz(black) in a steady state. It can be\nseen from the figure that mzis almost temporally con-\nstantbut slightlyoscillatesarounda certainvalue. These\nresults will be used for comparison with the dynamics in\nthe presence of the feedback current, as well as for the\ndevelopment of an analytical theory, below.\nB. Description of feedback effect\nThe strength of the spin-transfer torque, Eq. (2), in-\ncludesthe feedback currentgivenby χIm(t−τ)·p, where\nχis the rate of the feedback current with respect to\nthe direct current I, whereas τis the delay time. Due\nto tunnel magnetoresistance effect, the feedback current\ndepends on the relative direction of the magnetizations,\nm·p[10]. The feedback current brings the past infor-\nmation of the magnetization state, and extends the di-\nmension of the phase space, which presents a possibility\nto excite chaos in STO.\nLet us give brief comments on experiment to measure\nchaos in an STO. An experimental work injecting the\nfeedback current to a vortex STO and measuring the\noutput power was already reported [35]. The feedback\ncurrent can be injected to the STO independently from3\nthe direct currentby usinga bias-Teeanddelayline. The\nnumerical analyses shown below, as well as the analytical\ntheory developed in Sec. IV, predict that chaos appears\nfor a large feedback rate χand/or long delay time τcom-\nparedtotypicaltime scalesofthe STO.The typicalvalue\nofthe delay time possible in experiment is on the orderof\n10 ns [35]. On the other hand, the oscillation period ( ∼3\nns) of the vortex STO used in the previous work [35] is\nonly 10 times shorter than the delay time. This might be\nthe reason why chaos was not confirmed in the previous\nworks. Regarding this point, two approaches are taken\ninto account to observe chaos in STO. The first one is to\nmake the delay time long. A long delay time is realized\nby using a long electric cable. The second approach is\nto use an STO having a short oscillation period. In fact,\nthe STO studied in this work has a short period because\nof macrospin structure of the magnetization. Therefore,\nthe theoretical analyses shown below will possibly be ex-\namined experimentally. A possible remaining issue, how-\never, may be an energy loss in a cable, which should be\noptimized in experiments.\nWe also give a comment on the method to identify\nchaos by experiments. The experimental methods to\nidentifychaosare,forexample, theestimationoftheLya-\npunov exponent from time series of data and/or Fourier\nanalysis. The former method requires to measure the dy-\nnamical trajectory in the system, and estimate the Lya-\npunov exponent from a discrete set of time series data\nby evaluating the principal axis of the expansion [43]. A\npossible problem in applying this method to STO is the\nlimitation of the information on the dynamical trajec-\ntory obtained. The magnetization dynamics in the STO\nis measured through the magnetoresistance effect. Both\ngiant and tunnel magnetoresistances are proportional to\nm·p. Therefore, we can measure only the component\nof the magnetization mprojected to the direction of p.\nThis fact might makeit difficult to reproducethe dynam-\nical trajectory and identify chaos from the time series of\ndata. The Fourier analysis, on the other hand, indicates\nchaos from the shape of the spectrum. The Fourier spec-\ntrum shows a sharp peak for a non-chaotic dynamics,\nwhereas it has a broad structure without a unique peak\nin chaos state; see also Sec. IIIA. Therefore, the Fourier\nspectrum provides an evidence to identify chaos.\nC. Numerical method\nHere, let us provide a brief description of the numer-\nical technique used in the next section. The LLG equa-\ntion, Eq. (1), with the feedback current is solved by\na fourth-order Runge-Kutta scheme accompanied with\ncontinuation method. The details of this algorithm are\nsummarized in the Supplemental Material [37]. We also\nevaluate the bifurcation diagram, which is defined as the\nlocalmaximumof mz(t)afterthemagnetizationmovesto\nan attractor. In this work, chaosis defined as the dynam-\nics with the positive Lyapunov exponent. The Lyapunovexponent in this work is defined as an average of the in-\nstantaneous expansion rate of the dynamical trajectory\nin the phase space with respect to a small perturbation\nǫas\n˜λ= lim\nNL→∞1\nNL∆tNL/summationdisplay\ni=1log/vextendsingle/vextendsingle/vextendsingle/vextendsingle˜ǫi\nǫ/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (4)\nwhere ∆ tis the time step of the LLG simulation. The\nnumber of the perturbation applied to the STO is NL,\nwhereas ˜ ǫiis the expansion of the dynamical trajectory\nwith respect to the ith perturbation. The detail of the\nalgorithmtoevaluatetheLyapunovexponentisalsosum-\nmarized in the Supplemental Material [37].\nIII. NUMERICAL ANALYSIS\nIn this section, we show the results of the numerical\nsimulation of the LLG equation, as well as the Lyapunov\nexponent and bifurcation diagram.\nA. Lyapunov exponent as a function of feedback\nrate\nHere, we show the Lyapunov exponent and the bir-\nfurcation diagram as a function of the feedback rate χ.\nThe value of τin this section is set to be 30 ns. Figures\n2(a)-2(c) show the time evolutions of mz(t) forχ= 0.02,\nχ= 0.50, andχ= 0.89, respectively. Note that the time\nrange of each figure is different to understand the char-\nacteristics of each dynamics. In the presence of a small\nfeedback current shown in Fig. 2(a), although the ampli-\ntude of the oscillation is modulated, the dynamics in the\nsteadystateisstillperiodic. Ontheotherhand, whenthe\nfeedback rate becomes relatively large, chaotic behavior\nappears, as shownin Fig. 2(b). In this case, non-periodic\nand highly nonlinear dynamics appears over a wide time\nrange. The value of mzoscillates almost over its possi-\nble value, |mz| ≤1. A further increase of the feedback\nrate leads to a transition of the magnetization dynam-\nics from chaotic to non-chaotic, as shown in Fig. 2(c).\nThe chaotic dynamics suddenly disappears after a com-\nparatively long period, i.e., longer than the oscillation\nperiod of the limit cycle in the absence of the feedback\ncurrent. As mentioned below, the Lyapunov exponents\nof the dynamics in Figs. 2(a) and 2(c) are zero, whereas\nit is positive for the dynamics in Fig. 2(b).\nNote that evaluating the perpendicular component mz\nin time domain is useful for theoretical analysis because\nit is approximately constant in the absence of the feed-\nback effect, whereas it becomes complex by the feedback\nforce, as mentioned above. On the other hand, evalu-\nating the in-plane component mxwill be useful for ex-\nperiments because it directly relates to the experimen-\ntally observed signal through magnetoresistance effect.\nTherefore, we also show the Fourier spectra of mxfor4\ntime (ms)0 0.5 1.0 1.51.0\n0.5\n0(c)mz\nχ=0.89\ntime (μs)0 0.5 1.0 1.51.0\n0.5\n0(b)mz\nχ=0.50\ntime (ns)0 200 100 300 400 500 6001.0\n0.5\n0(a)\n(d)mz\nχ=0.02|mx(f)| (arb. unit)\n050100150\nfrequency (GHz)6.0 6.1 6.2 6.3 6.4 6.5 6.6(e)\n|mx(f)| (arb. unit)\n050100\nfrequency (GHz)5.6 5.8 6.0 6.2 6.4 6.6(f)\n|mx(f)| (arb. unit)\n0600\n500\n400\n300\n200\n100\nfrequency (GHz)5.6 5.9\nFIG. 2: Time evolutions of the perpendicular component mz(t) for the feedback rates of (a) χ= 0.02, (b) 0 .50, and (c) 0 .89.\nThe current and the delay time are I= 2.5 mA and τ= 30 ns. Note that the time range of each figure is different. Fou rier\nspectra of the in-plane component mx(t) for (d) χ= 0.02, (e) 0 .50, and (f) 0 .89 are also shown.\nχ= 0.02, 0.50, and 0 .89 in Figs. 2(d)-2(f), respectively.\nTheFourierspectrumhasasharppeakwith subpeaksfor\nχ= 0.02, which is a typical spectrum of the oscillation\nwith the amplitude modulation. The Fourier spectrum\nforχ= 0.50, on the other hand, shows a broad structure\nover a relatively wide range of the frequency. A main\npeak is not uniquely determined. The structure implies\nthat the dynamics is chaos. The Fourier spectrum for\nχ= 0.89 shows a sharp peak, corresponding to the os-\ncillation frequency after the transition from chaotic to\nlimit cycle oscillation. The oscillation frequency is dif-\nferent from that in the absence of the feedback because\nthe oscillation amplitude is modified due to the feedback\neffect. Regarding these results, the Fourier analysis will\nbe a possible tool to experimentally identify chaos.\nFigures 3(a) and 3(b) show the Lyapunov exponent\nand the bifurcation diagram as a function of the feed-\nback rate in a small range χ≤0.10. The Lyapunov\nexponent remains zero for χ/lessorsimilar0.024, where the dynam-\nics is a limit cycle, such as shown in Fig. 1(b), or the\noscillation with an amplitude modulation as shown in\nFig. 2(a). In the limit cycle state, the local maximum\nofmzis a single value, whereas it takes several values\nand shows symmetric distributions around its center in\nthe modulated dynamics, as can be seen in Fig. 3(b).\nThe Lyapunov exponent becomes positive for χ/greaterorsimilar0.025,\nwhere the bifurcation diagram shows an inhomogeneous\n(asymmetric) structure. The Lyapunov exponent and\nthe bifurcation diagram for a wide range of the feed-\nback rate, χ≤1.00, are shown in Figs. 3(c) and 3(d),\nrespectively. The positive Lyapunov exponent indicates\nthe existence of chaos in STO. The Lyapunov exponent\nmz\n00.2\n0.10.30.40.50.60.70.80.91.0\n0 0.2 0.1 0.3 0.4 0.5 0.7 0.9 0.6 0.8 1.0 0 0.2 0.1 0.3 0.4 0.5 0.7 0.9 0.6 0.8 1.00.06\n0.05\n0.04\n0.03\n0.02\n0.01\n0\n-0.01(c) (d)Lyapunov exponent (1/ns)\nmz\n0.40.50.60.70.8\nfeedback rate, χ\nfeedback rate, χ0 0.020.01 0.03 0.04 0.05 007 0.09 0.06 0.08 0.1\nfeedback rate, χ\nfeedback rate, χ0 0.020.01 0.03 0.04 0.05 007 0.09 0.06 0.08 0.10\n-0.010.010.020.030.04(a) (b)Lyapunov exponent (1/ns)\nFIG. 3: (a) (Maximum) Lyapunov exponent and (b) bi-\nfurcation cascade (local maximum of mz) as a function of\nthe feedback rate χ≤0.10. The current and delay time are\nI= 2.5 mA and τ= 30 ns, respectively. The range of χis\nextended to χ≤1.00 in (c) and (d).\nbecomes zero again when the feedback rate is further in-\ncreased to χ≃0.87. The magnetization dynamics shown\nin Fig. 2(c), corresponding to this parameter region, can\nbe regarded as transient chaos, which can be found in,\nfor example, a spatially extended turbulence model [44],\nwhere the dynamical system finally arrives at an attrac-\ntor with zero or negative Lyapunov exponent long time\nafter showing chaotic behavior [21]. For example, the\ntransient time observed in Fig. 2(c) is on the order of5\n0.1 ms, which is sufficiently longer than the period of the\nauto-oscillation in the absence of the feedback current\n(0.16 ns) but is measurable because it is shorter than the\nexperimentally available time range for STO dynamics\nreported up to date, 1.6 ms [45].\nB. Lyapunov exponent as a function of delay time\nHere, we show the Lyapunov exponent and the birfur-\ncation diagram as a function of the delay time τ. The\nvalue of χin this section is set to be 0 .20. Figures 4(a)\nand 4(b) show the time evolutions of mzfor short delay\ntimes,τ= 0.03 and 0.3 ns, respectively. For such a suffi-\nciently short delay time, the current necessary to excite\nan auto-oscillation of the magnetization is given by (see\nalso Appendix A for derivation)\n˜Ic=4αeMV\n/planckover2pi1ηλp0(Happl+HK−4πM),(5)\nwherep0=p(χ,τ,θ= 0) is\np0= 1−χ\nλcos2πfFMRτ, (6)\nwherefFMR=γ(Happl+HK−4πM)/(2π) is the ferro-\nmagnetic resonance (FMR) frequency. In the absence of\nthe feedback current ( χ→0), Eq. (5) becomes identical\nto Eq. (3). According to Eqs. (5) and (6), the threshold\ncurrent to move the magnetization from the energetically\nstable state ( θ= 0) is anoscillatingfunction of τ. Forex-\nample,Icgiven by Eq. (5) becomes 1 .9 mA for τ= 0.03\nns, which is smaller than the applied current, I= 2.5\nmA. Therefore, the magnetization can move from the\ninitial state, as shown in Fig. 4(a). On the other hand,\nIcbecomes 4 .4 mA for τ= 0.3 ns, and, therefore, the\nmagnetization stays in the energetically stable state in\nFig. 4(b). Such a modification of the instability thresh-\nold was studied in a vortex oscillator both theoretically\nand experimentally [34,35]. For a sufficiently long delay\ntime, the magnetization dynamics becomes highly com-\nplex, and Eq. (5) does not work. The periodic oscillation\nwith the amplitude modulation is found for τ= 9.3 ns,\nwhereas non-periodic dynamics appears for τ= 9.6 ns,\nas shown in Figs. 4(c) and 4(d).\nFigures 4(e) and 4(f) summarize the Lyapunov expo-\nnent and bifurcation diagram as a function of the delay\ntime, respectively. Note that the magnetization stays in\nthe energetically equilibrium state for 0 .3≤τ <1.2 ns,\nas in the case shown in Fig. 4(b). In such a case, the\nLyapunov exponent is negative, indicating that the mag-\nnetization saturates to a fixed point. On the other hand,\nchaos appears with increasing the delay time, whereas\nthe periodic oscillations with the amplitude modulation\nappear for specific values of τ. The negative Lyapunov\nexponent for a short delay time is approximately esti-\nmated from a linearized LLG equation [46] as\n˜λ≃ −2παfFMR/parenleftbigg\n1−I\n˜Ic/parenrightbigg\n. (7)time ( μs)0 100 50 150 200 250 300\ntime ( μs)0 100 50 150 200 250 3001.0\n0.5\n0(c)mz\nτ=9.3 nsτ=0.03 ns\nτ=9.6 nsτ=0.3 ns\n1.0\n0.5\n0(d)mz mztime (ns)0 0.5 1.0 1.51.0\n0.5\n0(a)mz\ntime (ns)0 40 20 60 80 1001.0\n0.5\n0(b)mz\n00.2\n0.10.30.40.50.60.70.80.91.0\ndelay time, τ (ns)0 5 10 15 20 25 30\ndelay time, τ (ns)0 5 10 15 20 25 300.10\n0.05\n0\n-0.15-0.10-0.05(e) (f)Lyapunov exponent (1/ns)\nFIG. 4: Time evolutions of mz(t) for the delay times of (a)\nτ= 0.03, (b) 0 .3, (c) 9.3 ns, and (d) 9 .6 ns. The current\nand the feedback rate are I= 2.5 mA and χ= 0.20. (e)\nThe Lyapunov exponent and (f) bifurcation cascade (local\nmaximum of mz) as a function of the delay time.\nFor example, for τ= 0.3, Eq. (7) is −0.09 GHz, which\nis close to the numerically estimated value, −0.11 GHz.\nWe simultaneously emphasize that the limit of τ→0\ndoes not correspond to the zero-feedback limit (the zero-\nfeedback limit corresponds to χ→0). Even in the limit\nofτ→0, the feedback current exists and affects the dy-\nnamics. For example, for τ= 0.03, the magnetization\nshows a limit cycle oscillation, and the Lyapunov expo-\nnent is zero. Equation (7) works when the magnetization\nstays at a fixed point, and the delay time τis short.\nIV. THEORETICAL ANALYSIS\nThe above numerical results indicate the existence of\nrich variety of nonlinear dynamics, including chaos, in\nan STO. Although it is difficult to solve the LLG equa-\ntion exactly due to its nonlinearity, let us investigate the\nphysical origin of the complex dynamics with help of an\napproximated theory, which has been known to be useful\nto analyze nonlinear dynamics such as auto-oscillation\n(limit cycle) [28,42] and synchronization [47]. An auto-\noscillation in an STO is excited when the spin-transfer\ntorque balances with the damping torque, and the field\ntorque,−γm×H, remains finite. The field torque leads\ntoasustainableoscillationofthe magnetizationonacon-\nstantenergycurveofthe magneticenergydensitydefined6\nasE=−M/integraltext\ndm·H. In the present system, the con-\nstant energy curve corresponds to the trajectory with\na constant zenith angle θ= cos−1mz, where the oscil-\nlation frequency, f(θ), on the constant energy curve is\nf(θ) =γ[Happl+(HK−4πM)cosθ]/(2π). It should be,\nhowever, emphasized that there is often an instantaneous\nimbalance between the spin-transfer torque and damping\ntorque because of their different angular dependencies.\nTherefore, strictly speaking, θ(ormz) in the present\nsystem is not a constant variable [42]; see also the inset\nof Fig. 1(b). However, for a sufficiently small damp-\ning constant α, the real trajectory of the auto-oscillation\nis practically close to a constant energy curve. In such\na case, it is useful to derive the equation of motion of\nθaveraged over the precession period T(θ) = 1/f(θ)\nasdθ/dt≡(1/T)/contintegraltext\ndt(dθ/dt) (see also Appendix A for\nderivation),\ndθ\ndt=−αγ[Happl+(HK−4πM)cosθ]sinθ\n+γHs0\nλtanθ/parenleftBigg\n1/radicalbig\n1−λ2sin2θ−1/parenrightBigg\np(χ,τ,θ),(8)\nwhereHs0=/planckover2pi1ηI/(2eMV), whereas p(χ,τ,θ) is given by\np(χ,τ,θ) = 1−χ\nλcos2πf(θ)τ. (9)\nThe angle θsatisfying dθ/dt= 0 and d(dθ/dt)/dθ <\n(>)0 corresponds to a stable (unstable) fixed point in\nthe reduced phase space [1]. In the absence of feedback\ncurrent, there is only one stable fixed point (attractor),\ncorresponding to auto-oscillation state in real space, in\nthe present STO [42]. On the other hand, Fig. 5(a)\nshows an example of dθ/dtin the presence of the feed-\nback. As shown, several attractors satisfying dθ/dt= 0\nandd(dθ/dt)/dθ <0 appear due to the feedback current.\nFigures 5(b) and 5(c) show the attractors mz= cosθas\na function of the feedback rate χand the delay time τ,\nrespectively. It canbe understoodfromthese figuresthat\nthe number of the attractor increases with increasing the\nfeedback rate and/or delay time. Let us here call such\nstructures as multiple attractors. Although these results\nare obtained with an approximation mentioned above,\nthey are useful to understand the origin of the complex\nmagnetization dynamics found by numerical simulation,\nas discussed below.\nThe multiple attractors originate from the function\np(χ,τ,θ) givenby Eq. (9). In the absenceofthe feedback\ncurrent ( χ= 0), the function p(χ,τ,θ) = 1 is indepen-\ndent of the angle θ. On the other hand, in the presence\nof the feedback current ( χ/negationslash= 0), several values of the\nangleθgive an identical value of p(χ,τ,θ) because the\nfunction includes a periodic (cosine) function depending\nonθ. As a result, several θcan simultaneously satisfy\nthe conditions of the stable fixed point.\nThe origin of the complex dynamics found in the nu-\nmerical simulation is considered to be the existence ofmultipleattractors. Sincetheattractorslocatediscretely,\nas shown in Fig. 5, one might consider that once the\nmagnetization is trapped by one of the attractors, it can-\nnot move to the others. It should be, however, reminded\nthat the assumption of a constant angle θwasused in the\nderivation of Eq. (8). As emphasized above, the real an-\ngleθ= cos−1mzin alimit cycleslightlyoscillatesaround\nthe fixed point estimated analytically by Eq. (8) because\nof the instantaneous imbalance between the spin-transfer\ntorque and damping torque. As a result, the magnetiza-\ntion can move from one attractor to the other when the\ndistance between the attractors is smaller than the oscil-\nlation amplitude of the angle θ. The transition between\nthe attractorscausesthe highlycomplexdynamicsshown\nin Fig. 2, contrary to the system without feedback in\nwhich an auto-oscillation state is uniquely determined.\nIt is considered that the above analytical theory can\nbe applied to any type of STO, although Eq. (8) was\nderived for its specific type. For example, the complex\ndynamics found in an in-plane magnetized STO [36] may\nbe causedby the samemechanism, i.e., the appearanceof\nmultiple attractors due to the existence of feedback cur-\nrent. The periodicity of the multiple attractors in this\ntype of STO is described by elliptic functions in contrast\nwith Eq. (9) where the periodicity is described by a sim-\nple trigonometric function; see Appendix B.\nA. Conclusion\nIn conclusion, the nonlinear magnetization dynamics\nin a spin-torque oscillator was studied by taking into ac-\ncount the effect of spin-transfer torque excited by the\nfeedback current. The numerical simulation reveals rich\nvariety of the nonlinear magnetization dynamics, which\ncan be controlled by the feedback parameter. The posi-\ntive Lyapunov exponent for a certain range of the feed-\nback rate indicated the existence of chaos in the spin-\ntorque oscillator, whereas transient behavior from the\nchaotic to the steady state was also observed in another\nrange of the feedback parameter. The analytical the-\nory based on the averaged equation of motion revealed\nthat the feedback current results in the multiple attrac-\ntors in the phase space. The number of the attractors\nincreased with increasing the feedback rate and/or de-\nlay time. An instantaneous imbalance between the spin-\ntransfer torque and damping torque caused a transition\nbetween the attractors, and induces the complex magne-\ntization dynamics.\nAcknowledgement\nThe authors are thankful to Joo-Von Kim, Take-\nhiko Yorozu, Sumito Tsunegi, and Shinji Miwa for\nvaluable discussion. T. T. is grateful to Satoshi\nIba, Aurelie Spiesser, Hiroki Maehara, and Ai Emura\nfor their support and encouragement. The results7mz\n00.2\n0.10.30.40.50.60.70.80.91.0\nfeedback rate, χ0 0.20.1 0.3 0.4 0.5 0.7 0.9 0.6 0.8 1.0angle, θ (deg)-0.2-0.10.2\n00.1\n0 10 20 30 40 50 60 70 80 90(a) (b)dθ/dt (1/ns)(c)mz\n00.2\n0.10.30.40.50.60.70.80.91.0\ndelay time, τ (ns)0 5 10 15 20 25 30\nFIG. 5: (a) The averaged dθ/dtgiven by Eq. (8) solved in the phase space as a function of θ= cos−1mz. The current,\nfeedback rate, and delay time are I= 2.5 mA,χ= 0.10, and τ= 30 ns. (b), (c) Stable fixed points mz= cosθestimated\nanalytically as a function of (b) the feedback rate χwithτ= 30 ns and (c) the delay time τwithχ= 0.10.\nwere partially obtained from a project (Innovative AI\nChips and Next-Generation Computing Technology De-\nvelopment/(2) Development of next-generation comput-\ning technologies/Exploration of Neuromorphic Dynam-\nics towards Future Symbiotic Society) commissioned by\nNEDO. K. N. is supported by JSPS KAKENHI Grant\nNumbers JP18H05472, and JP16KT0019. H. N. is sup-\nported by JSPS KAKENHI Grant Number JP18H01135,\nand JST PRESTO Grant Number JPMJPR16EA. M.\nK. is supported by JSPS KAKENHI Grant Numbers\nJP16H02155, JP17H02857.\nAppendix A: Averaged LLG equation of\nperpendicularly magnetized STO\nIntroducing the zenith and azimuth angles ( θ,ϕ) as\nm= (sinθcosϕ,sinθsinϕ,cosθ), the LLG equation (1),\nforθis given by\ndθ\ndt=−γ/planckover2pi1ηI[1+χm(t−τ)·p]\n2e(1+λsinθcosϕ)MVcosθcosϕ\n−αγ[Happl+(HK−4πM)cosθ]sinθ,(A1)\nwhere the higher order terms of αare neglected. As\nmentioned in the main text, an auto-oscillationis excited\nwith a trajectory depicting practically on a constant en-\nergy curve of E=−M/integraltext\ndm·H=−MHapplcosθ−\n[M(HK−4πM)/2]cos2θ. The dynamical trajectory\non the constant energy curve, which is the solution of\ndm/dt=−γm×H, is given by mx= sinθcosω(θ)t,\nmy= sinθsinω(θ)t, andmz= cosθ, whereθis constant\nwhereas\nω(θ) =γ[Happl+(HK−4πM)cosθ].(A2)\nThe frequency and period of the auto-oscillation are\nf(θ) =ω(θ)/(2π) andT(θ) = 1/f(θ), respectively. Sub-\nstituting these solutions, mx,my, andmz, into Eq. (A1),we find that\n1\nT(θ)/contintegraldisplay\ndtdθ\ndt\n=−γ/planckover2pi1ηI\n2eMVT(θ)/integraldisplayT(θ)\n0dt[1+χsinθcosω(t−τ)]cosθcosωt\n1+λsinθcosωt\n−αγ\nT(θ)/integraldisplayT(θ)\n0dt[Happl+(HK−4πM)]sinθ.\n(A3)\nUsing the integral formulas, we find that\n1\nT(θ)/contintegraldisplay\ndtdθ\ndt=γ/planckover2pi1ηI\n2eλMVtanθ/parenleftBigg\n1/radicalbig\n1−λ2sin2θ−1/parenrightBigg\np(χ,τ,θ)\n−αγ[Happl+(HK−4πM)cosθ]sinθ,\n(A4)\nwherep(χ,τ,θ) isgivenbyEq.(9). Equation(A4)isiden-\ntical to Eq. (8). The threshold current given by Eq. (5)\nis the current satisfying lim θ→0dθ/dt= 0, whereas Eq.\n(3) is Eq. (5) in the limit of χ→0.\nAppendix B: Averaged LLG equation of in-plane\nmagnetized STO\nIn the main text, the multiple attractors are investi-\ngated for an STO consisting of a perpendicularly mag-\nnetized free layer and an in-plane magnetized reference\nlayer. On the other hand, previous works had focused on\nan STO consisting of in-plane magnetized free and refer-\nence layers[29–31,36]. Therefore, let us show that the in-\nplane magnetized STOalso showsthe multiple attractors\nstructurewhen thespin-transfertorqueincludes thefeed-\nback current. In this Appendix, the values of the param-\neters are derived from Refs. [47–49], The magnetic field\nandthestrengthofthespin-transfertorqueofanin-plane\nmagnetized STO are given by H=HKmyey−4πMmzez\nandHs=/planckover2pi1ηJ/(2eMd), respectively, where HK= 200Oe\nis an in-plane anisotropy field along the easy ( y) axis,J\nis the current density, and d= 2.0 nm is the thickness\nof the free layer. The saturation magnetization and the8\nGilbert damping constant are M= 1500 emu/c.c. and\n0.01, respectively. The spin polarization ηis 0.5, whereas\nthe spin-transfer torque asymmetry λis assumed to be\nzero, for simplicity. The spin-polarization direction pis\nparallel to the easy axis, p=ey.\n1. Energy range of in-plane auto-oscillation\nAs mentioned in the main text, the averaged LLG\nequation is derived by assuming an auto-oscillation on\na constant energy curve. The energy density of an in-\nplane magnetized ferromagnet is given by\nE=−MHK\n2m2\ny+4πM2\n2m2\nz. (B1)\nThe minimum, saddle, and maximum energy densities\nareEmin=−MHK/2,Es= 0, and Emax= 4πM2/2,\ncorresponding to the magnetization states of m=±ey,\n±ex, and±ez, respectively. Here, we focus on the auto-\noscillation around the easy axis, where the corresponding\nenergy density Eis in the range of Emin< E < E s. The\nauto-oscillation is excited when the current density is in\nthe range of Jc< J < J∗[47–49], where JcandJ∗are\nthe critical and switching current densities given by\nJc=2αeMd\n/planckover2pi1η(HK+2πM), (B2)\nJ∗=4αeMd\nπ/planckover2pi1η/radicalbig\n4πM(HK+4πM).(B3)\n2. Averaged LLG equation in the absence of\nfeedback current\nThe LLG equation averaged over the constant energy\ncurve of Ein the in-plane magnetized ferromagnet with-\nout the feedback current is given by [47]\n/contintegraldisplay\ndtdE\ndt=Ws+Wα, (B4)\nwhere WsandWαare the work done by the spin-transfer\ntorque and the energy dissipation by the damping torque\nduring a precession on a constant energy curve,\nWs=γM/contintegraldisplay\ndtHs[p·H−(m·p)(m·H)]\n= 2πMHs2E/M+HK/radicalbig\nHK(HK+4πM),(B5)\nWα=−αγM/contintegraldisplay\ndt/bracketleftBig\nH2−(m·H)2/bracketrightBig\n=−4αM/radicalBigg\n4πM−2E/M\nHK/bracketleftbigg2E\nMK(k)+HKE(k)/bracketrightbigg\n,\n(B6)E/(MHK/2)dE/(MHK/2)\n-1.0 -0.9 -0.8 -0.6 -0.7 -0.5 -0.4 -0.3 -0.2 -0.1 080\n40\n0\n-40\n-80\n-120(a)\nE/(MHK/2)dE/(MHK/2)\n-1.0 -0.9 -0.8 -0.6 -0.7 -0.5 -0.4 -0.3 -0.2 -0.1 080\n40\n0\n-40\n-80\n-120(b)\nFIG. 6: The averaged energy change, dE≡/contintegraltext\ndt(dE/dt), an\nin-plane magnetized ferromagnet as a function of the energy\ndensityE. The vertical and horizontal axes are renormalized\nbyMHK/2. The feedback current is (a) zero and (b) χ= 0.10\nwithτ= 3 ns.\nwhereK(k) =/integraltext1\n0dx//radicalbig\n(1−x2)(1−k2x2) andE(k) =/integraltext1\n0dx/radicalbig\n(1−k2x2)/(1−x2) are the first and second kind\nof complete elliptic integral with the modulus k:\nk=/radicalBigg\n4πM(HK+2E/M)\nHK(4πM−2E/M). (B7)\nThe precession period T(E) on a constant energy curve\nofEis\nT(E) =4K(k)\nγ/radicalbig\nHK(4πM−2E/M).(B8)\nFigure 6(a) shows an example of dE≡/contintegraltext\ndt(dE/dt) in\nthe absence of the feedback current, where the current\ndensity is chosen to be J= (Jc+J∗)/2. The energy den-\nsityEsatisfying dE= 0 and d(dE)/dE <0 corresponds\nto a stable attractor. As in the case of the STO in the\nmain text, there is only one attractor in this system.\n3. Work done by feedback current\nNow let us consider the role of the feedback current.\nIn the presence of the feedback current, the spin-transfer\ntorque performs an additional work given by\nWχ\ns≡γM/contintegraldisplay\ndtHsχm(t−τ)·p[p·H−(m·p)(m·H)],\n(B9)\nwhere we assume that the feedback current density is\ngiven by χJm(t−τ)·p. The averaged LLG equation in\nthe presence of the feedback current becomes\n/contintegraldisplay\ndtdE\ndt=Ws+Wχ\ns+Wα. (B10)\nTo evaluate Wχ\ns, it is useful to note that the solution\nof the magnetization oscillating around the easy axis on\na constant energy curve of Eis given by [47]\nmx(t) =/radicalbigg\n1+2E\nMHKsn/bracketleftbigg4K(k)\nT(E)t,k/bracketrightbigg\n,(B11)9\nmy(t) =/radicalBigg\n4πM−2E/M\nHK+4πMdn/bracketleftbigg4K(k)\nT(E)t,k/bracketrightbigg\n,(B12)\nmz(t) =/radicalBigg\nHK+2E/M\nHK+4πMcn/bracketleftbigg4K(k)\nT(E)t,k/bracketrightbigg\n,(B13)where sn( u,k), dn(u,k), and cn( u,k) arethe Jacobiellip-\ntic functions with u= 4K(k)t/T(E). Introducing a new\nvariablex= sn(u,k), Eq. (B9) becomes\nWχ\ns=4χMHs/radicalbig\nHK(4πM−2E/M)/integraldisplay1\n0dx/bracketleftbig\nHKmy−my(HKm2\ny−4πMm2\nz)/bracketrightbig\n/radicalbig\n(1−x2)(1−k2x2)my(t−τ), (B14)\nwhere dn( u,k) and cn( u,k) inmy(t) andmz(t) are re-\nplaced by√\n1−k2x2and√\n1−x2, respectively. On theother hand, my(t−τ) in Eq. (B14) is given by [50]\nmy(t−τ) =/radicalBigg\n4πM−2E/M\nHK+4πMdn(u,k)dn(v,k)+k2sn(u,k)sn(v,k)cn(u,k)cn(v,k)\n1−k2sn2(u,k)sn2(v,k)\n=/radicalBigg\n4πM−2E/M\nHK+4πMdn(v,k)√\n1−k2x2+k2sn(v,k)cn(v,k)x√\n1−x2\n1−k2sn2(v,k)x2,(B15)\nwherev= 4K(k)τ/T(E). Equation (B15) indicates that\nthe multiple attractors originate from the periodicity of\nthe elliptic function. In contrast with Eqs. (B5) and\n(B6), the analytical expression of Eq. (B14) is complex;\nsee next section. Therefore, we evaluate Eq. (B14) nu-\nmerically.\nFigure 6(b) shows/contintegraltext\ndt(dE/dt) in the presence of the\nfeedback current, where χ= 0.10 andτ= 3 ns. As\nshown, the multiple attractors appear, as in the STO\nstudied in the main text. Therefore, we consider that\nthe chaotic dynamics studied in Ref. [36] might be also\nrelated to the multiple attractors.\n4. Analytical expression of Wχ\ns\nSubstituting Eq. (B15) into Eq. (B14), Wχ\nsis rewrit-\nten as\nWχ\ns=4χMHs/radicalbig\nHK(4πM−2E/M)5/summationdisplay\nℓ=1Iℓ,(B16)where we introduce Iℓas\nI1=c2\nyHKdn(v,k)/integraldisplay1\n0dx√\n1−k2x2\n√\n1−x2[1−k2sn2(v,k)x2]\n≡c2\nyHKdn(v,k)˜I1,\n(B17)\nI2=−c4\nyHKdn(v,k)/integraldisplay1\n0dx(1−k2x2)3/2\n√\n1−x2[1−k2sn2(v,k)x2]\n≡ −c4\nyHKdn(v,k)˜I2,\n(B18)\nI3=c2\nyc2\nz4πMdn(v,k)/integraldisplay1\n0dx/radicalbig\n(1−x2)(1−k2x2)\n1−k2sn2(v,k)x2\n≡c2\nyc2\nz4πMdn(v,k)˜I3,\n(B19)\nI4=c2\ny/bracketleftbig/parenleftbig\n1−c2\ny/parenrightbig\nHK+c2\nz4πM/bracketrightbig\nk2sn(v,k)cn(v,k)/integraldisplay1\n0dxx\n1−k2sn2(v,k)x2\n≡c2\ny/bracketleftbig/parenleftbig\n1−c2\ny/parenrightbig\nHK+c2\nz4πM/bracketrightbig\nk2sn(v,k)cn(v,k)˜I4,(B20)10\nI5=c2\ny/parenleftbig\nc2\nyHKk2−c2\nz4πM/parenrightbig\nk2sn(v,k)cn(v,k)/integraldisplay1\n0dxx3\n1−k2sn2(v,k)x2\n≡c2\ny/parenleftbig\nc2\nyHKk2−c2\nz4πM/parenrightbig\nk2sn(v,k)cn(v,k)˜I5.(B21)\nHere, we introducethe followingnotations, forsimplicity.\ncy=/radicalBigg\n4πM−2E/M\nHK+4πM, cz=/radicalBigg\nHK+2E/M\nHK+4πM.(B22)The integrals ˜Iℓ(ℓ= 1−5) can be performed as\n˜I1=/integraldisplay1\n0dx√\n1−k2x2\n√\n1−x2[1−k2sn2(v,k)x2]\n=1\nsn2(v,k)/integraldisplay1\n0dx/radicalbig\n(1−x2)(1−k2x2)−1−sn2(v,k)\nsn2(v,k)/integraldisplay1\n0dx/radicalbig\n(1−x2)(1−k2x2)[1−k2sn2(v,k)x2]\n=K(k)\nsn2(v,k)−cn2(v,k)\nsn2(v,k)Π[k2sn2(v,k),k],(B23)\n˜I2=/integraldisplay1\n0dx(1−k2x2)3/2\n√\n1−x2[1−k2sn2(v,k)x2]\n=1\nsn2(v,k)/integraldisplay1\n0dx/radicalbigg\n1−k2x2\n1−x2−cn2(v,k)\nsn2(v,k)/integraldisplay1\n0dx√\n1−k2x2\n√\n1−x2[1−k2sn2(v,k)x2]\n=E(k)\nsn2(v,k)−cn2(v,k)\nsn2(v,k)˜I1,(B24)\n˜I3=/integraldisplay1\n0dx/radicalbig\n(1−x2)(1−k2x2)\n1−k2sn2(v,k)x2\n=1\nk2sn2(v,k)/integraldisplay1\n0dx/radicalbigg\n1−k2x2\n1−x2−dn2(v,k)\nk2sn2(v,k)/integraldisplay1\n0dx√\n1−k2x2\n√\n1−x2[1−k2sn2(v,k)x2]\n=E(k)\nk2sn2(v,k)−dn2(v,k)\nk2sn2(v,k)˜I1,(B25)\n˜I4=/integraldisplay1\n0dxx\n1−k2sn2(v,k)x2\n=−log[1−k2sn2(v,k)]\n2k2sn2(v,k)\n=−logdn(v,k)\nk2sn2(v,k),(B26)\n˜I5=/integraldisplay1\n0dxx3\n1−k2sn2(v,k)x2\n=−1\nk2sn2(v,k)/integraldisplay1\n0dxx+1\nk2sn2(v,k)/integraldisplay1\n0dxx\n1−k2sn2(v,k)x2\n=−1\n2k2sn2(v,k)+˜I4\nk2sn2(v,k),(B27)11\nwhere Π( a2,k) =/integraltext1\n0dx/[(1−a2x2)/radicalbig\n(1−x2)(1−k2x2)]is the third kind of complete elliptic integral.\n1S. 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Friedman, Handbook of Elliptic In-\ntegrals for Engineers and Scientists (Springer, 1971), 2nd\ned."    },    {        "title": "2008.09390v1.Integration_and_characterization_of_micron_sized_YIG_structures_with_very_low_Gilbert_damping_on_arbitrary_substrates.pdf",        "content": "Integration and characterization of micron-sized YIG structures with very low Gilbert damping on\narbitrary substrates\nP. Trempler, R. Dreyer, P. Geyer, C. Hauser, and G. Woltersdorf\nInstitut für Physik, Martin-Luther-Universität Halle-Wittenberg, D-06120 Halle, Germany\nG. Schmidt\u0003\nInstitut für Physik, Martin-Luther-Universität Halle-Wittenberg, D-06120 Halle, Germany and\nInterdisziplinäres Zentrum für Materialwissenschaften, Martin-Luther-Universität Halle-Wittenberg, D-06120 Halle, Germany\n(Dated: November 28, 2021)\nWe present a novel process that allows the transfer of monocrystalline yttrium-iron-garnet microstructures\nonto virtually any kind of substrate. The process is based on a recently developed method that allows the fabri-\ncation of freestanding monocrystalline YIG bridges on gadolinium-gallium-garnet. Here the bridges’ spans are\ndetached from the substrate by a dry etching process and immersed in a watery solution. Using drop casting the\nimmersed YIG platelets can be transferred onto the substrate of choice, where the structures finally can be reat-\ntached and thus be integrated into complex devices or experimental geometries. Using time resolved scanning\nKerr microscopy and inductively measured ferromagnetic resonance we can demonstrate that the structures\nretain their excellent magnetic quality. At room temperature we find a ferromagnetic resonance linewidth of\nm0DHHWHM \u0019195\u0016T and we were even able to inductively measure magnon spectra on a single micron-sized\nyttrium-iron-garnet platelet at a temperature of 5 K. The process is flexible in terms of substrate material and\nshape of the structure. In the future this approach will allow for new types of spin dynamics experiments up to\nnow unthinkable.\nI. INTRODUCTION\nThe growth of high quality thin film yttrium-iron-garnet\n(YIG) is very challenging. Even today very low Gilbert\ndamping ( a\u00145\u000210\u00004) is only achieved for deposition on\ngadolinium-gallium-garnet (GGG) which is almost perfectly\nlattice matched to YIG (see overview in Schmidt et. al.1).\nNevertheless, for many experiments a GGG substrate is not\nsuitable. GGG exhibits a strong paramagnetism that even2in-\ncreases below 70K. This results in an enlarged Gilbert damp-\ning in a thin YIG film due to the coupling of the YIG with the\nsubstrate. As a consequence, many experiments which aim\nfor example for the investigation of the strong coupling of\nmagnons and microwave photons3,4are limited to bulk YIG\nfabricated by liquid phase epitaxy (LPE) or to macroscopic\nYIG spheres. Up to now, this problem prevents experiments in\nhybrid quantum magnonics on YIG microstructures. Further-\nmore, experiments using YIG microstructures and integrated\nmicrowave antennae on GGG are difficult because of its large\ndielectric constant ( e\u001930). Unfortunately, there also hasn’t\nbeen any successful attempt to grow high quality YIG with\nreasonably low Gilbert damping on other substrates. Thus,\na method to fabricate thin high quality YIG microstructures\non GGG along a subsequent transfer on a different substrate\nwould lead the way towards many new promising experiments\nand applications. We have developed a process that allows us\nto transfer YIG microstructures from GGG onto other sub-\nstrates. Although the process is not suitable for mass fabrica-\ntion it nonetheless enables a new class of experiments which\nuntil today seemed unthinkable.\nFIG. 1. Patterning process flow: (a) Array of monocrystalline YIG\nbridges5. (b) The AlOx mask is deposited by e-beam lithography,\nevaporation, and lift-off. (c) The bridges are detached from the sub-\nstrate by argon ion milling. (d) The AlOx is dissolved in ammonia\nwater releasing the remaining YIG platelets into the liquid.\nII. PROCESSING\nOur method is based on a fabrication process5using room-\ntemperatue (RT) pulsed laser deposition (PLD), lift-off and\nannealing, which yields freely suspended YIG structures,\nwhereby we apply the process in order to fabricate bridges or\ndoubly clamped beams. The suspended parts of these struc-\ntures exhibit extraordinary magnetic properties. For these\nstructures a ferromagnetic resonance (FMR) linewidth at 9.6\nGHz of m0DHHWHM=140\u0016T and a Gilbert damping of a\u0019\n2\u000210\u00004were demonstrated. Using this process we fabri-\ncate an array of 500,000 bridges of 1 :5\u00025\u0016m2span-size on a\nGGG substrate. We then mask the spans of the bridges by alu-\nminum oxide using electron beam lithography, e-beam evap-arXiv:2008.09390v1  [cond-mat.mes-hall]  21 Aug 20202\noration, and lift-off. [Fig. 1 (b)]. Using argon ion milling al-\nlows to remove the part of the bridge that connects the span\nto the substrate leaving the masked YIG as a micro slab like\nplatelet embedded in the aluminum oxide (AlOx). [Fig. 1 (c)].\nDissolving the mask in ammonia water lifts the 500,000 YIG\nmicro platelets from the substrate and immerses them in the\nsolution. The wet etchant is then stepwise replaced by water\nyielding a watery suspension of uniform monocrystalline YIG\nplatelets [Fig. 1 (d)]. By drop-casting the YIG platelets can\nnow be transferred to any substrate. After drying, the platelets\nstick to the substrate and even stay in place during subsequent\nspin-coating of further resist layers. With the help of addi-\ntioanl lithography the platelets can be integrated in complex\ndevices or applications.\nHere we show one example how a YIG platelet can be inte-\ngrated into a coplanar waveguide geometry to achieve in-plane\nexcitation and high sensitivity in FMR. As a substrate we use\nsapphire onto which 150nm of Au with a Ti adhesion layer\nwere deposited by electron beam evaporation. Sapphire is\nchosen because of its excellent properties for high frequency\nmeasurements. Before the drop-casting, a layer of PMMA is\nspun onto the sample. The suspension is exposed for a few\nseconds to ultrasonic agitation to ensure a homogeneous sus-\npension of the YIG platelets and by using a pipette a single\ndrop of the suspension is then put onto the sample. After the\ndrop-casting the YIG platelets are typically flat on the sample\nsurface but randomly oriented. Once a suitable YIG platelet is\nidentified we heat the sample up to 250\u000eC which is well above\nthe glass transition temperature of the PMMA6causing the\nYIG platelet to slightly sink into the PMMA film [Fig. 2 (a)].\nBy electron beam lithography we then crosslink the PMMA\nat the end of the bridge, defacto welding the bridge to the Au\nsurface [Fig. 2 (b)]. Using the PMMA layer under the YIG\nhas several advantages compared to direct deposition on the\nAu surface. No spin coating is required before the bridge\nis fixed and after removing the non-crosslinked PMMA the\nsample surface is now also clean from possible residue of the\ndrop-casting process. It should be noted that there is most\nlikely a gap of 10 \u000040nm between YIG and Au so the system\ncorresponds rather to a bridge with a YIG platelet as a span\nand two pedestals of PMMA as posts. To realize the final test\nstructure we now use electron beam lithography, AlOx evap-\noration and lift-off to mask the intended area of the CPW and\nthe YIG platelet itself. By Argon ion milling we remove the\nunmasked Au and Ti. After removing the AlOx mask we end\nup with a CPW perfectly aligned with the YIG platelet and\nideally suited in terms of size and shape for the FMR charac-\nterization of the YIG platelet [Fig. 2 (c)]. The final structure is\nshown in Fig. 3 as an false-color SEM image.\nIII. MAGNETIC PROPERTIES\nIn order to assess the sensitivity of our experiment we now\nperform FMR measurements. The samples are bonded onto a\nsample holder that fits into a4He bath cryostate. The cryostate\nis placed inside an electromagnet that can be rotated in the\nsample plane. The external magnetic field can be modulated\nFIG. 2. (a) The YIG drop-cast on the PMMA sinks into the poly-\nmer during heating. (b) The PMMA at the ends of the platelet is\ncrosslinked to fix the YIG to the Au. (c) Electron beam lithography\nand dry etching are used to pattern the CPW.\nFIG. 3. False-color SEM image of a transferred YIG platelet (ma-\ngenta) fixed with crosslinked PMMA (green) on top of a Ti/Au CPW\n(yellow). The bridge has a span length of 4 :5\u0016m, a width of 1 :5\u0016m\nand a nominell YIG layer thickness of approximately 160nm.\nusing an air coil of a few turns of Cu wire wound around the\nsample holder inside the cryostate. For our measurements the\nexternal magnetic field is oriented along the long side of the\nplatelet. RF excitation is done by applying an RF signal with a\npower of \u000021dBm. Measurements are performed by sweep-\ning the magnetic field at constant RF frequency. The transmit-\nted RF signal is rectified and the modulation of the external\nfield allows for lock-in detection to increase sensitivity. With\nthe YIG platelet centered on the waveguide the exciting RF\nfield is oriented in the sample plane and homogeneous over\nthe YIG platelet. As a consequence we can only excite stand-\ning spin wave modes with an uneven number of antinodes that\nhave non-zero magnetization.\nFig. 4 shows two resonance curves obtained at 4GHz at\nroom temperature and at 5K respectively. In both cases we\nobserve an extended spin-wave spectrum with a large number\nof backward-volume modes (BVMs). These discrete modes\nare caused by the finite size of the YIG platelet and corre-\nspond to standing spin wave modes as observed in a previ-\nous experiment5. Because of the complexity of the spectrum\nand the overlap of multiple modes it is difficult to obtain a\nlinewidth or even extract a Gilbert damping from measure-3\nFIG. 4. FMR spectra for a frequency of 4GHz at (a) 5K and (b)\n295K showing the occurance of several spin wave modes in the YIG\nbridge. The extended spin-wave spectra even for low temperatures\nsuggests a very low Gilbert damping.\nFIG. 5. Spatial resolved measurements acquired at a frequency of\n4GHz at different respective magnetic fields. The TRMOKE im-\nages show standing BVMs in the span of the bridge for m0Hextof\n(a) 74mT, (b) 80mT, (c) 84 :5mT and (d) 88 :5mT. The dotted lines\nserve as a guide to the eye to indicate the approximate sample posi-\ntion. m0Hextis applied along the x-direction.\nments at different respective frequencies. A closer look at the\nshape of the main resonance line indicates that it is not a sin-\ngle line but composed from at least two separate lines if not\nmore [Fig 6]. At 5K the spectrum is more noisy than at room\ntemperature but still the details of the spectrum are similar to\nthose at room temperature. The major difference to the room\ntemperature measurement is the change in resonance field that\ncan be attributed to the change in saturation magnetization7.\nWe perform TRMOKE experiments on the YIG in order\nto obtain more detailed information about the local struc-\nture of the excited modes. Further details of this technique\nare described in the work of Tamaru et. al.8and Neudecker\net. al.9. Again the measurements are performed with the exter-\nnal magnetic field oriented along the long side of the platelet.\nTRMOKE allows to locally image magnon modes in terms of\nboth intensity and phase5. To perform the spatially resolved\nimaging the frequency was set to 4 GHz at an RF amplitude of\n-25 dBm. The real and imaginary part of the dynamic suscep-\ntibility were detected in pointwise fashion while the magnetic\nfield was kept constant for each picture [Fig 5].\nThe spatially resolved measurements show several stand-\ning BVM with the fundamental mode with only one antin-\node [Fig. 5 (a)] and three standing BVMs with antinodes dis-\ntributed along the bridge in Fig. 5 (b)-(d)5. As expected all ob-\nserved modes exhibit an uneven number of antinodes. Again,\nFIG. 6. Main FMR line as composition of two separate lines for\na single transferred YIG platelet of 1 :5\u00024:5\u0016m2. The linewidth is\nm0DHHWHM=195\u0016T.\nit is not possible to extract a precise value for the line width for\nthis sample. Another platelet from the same batch was trans-\nferred into the gap of a coplanar waveguide. In this geometry\nthe out-of-plane RF field allows for TRMOKE measurements\nwith the external field applied perpendicular to the long side\nof the platelet. This results in a larger spacing between the\nresonance lines and yields the spectrum shown in Fig. 6. The\nresonance field is slightly shifted compared to the measure-\nments shown in Fig. 5. At 4 GHz we observe two superim-\nposed lines which can be fitted by two lorentzian line shapes.\nWe obtain a linewidth of m0DHHWHM\u0019195\u0016T. To the best of\nour knowledge even for large area thin films there are only two\npublications from other groups that show a smaller linewidth\nat this frequency10,11. For untransferred bridges (on GGG)\nwe have already measured a smaller linewidth, however, it is\nunclear whether the original sample produced for the drop-\ncasting was of similar quality. In any case the magnetic qual-\nity is only weakly affected by the transfer, if at all.\nIV . OUTLOOK\nThe presented process opens up a large number of options.\nAs we have shown in5the 3D patterning process is not limited\nto linear bridges. Besides we can also make frames, rings, cir-\ncular drums, tables, or other arbitrariliy shaped flat structures\nwhich would allow us to use the transfer technique presented\nhere. The main restriction is merely the size. With increasing\nstructure size the yield of the initial 3D patterning process is\nreduced and also the writing time increases linearly with the\narea. On the other hand we need a large number of structures\nto have enough statistical hits in the drop-casting process. A\nlow concentration of YIG structures in the suspension would\nmake the drop-casting a hopeless procedure. Beyond that, are\neven more options. Before the masking with AlOx we can\nperform additional processing on the bridges. We can for ex-4\nample deposit a thin metal film on top. After detachment and\ndrop-casting we have a 50:50 chance that the metal film ends\nup at the bottom of our platelet. A second evaporation step\ncould then be used to create a double side metallized YIG\nfilm as has been used in12for the demonstration of magnon\ndrag. In our case, however, we have no limitations as to the\nmetals that we want to use and their respective thicknesses.\nFurthermore we may even be able to nanopattern the metal\nbefore detaching the bridges and finally achieve a piece of\nYIG thin film with lithographically nanopatterned metal on\nboth sides. Our structures may even be suitable for hybrid\nquantum magnonics at mK temperatures. As van Loo et. al.13\nand Mihalceanu et. al.14have shown, the damping of thin film\nYIG increases at low temperatures, mainly because of inter-\naction with the GGG substrate. In our case the YIG platelet\nis no longer on the substrate. Even more it has never been in\ndirect contact with GGG so also contamination effects can be\nexcluded, making high performance at mK temperatures even\nmore likely. And finally these isolated structures may also\nbe suitable for the formation of magnon-based Bose-Einstein\ncondensates15.V . CONCLUSION\nWe have demonstrated that it is possible to transfer high\nquality thin film YIG microstructures onto other substrates\nand to integrate them in complex experiments. The magnetic\nquality is only slightly affected by the process, if at all.\nNotably, we are able to measure FMR spectra at 5K with\nmany details. This process opens up new routes towards a\nmultitude of experiments which formerly seemed completely\nout of reach.\nVI. DATA A VAILABILITY\nThe data that support the findings of this study are available\nfrom the corresponding author upon reasonable request.\nACKNOWLEDGMENTS\nWe wish to acknowledge the support of TRR227 project\nB02 WP3 and project B01.\n\u0003georg.schmidt@physik.uni-halle.de\n1G. Schmidt, C. Hauser, P. Trempler, M. Paleschke, and E. T.\nPapaioannou, physica status solidi (b) 257, 1900644 (2020).\n2V . Danilov, Y . V . Lyubon’ko, A. Y . Nechiporuk, S. Ryabchenko,\net al. , Soviet Physics Journal 32, 276 (1989).\n3H. Huebl, C. W. Zollitsch, J. Lotze, F. Hocke, M. Greifenstein,\nA. Marx, R. Gross, and S. T. Goennenwein, Physical Review\nLetters 111, 127003 (2013).\n4Y . Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami, and\nY . Nakamura, Phys. Rev. Lett. 113, 083603 (2014).\n5F. Heyroth, C. Hauser, P. Trempler, P. Geyer, F. Syrowatka,\nR. Dreyer, S. Ebbinghaus, G. Woltersdorf, and G. Schmidt, Phys.\nRev. Applied 12, 054031 (2019).\n6M. Mohammadi, H. fazli, M. karevan, and J. Davoodi, European\nPolymer Journal 91, 121 (2017).\n7P. Hansen, P. Röschmann, and W. Tolksdorf, Journal of Applied\nPhysics 45, 2728 (1974), https://doi.org/10.1063/1.1663657.8S. Tamaru, J. Bain, R. Van de Veerdonk, T. Crawford, M. Coving-\nton, and M. Kryder, Journal of Applied Physics 91, 8034 (2002).\n9I. Neudecker, G. Woltersdorf, B. Heinrich, T. Okuno, G. Gub-\nbiotti, and C. Back, Journal of Magnetism and Magnetic Materi-\nals307, 148 (2006).\n10O. d’Allivy Kelly, A. Anane, R. Bernard, J. Ben Youssef,\nC. Hahn, A. H. Molpeceres, C. Carrétéro, E. Jacquet, C. Deranlot,\nP. Bortolotti, et al. , Applied Physics Letters 103, 082408 (2013).\n11J. Ding, T. Liu, H. Chang, and M. Wu, IEEE Magnetics Letters\n11, 1 (2020).\n12J. Li, Y . Xu, M. Aldosary, C. Tang, Z. Lin, S. Zhang, R. Lake,\nand J. Shi, Nature communications 7, 1 (2016).\n13A. F. van Loo, R. G. E. Morris, and A. D. Karenowska, Phys.\nRev. Applied 10, 044070 (2018).\n14L. Mihalceanu, V . I. Vasyuchka, D. A. Bozhko, T. Langner, A. Y .\nNechiporuk, V . F. Romanyuk, B. Hillebrands, and A. A. Serga,\nPhys. Rev. B 97, 214405 (2018).\n15S. O. Demokritov, V . E. Demidov, O. Dzyapko, G. A. Melkov,\nA. A. Serga, B. Hillebrands, and A. N. Slavin, Nature 443, 430\n(2006)."    },    {        "title": "1305.0714v1.Co2_FeAl_thin_films_grown_on_MgO_substrates__Correlation_between_static__dynamic_and_structural_properties.pdf",        "content": "Co 2FeAl thin films grown on MgO substrates: Correlation between static, dynamic\nand structural properties\nM. Belmeguenai1, H. Tuzcuoglu1, M. S. Gabor2, T. Petrisor jr2, C. Tiusan2;3,\nD. Berling4, F. Zighem1, T. Chauveau1, S. M. Chérif1and P. Moch1\n1Laboratoire des Sciences des Procédés et des Matériaux, CNRS - Université Paris 13, France\n2Center for Superconductivity, Spintronics and Surface Science, Technical University of Cluj-Napoca, Romania\n3Institut Jean Lamour, CNRS-Université de Nancy, France and\n4Institut de Science des Matériaux de Mulhouse, CNRS-Université de Haute-Alsace, France\nCo2FeAl (CFA) thin films with thickness varying from 10 nm to 115 nm have been deposited on\nMgO(001)substratesbymagnetronsputteringandthencappedbyTaorCrlayer. X-raysdiffraction\n(XRD) revealed that the cubic [001]CFA axis is normal to the substrate and that all the CFA films\nexhibit full epitaxial growth. The chemical order varies from the B2phase to the A2phase when\ndecreasing the thickness. Magneto-optical Kerr effect (MOKE) and vibrating sample magnetometer\nmeasurements show that, depending on the field orientation, one or two-step switchings occur.\nMoreover, the films present a quadratic MOKE signal increasing with the CFA thickness, due\nto the increasing chemical order. Ferromagnetic resonance, MOKE transverse bias initial inverse\nsusceptibility and torque (TBIIST) measurements reveal that the in-plane anisotropy results from\nthe superposition of a uniaxial and of a fourfold symmetry term. The fourfold anisotropy is in\naccord with the crystal structure of the samples and is correlated to the biaxial strain and to the\nchemical order present in the films. In addition, a large negative perpendicular uniaxial anisotropy\nis observed. Frequency and angular dependences of the FMR linewidth show two magnon scattering\nand mosaicity contributions, which depend on the CFA thickness. A Gilbert damping coefficient as\nlow as 0.0011 is found.\nI. INTRODUCTION\nThe performances of spintronic devices depend on the\nspin polarization of the current. Therefore, half metallic\nmaterials should be ideal compounds as high spin polar-\nized current sources to realize a very large giant magne-\ntoresistance, a low current density for current induced\nmagnetization reversal, and an efficient spin injection\nintosemiconductors. Theoretically, differentkindsofma-\nterials, such as Fe 3O4[1, 2], CrO 2[3], mixed valence per-\novskites [4] and Heusler alloys [5, 6], have been predicted\nto be half metals. Moreover, the half metallic proper-\nties in these materials have been experimentally demon-\nstrated at low temperature. However, oxide half metals\nhavelowCurietemperature( TC)andthereforetheirspin\npolarization is miserably low at room temperature. From\nthis point of view, Heusler alloys are promising materials\nfor spintronics applications, because a number of them\nhave generally high TC[7] and therefore they may offer\nan alternative material choice to obtain half metallicity\neven at room temperature. Furthermore, their structural\nand electronic properties strongly depend on the crystal\nstructure. Recently, Heusler compounds have attracted\nconsiderable experimental and theoretical interest, not\nonly because of their half metallic behaviour but also due\nto magnetic shape memory and inverse magneto-caloric\nproperties that they exhibit. One of the most important\nCo-based full-Heusler alloys is Co 2FeAl (CFA). It has a\nhighTC(TC= 1000K) [7] and, therefore, it is promis-\ning for practical applications. Indeed, it can provide\ngiant tunnelling magnetoresistance ( 360%at RT) [8,9]\nwhen used as an electrode in magnetic tunnel junctions.\nFurthermore, as we illustrate in our present study, CFApresents the lowest magnetic damping parameter among\nHeuslers. This low damping should provide significantly\nlower current density required for spin-transfer torque\n(STT) switching, particularly important in prospective\nSTT devices. However, the integration of CFA as a\nferromagnetic electrode in spintronic devices requires a\ngood knowledge allowing for a precise control of its mag-\nnetic properties, such as its saturation magnetization, its\nmagnetic anisotropy, the exchange stiffness parameter,\nthe gyromagnetic factor and the damping mechanisms\nmonitoring its dynamic behaviour. In this paper we\nused X-rays diffraction (XRD), ferromagnetic resonance\nin microstrip line (MS-FMR) under in-plane and out of\nplane applied magnetic field, combined with transverse\nbiased initial inverse susceptibility and torque (TBIIST)\nmethod, in order to perform a complete correlated analy-\nsis between structural and magnetic properties of epitax-\nial Co 2FeAl thin films grown on MgO(001) substrates.\nIn addition, a detailed study of the different relaxation\nmechanisms leading to the linewidth broadening is pre-\nsented.\nII. SAMPLES PREPARATION AND\nEXPERIMENTAL METHODS\nCFA films were grown on MgO(001) single-crystal sub-\nstrates using a magnetron sputtering system with a base\npressure lower than 3\u000210\u00009Torr. Prior to the deposi-\ntion of the CFA films, a 4 nm thick MgO buffer layer was\ngrown at room temperature (RT) by rf sputtering from\na MgO polycrystalline target under an Argon pressure\nof 15 mTorr. Next, the CFA films, with variable thick-arXiv:1305.0714v1  [cond-mat.mtrl-sci]  3 May 20132\n30 40 50 60 70050100150200250300\n42.5 45.0 47.50100200\n115 nm\n70 nm\n45 nm\n20 nm\n2(degrees)Intensity (arb. units)(002) CFA\n(004) CFA\n10 nm(a) (220) CFA\n30 40 50 60 70 80 90050100150200250300350400450500(b)\n20 nm50 nmCFA(002)\nCFA(004)Intensity (arb. units)\n2 (degrees)\nFigure 1: (Colour online) (a) X-ray 2\u0012\u0000!(out-of-plane)\ndiffractionpatternusing(CuX-rayssource)fortheCr-capped\nand (b)\u0012\u00002\u0012pattern (Co X-rays source) for the Ta-capped\nCo2FeAl of different thicknesses. The inset shows selected\narea in plane diffraction patterns around (220) Co 2FeAl re-\nflection.\nnesses (10 nm\u0014d\u0014115nm), were deposited at RT by dc\nsputtering under an Argon pressure of 1 mTorr, at a rate\nof 0.1 nm/s. Finally, the CFA films were capped with\na MgO(4nm)/Cr(10nm) or with a MgO(4nm)/Ta(10nm)\nbilayer. Afterthegrowthofthestack, thestructureswere\nex-situ annealed at 600oC during 15 minutes in vacuum\n(pressure lower than 3\u000210\u00008Torr). The structural prop-\nerties of the samples have been characterized by XRD us-\ning a four-circle diffractometer. Their magnetic dynamic\nproperties have been studied by microstrip ferromagnetic\nresonance (MS-FMR).\nThe MS-FMR characterization was done with the help\nof a field modulated FMR setup using a vector network\nanalyzer (VNA) operating in the 0.1-40 GHz frequency\nrange. The sample (with the film side in direct con-\ntact) is mounted on 0.5 mm microstrip line connected\nto the VNA and to a lock-in amplifier to derive the field\nmodulated measurements via a Schottky detector. Thissetup is piloted via a Labview program providing flexi-\nbility of a real time control of the magnetic field sweep\ndirection, step and rate, real time data acquisition and\nvisualization. It allows both frequency and field-sweeps\nmeasurements with magnetic fields up to 20 kOe applied\nparallel or perpendicular to the sample plane. In-plane\nangular dependence of resonance frequencies and fields\nare used to measure anisotropies. The complete analy-\nsis of in-plane and perpendicular field resonance spectra\nexhibiting uniform precession and perpendicular stand-\ning spin wave (PSSW) modes leads to the determination\nof most of the magnetic parameters: effective magneti-\nzation, gyromagnetic factor, exchange stiffness constant\nand anisotropy terms. In addition, the angular and the\nfrequency dependences of the FMR linewidth are used\nin order to identify the relaxation mechanisms responsi-\nble of the line broadening and allow us for evaluating the\nparameters which monitor the intrinsic damping (Gilbert\nconstant) and the extrinsic one (two magnon scattering,\ninhomogeneity, mosaïcity).\nMagnetization at saturation and hysteresis loops for\neach sample were measured at room temperature using a\nvibrating sample magnetometer (VSM) and a magneto-\noptical Kerr effect (MOKE) system. Transverse biased\ninitialinversesusceptibilityandtorquemethod(TBIIST)\n[10] has been used to study the in-plane anisotropy for\ncomparison with MS-FMR. In this technique both a lon-\ngitudinal magnetic sweep field HL(parallel to the inci-\ndenceplane)andastatictransversefield HB(perpendic-\nulartotheincidenceplane)areappliedintheplaneofthe\nfilm and the longitudinal reduced magnetization compo-\nnentmLis measured versus HLfor various directions of\nHLwithconventionalmagneto-opticalKerrsetup. From\nthe measured hysteresis loops mL(HL)under transverse\nbiased field, the initial inverse susceptibility ( \u001f\u00001) and\nthe field offset ( \u000eH) which are related to the second and\nfirst angle-derivative of the magnetic anisotropy, respec-\ntively, are derived. Fourier analysis of \u001f\u00001and\u000eHversus\nthe applied field direction then easily resolves contribu-\ntions to the magnetic anisotropy of different orders and\ngives the precise corresponding values of their amplitude\nand of their principal axes.\nIn order to obtain the desirable accuracy or even sim-\nply meaningful results higher-order nonlinear in mLcon-\ntributions (quadratic or Voigt effect) as well as polar or\nother contributions to the Kerr signal are carefully deter-\nmined and corrected [10]. TBIIST method surely does\nnot have the same recognition than FMR techniques but\nseems to be complementary, especially for samples with a\nweak magnetic signal detectable with difficulty by FMR\nmethods.\nIII. STRUCTURAL CHARACTERIZATION\nFigure 1 shows the X-rays 2\u0012\u0000!diffraction patterns\nfor CFA of different thicknesses. These XRD patterns\nshow that, in addition to the feature arising from the3\nFigure 2: (Colour online) Pole figures around the Co 2FeAl\n(022) type reflection, for the 45 nm thick film, indicat-\ning the growth of Co 2FeAl on MgO with the Co 2FeAl\n(001)[110]kMgO (001)[100] epitaxial relation. The 0 and 90\ndegrees axis of the graph correspond to the MgO [100]and\n[010] crystalline directions.\n(002) peak of the MgO substrate, the Cr-capped samples\n(Fig. 1a: Cu X-rays source ( \u0015= 0:15406nm)) exhibit\nonly two peaks which are attributed to the (002) and\n(004) diffraction lines of CFA. The Ta-capped films (Fig.\n1b: Co-X-rays source ( \u0015= 1:7902)) show an additional\npeak (around 2\u0012= 63 °) arising from the (002) line is-\nsued from the Ta film. Pole figures (Fig. 2) allow to\nassert an epitaxial growth of the CFA films according\nto the expected CFA(001)[110]//MgO(001)[100] epitax-\nial relation. Using scans of various different orientations\nwe evaluated the out-of-plane ( a?) and the in-plane ( ak)\nlattice parameters (Fig. 3). A simple elastic model al-\nlowed us for deriving the unstrained a0 cubic parameter\naswellasthein-plane \"kandtheout-of-plane \"?strains:\na0=\u0000\nC11a?+ 2C12ak\u0001\n(C11+ 2C12);\n\"k=C11\n(C11+ 2C12)\u0000\nak\u0000a?\u0001\na0;\n\"?=2C12\n(C11+ 2C12)\u0000\nak\u0000a?\u0001\na0(1)\nwhere the values of the elastic constants C11= 253\nGPa andC12= 165GPa have been calculated previously\n[11]. Introducing the Poisson coefficient \u0017=C12=(C11+\nC12)the above parameters write as:\n05 0 1 0 00.5650.5700.575\n out-of-plane\n in-planelattice parameter (nm)\nthickness (nm)-10-50510152025 A002/A004\nratio (%)Figure 3: (Colour online) Evolution of the out-of-plane and\nin-plane lattice parameters and of the ratio of the integral in-\ntensitiesofthe (002)and(004)Co2FeAlpeaksA(002)/A(004)\nwith respect to the Co 2FeAl films thickness.\na0=\u0000\n(1\u0000\u0017)a?+ 2ak\u0001\n+ 2\u0017ak\n(1 +\u0017);\n\"?=(1\u0000\u0017)\n(1 +\u0017)\u0000\nak\u0000a?\u0001\na0;\n\"k=\u00002\u0017\n1 +\u0017\u0000\nak\u0000a?\u0001\na0(2)\nThecubiclatticeconstant a0doesnotdependuponthe\nthickness, except for the thinner 10 nm film (Fig. 4a),\nwhich shows a significant reduction; its value, 0:5717\u0006\n0:0005nm, is slightly smaller than the reported one in\nthe bulk compound with the L2 1structure (0.574 nm).\nThe in-plane strain \"kreveals a tensile stress originat-\ning from the mismatch with the lattice of the MgO sub-\nstrate: however, its value does not exceed a few°/°°, well\nbelow the Heussler/MgO mismatch, thus excluding an\nefficient planar clamping. The strain \"kdecreases versus\nthe thickness, at least above 40 nm (Fig. 4b).\nOdd Miller indices (e.g.: (111);(311);...) are allowed\nfor diffraction in the L2 1phase [12]. In contrast, they\nare forbidden in the B2 phase, which is characterized by\na total disorder between Al and Fe atoms but a regular\noccupation of the Co sites. In the A2 phase the chemical\ndisorder between Fe, Co and Al sites is complete: (hkl)\ndiffraction is only allowed for even indices subjected to\nh+k+l= 4n. We do not observe (111)or(311)lines\nand then conclude to the absence of the L2 1phase in\nthe studied films. In contrast, a (002)peak is observed,\nthus indicating that the samples do not belong to the A2\nphase. However, the ratio I002=I004of the integrated in-\ntensities of the (002)and of the (004)peaks increases ver-\nsus the film thickness (Fig. 3). This ratio is proportional\nto(1\u00002c)2, wherecis the chemical disorder. Assuming\nthat the thickest film belongs to the B2 phase ( c= 0)\nthe dependence of cupon the film thickness is shown in4\n0 2 04 06 08 0 1 0 0 1 2 00.00.20.40.0050.0060.0070.5700.5710.572\n(c)c\nFilm thickness (nm)(b)(a)a0 (nm)\nFigure 4: (Colour online) Thickness dependence of (a) the\nlattice cubic parameter a0, the in-plane strain \"kand (c) the\nchemical order cof Co 2FeAl thin films.\nfigure 4c: the A2 phase ( c= 0:5) is almost completely\nachieved for the 10 nm thick sample. The reduction of\na0in the thinner sample is probably due to its previously\nnoticed [13] smaller value in the A2 phase compared to\nthe B2 one.\nIV. MAGNETIC PROPERTIES\nThe experimental magnetic data have been analyzed\nconsidering a magnetic energy density which, in addition\ntoZeeman, demagnetizingandexchangeterms, ischarac-\nterized by the following effective anisotropy contribution\n[14]:\nEanis: =\u00001\n2(1 +cos(2('M\u0000'u))Kusin2\u0012M+\nK?sin2\u0012M\u00001\n8(3 + cos 4('M\u0000'4))K4sin4\u0012M(3)\nIn the above expression, \u0012Mand'Mrespectively rep-\nresent the out-of-plane and the in-plane (referring to the\nsubstrate edges) angles defining the direction of the mag-\nnetizationMS.'uand'4define the angles between an\neasy uniaxial planar axis or an easy planar fourfold axis,\nrespectively, with respect to this substrate edge. Ku,K4\nandK?are in-plane uniaxial, fourfold and out-of-plane\nuniaxialanisotropyconstants, respectively. Weintroduce\nthe effective magnetization Meff=Heff=4\u0019obtained\nby:\n4\u0019Meff=Heff= 4\u0019MS\u00002K?\nMS= 4\u0019MS\u0000H?(4)\nAs experimentally observed, the effective perpendicu-\nlar anisotropy term K?(and, consequently, the effective\nperpendicular anisotropy field H?), is thickness depen-\ndent:K?describes an effective perpendicular anisotropyterm which writes as:\nK?=K?V+2K?S\nd(5)\nwhereK?Srefers to the perpendicular anisotropy term\nof the interfacial energy density. Finally we define Hu=\n2Ku=MSandH4= 4K4=Msas the in-plane uniaxial and\nthe fourfold anisotropy fields respectively. The resonance\nexpressions for the frequency of the uniform and PSSW\nmodes assuming in-plane or perpendicular applied mag-\nneticfieldsaregivenbyequations(6)and(7)respectively\n[14, 15].\nFn:=\r\n2\u0019(Hcos('H\u0000'M) +2K4\nMScos 4('M\u0000'4)\n+2Ku\nMScos 2('M\u0000'u) +2Aex:\nMS\u0010n\u0019\nd\u00112\n)\u0002\n(Hcos('H\u0000'M) + 4\u0019Meff+K4\n2MS(3 + cos 4('M\u0000'4))\n+Ku\nMS(1 + cos 2('M\u0000'u)) +2Aex:\nMS(n\u0019\nd)2)(6)\nF?:=\r\n2\u0019(H\u00004\u0019Meff+2Aex:\nMS\u0010n\u0019\nd\u00112\n)\u0002(7)\nIn the above expressions \r=2\u0019=g\u00021:397\u0002106\ns\u00001.Oe\u00001is the gyromagnetic factor, nis the index of\nthe PSSW and Aexis the exchange stiffness constant.\nThe experimental results concerning the measured\npeak-to-peak FMR linewidths \u0001HPPare analyzed in\nthis work taking account of both intrinsic and extrinsic\nmechanisms. Therefore, in the most FMR experiments,\nthe observed magnetic field linewidth ( \u0001HPP) is usu-\nally analyzed by considering four different contributions\nas given by equation (8) [16-21].\n\u0001HPP= \u0001HGi+ (\u0001Hmos+ \u0001Hinh+ \u0001H2mag)(8)\nWhen the applied field and the magnetization are paral-\nlel, the intrinsic contribution is not angular dependent;\nit derives from the Gilbert damping and is given by:\n\u0001HGi=2p\n3\u000b\n\r2\u0019f (9)\n(9) wherefis the driven frequency and \u000bis the Gilbert\ncoefficient.\nThe relevant mechanisms [16] describing the extrinsic\ncontributions are:\n1- Mosaicity: the orientation spread of the crystallites\ncontributes to the linewidth. Its contribution is given by:\n\u0001Hmos=\f\f\f\f@Hres\n@'H\u0001'H\f\f\f\f=\f\f\f\f@H\n@'H\u0001'H\f\f\f\f\nres(10)5\nWhere \u0001'His the average spread of the easy axis\nanisotropy direction in the film plane. It is worth to men-\ntion that for frequency dependent measurements along\nthe easy and hard axes the partial derivatives are zero\nand thus the mosaicity contribution vanishes. The suffix\n“res” indicates that equation (10) should be evaluated at\nthe resonance. Therefore, using equation (6) for uniform\nmode (n= 0), the expression of@H\n@'His found and then\ncalculated using the corresponding value of Hand'M\nat the resonance.\n2- Inhomogeneous residual linewidth \u0001Hinhpresent\nat zero frequency. This contribution is frequency and\nangle independent inhomogeneity related to various local\nfluctuations such as the value of the film thickness.\n3- Two magnon scattering contribution to the\nlinewidth. This contribution is given by [22-24]:\n\u0001H2mag= \u00000+ \u00002cos 2('H\u0000'2)+\n\u00004cos 4('H\u0000'4) arcsin \nfp\nf2+f2\n0+f0!\n(11)\nwith:f0=\rMeff. The expected fourfold symmetry\ninduces the \u00000and\u00004coefficients; the coefficient \u00002is\nphenomenogically introduced.\nTheanalysisofthevariationoftheresonancelinewidth\n\u0001HPPversus the frequency and the in-plane field ori-\nentation allows for evaluating \u000b,\u0001'H,\u0001Hinh,\u00000,\u00002\n(and'2) and \u00004(and'4which, from symmetry consid-\nerations, is expected to have a 0°or45°value, depending\nupon the chosen sign of \u00004).\nA. Static properties\nThe magnetization at saturation measured by VSM,\naveraged upon all the samples has been found to be\nMS= 1000\u000650emu/cm3, thus providing a magnetic\nmoment of 5.05 ±0.25 Bohr magneton ( \u0016B) per unit for-\nmula, in agreement with the previously published values\nfor the B2 phase [7]. For all the studied films the hystere-\nsis loops were obtained by VSM and MOKE with an in-\nplane magnetic field applied along various orientations.\nFigure 5 shows representative behaviors of different CFA\nfilms. The observed shape mainly depends on the field\norientation, in agreement with the expected characteris-\ntics of the magnetic anisotropy. As confirmed below, in\nall the studied samples this anisotropy consists into the\nsuperposition of a fourfold and of a uniaxial term show-\ning parallel easy axes: this common axis coincides with\none of the substrate edges and, consequently, with one\nof the<110>crystallographic directions of the CFA\nphase. It results that if an orientation (say 'H= 0re-\nlated to [110]) is the easiest, the perpendicular direction\n(('H= 90\u000e) related to [110]) is less easy. A similar sit-\nuation was studied and interpreted previously [25]: it is\nexpected to provide square hysteresis loops for 'H= 0\u000e,\n-60 -40 -20 0 20 40 60-1.0-0.50.00.51.0(a)\nd=115 nmM/Ms\nMagnetic field (oe) H=0°\n H=45°\n H=90°\n-60 -40 -20 0 20 40 60-1.0-0.50.00.51.0(b)\nd=50 nmM/Ms\nMagnetic field (oe) H=0°\n H=45°\n H=90°\n0.00 0.02 0.04 0.06 0.08 0.1001020304050607080Coercive field (Oe)\n1/d(nm) Cr-capped\n Ta-capped(c)Figure 5: (Colour online) MOKE hysteresis loops of the (a)\n115 nm Cr-capped and (b) 50 nm Ta-capped Co 2FeAl thin\nfilms. Themagneticfieldisappliedparalleltothefilmsurface,\nat various angles ( 'H) with a respect to edges of the MgO\nsubstrate ( [100]or[010]). (c) Thickness dependence of the\ncoercive field, deduced from hysteresis loops along the easy\naxis, of Co 2FeAl Cr- and Ta-capped thin films.6\n0 50 100 150 200 250 300 350-2.50.022.525.0-505253035\nML2-MT2MLMTML\nCFA(50nm)/MgOKerr rotation (mdeg)\nSample orientation (degrees) Fit\n FitMLMT\nML2-MT2ML\nCFA(115nm)/MgOKerr rotation (mdeg) Fit\n Fit\n0 2 04 06 08 0 1 0 0 1 2 002420222426283032Amplitude, offset (mdeg)\nFilm thickness (nm) ML\n Amplitude of MLMT\n Amplitude of ML2-MT2\n Offset of MLMT\nFigure 6: (Colour online) (a) Separated quadratic MOKE\ncontributions as a function of the sample orientation at 46°\nincidence. The fits are obtained using equation (12). (b) The\nMLcontribution (at angle of incidence of 46°), the amplitudes\nandoffsetofthe MLMTcontributionandtheamplitudeofthe\n(M2\nL\u0000M2\nT) as a function of the Co 2FeAl thickness.\nas evidenced in figure 5, while in contrast, for 'H= 90\u000e\n, it leads to a two steps reversal, as can be seen in figure\n5. The intermediate step leads to a magnetization nearly\nperpendicular to the applied field. For all the studied\nfilms a two steps loop is observed for 'Hranging in the\nf55\u0000130°ginterval. In figure 5c the deduced coercive\nfields (HC) from hysteresis loops along the easy direction\n('H= 0\u000e) are compared for different thicknesses (10, 20,\n45, 50, 70, and 115 nm). For both Cr-capped and Ta-\ncapped films HC increases linearly with the inverse of the\nfilmthickness. TheCr-cappedsamplespresenthigherco-\nercive fields due to the different interface quality.\nOne can also observe that MOKE hysteresis loops are\nnot strictly centrosymmetrical (see for example Fig. 5b\nfor'H= 90\u000e) indicating the superposition of symmet-\nrical (even function of applied sweep field HL) and anti-\nsymmetrical (odd in HL) components in the Kerr signal.\nIt has been shown and confirmed [26, 27] that, for in-\nplane magnetized thin films, the antisymmetrical part\nobserved in the mL(HL)loops arises from the second or-der magneto-optical effects quadratic in magnetization.\nTherefore, the present study was not limited to the usual\nlinear MOKE. We have also investigated this quadratic\ncontribution through the study of the Kerr signal depen-\ndence upon the film orientation under a saturating in-\nplane field. Within the cubic approximation for a (001)\nsurface, the Kerr rotation angle writes as [27]:\n\u0012K=a1ML+a2(M2\nL\u0000M2\nT) sin(4 )+\n(b2+ 2a2cos(4 ))MLMT(12)\nWhereMLandMTstand for the longitudinal (i.e.:\nwithin the incidence plane) and the transverse (i.e.: nor-\nmal to the incidence plane) component of the magneti-\nzation, respectively, and where  is the angle of a cubic\n<110>axis with the plane of incidence. The first term\ndescribes the usual linear contribution while the follow-\ning ones correspond to the quadratic MOKE (QMOKE).\nThe experimental study was performed under an angle\nof incidence of 46°using a field magnitude large with re-\nspect to the anisotropy field. The different contributions\nto the Kerr signal, as functions of the film orientation  \nare extracted by applying a rotating field technique [10].\nRepresentative results obtained with 115 Cr- and 50 nm\nTa-capped films are shown in figure 6. Beside the lon-\ngitudinal component ( ML) of the Kerr rotation, which\nis dominant, the QMOKE signal, which is most proba-\nbly due to the second order spin-orbit coupling [26], is\npresent. The derived ( M2\nL\u0000M2\nT) andMLMTangular\nvariations show the behaviour expected from the above\nequation.\nThe values for the amplitudes of the 2MLMTand of\nthe (M2\nL\u0000M2\nT) contributions are the same within the\nexperimental error for each sample suggesting that the\napplied cubic model is correct. The offset of the MLMT\ncontributionissmallerthantheamplitudes,butgenerally\nit follows the same trend as the amplitudes. As the thick-\nness decreases the amplitudes and the offset decrease,\nsuggesting that the chemical order progressively changes\nfrom the B2 to the A2 phase, as discussed above. More-\nover, the amplitudes and offset values of CFA are compa-\nrable to those measured for Co 2MnSi, which presents the\nL21phase [28]. The TBIIST results are discussed in the\nfollowing section, in order to allow for a comparison with\nthe data derived from the FMR study of the dynamic\nproperties.\nB. Dynamic properties\n1. Exchange stiffness and effective magnetization\nTheuniformprecessionandthefirstPSSWmodeshave\nbeen observed in perpendicular and in-plane applied field\nconfigurations for samples thicknesses down to 50 nm.\nFor the thickest film (115 nm) it was even possible to ob-\nserve the second PSSW. For lower sample thickness, the7\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6246810121416d=115 nm\n Fit\n Fit\n Fit\n FitFrequency (GHz)\nMagnetic field (kOe) Uniform mode: H=90°\n PSSW mode: H=90°\n Uniform mode: H=45°\n PSSW mode: H=45°\n14 15 16 17 18 1924681012141618Frequency (GHz)\nMagnetic field (kOe) Uniform mode\n PSSW1\n PSSW2\n Fit\n Fit\n Fitd=115 nm\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6246810121416182022\n Fit\n Fit\n Fitd=50 nmFrequency (GHz)\nMagnetic field (kOe) Uniform mode: H=0°\n Uniform mode: H=45°\n PSSW mode: H=0°\n14 15 16 17 18 19 202468101214161820d=50nmFrequency (GHz)\nMagnetic field (kOe) Uniform mode\n PSSW mode\n Fit\n Fit\nFigure 7: (Colour online) Field dependence of the resonance frequency of the uniform precession and of the two first perpendic-\nular standing spin wave excited (PSSW) mode of 115 nm Cr-capped and 50 nm Ta-capped Co 2FeAl films. The magnetic field\nis applied perpendicular or in the film plane. The fits are obtained using equations (6) and (7) with the parameters indicated\nin the Table I.\n0.00 0.02 0.04 0.06 0.08 0.101213141516171819204Meff (kOe)\n1/d (nm-1) Cr-capped films\n Ta-capped films\n Fit\nFigure 8: (Colour online) Thickness dependence of the effec-\ntive magnetization ( 4\u0019Meff) extracted from the fit of FMR\nmeasurements. The solid lines are the linear fits.\nPSSW modes are not detected due their high frequencies\nover-passing the available bandwidth (0-24 GHz). Typ-\nical in-plane and perpendicular field dependences of theresonance frequencies of the uniform and PSSW modes\nare shown on figure 7 for the 115 nm Cr- and the 50\nnm Ta-capped films. By fitting the data in figure 7 to\nthe above presented model, the gyromagnetic factor ( \r),\nthe exchange stiffness constant ( Aex) and the effective\nmagnetization (4\u0019Meff) are extracted. The fitted \rand\nAexvalues are 2.92 GHz/kOe and 1.5 µerg/cm, respec-\ntively: they do not depend of the studied sample. The\nderived exchange constant is in good agreement with the\nreported one by Trudel et al. [7]. Meff=Heff=4\u0019\nFigure 8 plots out the extracted effective magnetiza-\ntion 4\u0019Meffversus the film thickness 1=d. It can be\nseen thatMefffollows a linear variation. This allows\nto derive the perpendicular surface anisotropy coefficient\nK?S:K?S=\u00001:8erg/cm2. The limit of 4\u0019Meffwhen\n1=dtends to infinity is equal to 12.2 kOe: within the\nabove mentioned experimental precision about the mag-\nnetization at saturation it does not differ from 4\u0019MS.\nWe conclude that the perpendicular anisotropy field de-\nrives from a surface energy term; being negative, it pro-\nvides an out-of-plane contribution. It may originate from\nthe magneto-elastic coupling arising from the interfacial\nstress due to the substrate.8\n0 50 100 150 200 250 300 35051015203690200400600800\nF=6GHz\n MeasurementsF=9GHzHPP (Oe)\nApplied field direction H (degrees) MeasurementsH=255OeH=597Oe\n Fit Measurements Measurements\n F(GHz)  FitF=6GHzF=9GHz\nMeasurements Measurementsd=50 nmHr (Oe)Fit\nFit\n Fit Fit\n0 50 100 150 200 250 300 3500100200300-40-2002040\n Measurements\n Fit(Oe)\nApplied field direction H (degrees)d=50 nm\n Measurements\n FitH (Oe)\n0 50 100 150 200 250 300 35012151821681012200400600\nF=8GHzd=20 nmHPP (Oe)\nApplied field direction H (degrees) Measurements H=828 Oe H=591 OeMeasurements Fit\n FitF(GHz)Measurements F=8GHz\n FitHr (Oe)\n Measurements\n Fit\n0 50 100 150 200 250 300 3500200400600-50050\n Measurements\n Fit(Oe)\nApplied field direction H (degrees) Measurements\n FitH (Oe)d=20 nm\nFigure 9: (Colour online) Angular dependence of the resonance frequency ( Fr), resonance field ( Hr), peak to peak field FMR\nlinewidth ( \u0001HPP), inverse susceptibility ( \u001f\u00001) and the field offset ( \u000eH) of 50 nm and 20 nm thick Co 2FeAl Ta-capped thin\nfilms. The TBIIST measurements were obtained using transverse static bias field HB= 200Oe and 225 Oe respectively for 50\nnm and 20 nm thick Co 2FeAl films. The solid lines refer to the fit suing the above mentioned models.\n2. Magnetic anisotropy\nFigure 9 shows the angular dependences of the reso-\nnance field (at fixed frequency) and of the resonance fre-\nquency (at fixed applied field) compared to the static\nTBIIST measurements for three different CFA films.\nBoth FMR and TBIIST data show that the angular be-\nhavior is governed by a superposition of uniaxial and\nfourfold anisotropy terms with the above-mentioned easy\naxes. As noticed above, the symmetry properties of the\nepitaxial observed films agree with the principal direc-\ntions of the fourfold contribution. The fourfold and uni-\naxialanisotropyfieldsextractedfromthefitoftheexperi-\nmentalTBIISTandFMRdatausingtheabove-presented\nmodel are drawn on figure 10 and summarized in Table I:\nthe compared results issued from the two techniques are\nin excellent agreement. For all the samples the fourfold\nanisotropy is dominant. While the uniaxial anisotropy\nfield (H2) of the Cr-capped films is small and does notseem to depend upon the thickness, in the Ta-capped\nfilmsH2is higher, maybe due to interface effects, and is a\ndecreasing function of the thickness (Figure 10). As sug-\ngested previously, we believe that the uniaxial anisotropy\nis induced by the stepping of the substrates, probably\nresulting from a small miscut along their [100]crystallo-\ngraphicdirectioncorrespondingtothe [110]studiedfilms.\nThe reduced effect of the steps of the substrate when the\nthickness increases could then explain the thickness de-\npendence of H2. However, up to now we have no com-\npletelysatisfyinginterpretationofthepresenceof H2and\nof its variations versus the nature of the film capping.\nThe fourfold anisotropy fields ( H4) are comparable for\nCr- and Ta-capped films and decrease when their thick-\nness increases, as seen in figure 10. For large values of\nd(45nm or higher) H4lies around 200 Oe and shows\na small linear variation versus the in-plane strain \"k, as\nshown in the insert of figure 10. This evolution confirms\na direct correlation between the H4 field and the in-plane\nbiaxial strain for the films with thicknesses above 45 nm.9\n0.00 0.02 0.04 0.06 0.08 0.10 0.120102030402004006008001000\n %H4 (Oe)Anistropy fields (Oe)\n1/d(nm-1) H2:Cr-capped\n H4:Cr-capped\n H2:Ta-capped\n H4:Ta-capped\nB2 A20.5 0.6 0.7200205210215220225\nFigure 10: (Colour online) Thickness dependence of the uni-\naxial (H2) and the fourfold anisotropy fields ( H4) extracted\nfrom the fit of FMR measurements. The solid lines are the\nlinear fits. The inset shows the evolution of the H4field, for\nthe 45, 70 and 115 nm thick samples, with the in-plane biaxial\nstrain.\nAt smaller values of d(10 or 20 nm) a large increase\nofH4, up to 920 Oe, is observed. It is presumably re-\nlated to the B2)A2 phase transition observed through\nX-rays diffraction. The observed symmetry argues for\na magneto-crystalline contribution, which, as previously\nobserved [29, 30], would be higher in phase A2 than in\nphase B2.\n3. FMR linewidth\nIn figure 9, the FMR peak to peak linewidth (( \u0001HPP)\nis plotted as a function of the field angle 'Hfor the 50\nnm and 20 nm Ta-capped CFA films using three driv-\ning frequencies: 6, 8, and 9 GHz. \u0001HPPis defined as\nthe field difference between the extrema of the sweep-\nfield measured FMR spectra. All the other samples show\na qualitatively similar behaviour to one of the samples\npresented here. The positions of the extrema depend on\nthe sample. The observed pronounced anisotropy of the\nlinewidth cannot be due to the Gilbert damping contri-\nbution, which is expected to be isotropic, and must be\ndue to additional extrinsic damping mechanisms. In the\n50 nm thick sample, the \u0001HPPangular variation shows\na perfect fourfold symmetry (in agreement with the vari-\nation of the resonance position). Such behaviour is char-\nacteristic of two magnon scattering. This effect is cor-\nrelated to the presence of defects preferentially oriented\nalong specific crystallographic directions, thus leading to\nanasymmetry(seeequation(11)). Concerningthe20nm\nthick film, the in-plane angular dependence of \u0001HPPis\nless simple and shows eight maxima, that is expected\nfrom a mosaicity driven linewidth broadening. It can be\n4 6 8 1 01 21 41 61 82 02 22 410203040506070\n10 nm, \nH=90°10 nm, \nH=45°20 nm\n50 nm70 nmFMR linewidth HPP (Oe)\nFrequency (GHz)Figure 11: (Colour online) Frequency dependence of the easy\naxis ('H= 0) peak to peak field FMR linewidth ( \u0001HPP) for\nCo2FeAl thin. The solid lines refer to the fit using equations\n(8-11).\nseen that a smaller fourfold symmetry (four maxima) is\nsuperimposed on the eight maxima, indicating that two\nmagnon scattering is still present. Therefore, the entire\nangular dependence of the FMR linewidth in our samples\ncan be explained as resulting of the four contributions\nappearing in equation (8).\nIn figure 11, \u0001HPPfor the field parallel to an easy axis\nand a hard axis ( 'H= 45 °for 10 nm thick sample) of\nthe fourfold anisotropy is plotted as a function of driving\nfrequency for all samples. An apparently extrinsic contri-\nbution to linewidth was observed, which increased with\ndecreasing film thickness. It should be mentioned that\nthe observed linear increase of the linewidth with fre-\nquency in figure 11 maybe due to Gilbert damping but\nother relaxation mechanisms can lead to such linear be-\nhaviour. Therefore, only an effective damping parameter\n\u000beffcan be extracted from the slope of the curves and\nranges between about 0.00154 for the easy axis of the 50\nnm thick film and 0.0068 for easy axis the thinnest film.\nThe pertinent parameters could thus be, in principle de-\nrived from the conjointly analysis of the frequency and\nangular dependence of \u0001HPP. However, due to the lim-\nited experimental precision, some additional hypotheses\nare necessary in order to allow for a complete determi-\nnation of the whole set of parameters describing the in-\ntrinsic Gilbert damping and the two magnon damping.\nA detailed analysis is presented in the appendix. Using\nthe previously reported value: \u000b= 1:1\u000210\u00003[31], which\nis in agreement with our experimental results, we were\nable to \u00000for each film. \u00000,\u00002,\u00004,'2,'4are listed in\nTable II which also contains the parameters describing\nthe damping effects of the mosaïcity ( \u0001'H) and of the\ninhomogeneity contribution ( \u0001Hinh).\nThe two magnon and the mosaïcity ( \u0001'H) contribu-\ntions to \u0001HPPincrease when the thickness decreases,\nprobably due to the progressive above reported loss of10\nchemical order. The increase of the residual inhomo-\ngeneities linewidth ( \u0001Hinh) with the thickness is most\nprobably due the increase of defects and roughness. The\nuniaxial term \u00002is observed only in the thinnest (20 and\n10 nm) samples. As expected, '4= 0, but the sign of\n\u00004is sample dependent. Finally, it is important to no-\ntice that the very low value of the intrinsic damping in\nthe studied samples allows for investigating the extrinsic\ncontributions.\nV. CONCLUSION\nCo2FeAl films of various thicknesses (10 nm \u0014d\u0014115\nnm)) were prepared by sputtering on a (001) MgO sub-\nstrate. They show full epitaxial growth with chemical\norder changing from B2 to A2 phase as thickness de-\ncreases. MOKE and VSM hysteresis loops obtained with\ndifferent field orientations revealed that, depending on\nthe direction of the in-plane applied field, two or one\njump switching occur, due to the superposition of uni-\naxial and fourfold anisotropies. The samples present a\nquadratic MOKE contribution with decreasing ampli-\ntudes as the CFA thickness decreases. The microstrip\nferromagnetic resonance (MS-FMR) and the transverse\nbiased initial inverse susceptibility and torque (TBIIST)\nmethods have been used to study the dynamic proper-\nties and the anisotropy. The in-plane anisotropy presents\ntwo contributions, showing a fourfold and a twofold ax-\nial symmetry, respectively. A good agreement concern-\ning the relevant in-plane anisotropy parameters deduced\nfrom the fit of MS-FMR and TBIIST measurements has\nbeen obtained. The fourfold in-plane field shows a thick-\nness dependence behavior correlated to the thickness\ndependence of the chemical order and strain in sam-\nples. The angular and frequency dependences of the\nFMR linewidth are governed by two magnon scattering,\nmosaïcity and by a sample independent Gilbert damping\nequal to 0.0011\nAppendix\nIn the section dealing with the discussion of the FMR\nlinewidth measurements we stated that the conjointly\nanalysis of the frequency and angular dependence of\n\u0001HPPdoes not allow for the determination of all the\nparameters given in equation (8) and additional hypoth-\nesisshouldbedone. Theaimofthisappendixistoclarify\nthemannerinwhichtheparameterssummarizedinTable\nII is done.\nFor most of the exploitable measurements the mi-\ncrowave frequency f during the \u0001HPPmeasurements is\nnot larger than f0and generally smaller ( f0varies from\n18.5 to 28.5 GHz, depending on the film thickness). It\nthen results that the two magnon damping is practically\nproportional to f and that the sum of the Gilbert and ofthe two magnon damping terms reads as (see equations\n(9) and (11)):\n\u0001HGi+2mag\u0018=((\u000bp\n3+\u00000\n2Heff)+\u00002\n2Heffcos 2('H\u0000'2)\n+\u00004\n2Heffcos 4('H\u0000'4))4\u0019\n\rf(13)\nIt is not possible to completely identify the respec-\ntive contributions of the Gilbert and of the two magnon\ndamping, only according to equation (13). The quasi-\nlinear variation versus the frequency (Fig. 11) observed\nfor\u0001HPPallows for defining an effective damping pa-\nrameter\u000beff, which, is angle dependent due to two\nmagnon scattering. The experimentally derived coeffi-\ncient\u000beff, from the linear fit of data presented in figure\n11, varies from 0.0068 to 0.00154. Furthermore, the mea-\nsured angular variation of the linewidth allows for evalu-\nating ( \u00002,'2) and ( \u00004,'4) but, concerning the isotropic\nterms appearing in equation (13), only the sum \u000b+p\n3\u00000\n2Heff\ncan be derived. However, remembering that \u000bcannot be\nnegative, the maximal available value of \u00000(correspond-\ning to\u000b= 0) is easily found. Moreover, a lowest value\ncan be obtained for \u00000noticing that equation (13) can\nalso be written:\n\u0001HGi+2mag\u0018=((\u000bp\n3+\u00000\u0000j\u00002j\u0000j\u00004j\n2Heff)+\nj\u00002j\n2Heff(1\u0006cos 2('H\u0000'2))+\nj\u00004j\n2Heff(1\u0006cos 4('H\u0000'4)))4\u0019\n\rf(14)\nwhere the adequate third and the fourth terms rep-\nresent the twofold and the fourfold contributions, which\ntake into account that both of them are necessarily non-\nnegative for any value of 'H. The additional residual\ntwo magnon isotropic contribution cannot be negative.\nHence: \u00000>j\u00002j+j\u00004j.\nIntroducingthisminimalaccessiblevalueof \u00000, (j\u00002j+\nj\u00004j), the maximal value of the Gilbert coefficient \u000bis\nthen easily obtained. To summarize, for each sample the\nexperimental data provide the allowed intervals for \u000band\nfor\u00000, respectively [0, \u000bmin] and [(j\u00002j+j\u00004j),\u00000Max],\nand indeed, the chosen value of \u000bwithin [0,\u000bmin] allows\nfor deducing \u00000. The smallest calculated interval for \u000b,\nequal to [0, 1:4\u000210\u00003] is obtained for the 70 nm film.\nA previous publication by Mizukami el al. [31] has con-\ncluded to a Gilbert coefficient equal to: \u000b= 1:1\u000210\u00003.\nWe then stated that \u000b= 1:1\u000210\u00003and, consequently,\nwe were able to deduce \u00000for each film. \u00000,\u00002,\u00004,'2,\n'4are listed in Table II which also contains the parame-\nters describing the damping effects of the mosaïcity and\nof the inhomogeneity.11\n[1] [1] A. Yanase and K. Siratori, J. Phys. Soc. Jpn. 53, 312\n(1984)\n[2] [2] Z. Zhang and S. Satpathy, Phys. Rev. B 44, 13319\n(1991)\n[3] [3] K. Schwarz, J. Phys. F: Met. Phys. 16, L211 (1986)\n[4] [4] J. H. 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Phys.105,\n07D306 (2009)"    },    {        "title": "1506.00780v2.Respective_influence_of_in_plane_and_out_of_plane_spin_transfer_torques_in_magnetization_switching_of_perpendicular_magnetic_tunnel_junctions.pdf",        "content": "Respective influence of in-plane and out-of-plane \nspin-transfer torques in magnetization switching of  \nperpendicular magnetic tunnel junctions \n \nA.A. Timopheev 1,2,3 , R.Sousa 1,2,3 , M.Chshiev 1,2,3 , L.D. Buda-Prejbeanu 1,2,3 ,  \nB. Dieny 1,2,3 \n \n 1. Univ. Grenoble Alpes, INAC-SPINTEC, F-38000 Gre noble, France \n 2. CEA, INAC-SPINTEC, F-38000 Grenoble, France \n 3. CNRS, SPINTEC, F- 38000 Grenoble, France \n  \nAbstract \n \nThe relative contributions of in-plane (damping-lik e) and out-of-plane (field-like) spin-\ntransfer-torques in the magnetization switching of out-of-plane magnetized magnetic tunnel \njunctions (pMTJ) has been theoretically analyzed us ing the transformed Landau-Lifshitz (LL) \nequation with the STT terms. It is demonstrated tha t in a pMTJ structure obeying macrospin \ndynamics, the out-of-plane torque influences the pr ecession frequency but it does not \ncontribute significantly to the STT switching proce ss (in particular to the switching time and \nswitching current density), which is mostly determi ned by the in-plane STT contribution. This \nconclusion is confirmed by finite temperature and f inite writing pulse macrospin simulations \nof the current-field switching diagrams. It contras ts with the case of STT-switching in in-plane \nmagnetized MTJ in which the field-like term also in fluences the switching critical current. This \ntheoretical analysis was successfully applied to th e interpretation of voltage-field STT \nswitching diagrams experimentally measured on perpe ndicular MTJ pillars 36 nm in diameter, \nwhich exhibit macrospin-like behavior. The physical  nonequivalence of Landau and Gilbert \ndissipation terms in presence of STT-induced dynami cs is also discussed. \n \n  1.  Introduction  \nFully perpendicular magnetic tunnel junctions (pMTJ ) constitute the storage element \nof spin-transfer-torque magnetoresistive random acc ess memory (STT-MRAM) [1-6]. STT-\nMRAM are very promising emerging non-volatile memor ies since they combine non-volatility, \nlow energy consumption, high thermal stability and almost unlimited endurance. The \nstrongest research and development efforts are nowa days focused on out-of-plane \nmagnetized MgO-based MTJs. Indeed, the latter combi ne several advantages.They exhibit a \nhigh tunnel magnetoresistance effect [7] amplitude due to a very efficient spin-filtering \nphenomenon associated with the symmetry of the tunn eling electron wave function [8,9]. \nFurthermore, they present a very large interfacial perpendicular anisotropy at the interface \nbetween the magnetic electrode and the MgO oxide ba rrier (Ks~1.4erg/cm²) [10] which allows \nto achieve a quite high thermal stability of the st orage layer magnetization and therefore a \nlong memory retention. In addition, a remarkable pr operty of this interfacial anisotropy is that \nit exists in materials having weak spin-orbit coupl ing and therefore relatively low Gilbert \ndamping α (α<0.01). This is very important in STT-MRAM since th e critical current for STT-\ninduced switching [11,12] of the storage layer magn etization is directly proportional to the \nGilbert damping. The advantage of using out-of-plan e magnetized MTJs in STT-MRAM rather \nthan in plane ones is twofold: firstly, the interfa cial perpendicular anisotropy at CoFeB/MgO \ninterface provides higher thermal stability at smal ler dimensions (sub-60nm) than the usual \nshape anisotropy provided by giving elliptical shap e to in-plane magnetized MTJs. Secondly, \nfor a given retention i.e. a given thermal stabilit y factor, the critical current for STT-induced \nswitching is lower with out-of-plane magnetized sto rage layer than it is for an in-plane \nmagnetized one [13,14].  \nFrom a theoretical point of view, a first approach to STT-induced switching can be \ndeveloped by solving the Landau Lifshitz Gilbert eq uation under the assumptions of zero kelvin \nmacrospin approximation under stationary applied sp in-polarized current. The equilibrium \nconfigurations of the system can thus be calculated  and the precessional dynamics of the \nsystem submitted to a small perturbation from the s tatic equilibrium can be studied. This \nallows to derive the threshold current required to achieve STT switching, as it was done in \nRefs. [13-15]. Thermal fluctuations can be taken in to account in several limiting cases using \nFokker-Planck equation. Thermal activation mainly d ecreases the threshold current value and \nthe switching time introducing an undesirable effec t of stochasticity in magnitude of both \nparameters [16,17]. The influence of the writing pu lse duration was also theoretically studied \n[16,18-21]. \nDespite the numerous experimental results [22, 23] and micromagnetic simulations \n[24-26] generally pointing on quantitative disagree ments with the macrospin-based \nestimations, usage of the macrospin approach is sti ll justified for at least for two reasons. First \nof all, it gives a simple but solid picture of the physical processes involved in the STT switching \nthat creates a common basis for qualitative analysi s of the different magnetic multilayered \nsystems, while most of the conclusions derived from  micromagnetic approaches are rather of \nparticular character. Micromagnetic behavior can be  mimicked, for example, by introduction \nof an effective activation volume instead of Stoner -Wohlfarth behavior, but still using a thermal activation model for the subvolume [22]. Se condly, considering the general trend to \nreduce the volume of the storage element (and, cons equently, the energy needed per \nwrite/read cycle), magnetic memory elements will ev entually behave in a macrospin manner. \nBased on these viewpoints, we investigated the STT switching in fully perpendicular \nmagnetic tunnel junction systems, where in addition  to Slonczewski STT term (sometime \ncalled in-plane torque since it lies in the plane d efined by the local magnetization and that of \nthe spin-polarization usually defined by the magnet ization direction of the reference pinned \nlayer), having damping-like structure, an out-of-pl ane, or field-like term exists. Several \ntheoretical works predicted that the torque produce d by out-of-plane STT term could reach \nan amplitude comparable to that of in-plane torque [27-29]. Several experimental works \ncarried out on in-plane MTJ structures have already  estimated it to be in the range of 30-40 % \nof the in-plane torque [30-33]. It was mentioned [3 4] that its presence may lead to a \nbackswitching process, a very undesirable effect in  magnetic memory applications causing \nwrite errors.  \nIn this study, after having analyzed the Landau-Lif shitz-Gilbert-Slonczewski equation \nmathematically transformed into Landau-Lifshitz for m, we show that in fully perpendicular \nMTJ structures, the field-like torque plays a negli gible role in the switching process. In contrast \nto in-plane MTJ systems [30-34], it only influences  the precessional frequency preceding the \nswitching but the switching current density is prim arily determined by the in-plane STT term. \nThe experiment carried out on 36nm diameter pMTJ pi llar supports our conclusions. \n \n2.  Phase boundaries from LLG equation transformed into  LL equation \nThe most accepted form of LLG equation describing d ynamics of a macrospin under constant \nspin polarized current can be presented as follows:   \n\u0001\t\u0003\u0004\n\u0001\t\u0005=−\b\t\u0003\u0004×\t\u000b\f\f\f\r\u000e\u000f\u000f \u0010+\u0012\u0013\u0003\u0004×\u0001\t\u0003\u0004\n\u0001\t\u0005\u0014−\b\t\u0003\u0004×\u0015\u0003\u0004×\u0016∥\t\u0018\u0004\u0019+\b\t\u0003\u0004×\u0016\u001a\t\u0018\u0004 ,  (1)  \nhere \u0003\u0004=\t\u001b\f\f\f\r\n\u001c\u001d – unit vector along the free layer magnetization d irection (M\u001f – free layer’s \nvolume magnetization saturation parameter), \t\u000b\f\f\f\r\u000e\u000f\u000f \teffective field (comprising applied field, \nanisotropy field, demagnetizing field),   \u0018\u0004 – unit vector along the polarizer layer magnetizat ion \ndirection, α Gilbert damping, γ gyromagnetic ratio. \u0016∥ and \u0016\u001a are, respectively, in-plane \n(damping-like) and out-of-plane (field-like) spin-t ransfer-torque prefactors. Both prefactors \ncan be phenomenologically represented as functions of the spin polarization in the magnetic \nelectrodes, current density or voltage bias applied  to the tunneling barrier as will be done later \nin the text.  \nIn-plane and out-of-plane STT terms as written in E q. (1) are geometrically equivalent \nto the precession and damping terms of Landau-Lifsh itz equation. One can therefore \ntransform Eq. (1) into Landau-Lifshitz form using t he standard technique, i.e. by making a \u0003\u0004× \nproduct on both sides of equation: \u0003\u0004×\u0001\t\u0003\u0004\n\u0001\t\u0005=−\b\t\u0003\u0004×\t\u0003\u0004×\t\u000b\f\f\f\r\u000e\u000f\u000f \u0010+\u0012\t\u0003\u0004×\u0013\u0003\u0004×\u0001\t\u0003\u0004\n\u0001\t\u0005\u0014−\b\t\u0016∥\t\u0003\u0004×\t\u0003\u0004×\u0015\u0003\u0004×\u0018\u0004\u0019\u0010+\n\b\t\u0016\u001a\u0003\u0004×\u0015\u0003\u0004×\t\u0018\u0004\u0019, \nand putting obtained in a replacement of the dampin g term in Eq.(1). This yields, \n\t !\"#\u0010\n$\u0001\t\u0003\u0004\n\u0001\t\u0005=−\t\u0003\u0004×\t\t\u000b\f\f\f\r\u000e\u000f\u000f −\t\u0015\u0016\u001a+\u0012\t\u0016∥\u0019\t\u0018\u0004\u0010−\u0003\u0004×\u0013\u0003\u0004×\t\t\u0012\t\u000b\f\f\f\r\u000e\u000f\u000f −\u0015\u0012\t\u0016\u001a−\u0016∥\u0019\t\u0018\u0004\u0010\u0014. (2) \nTo this moment, all the transformations born only a  character of mathematical identities and \nEq. 2 is valid for any system with any configuratio n of \u000b\f\f\f\r\u000e\u000f\u000f   and \u0018\u0004. Rewritten in such a way, it \nacquires more suitable form for further analytical treatment because dynamics in this system \nis fully determined by two vectors, namely \t\t\u000b\f\f\f\r\u000e\u000f\u000f −\t\u0015\u0016\u001a+\u0012\t\u0016∥\u0019\t\u0018\u0004\u0010 and \t\u0012\t\u000b\f\f\f\r\u000e\u000f\u000f −\u0015\u0012\t\u0016\u001a−\n\u0016∥\u0019\t\u0018\u0004\u0010, which have many similarities and whose form can b e significantly simplified as soon as \nactual geometry of \u000b\f\f\f\r\u000e\u000f\u000f   and  \u0018\u0004 has been set.  Also Eq. (2) is more convenient to u se in \nnumerical integration schemes. Further analysis wil l be focused on the case of pMTJ structure \nassuming macrospin dynamics of the storage layer de scribed by Eq.(2). \nWe consider fully perpendicular magnetic tunnel jun ctions submitted to an out-of-\nplane external magnetic field \u000b\f\f\f\r\u000e%&  therefore applied parallel to the symmetry axis. T his \nsituation allows analytical analysis wherein the qu antities \u000b\f\f\f\r\u000e\u000f\u000f ,\t\u0018\u0004\t,\t\u000b\f\f\f\r\u000e%& \t,( ) remain collinear \nindependently of the instantaneous direction of \u0003\u0004. The magnetic free energy density \nfunctional U of such system depends only on one var iable * – the angle between \nmagnetization vector \u0003\u0004 and quantization axis ( )  (see Fig.1). It writes: \n+ = \u0015,\u001a−2./01\u0019sin 1*−/0567\u0005 cos*.     (3) \nWhen |567\u0005 |<5\u001a, 5\u001a=1=>\n?@−4./0,\t567\u0005 =\u000b\f\f\f\r\u000e%& ∙( ), there are two stable magnetic \nmoment orientations which are independent of 567\u0005  and always collinear with ( ): \nCDE \nDF =0,\tDHE\nDFH >0,\n−5\u001a<567\u0005 <5\u001a,\n5\u001a>0.\t\t→\t\t  *K=0,  *K=..\t    (4) \n  \nThe collinearity of the four vectors \u000b\f\f\f\r\u000e\u000f\u000f ,\t\u0018\u0004\t,\t\u000b\f\f\f\r\u000e%& \t,( ) yields great simplifications in equation \n(2), allowing to work only with the magnitudes \t\u0016\u001a, \u0016∥ and \t56LL :  \n\t !\"#\u0010\n$\u0001\t\u0003\u0004\n\u0001\t\u0005=−\t\u0003\u0004×M\t( )−\u0003\u0004×\u0015\u0003\u0004×N( )\u0019,\nM=\t56LL −\t\t\u0016\u001a+\u0012\t\u0016∥\u0010,\nN =\u0012\t5 6LL −\t\u0012\t\u0016\u001a−\u0016∥\u0010,\n\t56LL =\u000b\f\f\f\r\u000e\u000f\u000f ∙( ) =−DE \nD\u001b\f\f\f\r∙( )=5\u001a\u0013cos*K+567\u0005 5\u001aO \u0014.   (5) \nHere, two scalar parameters M,N  are introduced, which represent the direction and \nmagnitude of the perpendicular and in-plane (the pl ane is formed by \u0003\u0004 and  \t\u0018\u0004) effective torques (see Fig.1) acting on the magnetization whe n the latter departs from its equilibrium \nposition *K (0 or π) because of thermal fluctuations.  \nImportant specifics of the considered system is tha t M-parameter cannot change the \norbit (i.e. the angle * ), it only influences the frequency of the precess ion. One can derive \nferromagnetic resonance (FMR) condition, which is j ust a modified “easy-axis” Kittel’s  formula \nfor this case: \nP\bO=\t56LL −\t\t\u0016\u001a+\u0012\t\u0016∥\u0010,    (6) \nwere P  is the angular frequency of the resonance precess ion. One can see, that if \t\u0016\u001a>\n\t56LL −\u0012\t\u0016∥   the precession direction will be changed, while increase or decrease of * is \nexclusively determined by the sign of N -parameter, wherein the damping-like STT-term is \ndominating since α is usually small ( typically in the range 0.007 to  0.02). The precessional \nresponse of the system before the switching could b e measured for instance by measuring P \nversus the DC applied voltage bias, QRSTU , on a single pMTJ pillar either by RF voltage freq uency \ndetection, noise measurements [35], spin-torque exp eriments or by microfocused BLS FMR \ntechnique. The excitation frequency would give acce ss to \u0016\u001a\u0015QRSTU \u0019  dependence, while the \nFMR linewidth parameter change versus QRSTU   would reflect mostly the \u0016∥\t\u0015QRSTU \u0019 \ndependence.  \n Turning back to the analysis of Eq. (5) and Fig. 1 , one can note that only the damping \nterm,  \u0003\u0004×\u0015\u0003\u0004×N( )\u0019, can change the precession angle *. It is therefore possible to derive the \nboundary conditions for a current-magnetic field st ability phase diagram. The magnetization \nswitching process starts when N -parameter changes sign. This condition yields the  threshold \ncriterion for the STT-induced magnetization switchi ng:    \n\u0012\t56LL +\t\u0016∥−\u0012\t\u0016\u001a=0   (7) \nOne can see from Eq. (7) that the contribution from  the in-plane STT term ( \u0016∥) is largely \ndominating the switching process. Indeed, the in-pl ane torque is of the order of \u001256LL while \nthe contribution of the perpendicular torque is wei ghted by the Gilbert damping resulting in a \nmuch weaker influence in the switching process. Her e one can note again that the best \nmethod to determine experimentally \u0016\u001a is through FMR measurements, and not from the \ninfluence of \u0016\u001a on the (current, field) phase diagram boundaries s ince the latter is very weak. \nIndeed, from the above discussion, being able to se e an influence of \u0016\u001a on the phase diagram \nboundaries would require to have \t\u0016\u001a≈\u0016∥\u0012O which seems to be physically unachievable in \nstandard pMTJ systems [27-34]. Also, as it will be shown in Sec.6, the \u0012\t\u0016\u001a term in the Eq. (7) \ndisappears if one chooses the dissipation term in t he Landau-Lifshitz formulation. In any case, \nEqs.(6,7) are quite useful for the analysis of STT switching experiments performed on pMTJ \nsystems.  \n3.  Stability phase diagram boundaries  \nHaving set the relations between electric current f lowing through pMTJ and the spin-\ntorque prefactors magnitudes, one can construct the  stability phase diagram explicitly from \nEq. (7) assuming that the spin-polarized current pu lse is long enough to complete any STT induced switching while influence of the thermal fl uctuations is limited to setting small initial \nmisalignment angle *K, so that |cos*K|≈1\t  . Modification of the phase boundaries due to \nthermal fluctuations and under short pulse writing regime, which are essential in real magnetic \nmemory applications, will be analyzed in the follow ing sections, while in this section the \nconditions of long-pulse and low-temperature regime  are assumed. \nIn most investigated pMTJs, one can expect the cond ition \t\u0016\u001a<\u0016∥ and \u0012\t\u0016\u001a≪\u0016∥ to \nbe fulfilled. In this case, one can set \t\u0016\u001a=0 and build up the boundaries of the (current, field ) \nstability phase diagram. In absence of the spin-pol arized current ( \u0016∥=0,\t\u0016\u001a=0 ), the \nswitching occurs when \u0012\t56LL  changes sign, i.e. when 567\u0005 =−5\u001a for *K=0 and 567\u0005 =5\u001a \nfor *K=.. This defines the vertical boundaries on the diagr am shown in Fig. 2a, depicted by \ndashed vertical lines. For 567\u0005 =0  and setting \u0016∥=Y\u0005∥\tZ[\tQRSTU    (Y\u0005∥=\tℏ\n16∙]\n\u0005^?_  = STT \nconversion efficiency factor, in units of Oe /(A· cm a1\u0019 ; b  – effective spin polarization \nparameter; Z[ – tunneling conductance factor, generally dependen t on  * and QRSTU , in units \nof Ωa cm a1, representing in the simplest interpretation the i nverse of the RxA product) one \ncan obtain that the switching current density dUe  is proportional to \u0012\t5\u001a:  \ndUeK=Z[QUeK=\"\tf>\n\tUg∥=16\n\tℏ∙\u0005^\t\"\tf>?_\n].   (8) \nIn the case 567\u0005 ≠0, relation (8) leads to a linear dependence between  the switching current \nand external magnetic field, yielding a linear slop e on the switching phase diagram given by: \n\u0001\ti_j \n\u0001\tfklg =\"\t\n\tUg∥=16\n\tℏ∙\u0005^\t\"\t?_\t\n].    (9) \nOne can conclude that if the effective spin polariz ation parameter b is constant (i.e. weakly \ndependent on the bias voltage QRSTU ), then the STT driven parts of the switching diagr am is \nlinearly dependent on the applied field with the sl ope proportional to the intrinsic damping \nparameter \u0012 and inversely proportional to the STT efficiency p refactor \tY\u0005∥ and with the zero-\nfield switching current magnitude being proportiona l to the effective perpendicular anisotropy \n5\u001a . One should also note that Eq. (8) is in full agr eement with other previously obtained \nexpressions [13-15, 36] for the zero field threshol d switching current derived from the analysis \nof precessional response of the system, assuming li near dependence of the damping-like STT \nprefactor versus the applied current. In our case, Eq.(7) allows one to calculate I-H stability \nphase diagram boundaries for any \u0016∥,\t\u0016\u001a  prefactors with arbitrary bias current (voltage) \ndependence, or by choosing it from the theoretical estimations made for the concrete MTJ \nsystem [28,29]. \nSimultaneous influence of both in-plane and out-of- plane STT terms on the phase \nboundaries is shown on Fig.2b. We have chosen reali stic values for the magnetic system (see \nthe figure caption) letting the in-plane prefactor be linearly dependent on bias voltage with  \nY\u0005∥ = 67 ∙Z[a  Oe/Volt. As for the out-of-plane prefactor, \t\t\u0016\u001a, we show three different cases:   \nzero, quadratic dependence with Y\u0005\u001a1 = 154 ∙Z[a1 Oe/(Volt)2 and the third one – quadratic + \nlinear dependence (which mimics features of an asym metric MTJ structure, see expression in \ncaption of Fig.2) with an unreasonably large STT co nversion coefficients. One can see, that within −5\u001a<567\u0005 <5\u001a the difference between the phase boundaries in all  three cases is \nnegligible. The second case uses exactly the same p arameters as the ones used in Ref.[15] in \nFig. 3. We can see that the boundaries calculated a nd simulated there are identical to all our \nthree cases: no matter what kind of prefactor depen dence is introduced for the out-of-plane \nSTT term. This confirms that the out-of-plane STT t erm has a negligible influence on the STT \nswitching diagram. Parabolic shape of the boundarie s starts being observed only in the third \ncase and it becomes noticeably different only for c urrent magnitudes several times larger than \nthe threshold switching current. Thus, one can conc lude that under long-pulse/low-\ntemperature conditions, the STT switching in fully perpendicular MTJ structures obeying \nmacrospin dynamics is almost not influenced by the out-of-plane STT term and by its actual \nprefactor bias voltage or current dependence. Below , we will show that this statement is still \nvalid at finite temperature and reasonably short wr iting pulses. \n \n4.  Macrospin simulations  \nAiming at extending the conclusions made in the pre vious sections to the case of finite \ntemperatures and finite writing pulse regime, a ser ies of macrospin simulations were \nperformed using Eq. (2) (i.e. with Gilbert damping) . The simulations were carried out with a \nfixed writing pulse duration of 40 ns and a cumulat ive integration time of 1 µs for each field \npoint. The following assumptions of bias voltage de pendences for the STT prefactors were \nused: \t\u0016\u001a=Y\u0005\u001a1\tZ[1\tQRSTU 1  and \u0016∥\t=Y\u0005∥\tZ[QRSTU \t , which is the case of symmetrical MTJ \nsystems with high spin polarization parameter. For convenience, the parameter Z[  was set \nconstant equal to 1 Ωa cm a1. The temperature was included in the form of stoch astic thermal \nfield Hth  with Gaussian distribution [37], added directly to  the effective field Heff . Statistical \nproperties of these thermal fluctuations are given by the following relations: \n 0) (, th =tHi  \nand \n ) '(2) ' () (\nSB\n, th , th ttVMTktHtHij \npj i − = δδγα  \n \nwhere kB is the Boltzman constant, and Vp  the free layer volume. The chosen LLG equation is \nintegrated with a (predictor-corrector) Heun scheme  [38].  Here we used Q[=2.07x10 -17  \ncm 3,\t5\u001a = 200 Oe,  /U= 1000 emu/cm 3, which gives the effective stability factor at T =  300 K: \n∆=f>?@no\n1\tpqr=50 . \nThis set of the parameters was chosen to mimic work ing conditions of an actual STT-MRAM \ndevice. Two sets of macrospin simulations, at T=0K and T=300K respectively, presented on \nFig.3 show how the phase boundaries are changed for  the different combinations of in-plane \nand out-of-plane STT-term prefactors magnitudes. We  will discuss firstly the results shown in \nFig. 3a corresponding to the case with finite pulse  duration and no thermal fluctuations (T=0K).  The finite duration of the writing pulse brings two  main effects. Firstly, the STT-driven \nboundaries are shifted towards much higher voltages  (currents). Evidently, to achieve \nswitching within the considered finite time period,  one has to apply higher amplitudes for the \nwriting pulses. On the initial stage, when \u0003\u0004 is almost collinear with the symmetry axis ( ), the \ntorque is very weak which results in a very slow ST T induced dynamics in the system. It is \nevident that in absence of thermal fluctuations, th e switching time from \u0003\u0004∥( ) initial would \nbe infinite for any spin-polarized current magnitud e [13,14]. To avoid this in the T=0K \nsimulations, a small misorientation (0.1°) between  \u0018\u0004 and  Hext  was introduced in the system. \nThe second effect is nonlinearities of the phase bo undaries which are seen even on the \ndiagrams with the in-plane STT-term only.  This eff ect is linked with non-linear dependence of \nthe time necessary for STT switching versus the app lied magnetic field. Both effects are \nentirely of dynamical nature and their influence on  the phase boundaries can be theoretically \ndescribed using the formalism developed in Ref. [16 ]. Renormalization of the effective dynamic \ntime allows one to link dependence between the crit ical current, pulse width and finite \ntemperature. This will be also done in the next sec tion, while here the discussion will be \nfocused on a qualitative analysis of the relative c ontributions of the in-plane and out-of-plane \nSTT terms to phase boundaries shapes. \nOne can see from Fig.3a that the general behavior o f the phase boundaries modification \non the simulated phase diagrams under finite writin g pulse regime is in agreement with the \nconclusions made in the previous sections for the D C regime. For the case of Y\u0005\u001a1 = 400 ∙Z[a1  \nOe/(Volt)2 and Y\u0005∥  = 0 ∙Z[a Oe/Volt, the simulated phase diagram demonstrates a \nunidirectional STT switching due to quadratic depen dence of \t\u0016\u001a  versus applied voltage. In \nother words, only switching to the antiparallel con figuration is possible for Y\u0005\u001a1> 0, Y\u0005∥ = 0.  \nZero-field ( 567\u0005 =0) STT switching voltage for this diagram is =+/-1.6 V. This voltage induces \nan effective STT field in the damping term of Eq.(2 ) ~1000 Oe, which is five times higher than \nthe effective perpendicular anisotropy field 5\u001a  =200 Oe.  At the same time, if one adds a \nrelatively small damping-like prefactor Y\u0005∥ =30 ∙Z[a   Oe/Volt  it completely removes any \napparent influence of the field-like STT term from the phase diagram despite of the huge value \nchosen for its prefactor. When the effective contri butions from both prefactors are \ncomparable, the phase diagram acquires a noticeable  asymmetry, as can be seen for the last \ntwo diagrams in the middle column. However, such co mbination of Y\u0005∥ and Y\u0005\u001a1 can be already \nphysically unrealistic. \nFigure 3b shows the same set of simulations made un der T=300K conditions. Several \ntemperature-induced effects are observed there: i) Decrease of the coercive field showing \nthat thermally activated magnetization reversal tak es place when the external magnetic field \nsubstantially lowers the effective barrier height i n the system; ii ) Shift of the voltage-driven \nparts of the boundaries towards lower switching vol tages. Thermal fluctuations of the \nmagnetic moment direction increases the probability  to launch STT switching thanks to a \nthermally-induced misorientation between \u0003\u0004 and \t\u0018\u0004. This increases the initial STT amplitude \nand substantially decreases the switching time for a given writing pulse amplitude. This is \nconsistent with earlier observations in STT-MRAM ce lls and with theoretical expectation of a \n\n\n\n\n\nΔ−=\n001ττLn ET kIIB\ncc  dependence of the switching current on the pulse duration under \nfinite temperature [39]. Therefore, Fig.3b Indicates that the general features observed in the  \nswitching phase diagram at 0K  (i.e. Fig.3a) are conserved at finite temperature and illustrates \nagain the negligible role of the out-of-plane STT t erm in the switching process (see in particular \nthe last column in Fig.3b).  \n \n5.  Experimental measurements of the (I,H) switching di agram \nIn this section, the STT efficiency and other magne tic parameters of pMTJ pillars are \ndirectly extracted from the measured diagram. Nomin al 50 nm diameter pMTJ pillars were \nfabricated from an MTJ stack grown by magnetron spu ttering. The stack contains a 1.7nm \nthick Co 20 Fe 60 B20  free layer sandwiched between two MgO barriers. Ma gnetization saturation \nparameter of the free layer was measured to be 1030  emu/cm 3. Current in-plane \nmagnetotransport measurements (CIPTMR) yielded RxA = 5.7 Ω µm2 and TMR=126 %. The \nsecond MgO barrier was introduced to increase the p erpendicular anisotropy of the free layer. \nIt has a negligible resistance-area (RA) product co mpared to the main tunnel barrier.  The \nbottom fixed layer is a synthetic antiferromagnetic -based perpendicularly magnetized \nmultilayer and the polarizer material has the same composition as for the free layer. The \nmetallic electrode above the second MgO barrier is non-magnetic. Experimentally, it was \nfound that the actual pillar diameter slightly diff ers from its nominal value due to the \nnanofabrication technology (36nm instead of 50nm no minal). This was recalculated using the \nvalues of the low resistive state ( Rpp  = 5.6 kΩ) of the magnetoresistance curve (Fig.4a) and \nassuming that RxA value is preserved after the nano fabrication. Knowing the volume of the \nfree layer in the pillar, \tQ[ , its room temperature coercivity, measurement tim e (~1s) and \nattempt frequency tK = 10 10  s-1, one can recalculate the perpendicular magnetic an isotropy \nfrom Neel-Brown formula [40,37]: \n5u\u0015v\u0019=\t5 \u001aw1−x1\tpqr\tyz\t\u0015\u0005{\tL|\u0019\n?@\tf>no\t},   (10) \nwhich gives \t5\u001a=2.6 kOe and ∆ = 56. \nThe phase diagram measured at room temperature is s hown in Fig. 4b. At each magnetic \nfield point, a 100 ns writing pulse with fixed ampl itude was applied to the pMTJ pillar. \nSubsequently, the resistance was measured under sma ll DC bias current and the next \nmagnetic field point was set. To reduce the stochas ticity in the switching field values, the \nmagnetoresistance loop was measured 15 times and th eir average was used for switching \nfields determination. The same procedure was used f or all writing pulse amplitudes and the \nfinal phase diagram was constructed from these aver aged magnetoresistance loops. Magnetic \nfield loop repetition frequency was 2 Hz. \nThe extracted phase boundaries are shown in Fig.4c.  The coercive field of the free layer is \n940 Oe and the coupling field with the reference la yer is only 11 Oe and it is ferromagnetic. \nThe voltage driven parts are linear and almost para llel to each other. To reduce the influence of small nonlinearities at the edges of the boundar ies, only the central parts (within +/- 500 \nOe region) were used in the fitting. The extracted slopes are 1.27·10 -4 Volt/Oe and 1.23·10 -4 \nVolt/Oe, their difference is within the fitting err or. The zero field switching voltages are 0.359 \nVolt and 0.385Volt respectively. The difference is most probably due to the small DC bias \ncurrent used for the resistance measurements.  \nThe phase diagram shape is similar to those obtaine d from the theoretical analysis (Sec.3) \nas well as from the simulations (Sec.4) where the o ut-of-plane STT term is not dominating. For \nthis system, we can choose the STT prefactors model  \t\u0016\u001a=0 , \u0016∥=Y\u0005∥\tZ[\tQRSTU  . It \ncorresponds to DC diagram shown in Fig.2 whose boun daries are described by Eqs. (8,9). To \nrecalculate Y\u0005∥\tparameter from the extracted diagram slopes, one ne eds firstly to remap the \nexperimental finite temperature – finite writing pu lse diagram to the model case of long pulse \n– low temperature diagram. Here, we will follow the  formalism described in Ref.[16]. Thermal \neffects in our case can be reduced to the regime of  thermally assisted ballistic STT switching. \nIn this regime, the main role of thermal fluctuatio ns is to increase the probability of STT \nswitching thanks to increased initial misorientatio n angle *K , |cos\t\u0015*K\u0019|≠1 . As already \nmentioned, STT switching dynamics starting from a t ilted state reduces the switching time ~ in \nagreement with [13,14]. The cone angle, 2\t*K, for which the equilibrium probability for the \nmagnetic moment orientation distribution is 0.5, is  determined by thermal stability parameter \n∆\t and applied magnetic field *K=\tln2∆O\u0010 /1\n\u00131+567\u0005 \t5\u001aO \u0014a /1\n, while the final angle, the \nextremum on the energy barrier \t*=arccos\t\u0015−5 67\u0005 /5⊥\u0019  (for *K<./2  ), is determined by \nmagnetic field (see Eq.77 in Ref.[16]). Having defi ned the initial *K and final \t* angles of the \nSTT-induced dynamics, one can calculate analyticall y the switching time ~  (see Eq. (58) in \nRef[16]):  \n\u0015−1\u0019\n=ln\u00137\n7|\u0014− \nS! ln\n!7#\n\n!7|#,   (11) \nhere K=tan* K , =tan*, ~=\t !\"#\u0010\n\"|$\tf>\t and, according to our formalism,  =dUe \ndUeKO−\nfklg \n\tf>\t . Having calculated *K = 6°, ~ =9.9·10 -9 s and assuming \u0012 =0.02 [41] and writing pulse \nduration ~ = 100·10 -9 s, we recalculated dUe \u0015567\u0005 \u0019 dependence from Eq.(11), which is blue line \nin Fig.5, and compared it with the case of the DC d iagram dUe \u0015567\u0005 \u0019 , which is shown by circles \nin Fig.5, derived from Eq.(8-9). One can conclude t hat 100ns writing pulses are long enough to \nremove the effect of dynamical distortion of the ph ase boundaries. For the measured device \nof Fig.5, we find \n=100.6 which is quite high. This gives the possibili ty to work directly with \nthe phase boundaries (Eq.(8-9)) derived from Eq.(7) . However, if \n< 10 (if the writing pulse \nwidth in the experiment would be lower than 10 ns) and/or *K  is too small, the phase \nboundaries remapping procedure is necessary before further analysis of the phase boundaries \ncan be made. Indeed, in the simulations shown in th e previous sections, the respective value \nof  \n is 1.54. Therefore, the switching currents are muc h higher and the linear slope is different \nfrom that expected from the model. One also should notice that this formalism works only in high-∆ approximation. Therefore, the parts of the phase b oundaries which are close to the \nregions where 567\u0005  approaches 5\u001ashould be removed from the analysis.   \nFrom extrapolation of the voltage driven boundaries  to V=0 one can estimate 5\u001a ~ 2.8-\n3.1 kOe, which is slightly higher than the correspo nding value extracted from Eq.10 (2.6kOe). \nNevertheless, the obtained 5\u001a values are in quite good agreement considering tha t these two \nvalues are derived from very different physical phe nomena (superparamagnetism vs STT \nswitching). The spin-torque efficiency prefactor, Y\u0005∥ , can be directly determined from the \nexperimental slope using Eq.(9): Y\u0005∥ =162∙Z[a   Oe/Volt . From this, assuming that Z[  =1/RxA, \nthe effective spin polarization parameter in the sy stem can be derived: b = 0.49. If one uses \nthe measured TMR value to estimate the polarization  factor assuming that b =\nv/\t\u0015v/+2\u0019/\u00152\u0015v/+1\u0019\u0019  [42] and TMR = 1.26, this would yield b =0.44, which is \nclose to the value extracted from the diagram bound ary slope. The zero-field switching \ncurrent, recalculated using Eq.(8) for obtained val ues of 5\u001a, Y\u0005∥ and known parameter \u0012 gives \ndUeK=0.35 Z[· Volt.  \nTherefore, one can conclude that the experiments ca rried out on 36 nm pMTJ system \ncan be well described within the macrospin approxim ation and thermally activated ballistic \nregime of STT switching. 5\u001a, Y\u0005∥  parameters extracted from the phase boundaries of  V bias -H \nstability diagram are in good agreement with those extracted independently from \nmagnetoresistance loop and Neel-Brown model.  \n \n6.  Landau vs Gilbert  \nIn this section, we emphasize an important issue na turally arising from the analysis \ncarried out in the previous sections. If the STT te rms are added directly into Landau-Lifshitz \n(LL) equation [43], then instead of Eq. 2 (obtained  with Gilbert dissipation term [44]) the \nfollowing modified equation is obtained: \n  \n$|\u0001\t\u0003\u0004\n\u0001\t\u0005=−\t\u0003\u0004×\u0013\t \n !\"#\u000b\f\f\f\r\u000e\u000f\u000f −\t\u0016\u001a\t\u0018\u0004\u0014−\u0003\u0004×w\u0003\u0004×\t\u0013\"\n !\"#\t\u000b\f\f\f\r\u000e\u000f\u000f +\u0016∥\t\u0018\u0004\u0014}.   (12) \nStill preserving the main features and general beha vior of the STT switching in fully \nperpendicular structures, Eq. 10 forbids the switch ing only by the out-of-plane STT-term, in \ncontrast to Eq. 2 where the \u0015\u0012\t\u0016\u001a\u0003\u0004×\u0015\u0003\u0004×\t\u0018\u0004\u0019\u0019 component allows the system to change its \nenergy even if \u0016∥=0. That turns us to the still open discussion [45-52 ] of physical validity of \nGilbert-like damping and Landau-like damping formul ation in the equation of magnetization \ndynamics. Although it is generally claimed that LL and LLG equations are mathematically \nequivalent, we can see a significant difference whe n the STT terms are added: field-like STT \nterm written in LL equation is fully conservative  and it cannot change the system energy if \nEq.12 is chosen to describe the STT-induced dynamic s. Leaving this fact “as is”, one should \nnotice that in numerical simulations, it is more co mmon to use LL form instead of LLG form \nand different ways to introduce STT-terms (i.e. exp licitly into LL equation (Eq.12) or via \ntransformation of LLG+ STT (Eq.2)) can lead to sign ificantly different results.  Figure 6 demonstrates this important issue by compa ring examples of macrospin \nsimulations using either Landau-Lifshitz or Gilbert  damping terms to describe the dissipation \nduring STT-induced switching. Here, we adjusted the  relative magnitudes of the field-like and \ndamping-like STT prefactors to have comparable cont ributions in the second part of Eq. 2, \nwhich is LLG+STT case. As soon as the field-like ST T prefactor is set to have only a quadratic \nbias voltage dependence (the case of a symmetrical tunnel junction), the produced torque \nalways pulls the free layer magnetization in the an tiparallel configuration with the fixed layer. \nThe damping-like STT prefactor is set to be linear on the bias voltage and therefore the torque \ndirection is determined by the current polarity. Wh en a negative voltage is applied to the \nsystem, field-like torque helps the damping-like to rque to switch the magnetization in the \nantiparallel state. It shifts the phase boundary to wards lower switching voltages. However the \nexpected boundary shift is too small to be visible in our simulations considering the chosen \nstep for the voltage writing pulse amplitude. Also a quadratic dependence of the field-like STT \nprefactor allows it to compete with the damping lik e torque only at relatively high writing \npulse voltages. At the same time, for positive puls es, field-like torque works against the \ndamping-like torque, which shifts the phase boundar y to higher voltages. The higher the \nswitching voltage – the higher the relative contrib ution from the field-like torque. Finally, \nwhen the writing pulse is about 1.6 V, field-like t orque compensates the damping-like one and \nfurther increase of the writing pulse amplitude sta rts shifting the phase boundary back \ntowards negative fields, decreasing the field windo w of the bipolar STT switching. The same \neffect is observed at finite temperatures on Fig. 3 b for the bottom-middle diagram. This \ncompetition between the STT terms, however, is impo ssible in case of simulation with the \nLandau damping term because \u0012\t\u0016\u001a\u0003\u0004×\u0015\u0003\u0004×\t\u0018\u0004\u0019 term is absent in Eq.12.  \nFinally, it is traditionally accepted that Landau-L ifshitz-Gilbert and Landau-Lifshitz \nequations are geometrically equivalent and the math ematical transformation from one to \nanother ends up with  \n !\"# rescaling of the gyromagnetic ratio. This  \n !\"# correction in real \nphysical systems is very small and experimentally u ndetectable. However, this is not the case \nanymore if the STT terms are added to the LLG equat ion. The equations are now different.  \nThe same transformation (i.e. LLG+STT -> LL) leads to appearance of two additional STT \npseudo-torques ( \u0012\t\u0016\u001a\u0003\u0004×\u0015\u0003\u0004×\t\u0018\u0004\u0019,\t\u0012\t\u0016∥\t\u0003\u0004×\t\u0018\u0004) which are linearly proportional to the \ndamping constant \u0012 and in principle can be experimentally detected.  \nExperimentally, it should be possible to assess whi ch formulation of damping is correct by \nmeasuring the variation of the precession frequency  in the sub-switching threshold regime in \nsamples having various damping constants. Such samp les could be produced for instance by \ndepositing a wedge of Pt above the storage layer be fore patterning of the wafer. For this \nexperiment, it would be preferable to use symmetric  MTJs so that the field like torque has a \nquadratic dependence on bias voltage. If the LLG fo rmulation is correct, we expect a linear \nvariation of the frequency with damping constant un der fixed bias voltage whereas if the LL \nformulation is valid, no dependence of the frequenc y on damping should be observed. \n \n  \n7.  Conclusions \nIt has been shown that Landau-Lifshitz-Gilbert equa tion with the field-like and \ndamping-like STT terms transformed into the Landau- Lifshitz form considerably simplifies the \nanalysis of the STT switching process. In case of a  fully perpendicular MTJ system, the \nboundaries of the I-H stability phase diagram can b e directly obtained from the transformed \nequation (2). It was shown that the field-like term  has negligible influence on the STT switching \nprocess in pMTJs with low damping, influencing main ly the FMR precession frequency for the \nsmall oscillations near the equilibrium. Considerin g that in standard pMTJ structures its \neffective magnitude cannot be much higher than the magnitude of the in-plane torque, it \nwould be hard to track its bias voltage (current) d ependence from experimentally measured \nstability phase diagrams. Measuring the bias voltag e dependence of the frequency in the \nprecessional regime would certainly better reveal t he influence of the field-like STT term but \nstill the contribution of the field like term would  have to be separated from the non-linear \ninfluence of the oscillations amplitude on the freq uency. \nFinite temperature macrospin simulations in LLG-STT  formalism under finite writing \npulse duration have confirmed the negligible role o f the field-like term in the STT switching \nprocess of pMTJ structure. Limitations of the macro spin model are not expected to be \nimportant in the case of pMTJ pillars with diameter  comparable to or below the exchange \nlength. This is confirmed by the experiments which were carried out on 36 nm diameter pMTJ \npillars.  \nOne should note that the developed method for the p hase boundaries construction \ngives the same results as those obtained from the a nalysis of dynamical response of the \nsystem, carried out by different groups supposing t he linear dependence of the damping-like \nSTT prefactor versus applied bias voltage. However,  we believe that it will be more useful in \nthe interpretation of the experiment and simulation s, because it is much more flexible and it \nallows to introduce any desirable current (voltage)  dependences for the in-plane and out-of-\nplane spin-torque prefactors.  \nUsing the developed formalism, the spin-torque effi ciency and effective spin \npolarization parameters have been derived from the current-field stability diagram boundaries \nexperimentally measured on 36 nm pMTJ pillar. The o btained parameters have been cross-\nchecked by estimations from magnetoresistance curve s and from the thermally activated \nmagnetization reversal regime. Good agreement betwe en the values derived from the analysis \nof different physical principles strongly supports the assumption of macrospin-like behavior in \nthe measured sample. \nWe also showed that the different dissipation terms  (i.e. Landau-Lishfitz or Gilbert) give \nrise to different analytical expressions describing  the phase boundaries of I-H switching \ndiagrams, which can be important in heavily damped systems. If Landau damping term is \nphysically correct, the action of the field-like an d the damping-like torques in pMTJ system is \ncompletely separated in precession and dissipation terms in the equation of dynamics. While \nif Gilbert damping term is correct, then two additi onal torques ( \u0012\t\u0016\u001a\u0003\u0004×\u0015\u0003\u0004×\t\u0018\u0004\u0019 and \t\u0012\t\u0016∥\t\u0003\u0004×\t\u0018\u0004) are mixed up to the main STT contributors ( \u0016∥\t\u0003\u0004×\t\u0018\u0004 and \t\t\u0016\u001a\u0003\u0004×\u0015\u0003\u0004×\t\u0018\u0004\u0019 \nrespectively). An experimental way to assess which damping formulation is correct in \ncombination with STT was proposed. \n \nAcknowledgements \nThis work was supported by the Samsung Global MRAM Innovation Program and \nEUROTALENTS Program. The authors are also grateful to Ursula Ebels for fruitful discussions. \n \n \n \n \n \n  References \n \n1.  T. Liu, Y. Zhang, J. W. Cai and H. Y. Pan, Scientif ic Reports 4, 5895 (2014). \n2.  A.V. Khvalkovskiy, D. Apalkov, S. Watts, R. Chepuls kii, R.S. Beach, A. Ong, X. Tang, A. Driskill-\nSmith, W.H. Butler, P.B. Visscher, D. Lottis, E. Ch en, V. Nikitin and M. Krounbi, J. Phys. 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Lett. 102 , 137601 (2009). \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 \n \n \n \n \n \n   \n \nFig.1 Geometry of the fully perpendicular MTJ syste m. \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  \n \nFig.2 (a) - Stability phase diagram constructed fro m Eq.(7) assuming \u0016∥=Y\u0005∥\tZ[\tQRSTU  and \t\u0016\u001a=0; \n(b) – Modification of the phase boundaries for the same \u0016∥ prefactor ( \u0016∥=Y\u0005∥\tZ[\tQRSTU ,  Y\u0005∥ = 67 \n∙Z[a Oe/Volt) and different forms of \t\u0016\u001a prefactor: solid line \t\u0016\u001a=0 ; circles  \u0016\u001a=\nY\u0005\u001a1\t\tZ[\tQRSTU \u00101 with  Y\u0005\u001a1 = 154 ∙Z[a1   Oe/(Volt) 2; dashed line \t\u0016\u001a=Y\u0005\u001a \tZ[\tQRSTU +\nY\u0005\u001a1\t\tZ[\tQRSTU \u00101 with Y\u0005\u001a  = 500  ∙Z[a   Oe/Volt and  Y\u0005\u001a1 = 10000  ∙Z[a1   Oe/(Volt) 2; Other \nsystem parameters are: \u0012 = 0.05, 5\u001a\t= 200 Oe.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n   \n \nFig. 3. Finite writing pulse phase diagrams for dif ferent in-plane and out-of-plane STT prefactors \nmagnitudes. The model parameters 5\u001a = 200 Oe, g =2.20 (g-factor), α = 0.01. Integration time was \n1 microsecond in each field point and the writing p ulse width is 40 ns. a) T = 0K case, the axes scale  \nis the same for all diagrams: +/- 2 V from top to b ottom and +/- 300 Oe from right to left. b) T=300K,  \nthe axes scale is: +/- 1.5 V and +/- 250 Oe. \n□\n□\n/1□\n/2\n/3 /4 /4 /5 /4 /4 /4/5 /6 /7 /4/8/9 □ /10 /10□ □ /11 /12 /13 /14 /11 /15 □ /16/17/18 /18\n/8/9 □ □ /10 □ □ /19□ □ /11 /12 /13 /14 /11 /15 □ /16/17/18/19/18\n/2 /20 /21 /1 /22 /23 /24\n /25 /26\n□\n/28 /29 /30 /28 □\n/30 /28/28□□\n□ /33 /34 /35 /35 □ □ □ /33 /36 /35 /35 □ □ □ □ □ □ □ □ /35 □ □ □ □ □ □ /36 /35 /35 □ □ □ □ □ /34 /35 /35 □ □ □ □ /37 /35 /35\n□\n/38 □ /39 /40 /41 /42\n□\n□ □ □ □ □ □ □ □ □ □ □ /45 /46 /47 /47 □ □ □ /45 /48 /47 /47 □ □ □ □ □ □ □ /47 □ □ □ □ □ □ /48 /47 /47 □ □ □ □ □ /46 /47 /47 □ □ □ □\n□\n/49 □ /50 /51 /52 /53\n□\n/55 /56 /57 /57 □ □ □ /55 /59 /57 /57 □ □ □ /55 /60 /57 /57 □ □ □ □ □ □ □ □ /57 □ □ □ □ □ □ /60 /57 /57 □ □ □ □ □ /59 /57 /57 □ □ □\n□\n/61 □ /62 /63 /64 /65\n□\n/67 /68/67 /69/70/69/68/71 □ /73 /71 /74\n□□□□\n□\n/80 /81/82/81/83/84 □ /86 /84 /87\n□□□□\n□\n/93 /94/95/94/96/97 □ /99 /97 /100\n/101 □ /103 /104 /105 /106\n□\n/108 □ /110 /111 /112 /113 /114 □ /116 /117 /118 /119/120 □ /122 /120 /123\n/124 /125 /126 /126 □ □ □ □ □ □ /124 /128 /126 /126 □ □ □ □ □ □ □ □ □ /126 □ □ □ □ □ □ □ □ □ /128 /126 /126 □ □ □ □ □ □ □ /125 /126 /126/124 /128/126/128\n/129 /130 /131 /131 □ □ □ □ □ □ /129 /133 /131 /131 □ □ □ □ □ □ □ □ □ /131 □ □ □ □ □ □ □ □ □ /133 /131 /131 □ □ □ □ □ □ □ /130 /131 /131 /134 /135 /136 /136 □ □ □ □ □ □ /134 /138 /136 /136 □ □ □ □ □ □ □ □ □ /136 □ □ □ □ □ □ □ □ □ /138 /136 /136 □ □ □ □ □ □ □ /135 /136 /136/139 □ /141 /139 /142\n/143 /144/145/144/146 □ /148 /146 /149\n/150 /151/152/151\n/153 /154 /154 /155 /154 /154 /154\n/156/157/158 □ □ /160 □ □ /161□ □ /162 /163 /164 /165 /162 /166 □ /167/168/169/161/169□ □ /171 /172 /173 /174/175/176 □ /177 /177□ □ /178 /179 /180 /181 /178 /182 □ /183/184/185 /185/186 /187 /188 /189 /190 \n \n \n \n \n \n \n \n \n  \n \n \nFig. 4. Experiment carried out on pMTJ pillar at ro om temperature applying 100 ns writing pulses. \na) Examples of magnetoresistance loops measured wit h zero writing pulses; b) Stability phase \ndiagram; c) Extracted phase boundaries and their li near fittings. c a b\n-1,0 -0,5 0,0 0,5 1,0 -0,6 -0,4 -0,2 0,0 0,2 0,4         Experiment \n ,   Linear fit Vpulse  (V) \nHext  (kOe) -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 -0,6 -0,4 -0,2 0,0 0,2 0,4 0,6 \nVpulse  (V) \nHext  (kOe) -2 -1 0 1 26810 12 \n R (k Ω)\nHext  (kOe)  \n \n \n \n \n \n \n \n \n \n \n  \n-1,0 -0,5 0,0 0,5 1,0 012345Iτ\nsw  / Isw0 \nHext /HDynamic boundaries correction:\n For the experiment\n For the simulations on Fig.3 \n DC diagram from Eqs.(8-9) \n \n \nFig. 5. Finite pulse – finite temperature diagram b oundary for \n=100.6 (for experiment - blue) \nand \n=1.5 (for simulations - red). The dots are respectiv e boundary obtained from Eqs. (8-9).  \n \n \n \n \n \n \n  \n \nFig. 6. Two identical macrospin simulations of a st ability phase diagram carried at T = 0K: (a) using \nEq.12 (LLG + STT) ;  (b) using Eq.2 (LL + STT). STT  prefactors: Y\u0005∥=12 ∙Z[a  Oe/Volt and Y\u0005\u001a1= \n400 ∙Z[a1Oe/(Volt) 2. Other parameters are the same as used for the simu lations in Section 4. /0 /1 /2 /2 /0 /3 /2 /2 /0 /4 /2 /2 /2 /4 /2 /2 /3 /2 /2 /1 /2 /2/0 /3/0 /4/2/4/3\n□□/6 □ /7 /6 /8\n/9 □ /7 /10 /11 /8\n/12 /13 /14 /12 □\n/14 /12/12\n/0 /1 /2 /2 /0 /3 /2 /2 /0 /4 /2 /2 /2 /4 /2 /2 /3 /2 /2 /1 /2 /2/0 /3/0 /4/2/4/3\n□□/6 □ /7 /6 /8\n/9 □ /7 /10 /11 /8/16 /17 /18 /17"    },    {        "title": "1303.1192v1.Angle_Dependent_Spin_Wave_Resonance_Spectroscopy_of__Ga_Mn_As_Films.pdf",        "content": "arXiv:1303.1192v1  [cond-mat.mtrl-sci]  5 Mar 2013Angle-Dependent Spin-Wave Resonance Spectroscopy of (Ga, Mn)As Films\nL. Dreher,1,∗C. Bihler,1E. Peiner,2A. Waag,2W. Schoch,3W. Limmer,3S.T.B. Goennenwein,4and M.S. Brandt1\n1Walter Schottky Institut, Technische Universit¨ at M¨ unch en, Am Coulombwall 4, 85748 Garching, Germany\n2Institut f¨ ur Halbleitertechnik, Technische Universit¨ a t Braunschweig,\nHans-Sommer-Straße 66, 38023 Braunschweig, Germany\n3Institut f¨ ur Quantenmaterie, Universit¨ at Ulm, 89069 Ulm , Germany\n4Walther-Meißner-Institut, Bayerische Akademie der Wisse nschaften,\nWalther-Meißner-Straße 8, 85748 Garching, Germany\n(Dated: October 31, 2018)\nA modeling approach for standing spin-wave resonances base d on a finite-difference formulation\nof the Landau-Lifshitz-Gilbert equation is presented. In c ontrast to a previous study [Bihler et al.,\nPhys. Rev. B 79, 045205 (2009)], this formalism accounts for elliptical ma gnetization precession and\nmagnetic properties arbitrarily varying across the layer t hickness, including the magnetic anisotropy\nparameters, the exchange stiffness, the Gilbert damping, an d the saturation magnetization. To\ndemonstrate the usefulness of our modeling approach, we exp erimentally study a set of (Ga,Mn)As\nsamples grown by low-temperature molecular-beam epitaxy b y means of angle-dependent stand-\ning spin-wave resonance spectroscopy and electrochemical capacitance-voltage measurements. By\napplying our modeling approach, the angle dependence of the spin-wave resonance data can be re-\nproduced in a simulation with one set of simulation paramete rs for all external field orientations.\nWe find that the approximately linear gradient in the out-of- plane magnetic anisotropy is related\nto a linear gradient in the hole concentrations of the sample s.\nPACS numbers: 75.50.Pp, 76.50.+g, 75.70.-i, 75.30.Ds\nKeywords: (Ga,Mn)As; spin wave resonance; magnetic anisot ropy\nI. INTRODUCTION\nDue to their particular magnetic properties, in-\ncluding magnetic anisotropy,1–3anisotropic magneto-\nresistance4,5and magneto-thermopower,6in the past\nyears ferromagnetic semiconductors have continued to\nbe of great scientific interest in exploring new physics\nand conceptual spintronic devices.7–11The most promi-\nnentferromagneticsemiconductoris(Ga,Mn)As, wherea\nsmall percentage of Mn atoms on Ga sites introduces lo-\ncalizedmagneticmomentsaswellasitinerantholeswhich\nmediate the ferromagneticinteractionofthe Mn spins ( p-\ndexchange interaction).12Both theoretical and experi-\nmental studies have shown that the magnetic anisotropy,\ni.e., the dependence of the free energy of the ferromagnet\non the magnetization orientation, depends on the elas-\ntic strain and the hole concentration in the (Ga,Mn)As\nlayer,12,13openingupseveralpathwaystomanipulatethe\nmagnetic anisotropy of (Ga,Mn)As.14–16\nA common spectroscopic method to probe the mag-\nnetic anisotropy of ferromagnets and in particular\n(Ga,Mn)As, is angle-dependent ferromagnetic resonance\n(FMR),17–23where FMR spectra are taken as a func-\ntion of the orientation of the external magnetic field. If\nthe magnetic properties of the ferromagnet are homo-\ngeneous, a zero wave vector ( k= 0) mode of collec-\ntively, uniformally precessing magnetic moments couples\nto the microwavemagnetic field, e.g., in a microwavecav-\nity, allowing for a detection of the magnetization preces-\nsion. The resonance field of this mode, referred to as\nuniform resonance magnetic field, depends on the em-\nployedmicrowavefrequency and the magnetic anisotropyparameters. Thus, by recording FMR spectra at differ-\nent orientations of the external field with respect to the\ncrystal axes, the anisotropy parameters can be deduced\nfrom the experiment. However, if the magnetic prop-\nerties of a ferromagnetic layer are non-homogeneous or\nthe spins at the surface and interface of the layer are\npinned, non-propagating modes with k/negationslash= 0, referred to\nas standing spin-wave resonances (SWR), can be excited\nby the cavity field and thus be detected in an FMR ex-\nperiment. On one hand this can hamper the derivation\nof anisotropy parameters, on the other hand a detailed\nanalysis of these modes can elucidate the anisotropy pro-\nfile of the layer and the nature of spin pinning condi-\ntions. Furthermore, the excitation of spin waves is of\ntopical interest in combination with spin-pumping,24–27\ni.e., the generation of pure spin currents by a precessing\nmagnetization.28–30In this context, the exact knowledge\nof the magnetization precession amplitude as a function\nof the position coordinate within the ferromagnet is of\nparticular importance.24\nSeveral publications report on SWR modes in\n(Ga,Mn)As with a mode spacing deviating from what is\nexpected according to the Kittel model for magnetically\nhomogeneous films with pinned spins at the surface.31–36\nThese results have been attributed to an out-of-plane\nanisotropyfieldlinearly31,36orquadraticallyvarying33–35\nas a function of the depth into the layer, as well as to\nspecific spin pinning conditions at the surface and at the\ninterface to the substrate.35While most of these studies\nhave focused on the spacings of the resonancefields when\nmodeling SWR measurements, in Ref. 36 a more sophis-\nticated approach, based on a normal mode analysis,37,38\nwas employed to model resonance fields as well as rela-2\ntive mode intensities for the external field oriented along\nhigh-symmetry directions, assuming a circularly precess-\ning magnetization.\nIn this work, we present a more general modeling ap-\nproach for SWR, based on a finite-difference formulation\nof the Landau-Lifshitz-Gilbert (LLG) equation. This ap-\nproach holds for any orientation of the external mag-\nnetic field and accounts for elliptical magnetization pre-\ncession [Sec. II]. It allows for a simulation of arbitrar-\nily varying profiles of the magnetic properties across the\nthickness of the film, including vatiations of the mag-\nnetic anisotropy parameters, the exchange stiffness, and\nthe Gilbert damping parameter. As the result ofthe sim-\nulation, we obtain the Polder susceptibility tensor as a\nfunction of the depth within the ferromagnet. Based on\nthis result, the absorbedpowerupon spin waveresonance\nandthe magnetizationprecessionamplitude asafunction\nof the depth can be calculated for any orientation of the\nexternal magnetic field.\nWe apply our modeling approach to a set of four\n(Ga,Mn)As samples epitaxially grown with different\nV/III flux ratios [Sec. III], motivated by the obser-\nvation that V/III flux ratios of /lessorsimilar3 lead to a gra-\ndient in the hole concentration p[Ref. 39], which in\nturn is expected to cause non-homogeneous magnetic\nanisotropyparameters.31,36Electrochemicalcapacitance-\nvoltage (ECV) measurements revealed a nearly linear\ngradient in pacross the thickness of the layers investi-\ngated. To show that our modeling approach is capa-\nble of simulating SWR spectra for arbitrary magnetic\nfield orientations, angle-dependent SWR data were taken\nand compared with the model using one set of magnetic\nparameters for each sample, revealing gradients in the\nuniform resonance magnetic fields. We discuss the in-\nfluence of the gradient in pon the observed uniform\nresonance field gradients as well as possible influences\nof strain and saturation magnetization gradients on the\nobserved out-of-plane anisotropy profile. It should be\nemphasized, however, that the objective of this work is\nto show the usefulness of our modeling approach, while\na detailed investigation of the origin of the gradient in\nthe out-of-plane magnetic anisotropy profile and there-\nfore a detailed understanding of the particular materials\nphysics of (Ga,Mn)As is beyond the scope of this study.\nFinally, we summarize our results and discuss further po-\ntential applications of this work [Sec. IV].\nII. THEORETICAL CONSIDERATIONS\nIn this section, we provide the theoretical framework\nnecessary to describe the full angle dependence of the\nspin-wave resonance spectra presented in Sec. III. Refer-\nring to the coordinate system depicted in Fig. 1, we start\nfrom the canonical expression for the free enthalpy den-\nsity (normalized to the saturation magnetization M) forφ0Θ0\n123m2\nm1m3≈1\nm\nx||[100]y||[010]z||[001]\nSubstrateFerromag net\nFIG. 1: (color online) Relation between the two coordinate\nsystems employed. The ( x,y,z) frame of reference is spanned\nby the cubic crystal axes, while the (1 ,2,3) coordinate sys-\ntem is determined by the equilibrium orientation of the mag-\nnetization (3-direction) and two transverse directions, t he 2-\ndirection being parallel to the film plane; the latter system is\nzandµ0Hdependent, as described in the text.\na tetragonally distorted (Ga,Mn)As film13,20,40,41\nG= const −µ0H·m+B001m2\nz+B4⊥m4\nz\n+B4/bardbl(m4\nx+m4\ny)+1\n2B1¯10(mx−my)2.(1)\nHere,µ0His a static external magnetic field, B001\nis a uniaxial out-of-plane anisotropy parameter, re-\nflecting shape and second-order crystalline anisotropy,13\nB4⊥,B4/bardbl, andB1¯10are fourth-order crystalline and\nsecond-order uniaxial in-plane anisotropy parameters,\nrespectively;1mx,my,mzdenote the components of the\nnormalized magnetization vector m(z) =M(z)/M(z)\nalong the cubic axes [100], [010], and [001], respectively.\nWe assume the magnetic properties of the layer to be ho-\nmogeneouslaterally(within the xyplane) and inhomoge-\nneous vertically (along z); the anisotropy parameters in\nEq. (1) and the magnetization are consequently a func-\ntion of the spatial variable z. To obtain the anisotropy\nparameters from Eq. (1) in units of energy density, it\nwould therefore be required to know the zdependence\nand the absolute value of M.\nThe minimum of Eq. (1) determines the equilibrium\norientation of the magnetization, given by the angles\nθ0=θ0(z) andφ0=φ0(z), cf. Fig. 1. To describe the\nmagnetization dynamics, we introduce a new frame of\nreference (1 ,2,3) shown in Fig. 1, in which the equilib-\nrium orientation of the magnetization m0coincides with\nthe axis 3. For small perturbations, the magnetization\nprecesses around its equilibrium with finite transverse\ncomponents of the magnetization mi(i= 1,2) as illus-\ntrated in the inset in Fig. 1. The transformation between\nthe two coordinate systems is given in the Appendix A\nby Eqs. (A1) and (A2). We write for the (normalized)\nmagnetization3\nm=\n0\n0\n1\n\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nm0+\nm1\nm2\n0\n+O(m2\n1,m2\n2).(2)\nThe evolution ofthe magnetizationunder the influence\nof an effective magnetic field µ0Heffis described by the\nLLG equation42,43\n∂tm=−γm×µ0Heff+αm×∂tm,(3)\nwhereγis the gyromagnetic ratio and αa phenomeno-\nlogical damping parameter. The effective magnetic field\nis given by36\nµ0Heff=−∇mG+Ds\nM∇2M+µ0h(t),(4)\nwhere∇m= (∂m1,∂m2,∂m3) isthe vectordifferentialop-\neratorwith respect to the componentsof m,Ds= 2A/M\nis the exchange stiffness with the exchange constant A,\n∇2is the spatial differential operator ∇2=∂2\nx+∂2\ny+∂2\nz,\nandh(t) =h0e−iωtis the externally applied microwave\nmagnetic field with the angular frequency ω;h(t) is ori-\nented perpendicularly to µ0H. Since the magnetic prop-\nerties are independent of xandy, Eq. (3) simplifies to\n∂tm=−γm×[−∇mG+Dsm′′+µ0h(t)]+αm×∂tm,\n(5)\nwithm′′=∂2\nzm, neglecting terms of the order of m2\ni(for\ni= 1,2). By definition of the (1 ,2,3) coordinate system,\nthe only non-vanishing component of ∇mGin the equi-\nlibrium is along the 3-direction. For small deviations of\nmfrom the equilibrium we find44\n∇mG=\nG11m1+G21m2\nG12m1+G22m2\nG3\n, (6)\nwhere we have introduced the abbreviations Gi=\n∂miG|m=m0andGij=∂mi∂mjG|m=m0; the explicit ex-\npressions for these derivatives are given in the Appendix\nA.\nInthefollowing,wecalculatethetransversemagnetiza-\ntion components assuming a harmonic time dependence\nmi=mi,0e−iωt. The linearized LLG equation, consider-\ning only the transverse components, reads as\n/parenleftbigg\nH11H12\nH21H22/parenrightbigg/parenleftbigg\nm1\nm2/parenrightbigg\n−Ds/parenleftbigg\nm′′\n1\nm′′\n2/parenrightbigg\n=µ0/parenleftbigg\nh1\nh2/parenrightbigg\n,(7)\nwherewe haveintroducedthe abbreviations H11=G11−\nG3−iαω/γ,H12=H∗\n21=G12+iω/γ, andH22=G22−\nG3−iαω/γ. We have dropped all terms which are non-\nlinear inmiand products of miwith the driving field.\nResonant uniform precession of the magnetization\n(m′′\ni= 0) occurs at the so called uniform resonance fieldµ0Huni(z), which is found by solving the homogeneous\n(h= 0) equation\nH11(z)H22(z)−H12(z)H21(z) = 0\n⇔(G11−G3)(G22−G3)−G2\n12=/parenleftbiggω\nγ/parenrightbigg2\n(8)\nforµ0H, neglecting the Gilbert damping ( α= 0). Equa-\ntion(8)canbeusedtoderiveanisotropyparametersfrom\nangle-dependent FMR spectra. As extensively discussed\nby Baselgia et al.44, using Eq. (8) is equivalent to using\nthe method of Smit and Beljers, which employs second\nderivatives of the free enthalpy with respect to the spher-\nical coordinates.41,45,46\nTo illustrate the role of the uniform resonance field in\nthe context of spin-wave resonances, we consider the spe-\ncial case where magnetization is aligned along the [001]\ncrystal axis ( θ0= 0), before we deal with the general\ncase of arbitrary field orientations. Neglecting the uniax-\nial in-plane anisotropy( B1¯10= 0) since this anisotropyis\ntypically weaker than all other anisotropies,13,41we find\nG3=−µ0H+2B001+4B4⊥andG11=G22=G12= 0,\nresulting in the uniform resonance field\nµ0H001\nuni(z) =ω/γ+2B001(z)+4B4⊥(z).(9)\nTo find the eigenmodes of the system, we consider the\nunperturbed and undamped case, i.e., α= 0 and h= 0\nin Eq. (7). With m2=im1= ˜mwe find the spin-wave\nequation\nDs˜m′′+µ0H001\nuni(z)˜m=µ0H˜m (10)\nin agreementwith Ref. 36. The relationofthe anisotropy\nparameters defined in Ref. 36 to the ones used here\nis given by B001=K100\neff/M+B1¯10,B1¯10=−K011\nu/M,\n2B4⊥=−K⊥\nc1/M, and 2B4/bardbl=−K/bardbl\nc1/M. Equation (10)\nis mathematically equivalent to the one-dimensional\ntime-independent Schr¨ odinger equation, where the\nuniform resonance field corresponds to the potential,\n˜mto the wave function, µ0Hto the energy, and Dsis\nproportional to the inverse mass. To calculate the actual\nprecession amplitude of the magnetization, the coupling\nof the eigenmodes of Eq. (10) to the driving field is\nrelevant, which is proportional to the net magnetic\nmoment of the mode.36,38In analogy to a particle in a\nbox, the geometry of the uniform resonance field as well\nas the boundary conditions determine the resonance\nfields and the spatial form of the precession amplitude.\nFor the remainder of this work, we assume the spins\nto exhibit natural freedom at the boundaries of the\nfilm, i.e.,∂z˜m= ˜m′= 0 at the interfaces,36,47since\nthese boundary conditions have been shown to describe\nthe out-of-plane SWR data of similar samples well.36\nTo graphically illustrate the influence of the uniform\nresonance field on the SWR modes, we consider in Fig. 2\na ferromagnetic layer with a thickness of 50 nm with\nconstant magnetic properties across the layer (a) and\nwith a linearly varying uniform resonance field (c); in4\n(a) (b)\n(c) (d) m (arb. u.)~ m (arb. u.)~\nFIG. 2: Simulation to demonstrate the influence of the uni-\nform resonance field µ0H001\nunion theSWRmodes for m0||[001],\nassuming circular precession. In (a), µ0H001\nuniis set to be con-\nstant across the layer, while in (c) it varies linearly (blue ,\ndashed lines), in analogy to a square potential and a trian-\ngular potential, respectively. The dotted black lines are t he\nresonance fields, calculated assuming boundary conditions of\nnatural freedom, see text. The solid red lines show the eigen -\nmodes of the system, i.e., the precession amplitude ˜ mof the\nmagnetization; for each mode the dotted line corresponds to\n˜m= 0. As can be seen in (a), for a constant uniform reso-\nnancefieldthefirstmodeoccursattheuniformresonance field\nand exhibits a constant precession amplitude across the lay er,\ni.e., an FMR mode. The second and third mode (higher-order\nmodes are not shown) exhibit a non-uniform magnetization\nprofile. In order to couple to the driving field the modes need\nto have a finite net magnetic moment. As can be seen in\n(a), the positive and negative areas of the second and third\nmode are equal, thus these modes are not visible in the SWR\nspectrum (b). This is in contrast to the case of the linearly\nvarying uniform resonance field (c) where the mode profile is\ngiven by Airy functions, which have a nonzero net magnetic\nmoment also for the second and third mode, resulting in a\nfinite SWR intensity of these modes (d). The spectra in (b)\nand (d) were calculated by integrating over the eigenmodes\n˜mand convoluting the square of the result with Lorentzians.\nboth cases we assume Ds= 13 Tnm−2, a similar value\nas obtained in previous studies.36For these conditions,\nwe numerically solve Eq. (10) by the finite difference\nmethod described in the Appendix B1, in orderto obtain\nthe resonance fields (eigenvalues) and the zdependence\nof the transverse magnetic moments (eigenfunctions).\nTo which amount a mode couples to the driving field is\ndetermined by the net magnetic moment of the mode,\nwhich is found by integrating ˜ m(z) over the thickness of\nthe film. For the magnetically homogeneous layer, the\nonly mode that couples to the driving field is the uniformprecession mode at µ0H001\nuni, since modes of higher order\nhave a zero net magnetic moment [Fig. 2 (a)], resulting\nin one resonance at the uniform resonance field, cf. Fig. 2\n(b). For the non-uniform layer, with µ0H001\nuni(z) linearly\nvarying across the film, the mode profile is given by Airy\nfunctions31,36,38and various non-uniform modes couple\nto the driving field, resulting in several spin-wave reso-\nnances with their amplitude proportional to the square\nof the net magnetic moment36,38of the corresponding\nmode, cf. Fig. 2 (c) and (d).\nWe now turn to the general case of arbitrary field ori-\nentations. Due to the magnetic anisotropy profile, the\nmagnetizationorientationisaprioriunknownandafunc-\ntion ofzandµ0H. Furthermore, the assumption of a\ncircularly precessing magnetization is not generally jus-\ntified. To solve Eq. (7) for arbitrary field orientations,\nwe employ a finite difference method as outlined in the\nAppendix B2. BysolvingEq.(7), weobtainthe zdepen-\ndent generalized Polder susceptibility tensor ¯ χ(µ0H,z),\nwhich relates the transverse magnetization components\nMi(z) =M(z)mi(z) with the components of the driving\nfield by\n/parenleftbigg\nM1\nM2/parenrightbigg\n= ¯χ(µ0H,z)/parenleftbigg\nh1\nh2/parenrightbigg\n. (11)\nInamicrowaveabsorptionmeasurement, thecomponents\nMiwhich are out-of-phase with the driving field are de-\ntected. The absorbed power density is related to the\nimaginary part of ¯ χ(µ0H,z) and can be calculated by48\nP=ωµ0\n2z0Im/braceleftbigg/integraldisplay0\n−z0/bracketleftbigg/parenleftbigh∗\n1,h∗\n2/parenrightbig\n¯χ(µ0H,z)/parenleftbigg\nh1\nh2/parenrightbigg/bracketrightbigg\ndz/bracerightbigg\n,\n(12)\nwherez0is the thickness ofthe ferromagneticlayer. Note\nthat the position coordinate zis negative in the film,\ncf. Fig. 1.\nTo obtain an impression of how gradients in differ-\nent anisotropy parameters influence the SWR spectra,\nwe plot in Fig. 3 simulated SWR spectra together with\nthe magnetization precession cone as a function of depth\nin the ferromagnetic layer. We assume a constant sat-\nuration magnetization (its value is not relevant for the\noutcome of the simulation), a constant exchange stiff-\nnessDs= 35 Tnm2unless otherwise specified, α= 0.09,\nandB001= 90 mT, B4||=−50 mT,B4⊥= 15 mT.\nIn Fig. 3 (a), we assume B001to vary across the layer\nthickness according to B001(z) =B001−b001×zwith\nb001=−0.8 mT/nm. Figure 3 (a i) shows the simu-\nlated SWR spectra calculated by taking the first deriva-\ntive of Eq. (12) with respect to µ0Hfor different angles\nψdefined in the inset in Fig. 3 (c iv). We observe sev-\neral SWR modes for µ0H||[001] which become less as\nµ0His tilted away from [001]. At ψ= 40◦only one\nmode is visible while for ψ= 0◦we again observe mul-\ntiple SWR modes. This observation can be understood\nby considering the uniform resonance fields as a func-\ntion of the depth for these orientations. In Fig. 3 (a5\n90\n03060\n90\n03060\n90\n03060\nSWR Intensi ty (arb . units)ψ (deg.) ψ (deg.) ψ (deg.)\n200 600 400\nµ0H (mT)ψµ0H\n[110][001](a i) (a ii) (a iii) (a iv)\n(b i) (b ii) (b iii) (b iv)\n(c i) (cii) (ciii) (c iv)Im(m1m2-m1m2) (arb. u.) * *\nFIG. 3: Atlas illustrating the influence of gradients in the a nisotropy parameters on SWR spectra. In (a) all anisotropy\nparameters are kept constant with the values given in the tex t, exceptB001which is varied linearly. Correspondingly, in (b)\nand (c)B4⊥andB4||were varied linearly, respectively. Panels (i) show the firs t derivative of simulations using Eq. (12) with\nrespect toµ0Hand panels (ii)-(iv) show the precession cone Im( m∗\n1m2−m1m∗\n2) in a color plot together with the uniform\nresonance field µ0Huni(z) (dashed blue lines) at three different external field orient ations; the black dotted lines indicate the\nresonance field positions of the modes. Panel (a i) additiona lly shows the influence of a linear gradient in the exchange st iffness\nparameter on the spin-wave spectra, see text for further det ails and discussion.\nii)-(a iv), we show the uniform resonance field (dashed\nblue line) for ψ= 0◦,ψ= 30◦, andψ= 90◦, respec-\ntively, together with the magnetization precession cone\nIm(m∗\n1m2−m1m∗\n2) in a contour plot as a function of\ndepth andµ0H. Atψ= 90◦, the uniform resonance field\nvaries strongly across the film, which can be understood\nby considering Eq. (9). This results in several spin wave\nmodes with their resonance fields indicated by dotted\nlines.\nFor other field orientations, the formula for the uni-\nform resonance field can also be derived but results in a\nlonger, more complex equation than Eq. (9). Important\nin this context is that positive values of B001lead to an\nincrease(decrease) ofthe resonancefield for the magneti-\nzation oriented perpendicular (parallel) to the film plane,\naccountingfor the reversedsign ofthe slopesof µ0Huniin\nFig. 3 (a ii) and (a iv). Consequently, in between those\ntwo extreme cases µ0Hunimust be constant across the\nlayer for some field orientation, in our case for ψ= 30◦,\nresulting in a single SWR mode, cf. Fig. 3 (a i) and (a\niii). In addition to the SWR simulations with constantDs, weplotinFig.3(ai)simulatedSWRspectrawith Ds\nvarying linearly across the film with Ds= 35−65 Tnm2\n(blue, dotted lines) and Ds= 35−5 Tnm2(green, dotted\nlines). A decreasing Dsleads to a decreasing spacing in\nthe modes and vice versa for an increasing Dsas can be\nseen, e.g., for µ0H||[001].\nIn Fig. 3 (b), we consider the case where all magnetic\nparameters are constant with the values given above, ex-\nceptB4⊥(z) =B4⊥−b4⊥×zwithb4⊥=−0.4 mT/nm.\nAs evident from Eq. (9), this results in the same slope of\nµ0Huniforψ= 90◦as in the case above where we varied\nB001only, cf. Fig. 3 (a iv) and (b iv). In contrast to the\ncasedepictedin(a), however,herefor ψ= 0◦theuniform\nresonancefield is constant. This can be understood when\nevaluating the parametersthat enter in the calculation of\nthe uniform resonance field [Eq. (8)]. If mis in the film\nplane, none of the parameters in Eqs. (A4)-(A6) depends\nonB4⊥, resulting in a constant uniform resonance field\nforψ= 0◦. Asmis tilted away from the film plane,\nB4⊥enters in some of the terms Eqs. (A4)-(A6). As a\nconsequence, µ0Hunivaries, first such that it increases6\n[cf. Fig. 3 (b iii)] and finally, such that it decreases as a\nfunction of depth [cf. Fig. (b iv)].\nFinally, we discuss the case where all parameters are\nconstant except B4||(z) =B4||−b4||×zwithb4||=\n−0.4 mT/nm [Fig. 3 (c)]. Here, µ0Huniis constant for\nψ= 90◦aspredictedbyEq.(9). As mistiltedawayfrom\n[001] a varying B4||leads to a varying uniform resonance\nfield as shown in Fig. 3 (c ii) and (c iii). Here, a sign\nreversal of the slope as it was the case in Fig. 3 (a) and\n(b) does not take place and multiple resonances occur,\nstarting from ψ= 60◦[Fig. 3 (c i)].\nIII. EXPERIMENTAL RESULTS AND\nDISCUSSION\n(Ga,Mn)As samples with a nominal Mn concentration\nof≈4% were grown on (001)-oriented GaAs substrates\nby low-temperature molecular-beam epitaxy at a sub-\nstrate temperature of 220◦C using V/III flux ratios of\n1.1, 1.3, 1.5, and 3.5, referred to as sample A, B, C, and\nD, respectively. The layer thickness was 210-280 nm as\ndetermined from the ECV measurements, cf. Fig. 4. For\nsamples with V/III flux ratios /lessorsimilar3 a gradient in the hole\nconcentration has been reported,39hence this set of sam-\nples was chosen to study the influence of a gradient in p\non the out-of-plane magnetic anisotropy. Further details\non the sample growth can be found in Refs. 39 and 41.\nThe hole concentration profile of the as-grown\n(Ga,Mn)As layers were determined by ECV profiling us-\ning a BioRad PN4400 profiler with a 250 ml aqueous\nsolution of 2.0 g NaOH+9.3 g EDTA as the electrolyte.\nFor further details on the ECV analysis see Ref. 39. The\nresults of the ECV measurements for the layers investi-\ngated areshown in Fig. 4(a). Except for the sample with\nV/III=3.5, they reveala nearlylinearly varying hole con-\ncentration across the layer thickness with different slopes\nand with the absolute value of the hole concentration\nat the surface of the layer varying by about 20%. The\nprofiles are reproducible within an uncertainty of about\n15%.\nTo investigate the magnetic anisotropy profiles of the\nsamples, weperformedcavity-basedFMRmeasurements,\nusing a Bruker ESP300 spectrometer operating at a mi-\ncrowave frequency of 9.265 GHz ( X-band) with a mi-\ncrowave power of 2 mW at T= 5 K; we used magnetic\nfield modulation at a frequency of 100 kHz and an ampli-\ntude of3.2mT. Since wearemainlyinterested in the out-\nof-plane magnetic anisotropy, we recorded spectra for ex-\nternalmagneticfieldorientationswithinthe crystalplane\nspanned by the [110] and [001] crystal axes in 5◦steps,\ncf. the inset in Fig. 5. For each orientation, the field was\nramped to 1 T in order to saturate the magnetization\nand then swept from 650 mT to 250 mT; the spectra for\nthe samples investigated are shown in Fig. 5.\nWe start by discussing qualitative differences in the\nspectra. The samples A and B exhibit several pro-\nnounced resonances for the external field oriented along(a)\n(b)\nFIG.4: (a)Theholeconcentration inthedifferent(Ga,Mn)As\nsamples is shown as a function of the depth within the layers\nas determined by ECV profiling. (b) The uniform resonance\nfieldsµ0H001\nuni(z) for the four samples obtained from the sim-\nulations for the out-of-plane orientation of the external fi eld\n(ψ= 90◦) as a function of the depth.\n[001], which we attribute to standing spin-wave reso-\nnances [Fig. 5 (a) and (b)]. For these samples, the [001]\ndirection is the magnetically hardest axis since at this\norientation the resonance field of the fundamental spin-\nwavemode is largerthan at all other orientations. As the\nexternal field is rotated into the film plane, the resonance\nposition of this mode gradually shifts to lower field val-\nues as expected for a pronounced out-of-plane hard axis.\nIn contrast, the samples C and D exhibit the largest res-\nonance fields for a field orientation of 50-60◦[Fig. 5 (c)\nand (d)] pointing to an interplay of second- and fourth-\norder out-of-plane anisotropy with different signs of the\ncorresponding anisotropy parameters. These samples ex-\nhibit spin-wave resonances as well, however they are less\npronounced than for samples A and B.\nTo quantitatively model the spin-wave spectra we nu-\nmerically solve for each magnetic field orientation the\nspin-wave equation (7) by the finite difference method as\noutlined in the Appendix B2. Although this method al-\nlows for the modeling of the SWR for arbitrary profiles\nof the anisotropy parameters, the exchange stiffness, the\nGilbert damping parameter, and the saturation magne-\ntization, we assume the parameters to vary linearly as\na function of z. This approach is motivated by the lin-\near gradient in the hole concentration, which in first ap-\nproximation is assumed to cause a linear gradient in the7\nψ[001]\n[110]µ0Hext\nSimulationExperiment(a) (b)\n(d) (c)\nV/III=1.5 V/III=3.5V/III=1.3 V/III=1.1\nFIG. 5: The spin-wave resonance data (dotted, blue lines) ar e shown together with simulations (red, solid lines) using t he\nnumerical procedure described in the text and in the Appendi x B2. The data were obtained as a function of the external\nmagnetic field orientation and magnitude for samples with a V /III flux ratio of (a) 1.1, (b) 1.3, (c) 1.5, and (d) 3.5. The\nrotation angle ψis defined in the inset and the parameters used for the simulat ions are summarized in Tab. I.8\nanisotropy parameters, resulting in the spin-wave reso-\nnances observed in the samples.31,36In Tab. I, we have\nsummarized the parameters used in the simulation for\nthe different samples. The parameters in capital let-\nters denote the value at the surface of the sample while\nthe ones in lower-case letters denote the slope of this\nparameter; e.g., the zdependence of the second-order,\nuniaxial out-of-plane anisotropy parameter is given by\nB001(z) =B001−b001×z. The layer thickness used for\nthe simulationcanbe inferredfromFig. 4(a) andwasde-\ntermined from the ECV data under the assumption that\nat the position where the hole concentration rapidly de-\ncreases the magnetic properties of the layer abruptly un-\ndergo a transition from ferromagnetic to paramagnetic.\nFor the simulations, we divided each film into n= 100\nlayers with constant magnetic properties within each\nlayer. For the gyromagnetic ratio we used γ=gµB//planckover2pi1\nwithg= 2.21\nAs result of the simulation we obtain the Polder sus-\nceptibility tensor ¯ χ(µ0H,z) and the transverse magneti-\nzationcomponentsasafunctionof zandµ0H. Addition-\nally, weobtainthe zdependenceoftheuniformresonance\nfield by solving Eq. (8) for each field orientation. In an\nSWR absorption experiment with magnetic field mod-\nulation, the obtained signal is proportional to the first\nderivative of the absorbed microwave power with respect\nto the magnetic field. Thus, we calculate the absorbed\npowerusingthesimulatedsusceptibilityandEq.(12) and\nnumerically differentiate the result in order to compare\nthe simulated SWR spectra with the experiment. Addi-\ntionally, we use a global scaling factor, accounting, e.g.,\nfor the modulation amplitude, which is the same for all\nfield orientations, and we multiply all the simulated data\nwith this factor. In Fig. 5, we plot the experimental data\ntogether with the simulations using the parameters given\nin Tab. I, demonstrating that a reasonableagreementbe-\ntweentheoryandexperimentcanbefoundwithonesetof\nsimulation parameters for all magnetic field orientations\nfor each sample.\nWe will now exemplarily discuss the angle dependence\nof the SWR spectrum of sample A shown in Fig. 5(a)\nbased on the uniform resonance field and the resulting\nmagnetizationmodeprofileobtainedfromthesimulation.\nTo this end, we plot in Fig. 6 (a)-(c) the magnetization\nprecession amplitude Im( m∗\n1m2−m1m∗\n2) for selected ex-\nternal field orientations as a function of depth and ex-\nternal magnetic field in a contour plot, together with the\ncorresponding uniform resonance field. In Fig. 6 (d)-(f),\nwe show for each external field orientation a magnifica-\ntion of the corresponding SWR spectrum together with\nthesimulation. Notethatincontrasttothenormal-mode\napproach (Appendix B1) used to calculate the modes in\nFig. 2, where the coupling of each mode to the cavity\nfield has to be found by integration, the approach elabo-\nrated in the Appendix B2 directly yields the transverse\nmagnetization components, already accounting for the\ncoupling efficiency and the linewidth. Further, the ap-\nproach presented in the Appendix B2 is also valid whenthe differencein the resonancefields oftwomodesis com-\nparable with or smaller than their linewidth, in contrast\nto the normal-mode approach38.\nIf the external field is parallel to the surface normal\n(ψ= 90◦) the uniform resonance field varies by about\n350 mT across the film thickness [cf. the dashed line\nin Fig. 6 (a)], resulting in several well-resolved stand-\ning spin-wave modes. The spin-wave resonance fields are\nplotted as dotted lines in Fig. 6 (a); since the spacing of\nthe resonance fields is larger than the SWR linewidth,\nthe modes are clearly resolved, cf. Fig. 6 (a) and (d).\nIn the simulation two regions with different b001values\nwereused in orderto reproducethe spacingofthe higher-\norder spin-wave modes found in the experiment. Using\nthe same slope as in the first 100 nm for the entire layer\nwould lead to a smaller spacing between the third and\nhigher order modes. Instead of defining two regions with\ndifferent slopes b001, a gradient in the exchange stiffness\nwith positiveslopecouldalsobe usedto modelthe exper-\nimentally found mode spacing as discussed in the context\nof Fig. 3. Since the exchange interaction in (Ga,Mn)As is\nmediated by holes12andpdecreases across the layer, we\nrefrainfrom modeling ourresultswith a positivegradient\ninDs. Further, the results in Ref. 36 rather point to a\nnegative gradient in Dsin a similar sample. However, a\ndecreasing Mn concentration as a function of the depth\ncould lead to an increase of Ds.34\nFinally, we note, since B1¯10= 0 in the simulations,\nthe magnetization precesses circularly for ψ= 90◦and\nthus Im(m∗\n1m2−m1m∗\n2) = 2sin2τ,49with the precession\ncone angle τ. For all other orientations, mprecesses\nelliptically which is accounted for in our simulations. In\nthe simulations of the precession amplitudes, we have\nassumed an externally applied microwave magnetic field\nwithµ0h= 0.1 mT.\nAt an external field orientation of ψ= 50◦the uni-\nform resonance field is nearly constant across the layer,\nand consequently only one SWR mode is observed with\nan almost uniform magnetization precession across the\nlayer, cf. Fig. 6 (b). The precession amplitude is a mea-\nsureforthe SWR intensity. While the fundamental mode\natψ= 90◦exhibits a larger precession cone at the in-\nterface, it rapidly decays as a function of the depth, in\ncontrast to the nearly uniform precession amplitude for\nψ= 50◦. Since the entire layer contributes to the power\nabsorption, consequently, the SWR mode at ψ= 50◦is\nmore intense than the fundamental mode for ψ= 90◦,\nwhich is indeed observed in the experiment [cf. Fig. 6 (d)\nand (e)].\nFor the magnetic field within the film plane [ ψ= 0◦,\ncf.Fig.6(c)], the uniformresonancefieldagainvarieslin-\nearly across the film, however in a less pronounced way\nthan for the out-of-plane field orientation and with an\nopposite sign of the slope. The sign reversal of the slope\ncan be understood in terms of the uniaxial out-of-plane\nanisotropy parameter B001: positive values of these pa-\nrameters lead to an increase (decrease) of the resonance\nfield for the magnetization oriented perpendicular (par-9\nTABLE I: Simulation parameters and their zdependence of the samples under study as obtained by fitting t he simulations to\nthe SWR measurements. For the anisotropy parameters the cap ital letters denote the value at the surface of the film and the\nlower case letters the slope as described in the text. For sam ple A, the first value of b001was used for the first 100 nm and the\nsecond one for the remaining layer. In addition to the anisot ropy parameters, the saturation magnetization is also assu med to\nvary linearly across the layer, while its absolute value is u nknown and not important for the SWR simulations.\nSample V/III B001 b001 B4/bardblb4/bardblB4⊥b4⊥Dsα∂M(z)\n∂zM(0)\n(mT) (mT\nnm) (mT) (mT\nnm) (mT) (mT\nnm) (Tnm2) (1\nµm)\nA 1.1 90 -0.1, -0.3 -50 0.05 25 -0.3 35 0.09 -3\nB 1.3 130 -0.5 -50 0 0 0 20 0.06 -4\nC 1.5 75 -0.4 -55 -0.04 -15 0 40 0.11 -4\nD 3.5 91 -0.3 -55 -0.04 -15 0 20 0.09 -3\nallel) to the film plane, accounting for the slopes of the\nuniform resonance fields in Fig. 6. Since the gradient in\nthe uniform resonance field is less pronounced for ψ= 0◦\nthan forψ= 90◦, the spin-wave modes are not resolved\nforψ= 0◦, since their spacing is smaller than the SWR\nlinewidth, leading to one rather broad line [cf. Fig. 6 (c)\nand (f)]. A steeper gradient in B4||in combination with\na different Gilbert damping (or with an additional inho-\nmogeneous damping parameter) and amplitude scaling\nfactor, could improve the agreement of simulation and\nexperiment in the in-plane configuration, as discussed\nlater. A detailed study of the in-plane anisotropy pro-\nfile is however beyond the scope of this work. Given that\nthe presented simulations were obtained with one set of\nparameters, the agreement of theory and experiment is\nreasonably good also for the in-plane configuration, since\nsalient features of the SWR lineshape are reproduced in\nthe simulation.\nHaving discussed the angle-dependence of the SWR\nspectra, we turn to the zdependence of the out-of-plane\nanisotropy of sample A. Our simulations reveal that it\nis governed by the zdependence of both B001(z) and\nB4⊥(z). Assuming only a gradient in B001results in a\nreasonable agreement of theory and experiment for the\nexternal field oriented along [001] and [110], but fails to\nreproduce the spectra observed for the intermediate field\norientations, e.g., ψ= 50◦. This is illustrated by the\ndashed black line in Fig. 6 (e), which represent simu-\nlations with a constant B4⊥(z) forψ= 50◦. As can\nbe seen, this simulation produces several spin-wave res-\nonances, whereas in the experiment only one resonance\nis present, which is better reproduced by the simulation\nwith bothB001(z) andB4⊥(z) varying across the layer.\nWe will now discuss the anisotropy parameters of all\nsamples. In contrast to sample A, the out-of-plane\nanisotropy profile of all other samples appears to be gov-\nerned by a gradient in B001(z). As already discussed\nqualitatively, the hard axis of the samples is determined\nby an interplay of B001andB4⊥. For sample A and B\nB4⊥is positive and zero, leading to an out-of-plane hard\naxis. Incontrast,sampleCandDexhibitanegative B4⊥,\nleading to a hard axis between out-of-plane and in-plane.\nTheB4||parameter is negative and of similar magnitude\nfor all samples.Since the out-of-plane anisotropy profile of sample A\nis governed by B001(z) andB4⊥(z), a comparison of the\nout-of-plane anisotropy profile between all samples based\non anisotropy parameters is difficult. We therefore com-\npare the uniform resonance fields, where both anisotropy\nparameters enter. As evident from Fig. 6, the strongest\ninfluence of the magnetic inhomogeneity of the layers on\nthe uniform resonance fields is observed for the exter-\nnal field along [001]. To compare the hole concentration\nprofile in Fig. 4 (a) with the anisotropy profile, we there-\nfore plot in Fig. 4 (b) the zdependence of the uniform\nresonance field µ0H001\nunifor this field orientation. The\nfigure demonstrates that the gradient in µ0H001\nuniis cor-\nrelated with the gradient in p. For the sample with the\nstrongest gradient in pthe gradient in µ0H001\nuniis also\nmost distinct while the samples with a weaker gradient\ninpexhibit a less pronounced gradient in µ0H001\nuni. How-\never, for sample D, exhibiting a nearly constant p, we\nstillobservestandingspinwaveresonancesfor µ0H||[001]\n[Fig. 5 (d)], reflected in a slight gradient of µ0H001\nuni. This\nobservation suggests that aditionally other mechanisms\nlead to a variation of the anisotropy profile. One possi-\nbilitywouldbeagradientintheelasticstrainofthelayer,\ndue to a non-homogeneous incorporation of Mn atoms in\nthe lattice. However, x-ray diffraction measurements of\nthis sample, in combination with a numerical simulation\nbased on dynamic scattering theory, reveal a variation of\nthe vertical strain ∆ εzzas small as 3 ×10−5across the\nlayer. According to the measurements in Ref. 13, such a\nvariation in strain would lead to a variation of the B001\nparameter by a few mT only, insufficient to account for\nthe variation of µ0Huniby almost 100 mT across the\nlayer. A more likely explanation seems to be a varia-\ntion of the saturation magnetization, which should also\ninfluence the anisotropy parameters. In the simulation,\na non-homogeneous saturation was assumed, potentially\nexplaining also the observed gradient in the anisotropy\nparameters and therefore in the uniform resonance field.\nIn contrast to the out-of-plane anisotropy parameters,\nB4||was found to depend only weakly on z, for all sam-\nples except sample B where it was constant. Addition-\nally,B1¯10, typically of the order of a few mT,13might\nhave an influence and interplay with B4||in determining\nthe in-plane anisotropy. We here however focus on the10\n0°\n(b) (c)\n(a)Im(m1m2-m1m2) (10-5) * *\n0 1.2Im(m1m2-m1m2) (10-5) * *\n0 0.3Im(m1m2-m1m2) (10-5) * *\n0 0.53\n(d) (e) (f)001arb.(a)\nFIG. 6: Simulated magnetization mode profile and uniform res onance field of sample A. The contour plots show the magneti-\nzation precession amplitude Im( m∗\n1m2−m1m∗\n2) as a function of the position within the film and the external magnetic field\nfor the external field aligned (a) along [001], (b) at an angle of 50◦with respect to [110] (cf. the inset in Fig. 5) and (c) along\n[110]. The blue, dashed lines in (a)-(c) show the uniform res onance field, obtained by numerically solving Eq. (8) for eac h given\nfield orientation. The dotted black lines in (a) indicate the resonance magnetic fields. In (d)-(f), a magnification of the data\n(blue dotted lines) and simulation (red solid lines) from Fi g. 5 (a) is shown using the same scale for all orientations. In (e), a\nsimulation with a different set of parameters is shown for com parison (black, dashed line), see text.11\nout-of-plane anisotropy and therefore neglect B1¯10in our\nsimulations. An in-plane rotation of the external field\nwould be required for a more accurate measurement of\nB4||andB1¯10, but is outside the scope of this work.\nAccording to the valence-band model in Ref. 12, an\noscillatory behavior of the magnetic anisotropy parame-\nters is expected as a function of p. Therefore, depend-\ning on the absolute value of p, different values for, e.g.,\n∂B001/∂pare expected. In particular, there are regions\nwhere a anisotropy parameter might be nearly indepen-\ndent ofpand other regions with a very steep pdepen-\ndence. Since the absolute value of pis unknown, a quan-\ntitative discussion of the pdependence of the obtained\nanisotropy parameters based on the model in Ref. 12\nis not possible. In addition to p, thep-dexchange\nintegral,12which mayalsovary asa function ofthe depth\nin a non-homogneous film, also influences the anisotropy\nparameters,12further complicating a quantitative analy-\nsis.\nFor all samples, we used a constant exchange stiffness\nDsin our modeling. As alluded to above, there is some\nambiguity in this assumption, since the exchange stiff-\nness and the gradient in the anisotropy both influence\nthe mode spacing. For simplicity, however, we intended\nto keep as many simulation parameters as possible con-\nstant. The absolute values obtained for the exchange\nstiffness agree within a factor of 2 with the ones obtained\nin previous experiments36,50but are a factor of 2-4 larger\nthan theoretically predicted.51For the reasons discussed\nabove,thereisalargeuncertaintyalsointhederivationof\nthe absolute value of Dsfrom standing spin-wave modes\nin layers with a gradient in the magnetic anisotropy con-\nstants.\nIn order to use one parameter set for all field-\norientations, the Gilbert damping parameter was as-\nsumed to be isotropic in the simulations. The modeling\nof the SWR data could be further improved by assum-\ning a non-isotropic damping, its value being larger for\nµ0H||[110] than for µ0H||[001] [cf. Fig. 5]. This how-\never, only improves the result when assuming a field\norientation-dependent scaling factor for the amplitude,\nwhich could be motivated, e.g., by the assumption that\nthe microwave magnetic field present at the sample po-\nsition depends on the sample orientation within the\ncavity. The absolute values of αdetermined here are\ncomparable with the ones obtained by ultra-fast opti-\ncal experiments,52but are larger than the typical α=\n0.01...0.03 values found by frequency-dependent FMR\nstudies.53,54As already alluded to, inhomogeneous line-\nbroadening mechanisms may play a dominant role,54in\nparticular for as-grown samples.55We therefore assume\nthat the values for αobtained in this study overesti-\nmate the actual intrinsic Gilbert damping. A frequency-\ndependent SWR study would be required to determine\nthe intrinsic α. Such a study could possibly also reveal a\np-dependent αas theoretically predicted.55In our study,\nassuming a zdependent αdid not improve the agree-\nment between simulation and experiment, corroboratingthe conjecture that inhomogeneous broadening mecha-\nnisms dominate the linewidth and therefore obscure a\npossiblezdependence of α.\nIV. SUMMARY\nWehavepresentedafinitedifference-typemodelingap-\nproach for standing spin-wave resonances based on a nu-\nmerical solution of the LLG equation. With this generic\nformalism, SWR spectra can be simulated accounting for\nelliptical magnetization precession, for arbitrary orienta-\ntionsofthe externalmagneticfield, andforarbitrarypro-\nfiles of all magnetic properties, including anisotropy pa-\nrameters, exchange stiffness, Gilbert damping, and sat-\nuration magnetization. The approach is applicable not\nonly to (Ga,Mn)As but to all ferromagnets.\nFour(Ga,Mn)Assamples, epitaxiallygrownwithV/III\nflux ratios of 1.1, 1.3, 1.5, and 3.5 were investigated by\nECV and spin-wave resonance spectroscopy, revealing a\ncorrelation of a linear gradient in the hole concentration\nwith the occurrence of standing spin wave resonances, in\nparticularfortheexternalfieldorientedout-of-plane. Us-\ning the presented modeling approach, the SWR spectra\ncould be reproduced in a simulation with one parameter\nset for all external field orientations. The simulation re-\nsults demonstrate that the profileof the out-of-planeuni-\nformresonancefieldiscorrelatedwiththeholeconcentra-\ntion profile. However, our measurements and simulations\nshow,that anon-uniformholeconcentrationprofileisnot\nthe only cause that leads to the observed non-uniform\nmagnetic anisotropy; possibly, a variation in the satura-\ntion magnetization also influences the anisotropy param-\neters. To gain a quantitative understanding of this issue,\nmore samples with known hole concentrations would be\nrequired, where both the absolute values and the profiles\nofpare varied. Such a study was, however, outside the\nscope of this work.\nBesides the modeling of SWR intensities and\nlinewidths, the presented formalism yields the magne-\ntization precession amplitude as a function of the po-\nsition within the ferromagnet. It can therefore be used\nto investigate spin-pumping intensities in (Ga,Mn)As/Pt\nbilayers.27The spin-pumping signal, detected as a volt-\nage across the Pt layer, should be proportional to\nthe magnetization precession cone in the vicinity of\nthe (Ga,Mn)As/Pt interface. By measuring the spin-\npumping signal as well as the SWR intensities of\n(Ga,Mn)As/Pt and by using our modeling approach, it\nshould be possible to investigate to which extent a mag-\nnetization mode which is localized at a certain posi-\ntion within the (Ga,Mn)As layer contributes to the spin-\npumping signal.12\nAcknowledgments\nThis work was supported by the Deutsche Forschungs-\nGemeinschaft via Grant No. SFB 631 C3 (Walter Schot-\ntky Institut) and Grant No. Li 988/4 (Universit¨ at Ulm).\nAppendix A: Coordinate Transformation and Free\nEnthalpy derivatives\nThe transformation between the crystallographiccoor-\ndinate system ( x,y,z) and the equilibrium system (1,2,3)\nis given by\n\nmx\nmy\nmz\n=T\nm1\nm2\nm3\n, (A1)\nwith\nT=\ncosθ0cosφ0−sinφ0sinθ0cosφ0\ncosθ0sinφ0cosφ0sinθ0sinφ0\n−sinθ00 cos θ0\n.(A2)\nThe derivatives of the free enthalpy density Eq. (6) with\nrespect to the magnetization components are\nG3=∂m3G|m=m0=−µ0H3+2B001cos2θ0\n+B1¯10(sinθ0cosφ0−sinθ0sinφ0)2\n+ 4B4⊥cos4θ0\n+ 4B4/bardblsin4θ0(cos4φ0+sin4φ0) (A3)\nG21=G12=∂m1∂m2G|m=m0\n= cosθ0(1−2cos2φ0)[B1¯10\n+ 12B4/bardblsin2θ0cosφ0sinφ0] (A4)\nG11=∂m1∂m1G|m=m0= 2B001sin2θ0\n+ 12cos2θ0sin2θ0[B4⊥\n+B4/bardbl(cos4φ0+sin4φ0)]\n+B1¯10cos2θ0(cosφ0−sinφ0)2(A5)\nG22=∂m2∂m2G|m=m0= 2B1¯10(sinφ0+cosφ0)2\n+ 24B4/bardblsin2θ0cos2φ0sin2φ0. (A6)Appendix B: Finite Difference Method\nIn this Appendix, we describe how the spin-waveequa-\ntion can be numerically solved by the finite difference\nmethod. We start with the simple case of a circulary pre-\ncessing magnetization, neglecting Gilbert damping and\nthe driving field (Sec. B1). Then we turn to the gen-\neral case, where the magnetization precesses elliptically\nand the Gilbert damping as well as the driving field are\nincluded (Sec. B2).\n1. The One-Dimensional, Homogeneous,\nUndamped Case\nHere, we describe how the resonance fields and the\nspin-wave modes can be found, assuming a circularly\nprecessing magnetization m2=im1= ˜m, a constant ex-\nchangestiffness, and a zindependent equilibrium magne-\ntization. This case has been considered in Ref. 36 using a\nsemi-analytical approach to solve the spin-wave equation\nEq. (10). The approach considered here, is slightly more\ngeneral, as it is straight forward to determine resonance\nfields and eigenmodes of the system for an arbitrary z\ndependence of the uniform resonance field. To solve\nEq. (10), we divide the ferromagnetic film into a finite\nnumbernof layers with equal thickness land constant\nmagnetic properties within each of these layers. The z\ndependence of ˜ mandµ0H001\nuniis thus given by an index\nj= 1...n. Within each of these layers the uniform reso-\nnance field and ˜ m(z) are thus constant and given by the\nvaluesµ0H001,j\nuni=:Kjand ˜mj, respectively. The second\nderivative of ˜ mis approximated by\n˜m′′(z=j·l)≈˜mj−1−2˜mj+ ˜mj+1\nl2.(B1)\nConsequently, Eq. (10) is converted to the homogeneous\nequation system\n\n.........\n... Kj−1+2d−d 0...\n...−d Kj+2d−d ...\n...0 −d Kj+1+2d ...\n.........\n\n...\n˜mj−1\n˜mj\n˜mj+1\n...\n=µ0H\n...\n˜mj−1\n˜mj\n˜mj+1\n...\n, (B2)\nwith the abbreviation d=−Ds/l2. The boundary condi-\ntion of natural freedom36(von Neumann boundary con-dition) reads as ˜ m0= ˜m1and ˜mn−1= ˜mnand can be\nincorporated in Eq. (B2). Since the matrix on the left13\nhand side of Eq. (B2) is sparse, it can be efficiently diag-\nonalized numerically, yielding the resonancefields (eigen-\nvalues) and the corresponding modes (eigenvectors). Af-\nter diagonalizingthe matrix, the relevantresonancefields\nare found by sorting the eigenvalues and considering only\nthe modes with positive resonance fields, corresponding\nto the bound states in the particle-in-a-box analogon.\nThe SWR amplitude of each mode is proportional to its\nnet magnetic moment; thus, the amplitudes can be found\nby integrating the (normalized) eigenmodes. The mode\nprofile, the resonance fields, and the SWR intensities are\nillustrated in Fig. 2 for a constant and a linearly varying\nuniform resonance field. The finite linewidth of the SWR\nmodes can be accounted for by assuming a Lorentzian\nlineshape for each mode with a certain linewidth and\nwith the resonance fields and intensities calculated as\ndescribed above36. Note that this approach to derive\nresonance fields and intensities is only valid if the mode\nseparation is large compared with the linewidth of the\nmodes; this restriction does not apply to the model pre-\nsented in the Appendix B2.\n2. The General Case\nTosolveEq.(7)forarbitrary µ0Handarbitrarilyvary-\ning magnetic properties, we again divide the ferromag-netic film into a finite number nof layers with equal\nthicknessland constant magnetic properties within each\nof these layers. In contrast to the case in the Appendix\nB1, where only the uniform resonance field was varied\nacross the layer, here potentially all magnetic proper-\nties entering Eq. (7) can be assumed to be zdependent.\nAdditionally, the components of the driving field µ0hi\n(i= 1,2), can also vary as a function of z, since the\n(1,2,3) frame of reference is zdependent and thus the\nprojections of the driving field have to be calculated for\neach layer. The zdependence of the components mi\n(i= 1,2), of the parameters H11,H12,H21,H22(de-\nfined in Sec. II) and the exchange stiffness is thus given\nby the index j= 0...n; the second derivative of each of\nthe components miis approximated as in Eq. (B1).\nThe linearized LLG equation Eq. (7), is thus converted\ninto the inhomogeneous equation system\n\n..................\n...Hj−1\n11−2dj−1Hj−1\n12dj−10 0 0 ...\n... Hj−1\n21Hj−1\n22−2dj−10dj−10 0 ...\n... dj0Hj\n11−2djHj\n12dj0...\n... 0 djHj\n21Hj\n22−2dj0 dj...\n... 0 0 dj+10Hj+1\n11−2dj+1Hj+1\n12...\n... 0 0 0 dj+1Hj+1\n21Hj+1\n22−2dj+1...\n..................\n\n...\nmj−1\n1\nmj−1\n2\nmj\n1\nmj\n2\nmj+1\n1\nmj+1\n2...\n=µ0\n...\nhj−1\n1\nhj−1\n2\nhj\n1\nhj\n2\nhj+1\n1\nhj+1\n2...\n,(B3)\nwith the abbreviation dj=−Dj\ns/l2. 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B 69,\n085209 (2004)."    },    {        "title": "2303.03852v1.Electrically_tunable_Gilbert_damping_in_van_der_Waals_heterostructures_of_two_dimensional_ferromagnetic_metals_and_ferroelectrics.pdf",        "content": "Page 1 of 15 \n Electrically tunable Gilbert damping in van der Waals heterostructures of two-\ndimensional ferromagnetic meta ls and ferroelectrics \nLiang Qiu,1 Zequan Wang,1 Xiao-Sheng Ni,1 Dao-Xin Yao1,2 and Yusheng Hou 1,* \nAFFILIATIONS \n1 Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices, State \nKey Laboratory of Optoelectronic Materials and Technologies, Center for Neutron \nScience and Technology, School of Physics, Sun Yat-Sen University, Guangzhou, \n510275, China  \n2 International Quantum Academy, Shenzhen 518048, China \n \nABSTRACT  \nTuning the Gilbert damping of ferromagnetic (FM) metals via a nonvolatile way  is \nof importance to exploit and design next-generation novel spintronic devices. Through \nsystematical first-principles calculations, we study the magnetic properties of the van \nder Waals heterostructure of two-dimensional FM metal  CrTe 2 and ferroelectric (FE) \nIn2Te3 monolayers. The ferromagnetism of CrTe 2 is maintained in CrTe 2/In2Te3 and its \nmagnetic easy axis can be switched from in-plane to out- of-plane by reversing the FE \npolarization of In 2Te3. Excitingly, we find that the Gilbert damping of CrTe 2 is tunable \nwhen the FE polarization of In 2Te3 is reversed from upward to downward. By analyzing \nthe k-dependent contributions to the Gilbert damping, we unravel that such tunability \nresults from the changed intersections between the bands of CrTe 2 and Fermi level on \nthe reversal of the FE polarizations of In 2Te3 in CrTe 2/In2Te3. Our work provides a n \nappealing way to electrically tailor Gilbert dampings of two-dimensional FM metals by \ncontacting them with ferroelectrics.  \n \n*Authors to whom correspondence should be addressed:  \n[Yusheng Hou, houysh@mail.sysu.edu.cn]  \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: 10.1063/5.0145401Page 2 of 15 \n Since the atomically thin long-range ferromagnetic ( FM) orders at finite \ntemperatures are discovered in CrI 31 monolayer (ML) and Cr 2Ge2Te62 bilayer, two-\ndimensional (2D) van der Waals (vdW) FM materials have attracted intensive \nattention.3-5 Up to now, many novel vdW ferromagnets such as Fe 3GeTe 2,6 Fe5GeTe 2,7 \nVSe 28,9 and MnSe 210 have been synthesized in experiments. Due to the intrinsic \nferromagnetism in these vdW FM materials, it is highly fertile to engineer emergent \nphenomena through magnetic proximity effect in their heterostructures.11 For instance , \nan unprecedented control of the spin and valley pseudospins in WSe 2 ML is reported in \nCrI 3/WSe 2.12 By contacting the thin films of three-dimensional topological insulators \nand graphene with CrI 3, high-temperature quantum anomalous Hall effect and vdW spin \nvalves are proposed in CrI 3/Bi2Se3/CrI 313 and CrI 3/graphene/CrI 3,14 respectively. On the \nother hand, the magnetic properties of these vdW FM materials can also be controlled \nby means of external perturbations such as gating and moiré patterns.3 In CrI 3 bilayer, \nHuang et al. observed a voltage-controlled switching between antiferromagnetic (AFM) \nand FM states.15 Via an ionic gate, Deng et al. even increased the Curie temperature \n(TC) of the thin flake of vdW FM metal Fe 3GeTe 2 to room temperature, which is much \nhigher than its bulk TC.6 Very recently, Xu et al. demonstrated a coexisting FM and \nAFM state in a twisted bilayer CrI 3.16 These indicate that vdW FM materials are \npromising platforms to design and implement spintronic devices in the 2D limit.4,11  \nRecently, of great interest is the emergent vdW magnetic material CrTe 2 which is \na new platform for realizing room-temperature intrinsic ferromagnetism.17,18 Especially, \nCrTe 2 exhibits greatly tunable magneti sm. In the beginning, its ground state is believed \nto be the nonmagnetic 2 H phase,19 while several later researches suggest that either the \nFM or AFM 1 T phases should be the ground state of CrTe 2.17,18,20- 23 Currently, the \nconsensus is that the structural ground state of CrTe 2 is the 1 T phase. With respect to its \nmagnetic ground state, a first-principles study shows that the FM and AFM ground \nstates in CrTe 2 ML depend on its in-plane lattice constants.24 It is worth noting that the \nTC of FM CrTe 2 down to the few-layer limit can be higher than 300 K,18 making it have \nwide practical application prospects in spintronics.  \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: 10.1063/5.0145401Page 3 of 15 \n Building heterostructures of FM and ferroelectric (FE) materials offers an effective \nway to control nonvolatile magnetism via an electric field. Experimentally, Eerenstein \net al. presented an electric-field-controlled nonvolatile converse magnetoelectric effect \nin a multiferroic heterostructure La0.67Sr0.33MnO 3/BaTiO 3.25 Later, Zhang et al. reported \nan electric-field-driven control of nonvolatile magnetization in a heterostructure of FM \namorphous alloy Co40Fe40B20 and FE Pb(Mg 1/3Nb2/3)0.7Ti0.3O3.26 Theoretically, Chen et \nal. demonstrated based on first-principles calculations that the interlayer magnetism of \nCrI 3 bilayer in CrI 3/In2Se3 is switchable between FM and AFM couplings by the \nnonvolatile control of the FE polarization direction of In 2Se3.27 In spite of these \ninteresting findings, using FE substrates to electrically tune the Gilbert damping of \nferromagnets, an important factor determining the operation speed of spintronic devices, \nis rarely investigated in 2D FM/FE vdW heterostructures. Therefore, it is of great \nimportance to explore the possibility of tuning the Gilbert damping in such kind of \nheterostructures.  \nIn this work, we first demonstrate that the magnetic ground state of 1 T-phase CrTe 2 \nML will change from the zigzag AFM (denoted as z-AFM) to FM orders with increasing \nits in-plane lattice constants. By building a vdW heterostructure of CrTe 2 and FE In2Te3 \nMLs, we show that the magnetic easy axis of CrTe 2 can be tuned from in-plane to out-\nof-plane by reversing the FE polarization of In 2Te3, although its ferromagnetism is kept . \nImportantly, we find that the Gilbert damping of CrTe 2 is tunable with a wide range on \nreversing the FE polarization of In 2Te3 from upward to downward. Through looking \ninto the k-dependent contributions to the Gilbert damping, we reveal that such tunability \noriginates from the changed intersections between the bands of CrTe 2 and Fermi level \nwhen the FE polarizations of In 2Te3 is reversed in CrTe 2/In2Te3. Our work demonstrates \nthat putting 2D vdW FM metals on FE substrates is an attractive method to electrically \ntune their Gilbert dampings.  \nCrTe 2, a member of the 2D transition metal dichalcogenide family, can potentially \ncrystalize into several different layered structures such as 1 T, 1Td, 1H and 2 H phases.28 \nIt is believed that the 1 T phase is the most stable among all of the se possible phases 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: 10.1063/5.0145401Page 4 of 15 \n both bulk and ML. This phase has a hexagonal lattice and belongs to the P3_m1 space \ngroup, with each Cr atom surrounded by the octahedrons of Te atoms (Fig. 1a). In view \nof the hot debates on the magnetic ground state in CrTe 2 ML, we establish a 2× 2√3 \nsupercell and calculate the total energies of several different magnetic structures (Fig. \nS1 in Supplementary Materials) when its lattice constant varies from 3.65 to 4.00 Å. As \nshown in Fig. 1b, our calculations show that z-AFM order is the magnetic ground state \nwhen the lattice constant is from 3.65 to 3.80 Å. By contrast, the FM order is the \nmagnetic ground state when the lattice constant is in the range from 3.80 to 4.00 Å. \nNote that our results are consistent with the experimentally observed z- AFM23 and \nFM29 orders in CrTe 2 with a lattice constant of 3.70 and 3.95 Å, respectively. Since we \nare interested in the Gilbert damping of ferromagnets and the experimentally grow n \nCrTe 2 on ZrTe 2 has a lattice constant of 3.95 Å,29 we will focus on CrTe 2 ML with this \nlattice constant hereinafter.  \n \n \nFIG. 1. (a) Side (the top panel) and top (the bottom panel) views of CrTe 2 ML. The NN \nand second- NN exchange paths are shown by red arrows in the top view. (b) The phase \ndiagram of the magnetic ground state of CrTe 2 ML  with different lattice constants. Insets \nshow the schematic illustrations of the z-AFM and FM orders. The up and down spins \nare indicated by the blue and red balls, respectively. The stars highlight the experimental \nlattice constants of CrTe 2 in Ref.23 and Ref.29.  \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: 10.1063/5.0145401Page 5 of 15 \n To obtain an deeper understanding on the ferromagnetism of CrTe 2 ML, we adopt \na spin Hamiltonian consisting of Heisenberg exchange couplings and single-ion \nmagnetic anisotropy (SIA) as follows:30  \n𝐻 = 𝐽 1∑ 𝑆𝑖⋅ 𝑆𝑗 ⟨𝑖𝑗⟩ + 𝐽 2∑ 𝑆𝑖⋅ 𝑆𝑗 ⟨⟨𝑖𝑗⟩⟩ − 𝐴 ∑ ( 𝑆𝑖𝑧)2\n𝑖  (1) \nIn Eq. (1), J1 and J2 are the nearest neighbor (NN) and second- NN Heisenberg exchange \ncouplings. Note that a negative (positive) J means a FM (AFM) Heisenberg exchange \ncouples. Besides, A parameterizes the SIA term. First of all, our DFT calculations show \nthat the magnetic moment of CrTe 2 ML is 3.35 μB/Cr, consistent with previous DFT \ncalculations.31 As shown in Table I, the calculated J1 and J2 are both FM and J1 is much \nstronger than J2. Both FM  J1 and J2 undoubtedly indicate that CrTe 2 ML has a FM \nmagnetic ground state. Finally, the SIA parameter A is obtained by calculating the \nenergy difference between two FM states with out-of-plane and in-plane magnetizations. \nOur calculations obtain A=1.81 meV/Cr, indicating that CrTe 2 ML has an out-of-plane \nmagnetic easy axis. Hence, our calculations show that CrTe 2 ML exhibits an out-of-\nplane FM order, consistent with experimental observations.29  \n \nTABLE I. Listed are the in-plane lattice constant s a, Heisenberg exchange couplings J \n(in unit of meV) and SIA (in unit of meV/Cr) of CrTe 2 ML and CrTe 2/In2Te3. \nSystem  a (Å) J1 J2  A \nCrTe 2 3.95 -24.56  -0.88  1.81 \nCrTe 2/In2Te3(↑) 7.90 -20.90  -1.80  -1.44  \nCrTe 2/In2Te3(↓) 7.90 -19.33  -0.88   0.16  \n \nTo achieve an electrically tunable Gilbert damping in CrTe 2 ML, we establish its \nvdW heterostructure with F E In2Te3 ML. In building this heterostructure, w e stack a \n2×2 supercell of CrTe 2 and a √3 ×√3  supercell of In2Te3 along the (001) direction. \nBecause the magnetic properties of CrTe 2 ML are the primary topic and the electronic \nproperties of In2Te3 ML are basically not affected by a strain (Fig. S2), we stretch the \nlattice constant of the latter to match that of the former. Fig. 2a shows the most stable \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: 10.1063/5.0145401Page 6 of 15 \n stacking configuration in CrTe 2/In2Te3 with an upward FE polarization [denoted as \nCrTe 2/In2Te3(↑)]. At the interface in this configuration, one of four Cr atoms and one of \nfour Te atoms at the bottom of CrTe 2 sits on the top of the top-layer Te atom s of In2Te3. \nIn CrTe 2/In2Te3 with a downward FE polarization [denoted as CrTe 2/In2Te3(↓ )], the \nstacking configuration at its interface is same as that in CrTe 2/In2Te3(↑ ). The only \ndifference between CrTe 2/In2Te3(↑ ) and CrTe 2/In2Te3(↓ ) is that the middle-layer Te \natoms of In 2Te3 in the former is farther to CrTe 2 than that in the latter (Fig. 2a and 2c). \nIt is noteworthy that the bottom-layer Te atoms of CrTe 2 do not stay at a plane anymore \nin the relaxed CrTe 2/In2Te3 (see more details in Fig. S3), suggesting non-negligible \ninteractions between CrTe 2 and In 2Te3. \n \n  \nFIG. 2. (a) The schematic stacking configuration and (b) charge density difference 𝛥ρ \nof CrTe 2/In2Te3(↑). (c) and (d) same as (a) and (b) but for CrTe 2/In2Te3(↓). In (b) and \n(d), color bar indicates the weight of negative (blue) and positive (red) charge density \ndifferences. (e) The total DOS of CrTe 2/In2Te3. (f) and (g) show the PDOS of CrTe 2 and \nIn2Te3 in CrTe 2/In2Te3, respectively. In (e)-(g), upward and downward polarizations are \nindicated by black and red lines, respectively.  \n \nTo shed light on the effect of the FE polarization of In 2Te3 on the electronic \nproperty of CrTe 2/In2Te3, we first investigate the spatial distribution of charge density \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: 10.1063/5.0145401Page 7 of 15 \n difference \n2 2 3 2 2 3 CrTe In Te CrTe In Te     = − −  with different FE polarization directions. \nAs shown in Fig. 2b and 2d, we see that there is an obvious charge transfer at the \ninterfaces of both CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓), which is further confirmed by \nthe planar averaged 𝛥ρ (Fig. S4). Additionally, the charge transfer in CrTe 2/In2Te3(↑) \nis distinctly less than th at in CrTe 2/In2Te3(↓). Fig. 2e shows that the total density of \nstates (DOS) near Fermi level are highly different in CrTe 2/In2Te3(↑ ) and \nCrTe 2/In2Te3(↓). By projecting the DOS onto CrTe 2 and In 2Te3, Fig. 2f shows that the \nprojected DOS (PDOS) of CrTe 2 in CrTe 2/In2Te3(↑) is larger than that in CrTe 2/In2Te3(↓) \nat Fermi level. Interestingly, the PDOS of In 2Te3 in CrTe 2/In2Te3(↑) is larger than that \nin CrTe 2/In2Te3(↓) below Fermi level while the situation is inversed above Fermi level \n(Fig. 2g). By looking into the five Cr- d orbital projected DOS in CrTe 2/In2Te3(↑) and \nCrTe 2/In2Te3(↓) (Fig. S5), we see that there are obviously different occupations for xyd, \n22xyd− and 223zrd− orbitals near Fermi level. All of these imply that the reversal of the \nFE polarization of In 2Te3 may have an unignorable influence on the magnetic properites \nof CrTe 2/In2Te3.   \nDue to the presence of the FE In 2Te3, the inversion symmetry is inevitably broken \nand nonzero Dzyaloshinskii-Moriya interactions (DMIs) may exist in CrTe 2/In2Te3. In \nthis case, we add a DMI term into Eq. (1) to investigate the magnetism of CrTe 2/In2Te3 \nand the corresponding spin Hamiltonian is in the form of32  \n𝐻 = 𝐽 1∑ 𝑆𝑖⋅ 𝑆𝑗 ⟨𝑖𝑗⟩ + 𝐽 2∑ 𝑆𝑖⋅ 𝑆𝑗 ⟨⟨𝑖𝑗⟩⟩ +∑ 𝑫𝑖𝑗⋅ (𝑆 𝑖× 𝑆 𝑗) ⟨𝑖𝑗⟩ − 𝐴 ∑ ( 𝑆𝑖𝑧)2\n𝑖  (2). \nIn Eq. (2), Dij is the DMI vector of the NN Cr-Cr pairs. As the C6-rotational symmetry \nwith respect to Cr atoms in CrTe 2 is reduced to the C3-rotational symmetry, the NN \nDMIs are split into four different DMIs (Fig. S6). For simplicity, the J1 and J2 are still \nregard ed to be six-fold. From Table I, we see that the NN J1 of both CrTe 2/In2Te3(↑) and \nCrTe 2/In2Te3(↓) are still FM but slightly smaller than that of free-standing CrTe 2 ML. \nMoreover, the second- NN FM J2 is obviously enhanced in CrTe 2/In2Te3(↑) compared \nwith CrTe 2/In2Te3(↓) and free-standing CrTe 2 ML. To calculated the NN DMIs, we build \na √3×√3 supercell of CrTe 2/In2Te3 and the four-state method33 is employed here. As \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: 10.1063/5.0145401Page 8 of 15 \n listed in Table S1, the FE polarization direction of In 2Te3 basically has no qualitative \neffect on the DMIs in CrTe 2/In2Te3 although it affects their magnitudes. More explicitly, \nthe magnitudes of the calculated DMIs range from 1.22 to 2.81 meV, which are about \none order smaller than the NN J1. Finally, we find that the SIA of CrTe 2/In2Te3 is \nstrongly dependent on the FE polarization of In 2Te3. When In 2Te3 has an upward FE \npolarization, the SIA of CrTe 2/In2Te3(↑) is negative, indicating an in-plane magnetic \neasy axis. However, when the FE polarization of In 2Te3 is downward, CrTe 2/In2Te3(↓) \nhas a positive SIA, indicating an out-of-plane magnetic easy axis. It is worth noting that \nCrTe 2/In2Te3(↓) has a much weak SIA than the free-standing CrTe 2 ML, although they \nboth have positive SIAs. The different Heisenberg exchange couplings, DMIs and SIAs \nin CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓) clearly unveil that the magnetic properties of \nCrTe 2 are tuned by the FE polarization of In2Te3.  \nTo obtain the magnetic ground state of CrTe 2/In2Te3, MC simulations are carried \nout. As shown in Fig. S7, CrTe 2/In2Te3(↑) has an in-plane FM magnetic ground state \nwhereas CrTe 2/In2Te3(↓ ) has an out-of-plane one. Such magnetic ground states are \nunderstandable. Firstly, the ratios between DMIs and the NN Heisenberg exchange \ncouplings are small and most of them are out of the typical range of 0.1–0.2 for the \nappearance of magnetic skyrmions.34 Secondly, the SIAs of the CrTe 2/In2Te3(↑) and the \nCrTe 2/In2Te3(↓ ) prefer in-plane and out-of-plane magnetic easy axes, respectively . \nTaking them together, we obtain that the FM Heisenberg exchange couplings dominate \nover the DMIs and thus give rise to a FM magnetic ground state with its magnetization \ndetermined by the SIA,35 consistent with our MC simulated results.  \nFigure 3a shows the Γ-dependent Gilbert dampings of CrTe 2/In2Te3 with upward \nand downward FE polarizations of In2Te3. Similar to previous studies,36,37 the Gilbert \ndampings of CrTe 2/In2Te3 decrease first and then increase as the scattering rate Γ \nincreases. Astonishingly, the Gilbert dampings of CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓) \nare distinctly different at the same scattering rate Γ ranging from 0.001 to 1.0 eV . To \nhave a more intuitive sense on the effect of the FE polarizations of In 2Te3 on the Gilbert \ndampings in CrTe 2/In2Te3, we calculate the ratio   =  at any given Γ, where \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: 10.1063/5.0145401Page 9 of 15 \n ( ) is the Gilbert damping of CrTe 2/In2Te3(↑) [CrTe 2/In2Te3(↓)]. As shown in Fig. \n3b, the ratio 𝜂 ranges from 6 to around 1.3 with increasing Γ. As the FE polarization \nof In 2Te3 can be switched from upward to downward by an external electric field, the \nGilbert damping of CrTe 2/In2Te3 is electrically tunable in practice.  \n \n \nFIG. 3. (a) The Γ-dependent Gilbert dampings of CrTe 2/In2Te3 with upward (black line) \nand downward (red line) FE polarizations of In 2Te3. (b) The Gilbert damping ratio 𝜂 \nas a function of the scattering rate Γ.  \n \nTo gain a deep insight into how the FE polarization of In 2Te3 tunes the Gilbert \ndamping in CrTe 2/In2Te3, we investigate the k-dependent contributions to the Gilbert \ndampings of CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓). As shown in Fig. 4a and 4b, the bands \naround Fermi level have qiute different intermixing between CrTe 2 and In 2Te3 states \nwhen the FE polarizaiton of In 2Te3 is reversed. Explicitly, there are obvious intermixing \nbelow Fermi leve in CrTe 2/In2Te3(↑) while the intermixing mainly takes place above \nFermi level in CrTe 2/In2Te3(↓). Especially, the bands intersected by Fermi level are at \ndifferent k points in CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓). Through looking into the k-\ndependent contributions to the ir Gilbert dampings (Fig. 4c and 4d), we see that large \ncontributions are from the k points (highlighted by arrows in Fig. 4) at which the bands \nof CrTe 2 cross Fermi level. In addition, these large contributions are different. Such k-\ndipendent contribution to Gilbert dampings is understandable. Based on the scattering \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: 10.1063/5.0145401Page 10 of 15 \n theory of Gilbert damping 38, Gilbert damping parameter is calculated using the \nfollowing Eq. (3) 36  \n( ) ( ) , , , , , , (3)kk\nk i k j k j k i F k i F k j\nk ij SE E E EM u u\n      = − − −\nHH, \nwhere EF is Fermi level and Ek,i is the enery of band i at a given k point. Due to the delta \n( ) ( ) ,, F k i F k jE E E E −−  , only the valence and conduction bands near Fermi level \nmake dominant contribution to the Gilbert damping. Additionally, their contributions \nalso depend on factor , , , ,kk\nk i k j k j k iuu   \nHH. Overall , through changing the \nintersections between the bands of CrTe 2 and Fermi level, the reversal of the FE \npolarization of In 2Te3 can modulate the contributions to Gilbert damping. Consequently, \nthe total Gilbert dampings are different in CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓).  \n \n \nFIG. 4. (a) Band structure calculated with spin-orbit coupling and (c) the k-dependent \ncontributions to the Gilbert damping in CrTe 2/In2Te3(↑). (b) and (d) same as (a) and (c) \nbut for CrTe 2/In2Te3(↓). In (a) and (b), Fermi levels are indicated by horizontal dash \nlines and the states from CrTe 2 and In 2Te3 are shown by red and blue, respectively.  \n \n    From experimental perspectives, the fabrication of CrTe 2/In2Te3 vdW \nheterostructure should be feasible. On the one hand, CrTe 2 with the lattice constant of \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: 10.1063/5.0145401Page 11 of 15 \n 3.95 Å has been successfully grown on ZrTe 2 substrate by the molecular beam epitaxy.29 \nOn the other hand, In 2Te3 is also synthesized.39 Taking these and the vdW nature of \nCrTe 2 and In 2Te3 together, a practical scheme of growing CrTe 2/In2Te3 is sketched in \nFig. S8 : first grow CrTe 2 ML on ZrTe 2 substrate29 and then put In2Te3 ML on CrTe 2 to \nform the desired CrTe 2/In2Te3 vdW heterostructure.     \nIn summary, by constructing a vdW heterostructure of 2D FM metal CrTe 2 and FE \nIn2Te3 MLs, we find that the magnetic properties of CrTe 2 are engineered by the reversal \nof the FE polariton of In 2Te3. Although the ferromagnetism of CrTe 2 is maintained in \nthe presence of the FE In2Te3, its magnetic easy axis can be tuned from in-plane to out-\nof-plane by reversing the FE polarization of In 2Te3. More importantly, the Gilbert \ndamping of CrTe 2 is tunable with a wide range when reversing the FE polarization of \nIn2Te3 from upward to downward. Such tunability of the Gilbert damping in \nCrTe 2/In2Te3 results from the changed intersections between the bands of CrTe 2 and \nFermi level on reversing the FE polarizations of In 2Te3. Our work introduces a \nremarkably useful method to electrically tune the Gilbert dampings of 2D vdW FM \nmetals by contacting them with ferroelectrics, and should stimulate more experimental \ninvestigations in this realm.  \n \nSee the supplementary material for the details of computational methods31,36,40- 50 \nand other results mentioned in the main text.  \n \nThis project is supported by National Nature Science Foundation of China (No. \n12104518, 92165204, 11974432), NKRDPC-2018YFA0306001, NKRDPC-\n2022YFA1402802, GBABRF-2022A1515012643 and GZABRF-202201011118 . \nDensity functional theory calculations are performed at Tianhe- II. \n \nAUTHOR DECLARATIONS  \nConflict of Interest  \nThe authors have no conflicts to disclose.  \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: 10.1063/5.0145401Page 12 of 15 \n Author Contributions  \nLiang Qiu : Investigation (equal); Methodology (equal); Writing –original draft (equal). \nZequan Wang : Methodology (equal). Xiao -sheng Ni : Investigation (equal); \nMethodology (equal).  Dao-Xin Yao : Supervision (equal); Funding acquisition (equal); \nInvestigation (equal); Writing – review &editing (equal). Yusheng Hou : \nConceptualization (equal); Funding acquisition (equal); Investigation (equal); Project \nadministration(equal); Resources (equal); Supervision (equal); Writing – review \n&editing (equal).  \n \nDATA A V AILABILITY  \nThe data that support the findings of this study are available from the \ncorresponding authors upon reasonable request.  \n \n \nREFERENCES  \n1 B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, \nD. Zhong, E. 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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: 10.1063/5.0145401This 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: 10.1063/5.0145401This 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: 10.1063/5.0145401This 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: 10.1063/5.0145401This 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: 10.1063/5.0145401"    },    {        "title": "1511.04227v1.Magnified_Damping_under_Rashba_Spin_Orbit_Coupling.pdf",        "content": "Magnified Damping under Rashba Spin Orbit Coupling  \nNovember 1, 2015  \n \n \n1 \n Magnified Damping  under Rashba Spin Orbit Coupling  \n \nSeng  Ghee  Tan† 1,2 ,Mansoor  B.A.Jalil1,2   \n(1) Data Storage Institute, Agency for Science, Technology and Research (A*STAR)  \n2  Fusionopolis Way , #08-01 DSI , Innovis , Singapore 138634  \n \n(2) Department of Electrical  Engineering, National University of Singapore,  \n             4 Engineering Drive 3, Singapore 117576  \n \n \nAbstract  \nThe spin orbit coupling spin torque consists of the field -like [REF: S.G. Tan et al.,  \narXiv:0705.3502, (2007). ] and the damping -like terms [REF:  H. Kurebayas hi et al., Nature \nNanotechnology 9, 211 (2014). ] that have been widely studied for applications in magnetic \nmemory. We focus , in this article,  not on the spin orbit effect producing the above spin \ntorques, but on its magnifying the damping constant of all field like spin torques. As first \norder precession leads to second order damping, the Rashba constant is naturally co -opted, \nproducing a magnified field -like dam ping effect. The Landau -Liftshitz -Gilbert equations are \nwritten separately for the local magnet ization and the itinerant spin, allowing the \nprogression of magnetization to be self -consistently locked to  the spin.   \n \n  \n \n \n \nPACS: 03.65.Vf, 73.63. -b, 73.43. -f \n† Correspondence author:  \nSeng Ghee Tan  \nEmail: Tan_Seng_Ghee@dsi.a -star.edu.sg  \n Magnified Damping under Rashba Spin Orbit Coupling  \nNovember 1, 2015  \n \n \n2 \n 1. Introduction  \n      In spintronic and magnetic physics, magnetization switching and spin torque [1] have \nbeen well -studied. The advent of the Rashba spin-orbit coupling ( RSOC) [2,3]  due to \ninversion asymmetry at the i nterface of the ferromagnetic/heavy atom (FM/HA) \nheterostructure introduces new spin torque to the FM magnetization. The field -like [4-6] \nand the damping -like [7] SOC spin torque had been  theoretically derived  based on the gauge \nphysics and the Pancharatna m-Berry’s phase , as well as  experimentally verified  and resolved . \nThe numerous  observation s of spin -orbit generation of spin torque [8-10], are all related to \nthe experimental resolutions [6,7] of their field -like and damping -like nature, thus ushering \nin the possibility of spin -orbit based magnetic memory. While the damping -like spin torque  \ndue to Kurebayashi et al. [7] is dissipative in nature, the field -like due to Tan et al. [4,11], is \nnon-dissipative , and precession causing . Recent studies have even mo re clearly \ndemonstrated the physics and application promises of both the field -like and the damping -\nlike SOC spin torque [12-14]. Besides , similar SOC spin torque have also been studied \ntheoretically  in FM/3D -Rashba [15] and FM -topological -insulator materi al [16,17] , and \nexperimentally shown [18, 19 ] in topological insulator materials.  \n      The dissipative physics of all field -like magnetic torque terms have been  derived in \nsecond -order manifestation in a manner introduced by Gilbert  in the 1950 ’s. Conven tional \nstud y of magnetization dynamics is based on a Gilbert damping constant  which is  \nincorporated manually into the Landau -Lifshit z-Gilbert  (LLG)  equation. In this paper, we will \nfocus our attention not so much on the spin-orbit effect producing the SOC spin torque, as \non the spin -orbit effect magnifying the damping constant of all field -like spin torques. As \nfield -like spin torques,  regardless of origin s, generate first-order precession , the Rashba \nconstant will be co -opted in to the second -order damping effect, producing a mag nified \ndamping  constant . On the other hand, c onventional incorporation of the dissipative \ndamping physics into the LLG would fail t o account for the spin-orbit magnification of  the \ndamping strength . It would therefore be necessary to  deriv e the LLG equations from a \nHamiltonian  which  describes electron  due to  the local  FM magnetization (𝒎), and those \nitinerant (𝒔) and injected from external parts . We present a set of modified LLG  equation s \nfor the 𝒎 and the 𝒔. This will be necessary  for a more precise modeling of the 𝒎 trajectory  \nthat simultaneously tracks the  𝒔 trajectory. In summary, the two central themes of this Magnified Damping under Rashba Spin Orbit Coupling  \nNovember 1, 2015  \n \n \n3 \n paper is our presentation of  a self-consistent set of LLG equations under the Rashba SOC \nand the  derivation of the Ra shba -magnified d amping constant in the  second -order damping -\nlike spin torque .  \n \n2. Theory of Magnified Damping  \n    The system under consideration is a FM/HA hetero -structure  with inversion asymmetry \nprovided by the interface. F ree electron denoted by  𝒔, is injected in an in -plane manner into \nthe device . The FM equilibrium  electron  is denoted by  𝒎. One considers the external \nsource -drain bias  to inject  electron  of free-electron  nature 𝒔 into the FM with kinetic, \nscattering , magnetic, and spin -orbit  energ ies.  The Hamiltonian is \n𝐻𝑓=𝑝2\n2𝑚+𝑉𝑖𝑚𝑝𝑠+𝐽𝑠𝑑𝑺.𝑴 +𝜇0𝑴.𝑯𝒂𝒏𝒊+(2(𝜆+𝜆′)\nℏ)(𝒔+𝒎).(𝒑×𝑬𝒕)\n−𝑖(𝜆+𝜆′)(𝒔+𝒎).(∇×𝑬𝒕) \n(1) \nwhere  𝒔,𝒎  have  the unit s of angular momentum i.e.  𝑛ℏ\n2 ,  while 𝑴=(𝑔𝑠𝜇𝐵\nℏ)𝒎 has the unit \nof magnetic moment , and 𝜇𝐵=𝑒ℏ\n2𝑚 is the Bohr magneton . Note that  (2𝜆\nℏ) is the vacuum SOC \nconstant, while  (2𝜆′\nℏ=2𝜂𝑅\nℏ2𝐸𝑖𝑛𝑣) is the Rashba  SOC constant. The SOC part of the Hamiltonian \nillustrates th e simultaneous presence of vacuum and Rashba SOC. The proportion of the \nnumber of electron subject to each coupling would depend on the degree of hybridization.  \nBut s ince 𝜆′≫𝜆, the above can be written with just the Rashba SOC effect. Care is taken t o \nensure  𝜆,𝜆′ share the same dimension of 𝑇𝑒𝑠𝑙 𝑎−1, and  𝑬𝒕 is the total electric field ,  𝐽𝑠𝑑 is \nthe s -d coupling constant, 𝑉𝑖𝑚𝑝𝑠 denotes the spin flip scattering potential, 𝑯𝒂𝒏𝒊 denotes the \naniso tropy field of the FM  material. On the other hand, one needs to be aware that the \nabove is an e xpanded SOC expression that c omprises a momentum part as well as a \ncurvature part [20]. One can then consider  the physics of the electric  curvature as related to  \nthe time dynamic of the spin moment , which bears  a similar origin to the Faraday effect.  In \nthe modern context of  Rashba physics  [21], one considers electron spin to lock to the orbital \nangular momentum  𝑳 due to intrinsic spin orbit coupling at the  atomic level. Due to  broken  Magnified Damping under Rashba Spin Orbit Coupling  \nNovember 1, 2015  \n \n \n4 \n inversion symmetry , electric field (𝑬𝒊𝒏𝒗) points perpendicular to the plane of the FM/HA \nhost . Because of hybridization,  the 𝒔,𝑳,𝒑 of an electron is coupled in a complic ated way by \nthe electric field. In a simple way, one first considers 𝑳 to be  coupled as 𝐻=(2𝜆\nℏ)𝑳.(𝒑×\n𝑬𝒊𝒏𝒗). As spin 𝒔 is coupled via atomic spin orbit locking to 𝑳, an effective coupling of 𝒔 to \n𝑬𝒊𝒏𝒗 can be expected  to occur with strength as determined by the atomic electric field.   We \nwill now take things a step further to make an assumption that 𝒔 is also coupled  via 𝑳  to \nother sources of electric f ields  e.g. those  arising from spin dynamic  (𝒅 𝑴\n𝒅𝒕,𝒅 𝑺\n𝒅𝒕), in the same \nway that it is coupled to 𝑬𝒊𝒏𝒗 . The actual extent of coupling will , however,  be an \nexperimental parameter that measures  the efficiency of Rashba coupling to 𝑬𝒊𝒏𝒗 as opposed \nto electric fields (𝑬𝒎 ,𝑬𝒔) arising due to spin dynamic . The total electric field  in the system \nis now  𝑬𝒕=𝑬𝒊𝒏𝒗 +𝑬𝒎+𝑬𝒔 , where 𝑬𝒎 ,𝑬𝒔 arise due to 𝒅 𝑴\n𝒅𝒕,𝒅 𝑺\n𝒅𝒕, respectively. On the \nmomentum part of the Hamiltonian  2 𝜆′𝒔.(𝒌×𝑬𝒕), we only need to  consider  that \n𝑬𝒕=𝑬𝒊𝒏𝒗 as one can, for simplicity, consider 𝑬𝒎 and  𝑬𝒔 to simply vanish on average. Thus \nin this renewed treatment, the momentum part is : \n2𝜆′\nℏ𝒔.(𝒑×𝑬𝒊𝒏𝒗)=𝜂𝑅𝝈.(𝒌×𝒆𝒊𝒏𝒗) \n(2) \nwhere 𝜂𝑅=𝜆′ℏ𝐸𝑖𝑛𝑣 is the Rashba constant that has been vastly measured  in many material \nsystems  with  experimental values ranging from 0.1 to 2 𝑒𝑉𝐴̇. On the curvature part, one \nconsiders   𝑬𝒕=𝑬𝒎+𝑬𝒔 without the 𝑬𝒊𝒏𝒗 as 𝑬𝒊𝒏𝒗  is spatially uniform and thus would have \nzero curvature.  In summary, the theory of this paper has it that the time -dynamic of the \nspin in a Rashba system produces a curvature part o f  𝑖𝜆′(𝒔+𝒎).(∇×𝑬𝒕). Without the \nRashba effect, this energy term  would just  take on the vacuum constant of (2𝜆\nℏ) instead of \nthe magnified (2𝜆′\nℏ).  The key physics is that in a Rashba FM/HA system , curvature \n𝑖𝜆′(𝒔+𝒎).(∇×𝑬𝒕) is satisfied by the first-order precession due to 𝒅𝑴\n𝒅𝒕,𝒅𝑺\n𝒅𝒕  which provide \nthe electric field curvature  in the form of −𝜇0(1+𝜒𝑚)𝑑 𝑴\n𝑑𝑡=∇×𝑬𝒎 , and  −𝜇0(1+\n𝜒𝑠)𝑑𝑺\n𝑑𝑡=∇×𝑬𝒔 , where we remind reader again that 𝑴,𝑺 have the unit of magnetic \nmoment.  This results in  spin becoming couple d to its own time dynamic, producing a spin -Magnified Damping under Rashba Spin Orbit Coupling  \nNovember 1, 2015  \n \n \n5 \n orbit second -order damping -like spin torque.  The electric field effect is illustrated in Fig. 1 \nbelow:  \n         \n \n \n \n \n \n \n \n \nFig.1 . Magnetic precession under the effect of electric fields due to inv ersion asymmetry, self -dynamic of  𝑑𝑴\n𝑑𝑡 \nand the spin dynamic of  𝑑𝑺\n𝑑𝑡 . Projecting 𝑑𝑀 to the heterostructure surface, one could visualize the emergence \nof an induced electric field in the form of 𝛻𝑋𝐸 in such orientation as to satisfy the law of electromagnetism.  \n \n \n    One notes that the LLG equation is normally derived by letting 𝑺 satisfy the physical \nrequirements of spin transport . One example  of these requirements is assumed and \ndiscussed in REF 1 , with definitions  contained  therein :   \n𝑺(𝒓,𝑡)=𝑆0𝒏+𝜹𝑺 \n𝑱(𝒓,𝑡)=−𝜇𝐵𝑃\n𝑒 𝑱𝒆⊗𝒏−𝐷0∇𝜹𝑺 \n(3) \nwhere 𝒏  is the unit vector of 𝑴, and 𝐷0 is the spin diffusion constant.  Thus 𝑺=𝑺𝟎+𝜹𝑺 \nwould be the total spin density that contains , respectively,  the equilibrium, the non-\nequilibrium adiabatic,  non-adiabatic , and Rashba field -like terms , i.e. 𝜹𝑺=𝜹𝑺𝒂+𝜹𝑺𝒏𝒂+\n𝜹𝑺𝑹.  One notes that 𝑺𝟎 is the equilibrium part of 𝒔 that is aligned to 𝒎, meaning 𝒔𝟎 could  \nexist  in the absence of external field and current in the system. The conditions to satisfy are \nrepresented explicitly by the equations  of:   𝑑𝑀 \n𝐸 𝑓𝑖𝑒𝑙𝑑  𝑑𝑢𝑒 𝑡𝑜 \n𝑖𝑛𝑣𝑒𝑟𝑠𝑖𝑜𝑛  𝑎𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑦  \n𝛻𝑋𝐸 \n𝑑𝑀 Magnified Damping under Rashba Spin Orbit Coupling  \nNovember 1, 2015  \n \n \n6 \n 𝜕𝜹𝑺 \n𝜕𝑡=0, 𝐷0∇2𝜹𝑺=0,−𝜇𝐵𝑃\n𝑒 𝛁.𝑱𝒆𝑴\n𝑀𝑠=0, 𝑠0𝑴(𝒓,𝑡)\n𝑡𝑓𝑀𝑠=0 \n (4) \nIn the steady state treatment where  𝜕  𝜹𝑺\n𝜕𝑡=0, one recover s the adiabatic component of \n𝜹𝑺𝒂=𝒏×𝒋𝒆.𝛁𝒏 , and the non -adiabatic component of 𝜹𝑺𝒏𝒂=𝒋𝒆.𝛁𝒏. We also take the \nopportunity  here  to reconcile this with  the gauge physics of spin torque, in which case , the \nspin potential 𝐴𝜇𝑠𝑚=𝑒 [𝛼 𝑈𝐸𝑖𝜎𝑗𝜀𝑖𝑗𝜇𝑈†+𝑖ℏ\n𝑒𝑈𝜕𝜇𝑈†] would  correspond , respectively, to  \n𝜹𝑺𝑹+ 𝜹𝑺𝒂. In fact, t he emergent spin p otential [22, 23] can be considered to encapsulate  \nthe physics of electron interaction with the local magnetization under the effect of SOC [4, 5 , \n24-26]. Here  we caution that 𝜹𝒔𝑹 is restricted to the field -like spin -orbit effect  only .  \n    However, in this paper ,  𝑺 is defined to satisfy the transport equations in Eq.(4) except for \n𝜕𝜹𝑺 \n𝜕𝑡=0. Keeping the dynamic property of 𝑺 here allows  a self -consistent equation set \n𝑑𝑺\n𝑑𝑡,𝑑𝑴\n𝑑𝑡 to be introduced . The energy as experienced by the 𝑺,𝑴 electron are, respectively,  \n𝐻𝑓𝑠=𝑺.𝛿𝐻𝑓\n𝛿𝑺 ,       𝐻𝑓𝑚=𝑴.𝛿𝐻𝑓\n𝛿𝑴 \n(5) \nwith caution that  𝐻𝑓𝑠≠𝐻𝑓𝑚 . Upon rearrangement, the 𝒔,𝒎 centric energ ies are, \nrespectively,   \n𝐻𝑓𝑠=(𝑝2\n2𝑚+𝑉𝑖𝑚𝑝𝑠+𝐽𝑠𝑑𝑺.𝑴+𝑺.𝑩𝑹−𝒊𝜆′𝒔.(∇×𝑬𝒕)) \n𝐻𝑓𝑚=(𝐽𝑠𝑑𝑴.𝑺+𝜇0𝑴.𝑯𝒂−𝒊𝜆′𝒎.(∇×𝑬𝒕) ) \n(6) \nwhere 2𝜆′\nℏ𝒔.(𝒑×𝑬𝒕)=𝑺.𝑩𝑹, while  2𝜆′\nℏ𝒎.(𝒑×𝑬𝒕)  vanishes . We particularly note that \nthere have been recent discussions on the field -like [4,6,11 ] spin orbit torque as well as the \ndamping [7] version. With 𝑑𝒔\n𝑑𝑡=−𝟏\n𝒊ℏ[𝒔,𝐻𝑓𝑠] ,𝑑𝒎\n𝑑𝑡=−𝟏\n𝒊ℏ[𝒎,𝐻𝑓𝑚], one would now  have four \ndissipative torque t erms experienced by electron 𝒔,𝒎 as shown below : Magnified Damping under Rashba Spin Orbit Coupling  \nNovember 1, 2015  \n \n \n7 \n ( 𝝉𝑺𝑺 𝝉𝑺𝑴\n𝝉𝑴𝑺 𝝉𝑴𝑴)=𝑖𝜆′𝜇0(𝒔×(1+𝜒𝑠−1)𝑑𝑺\n𝑑𝑡𝒔×(1+𝜒𝑚−1)𝑑𝑴\n𝑑𝑡\n𝒎×(1+𝜒𝑠−1)𝑑𝑺\n𝑑𝑡𝒎×(1+𝜒𝑚−1)𝑑𝑴\n𝑑𝑡) \n(7) \nTo be consistent with conventional necessity to preserve magnetization norm in the physics \nof the LLG equation, we will drop the  off-diagonal terms which are  norm -breaking (non -\nconservation) . This is in order to keep the LLG equation in its conventional norm -conserving  \nform, simplifying physics and calculation therefrom. Nonetheless, the non -conserving parts \nrepresent new dynamic physics that can be analysed in the future with techniques other \nthan the familiar LLG  equations.  The self -consistent pair of spin torque equations in their \nopen forms are: \n𝜕𝑺\n𝜕𝑡=−(𝑺× 𝑩𝑹+𝑺\n𝑡𝑓)−1\n𝑒𝛻𝑎(𝑗𝑎𝒔 𝑺)−(𝑺×𝑴\n𝑚𝑡𝑒𝑥)−𝝉𝑺𝑺 \n𝜕𝑴\n𝜕𝑡=−𝛾𝑴×𝜇0𝑯𝒂−𝑴×𝑺\n𝑚 𝑡𝑒𝑥−𝝉𝑴𝑴  \n(8) \nwhere 𝐽𝑠𝑑=1\n𝑚𝑡𝑒𝑥 has been applied, 𝛾 is the gyromagnetic ratio, 𝜒𝑚 is the susceptibility.  For \nthe stud y of Rashba -magnified damping i n this paper, we only need to keep  the most \nrelevant term which is 𝝉𝑴𝑴=𝑖 𝜂𝑅\nℏ𝐸𝑖𝑛𝑣𝜇0(1+𝜒𝑚−1) 𝒎×𝑑 𝑴\n𝑑𝑡. In the phenomenological physics \nof Gilbert,  the first-order precession leads inevitably to the second -order dissipative terms \nvia  𝒔.𝒅𝑺\n𝒅𝒕 ,𝒎.𝒅𝑴\n𝒅𝒕. But in this paper, the general SOC physics had been expanded  as shown in \nearlier sections,  so that the dissipative terms  are to naturally arise fr om such expansion. The \nadvantage of the non -phenomenological approach is  that, as said earlier, the Rashba \nconstant will be co -opted into the second -order damping effect, resulting in the \nmagnification of the damping constant associated with all field -like spin torque.   \n \n Magnified Damping under Rashba Spin Orbit Coupling  \nNovember 1, 2015  \n \n \n8 \n 3. Conclusion  \n     The im portant result in this paper is that the damping constants have been magnified by \nthe Rashba effect. This would not be possible if the damping constant was incorporated \nmanually by standard means of Gilbert. As the Rashba constant is larger than the vacuum \nSOC constant  as can be deduced from Table 1 and shown below  \n𝛼𝑅=𝛼𝜆′\n𝜆 ,  \n(9) \nmagnetization dynamics in FM/HA hetero -structure with inversion asymmetry (interface, or \nbulk) might have to be modelled with the new equations. It is important to remind  that all \npreviously measured 𝜂𝑅 has had 𝐸𝑖𝑛𝑣 captured in the measured value.  But w hat is needed in \nour study is the coupling of 𝑺 to a dynamic electric field, and that requires the value of just \nthe coupling strength (𝜆′). As most measurement is carr ied out for 𝜂𝑅, the exact knowledge \nof 𝐸𝑖𝑛𝑣 corresponding to a specific 𝜂𝑅 will have a direct impact on the actual value of 𝜆′.  We \nwill, nonetheless, provide a quick, possibly exaggerated estimate. Noting that 𝜆=𝑒ℏ\n4𝑚2𝑐2  \nand 𝜆′=𝜂𝑅\nℏ𝐸𝑖𝑛𝑣, and taking one measured value  of 𝜂𝑅=1×10−10𝑒𝑉𝑚 , corresponding to a \n𝐸𝑖𝑛𝑣=1010𝑉/𝑚, the magnification of 𝛼 works out to 104 times in magnitude , which may \nseem unrealistically strong . The caveat lies  in the exact correspondence of 𝜂𝑅 to 𝐸𝑖𝑛𝑣, which \nremains to be determined experimentally.  For example, if an experimentally determined   \n𝜂𝑅 actual ly corresponds to a much larger 𝐸𝑖𝑛𝑣, that would mean that 𝜆′=𝜂𝑅\nℏ𝐸𝑖𝑛𝑣 which \nmagnifies the damping constant through  𝛼𝑅=𝛼𝜆′\n𝜆 might actually be much lower than \npres ent estimate.  Therefore, it is worth remembering, for simplicity sake  that 𝛼𝑅 actually \ndepends on the ratio of 𝜂𝑅\n𝐸𝑖𝑛𝑣 but not 𝜂𝑅.  It has  also been assumed that  𝑳 couples to  𝑬𝒔,𝑬𝒎 \nwith the same efficiency that it couples to 𝑬𝒊𝒏𝒗.  This is still uncertain as th e Rashba \nconstant with respect to 𝑬𝒔,𝑬𝒎 might actually be lower than  those  𝜂𝑅 values that have \nbeen experimentally measured mostly with respect to 𝑬𝒊𝒏𝒗. Last, we note that as damping \nconstant has been magnified here, and as increasingly high -precision, live monitoring of \nsimult aneous 𝒔,𝒎 evolution is no longer redundant in smaller devices, care has been taken Magnified Damping under Rashba Spin Orbit Coupling  \nNovember 1, 2015  \n \n \n9 \n to present the LLG equations  in the form of a self-consistent  pair of dynamic equations \ninvolving 𝑴 and 𝑺. This will be necessary for the accurate modeling of the simultaneous \ntrajectory of both 𝑴 and 𝑺.   \n \nTable 1. Summary of damping torque and damping con stant with and without Rashba effects.   \n Hamiltonian  Torque  Damping constant  \n1. 𝐻=(2𝜆\nℏ)𝒔.(𝒑×𝑬𝒕) \n𝜆=𝑒ℏ\n4𝑚2𝑐2  \n𝜕𝒎\n𝑑𝑡=𝑖𝜆𝜇0𝒎×(1+𝜒𝑚−1)𝜕𝑴\n𝑑𝑡  \n𝛼=𝑖𝜆\n2𝜇0𝑀𝑠(1+𝜒𝑚−1) \n2. 𝐻𝑅=(2𝜆′\nℏ)𝒔.(𝒑×𝑬𝒕) \n𝜆′=𝜂𝑅\nℏ𝐸𝑖𝑛𝑣  \n𝜕𝒎\n𝑑𝑡=𝑖𝜆′𝜇0𝒎×(1+𝜒𝑚−1)𝑑𝑴\n𝑑𝑡  \n𝛼𝑅=𝑖𝜆′\n2𝜇0𝑀𝑠(1+𝜒𝑚−1) \n    \n \n \n \n \nREFERENCES  \n \n \n[1] S. Zhang & Z. Li, “Roles of Non -equilibrium conduction electrons on the  magnetization dynamics \nof ferromagnets”, Phys. Rev. Letts 93, 127204 (2004).  \n[2] F.T. Vasko, “Spin splitting in the spectrum of two -dimensional electrons due to the surface \npotential”, Pis’ma Zh.  Eksp. Teor. Fiz.  30, 574 (1979) [ JETP Lett. , 30, 541].  \n[3] Y.A. Bychkov  & E.I. Rashba, “Properties of a 2D electron gas with lifted spectral degeneracy”, \nPis’ma Zh. Eksp. Teor. Fiz. , 39, 66 (1984) [ JETP Lett. , 39, 78].  \n[4] S. G. Tan, M. B. A. 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Chapter -5 (2012).  \n \n  "    },    {        "title": "2211.07744v2.Magnetization_Dynamics_in_Synthetic_Antiferromagnets_with_Perpendicular_Magnetic_Anisotropy.pdf",        "content": "1 \n Magnetization Dynamics in Synthetic Antiferromagnets with Perpendicular \nMagnetic Anisotropy  \n \nDingbin Huang1,*, Delin Zhang2, Yun Kim1, Jian-Ping Wang2, and Xiaojia Wang1,* \n1Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455, USA  \n2Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN \n55455, USA  \n \n \nABSTRACT:   \nUnderstanding the rich physics of magnetization dynamics in  perpendicular  synthetic \nantiferromagnets  (p-SAFs)  is crucial  for developing next-generation  spintronic devices.  In this \nwork, we systematically investigate  the magnetization dynamics in p-SAFs  combining time-\nresolved magneto -optical Kerr effect (TR -MOKE)  measurements with theoretical modeling . \nThese model analyses, based on a Landau -Lifshitz -Gilbert approach incorporating exchange \ncoupling , provide detail s about the  magnetization dynamic characteristics  including  the amplitude s, \ndirections, and phases of the precession  of p-SAFs under varying magnetic fields . These model -\npredicted characteristics are in excellent quantitative agreement with TR-MOKE measurements \non an asymmetric p -SAF. We further reveal the damping mechanisms of two precession modes  \nco-existing in the p -SAF and successfully identify individual contributions from different sources , \ninclud ing Gilbert damping of each ferromagnetic layer , spin  pumping, and inhomogeneous \nbroadening . Such a comprehensive understanding of magnetization dynam ics in p -SAFs, obtained \n                                                 \n*Author s to whom correspondence should be addressed : huan1746@umn.edu  and wang4940@umn.edu  2 \n by integrating high -fidelity TR -MOKE measurements and theoretical modeling, can guide  the \ndesign of p-SAF-based architectures for spintronic  applications . \n \nKEYWORDS:  Synthetic antiferromagnets; Perpendicular magnetic anisotropy; Magnetization \nDynamics; Time -resolved magneto -optical Kerr effect; Spintronics   3 \n 1 INTRODUCTION  \nSynthetic antiferromagnet ic (SAF) structures have attracted considerable  interest for \napplications in spin mem ory and logic devices  because of their unique magnetic  configuration s [1-\n3]. The SAF structures are composed of two ferromagnetic (FM) layers anti -parallelly coupled \nthrough a non -magnetic (NM) spacer, offer ing great  flexibilit ies for the  manipulat ion of magnetic \nconfigurations  through  external stimuli  (e.g., electric -field and spin-orbit torque , SOT) . This \npermit s the design of new architecture s for  spintronic applications , such as magnetic tunnel \njunct ion (MTJ), SOT  devices, domain wall devices, sky rmion devices, among others  [4-7]. The \nSAF structures  possess many advantages for such applications , including  fast switching speeds \n(potentially in the THz regimes),  low off set fields, small switching current s (and thus low energy \nconsumption) , high thermal stability,  excellent resilience  to perturbations from external magnetic  \nfields, and large turnabilit y of magnetic properties  [3,8-16].  \nA comprehensive study  of the magnetization dynami cs of SAF structures  can facilitate the \nunderstanding of the switching behavior of spintronic  devices , and ultimately guide the design of \nnovel device architectures . Different from a single FM free layer, magnetization dynamics of the \nSAF structures involv es two modes of precession , namely high -frequency (HF) and low -frequency  \n(LF) modes,  that result from the hybridization of magnetizations  precession  in the two FM layers . \nThe relative phase and precession amplitude in two FM layers can significantly affect the spin-\npumping enhancement of magnetic damping  [17], and thus play an important role in determining \nthe magnetization dynamic behaviors in SAFs. Heretofore, the exchange -coupling strength and \nmagnetic damping constant of SAFs have been studied by ferromagnetic resonance (F MR)  [18-\n21] and optical metrolog y [22-25]. Most FMR -based experimental studies were limited to SAFs \nwith in -plane magnetic anisotropy (IM A). For device applications, perpendicular magnetic 4 \n anisotropy (PMA) gives  better scalability  [3,26] . Therefore , the characteristics of magnetization \ndynamics of perpendicular SAF (p-SAF) structures  are of much valu e to investigat e. In addition, \nprior  studies mainly focus ed on the mutual  spin pumping between two FM layers  [22,27,28] . A \nmore thorough understanding of the contribution s from various sources, including inhomogeneous \nbroadening  [29], remains elusive . \nIn this paper, we report  a comprehensive study of the magnetization dynamics of p -SAFs  by \nintegrating high -fidelity experiments and theoretical modeling to detail the characteristic \nparameters. These parameters describe the amplitude, phase, and direction of magnetization \nprecess ion of both the HF and LF modes for the two exchange -coupled FM layers in a p -SAF.  We \nconduct all -optical time -resolved magneto -optical Kerr effect (TR -MOKE) measurements  [30-33] \non an asymmetric p -SAF structure with two different FM layers. The field-dependent  amplitude \nand phase of TR -MOKE signal s can be well captured by our theoretical  model, which in turn \nprovid es comprehensive physical insights into the magnetization dynamics of p -SAF structures.  \nMost importantly, we show that inhomogeneous broadening  plays a  critical  role in determining \nthe effective damping of both HF and LF modes, especially at low fields.  We demonstrate the \nquantification of contributions from inhomogeneous broadening  and mutual spin pumping  (i.e., \nthe exchange of angular momentum between two FM layers via pumped spin currents ) [21] to the \neffect ive damping, enabl ing accurate determination of the Gilbert damping for individual FM \nlayers. Results of this work  are beneficial  for designing  p-SAF-based architectures  in spintronic  \napplication s. Additionally, this work also serves as a successful example demonstrating that  TR-\nMOKE, as an all -optical met rology, is a powerful tool to capture the magnetization dynamics and \nreveal the rich physics of complex structures that involve multilayer coupling . \n 5 \n 2 METHODOLOTY  \n2.1 Sample preparation and characterization  \nOne SAF  structure  was deposited onto thermally oxidize d silicon  wafers with a 300 -nm SiO 2 \nlayer by magnetron sputtering at room  temperature (RT) in a six -target ultra -high vacuum (UHV) \nShamrock sputtering system. The  base pressure is below  5×10−8 Torr. The stacking structure of \nthe SAF is:  [Si/SiO 2]sub/[Ta(5)/Pd(3)] seed/[Co(0.4)/Pd(0.7)/Co(0.4)] FM1/[Ru(0.6)/Ta(0.3)] NM/ \nCoFeB(1) FM2/[MgO(2)/Ta(3)] capping . The numbers in parentheses denote the layer thicknesses in \nnanometers. After deposition, the sample was annealed at 250 ℃ for 20 minutes by a rapid -\nthermal-annealing process.  The two FM layers are CoFeB and Co/Pd/Co layers, separated by a \nRu/Ta spacer, forming an asymmetric p -SAF structure ( i.e., two FM layers having different \nmagnetic properties). The M-Hext loops were characterized by a physical propert y measurement \nsystem (PPMS) with a vibrating -sample magnetometer (VSM) module. The resulting M-Hext loops \nare displayed in Fig. 1(a). Under low out -of-plane fields ( Hext < 500 Oe), the total magnetic \nmoments in two FM layers of the SAF stack perfectly cancel out each other: M1d1 = M2d2 with Mi \nand di being the magnetization and thickness of each FM layer ( i = 1 for the top CoFeB layer and \ni = 2 for the bottom Co/Pd/Co laye r). The spin-flipping field ( Hf  ≈ 500 Oe ) in the out -of-plane \nloop indicates the bilinear interlayer -exchange -coupling (IEC) J1 between the two FM layers : J1 = \n−HfMs,1d1 ≈ −0.062 erg cm-2 [34]. The values of Ms,1, Ms,2, d1, and d2 can be found in Table SI of \nthe Supplemental Material (SM)  [35]. \n \n2.2 Theoretical foundation  of magnetization dynamics for a p -SAF structure  \nThe magnetic free energy per unit area for a p -SAF structure with uniaxial PMA can be \nexpressed as [36]: 6 \n 𝐹=−𝐽1(𝐦1⋅𝐦2)−𝐽2(𝐦1⋅𝐦2)2\n+∑2\n𝑖=1𝑑𝑖𝑀s,𝑖[−1\n2𝐻k,eff,𝑖(𝐧⋅𝐦𝑖)2−𝐦𝑖⋅𝐇ext] (1) \nwhere J1 and J2 are the strength of the bilinear and biquadratic IEC. mi = Mi / Ms,i are the normalized \nmagnetization vectors for individual FM layers ( i = 1, 2). di, Ms,i, and Hk,eff, i denote, respectively, \nthe thickness, saturation magnetization, and the effective anisotropy field of the i-th layer. n is a \nunit vector indicating the sur face normal direction of the film. For the convenience of derivation \nand discussion, the direction of mi is represented in the spherical coordinates by the polar angle θi \nand the azimuthal angle φi, as shown in Fig. 1 (b). \nThe equilibrium direction of magne tization in each layer (𝜃0,𝑖,𝜑0,𝑖) under a given Hext is \nobtained by minimizing F in the (𝜃1,𝜑1,𝜃2,𝜑2) space. The magnetization precession is governed \nby the Landau -Lifshitz -Gilbert  (LLG)  equation considering the mutual spin pumping between two \nFM layers [27,37 -40]: \n𝑑𝐌𝑖\n𝑑𝑡=−𝛾𝑖𝐌𝑖×𝐇eff,𝑖+(𝛼0,𝑖+𝛼sp,𝑖𝑖)\n𝑀s,𝑖𝐌𝒊×𝑑𝐌𝒊\n𝑑𝑡−𝛼sp,𝑖𝑗\n𝑀s,𝑖𝐌𝒊×(𝐦𝐣×𝑑𝐦𝒋\n𝑑𝑡)×𝐌𝒊 (2) \nOn the right -hand side of Eq. (2), the first term describes the precession with the effective field \nHeff,i in each layer, given by the partial derivative of the total free energy in the M space via 𝐇eff,𝑖=\n−∇𝐌𝑖𝐹. The second term represents the relaxation induced by Gilbert damping ( α) of the i-th layer, \nwhich includes the intrinsic ( 𝛼0,𝑖) and spin -pumping -enhanced ( 𝛼sp,𝑖𝑖) damping. For TR -MOKE \nmeasurements, 𝛼0,𝑖 and 𝛼sp,𝑖𝑖 are indistinguishable. Hence, we def ine 𝛼𝑖=𝛼0,𝑖+𝛼sp,𝑖𝑖 to include \nboth terms. The last term in Eq. (2) considers the influence of pumped spin currents from the layer \nj on the magn etization dynamics of the layer  i.  7 \n The time evolution of Mi can be obtained by solving the linearized Eq. (2). Details are provided \nin Note 1  of the SM  [35]. The solutions to Eq. (2) in spherical coordinates are:  \n[𝜃1(𝑡)\n𝜑1(𝑡)\n𝜃2(𝑡)\n𝜑2(𝑡)]=[𝜃0,1\n𝜑0,1\n𝜃0,2\n𝜑0,2]+[Δ𝜃1(𝑡)\nΔ𝜑1(𝑡)\nΔ𝜃2(𝑡)\nΔ𝜑2(𝑡)]=[𝜃0,1\n𝜑0,1\n𝜃0,2\n𝜑0,2]+\n[    𝐶𝜃,1HF\n𝐶𝜑,1HF\n𝐶𝜃,2HF\n𝐶𝜑,2HF]    \nexp(𝑖𝜔HF𝑡)+\n[    𝐶𝜃,1LF\n𝐶𝜑,1LF\n𝐶𝜃,2LF\n𝐶𝜑,2LF]    \nexp(𝑖𝜔LF𝑡) (3) \nwith Δ𝜃𝑖 and Δ𝜑𝑖 representing the deviation angles of magnetization from its equilibrium direction \nalong the polar and azimuthal directions . The last two terms are the linear combination of two \neigen -solutions, denoted by  superscripts HF (high -frequency mode) and LF (low -frequency mode). \nω is the complex angular frequencies of two modes, with the real and imaginary parts representing \nthe precession angular frequency ( 𝑓/2𝜋) and relaxation rate (1/ τ), respectively. For each mode, \nthe complex prefactor vector [𝐶𝜃,1,𝐶𝜑,1,𝐶𝜃,2,𝐶𝜑,2]𝑇 contains detailed information about the \nmagnetization dynamics. As illustrated in Fig. 1 (c), the moduli, |𝐶𝜃,𝑖| and |𝐶𝜑,𝑖| correspond to the \nhalf cone angles of t he precession in layer i along the polar and azimuthal directions for a given \nmode immediately after laser heating, as shown by Δ𝜃 and Δ𝜑 in Figs. 1 (b-c). The phase \ndifference between Δ𝜃𝑖 and Δ𝜑𝑖, defined as Arg(Δ𝜃𝑖/Δ𝜑𝑖)=Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖) with Arg  \nrepresenting the argument of complex numbers, determines the direction of precession. If Δ𝜃𝑖 \nadvances Δ𝜑𝑖 by 90°, meaning Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖)=90°, the precession is counter -clockwise (CCW) \nin the θ-φ space  (from a view against Mi).Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖)=−90°, on the contrary, suggests \nclockwise (CW) precession [ Fig. 1 (d)]. Further, the argument of 𝐶𝜃,2/𝐶𝜃,1 provides the relative \nphase in two FM layers. Arg(𝐶𝜃,2/𝐶𝜃,1)=0° corresponds to the precession motions in two FM \nlayers that are in -phase (IP) in terms of θ for a given mode. While the out -of-phase (OOP) \nprecession in terms of θ is represented by Arg(𝐶𝜃,2/𝐶𝜃,1)=180°  [Fig. 1 (e)]. Given the precession 8 \n direction in each  layer and the phase difference between the two FM layers in terms of θ, the phase \ndifference in terms of φ can be automatically determined.   \n \n \nFIG. 1 (a) Magnetic hysteresis ( M-Hext) loops of the p -SAF stack. The magnetization is n ormalized \nto the saturation magnetization ( M/Ms). (b) Schematic illustration of the half cone angles (Δ θ and \nΔφ) and precession direction of magnetization. The precession direction is defined from a view \nagainst the equilibrium direction ( 0, φ0) of M. The representative precession direction in the \nschematic is counterclockwise (CCW). (c) The relation between precession half cone angles and \nthe prefactors. (d) The relation between precession direction and the prefactors. (e) The relative \nphase between two FM layers for different prefactor values.  \n \nAs for the effective damping 𝛼eff=1/2𝜋𝑓𝜏, in addition to the intrinsic damping ( α0,i) and the \nspin-pumping contribution ( αsp,ii and αsp,ji) considered in Eq. (2), inhomogeneities can also bring \nsubstantial damping enhancement [32,33,41,42] . Here, we m odel the total relaxation rate as \nfollows:  \n9 \n 1\n𝜏Φ=−Im(𝜔Φ)+1\n𝜏inhomoΦ (4) \nThe superscript Φ = HF or LF, representing either the high -frequency or low -frequency precession \nmodes. 𝜔Φ includes both the intrinsic and spin -pumping contributions. The inhomogeneous \nbroadening is calculated as:  \n1\n𝜏inhomoΦ=∑1\n𝜋|𝜕𝑓Φ\n𝜕𝐻k,eff,𝑖|\n𝑖Δ𝐻k,eff,𝑖+∑1\n𝜋|𝜕𝑓Φ\n𝜕𝐽𝑖|\n𝑖Δ𝐽𝑖 (5) \nwhere the first summation represents the contrib ution from the spatial variation of the effective \nanisotropy field of individual FM layers (Δ Hk,eff, i). The second summation denotes the contribution \nfrom the spatial fluctuations of the bilinear and biquadratic IEC (Δ J1 and Δ J2). According to \nSlonczewski’s “thickness fluctuations” theory, Δ J1 generates J2 [43,44] . Therefore,  the fact that  J2 \n= 0 for our sample suggests that  ΔJ1 is sufficiently small, allowing us to neglect  the inhomogeneous \nbroadening from th e fluctuations of both the bilinear and biquadratic IEC  in the following analyses . \n \n2.3 Detection of magnetization dynamics  \nThe magnetization dynamics of the p -SAF sample is detected by TR -MOKE, which is \nultrafast -laser -based metrology utilizing a pump -probe configuration. In TR -MOKE, pump laser \npulses interact with the sample, initiating magnetization dynamics in magnetic layers via inducing \nultrafast thermal demagnetization.  The laser -induced heating brings a rapid decrease to  the \nmagnetic anisotropy  fields  and IEC [45,46] , which changes 𝜃0,𝑖, 𝜑0,𝑖 and initiates the precession.  \nThe magnetizati on dynamics due to pump excitation is detected by a probe beam through the \nmagneto -optical Kerr effect. In our setup, the incident probe beam is normal to the sample surface \n(polar MOKE); therefore, the Kerr rotation angle ( 𝜃K) of the reflected probe beam  is proportional \nto the z component of the magnetization [47]. More details about the experimental setup can be 10 \n found in Refs. [30,32] . For p -SAF, TR -MOKE signals contain two oscillating frequencies that \ncorrespond to the HF and LF  modes  (𝑓HF>𝑓LF). The signals  are proportional to the change in \n𝜃K and can be analyzed as follows:  \nΔ𝜃K(𝑡)=𝐴+𝐵𝑒−𝑡/𝜏T+𝐶HFcos(2𝜋𝑓HF𝑡+𝛽HF)𝑒−𝑡/𝜏HF+𝐶LFcos(2𝜋𝑓LF𝑡+𝛽LF)𝑒−𝑡/𝜏LF (6) \nwhere the exponential term  𝐵𝑒−𝑡/𝜏T is related to the thermal background  with 𝜏T being the time \nscale of heat dissipation . The rest two terms on the right -hand side are the precession terms  with \nC, f, β, and τ denoting , respectively, the amplitude, frequency, phase, and relaxation time of the \nHF and LF  modes.  \nAfter excluding the thermal background from TR -MOKE signals, the precession is modeled \nwith the initial conditions of step -function de creases in 𝐻k,eff,𝑖 and 𝐽𝑖, following the ultrafast laser \nexcitation [48]. This is a reasonable  approximation since the precession period (~15 -100 ps for \nHext > 5 kOe) is much longer than the time scales of the laser excitation (~1.5 ps) and subsequent \nrelaxations among electrons, magnons, and lattice (~ 1 -2 ps) [49], but much shorter than the time \nscale of heat dissipation -governed recovery (~400 ps). With these  initial conditions , the prefactors \nin Eq. ( 3) can be determined  (see m ore details in Note 1  of the SM  [35]).  \nFor our SAF structure, 𝜃K detected by the probe beam contain s weighted contributions from \nboth the top and  bottom FM layers:  \n𝜃K(𝑡)\n𝜃K,s=𝑤cos𝜃1(𝑡)+(1−𝑤)cos𝜃2(𝑡) (7) \nwhere 𝜃K,s represents the Kerr rotation angle when the SAF s tack is saturated along the positive \nout-of-plane ( z) direction. w is the weighting factor, considering the different contributions to the \ntotal MOKE signals from two FM layers. w can be obtained from static MOKE measurements [50], \nwhich gives 𝑤= 0.457 (see more details in Note 2  of the SM  [35]). 11 \n  \n3 RESULTS AND DISCUSSION  \n3.1 Field -dependent p recession frequencies and equilibrium magnetization directions  \nTR-MOKE signals measured at varying  Hext are depicted in Fig. 2 (a). The external field is \ntilted 15 ° away from in-plane [θH = 75°, as defined by Fig. 2 (c)] to achieve larger amplitdues of  \nTR-MOKE signals [51]. The signals can be fitted to Eq.  (6) to extract the LF and HF precession \nmodes. The field -dependen t precession frequenc ies of both modes  are summarized in Fig. 2 (b). \nFor simplicity, when analyzing precession frequencies, magnetic damping  and mutual spin \npumping  are neglected due to its insignificant impacts on  precession frequencies.  By comparing \nthe experimental data and the prediction of ωHF/2π and ωLF/2π based on E q. (3), the effective \nanisotropy fields and the IEC strength  are fitted as Hk,eff, 1 = 1.23 ± 0.28  kOe, Hk,eff, 2 = 6.18 ± 0.13  \nkOe, J1 = −0.050 ± 0.020  erg cm−2, and J2 = 0. All parameters and their determination methods are \nsummarized in Table SI of the SM  [35]. The fitted J1 is close to that obtained from the M-Hext \nloops (~−0.062 erg cm-2). The inset of Fig 2 (b) shows the zoom ed-in view of field -dependent \nprecession frequencies around Hext = 8 kOe, where a n anti -crossing feature is observed: a  narrow \ngap (~2  GHz) open s in the frequency dispersion curves of the HF and LF modes owing to the weak \nIEC between two FM layers. Without a ny IEC, the precession frequencies of two FM layers would \ncross at Hext = 8 kOe, as indicated by the green dashed line and blue dashed line in the figure. We \nrefer to t hese two sets of crossing frequencies as the single -layer natural frequencies of two FM \nlayers (FM 1 and FM 2) in the following discussions .  12 \n  \nFIG. 2 (a) TR -MOKE signals under varying Hext when θH = 75° [as defined in panel (c)]. Circles \nare the experimental data and black lines are the fitting curves based on Eq. (6). (b) The precession \nfrequencies of the HF and LF modes as functions of Hext. Circles are experimental data and solid \nlines are fitting curves. The  inset highlights the zoomed -in view of the field -dependent frequencies \naround 8 kOe, where the green dashed line and blue dashed line are the single -layer (SL) \nprecession frequencies of FM 1 and FM 2 without interlayer exchange coupling. (c) Schematic \nillustration of the definition of the equilibrium polar angles ( θ0,1 and θ0,2), and the direction of the \nexternal magnetic field ( θH). The illustration is equivalent to Fig. 1(b) due to symmetry. (d) θ0,1 \nand θ0,2 as functions of Hext. The dash -dotted line plots the difference between the two equilibrium \npolar angles.  \n \n13 \n Based on the fitted stack properties ( Hk,eff,1, Hk,eff,2, J1, and J2), the equilibrium magnetization \ndirections in the two layers can be calculated. For SAFs with weak IEC compared with uniaxial \nPMA, the azimuthal angles of the magnetization in two FM layers are always the same as that of \nthe external field at equilibrium status. Therefore, two polar angles will be sufficient to describe \nthe equilibrium magnetization con figuration. Figure 2(c) illustrates the definition of the \nequilibrium polar angles of two FM layers ( θ0,1, θ0,2) and the external field ( θH). The values of θ0,1, \nθ0,2, and the difference between these two polar angles as functions of Hext are shown in Fig. 2(d). \nWhen Hext is low (<  1.6 kOe), magnetic anisotropy and antiferromagnetic coupling are dominant \nand |θ0,1 − θ0,2| is larger than 90 °. As Hext increases, both θ0,1 and θ0,2 approach θH. When Hext is \nhigh (> 15 kOe), the Zeeman energy becomes dominant and both M1 and M2 are almost aligned \nwith Hext. \n \n3.2 Cone angle, direction, and phase of magnetization  precession revealed by modeling  \nBesides the equilibrium configuration, using sample properties extracted from Fig. 2 (b) as \ninput parameters, the LLG -based modeling (described in section 2.2) also provide s information o n \nthe cone angle, direction, and phase of magnetization precession for each mode ( Fig. 1 ). The \ndiscussion in this section  is limited to the case  without damping an d mutual spin pumping . They \nwill be  considered  in Note 4 of the SM  [35], sections 3.3, and 3.4. The calculation results are \nshown in  Fig. 3 , which are categorized into three regions. At high external fields ( Hext > 1.6 kOe, \nregions 2 and 3), both FM layers precess CCW [ Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖)=90°], and the polar angles of \nmagnetization in two layers are  in-phase [Arg(𝐶𝜃,2/𝐶𝜃,1)= 0°] for the HF mode and  out-of-phase \n[Arg(𝐶𝜃,2/𝐶𝜃,1) = 180° ] for the LF mode. This is the reason for the HF mode (LF mode) also \nbeing called the acoustic mode ( optical mode) in the literature  [23]. The criterion to differentiate 14 \n region 2 from region 3 is the FM layer that dominat es a given precessional mode  (i.e., the layer \nwith larger precession cone angles) . In region 2  (1.6 kOe < Hext < 8 kOe) , the HF mode is \ndominated by FM 2 because FM 2 has larger cone angles than FM 1. This is reasonable since the \nhigher precession frequency is closer to the natural frequency of FM 2 [see Fig. 2(b)] in region 2. \nSimilarly, in region 3, the HF mode is dominated by FM 1 with larger precession cone angles.  \nWhen Hext is low (region 1), the angle between two magnetizations is larger than 90° [ Fig. 2 (d)] \nowing to the  more  dominan t AF-exchange -coupling  energy as compared with the Zeeman energy . \nIn this region, magnetization dynamics exhibits some unique features.  Firstly, CW [ Arg(𝐶𝜃,𝑖/\n𝐶𝜑,𝑖)=−90°] precession emerges: for each mode, the dominant layer precesses CCW (FM 2 for \nthe HF mode and FM 1 for the  LF mode) and the subservient layer precesses CW (FM 1 for the HF \nmode and FM 2 for the LF mode). This is because the effective field for the subservient layer [ e.g., \nHeff,1 for the HF mode, see Eq.  (2)] precesses CW owing to the CCW precession of the dominant \nlayer when |𝜃0,1−𝜃0,2|>90° [Fig. 2(d)] . In other words, a low Hext that makes |𝜃0,1−𝜃0,2|>\n90° is a necessary condition for the CW precession.  However, it is not a sufficient condition. In \ngeneral, certain degrees  of symmetry breaking ( Hk,eff,1 ≠ Hk,eff,2 or the field is tilted away from the \ndirection normal to the easy axis ) are also needed to generate CW precession.  For example, for \nsymmetric a ntiferromagnets ( Hk,eff,1 = Hk,eff,2) under fields perpendicular to the easy axis, CW \nprecession does not appear even at low fields (Fig. 2(a) in Ref. [52]). See Note 5 of the SM  [35] \nfor more details.  Secondly, as shown in  Fig. 3 , the precession motions in two FM layers are always \nin-phase for both HF and LF modes; thus, there is no longer a clear differentiation between \n“acoustic mode” and “optical mo de”. Instead, the two modes can be differentiated as “right -handed” \nand “left -handed” based on the chirality [53]. Here, we define the chirality with respect to a \nreference direction taken as the projection of Hext or M2 (magnetization direction of the layer with 15 \n a higher Hk,eff) on the easy axis [ -z direction in Fig. 3 (c)]. Lastly, the shape of the precession cone \nalso varies in different regions. Δ θi and Δφi are almost the same for both modes in region 3, \nindic ating the precession trajectories are nearly circular. While in regions 1 and 2, Δ θi and Δφi are \nnot always equal, suggesting the precession trajectories may have high ellipticities.  \n \n \nFIG. 3 The calculated  half cone angle, direction, and phase of magnetization precession for (a) the \nHF mode and (b) the LF mode. In the top row, four curves represent the polar and azimuthal  half \ncone angles of precession in two FM layers. All  half cone angles are normalized with r espect to \nΔθ1. The middle row shows the value of Arg(𝐶𝜃,𝑖/𝐶𝜑,𝑖) under different Hext. A value of 90°  (−90°) \nrepresents CCW (CW) precession. The bottom row is the phase difference of the polar angles in \ntwo layers. A value of 0° (180°) corresponds to the polar angles of the magnetization in two layers \nare IP (OOP ) during precession.  Dashed lines correspon d to the reference case where damping is \nzero in both layers. (c) Schematic illustrations of the cone angle, direction, and phase of \n16 \n magnetization precession for the HF and LF modes in different regions, and their corresponding \ncharacteristics regarding ch irality and phase difference.  \n \n3.3 Amplitude and phase of TR -MOKE signals  \nActual magnetization dynamics is resolvable  as a linear combination of the two eigenmodes  \n(the HF and  the LF modes ). By taking into account the initial conditions (i.e., laser excitation , see \nNote 1  of the SM  [35]), we can determine the amplitude and phase of the two modes in TR -MOKE \nsignals . Figure 4 (a) summarizes the amplitudes of both HF and LF modes [CHF and CLF in Eq.  (3)] \nunder different  Hext. Noted that  the y-axis represents Kerr angle ( θK) instead of the cone angle of \nprecession.  The LF mode  has a local minimum near 8 kOe, where the two FM layers have similar \nprecession cone angles but opposite phase s for the LF mode [ Fig. 3 (b)]. The amplitude s of both \nmodes decrease with Hext in the high -field region. This is similar to the single -layer case, where \nthe amplitudes of TR -MOKE signals decrease with Hext because the decrease  in Hk,eff induced by \nlaser heating is not able to significantly  alternate  the equilibrium magnetization direction when the \nZeeman energy dominates  [51]. The LF mode also has an amplitude peak at low fields ( Hext < 3 \nkOe), where the dominant layer of FM 1 changes its equilibrium direction dramatically with Hext \n(from ~75° to 170°) as shown in Fig. 2(d).  \nTo directly compare the amplitudes of TR -MOKE signals and the LLG -based calculations , the \nweighting factor w and the initial conditions are needed. The initial conditions are determined by \n𝐻k,eff,1′,𝐻k,eff,2′, and 𝐽1′, representing the instantan eous effective anisotropy fields and IEC strength \nupon laser heating. These instantaneous properties are different from their corresponding room -\ntemperature values ( Hk,eff,1, Hk,eff,2, and J1). The accurate determination of𝐻k,eff,1′,𝐻k,eff,2′, and 𝐽1′ \ndemands the modeling of the laser heating process as well as the temperature dependence of stack \nproperties, which are challenging. Here, we treat these three variables as adjustable parameters and 17 \n determine their values by fitting the field -dependent amp litudes of TR -MOKE signals , which \nyields 𝐻k,eff,1′𝐻k,eff,1⁄=0.90±0.01, 𝐻k,eff,2′𝐻k,eff,2⁄=0.95±0.01, and 𝐽1′𝐽1⁄=0.83±0.01. \nIt is apparent that the field dependence of TR -MOKE signal amplitude is in excellent agreement \nwith the theoretical modeling , as s hown in Fig. 4 (a). \nFigure 4 (b) shows the calculated  half polar cone angles for each mode in each FM layer. In \nTR-MOKE signals, the optical mode (the LF mode in regions 2 and 3) tends to be partially \ncanceled out  because the two layers precess out -of-phase.  Therefore, compared with Fig. 4 (a), the \ninformation in Fig. 4 (b) better reflects the actual intensity of both modes in FM 1 and FM 2. In Fig. \n4(b), the precession cone angles of both modes in FM 1 (Δ𝜃1HF,Δ𝜃1LF) have local maxima at the \nanti-crossing field (Hext ≈ 8 kOe). On the contrary, Δ𝜃2LF and Δ𝜃2HF of FM 2 have their maxima \neither above or below  the anti-crossing field. This is because FM 2 has larger precession amplitudes \n(cone angles) than FM 1 at the anti-crossing field if there is no IEC [the dotted  lines of FM 1 (SL) \nand FM 2 (SL) in Fig. 4 (b)]. With IEC, FM 2 with larger cone angles can drive the precession motion \nin FM 1 significantly near the anti-crossing field, where IEC is effective. Subsequently, the \nprecession amplitudes of FM 1 exhibit local maxima as its cone angle peaks at the anti-crossing  \nfield [solid lines in Fig. 4(b)]. Also, compared with the uncoupled case [FM 1 (SL) in Fig. 4(b)], \nFM 1 in the SAF structure has a much larger cone angle at the boundary between regions 1 and  2 \n(Hext ≈ 1.6 kOe).  This corresponds to the case where FM 1 fast switch ing is driven by Hext, as shown \nin Fig. 2( d). The energy valley of FM 1 created by IEC and uniaxial anisotropy is canceled out by \nHext. As a result, any perturbation in Hk,eff,1 or IEC can induce a large change in 𝜃1. \nBesides amplitude, the phase of TR -MOKE signals [ HF and LF in Eq. (6)] also provides \nimportant information about the magnetization dynamics in SAF  [Fig. 4 (c)]. In Fig. 4 (c), the phase \nof the HF mode stays constant around π. However, the LF mode goes through a π-phase shift at 18 \n the transition from region 2 to region 3. Th is phase shift can be explained by the change of the \ndominant layer from region 2 to region 3 for the LF mode [ Fig. 3(c)]. As illustrated in Fig. 4 (d), \nthe LF  mode (optical mode in regions 2 and 3) has opposite phases in FM 1 (~0°) and FM 2 (~180°). \nConsidering the two FM layers have comparable optical contributions to TR -MOKE signals ( w ≈ \n0.5), TR -MOKE signals will reflect the phase of the dominant layer for each mode. In region 3, \nFM 2 has larger p recession cone angles than FM 1 for the LF  mode ; therefore, LF TR-MOKE signals \nhave the same phase as FM 2 (~180°). However, in region 2, the dominant layer shifts from FM 2 to \nFM 1 for the LF mode. Hence, the phase of LF TR-MOKE signals also change s by ~180° t o be \nconsistent with the phase of FM 1 (~0°). As for the HF mode, since the two layers always have  \nalmost  the same phase ( ~180°), the change of the dominant layer does not cause a shift in the phase \nof TR -MOKE signals.   \nBy comparing Fig. 4 (d) and Fig. 3 (a-b), one can notice that the phase difference between two \nFM layers could deviate from 0° or 180° when damping  and mutual spin pumping  is considered \n[Fig. 4(d)]. The deviation of phase allows energy  to be transferred from one FM layer to the other \nduring precession via exchange coupling  [54]. In our sample system, FM 2 has a higher damping \nconstant ( 𝛼1= 0.020 and 𝛼2=0.060); therefore, the net transfer of energy is from FM 1 to FM 2. \nMore details can be found in Note 4 of the SM  [35], which shows the phase of TR -MOKE signals \nis affected by Gilbert damping in both layers and the mutual spin pumping . By fitting the phase \n[Fig. 4(c)] and the damping [ Fig. 5(a) ] of TR -MOKE signals simultaneously, we obtained 𝛼sp,12 \n= 0.010 ± 0.004, 𝛼sp,21=0.007−0.007+0.009, 1= 0.020 ± 0.002, and 2 = 0.060 ± 0.008. Nonreciprocal \nspin pumping damping ( 𝛼sp,12≠𝛼sp,21) has been reported in asymmetric FM 1/NM/FM 2 trilayers \nand attributed to the different spin -mixing conductance ( 𝑔𝑖↑↓) at the two FM/NM interfaces [27], \nfollowing 𝛼sp,𝑖𝑗=𝑔𝑖𝜇B𝑔𝑗↑↓/(8𝜋𝑀s,𝑖𝑑𝑖), with 𝑔𝑖 the 𝑔-factor of the i-th layer and 𝜇B the Bohr 19 \n magneton [55]. The above equation neglects the spin -flip scattering in NM and assumes that the \nspin accumulation in the NM spacer equally flows back to FM 1 and FM 2  [37]. However, the \nuncertainties of our 𝛼sp,𝑖𝑗 are too high to justify the nonreciprocity of 𝛼sp,𝑖𝑗 (see Note 3 of the SM  \n[35] for detailed uncertainty analyses). In fact, if the spin backflow to FM i is proportional  to 𝑔𝑖↑↓, \nthen 𝛼sp,𝑖𝑗=𝑔𝑖𝜇B𝑔𝑖↑↓𝑔𝑗↑↓/[4𝜋𝑀s,𝑖𝑑𝑖(𝑔𝑖↑↓+𝑔𝑗↑↓)] (Eq. 1.14 in Ref. [56]). In this case, the different \nspin-mixing conductance at two FM/NM interfaces ( 𝑔1↑↓≠𝑔2↑↓) will not lead to nonreciprocal \n𝛼sp,𝑖𝑗. Although differences in 𝑔𝑖 and magnetic moment per area ( 𝑀s,i𝑑𝑖) can potentially  lead to \nnonreciprocal 𝛼sp,𝑖𝑗, the values of 𝑔𝑖 and 𝑀s,i𝑑𝑖 for the two FM layers are expected to be similar \n(the net magnetization of SAF is zero without external fields). Therefore, nearly reciprocal 𝛼sp,𝑖𝑗 \nare plausible for our SAF stack. Assu ming 𝑔𝑖↑↓ values are similar at the two FM/NM interfaces \n(𝑔1↑↓≈𝑔2↑↓=𝑔↑↓), this yields 𝑔↑↓ =8𝜋𝑀s,𝑖𝑑𝑖𝛼sp,𝑖𝑗/(𝑔𝑖𝜇B) = 1.2 ~ 1.7 × 1015 cm−2. 𝑔↑↓ can also \nbe estimated from the free electron density per spin ( n) in the NM layer: 𝑔↑↓ ≈ 1.2𝑛2/3 [57]. With \nn = 5.2 × 1028 m−3 for Ru [58] (the value of n is similar for Ta [59]), 𝑔↑↓ is estimated to be 1.7  × 1015 \ncm−2, the same order as the 𝑔↑↓ value from TR -MOKE measurements, which justifies the 𝛼sp,𝑖𝑗 \nvalues derived from TR -MOKE are within a reasonable range. The values of 𝛼1 and 𝛼2 will be \ndiscussed in section 3.4.  \n 20 \n  \nFIG. 4 (a) Amplitudes of TR -MOKE signals a s functions of Hext. The circles and curves represent \nexperimental data and modeling fitting , respectively. (b) The calculated precession  half cone \nangles at different Hext. Red curves and black curves represent the cone angles of the HF  mode and \nthe LF mode in FM 1 (solid lines) and FM 2 (dash ed lines). Dotted lines are the precession cone \nangles of single -layer (SL) FM 1 and FM 2 without IEC. (c) Phases of TR -MOKE signals at  varying  \nHext. Circles and curves are experimental data and modeling fitting (𝛼sp,12=0.010, 𝛼sp,21=\n0.007, 𝛼1=0.020, 𝛼2=0.060). (d) Simulated precession phase of the HF mode (red curves) and \nthe LF mode (black curves) in FM 1 (solid lines) and FM 2 (dash ed lines).  \n \n3.4 Magnetic damping of the HF and LF precession modes  \nIn addition to the amplitude and phase of TR -MOKE signals for the p -SAF stack, the model \nanalyses also provide a better understanding of magnetic damping. Figure 5 (a) shows the effective \ndamping constant ( 𝛼eff=1/2𝜋𝑓𝜏) measured at different Hext (symbols), in comparison with \n21 \n model ing fitting (solid lines). The general Hext dependence of αeff can be well captured by the \nmodel. The fitted Gilbert damping, 1= 0.020 ± 0.002 and 2 = 0.060 ± 0.008 are close to the \nGilbert damping of Ta/CoFeB(1 nm)/MgO thin films (~0.017) [41,60]  and Co/Pd multilayers with \na similar tCo/tPd ratio (~0.085)  [61]. Other fitted parameters are  Δ𝐻k,eff,1=0.26±0.02 kOe, \nΔ𝐻k,eff,2= 1.42±0.18 kOe, 𝛼12sp=0.010±0.004  𝛼21sp=0.007−0.007+0.009. Δ𝐽1 and Δ𝐽2 are set to be \nzero, as explained in Sec. 2.2. More details regarding the values and determination methods of all \nparameters involved in our data reduction are provided in Note 3 of the SM  [35]. Dashed lines \nshow the calculated 𝛼eff without inhomogeneous broadening. At high Hext, the difference between \nthe solid lines  and dashed lines approaches zero because the inhomogeneous broadening is \nsuppressed. At low Hext, the solid lines are  significantly higher than the dashed lines , indicating \nsubstantial inhomogeneous broadening contributions .  \nThe effective  damping shows interesting features  near the anti-crossing field. As shown in \nFig. 5(b), due to the effective coupling  between two FM layers near the anti-crossing field, the \nhybridization of precession in two FM layers leads to a mix of damping with contributions from \nboth layers. The effective damping  of the FM 1-dominant mode  reaches a maximum within the \nanti-crossing region ( 7  Hext  10 kOe) and is higher than the single -layer (SL)  FM 1 case. \nSimilarly, the hybridized HF and LF  modes at 8.5 kOe exhibit  a lower 𝛼eff (~0.073) compared to \nthe SL FM 2 case. eff consists of contributions from Gilbert damping ( 𝛼𝑖), mutual spin pumping \n(𝛼sp,𝑖𝑗, 𝑖≠𝑗), and inhomogeneous broadening ( Δ𝐻k,eff,𝑖 and Δ𝐽𝑖). To better understand the mixing \ndamping behavior, Fig. 5 (c) shows eff after excluding the inhomogeneous contribution ( 𝛼effinhomo). \nCompared to the SL layer c ase (green and blue dashed lines), the HF and LF modes (red and black \ndashed lines) clearly suggest that IEC effectively mixes the damping in two layers around the anti -\ncrossing field. Without the IEC, precession in FM 2 with a higher damping relaxes faster  than that 22 \n in FM 1. However, the IEC provides a channel to transfer energy from FM 1 to FM 2, such that the \ntwo layers have the same precession relaxation rate for a given mode. Near the anti -crossing field, \ntwo layers have comparable precession cone angles; therefore, the damping values of the \nhybridized modes are roughly the average of two FM layers. In addition to the static IEC, dynamic \nspin pumping can also modify the damping of individual modes. The black and red solid lines \nrepresent the cases with mutu al spin pumping ( 𝛼sp,12 = 0.01 and 𝛼sp,21 = 0.007). Generally, in \nregions 2 & 3, mutual spin pumping reduces the damping of the HF mode and increases the \ndamping of the LF mode because the HF (LF) mode is near in -phase (out -of-phase). Overall, the \nstatic  IEC still plays the essential role for the damping mix near the anti -crossing field.  \n \n \nFIG. 5 (a) Effective damping constant under varying Hext. Circles are experimental data. Solid \nlines are fitting curves based on Eqs. (4 -5). Dashed lines denote eff after the removal of \ninhomogeneous -broadening contribution. (b) A zoomed -in figure of panel (a) between 5 kOe and \n15 kOe. Blue and green circles are measured effective damping of the mode dominated by FM 1 \nand FM 2, respectively. Blu e and green dashed lines are the 𝛼eff of FM 1 and FM 2 single layer \nwithout IEC. (c) Effective damping after excluding the inhomogeneous contribution as a function \nof Hext. The HF mode (red curves) and the LF mode (black curves) are represented by solid (o r \ndashed) curves when the mutual spin pumping terms ( 𝛼sp,12 and 𝛼sp,21) are considered (or \nexcluded). The d ashed green and blue lines are the SL cases for FM 1 and FM 2, respectively.  \n \n \n \n23 \n 4 CONCLUSION  \nWe systematically investigate d the magnetization dynamics excited by ultrafast laser pulses in \nan asymmetric p -SAF sample both theoretically and experimentally. We obtained d etailed \ninformation regarding magnetization dynamics, including the cone angles, directions, and phases \nof spin precession in each layer under different Hext. In particular, the dynamic features in the low -\nfield region  (region 1)  exhibiting  CW precession, were  revealed. The r esonance between the \nprecession of two FM layers occurs at the boundary between regions 2 an d 3, where an anti -\ncrossing feature is present in the frequency vs. Hext profile . The dominant FM layer for a given \nprecession mode also switches from region 2 to region 3. The amplitude and phase of TR -MOKE \nsignals are well captured by theoretical modeling . Importantly , we successfully quantified the \nindividual contributions from various sources to the effective damping , which enables the \ndetermination of Gilbert damping for both FM layers.  At low Hext, the contribution of \ninhomogeneous broadening to the effective damping is significant. Near the anti-crossing field, \nthe effective damping of  two coupled  modes contains substantial contributions from both FM \nlayers owing to the strong hybridization  via IEC . Although the analyses were made for an \nasymme tric SAF sample, this approach can be directly  applied to study magnetization dynamics \nand magnetic properties of  general complex material systems with  coupled multilayers , and thus  \nbenefits the design and optimization of spintronic materials via structural engineering.  \n \nAcknowledgements  \nThis work is  primarily  supported by the National Science Foundation ( NSF, CBET - 2226579). \nD.L.Z gratefully acknowledges the funding support from  the ERI program (FRANC) “Advanced \nMTJs for computation in and near ra ndom access memory” by DARPA, and ASCENT, one of six 24 \n centers in JUMP (a Semiconductor Research Corporation program, sponsored by MARCO and \nDARPA).  J.P.W  and X.J.W also appreciate the partial support from the UMN MRSEC Seed \nprogram (NSF, DMR -2011401 ). D.B.H . would like to thank the support from the UMN 2022 -2023 \nDoctoral Dissertation Fellowship. 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Magn. 48, 3288 (2012).  \n 1 \n Supplement al Material  for \nMagnetization Dynamics in Synthetic Antiferromagnets  with Perpendicular \nMagnetic Anisotropy   \n \nDingbin Huang1,*, Delin Zhang2, Yun Kim1, Jian -Ping Wang2, and Xiaojia Wang1,* \n1Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455, USA  \n2Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN \n55455, USA  \n \nSupplement al Note 1: Analyses of the magnetization precession in each  Ferromagnetic  (FM) \nlayer  \nFor the convenience of derivation, mi is represented in  the spherical coordinate s with the polar \nangle θi and the azimuthal angle φi, as shown in Fig. 1(b): \n𝐦𝑖=(sin𝜃𝑖cos𝜑𝑖,sin𝜃𝑖sin𝜑𝑖,cos𝜃𝑖) (S1) \nAccordingly, t he expressi on of  Eq. ( 2) in the spherical coordinate s is:  \n{        𝜃̇1=−𝛾1\n𝑑1𝑀s,1sin𝜃1∂𝐹\n∂𝜑1−𝛼1sin𝜃1𝜑̇1+𝛼sp,12sin𝜃2cos(𝜃2−𝜃1)𝜑̇2\n𝜑̇1=𝛾1\n𝑑1𝑀s,1sin𝜃1∂𝐹\n∂𝜃1+𝛼1\nsin𝜃1𝜃̇1−𝛼sp,12\nsin𝜃1𝜃̇2\n𝜃̇2=−𝛾2\n𝑑2𝑀s,2sin𝜃2∂𝐹\n∂𝜑2−𝛼2sin𝜃2𝜑̇2+𝛼sp,21sin𝜃1cos(𝜃1−𝜃2)𝜑̇1\n𝜑̇2=𝛾2\n𝑑2𝑀s,2sin𝜃2∂𝐹\n∂𝜃2+𝛼2\nsin𝜃2𝜃̇2−𝛼sp,21\nsin𝜃2𝜃̇1 (S2) \n \n*Author s to whom correspondence should be addressed : huan1746@umn.edu  and wang4940@umn.edu  2 \n where, a dot over variables  represents  a derivative with respect to time. When Mi  precesses  around \nits equilibrium direction:  \n{𝜃𝑖=𝜃0,𝑖+Δ𝜃𝑖\n𝜑𝑖=𝜑0,𝑖+Δ𝜑𝑖 (S3) \nwith \ni  and \ni  representing the deviation angles of  Mi from  its equilibrium direction  along the \npolar and azimuthal directions.  Assuming the deviation is small, under the first -order \napproximation, the first -order partial derivative of F in Eq. (S2) can be expanded as:  \n{    ∂𝐹\n∂𝜃𝑖≈∂2𝐹\n∂𝜃𝑖2Δ𝜃𝑖+∂2𝐹\n∂𝜑𝑖∂𝜃𝑖Δ𝜑𝑖+∂2𝐹\n∂𝜃𝑗∂𝜃𝑖Δ𝜃𝑗+∂2𝐹\n∂𝜑𝑗∂𝜃𝑖Δ𝜑𝑗\n∂𝐹\n∂𝜑𝑖≈∂2𝐹\n∂𝜃𝑖𝜕𝜑𝑖Δ𝜃𝑖+∂2𝐹\n∂𝜑𝑖2Δ𝜑𝑖+∂2𝐹\n∂𝜃𝑗∂𝜑𝑖Δ𝜃𝑗+∂2𝐹\n∂𝜑𝑗∂𝜑𝑖Δ𝜑𝑗 (S4) \nBy substituting Eq. ( S4), Equation ( S2) is linearized  as [1]: \n[   Δ𝜃̇1\nΔ𝜑̇1\nΔ𝜃̇2\nΔ𝜑̇2]   \n=𝐊[Δ𝜃1\nΔ𝜑1\nΔ𝜃2\nΔ𝜑2] (S5) \nwhere, K is a 4×4 matrix, con sisting of  the properties  of individual FM layers  and the second -\norder derivatives of F in terms of 𝜃1,𝜑1,𝜃2,and𝜑2. Equation (S5) has four eigen -solutions, in the \nform of 𝐶exp(𝑖𝜔𝑡), corresponding to four precession frequencies:  ±𝜔HF and ±𝜔LF. A pair of \neigen -solutions with the same absolute precession frequency are physically equivalent. Therefore, \nonly two eigen -solutions need to be considered:  \n{Δ𝜃𝑖=𝐶𝜃,𝑖HFexp(𝑖𝜔HF𝑡)\nΔ𝜑𝑖=𝐶𝜑,𝑖HFexp(𝑖𝜔HF𝑡) and {Δ𝜃𝑖=𝐶𝜃,𝑖LFexp(𝑖𝜔LF𝑡)\nΔ𝜑𝑖=𝐶𝜑,𝑖LFexp(𝑖𝜔LF𝑡) (S6) \nAfter r earrange ment , the full solutions in the spherical coordinates are expressed as below (also \nEq. (3) in the main paper).  3 \n [𝜃1(𝑡)\n𝜑1(𝑡)\n𝜃2(𝑡)\n𝜑2(𝑡)]=\n[   𝜃0,1\n𝜑0,1\n𝜃0,2\n𝜑0,2]   \n+[Δ𝜃1(𝑡)\nΔ𝜑1(𝑡)\nΔ𝜃2(𝑡)\nΔ𝜑2(𝑡)]=[𝜃0,1\n𝜑0,1\n𝜃0,2\n𝜑0,2]+\n[    𝐶𝜃,1HF\n𝐶𝜑,1HF\n𝐶𝜃,2HF\n𝐶𝜑,2HF]    \nexp(𝑖𝜔HF𝑡)+\n[    𝐶𝜃,1LF\n𝐶𝜑,1LF\n𝐶𝜃,2LF\n𝐶𝜑,2LF]    \nexp(𝑖𝜔LF𝑡)    (S7) \nThe prefactors of these  eigen -solutions  provide information about magnetization dynamics of both \nthe HF and LF modes. Directly from solving Eq. (S2), one can obtain the relative ratios of these \nprefactors , which are [𝐶𝜑1HF/𝐶𝜃1HF,𝐶𝜃2HF/𝐶𝜃1HF,𝐶𝜑2HF/𝐶𝜃1HF] and [𝐶𝜑1LF /𝐶𝜃1LF ,𝐶𝜃2LF /𝐶𝜃1LF ,𝐶𝜑2LF /𝐶𝜃1LF ]. \nThese ratios provide  precession information of each mode, as presented in Fig. 3.   \nObtaining the absolute values of [𝐶𝜃,1,𝐶𝜑,1,𝐶𝜃,2,𝐶𝜑,2]𝑇 for each mode requires the initial \nconditions of precession , which i s necessary for fitting the actual precession amplitudes in TR -\nMOKE signals . In TR -MOKE measurements, magnetization precession is initiated by laser \nheating, which reduces the magnetic anisotropy of each  FM layer and the interlayer exchange \ncoupling streng th between two FM layers  [2]. Considering the laser heating process is ultrafast \ncompared with magnetization precession while the following cooling due to heat dissipation is \nmuch slower than magnetization dynamics, we approximately model the temporal profiles of \neffective anisotropy fields and exchange coupling as step functions.  Owing to the sudden change \nin magnetic properties  induced by laser heating , magnetization in each layer will establish a new \nequilibrium direction (𝜃0,𝑖′,𝜑0,𝑖′). In other words, M i deviates from its new eq uilibrium direction \nby Δ𝜃𝑖=𝜃0,𝑖−𝜃0,𝑖′, Δ𝜑𝑖=𝜑0,𝑖−𝜑0,𝑖′. Substituting 𝑡=0 to Eq. ( S7), one can get the initial \nconditions for magnetization dynamics:  \nΔ𝜃𝑖(𝑡=0)=𝐶𝜃,𝑖HF+𝐶𝜃,𝑖LF=𝜃0,𝑖−𝜃0,𝑖′  \nΔ𝜑𝑖(𝑡=0)=𝐶𝜑,𝑖HF+𝐶𝜑,𝑖LF=𝜑0,𝑖−𝜑0,𝑖′=0 (S8) \nOnce the initial conditions are set, the absolute values of all prefactors can be obtained . \n 4 \n Supplementa l Note 2: Estimation of each layer’s contribution to total TR -MOKE signals  \nThe contribution from each FM layer is estimated by static MOKE measurement. According \nto Ref. [3], the resu lt from this method matches well with that from the optical calculation. The \nsample is perpendicularly saturated before the static MOKE measurement. Then the out -of-plane \nM-Hext loop ( Fig. S1) is measured by static MOKE. As shown in the figure, two different \nantiferromagnetic (AF) configurations have different normalized MOKE signals, indicating the \ndifferent contribution s to the total signals by two layers.  The weighting factor is calculated by:   \n−𝑤+(1−𝑤)=0.085 (S9) \nwhich gives 𝑤=0.457. Considering the relatively small layer thicknesses [FM 1: CoFeB(1), spacer: \nRu(0.6)/Ta(0.3), and FM 2: Co(0.4)/Pd(0.7)/Co(0.4)], it is reasonable that FM 1 and FM 2 make \ncomparable contributions to the total TR -MOKE signals ( i.e., w ≈ 0.5). \n \nFIG. S1 Static MOKE hysteresis loop. Magnetic fields are applied along the out -of-plane direction.  \n \n \n \n5 \n Supplemental Note 3: Summary of the parameters and uncertainties for data reduction  \nGiven that a number of variables are involved in the analysis, TABLE SI summarizes the major \nvariables discussed in the manuscript, along with their values and determinatio n methods.  \nTABLE SI. Summary of the values and determination methods of parameters used in the data \nreduction. The reported uncertainties are one -sigma uncertainties from the mathematical model \nfitting to the TR -MOKE measurement data.  \nParameters  Values  Determination Methods  \nHf ~500 Oe  VSM  \nMs,1 1240 emu cm−3 VSM  \nMs,2  827 emu  cm−3 VSM  \nd1 1 nm  Sample structure  \nd2 1.5 nm  Sample structure  \nHk,eff,1  1.23 ± 0.28 kOe  Fitted from f vs. Hext [Fig. 2(b)]  \nHk,eff,2 6.18 ± 0.13 kOe  Fitted from f vs. Hext [Fig. 2(b)]  \nγ1 17.79 ± 0.04  \nrad ns−1 kOe−1 Fitted from f vs. Hext [Fig. 2(b)]  \nγ2 17.85 ± 0.04  \nrad ns−1 kOe−1 Fitted from f vs. Hext [Fig. 2(b)]  \nJ1 −0.050 ± 0.020  \nerg cm−2 Fitted from f vs. Hext [Fig. 2(b)]  \nJ2 0 Fitted from f vs. Hext [Fig. 2(b)]  \nw 0.457  Static MOKE  \n𝐻k,eff,1′/𝐻k,eff,1 0.90 ± 0.01  Fitted from Amp vs. Hext [Fig. 4(a)]  \n𝐻k,eff,2′/𝐻k,eff,2 0.95 ± 0.01  Fitted from Amp vs. Hext [Fig. 4(a)]  \n𝐽1′/𝐽1 0.83 ± 0.01  Fitted from Amp vs. Hext [Fig. 4(a)]  \n𝛼1 0.020 ± 0.002  Fitted from eff vs. Hext [Fig. 5(a)] and  vs. Hext [Fig. 4(c)]  \n𝛼2 0.060 ± 0.008  Fitted from eff vs. Hext [Fig. 5(a)] and  vs. Hext [Fig. 4(c)]  \nΔ𝐻k,eff,1 0.26 ± 0.02 kOe  Fitted from eff vs. Hext [Fig. 5(a)]  \nΔ𝐻k,eff,2 1.42 ± 0.18 kOe  Fitted from eff vs. Hext [Fig. 5(a)]  \n𝛼sp,12 0.010 ± 0.004  Fitted from eff vs. Hext [Fig. 5(a)] and  vs. Hext [Fig. 4(c)]  \n𝛼sp,21 0.007−0.007+0.009 Fitted from eff vs. Hext [Fig. 5(a)] and  vs. Hext [Fig. 4(c)]  \n 6 \n Supplemental Note  4: Impacts of 𝜶𝟏, 𝜶𝟐, and mutual spin pumping on the phase  \nWithout damping, the phase difference in the precession polar angles of two FM layers \n[Arg(𝐶𝜃2/𝐶𝜃1)] is always 0° or 180°, as shown in Fig. 3 of the main article. However, this does not \nnecessarily hold if either the damping or mutual spin pumping is considered. The changes in the \nphase difference due to damping are depicted in Fig. S2. When 1 = 2, the phase difference \nbetween two layers stays at 0° or 180° [ Fig. S2(a)], identical to the lossless case ( 1 = 2 = 0) in \nFig. 3. As a result, the initial phase of TR -MOKE signals ( ) also stays at 0° or 180° [ Fig. S2(b)]. \nHowever, when 𝛼1≠𝛼2, Arg(𝐶𝜃2/𝐶𝜃1) deviates from 0° or 180° especially at high fields ( Hext > \n5 kOe) [ Fig. S2(c,e)]. The layer with a higher damping [FM 1 in (c) or FM 2 in (e)] tends to have a \nmore advanced phase at high fields (regions 2 and 3). For example, in Fig. S2(e), 0° < Arg( 𝐶𝜃2/𝐶𝜃1) \n< 180° for both HF and LF modes in regions 2 and 3. The deviation from the perfect in -phase (0°) \nor out -of-phase (180°) condition allows the IEC to transfer energy from the low -damping layer to \nthe high -damping layer, such that the precession in  both layers can damp at the same rate [4]. As \na result, the initial phase of the TR -MOKE signals also changes, which opens a negative or positive  \ngap at high fields (> 10 kOe) for both modes, as shown in Fig. S2(d,f). This enables us to determine \nthe difference between 1 and 2 by analyzing the in itial phase of TR -MOKE signals.  7 \n  \nFIG. S 2 Impact of 𝛼1 and 𝛼2 on the phase without mutual spin pumping. (a,c,e) The phase \ndifference between the polar angles in two layers for HF and LF modes. (b,d,f) The calculated \ninitial phase of TR -MOKE signals for each mode with 1 = 2 = 0.02 (a,b), 1 = 0.06 and 2 = \n0.02 (c,d), and 1 = 0.02 and 2 = 0.06 (e,f). The mutual spin pumping is set as 𝛼sp,12=𝛼sp,21 = \n0 for all three cases. The rest of the parameters used in this calculation can be found in TABLE SI.  \n \nThe impact of mutual spin pumping on the precession phase is illustrated in Fig. S3, where \nthree different cases of either the one -way (𝛼sp,12 or 𝛼sp,21) or two -way (both 𝛼sp,12 and 𝛼sp,21) \nspin pumping are considered. A reference case without the consideration of mutual spin pumping \n(1 = 0.02, 2 = 0.06, and 𝛼sp,12= 𝛼sp,21 = 0) is also plotted (dashed curves) for the ease of \ncomparison. In general, it can be seen that mutual spin pumping could also change the phase \ndifference in the precession polar angles of two layers, and thus the initial phase o f TR -MOKE \nsignals noticeably. This can be explained by the damping modification resulting from spin \npumping. In regions 2 and 3, Eq. (2) can be approximately rearranged as:  \n \n8 \n 𝑑𝐦𝑖\n𝑑𝑡≈−𝛾𝑖𝐦𝑖×𝐇eff,𝑖+𝛼𝑖𝐦𝑖×𝑑𝐦𝑖\n𝑑𝑡−𝐶𝑗\n𝐶𝑖𝛼sp,𝑖𝑗cos(𝜃0,1−𝜃0,2)𝐦𝑖×𝑑𝐦𝑖\n𝑑𝑡\n≈−𝛾𝑖𝐦𝑖×𝐇eff,𝑖+[𝛼𝑖−𝐶𝑗\n𝐶𝑖𝛼sp,𝑖𝑗cos(𝜃0,1−𝜃0,2)]𝐦𝑖×𝑑𝐦𝑖\n𝑑𝑡\n=−𝛾𝑖𝐦𝑖×𝐇eff,𝑖+𝛼̅𝑖𝐦𝑖×𝑑𝐦𝑖\n𝑑𝑡 (S10) \nwhere 𝐶𝑗/𝐶𝑖 represents the ratio of the cone angles in the j-th FM layer to the i-th FM layer. 𝐶𝑗/𝐶𝑖 \nis positive for the in -phase mode and negative for the out -of-phase mode.  θ0,1 and θ0,2 are the \nequilibrium polar angle s of M1 and M2, as defined in Fig. 2(c).  Therefore, the mutual spin -pumping \nterm either enhances or reduces the damping depending on the mode. 𝛼̅𝑖 = 𝛼𝑖−\n𝐶𝑗\n𝐶𝑖𝛼sp,𝑖𝑗cos(𝜃0,1−𝜃0,2) represents the effective Gilbert damping in the i-th FM layer after \nconsidering the mutual spin -pumping effect. This modification to damping is more significant \nwhen the i-th layer is subservient with a smaller cone angle ( e.g., FM 2 for the HF mode in region \n3), while the j-th layer is dominant with a mu ch larger precession cone angle ( e.g., FM 1 for the LF \nmode in region 3), leading to a large ratio of |𝐶𝑗/𝐶𝑖|. \nIn Fig. S 3(a), only the spin current injected from FM 1 to FM 2 is considered. According to the \nabove analysis, 𝛼sp,21 can only bring noticeable modifications to the damping of FM 2 when FM 1 \nis the dominant layer. Based on Fig. 3 in the main article, the LF mode in region 2 and HF mode \nin region 3 satisfy this condition (FM 1 dominant and FM 2 subservient). As shown in Fig. S 3(a), \nthe phase difference  noticeably deviates from the reference case without mutual spin pumping \n(dashed curves) in region 2 for the LF mode (black curves) and in region 3 for the HF mode (red \ncurves). For the LF mode in region 2, the precession motions in two layers are nearly o ut-of-phase \n(negative C1/C2); therefore, the spin pumping from FM 1 enhances the damping in FM 2. Since 1 \n(0.02) is less than 2 (0.06), the spin pumping from FM 1 to FM 2 further increases |  𝛼̅1 − 𝛼̅2| \nbetween the two layers. Consequently, the phase difference shifts further away from 180°. While 9 \n for the HF mode in region 3, 𝛼sp,21 reduces the damping of FM 2 because C1/C2 is positive resulting \nfrom the near in -phase feature of this mode. Hence, |  𝛼̅1 − 𝛼̅2| becomes smaller and the phase \ndifference gets closer to 0°. In Fig. S 3(c), only 𝛼sp,12 is considered, which requires FM 2 as the \ndominant layer (the HF mode in region 2 and LF mode in region 3) for noticeable changes in |  𝛼̅1 \n− 𝛼̅2|. For the HF mode  in region 2, spin pumping from FM 2 reduces 𝛼̅1 given that the precession \nmotions in two layers are nearly in phase (positive C2/C1). Therefore, |  𝛼̅1 − 𝛼̅2| increases and the \nphase difference in Fig. S 3(c) shifts further away from 0° in region 2. However,  for the LF mode \nin regions 3, the nearly out -of-phase precession in two FM layers (negative C1/C2) increases 𝛼̅1 \nand reduces |  𝛼̅1 − 𝛼̅2|. As a result, the phase difference in Fig. S 3(c) shifts toward 180°. When \nboth 𝛼sp,12 and 𝛼sp,21 are considered [ Fig. S 3(e)], a combined effect is expected for the phase \ndifference with noticeable changes for both the HF and LF modes in regions 2 and 3.  \nThe impacts of mutual spin pumping on the phase difference between the HF and LF modes \nare reflected by the initial phase of TR -MOKE signals [  in Fig. S 3(b,d,f)]. Compared with the \nreference case without mutual spin pumping (dashed curves), the introduction of mutual spin \npumping tends to change the gap in  between the two modes. As shown in Fig. S3(e,f), the values \nof two mutual -spin-pumping induced damping terms are chosen as 𝛼sp,12 = 0.013 and 𝛼sp,21 = \n0.004, such that the  gap of the initial phase of TR -MOKE signals is closed at high fields \n(region  3). Therefore, the initial phase of TR -MOKE signals provides certain measurement \nsensitivities to 𝛼sp,12 and 𝛼sp,21, which enables us to extract the values of 𝛼sp,𝑖𝑗 from \nmeasurement fitting. Here, we acknowledge that the measurement sensitivity to 𝛼sp,𝑖𝑗 from TR -\nMOKE is limited, which subsequently leads to relatively large error bars for 𝛼sp,𝑖𝑗 (see Table SI).  \n 10 \n  \nFIG. S3 Impact of mutual spin pumping on the phase with fixed damping values of 1 = 0.02 and \n2 = 0.06. (a,c,e) The phase difference between the polar angles in two layers for HF and LF modes. \n(b,d,f) The calculated initial phase of TR -MOKE signals ( ) for each mode with 𝛼sp,12= 0 and \n𝛼sp,21 = 0.01 (a,b), 𝛼sp,12 = 0.01 and 𝛼sp,21 = 0 (c,d), and  𝛼sp,12 = 0.013 and 𝛼sp,21 = 0.004 (e,f). \nFor the third case (e,f), the values of mutual spin pumping are chosen to close the  gap in panel \n(f) for Hext > 15 kOe. The rest of the parameters used in this calculation can be found in TABLE \nSI. Dashed lines  represent the reference case without mutual spin pumping ( 1 = 0.02, 2 = 0.06, \nand 𝛼sp,12= 𝛼sp,21 = 0).  \n \nSupplemental Note 5: Region diagram s for p -SAFs with different degrees of asymmetries  \nFigure S4 shows the region diagrams for p -SAFs with different degrees of asymmetries , \nrepresented by the difference of Hk,eff in two FM layers. Hk,eff,1 = Hk,eff,2 corresponds to the \nsymmetric case (lowest asymmetry), as shown by Fig. S4(c). While  the SAF  in Fig. S4(a) has the \nhighest asymmetry: Hk,eff,1 = 2 kOe, Hk,eff,2 = 6 kOe. Figure S4 clearly shows that |𝜃0,1−𝜃0,2|>\n90° is a necessary but not sufficient condition for region 1 (CW precession). Because regions 2 or \n3 also appear to the left of the red cu rve (where |𝜃0,1−𝜃0,2|>90°), especially when θH is close \nto 90° and Hk,eff,1 is close to Hk,eff,2. \n11 \n  \nFIG. S4 Region diagrams of p -SAFs with different  degrees of  asymmetries: Hk,eff,1 = 2 kOe, Hk,eff,2 \n= 6 kOe (a), Hk,eff,1 = 4 kOe, Hk,eff,2 = 6 kOe (b), Hk,eff,1 = 6 kOe, Hk,eff,2 = 6 kOe (c). The blue \nbackground represents region 1. The green background covers regions 2 and 3. The red curve \nshows the conditions where |𝜃0,1−𝜃0,2|=90°. |𝜃0,1−𝜃0,2|>90° to the left of the red curve. 𝛼1, \n𝛼2, 𝛼sp,12, and 𝛼sp,21 are set as zero. 𝛾1=𝛾2=17.8 rad ns−1 kOe−1. Values of the rest parameters are \nthe same as those in Table SI.  \n \nReferences  \n[1] Z. Zhang, L. Zhou, P. E. Wigen, and K. Ounadjela, Angular dependence of ferromagnetic \nresonance in exchange -coupled Co/Ru/Co trilayer structures, Phys. Rev. B 50, 6094 (1994).  \n[2] W. Wang, P. Li, C. Cao, F. Liu, R. Tang, G. Chai, and C. Jiang, Temperature dependence \nof interlayer exchange coupling and Gilbert damping in synthetic antiferromagnetic \ntrilayers investigated using broadband ferromagnetic resonance, Appl. Phys. Lett. 113, \n042401 (2018).  \n[3] G. Malinowski, F. Dalla Longa, J. H. H. Rietjens, P. V. Paluskar, R. Huijink, H. J. M. \nSwagten, and B. Koopmans, Control of speed and efficiency of ultrafast demagnetization \nby direct transfer of spin angular momentum, Nat. Phys. 4, 855 (2008).  \n[4] D. H. Zanette, Energy exchange between coupled mechanical oscil lators: linear regimes, J. \nPhys. Commun. 2, 095015 (2018).  \n \n"    },    {        "title": "2307.13876v1.Oscillatory_Edge_Modes_in_Two_Dimensional_Spin_Torque_Oscillator_Arrays.pdf",        "content": "Oscillatory Edge Modes in Two Dimensional Spin-Torque Oscillator Arrays\nShivam Kamboj,1, 2Rembert A. Duine,3, 4Benedetta Flebus,5and Hilary M. Hurst1\n1Department of Physics and Astronomy, San Jos´ e State University, San Jos´ e, California, 95192, USA\n2Department of Physics, University of Califonia, Merced, California, 95343, USA\n3Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena,\nUtrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands\n4Department of Applied Physics, Eindhoven University of Technology,\nP.O. Box 513, 5600 MB Eindhoven, The Netherlands\n5Department of Physics, Boston College, 140 Commonwealth Avenue, Chestnut Hill, Massachusetts 02467, USA\nSpin torque oscillators (STOs) are dissipative magnetic systems that provide a natural platform\nfor exploring non-Hermitian phenomena. We theoretically study a two-dimensional (2d) array of\nSTOs and show that its dynamics can be mapped to a 2d, non-Hermitian Su-Schrieffer-Heeger\n(SSH) model. We calculate the energy spectrum and identify the one-dimensional (1d) edge states\nof our model, corresponding to auto-oscillation of STOs on the boundary of the system while the\nbulk oscillators do not activate. We show that tuning the Gilbert damping, injected spin current,\nand coupling between STOs allows for exploring the edge state properties under different parameter\nregimes. Furthermore, this system admits 1d edge states with non-uniform probability density, and\nwe explore their properties in systems of different sizes. Additional symmetry analysis indicates\nthat these states are not topologically protected but are nevertheless confined to the edge of the\nsystem, as the bulk is protected by PT-symmetry. These results indicate that 2d arrays of STOs\nmay be useful to explore novel edge state behavior in dissipative systems.\nI. INTRODUCTION\nTopology and its connection to condensed matter sys-\ntems has been the subject of intense research for nearly\nfifty years, since the discovery of quantized Hall resis-\ntance and its topological origin [1, 2]. Topology is now\nunderstood as a critical underlying feature of many ma-\nterials that can affect global transport properties, result-\ning in, e.g., the quantum anomalous and spin Hall ef-\nfects [3, 4] and leading to entirely new classes of topo-\nlogical materials [5–7]. The effects of topology in sys-\ntems with dissipation, i.e. non-Hermitian systems, can be\nmarkedly different from their Hermitian counterparts [8–\n14]. Non-Hermitian systems can exhibit exceptional\npoints [15–18], i.e. the coalescence of two or more eigen-\nvectors, as well as the non-Hermitian skin effect, a phe-\nnomenon where bulk eigenstates localize on the edge of\nthe system [19–21]. The edge states in non-Hermitian\nsystems can also exist despite the breakdown of the bulk-\nboundary correspondence [9, 10] and exhibit lasing be-\nhavior [22, 23]. While several experimental realizations\nof non-Hermitian phenomena have been observed in pho-\ntonic [22], acoustic [24], and electronic circuits [25, 26],\ntheir experimental exploration in magnonic systems is\nstill in its infancy. Magnonic systems are, however, a nat-\nural platform in which to realize non-Hermitian physics\nbecause they are always coupled to a surrounding envi-\nronment and exhibit lossy dynamics [27, 28].\nSpin torque oscillators (STOs) have recently emerged\nas a promising platform for harboring non-Hermitian\nphenomena [27, 29, 30]. These magnetic nanopillars are\nnanometer sized devices that conduct spin currents via\nspin transfer torque [31]. STOs are dissipative systems\nbecause, like all magnetic systems, they are subject to\nubiquitous spin non-conserving interactions parameter-ized by Gilbert damping [32]. The magnetic dynamics\nof 1d STO arrays was successfully mapped to a non-\nHermitian Su-Schrieffer-Heeger (SSH) model with topo-\nlogically protected lasing edge states in Ref. [29]. In a\nsubsequent numerical study these edge states were shown\nto be robust in the presence of additional terms such\nas dipolar interactions and nonlinear STO behavior [33].\nHere, we examine whether these lasing edge states can\nbe realized in higher-dimensional arrays of STOs.\nThe 1d SSH model and it’s non-Hermitian variants are\nwidely used in condensed matter physics to study topo-\nlogical systems [10, 34]; 2d non-Hermitian SSH mod-\nels have been studied in a variety of contexts, where a\nnumber of interesting properties such as in-gap topolog-\nical states and non-trivial bulk-band topology have been\nfound [25, 26, 35–37]. However, most of these models lack\na clear experimental implementation, with topological\ncircuits being a notable exception [25, 26]. The sources of\nnon-Hermitian terms (i.e. energy non-conserving terms)\ncan be difficult to quantify and characterize in experi-\nmental platforms. Here, we propose STO arrays as a\nplatform for experimental realization of a non-Hermitian\n2d SSH model. Specifically, we focus on a geometry con-\nsisting of several 1d chains that are weakly coupled to\nform a 2d STO array. By introducing this new type of\n‘vertical coupling’, we derive a non-Hermitian 2d SSH\nmodel that exhibits 1d lasing edge states.\nThe manuscript is organized as follows: In Section II\nwe map the linearized Landau-Lifshitz-Gilbert (LLG)\nequation for the magnetization dynamics into a non-\nHermitian tight-binding Hamiltonian. In Sec. III we dis-\ncuss the properties of our model including the oscillatory\nedge states and the symmetry properties of the Bloch\nHamiltonian. Finally, we conclude and discuss directions\nfor future work in Sec. IV.arXiv:2307.13876v1  [cond-mat.mes-hall]  26 Jul 20232\nII. MODEL\nWe consider a 2d array of M×2NSTOs which could be\nfabricated from individual nanopillars [29] or a multilayer\nstructure [33]. Here, Mis the total number of rows in\nthe array and Nindicates the number of unit cells per\nrow, where there are two STOs per unit cell. A single\nSTO consists of a layer of fixed magnetic polarization\nand a ‘free’ magnetic layer without fixed polarization,\nseparated by a thin metallic spacer. By injecting spin\ncurrent into the free layer, the fixed layer is driven to\nprecess about its equilibrium direction, which is set by\nan external applied magnetic field. The dynamics of an\nisolated STO subjected to a magnetic field H0=H0ˆz\nand spin current JS\nη=JS\nηˆzare described by the LLG\nequation for the magnetization vector mη,ij[31]. Here,\nthe index η=A, B denotes the AandBsublattices and\nthe indices i, jlabel the sites using the (row, column)\nconvention. The resulting LLG equation is\n˙mη,ij|0=ωη,ijˆz×mη,ij+αη,ijmη,ij×˙mη,ij\n+JS\nηmη,ij×(mη,ij׈z). (1)\nThe STO ferromagnetic resonance frequency is given\nbyωη,ij=γη,ij(H0−4πMη,ij) where γη,ijis the gyro-\nmagnetic ratio and Mη,ijis the saturation magnetization;\nαη,ij≪1 is the dimensionless Gilbert damping parame-\nter. We assume the resonance frequency and the Gilbert\ndamping to the be same for all STOs in the array and\ndrop the subscripts going forward, i.e. ωη,ij→ωand\nαη,ij→α. The third term in Eq. (1) is the Slonczewski-\nBerger spin-transfer torque [38, 39]. The injected spin\ncurrent JS\nηis assumed to the be same for all sites in a\ngiven sublattice.\nInteractions between adjacent STOs are mediated by a\nreactive Ruderman- Kittel-Kasuya-Yosida (RKKY)-type\nmagnetic exchange coupling, which is given by [29]\n˙mA,ij|coup=−mA,ij×(JmB,ij+˜JmB,ij−1)\n−mA,ij×J2(mη′,i+1j+mη′,i−1j) (2)\n˙mB,ij|coup=−mB,ij×(JmA,ij+˜JmA,ij+1)\n−mB,ij×J2(mη′′,i+1j+mη′′,i−1j) (3)\nThe coupling strengths J,˜Jare the intracell and in-\ntercell coupling for the 1d unit cell, which contains two\nSTOs, and J,˜J, J 2>0 indicate ferromagnetic coupling.\nWe consider the geometry depicted in Fig. 1, for which\nη′=A, η′′=B.\nWe now derive the linearized equations of motion for\nthe STO magnetization mA(B). That is, we start in\nthe strong-field regime where the magnetic moment is\nmostly aligned along the ˆzdirection and the energy scale\ngiven by ωis the largest one in the problem. Starting\nFIG. 1. Schematic of the effective tight-binding model of a 2d\nSTO array, showing a system of size 4 ×8, where there are 4\nunit cells per row. Red (dark color) sites are the A sublattice\nand green (light color) sites are the B lattice.\nfrom Eq. (1), we can derive an effective Hamiltonian in\nthe following way: First, we linearize the equations of\nmotion about the equilibrium field direction by writing\nmη,ij= (mx\nη,ij, my\nη,ij,1)Twhere |mx\nη,ij| ≃ |my\nη,ij| ≪ 1,\nand|mz| ≃1 is assumed to be a constant. We then\nintroduce the variable 2 m−\nη,ij=mx\nη,ij−imy\nη,ij, and\nmake the Holstein-Primakoff approximation m−\nA(B),ij=\n⟨aij(bij)⟩e−iωtwhere the second quantized bosonic oper-\nators aij,bijannihilate a magnon on the STO at site\n(i, j) [40]. From the Heisenberg equations of motion\n˙aij=i/ℏ[H, a ij], we derive an effective Hamiltonian cor-\nresponding to the linearized equations of motion. In the\nfollowing we set ℏ= 1.\nFor the 2d STO array, H=P\nijHijis\nHij=ω(a†\nijaij+b†\nijbij)\n+i(JS\nA−αω)a†\nijaij+i(JS\nB−αω)b†\nijbij\n−J(a†\nijbij+h.c.)−˜J(a†\nijbij−1+h.c.)\n−J2(a†\nijai−1j+b†\nijbi−1j+h.c). (4)\nWe see that non-Hermiticity arises due to the onsite\nspin current injection and Gilbert damping, resulting\nin onsite terms ∝i(JS\nη−αω), and the degree of non-\nHermiticity can be tuned by balancing the injected spin\ncurrent and Gilbert damping. Given the non-Hermitian\nlattice model, we can now explore its energy spectrum\nand edge state properties. In this manuscript we use the\nterm ‘energy spectrum’ and the symbol Eto denote the\ncomplex eigenvalues of H; Re(E) can be thought of as an\nenergy while Im( E)>0 (<0) is an indication of lasing\n(damping).\nHere we consider the effect of different coupling\nstrengths J2. We confine the system to the parity-time\n(PT) symmetric regime where the injected spin current\nisJSA= 2αω,JSB= 0; further symmetry analysis is\nperformed in Sec. III C. This is a 2d extension of the\n1dPT-symmetric, non-Hermitian SSH model analyzed3\n−2.50.02.5Re(E) - ω\n [arb. units](a) (b) (c) (d)\n−2 −1 0 1 2\nJ/|̃J|−0.10.00.1Im(E) [arb. units]\n(e)\n−2 −1 0 1 2\nJ/|̃J|(f)\n−2 −1 0 1 2\nJ/|̃J|(g)\n−2 −1 0 1 2\nJ/|̃J|(h)\nFIG. 2. The dependence of the real (a-b) and imaginary (e-f) parts of the energy spectrum of HonJ/|˜J|for a system of 10 ×20\nSTOs for ω= 1˜J,α= 0.2, and J2= 0.01˜J(a,e) and J2= 0.1˜J(b,f). Modes with Im( E)>0 correspond to auto-oscillation of\nSTOs on the edge of the system, appearing for |J|<˜J. These modes correspond to the flat in-gap bands at Re( E)−ω≈0\nin the real spectrum. Panels (c-d, g-h) show the real (c-d) and imaginary (g-h) parts of the energy spectrum for a system of\n50×100 STOs, with J2= 0.01˜J(c,g) and J2= 0.1˜J(d,h). Apart from the different system size, all the other parameters are\nsame as in (a-b) and (e-f). In the larger system there is some activation of bulk STOs in the PT-broken regime, indicated in\n(g-h) by the additional states with Im( E)>0 in the region αω >|˜J−J|.\nin Ref. [29]. We note that other versions of the non-\nHermitian 2d SSH model studied in the literature have\nalternating A and B sublattice sites in the vertical di-\nrection [35–37], resulting in a four site unit cell. Here,\nhowever, we introduce the coupling such that each col-\numn is all A or B sublattice sites, thus reducing the unit\ncell to two STOs. Our reasoning for this geometry is that\nit would be easier to inject spin current into an entire col-\numn of STOs, for example using a metallic strip, rather\nthan having to individually address each A or B site STO\nin the grid.\nIII. RESULTS\nOur goal is to understand how the vertical coupling\nJ2between rows affects the properties of the original\n1d system, thus we briefly review the results for a 1d\nSTO chain [29, 33]. The real energy spectrum of the 1d\nSTO model Hamiltonian has two degenerate flat bands\nforJ <|˜J|and admits a real line gap in k-space [12].\nThe flat bands with degenerate energy eigenvalues are\nan indication of topologically protected edge states of\nthe Hamiltonian. The eigenstates corresponding to the\nflat-bands also have non-zero imaginary eigenvalues, in-\ndicating lasing (Im( E)>0) and damped (Im( E)<0)\nstates. The lasing states correspond to an auto oscilla-\ntion of STOs, which occurs only at the edge of the sys-\ntem for αω < |J−˜J|. In the regime αω > |J−˜J|, thebulk Hamiltonian also has complex eigenvalues which can\nlead to oscillation of the bulk STOs; this is the so-called\nPT-broken regime of the model where the Hamiltonian\nrespects PT-symmetry but its eigenstates do not [27].\nSymmetry analysis of the Bloch Hamiltonian for the 1d\nmodel confirms the auto-oscillation of the edge STOs to\nbe a zero-dimensional ‘edge state’ with topological pro-\ntection.\nIn the 2d case, the non-Hermitian Hamiltonian Eq. (4)\nalso has complex eigenvalues. The energy spectrum of\nthe 2d model looks similar to the 1d model in the case of\nweak vertical coupling J2/˜J≲0.01, but the degeneracy\nin the flat bands breaks immediately even with infinites-\nimal vertical coupling. In Fig. 2, we show the energy\nspectrum of the Hamiltonian considering different verti-\ncal coupling strengths J2/˜J= 0.01,0.1 for two different\nsystem sizes. We found that as J2increases with respect\nto˜J, the energy separation between flat bands increases\nand they hybridize with the bulk states. We simulated\nresults for a 2d array 20 sites wide and 10 sites in the ver-\ntical direction, as well as a larger system 100 sites wide\nand 50 sites in the vertical direction. In the upcoming\nsubsections, we respectively discuss the numerical results\nfor the edge states, analyze the Hamiltonian in momen-\ntum space, and present a symmetry analysis of the model.4\n0\n4\n8Site no.(a)\n|ψ|205\n(b)\n|ψ|205\n0 8 16\nSite no.0\n4\n8Site no.(c)\n|ψ|205\n0 8 16\nSite no.(d)\n|ψ|205\n0.000.050.100.15|ψ|2\n0 2 4 6 8\n Site no. 0.050.100.15|ψ|2(e)Re(E) -ω\n-0.19\n-0.17\n-0.13\n-0.08\n-0.03\n0.03\n0.08\n0.13\n0.17\n0.19\nFIG. 3. Spatial distribution of 1d edge modes in a 2d STO\narray. (a-d) Color density plots showing |ψ|2for four of the\nthe flat-band modes with Im( E)>0. The states are localized\non the left edge, with |ψ|2= 0 everywhere in the bulk. The\ninsets show the 1d distribution along the edge. (e) 1d plot\nshowing the spatial distribution of all 10 lasing edge states\nand the corresponding real part of the energy, Re( E)−ω.\nThe edge states are not uniform across the system due to the\ncoupling J2between rows. States appear in pairs where states\nwith equal magnitude and opposite sign of Re( E)−ωhave\nthe same spatial distribution (dashed and solid lines overlap).\nA. Edge States\nUsing exact diagonalization, we find that the real-\nspace Hamiltonian Eq. (4) exhibits one-dimensional ‘las-\ning’ edge states where Im( E)>0. In Fig. 3 we show\nthe spatial distribution of edge states for a system of\n10×20 STOs. Since there are 10 rows, the system\nexhibits 10 lasing edge modes, all with Im( E) = 0 .09.\nWe have confirmed that the corresponding damped edge\nmodes with Im( E) =−0.09 occur on the opposite edge\nof the system, as expected (not pictured here). We set\nthe intercell coupling to ˜J= 1 and consider the regime\nwhere intracell coupling J/˜J= 0.2 and vertical cou-\npling J2/˜J= 0.1. The physical manifestation of these\nedge states is an auto-oscillation of STOs that is non-\nuniform at the edge of the sample. Thus, the STOs will\nexhibit spatially varying microwave emission which can\nbe tuned based on the magnon population in each edge\nmode. We also diagonalized a larger STO array to\nsee how the results vary if we scale up the system size.\nThe Hamiltonian matrix has a block banded structure,\n0.000.010.020.03|ψ|2\n0.000.010.020.03|ψ|2\n0.000.010.020.03|ψ|2\n0.000.010.020.03|ψ|2\n0 20 40\nSite no.0.000.010.02|ψ|2\n0 20 40\nSite no.Re(E) -ω x 10−1\n0.06\n0.180.31\n0.430.55\n0.660.78\n0.891.0\n1.1FIG. 4. Example of the spatial distribution of 10 unique las-\ning edge states (out of 50) and the corresponding real part of\nthe energy, Re( E)−ω×10−1for a system of 50 ×100 sites.\nThe larger system shows greater heterogeneity in the spatial\ndistribution of edge modes. The top row of the legend corre-\nsponds to the left column and the bottom row corresponds to\nthe right column; values increase downward in each column.\nFor example Re( E)−ω= 1.0 corresponds to the bottom-left\npanel.\ncalled a block Toeplitz-tridiagonal (TT) matrix. For a\nlarger system i.e. 50x100 sites, the matrix becomes 5000\nx 5000 for which the diagonalization is computationally\nexpensive for different values of J2. To diagonalize this\nlarge sparse matrix efficiently, we used the method given5\nby Ref. [41]. The energy spectrum for the larger system -\nas shown in Fig. 2 (c-d), (g-h) - doesn’t show any devia-\ntion apart from very small bulk oscillations in the region\nwhere αω >|J−˜J|. This proves the 1d edge states can\nexist in a significantly larger system. As shown in Fig. 4,\nthe edge states in the larger system also exhibit a non-\nuniform spatial distribution, with increased oscillations\nand beating behavior visible in some modes. Here, we\nshow the first 10 out of 50 total edge states ranked by\nincreasing Re( E).\nB. Bloch Hamiltonian\nHere we consider the PT-symmetric regime where\nJSA= 2αωandJSB= 0, leading to balanced gain (loss)\nterms ±iαωon the A(B) sites. To analyze the Hamilto-\nnian in momentum space, we consider periodic boundary\nconditions and Fourier transform Eq. (4), which is writ-\nten as\nˆH=X\nk\u0000\na†\nkb†\nk\u0001\n(Hk−ω)\u0012\nak\nbk\u0013\n(5)\nwhere\nHk=\u0012\niαω−2J2cosky−J−˜Jeikx\n−J−˜Je−ikx−iαω−2J2cosky\u0013\n(6)\nThe resonant frequency simply provides an overall shift\nof the energy spectrum, therefore we redefine ωas the\nzero energy point.\nThe Bloch Hamiltonian can be written\nHk=d0(ky)1+d(kx)·σ (7)\nwhere 1is the 2 ×2 identity matrix and σ= (σx, σy, σz)\nis the vector of Pauli matrices. We define the functions\nd0(ky) =−2J2cos(ky) and\nd(kx) =\n−J−˜Jcos(kx)\n˜Jsin(kx)\niαω\n. (8)\nWe find the two-band energy spectrum\nϵ±(k) =−2J2cos(ky)±q\nJ2+˜J2+ 2J˜Jcos(kx)−α2ω2.\n(9)\nHere we see that the eigenvalues are real for αω <\n|J+˜Jeikx|, i.e. the system remains in the PT-unbroken\nregime exhibiting real eigenvalues as long as the Gilbert\ndamping is relatively small. This condition is satisfied for\nallkifαω < |J−˜J|. Furthermore, the spectrum is lin-\near in the vertical coupling J2; thus the effect of coupling\nadjacent 1d STO chains together is to shift the spectrum\naway from the resonant frequency ω. We can see this\nclearly for example in Fig. 2 (b), where the flat bands of\nadjacent 1d chains hybridize and are vertically shifted.\nFurthermore, J2can cause the energy gap to close, how-\never in the regime J2≲q\n(J−˜J)2−α2ω2the energy\ngap is open and the edge states remain well separated\nfrom the bulk.C. Symmetry Analysis\nSymmetry analysis can help determine whether the\n1d edge states displayed in Figs. 3 and 4 are topolog-\nically protected. We investigate the following symme-\ntries of the Bloch Hamiltonian, Eq. (6): chiral sym-\nmetry, chiral-inversion symmetry, sublattice symmetry,\nand parity-time ( PT) symmetry. Systems obeying chi-\nral, chiral-inversion, or sublattice symmetry can exhibit\ntopologically protected edge modes [11, 12, 42], therefore\nit is important to check whether these symmetries are\npreserved for our model. PT-symmetry ensures there is\nalways a regime in which the Hamiltonian has real eigen-\nvalues [43]. We note that for a Hermitian system, chi-\nral and sublattice symmetries are equivalent, however for\na non-Hermitian Hamiltonian this is no longer the case\nand some care must be taken. Here we use the symmetry\nnaming conventions from Ref. [12].\nTo have chiral symmetry (CS), the Hamiltonian must\nsatisfy the condition σzH†\nkσz=−Hk. We find that the\nvertical coupling term J2cos(ky) breaks chiral symmetry\nin general, however for the special values ky=±π/2 chi-\nral symmetry is preserved. To have chiral-inversion (CI)\nsymmetry, the Hamiltonian must satisfy the condition\nσyHkσy=−H−k. Like the case of CS, CI is in general\nbroken by J2and only preserved for ky=±π/2. For\nsublattice symmetry, the Hamiltonian must satisfy the\ncondition σzHkσz=−Hk. This model does not posses\nsublattice symmetry for any parameter regime due to the\nnon-Hermitian terms, as is the case in 1d [10, 29].\nTo have PT-symmetry, the Hamiltonian must satisfy\nthe condition σxH∗\nkσx=Hk. This condition is satisfied\nforJSA= 2αωandJSB= 0, thus as with the 1d case\nthe system can be tuned to the PT-symmetric regime by\naltering the injected spin current on A and B sublattice\nsites.PT-symmetry alone does not guarantee topological\nprotection [10], and the results from the symmetry anal-\nysis indicate that the edge states observed in this model\nare not topologically protected. However, the confine-\nment of the oscillatory modes to the edge can be under-\nstood as a result of PT-symmetry in the bulk, which is\nbroken spontaneously by the edge of the system. Fur-\nthermore, PT-symmetry guarantees that the bulk STOs\ndo not have any lasing modes as long as αω <|J−˜J|.\nIV. CONCLUSION AND OUTLOOK\nIn this work we have examined a novel realization of a\nnon-Hermitian 2d SSH model which can be constructed\nfrom an array of STOs. Using exact diagonalization and\nanalysis of the Bloch Hamiltonian, we have shown that\nthis model exhibits 1d lasing edge states with a non-\nuniform spatial distribution. The physical manifestation\nof these modes is an auto-oscillation of STOs along one\nedge of the system which is spatially varying. The exten-\nsion of the model from 1d to 2d via the addition of vertical\ncoupling between individual 1d STO chains breaks chiral-6\ninversion, chiral, and sublattice symmetry, indicating a\nloss of topological protection for these modes. However,\nthe vertical coupling preserves PT-symmetry for the bulk\nstates, thereby guaranteeing that the bulk oscillators do\nnot activate, even in the presence of spin current injected\ninto the bulk.\nHere we have considered an injected spin current such\nthat the system remains at the PT-symmetric point. Fu-\nture works could investigate the robustness of these edge\nstates in the presence of additional terms such as dipo-\nlar interactions as well as dissipative coupling between\nSTOs. However, results from studies of the analogous 1d\nmodel indicate that if these terms are small compared\nto the reactive RKKY coupling studied here, they would\nnot strongly affect the presence of edge states [29, 33].\nAnother interesting extension of this model would be\nto explore possible application in devices. The dispersionrelation in Eq. (9) indicates that the 1d edge states have\nnonzero group velocity due to the coupling J2. 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Staunton\nDepartment of Physics, University of Warwick, Coventry, UK\nT. Gruenbaum, M. A. W. Schoen and C. H. Back\nDepartment of Physics, Regensburg University, Regensburg, Germany\nX. Z. Chen, C. Song\nKey Laboratory of Advanced Materials (MOE), School of Materials\nScience and Engineering, Tsinghua University, Beijing 100084, China.\n(Dated: April 3, 2018)\nThe \fnite-temperature transport properties of FeRh compounds are investigated by \frst-principles\nDensity Functional Theory-based calculations. The focus is on the behavior of the longitudinal re-\nsistivity with rising temperature, which exhibits an abrupt decrease at the metamagnetic transition\npoint,T=Tmbetween ferro- and antiferromagnetic phases. A detailed electronic structure inves-\ntigation for T\u00150 K explains this feature and demonstrates the important role of (i) the di\u000berence\nof the electronic structure at the Fermi level between the two magnetically ordered states and (ii)\nthe di\u000berent degree of thermally induced magnetic disorder in the vicinity of Tm, giving di\u000berent\ncontributions to the resistivity. To support these conclusions, we also describe the temperature\ndependence of the spin-orbit induced anomalous Hall resistivity and Gilbert damping parameter.\nFor the various response quantities considered the impact of thermal lattice vibrations and spin \ruc-\ntuations on their temperature dependence is investigated in detail. Comparison with corresponding\nexperimental data \fnds in general a very good agreement.\nPACS numbers: Valid PACS appear here\nFor a long time the ordered equiatomic FeRh alloy has\nattracted much attention owing to its intriguing temper-\nature dependent magnetic and magnetotransport prop-\nerties. The crux of these features of this CsCl-structured\nmaterial is the \frst order transition from an antiferro-\nmagnetic (AFM) to ferromagnetic (FM) state when the\ntemperature is increased above Tm= 320 K [1, 2]. In\nthis context the drop of the electrical resistivity that is\nobserved across the metamagnetic transition is of central\ninterest. Furthermore, if the AFM to FM transition is\ninduced by an applied magnetic \feld, a pronounced mag-\nnetoresistance (MR) e\u000bect is found experimentally with\na measured MR ratio \u001850% at room temperature [2{\n4]. The temperature of the metamagnetic transition as\nwell as the MR ratio can be tuned by addition of small\namounts of impurities [2, 5{8]. These properties make\nFeRh-based materials very attractive for future applica-\ntions in data storage devices. The origin of the large MR\ne\u000bect in FeRh, however, is still under debate. Suzuki et\nal. [9] suggest that, for deposited thin FeRh \flms, the\nmain mechanism stems from the spin-dependent scatter-\ning of conducting electrons on localized magnetic mo-\nments associated with partially occupied electronic d-\nstates [10] at grain boundaries. Kobayashi et al. [11]\nhave also discussed the MR e\u000bect in the bulk ordered\nFeRh system attributing its origin to the modi\fcation of\nthe Fermi surface across the metamagnetic transition. Sofar only one theoretical investigation of the MR e\u000bect in\nFeRh has been carried out on an ab-initio level [12].\nThe present study is based on spin-polarized, electronic\nstructure calculations using the fully relativistic multiple\nscattering KKR (Korringa-Kohn-Rostoker) Green func-\ntion method [13{15]. This approach allowed to calcu-\nlate the transport properties of FeRh at \fnite tempera-\ntures on the basis of the linear response formalism using\nthe Kubo-St\u0014 reda expression for the conductivity tensor\n[16, 17]\n\u001b\u0016\u0017=~\n4\u0019N\nTrace\n^j\u0016(G+(EF)\u0000G\u0000(EF))^j\u0017G\u0000(EF)\n\u0000^j\u0016G+(EF)^j\u0017(G+(EF)\u0000G\u0000(EF))\u000b\nc;(1)\nwhere \n is the volume of the unit cell, Nis the num-\nber of sites, ^j\u0016is the relativistic current operator and\nG\u0006(EF) are the electronic retarded and advanced Green\nfunctions, respectively, calculated at the Fermi energy\nEF. In Eq. (1) the orbital current term has been omit-\nted as it only provides small corrections to the prevailing\ncontribution arising from the \frst term in the case of a\ncubic metallic system [18{20].\nHere we focus on the \fnite temperature transport\nproperties of FeRh. In order to take into account\nelectron-phonon and electron-magnon scattering e\u000bects\nin the calculations, the so-called alloy analogy modelarXiv:1606.02072v1  [cond-mat.mtrl-sci]  7 Jun 20162\n[21, 22] is used. Within this approach the tempera-\nture induced spin (local moment) and lattice excitations\nare treated as localized, slowly varying degrees of free-\ndom with temperature dependent amplitudes. Using the\nadiabatic approximation in the calculations of transport\nproperties, and accounting for the random character of\nthe motions, the evaluation of the thermal average over\nthe spin and lattice excitations in Eq. (1) is reduced to\na calculation of the con\fgurational average over the lo-\ncal lattice distortions and magnetic moment orientations,\nh:::ic, using the recently reported approach [21, 22] which\nis based on the coherent potential approximation (CPA)\nalloy theory [23{25].\nTo account for the e\u000bect of spin \ructuations, which\nwe describe in a similar way as is done within the dis-\nordered local moment (DLM) theory [26], the angular\ndistribution of thermal spin moment \ructuations is cal-\nculated using the results of Monte Carlo (MC) simula-\ntions. These are based on ab-initio exchange coupling\nparameters and reproduce the \fnite temperature mag-\nnetic properties for the AFM and FM state in both the\nlow- (T < Tm) and high-temperature ( T > Tm) regions\nvery well [27]. Figure 1(a), inset, shows the temperature\ndependent magnetization, M(T), for one of the two Fe\nsublattices aligned antiparallel/parallel to each other in\nthe AFM/FM state, calculated across the temperature\nregion covering both AFM and FM states of the system.\nThe di\u000bering behavior of the magnetic order M(T) in the\ntwo phases has important consequences for the transport\nproperties as discussed below.\nFigure 1(a) shows the calculated electrical resistiv-\nity as a function of temperature, \u001axx(T), accounting\nfor the e\u000bects of electron scattering from thermal spin\nand lattice excitations, and compares it with experi-\nmental data. There is clearly a rather good theory-\nexperiment agreement especially concerning the di\u000ber-\nence\u001aAFM\nxx (Tm)\u0000\u001aFM\nxx(Tm) at the AFM/FM transition,\nTm= 320K. The AFM state's resistivity increases more\nsteeply with temperature when compared to that of the\nFM state, that has also been calculated for temperatures\nbelow the metamagnetic transition temperature (dotted\nline). Note that the experimental measurements have\nbeen performed for a sample with 1% intermixing be-\ntween the Rh and Fe sublattices leading to a \fnite resid-\nual resistivity at T!0 K, and as a consequence there is\na shift of the experimental \u001axx(T) curve with respect to\nthe theoretical one [28].\nWe can separate out the contributions of spin \ructua-\ntions and lattice vibrations to the electrical resistivities,\n\u001afluc\nxx(T) and\u001avib\nxx(T), respectively. These two compo-\nnents have been calculated for \fnite temperatures keep-\ning the atomic positions undistorted to \fnd \u001afluc\nxx(T) and\n\fxed collinear orientations of all magnetic moments to\n\fnd\u001avib\nxx(T), respectively. The results for the AFM and\nFM states are shown in Fig. 1(b), where again the FM\n(AFM) state has also been considered below (above) the\n(a)\n(b)\nFIG. 1. (a) Calculated longitudinal resistivity (closed cir-\ncles - AFM state, open circles - FM state) in comparison\nwith experiment [2]. The dashed line represents the results\nfor Fe 0:49Rh0:51, while the dash-dotted line gives results for\n(Fe-Ni) 0:49Rh0:51with the Ni concentration x= 0:05 to sta-\nbilize the FM state at low temperature). The inset represents\nthe relative magnetization of a Fe sub-lattice as a function\nof temperature obtained from MC simulations. (b) electrical\nresistivity calculated for the AFM (closed symbols) and FM\n(open symbols) states accounting for all thermal scattering ef-\nfects (circles) as well as e\u000bects of lattice vibrations (diamond)\nand spin \ructuations (squares) separately. The inset shows\nthe temperature dependent longitudinal conductivity for the\nAFM and FM states due to lattice vibrations only.\ntransition temperature Tm. For both magnetic states the\nlocal moment \ructuations have a dominant impact on\nthe resistivity. One can also see that both components,\n\u001afluc\nxx(T) and\u001avib\nxx(T), in the AFM state have a steeper\nincrease with temperature than those of the FM state.\nThe origin of this behavior can be clari\fed by refer-\nring to Mott's model [29] with its distinction between\ndelocalized sp-electrons, which primarily determine the\ntransport properties owing to their high mobility, and\nthe more localized d-electrons. Accordingly, the conduc-\ntivity should depend essentially on (see. e.g. [30]): (i)\nthe carrier (essentially sp-character) concentration nand\n(ii) the relaxation time \u001c\u0018[V2\nscattn(EF)]\u00001, whereVscatt\nis the average scattering potential and n(EF) the total\ndensity of states at the Fermi level. This model has been\nused, in particular, for qualitative discussions of the ori-\ngin of the GMR e\u000bect in heterostructures consisting of3\nmagnetic layers separated by non-magnetic spacers. In\nthis case the GMR e\u000bect can be attributed to the spin\ndependent scattering of conduction electrons which leads\nto a dependence of the resistivities on the relative ori-\nentation of magnetic layers, parallel or antiparallel, as-\nsuming the electronic structure of non-magnetic spacer\nto be unchanged. These arguments, however, cannot be\nstraightforwardly applied to CsCl-structured FeRh, even\nthough it can be pictured as a layered system with one\natom thick layers, since the electronic structure of FeRh\nshows strong modi\fcations across the AFM-FM transi-\ntion as discussed, for example, by Kobayashi et al. [11]\nto explain the large MR e\u000bect in FeRh.\n(a) (b)\n(c) (d)\nFIG. 2. Comparison of the temperature dependent densities\nof states (DOS) for the FM and AFM states of FeRh for\nT= 40\u0000 \u0000400 K : (a) Fe s-DOS, (b) Fe p-DOS, (c) Rh\ns-DOS, and (d) Rh p-DOS.\nWe use the calculated density of states at the Fermi\nlevel as a measure of the concentration of the conducting\nelectrons. The change of the carriers concentration at the\nAFM-FM transition can therefore be seen from the mod-\ni\fcation of the sp-DOS at the Fermi level. The element-\nprojected spin-resolved sp-DOS (nsp(E)), calculated for\nboth FM and AFM states at di\u000berent temperatures, is\nshown in Fig. 2. At low temperature, for both Fe and\nRh sublattices, the sp-DOS atEFis higher in the FM\nthan in the AFM state, nFM\nsp(EF)> nAFM\nsp (EF). This\ngives a \frst hint concerning the origin of the large dif-\nference between the FM- and AFM-conductivities in the\nlow temperature limit (see inset for \u001bvib\nxxin Fig. 1(b)).\nIn this case the relaxation time \u001cis still long owing to\nthe low level of both lattice vibrations and spin \ructu-\nations which determines the scattering potential Vscatt.\nFor both magnetic states the decrease of the conductiv-ity with rising temperature is caused by the increase of\nscattering processes and consequent decrease of the re-\nlaxation time. At the same time, the conductivity di\u000ber-\nence, \u0001\u001b(T) =\u001bvib;FM\nxx (T)\u0000\u001bvib;AFM\nxx (T), reduces with\nincrease in temperature. This e\u000bect can partially be at-\ntributed to the temperature dependent changes of the\nelectronic structure (disorder smearing of the electronic\nstates) re\rected by changes in the density of states at\nthe Fermi level [28] (see Fig. 2). Despite this, up to the\ntransition temperature, T=Tm, the di\u000berence \u0001 \u001b(T) is\nrather pronounced leading to a signi\fcant change of the\nresistivity at T=Tm.\n0.0 0.1 0.2 0.3 0.4 0.50.00.10.20.30.40.5\n(a)\n0.0 0.1 0.2 0.3 0.4 0.50.00.10.20.30.40.5\n(b)\n0.0 0.1 0.2 0.3 0.4 0.50.00.10.20.30.40.5\n(c)\nFIG. 3. (a) Bloch spectral function of FeRh calculated for\nthe AFM state at T= 300 K (a) and for the FM state re-\nsolved into majority spin (b) and minority spin (c) electron\ncomponents, calculated for T= 320 K. The \fnite width of\nthis features determine the electronic mean free paths.4\nOne has to stress that in calculating the contribution\nof spin moment \ructuations to the resistivity, the dif-\nferent temperature dependent behavior of the magnetic\norder in the FM and AFM states must be taken into ac-\ncount. This means, that at the critical point, T=Tm,\nthe smaller sublattice magnetization in the AFM state\ndescribes a more pronounced magnetic disorder when\ncompared to the FM state which leads to both a smaller\nrelaxation time and shorter mean free path. The result\nis a higher resistivity in the AFM state.\nThe di\u000berent mean free path lengths in the FM and\nAFM states at a given temperature can be analyzed using\nthe Bloch spectral function (BSF), AB(~k;E) [15], calcu-\nlated forE=EF, since the electronic states at the Fermi\nlevel give the contribution to the electrical conductivity.\nFor a system with thermally induced spin \ructuations\nand lattice displacements the BSF has features with \f-\nnite width from which the mean free path length of the\nelectrons can be inferred. Fig. 3 shows an intensity con-\ntour plot for the BSF of FeRh averaged over local moment\ncon\fgurations appropriate for the FM and AFM states\njust above and just below the FM-AFM transition respec-\ntively. Fig. 3(a) shows the AFM Bloch spectral function\nwhereas Figs. 3(b) and (c) show the sharper features of\nthe spin-polarized BSF of the FM state especially for\nthe minority spin states. This implies a longer electronic\nmean free path in the FM state in comparison to that\nin the AFM state which is consistent with the drop in\nresistivity.\nIn particular concerning technical applications of\nFeRh, it is interesting to study further temperature de-\npendent response properties. In Fig. 4(a) we show our\ncalculations of the total anomalous Hall resistivity for\nFeRh in the FM state, represented by the o\u000b-diagonal\nterm\u001axyof the resistivity tensor and compare it with\nexperimental data [11]. As the FM state is unstable in\npure FeRh at low temperatures, the measurements were\nperformed for (Fe 0:965Ni0:035)Rh, for which the FM state\nhas been stabilized by Ni doping. The calculations have\nbeen performed both, for the pure FeRh compound as\nwell as for FeRh with 5% Ni doping, (Fe 0:95Ni0:05)Rh,\nwhich theory \fnds to be ferromagnetically ordered down\ntoT=0 K. As can be seen the magnitude of \u001axy(T)\nincreases in a more pronounced way for the undoped\nsystem. Nevertheless, both results are in a rather good\nagreement with experiment.\nIn addition to the temperature dependent transport\nproperties the inclusion of relativistic e\u000bects into the ab-\ninitio theory enables us to present results for the Gilbert\ndamping, which plays a crucial role for spin dynamics.\nWe have calculated this quantity taking into account all\ntemperature induced e\u000bects, i.e. spin \ructuations and\nlattice vibrations [32, 33]. As one can see in Fig. 4(b), the\ncalculated results are in rather good agreement with the\nexperimental value (shown by diamond) \u000b= 0:0012 ob-\ntained for a thick \flm at T= 420 K [31] as well as new ex-\n(a)\n(b)\nFIG. 4. (a) The temperature dependence of the anomalous\nHall resistivity for the FM state of (Fe 0:95Ni0:05)Rh in com-\nparison with experimental data [11]; (b) Gilbert damping pa-\nrameter as a function of temperature: theory accounting for\nall thermal contributions (squares) in comparison with the\nexperimental results for thick-\flm system (50 nm) [31] (open\ndiamond) and for FeRh thin \flm deposited on MgO(001) sur-\nface (up- and down-triangles). Up- and down-triangles rep-\nresent data for a heating and cooling cycles, respectively (for\ndetails see supplementary materials). The inset represents the\nresults for the individual sources for the Gilbert damping, i.e.,\nlattice vibrations (circles) and spin \ructuations (diamonds).\nThe total\u000bvalues calculated for FeRh crystal without (c)\nand with tetragonal (t) distortions ( c=a= 1:016) are shown\nby open and closed squares, respectively.\nperimental data for thin \flms [15]. The separate contri-\nbutions to the Gilbert damping due to spin \ructuations\nand lattice vibrations are presented in the inset to Fig.\n4(b) for a given temperature window again arti\fcially ex-\ntended to low temperatures. These results allow to iden-\ntify the leading role of lattice vibrations (circles in the\ninset to Fig. 4(b)) at high temperature region where the\nelectron spin-\rip interband transitions are most respon-\nsible for dissipation due to the magnetization dynamics.\nIn the low-temperature region, where the T-dependence\nof\u000bis determined by intraband spin-conserving scatter-\ning events, it stems dominantly from electron scattering\ndue to thermally induced spin-\ructuations (diamonds in\nthe inset to Fig. 4(b)).\nThe experimental data shown in Fig. 4(b)) by trian-\ngles represent results for rather thin \flms ( d= 25 nm)5\ndeposited on top of a MgO(001) substrate [15]. The FeRh\nunit cell with a lattice constantp\n2 times smaller than\nthat of MgO, is rotated around zaxis by 45owith respect\nto the MgO cell. Because of this, a compressive strain in\nthe FeRh \flm occurs. As it follows from the experimen-\ntal data [34], this implies a tetragonal distortion of the\nFM FeRh unit cell with c=a= 1:016. Results of corre-\nsponding calculations for \u000bare given in the inset of Fig.\n4(b) by full squares, demonstrating a rather weak e\u000bect\nof this distortion. The smaller value of \u000bcompared to\nexperiment, has therfore to be attributed to the use of\nbulk geometry instead of the experimental \flm geometry\nwith a corresponding impact on the damping parameter.\nIn summary, we have presented ab-initio calculations\nfor the \fnite temperature transport properties of the\nFeRh compound. A steep increase of the electric resis-\ntivity has been obtained for the AFM state leading to a\npronounced drop of resistivity at the AFM to FM transi-\ntion temperature. This e\u000bect can be attributed partially\nto the di\u000berence of the electronic structure of FeRh in the\nFM and AFM states, as well as to a faster increase of the\namplitude of spin \ructuations caused by temperature in\nthe AFM state. Further calculated temperature depen-\ndent response properties such as the AHE resistivity and\nthe Gilbert damping parameter for the FM system show\nalso good agreement with experimental data. This gives\nadditional con\fdence in the model used to account for\nthermal lattice vibrations and spin \ructuations.\nACKNOWLEDGEMENTS\nFinancial support by the DFG via SFB 689\n(Spinph anomene in reduzierten Dimensionen) and from\nthe EPSRC (UK) (Grant No. EP/J006750/1) is grate-\nfully acknowledged.\n[1] J. S. Kouvel and C. C. Hartelius, J. Appl. Phys. 33\n(1962).\n[2] N. Baranov and E. Barabanova, Journal of Alloys and\nCompounds 219, 139 (1995), eleventh international con-\nference on solid compounds of transition elements.\n[3] P. A. Algarabel, M. R. Ibarra, C. Marquina, A. del Moral,\nJ. Galibert, M. Iqbal, and S. Askenazy, Applied Physics\nLetters 66, 3061 (1995).\n[4] M. A. de Vries, M. Loving, A. P. Mihai, L. H. Lewis,\nD. Heiman, and C. H. Marrows, New Journal of Physics\n15, 013008 (2013).\n[5] P. H. L. Walter, Journal of Applied Physics 35, 938\n(1964).\n[6] J. S. Kouvel, Journal of Applied Physics 37, 1257 (1966).\n[7] R. Barua, F. Jimnez-Villacorta, and L. H. Lewis, Ap-\nplied Physics Letters 103, 102407 (2013).[8] W. Lu, N. T. Nam, and T. Suzuki, Journal of Applied\nPhysics 105, 07A904 (2009).\n[9] I. Suzuki, T. Naito, M. Itoh, T. Sato, and T. Taniyama,\nJournal of Applied Physics 109, 07C717 (2011).\n[10] M. Kn ulle, D. Ahlers, and G. Sch utz, Solid State Com-\nmun.94, 267 (1995).\n[11] Y. Kobayashi, K. Muta, and K. Asai, Journal of Physics:\nCondensed Matter 13, 3335 (2001).\n[12] J. Kudrnovsk\u0013 y, V. Drchal, and I. Turek, Phys. Rev. B\n91, 014435 (2015).\n[13] H. Ebert et al., The Munich SPR-KKR package , version\n6.3,\nH. Ebert et al.\nhttp://olymp.cup.uni-muenchen.de/ak/ebert/SPRKKR\n(2012).\n[14] H. Ebert, D. K odderitzsch, and J. Min\u0013 ar, Rep. Prog.\nPhys. 74, 096501 (2011).\n[15] see supplementary materials, .\n[16] P. St\u0014 reda, J. Phys. C: Solid State Phys. 15, L717 (1982).\n[17] S. Lowitzer, D. K odderitzsch, and H. Ebert, Phys. Rev.\nB82, 140402(R) (2010).\n[18] T. Naito, D. S. Hirashima, and H. Kontani, Phys. Rev.\nB81, 195111 (2010).\n[19] S. Lowitzer, M. Gradhand, D. K odderitzsch, D. V. Fe-\ndorov, I. Mertig, and H. Ebert, Phys. Rev. Lett. 106,\n056601 (2011).\n[20] I. Turek, J. Kudrnovsk\u0013 y, and V. Drchal, Phys. Rev. B\n86, 014405 (2012).\n[21] H. Ebert, S. Mankovsky, D. K odderitzsch, and\nP. J. Kelly, Phys. Rev. Lett. 107, 066603 (2011),\nhttp://arxiv.org/abs/1102.4551v1.\n[22] H. Ebert, S. Mankovsky, K. Chadova, S. Polesya,\nJ. Min\u0013 ar, and D. K odderitzsch, Phys. Rev. B 91, 165132\n(2015).\n[23] B. Velick\u0013 y, Phys. Rev. 184, 614 (1969).\n[24] W. H. Butler, Phys. Rev. B 31, 3260 (1985).\n[25] I. Turek, J. Kudrnovsk\u0013 y, V. Drchal, L. Szunyogh, and\nP. Weinberger, Phys. Rev. B 65, 125101 (2002).\n[26] B. L. Gyor\u000by, A. J. Pindor, J. Staunton, G. M. Stocks,\nand H. Winter, J. Phys. F: Met. Phys. 15, 1337 (1985).\n[27] S. Polesya, S. Mankovsky, D. K odderitzsch, J. Min\u0013 ar,\nand H. Ebert, Phys. Rev. B 93, 024423 (2016).\n[28] J. B. Staunton, M. Banerjee, dos Santos Dias, A. Deak,\nand L. Szunyogh, Phys. Rev. B 89, 054427 (2014).\n[29] N. F. Mott, Adv. Phys. 13, 325 (1964).\n[30] E. Y. Tsymbal, D. G. Pettifor, and S. Maekawa, \\Gi-\nant magnetoresistance: Theory,\" in Handbook of Spin\nTransport and Magnetism , edited by E. Y. Tsymbal and\nI. Zuti\u0013 c (Taylor and Francis Group, New York, 2012).\n[31] E. Mancini, F. Pressacco, M. Haertinger, E. E. Fullerton,\nT. Suzuki, G. Woltersdorf, and C. H. Back, Journal of\nPhysics D: Applied Physics 46, 245302 (2013).\n[32] S. Mankovsky, D. K odderitzsch, G. Woltersdorf, and\nH. Ebert, Phys. Rev. B 87, 014430 (2013).\n[33] H. Ebert, S. Mankovsky, K. Chadova, S. Polesya,\nJ. Min\u0013 ar, and D. K odderitzsch, Phys. Rev. B 91, 165132\n(2015).\n[34] C. Bordel, J. Juraszek, D. W. Cooke, C. Baldasseroni,\nS. Mankovsky, J. Min\u0013 ar, H. Ebert, S. Moyerman, E. E.\nFullerton, and F. Hellman, Phys. Rev. Lett. 109, 117201\n(2012)."    },    {        "title": "2007.04372v2.Finite_frequency_spin_susceptibility_and_spin_pumping_in_superconductors_with_spin_orbit_relaxation.pdf",        "content": "Finite-frequency spin susceptibility and spin pumping in superconductors with\nspin-orbit relaxation\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\n(Dated: July 24, 2020)\nStatic spin susceptibility of superconductors with spin-orbit relaxation has been calculated in the\nseminal work of A.A. Abrikosov and L.P. Gor'kov [Sov. Phys. JETP, 15, 752 (1962)]. Surprisingly\nthe generalization of this result to \fnite frequencies has not been done despite being quite important\nfor the modern topic of superconducting spintronics. The present paper \flls this gap by deriving\nthe analytical expression for spin susceptibility. The time-dependent spin response is shown to\nbe captured by the quasiclassical Eilenberger equation with collision integrals corresponding to\nthe ordinary and spin-orbit scattering. Using the developed formalism we study the linear spin\npumping e\u000bect between the ferromagnet and the adjacent superconducting \flm. The consequences\nfor understanding recent experiments demonstrating the modi\fcation of Gilbert damping by the\nsuperconducting correlations are discussed.\nI. INTRODUCTION\nSpin transport and spin dynamics in superconductors\nhave attracted signi\fcant attention recently1{7. Quite in-\nteresting experimental results have been obtained for the\nspin pumping e\u000bects which in general play the central\nrole in spintronics8{10. Ferromagnet/ superconductor\nmultilayers were found recently to demonstrate changes\nof the ferromagnetic resonance (FMR) frequency and\nlinewidth11{19due to the superconducting correlations.\nDespite signi\fcant e\u000borts theoretical understanding of\nthese e\u000bects is not complete yet. For example, puz-\nzling experimental result has been obtained for the fer-\nromagnetic insulator/superconductor multilayers where\nthe pronounced peaks in the temperature dependence of\nGilbert damping have been observed16.\nThe enhancement of Gilbert damping due to the\nmetal spin sink can be calculated using the linear re-\nsponse approximation20which involves the momentum\nand frequency-dependent spin susceptibility \u001fh(k;\n) of\nthe metal spin sink. Hence, to understand the mod-\ni\fcation of Gilbert damping due to the spin pumping\nin superconducting \flms it is necessary to the calculate\nthe corresponding function \u001fh(k;\n) in the presence of\nspin relaxation mechanisms like the spin-orbit scatter-\ning. Quite surprisingly, this calculation has not been\never performed correctly. Recent papers which have ad-\ndressed this topic in connection with spin pumping21,22\nreport \fnite zero-temperature dissipation at low frequen-\ncies: Im\u001fh(q;\n)=\n6= 0 at \n!0. This result contradicts\nphysical intuition because there can be no dissipation\nat \n<2\u0001 and in the absence of thermal quasiparti-\ncles which are frozen out in superconductors at T\u001c\u0001,\nwhere \u0001 is the superconducting energy gap. As we show\nbelow this inconsistency comes from neglecting the im-\nportant contributions while performing analytical contin-\nuation procedure.The \frst purpose of the present paper is to report the\nanalytical expression for the \fnite-frequency spin sus-\nceptibility of superconductors with spin-orbit relaxation\nmechanism. This result is a generalization of the classi-\ncal work of Abrikosov and Gor'kov23who have considered\nthe static spin susceptibility to explain the \fnite Knight\nshift in superconductors at T\u001c\u0001. We analyse di\u000berent\ncharacteristic regimes including large and strong spin re-\nlaxation as well as the behaviour for various values of the\nDynes parameter24.\nThe second purpose is to study the spin pumping in\nsuperconductor/ferromagnet systems in the framework\nof the interfacial exchange model20. The expressions for\nGilbert damping are derived for the \fnite thickness of\nthe spin sink layer. Also we consider the system with an\nadditional perfect spin absorber which can be realized\nexperimentally by adding the layer of material with very\nstrong spin relaxation. The derived general expressions\ncan be parametrized in terms of the dimensionless param-\neter characterizing the strength of the interfacial coupling\nbetween the ferromagnet and adjacent superconductor.\nSystems with elevated values of this parameter are pre-\ndicted to feature pronounced shift of the ferromagnetic\nresonance line induced by superconducting correlations.\nII. GENERAL FORMALISM\nA. Diagrammatic formalism\nWe describe the interaction of electrons with Zeeman\n\feldh=h(r;t) using the following Hamiltonian\n^VP=^\u001bh (1)\nwhere ^\u001b= (^\u001bx;^\u001by;^\u001bz) is the vector of spin Pauli matri-\nces. Besides that we assume the presence of disorder\ndescribed by the Gaussian impurity potential. It hasarXiv:2007.04372v2  [cond-mat.supr-con]  23 Jul 20202\n(a)\n(b)\nFIG. 1. (Color online) (a) Bubble diagram for the linear\nresponse of spin polarization generated by the time-dependent\nZeeman \feldh\nei\nt+iqrshown by the wavy line. Circles show\nspin vertices ^\u001b. The shaded region shows impurity ladder.\n(b) Diagrammatic equation for the impurity ladder. The blue\nand red dashed lines correspond to the ordinary and spin-\norbit scattering potentials averaged over the random impurity\ncon\fguration.\nboth the usual Vimpand the spin-orbit Vsoscattering\namplitudes\n^V(p;p0) = (2)\nu0X\nroeiro(p\u0000p0)+uso\np2\nF^\u001b\u0001(p\u0002p0)X\nrsoeirso(p\u0000p0);\nwhereroandrsodenote the random impurity coordi-\nnates corresponding to the ordinary and spin-orbit scat-\ntering respectively. We assume this coordinates to be\nindependent and thus neglect the magnetoelectric ef-\nfects arising from the combined ordinary and spin-orbit\nscattering25.\nThe spin polarization as a function of the imaginary\ntimet2[0;\f] where\f= 1=Tis given by\nS(t;r) =1\n4Tr[^\u001b^G](r;r;t1;2=t) (3)\nwhere ^G(r1;r2;t1;2) is the imaginary time Green's func-\ntion (GF). The stationary propagators depend only on\nthe relative time and coordinate. In the frequency and\nmomentum representation they are given by23,26\n^G0(!;p) =~\u0001^\u001c2\u0000i~!^\u001c0+\u0018p^\u001c3\n~\u00012+ ~!2+\u00182p(4)\n~!=!~s(!)\ns(!);~\u0001 = \u0001~s(!)\ns(!); (5)\nwhere\u0018p=p2=2m\u0000\u0016is the deviation of the kinetic en-\nergy from the chemical potential \u0016and ^\u001c1;2;3are the Pauli\nmatrices in Nambu space. We denote s=p\n!2+ \u00012and\n~s=s+1=2\u001cimpwhere the scattering time is given by the\nsuperposition \u001c\u00001\nimp=\u001c\u00001\no+\u001c\u00001\nso. We denote the usual\u001c\u00001\no= 2\u0019n\u0017u 0and spin-orbit \u001c\u00001\nso= 2\u0019n\u0017uso=3 scat-\ntering rates. The propagator (4) is averaged over the\nrandomly disordered point scatterers con\fgurations.\nWe are interested in the spin polarization induced by\nthe external Zeeman \feld h(t;r) =h\nei\nt+iqr. The in-\nduced spin polarization as given by the diagram shown\nin Fig.1a can be written as follows\nS\n=\u001fh(\n;q)h\n (6)\nThe linear spin susceptibility is de\fned by substituting\ninto the Eq.3 ^Ghwhich is the \frst-order correction to\nthe GF induced by the Zeeman \feld. The diagrammatic\nequation for this correction which includes the summa-\ntion of impurity ladder corrections is shown Fig.1b. The\nshaded region denotes impurity ladder corresponding to\nthe ordinary and spin-orbit impurity scattering averaged\nover the random disorder con\fguration. The red and\nblue dashed lines correspond to the spin-orbit and or-\ndinary impurity scattering potentials averaged over the\nrandomly distributed impurities. Analytical expression\nfor the diagrammatic equation in Fig.1b reads as follows\n^Gh(12) =\u0000^G0(1)^\u001bh\n^G0(2)+ (7)\n^G0(1)^\u001bh^ghi^\u001b^\u001c3\n6i\u001cso^G0(2) + ^G0(1)h^ghi^\u001c3\n2i\u001co^G0(2)\nwhere we have introduced the notation\n^gh=i\n\u0019 \nd\u0018p^\u001c3^Gh: (8)\nWe use the condensed notation ^G0(2) = ^G0(!;p) and\n^G0(1) = ^G0(!\u0000\n;p+q). The correction depends on the\ntwo frequencies and momenta ^Gh(12) = ^Gh(!1;p;!2;p+\nq). The angular brackets denote average over the mo-\nmentum directions on the Fermi sphere so that in total\nh^ghi= (i=\u0019\u0017)\u0001\nd3p^Gh, where\u0017is the density of states at\nthe Fermi level. Diagrammatically the equation for impu-\nrity ladder (7) is shown in Fig.1b. The second and third\nterms in Eq.7 corresponding to the spin-orbit and ordi-\nnary scattering are shown by blue and red dashed lines,\nrespectively. As we see below the momentum-integrated\ncorrection ^Ghcoincides with the solution of quasiclassi-\ncal Eilenberger equation27with collision integrals corre-\nsponding to the ordinary and spin -orbit scattering28.\nB. Quasiclassical formalism\nUnder quite general conditions the non-equilibrium\nstate of a metal involves perturbations of spectrum and\ndistribution function in the vicinity of the Fermi level.\nFor that the external \felds should have frequencies much\nsmaller than the Fermi energy and spatial scales much\nlarger than the Fermi wave length. Both these require-\nments are satis\fed for the spin pumping systems. Hence3\nwe can use the theory formulated in terms of the quasi-\nclassical propagator27\n^g(r;np;t;t0) =i\n\u0019 \nd\u0018p^\u001c3^G: (9)\nThe calculation can be performed either using the\nimaginary time formalism of the real-time formalism. In\nthe imaginary time domain the quasiclassical propagator\nis determined by the Eilenberger equation with collision\nintegrals describing the impurity scattering27\n(vFr)^g\u0000if^\u001c3@t;^ggt=i[^\u001c3^H;^g]t+ [(^\u0006o+^\u0006so)\u000e;^g]t\n(10)\n^\u0006so= (^\u001bh^gi^\u001b)=6\u001cso (11)\n^\u0006o=h^gi=2\u001co: (12)\nHere ^\u0006oand ^\u0006soare the self-energies corresponding to\nthe ordinary and spin-orbit scattering, respectively29and\n^H= \u0001^\u001c2+^\u001bh. We denote the commutators [ X;g]t=\nX(t1)g(t1;t2)\u0000g(t1;t2)X(t2) and the convolution h^gi\u000e\n^g=\u0001\f\n0dth^gi(t1;t)^g(t;t2). The angle-averaging over the\nFermi surface is given by hgi. The spin polarization is\ngiven by\nS(t;r) =\u0000i\u0019\u0017\n4Tr[^\u001c3^\u001bh^g(t;t;r)i] (13)\nThe quasiclassical equations are supplemented by the\nnormalization condition ^ g\u000e^g= 1.\nC. Analytical continuation\nIn order to \fnd the real-frequency response we need to\nimplement the analytic continuation of Eq. (13). The\n\frst-order correction to the quasiclassical GF can be\nwritten as ^gh(t1;t2) =TP\n!e\u0000i!1t1+i!2t2g(!1;!2) where\n!2=!and!1=!\u0000\n are the fermionic Matsubara fre-\nquencies shifted by the Bosonic frequency \n of the exter-\nnal Zeeman \feld. The analytic continuation of the sum\nis determined according to the general rule30\nTX\n!gh(!1;!2)! (14)\n\u0002d\"\n4\u0019in0(\"1)\u0002\ngh(\u0000i\"R\n1;\u0000i\"A\n2)\u0000gh(\u0000i\"A\n1;\u0000i\"A\n2)\u0003\n+\n\u0002d\"\n4\u0019in0(\"2)\u0002\ngh(\u0000i\"R\n1;\u0000i\"R\n2)\u0000gh(\u0000i\"R\n1;\u0000i\"A\n2)\u0003\nwheren0(\") = tanh(\"=2T) is the equilibrium distribution\nfunction. In the r.h.s. of (14) we substitute \"1=\"\u0000\n,\n\"2=\"and\"R=\"+i\u0000,\"A=\"\u0000i\u0000. Here the term with\n\u0000>0 is added to shift the integration contour into the\ncorresponding half-plane. At the same time, \u0000 can be\nused as the Dynes parameter31to describe the e\u000bect of\ndi\u000berent depairing mechanisms on spectral functions in\nthe superconductor. We implement the analytical contin-\nuation in such a way that s(\u0000i\"R;A) =\u0000ip\n(\"R;A)2\u0000\u00012assuming that the branch cuts run from (\u0001 ;1) and\n(\u00001;\u0000\u0001).\nEquilibrium GF in the imaginary frequency domain\nis given by ^ g0(!) = (^\u001c3!+ ^\u001c1\u0001)=s(!). The real-\nfrequency continuation reads ^ gR;A\n0(\") = (^\u001c3\"R;A+\ni^\u001c1\u0001)=p\n(\"R;A)2\u0000\u00012.\nThus the linear response spin polarization is given by\n\u001fh+ 1 = (15)\u0002d\"\n4\u0019i\u001f(\u0000i\"R\n1;\u0000i\"A\n2) [n0(\"1)\u0000n0(\"2)] +\n\u0002d\"\n4\u0019i\u0002\nn0(\"2)\u001f(\u0000i\"R\n1;\u0000i\"R\n2)\u0000n0(\"1)\u001f(\u0000i\"A\n1;\u0000i\"A\n2)\u0003\nwhere we denote \u001f(!1;!2) = (\u000e=\u000eh)Tr[\u001b^gh(!1;!2)]. In\nthe l.h.s. of Eq. 15 we subtract the o\u000b-shell contribution\nto the spin polarization due to the band edge shift by the\nZeeman \feld.\nIt is interesting to note that in the superconducting\nstate both the \frst and the second terms in the r.h.s. of\n(15) contribute to the dissipative part of spin suscepti-\nbility With that we obtain physically correct behaviour\nin the low-temperature limit Im \u001fh(\n)=\n!0 atT!0\nand small frequency \n \u001c\u0001. This is in contrast to pre-\nvious calculations21,22which take into account only the\n\frst term in (15) and obtained physically incorrect \fnite\ndissipation in the absence of quasiparticles at T= 0.\nIII. SPIN SUSCEPTIBILITY\nA. Diagram summation\nFirst, we demonstrate connection between response\nfunctions determined by the diagram Fig.1a and by the\nsolution of time-dependent Eilenberger equation (10). In-\nstead of using the usual approach of calculating the ver-\ntex function23we use the alternative route and solve di-\nrectly the equation for the \frst-order correction 7.\nWe use the general approach suggested recently32for\nderiving equation for the momentum-integrated propaga-\ntors ^ghstarting from the general equation for the exact\nGF (7). The key idea of this derivation is based on the\nfollowing trick. Let us multiply the function ^Gh(12) by\n^G\u00001\n0(1) from the left and by ^G\u00001\n0(2) from the right, sub-\ntract the results and integrate by \u0018p. We use that Eq.(4)\nyields the relations ^G\u00001\n0(j) =~\u0001j^\u001c2+i~!j^\u001c0+\u0018p(pj)^\u001c3and\n~\u0001j^\u001c2+i~!j^\u001c0=i(sj+ 1=2\u001cimp)^g0(!j)^\u001c3. Then we elim-\ninate o\u000b-shell contributions in the momentum integrals\nto express the result through quasiclassical propagators\n\u0002d\u0018p\n\u0019h\n^G\u00001\n0(1)^Gh\u0000^\u001c3^Gh^G\u00001\n0(2)^\u001c3i\n= (16)\n~s1^g0(1)^gh\u0000~s2^gh^g0(2)\u0000i(vFq)^gh\nNext let us derive the l.h.s. of the equation for ^ gh.\nUsing the diagram Fig.1b or the Eq.7 we get that4\n^G\u00001\n0(1)^Gh\u0000^\u001c3^Gh^G\u00001\n0(2)^\u001c3= (17)\n^\u001c3^G0(1)(^h\n+ih^ghi^\u001c3=2\u001co+i\u001bh^ghi\u001b^\u001c3=6\u001cso)^\u001c3\n\u0000(^h\n+ih^ghi^\u001c3=2\u001co+i\u001bh^ghi\u001b^\u001c3=6\u001cso)^G0(2)+\nwhere we denote ^h\n=^\u001bh\n. Then combining with Eq.16\nwe obtain the following equation with collision integrals\n^Ioand^Iso\ns1^g0(1)^gh\u0000s2^gh^g0(2)\u0000i(vFq)^gh (18)\n=\u0000i[^g0(1)^h\n^\u001c3\u0000^h\n^\u001c3^g0(2)] + ^Iso+^Io\n^Io= [^g0(1)h^ghi+h^ghi^g0(2)\u0000 (19)\nh^ghi^g0(2)\u0000^g0(1)^gh]=2\u001co\n^Iso= [^g0(1)\u001bh^ghi\u001b+ 3h^ghi^g0(2)\u0000 (20)\n\u001bh^ghi\u001b^g0(2)\u00003^g0(1)h^ghi]=6\u001cso\nThis Eq.(18) coincides with the Eilenberger Eq. (10) ex-\npanded for the \frst-order correction ^ gh. This proves that\nthe time-dependent spin response in metals is captured\nby the Eilenberger equation with corresponding collision\nintegrals.\nB. Susceptibility of the spatially homogeneous\nsystem\nFirst, we consider the spatially homogeneous system\nwhen the Zeeman \feld depends only on time and not\non the spatial coordinate so that q= 0. The spatial\ndispersion of susceptibility is discussed in in the di\u000busive\nlimit in Sec.III C. In the homogeneous case the ordinary\nscattering drops out from Eq.18 since ^Io= 0. Then Eq.18\ncan be solved analytically yielding the frequency-resolved\nsusceptibility \u001f(12) = (\u000e=\u000eh)Tr[\u001b^gh(12)]\n\u001f(12) =\u00012+s1s2\u0000!1!2\ns1s2(s1+s2+ 4=3\u001cso); (21)\nwhere!are fermionic Matsubara frequencies, !1=!\u0000\n,\n!2=!,s1;2=q\n!2\n1;2+ \u00012. Substituting this expression\nto the analytical continuation rule (15) we obtain the\nfrequency dependent spin susceptibility \u001fh=\u001fh(\n). It\nis interesting to note that this response function (21) is\nidentical to that which determines the \fnite-frequency\nconductivity of a superconductor.\nWe can obtain analytical results in several important\nlimiting cases. For the (i) normal metal \u0001 = 0\nEqs.(21,15) yield (see detailed calculation in Appendix\nSec.B)\n\u001fh(\n) =2(2=3\u001cso+ \u0000)\n2(2=3\u001cso+ \u0000)\u0000i\n(22)In this case the only contribution is provided by the \frst\nterm in Eq.15. As one can in the absence of spin re-\nlaxation \n\u001cso! 1 and \u0000!0 the susceptibility is\nvanishes. Physically this result is quite transparent be-\ncause without relaxation the spin projection on the os-\ncillating Zeeman \feld remains a good quantum number.\nLet us check that this result remains valid in the su-\nperconducting state. For that we consider the limit of\n(ii) superconductor without spin relaxation . In\nthis case using following relations s2\n1\u0000s2\n2=!2\n1\u0000!2\n2and\n2(!1!2\u0000\u00012\u0000s1s2) = (!1+!2)2\u0000(s1+s2)2Eq.21 can\nbe simpli\fed as follows, see details in Appendix A\n\u001f(12) =2\n\n\u0012!1\ns1\u0000!2\ns2\u0013\n(23)\nThus making the analytical continuation and neglecting\nterms of the order \u0000 =\n we obtain\n\u001fh(\n) =\u00001\u0000\u00021\n\u00001d\"\n2\n[N(\"1)n0(\"1)\u0000N(\"2)n0(\"2)]\n(24)\nwhereN(\") is the normalized DOS, \"1=\"\u0000\n and\"2=\n\". One can see that this expression yields \u001fh(\n) = 0\nirrespective of the particular energy dependence of DOS.\nThis result can be qualitatively explained by the fact that\nin the absence of spin relaxation spin projection on the\noscillating Zeeman \feld axis is a conserved quantity.\nForm this limiting case one can clearly see that to ob-\ntain the correct result it is necessary to take into ac-\ncount all parts in the Eq.15. Indeed, the contribution of\nthe \frst term in Eq.15 is proportional to\u0001\nd\"[^gR\n0(1)\u0000\n^gA\n0(2)]@\"n0\u00192\n=\u0001 at low temperatures. This contri-\nbution is cancelled by the second term in Eq.14 to yield\n\u001fh(\n) = 0 for \u001c\u00001\nso= 0.\nAs we have obtained in the normal metal limit, the\ncontribution of the \frst term in spin susceptibility (15)\nis of the order \n \u001csfor weak spin relaxation \n \u001cs\u001d1.\nThus when \u001cso\u0001\u001d1 the contribution of second term can\nbe neglected. For stronger spin-orbit relaxation such an\napproximation which has been used in as it has been done\nin previous works21,22is inaccurate. Below we con\frm\nthis conclusion by evaluation Eq.15 numerically.\nLet us now considered the opposite limit of (iii) su-\nperconductor with strong spin relaxation \u001cso\u0001\u001c1\nand small frequencies \n \u001c\u0001. In this case from the gen-\neral Eq.21 we obtain\n\u001f(12) =3\u001cso\n4\u0012\u00012\u0000!1!2\ns1s2+ 1\u0013\n; (25)\nSubstituting this expression into the analytical continu-\nation rule (15) after some algebra we get\n4\n3\u001csoIm\u001fh\n\n=\u00021\n\u00001d\"\u0000\n\u00012=\"2+ 1\u0001\nN2@\"n0 (26)\nFrom this expression one can see analytically that the\ndissipative part of the susceptibility vanishes in the zero-\ntemperature limit.5\n(a)\u001csoTc= 100\n (b)\u001csoTc= 10\n (c)\u001csoTc= 1\n (d)\u001csoTc= 0:1\nFIG. 2. Comparison of the contributions to the dissipative spin response Im \u001fhgiven by the both terms in Eq.15 (solid blue\nlines) and only the \frst term in Eq.15 (red dashed lines). The parameters are \u0000 = 0 :001Tc, \n = 0:01Tcand spin-orbit scattering\ntime\u001csoTcis (a)100, (b) 10, (c) 1, (d) 0 :1.\n(a)\u001csoTc= 100\n\u0000=Tc\n(b)\u001csoTc= 10\n (c)\u001csoTc= 1\n (d)\u001csoTc= 0:1\nFIG. 3. Temperature dependencies of the dissipative part of spin susceptibility Im \u001fhat small frequency \n = 0 :01Tc. In each\npanel curves from top to bottom correspond to the Dynes parameter values \u0000 =Tc= 0:0001; 0:01; 0:1. The spin-orbit scattering\ntime\u001csoTcis (a)100, (b) 10, (c) 1, (d) 0 :1.\nGeneral case. Now let us consider the behaviour\nof spin susceptibility in the wide range of parameters by\nevaluating numerically the integral in Eq.15. First, we\ncompare the results given by the full Eq.15 with the con-\ntribution of only the \frst term. The sequence of plots\nin Fig.2 show temperature dependence of the dissipative\npart Im\u001fhat \n = 0:01Tc, Dynes parameter \u0000 = 0 :0001Tc\nand several values of the spin-orbit scattering rate. The\ndependencies given by the full Eq.15 are shown by the\nblue solid curves while the dependencies given only by\nthe \frst term in Eq.15 are shown by the red dashed\ncurves. One can see that for weak spin-orbit scatter-\ning\u001csoTc\u001d1 these curves coincide, according to the\nconclusion we have made based on the analysis of lim-\niting cases above. However, there is a large discrepancy\nfor stronger spin-orbit relaxation \u001csoTc<1. Note that\nthe behaviour of dashed curves is similar to that which\nhas been obtained for the dissipation signal in previous\nworks21. That is, at \u001csoTc<1 they signi\fcantly deviate\nfrom zero at T!0. As we have noted, the \fnite value of\nIm\u001fhat in the low-temperature limit is physically incor-rect. On the other hand, the solid curves always demon-\nstrate the correct behaviour going to zero in the limit\nT!0. Thus, the numerical analysis also con\frms that\nboth terms in the Eq.15 contribute to the dissipative part\nof the spin response in the superconducting state.\nNext, let us consider how the temperature dependen-\ncies of Im\u001fhat \n = 0:01Tcchange with the Dynes pa-\nrameter. The sequence of plots for the three values of\n\u0000=Tc= 0:001; 0:01; 1 is shown in Fig.3 for di\u000berent val-\nues of the spin-orbit relaxation rate. One feature demon-\nstrated by these curves is that the peak in the temper-\nature dependencies becomes less pronounced and disap-\npears for weak spin relaxation. At the same time there\nrelative hight of the peak almost does not change between\nstrong\u001csoTc= 1 (Fig.3c) and very strong \u001csoTc= 0:1\n(Fig.3d) spin relaxation. Besides that, one can see that\nthe height of the peak is strongly suppressed by increas-\ning Dynes parameter. For the realistic value in the super-\nconductor NbN \u0000 = 0 :1Tcthe relative hight of the peak\nis about 0:2\u00000:5 of the normal metal value at T > Tc.\nThis increase is by the order of magnitude weaker than6\n(a) \u0000 = 0:1Tc\n\u001csoTc\n(b) \u0000 = 0:01Tc\nFIG. 4. Temperature dependencies of the dissipative part\nof spin susceptibility at \n = 0 :01Tcand di\u000berent values of\nthe Dynes parameter (a) \u0000 = 0 :1Tc; (b) \u0000 = 0 :01Tc. Curves\nfrom top to bottom in each panel correspond to the spin-orbit\nscattering times \u001csoTc= 0:1; 1; 5; 10.\nthe relative peak heights of 2 \u00003 observed in spin pump-\ning experiment in GdN/NbN bilayers16. Therefore one\ncan assume that there should be a di\u000berent explanation\nof the this experiment rather than the peaked behaviour\nof spin susceptibility21.\nNow let us consider the behaviour of spin susceptibility\nat larger frequencies comparable with superconducting\nenergy scales \n\u0018Tc. In this case it is interesting to con-\nsider both the dissipative and the non-dissipative parts\nof spin susceptibility. As we show below they are respon-\nsible for the damping and \feld-like spin torque contri-\nbutions to the spin dynamics. In Fig.5 we plot the rele-\nvant quantities Im \u001fh(\n)=\n which contributes to the ex-\ncess Gilbert damping and Re \u001fh(\n)\u0000Re\u001fh(0) which con-\ntributes to the shift of the ferromagnetic resonance cen-\ntral frequency. First, we notice that the non-monotonic\ntemperature dependence of the dissipative part (left pan-\nels in Fig.) disappear at the frequencies much larger\nthan the Dynes parameter \n \u001d\u0000. For such frequencies\nIm\u001fhmonotonically decreases with temperature and \f-\nnally disappears at T!0 provided that \n <2\u0001. For\n\n>2\u0001 there a non-zero signal even at T= 0 due to the\nexcitation of quasiparticles across the gap.\nC. Spatial dispersion of the susceptibility\nIn general, due to the presence of anisotropic term in\nEq.18 the analytical solution is not possible for q6= 0.\nHowever, we can still get the analytical solution in the\nexperimentally relevant di\u000busive limit when the ordinary\nscattering rate is very large ( Tc\u001co)\u00001\u001d1. In this case\nEq. 10 can be simpli\fed by averaging over momentum di-\nrections. The isotropic part of the GF satis\fes Keldysh-\nUsadel equation\n\u0000if^\u001c3@t;\u0014ggt+Dr(\u0014g\u000er\u0014g) =i[^\u001c3^H;\u0014g]t+[\u0014\u0006so\u000e;\u0014g]t(27)\nwhereD=\u001cov2\nF=3 is the di\u000busion coe\u000ecient.\n(a) \u0000 = 0:1Tc\n(b) \u0000 = 0:001tc\n(b) \u0000 = 0:001Tc\nFIG. 5. Imaginary (left row) and real (right row) parts of the\nspin susceptibility as functions of Tand \n, normalized to the\nzero-temperature gap \u0001( T= 0). The Dynes parameters are\n(a) \u0000 = 0:1Tcand (b) \u0000 = 0 :001Tc. The spin-orbit scattering\ntime is\u001csoTc= 1.\nThe spin response to the spatially-inhomogeneous Zee-\nman \feldh\nei\nt+iqzcan be calculated analytically in\nthe di\u000busive limit using Usadel Eq.27. Using the imag-\ninary time representation and searching the solution in\nthe form ^gh(12)eiqzei(!1t1\u0000!2t2)we obtain the linearized\nUsadel equation\n(s1+Dq2)^g0(1)^gh\u0000s2^gh^g0(2) = (28)\ni(h\n^\u001b)[^g0(1)^\u001c3\u0000^\u001c3^g0(2)]\nThe solution of this equation yields susceptibility in the\nform (21) with the substitution of e\u000bective spin relax-\nation time 4 =3\u001cso!4=3\u001cso+Dq2\n\u001f(12) =\u00012+s1s2\u0000!1!2\ns1s2(s1+s2+Dq2+ 4=3\u001cso)(29)\nThis expression together with Eq.15 can be used to\nstudy various phenomena related to the spin dynamics\nin superconductors with spin-orbit relaxation. For exam-\nple, it is possible to study the e\u000bect of spin relaxation on\nthe nuclear magnetic resonance33,34and electron param-\nagnetic resonance35in superconductors. It is interesting\nthat the peak in spin relaxation observed in these exper-\niments is robust against even the very strong spin-orbit\nscattering as it follows from Fig.3d and Fig.4. In the limit\nof weak spin relaxation there is no peak, i.e. the temper-\nature dependence is monotonous as shown in Fig.2a and\n3a.7\nD. Keldysh formalism and kinetic equations\nIn the general case the procedure of analytical continu-\nation is not possible and one has to consider the real time\nequations from the very start. This brings extra compli-\ncation related to the matrix structure of the contour-\nordered propagator \u0014 g=\u0012\n^gR^gK\n0 ^gA\u0013\nhaving the spectral\nretarded (advanced) ^ gR(A)and the Keldysh component\n^gK. The matrix GF satis\fes Keldysh-Usadel equation\nwhich is formally identical to the Eq.10 or 27 with the\nsubstitution @t!\u0000i@t. Using the normalization con-\ndition ^gR\u000e^gK+ ^gK\u000e^gA= 0 one can introduce the\nparametrization of the Keldysh component in terms of\nthe distribution function ^ gK= ^gR\u000e^f\u0000^f\u000e^gA. Local\nspin density given by\nS(t) =\u0000\u0019\u0017\n4Tr[^\u001b^\u001c3^gK(t;t)] (30)\nThe driven state of superconductor is described by the\ndeviation of the Keldysh function from equilibrium which\nconsists of the parts with perturbations of spectral func-\ntions\u000e^gR;Aand the non-equilibrium part of distribution\nfunction\u000e^f. In the linear response regime one can write\n\u000e^gK(12) = (31)\n[^gR\n0(1)\u0000^gA\n0(2)]\u000e^f+\u000egR(12)n0(2)\u0000\u000egA(12)n0(1)\nComparing expressions (31,37) with 15 one can see that\nthe \frst term here yields the \frst term in the r.h.s. of\nEq. 15 and ^f/n(\"1)\u0000n(\"2).\nIn the low-frequency limit one can calculate the correc-\ntions to distribution function using the kinetic equation2\nwith the driving term obtained from the gradient expan-\nsion of the mixed product in the analytical continuation\nof Eq.27\n[^H;^f]t=i^\u001b@th@\"n0 (32)\nParametrizing the spin-dependent distribution function\nas^f=^\u001bfwe get the kinetic equation which for the\nspatially homogeneous system is given by\n@tf+ (2\u0000 +\u001c\u00001\ns)f=@\"n0@th (33)\n\u001c\u00001\ns= (1=3\u001cso)N\u00001Tr(1\u0000^gR^gA) (34)\nwhereN= ReTr[^\u001c3^gR]=2 is the normalized density of\nstates. At the subgap energies j\"j<\u0001 the spin relaxation\nrate (34) is not de\fned if the density of states is strictly\nzeroN= 0. However, for the \fnite Dynes parameter\nN/\u0000 so that\u001c\u00001\ns/\u0000. The solution of the Eq.33 yields\nthe contribution to the spin density\n\u001fkin+ 1 =\n2\u00021\n\u00001d\"N@\"n0\n\n\u0000i(2\u0000 +\u001c\u00001s); (35)\nwhich coincides with the \frst term in Eq.(15) in the low-\nfrequency limit.IV. SPIN PUMPING IN SUPERCONDUCTING\nFILMS\n(a)\n(b)\nFIG. 6. (Color online) Schematic setup with the interface\nbetween metallic spin sink (M) and ferromagnetic \flm (F) of\nthe widths dManddF, respectively. The constant external\nmagnetic \feld is H0x. The magnetization precession m\nei\nt\nis driven by the external magnetic \feld H\nei\nty. It generates\nspin currenti\npumped from F to M. (a) M has interface with\nvacuum; (b) M has interface with the perfect spin absorber.\nWith the general expression for spin susceptibility in\nhand we can study e\u000bects of spin pumping from the fer-\nromagnet into the adjacent metallic \flm. The schematic\nsetups are shown in Fig.6. The metallic spin sink M has\nan interface with (a) vacuum and (b) perfect spin ab-\nsorber. The correposnding boundary conditions are (a)\nvanishing spin current and (b) vanishing non-equilibrium\nspin polarization at z=dM. To quantify the spin pump-\ning e\u000bect we consider the interfacial exchange interaction\nbetween the localized spins in F and conduction elections\nin M. Within this model the local spin polarization close\nto the interface S(t) acts as e\u000bective \feld for the local-\nized magnetic moments. This process can be taken into\naccount by introducing the additional term i(t) into the\nLandau-Lifshitz-Gilber equation\n(1 +\u000bm\u0002)@tm+\rm\u0002Heff=i=SF0dF (36)\ni(t) =JsdS(t)\u0002m(t) (37)\nHereSF0is the equilibrium spin density in F, dFis the\nFI \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 F and M.\nTo calculateS(t) we use the spin susceptibility (6) with\nthe Zeeman \feld determined by the interfacial exchange8\nh=Jsdm\u000e(z). In the linear regime the local spin polar-\nization near F interface can be written as follows\nS\n=\u0017heff\u001fmm\n (38)\n\u001fm(\n) =1X\nn=0\u001fh(qn;\n) (39)\nwhere we introduce the e\u000bective exchange \feld heff=\nJsd=dMand the local spin susceptibility \u001fMwhich deter-\nmines the response to the delta-functional Zeeman \feld.\nThe summation in Eq.39 runs over the discrete set of\nmomenta given by qn=n\u0019=dMfor the vacuum interface\nFig.(6a) which is determined by the zero boundary con-\ndition for the spin current at the interface with vacuum\nz=dM. For the strong spin sink interface Fig.6b we\nhaveqn= (n+ 1=2)\u0019=dMwhich is determined by the\nzero boundary of the non-equilibrium spin polarization\nwhich is suppressed by the strong spin sink at z=dM.\nDerivation of this result is given in Appendix D.\nTaking into account the Eq.29 one can see that the\nonly di\u000berence introduced by the spin absorber Fig.6b is\nthe modi\fcation of spin relaxation rate to \u001c\u00001\nso!\u001c\u00001\nso+\nD(pi=2dM)2. Therefore hereafter we will not distinguish\nthese two cases implying that the e\u000bective spin relaxation\nis used.\nThe Fourier components of the spin current (37) is\ngiven by\ni(\n) =\u0017h2\neffdM[\u001fm(\n)\u0000\u001fm(0)]m\u0002m\n (40)\nFor the con\fguration in Fig.6 the e\u000bective \feld is given\nbyHeff=H\nei\nty+B0xwhereB0=H0+ 4\u0019M. In\nthis case the eigen frequencies of LLG Eq.36 satisfy the\nequation\n\n =p\n(\rB0+\u000e!)(\rH0+\u000e!) (41)\n\u000e!=i\n\u000b+ [\u001fm(\n)\u0000\u001fm(0)]TcC (42)\nC=heff\nTc\u0017heff\nSF0dM\ndF(43)\nThe extra dissipation, that is the imaginary part of\n\n in (41) can be considered resulting from the e\u000bective\nGilbert damping constant increase\n\u000e\u000b=CTcIm\u001fm=\n (44)\nIn case if the \flm thickness is small dM< min (lso;\u0018)\nwherelso=pD\u001csois spin relaxation length and \u0018= p\nD=Tcis the zero-temperature coherence length , only\nthe contribution with n= 0 in the sum (39) is impor-\ntant. In this case the spin pumping e\u000bect is totally deter-\nmined by the homogeneous spin-orbit relaxation so that\n\u001fm(\n)\u0019\u001fh(\n;q= 0). For larger \flm thickness we need\nto take into account several terms in Eq.(39). Only for\nthe very large thickness dM\u001dmin(lso;\u0018) the expression\nused in previous works20,21is recovered in the form\n\u000e\u000b=\u0017J2\nsd\ndFSF0\u00021\n\u00001dq\n\u0019Im\u001fh(q;\n)\n\n: (45)Temperature dependencies of the normalized excess\nGilbert damping are shown in Fig.5. One can see that\nthese dependencies are qualitatively similar to that ob-\ntained in the absence of spin relaxation for in\fnite super-\nconducting \flms36. They are also qualitatively similar\nto the temperature dependencies of the NMR33,34and\nEPR37,38linewidths in superconductors. Note that for\nrelatively large Dynes parameter \u0000 = 0 :1Tcthe peak in\nthe temperature dependencies of Gilbert damping is al-\nmost absent (red curves in Fig.7) and superconductivity\nleads to the monotonous suppression of the spin pumping\ndissipative signal. This result reproduces theoretically\nthe behaviour observed in FMR experiments with Py/Nb\nbilayers11. Using large Dynes parameter \u0000 \u0018Tcone can\ndescribe qualitatively the e\u000bect of superconducting gap\nsuppression near the surface of metallic ferromagnet such\nas Fe or Ni. At the same time the Dynes parameter\n\u0000 = 0:1Tccorresponds to the superconductors with large\nelectron-phonon relaxation rate such as NbN. Therefore,\nprovided the mechanism of spin pumping between the FI\nand NbN superconductor is correctly described by the\nEq.45 or Eq. 44 the Gilbert damping behaviour should\ncorrespond to the red curves in Fig.7 with rather weak\npeaks. The amplitude of these peaks is much smaller\nthan has been observed in the experiment16. This dis-\ncrepancy shows the presence of some other yet unknown\nmechanism of spin pumping which can yield more pro-\nnounced peaks. The identi\fcation of such a mechanism\nis however beyond the scope of the present paper.\n(a)dM= 3\u0018\n (b)dM=\u0018=2\n\u0000=Tc\nFIG. 7. Temperature dependencies of the additional Gilbert\ndamping coe\u000ecient \u000e\u000bEq.44 at small frequency \n = 0 :01Tc.\nIn each panel curves corresponding to the Dynes parameter\nvalues \u0000=Tc= 0:001; 0:1 are shown. The spin-orbit scatter-\ning time\u001csoTc= 4 corresopnding to the normal state spin\nrelaxation length lso=\u0018=2. The metallic \flm thickness is (a)\ndM= 3\u0018, (b)dM= 0:5\u0018.\nQuite interestingly, spin relaxation and superconduct-\ning correlations lead to the pronounced frequency depen-\ndence of the real part of the susceptibility Re \u001fhas shown\nin Fig.5, right panels. This leads to the additional contri-\nbution to the spin pumping having the form of the \feld-\nlike spin torque, that is additional frequency-dependent\ne\u000bective \feld acting on the magnetization of the ferro-9\n(a)C= 0:01\n (b)C= 0:1\nFIG. 8. Normalized amplitude of the FMR response signal\nas a function of the constant external magnetic \feld H0and\ntemperature T. The magnetic \feld is measured in the units\nHp= \u0001(T= 0)=\r. The spin relaxation time is \u001csoTc= 1 and\nthe frequency is \n = Tc. We consider (a) weak C= 0:0:1 and\n(b) relatively large C= 0:1 values of the interfacial coupling\nparameter (43).\nmagnetm. This leads to the shift of the FMR central\nfrequency which can be obtained from Eq.41 as follows\n\u000e\n =CTcRe[\u001fm(\n)\u0000\u001fm(0)]\n2\n\r(B0+H0) (46)\nThis shift is negligible at small frequencies \n Tc\u001c1\nand \n\u001cso\u001c1 and small interfacial coupling between F\nand M \flms measured by the dimensionless parameter\n(43). However, it becomes signi\fcant for higher frequen-\ncies and larger C.\nTo quantify the superconductivity-induced FMR fre-\nquency shift we consider the system with not very strong\nspin relaxation \u001csoTc= 1. The normalized FMR response\nfunction which according to Eqs.(36,41) is proportional\nto [\n2\u0000(\rB0+\u000e!)(\rH0+\u000e!)]\u00001. In Fig.8 we nor-\nmalize this response function of its largest value at each\nfrequency, so that it is possible to see the transformation\nof the FMR line as a function of temperature.\nOne can see two pronounced e\u000bects which appear with\nincreasing the coupling parameter. First, comparing\nFig.8a and 8b at T >Tcone can see a signi\fcant growth\nof the normal state resonance linewidth. Given the fact\nthe in the experiment16with FMR in FI/S multilayers\nthe resonance is well-de\fned at \n \u00190:01Tcone can con-\nclude that the coupling parameter is C\u00180:01 corre-\nsponding to the Fig.8a. In this case there is no noticeable\nshift of the FMR resonance line as a function of temper-\nature.\nAs follows from its de\fnition (43) the coupling param-\neterC/(dFdM)\u00001can be increased by decreasing either\nthe thickness of the metal \flm dMor the ferromagnetic\n\flmdF. By doing so and reaching the value of C= 0:1\none would be able to see that the superconducting cor-\nrelations produce signi\fcant shifht oincrease of the tem-\nperature dependence of the resonant \feld H0.V. CONCLUSION\nWe have derived and analysed the general expression\nfor the time-dependent linear spin response in the super-\nconductor with spin-orbit relaxation. The homogeneous\nspin susceptibility is found for any amount of the ordi-\nnary disorder. In the spatially-inhomogeneous case the\ndi\u000busive limit is considered. We show that the e\u000bective\nspin relaxation rate is given by the sum of the spin-orbit\nscattering rate and the di\u000busive term. At low frequencies\n\n\u001cTcincreasing the e\u000bective spin relaxation leads to\nthe formation of the peak in the temperature dependence\nof the dissipative part of spin susceptibility. This peak is\nstrongly suppressed by increasing the Dynes parameter\nwhich models the smearing of the gap edge singularities\nin the superconductors due to the inhomogeneities or the\ninelastic phonon scattering.\nUsing this result and the model of interfacial exchange\ninteraction we examined the spin pumping from the fer-\nromagnet with magnetization precession into the adja-\ncent superconducting \flm. In the low-frequency regime,\ncorresponding to the recent experiments11{19we have\nanalysed the temperature dependence of the additional\nGilbert damping parameter induced by the spin pump-\ning. For realistic values of the Dynes parameter in such\nmaterials as NbN this temperature dependence is al-\nmost monotonic. This result indicates that there should\nexist some other mechanism for producing large peaks\nobserved recently in S/FI structures16. The regime of\nlarge Dynes parameters can be also considered to model\nthe spectral smearing which occurs due to the spatial\ninhomogenuity of the order parameter in systems with\nmetallic ferromagnets. The monotonic suppression of\nthe Gilbert damping parameter in this case corresponds\nto experimentally observed behaviour of FMR in Py/Nb\nsystems11. Similar behaviour is also reproduced by the\nscattering theory formalism39.\nFor larger frequencies, comparable with the supercon-\nducting gap and enhanced interfacial couplings we get\nsigni\fcant shifts of the FMR line. These shifts act to-\nwards increasing the resonant \feld H0at a given fre-\nquency. This behaviour is opposite to the one found in\nrecent experiments at low frequencies14,17{19.\nVI. ACKNOWLEDGEMENTS\nThis work was supported by the Academy of Finland\n(Project No. 297439) and Russian Science Foundation,\nGrant No. 19-19-00594.10\nAppendix A: Absence of spin response without spin\nrelaxation\nIn the absence of spin-orbit scattering \u001c\u00001\nso= 0 and\nq= 0 the susceptibility can be written as follows\n\u001fh(\n;q= 0) =\u0019TX\n!\u00012+s1s2\u0000!1!2\ns1s2(s1+s2)\nWe can use following relations s2\n1\u0000s2\n2=!2\n1\u0000!2\n2and\n2(!1!2\u0000\u00012\u0000s1s2) = (!1+!2)2\u0000(s1+s2)2so that\nX\n!(!1+!2)2\u0000(s1+s2)2\ns1s2(s1+s2)=\nX\n!\u0014(!1+!2)2\ns1s2(s1+s2)\u0000s1+s2\ns1s2\u0015\n=\nX\n!\u0014(!1+!2)\n(!1\u0000!2)\u0000\ns\u00001\n2\u0000s\u00001\n1\u0001\n\u0000s\u00001\n1\u0000s\u00001\n2\u0015\n=\n\u00001\n\nX\n!\u0002\n(!2\u0000!1)(s\u00001\n1+s\u00001\n2)\u0000(!1+!2)\u0000\ns\u00001\n2\u0000s\u00001\n1\u0001\u0003\n=\n2\n\nX\n!\u0002\n!1s\u00001\n1\u0000!2s\u00001\n2\u0003\n=2\n\nX\n![sgn(!1)\u0000sgn(!2)]\n(A1)\nThus after analytical continuation we can write\n\u001fh(\n) + 1 =\u00021\n\u00001d\"\n2\n[n0(\"+ \n)\u0000n0(\")] = 1\nso that\u001fh(\n) = 0 at \n6= 0.\nAppendix B: Normal metal limit\nIn the normal metal limit \u0001 = 0 and \u00181;2=j!1;2j.\nThen\n\u001fh+ 1 =\u0019TX\n!1\u0000sign(!1)sign(!2)\n(j!1j+j!2j+ 4=3\u001cso)(B1)\nAnalytical continuation is implemented as follows\n\u001fh+ 1 =\n\u00021\n\u00001d\"\n4i[n0(\"\u00001)\u0000n0(\")][1\u0000sign(!1)Rsign(!2)A]\n(j!1jR+j!2jA+ 4=3\u001cso)\nwhere we have used that j!1jR!s(\u0000i\"R\n1) =i(\"\u0000\n)+\u0000\nandj!2jA!s(\u0000i\"A\n2) =\u0000i\"+\u0000, so thatj!1jR+j!2jA!\n\u0000i(\"\u0000\n) +i\"+ 2\u0000 =i\n + 2\u0000\nThen we obtain\n\u001fh+ 1 =\u00021\n\u00001d\"\n2i[n0(\")\u0000n0(\"+ \n)]\n(i\n + 2\u0000 + 4=3\u001cso)=\n\n\n\u00002i(2=3\u001cso+ \u0000)(B2)\nFrom this we obtain Eq.22.Appendix C: Derivation of the strong spin\nrelaxation limit Eq.26\nSubstituting Eq.(21) obtained assuming the strong\nspin relaxation to the general analytical continuation rule\n(14) we obtain\n8\n3i\u001cso\u001fh= \u00012\u0002\nd\"\u0014F1(\"\u0000\n)\n\u0018A(\")+F1(\"+ \n)\n\u0018R(\")\u0015\n+ (C1)\n\u0002\nd\"\u0014F2(\"\u0000\n)\"\n\u0018A(\")+F2(\"+ \n)\"\n\u0018R(\")\u0015\n+\n\u0002\nd\"\u0014F2(\")(\"+ \n)\n\u0018A(\"+ \n)\u0000F2(\"\u0000\n)\"\n\u0018A(\")\u0015\nwhere\u0018R;A(\") =p\n(\"R;A)2\u0000\u00012,F1=n0(\")N(\")=\",\nF2=n0(\")N(\"), andN= Re(\"=\u0018R) is the DOS. The\ncontribution of last term can be calculated to be equal\n\u0000i\n using asymptotic F2(\"\u00061) =\u00061 and\"=\u0018A(\")!\u00001\nat large energies. The \frst two terms can be calculated\nusing expansions F(\"\u0006\n) =F(\")\u0006\n@\"Fwhich yields\n2\n3\u001csoIm\u001fh\n\n=\u00021\n\u00001d\"N\n\"(\u00012@\"F1+\"@\"F2)\u00001 (C2)\nIntegrating by parts this equation can be rewritten as\nEq.26 in the main text.\nAppendix D: Calculation of local spin susceptibility\nin the \flm of \fnite thickness\nTo take into account \fnite metallic \flm thickness we\nincorporate the interfacial exchange \feld as the boundary\nconditions to the non-stationary Usadel equations\nD\u0014g\u000e@z\u0014g(z= 0) =iJsd[\u001bm;^g]t (D1)\nMathematically it is more convenient to consider the\nequivalent problem incorporating the interfacial ex-\nchange \feld as the point source to the Usadel equation\n\u0000if^\u001c3@t;\u0014ggt+D@z(\u0014g\u000e@z\u0014g) = (D2)\ni[^\u001c3^\u001c2\u0001;\u0014g] + [\u0014\u0006so\u000e;\u0014g]t+iJsd\u000e(z)[ ^m;\u0014g]t\nThis equation is considered in the interval jzj< dM. In\ncase if atz=\u0006dMare the interfaces with vacuum the\ncurrent vanishes\n\u0014g\u000e@z\u0014g(z=\u0006dM) = 0 (D3)\nIn case if at z=dMare the interfaces with very strong\nspin sink the correction to GF vanishes\n\u0014gh(z=\u0006dM) = 0 (D4)\nWe assume that magnetization depends on time as\nm(t) =m\ne\u0000i\ntand search for the corrections to the\nGF in the form\n^g(t;t0) =TX\n![^g0(1)e\u0000i!1(t1\u0000t2)+ ^gh(12)e\u0000i(!1t1\u0000!2t2)]\n(D5)11\nwhere!2=!1\u0000\n and ^ghrepresents the correction to\nthe \frst order of the oscillating \feld m\n. To satisfy\nboundary conditions we search the solution in the form\n^gh(12) =1X\nn=0gqn(12) cos(qnz) (D6)\nwithqn=n\u0019=dMin case of the vacuum interface\n(D3) andqn= (n+ 1=2)\u0019=dMin case of the strong\nspin sink interface (D4) . 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Tserkovnyak, EPL (Europhysics Letters) 84, 57008\n(2008)."    },    {        "title": "2109.03684v2.Room_Temperature_Intrinsic_and_Extrinsic_Damping_in_Polycrystalline_Fe_Thin_Films.pdf",        "content": "Room-Temperature Intrinsic and Extrinsic Damping in\nPolycrystalline Fe Thin Films\nShuang Wu,1David A. Smith,1Prabandha Nakarmi,2Anish Rai,2Michael Clavel,3Mantu\nK. Hudait,3Jing Zhao,4F. Marc Michel,4Claudia Mewes,2Tim Mewes,2and Satoru Emori1\n1Department of Physics, Virginia Polytechnic Institute\nand State University, Blacksburg, VA 24061, USA\n2Department of Physics and Astronomy,\nThe University of Alabama, Tuscaloosa, AL 35487 USA\n3Department of Electrical and Computer Engineering,\nVirginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA\n4Department of Geosciences, Virginia Polytechnic Institute\nand State University, Blacksburg, VA 24061, USA\nAbstract\nWe examine room-temperature magnetic relaxation in polycrystalline Fe \flms. Out-of-plane fer-\nromagnetic resonance (FMR) measurements reveal Gilbert damping parameters of \u00190.0024 for Fe\n\flms with thicknesses of 4-25 nm, regardless of their microstructural properties. The remarkable\ninvariance with \flm microstructure strongly suggests that intrinsic Gilbert damping in polycrys-\ntalline metals at room temperature is a local property of nanoscale crystal grains, with limited\nimpact from grain boundaries and \flm roughness. By contrast, the in-plane FMR linewidths of\nthe Fe \flms exhibit distinct nonlinear frequency dependences, indicating the presence of strong\nextrinsic damping. To \ft our in-plane FMR data, we have used a grain-to-grain two-magnon scat-\ntering model with two types of correlation functions aimed at describing the spatial distribution of\ninhomogeneities in the \flm. However, neither of the two correlation functions is able to reproduce\nthe experimental data quantitatively with physically reasonable parameters. Our \fndings advance\nthe fundamental understanding of intrinsic Gilbert damping in structurally disordered \flms, while\ndemonstrating the need for a deeper examination of how microstructural disorder governs extrinsic\ndamping.\n1arXiv:2109.03684v2  [cond-mat.mtrl-sci]  24 Feb 2022I. INTRODUCTION\nIn all magnetic materials, magnetization has the tendency to relax toward an e\u000bective\nmagnetic \feld. How fast the magnetization relaxes governs the performance of a variety\nof magnetic devices. For example, magnetization relaxation hinders e\u000ecient precessional\ndynamics and should be minimized in devices such as precessional magnetic random access\nmemories, spin-torque oscillators, and magnonic circuits1{4. From the technological perspec-\ntive, it is important to understand the mechanisms behind magnetic relaxation in thin-\flm\nmaterials that comprise various nanomagnetic device applications. Among these materials,\nbcc Fe is a prototypical elemental ferromagnet with attractive properties, including high sat-\nuration magnetization, soft magnetism5, and large tunnel magnetoresistance6,7. Our present\nstudy is therefore motivated by the need to uncover magnetic relaxation mechanisms in Fe\nthin \flms { particularly polycrystalline \flms that can be easily grown on arbitrary substrates\nfor diverse applications.\nTo gain insights into the contributions to magnetic relaxation, a common approach is to\nexamine the frequency dependence of the ferromagnetic resonance (FMR) linewidth. The\nmost often studied contribution is viscous Gilbert damping8{13, which yields a linear increase\nin FMR linewidth with increasing precessional frequency. In ferromagnetic metals, Gilbert\ndamping arises predominately from \\intrinsic\" mechanisms14{16governed by the electronic\nband structure17. Indeed, a recent experimental study by Khodadadi et al.18has shown\nthat intrinsic, band-structure-based Gilbert damping dominates magnetic relaxation in high-\nquality crystalline thin \flms of Fe, epitaxially grown on lattice-matched substrates. However,\nit is yet unclear how intrinsic damping is impacted by the microstructure of polycrystalline\nFe \flms.\nMicrostructural disorder in polycrystalline Fe \flms can also introduce extrinsic magnetic\nrelaxation. A well-known extrinsic relaxation mechanism is two-magnon scattering, where\nthe uniform precession mode with zero wave vector scatters into a degenerate magnon mode\nwith a \fnite wave vector19{22. Two-magnon scattering generally leads to a nonlinear fre-\nquency dependence of the FMR linewidth, governed by the nature of magnon scattering\ncenters at the surfaces23,24or in the bulk of the \flm25{28. While some prior experiments\npoint to the prominent roles of extrinsic magnetic relaxation in polycrystalline ferromag-\nnetic \flms29{31, systematic studies of extrinsic relaxation (e.g., two-magnon scattering) on\n2polycrystalline Fe thin \flms are still lacking.\nHere, we investigate both the intrinsic and extrinsic contributions to magnetic relax-\nation at room temperature in polycrystalline Fe \flms. We have measured the frequency\ndependence of the FMR linewidth with (1) the \flm magnetized out-of-plane (OOP), where\ntwo-magnon scattering is suppressed25such that intrinsic Gilbert damping is quanti\fed re-\nliably, and (2) the \flm magnetized in-plane (IP), where two-magnon scattering is generally\nexpected to coexist with intrinsic Gilbert damping.\nFrom OOP FMR results, we \fnd that the intrinsic Gilbert damping of polycrystalline Fe\n\flms at room temperature is independent of their structural properties and almost identical\nto that of epitaxial \flms. Such insensitivity to microstructure is in contrast to disorder-\nsensitive Gilbert damping recently shown in epitaxial Fe at cryogenic temperature18. Our\npresent work implies that Gilbert damping at a su\u000eciently high temperature becomes a\nlocal property of the metal, primarily governed by the structure within nanoscale crystal\ngrains rather than grain boundaries or interfacial disorder. This implication refutes the\nintuitive expectation that intrinsic Gilbert damping should depend on structural disorder in\npolycrystalline \flms.\nIn IP FMR results, the frequency dependence of the FMR linewidth exhibits strong\nnonlinear trends that vary signi\fcantly with \flm microstructure. To analyze the nonlin-\near trends, we have employed the grain-to-grain two-magnon scattering model developed\nby McMichael and Krivosik25with two types of correlation functions for capturing inho-\nmogeneities in the \flm. However, neither of the correlation functions yields quantitative\nagreement with the experimental results or physically consistent, reasonable parameters.\nThis \fnding implies that a physical, quantitative understanding of extrinsic magnetic re-\nlaxation requires further corrections of the existing two-magnon scattering model, along\nwith much more detailed characterization of the nanoscale inhomogeneities of the magnetic\n\flm. Our study stimulates opportunities for a deeper examination of fundamental magnetic\nrelaxation mechanisms in structurally disordered ferromagnetic metal \flms.\nII. FILM DEPOSITION AND STRUCTURAL PROPERTIES\nPolycrystalline Fe thin \flms were deposited using DC magnetron sputtering at room\ntemperature on Si substrates with a native oxide layer of SiO 2. The base pressure of the\n3chamber was below 1 \u000210\u00007Torr and all \flms were deposited with 3 mTorr Ar pressure. Two\nsample series with di\u000berent seed layers were prepared in our study: subs./Ti(3 nm)/Cu(3\nnm)/Fe(2-25 nm)/Ti(3 nm) and subs./Ti(3 nm)/Ag(3 nm)/Fe(2-25 nm)/Ti(3 nm). In this\npaper we refer to these two sample series as Cu/Fe and Ag/Fe, respectively. The layer\nthicknesses are based on deposition rates derived from x-ray re\rectivity (XRR) of thick\ncalibration \flms. The Ti layer grown directly on the substrate ensures good adhesion of\nthe \flm, whereas the Cu and Ag layers yield distinct microstructural properties for Fe\nas described below. We note that Cu is often used as a seed layer for growing textured\npolycrystalline ferromagnetic metal \flms32,33. Our initial motivation for selecting Ag as an\nalternative seed layer was that it might promote qualitatively di\u000berent Fe \flm growth34,\nowing to a better match in bulk lattice parameter 𝑎between Fe ( 𝑎\u0019286\u0017A) and Ag\n(𝑎p\n2\u0019288\u0017A) compared to Fe and Cu ( 𝑎p\n2\u0019255\u0017A).\nWe performed x-ray di\u000braction (XRD) measurements to compare the structural properties\nof the Cu/Fe and Ag/Fe \flms. Figure 1(a,b) shows symmetric 𝜃-2𝜃XRD scan curves\nfor several \flms from both the Cu/Fe and Ag/Fe sample series. For all Cu/Fe \flms, the\n(110) body-center-cubic (bcc) peak can be observed around 2 𝜃=44°\u000045°(Fig. 1(a)). This\nobservation con\frms that the Fe \flms grown on Cu are polycrystalline and textured, where\nthe crystal grains predominantly possess (110)-oriented planes that are parallel to the sample\nsurface. For Ag/Fe (Fig. 1(b)), the (110) bcc peak is absent or extremely weak, from\nwhich one might surmise that the Fe \flms grown on Ag are amorphous or only possess\nweak crystallographic texture. However, we \fnd that the Ag/Fe \flms are, in fact, also\npolycrystalline with evidence of (110) texturing. In the following, we elaborate on our XRD\nresults, \frst for Cu/Fe and then Ag/Fe.\nWe observe evidence for a peculiar, non-monotonic trend in the microstructural properties\nof the Cu/Fe \flms. Speci\fcally, the height of the 𝜃-2𝜃di\u000braction peak (Fig. 1(a)) increases\nwith Fe \flm thickness up to \u001910 nm but then decreases at higher Fe \flm thicknesses. While\nwe do not have a complete explanation for this peculiar nonmonotonic trend with \flm\nthickness, a closer inspection of the XRD results (Fig. 1) provides useful insights. First, the\nFe \flm di\u000braction peak shifts toward a higher 2 𝜃value with increasing \flm thickness. This\nsigni\fes that thinner Fe \flms on Cu are strained (with the Fe crystal lattice tetragonally\ndistorted), whereas thicker Fe \flms undergo structural relaxation such that the out-of-plane\nlattice parameter converges toward the bulk value of \u00192.86 \u0017A, as summarized in Fig. 1(e).\n4354 04 55 05 5Ag/Fe2 nm6 nmIntensity [arb. unit]  \nCu/Febulk bcc Fe (110)1\n0 nm15 nm25 nm8\n nm(\na)\n4045502\nθ [deg]10 nm2\n5 nm \n2\nθ [deg]10 nm15 nm6\n nm2\n nm8 nm(\nb)\n16182022242628Ag/Fe2 nm6 nmIntensity [arb. unit]  \nCu/Febulk bcc Fe (110)1\n0 nm15 nm25 nm8\n nm(\nc)2\n5 nm \nθ\n [deg]10 nm15 nm6\n nm2\n nm8 nm(\nd)\n2.842.862.882.902.922.940\n5 10152025051015Bulk value 2.86 Cu/Fe \nAg/FeOut-of-planel\nattice parameter [Å](\ne)Crystallite size [nm]T\nhickness [nm](f)FIG. 1. (Color online) 𝜃-2𝜃X-ray di\u000braction scan curves for (a) Cu/Fe (blue lines) and (b) Ag/Fe\n(red lines) sample series. The inset in (b) is the grazing-incidence XRD scan curve for 10 nm thick\nAg/Fe \flm. Rocking curves for (c) Cu/Fe (blue lines) and (d) Ag/Fe (red lines) sample series.\n(e) Out-of-plane lattice parameter estimated via Bragg's law using the 2 𝜃value at the maximum\nof the tallest \flm di\u000braction peak. (f) Crystallite size estimated via the Scherrer equation using\nthe full-width-at-half-maximum of the tallest \flm di\u000braction peak. In (e) and (f), the data for the\nAg/Fe \flm series at a few thickness values are missing because of the absence of the bcc (110) peak\nin𝜃-2𝜃XRD scans.\nSecond, as the Fe \flm thickness approaches \u001910 nm, additional di\u000braction peaks appear to\nthe left of the tall primary peak. We speculate that these additional peaks may originate\nfrom Fe crystals that remain relatively strained (i.e., with an out-of-plane lattice parameter\nlarger than the bulk value), while the primary peak arises from more relaxed Fe crystals\n(i.e., with a lattice parameter closer to the bulk value). The coexistence of such di\u000berent\nFe crystals appears to be consistent with the rocking curve measurements (Fig. 1(c)), which\nexhibit a large broad background peak in addition to a small sharp peak for Cu/Fe \flms\nwith thicknesses near \u001910 nm. As we describe in Sec. IV, these \u001910 nm thick Cu/Fe samples\nalso show distinct behaviors in extrinsic damping (highly nonlinear frequency dependence of\n5the FMR linewidth) and static magnetization reversal (enhanced coercivity), which appear\nto be correlated with the peculiar microstructural properties evidenced by our XRD results.\nOn the other hand, it is worth noting that the estimated crystal grain size (Fig. 1(f)) {\nderived from the width of the 𝜃-2𝜃di\u000braction peak { does not exhibit any anomaly near the\n\flm thickness of\u001910 nm, but rather increases monotonically with \flm thickness.\nUnlike the Cu/Fe \flms discussed above, the Ag/Fe \flms do not show a strong (110) bcc\npeak in the 𝜃-2𝜃XRD results. However, the lack of pronounced peaks in the symmetric 𝜃-2𝜃\nscans does not necessarily signify that Ag/Fe is amorphous. This is because symmetric 𝜃-2𝜃\nXRD is sensitive to crystal planes that are nearly parallel to the sample surface, such that the\ndi\u000braction peaks capture only the crystal planes with out-of-plane orientation with a rather\nsmall range of misalignment (within \u00181°, dictated by incident X-ray beam divergence). In\nfact, from asymmetric grazing-incidence XRD scans that are sensitive to other planes, we\nare able to observe a clear bcc Fe (110) di\u000braction peak even for Ag/Fe samples that lack\nan obvious di\u000braction peak in 𝜃-2𝜃scans (see e.g. inset of Fig. 1(b)). Furthermore, rocking\ncurve scans (conducted with 2 𝜃\fxed to the expected position of the (110) Fe \flm di\u000braction\npeak) provide orientation information over an angular range much wider than \u00181°. As shown\nin Fig. 1(d), a clear rocking curve peak is observed for each Ag/Fe sample, suggesting that\nFe \flms grown on Ag are polycrystalline and (110)-textured { albeit with the (110) crystal\nplanes more misaligned from the sample surface compared to the Cu/Fe samples. The out-\nof-plane lattice parameters of Ag/Fe \flms (with discernible 𝜃-2𝜃di\u000braction \flm peaks) show\nthe trend of relaxation towards the bulk value with increasing Fe thickness, similar to the\nCu/Fe series. Yet, the lattice parameters for Ag/Fe at small thicknesses are systematically\ncloser to the bulk value, possibly because Fe is less strained (i.e., better lattice matched)\non Ag than on Cu. We also \fnd that the estimation of the crystal grain size for Ag/Fe {\nalthough made di\u000ecult by the smallness of the di\u000braction peak { yields a trend comparable\nto Cu/Fe, as shown in Fig. 1(f).\nWe also observe a notable di\u000berence between Cu/Fe and Ag/Fe in the properties of \flm\ninterfaces, as revealed by XRR scans in Fig. 2. The oscillation period depends inversely\non the \flm thickness. The faster decay of the oscillatory re\rectivity signal at high angles\nfor the Ag/Fe \flms suggests that the Ag/Fe \flms may have rougher interfaces compared to\nthe Cu/Fe \flms. Another interpretation of the XRR results is that the Ag/Fe interface is\nmore di\u000buse than the Cu/Fe interface { i.e., due to interfacial intermixing of Ag and Fe. By\n60.000.050.100.150.200.250.3010 nm2\n5 nmReflectivity [a.u.] \n(\na) Cu/Fe \nAg/Fe \nq\nz [Å-1](b)FIG. 2. (Color online) X-ray re\rectivity scans of 10 nm and 25 nm thick \flms from (a) Cu/Fe\n(blue circles) and (b) Ag/Fe (red squares) sample series. Black solid curves are \fts to the data.\n\ftting the XRR results35, we estimate an average roughness (or the thickness of the di\u000buse\ninterfacial layer) of .1 nm for the Fe layer in Cu/Fe, while it is much greater at \u00192-3 nm\nfor Ag/Fe36.\nOur structural characterization described above thus reveals key attributes of the Cu/Fe\nand Ag/Fe sample series. Both \flm series are polycrystalline, exhibit (110) texture, and\nhave grain sizes of order \flm thickness. Nevertheless, there are also crucial di\u000berences\nbetween Cu/Fe and Ag/Fe. The Cu/Fe series overall exhibits stronger 𝜃-2𝜃di\u000braction\npeaks than the Ag/Fe series, suggesting that the (110) bcc crystal planes of Fe grown on\nCu are aligned within a tighter angular range than those grown on Ag. Moreover, Fe grown\non Cu has relatively smooth or sharp interfaces compared to Fe grown on Ag. Although\nidentifying the origin of such structural di\u000berences is beyond the scope of this work, Cu/Fe\n7and Ag/Fe constitute two qualitatively distinct series of polycrystalline Fe \flms for exploring\nthe in\ruence of microstructure on magnetic relaxation.\nIII. INTRINSIC GILBERT DAMPING PROBED BY OUT-OF-PLANE FMR\nHaving established the di\u000berence in structural properties between Cu/Fe and Ag/Fe, we\ncharacterize room-temperature intrinsic damping for these samples with OOP FMR mea-\nsurements. The OOP geometry suppresses two-magnon scattering25such that the Gilbert\ndamping parameter can be quanti\fed in a straightforward manner. We use a W-band\nshorted waveguide in a superconducting magnet, which permits FMR measurements at high\n\felds ( &4 T) that completely magnetize the Fe \flms out of plane. The details of the mea-\nsurement method are found in Refs.18,37. Figure 3(a) shows the frequency dependence of\nhalf-width-at-half-maximum (HWHM) linewidth Δ𝐻OOP for selected thicknesses from both\nsample series. The linewidth data of 25 nm thick epitaxial Fe \flm from a previous study18\nis plotted in Fig. 3 (a) as well. The intrinsic damping parameter can be extracted from the\nlinewidth plot using\nΔ𝐻OOP=Δ𝐻0¸2𝜋\n𝛾𝛼OOP𝑓 (1)\nwhereΔ𝐻0is the inhomogeneous broadening38,𝛾=𝑔𝜇𝐵\nℏis the gyromagnetic ratio ( 𝛾2𝜋\u0019\n2.9 MHz/Oe [Ref.39], obtained from the frequency dependence of resonance \feld37), and\n𝛼OOP is the measured viscous damping parameter. In general, 𝛼OOP can include not only\nintrinsic Gilbert damping, parameterized by 𝛼int, but also eddy-current, radiative damping,\nand spin pumping contributions40, which all yield a linear frequency dependence of the\nlinewidth. Damping due to eddy current is estimated to make up less than 10% of the total\nmeasured damping parameter37and is ignored here. Since we used a shorted waveguide in\nour setup, the radiative damping does not apply here. Spin pumping is also negligible for\nmost of the samples here because the materials in the seed and capping layers (i.e., Ti, Cu,\nand Ag) possess weak spin-orbit coupling and are hence poor spin sinks31,41,42. We therefore\nproceed by assuming that the measured OOP damping parameter 𝛼OOP is equivalent to the\nintrinsic Gilbert damping parameter.\nThe extracted damping parameter is plotted as a function of Fe \flm thickness in Fig.\n3(b). The room-temperature damping parameters of all Fe \flms with thicknesses of 4-25\n80204060801001200306090120150180 \n25nm epitaxial Fe \n10nm Cu/Fe \n25nm Cu/Fe \n10nm Ag/Fe \n25nm Ag/FeΔHOOP [Oe]f\n [GHz](a)\n05101520250.0000.0010.0020.0030.004 epitaxial Fe \nCu/Fe \nAg/FeαOOPT\nhickness [nm](b)FIG. 3. (Color online) (a) OOP FMR half-width-at-half-maximum linewidth Δ𝐻OOPas a function\nof resonance frequency 𝑓. Lines correspond to \fts to the data. (b) Gilbert damping parameter\n𝛼𝑚𝑎𝑡ℎ𝑟𝑚𝑂𝑂𝑃 extracted from OOP FMR as a function of \flm thickness. The red shaded area\nhighlights the damping value range that contains data points of all \flms thicker than 4 nm. The\ndata for the epitaxial Fe sample (25 nm thick Fe grown on MgAl 2O4) are adapted from Ref.18.\nnm fall in the range of 0.0024 \u00060.0004, which is shaded in red in Fig. 3(b). This damping\nparameter range is quantitatively in line with the value reported for epitaxial Fe (black\nsymbol in Fig. 3(b))18. For 2 nm thick samples, the damping parameter is larger likely\ndue to an additional interfacial contribution43{45{ e.g., spin relaxation through interfacial\nRashba spin-orbit coupling46that becomes evident only for ultrathin Fe. The results in\nFig. 3(b) therefore indicate that the structural properties of the &4 nm thick polycrystalline\nbcc Fe \flms have little in\ruence on their intrinsic damping.\nIt is remarkable that these polycrystalline Cu/Fe and Ag/Fe \flms { with di\u000berent thick-\n9nesses and microstructural properties (as revealed in Sec. II) { exhibit essentially the same\nroom-temperature intrinsic Gilbert damping parameter as single-crystalline bcc Fe. This\n\fnding is qualitatively distinct from a prior report18on intrinsic Gilbert damping in single-\ncrystalline Fe \flms at cryogenic temperature, which is sensitive to microstructural disorder.\nIn the following, we discuss the possible di\u000berences in the mechanisms of intrinsic damping\nbetween these temperature regimes.\nIntrinsic Gilbert damping in ferromagnetic metals is predominantly governed by transi-\ntions of spin-polarized electrons between electronic states, within a given electronic band\n(intraband scattering) or in di\u000berent electronic bands (interband scattering) near the Fermi\nlevel15. For Fe, previous studies15,18,47indicate that intraband scattering tends to dominate\nat low temperature where the electronic scattering rate is low (e.g., \u00181013s\u00001); by contrast,\ninterband scattering likely dominates at room temperature where the electronic scattering\nrate is higher (e.g., \u00181014s\u00001). According to our results (Fig. 3(b)), intrinsic damping at\nroom temperature is evidently una\u000bected by the variation in the structural properties of the\nFe \flms. Hence, the observed intrinsic damping is mostly governed by the electronic band\nstructure within the Fe grains , such that disorder in grain boundaries or \flm interfaces has\nminimal impact.\nThe question remains as to why interband scattering at room temperature leads to Gilbert\ndamping that is insensitive to microstructural disorder, in contrast to intraband scattering\nat low temperature yielding damping that is quite sensitive to microstructure18. This dis-\ntinction may be governed by what predominantly drives electronic scattering { speci\fcally,\ndefects (e.g., grain boundaries, rough or di\u000buse interfaces) at low temperature, as opposed\nto phonons at high temperature. That is, the dominance of phonon-driven scattering at\nroom temperature may e\u000bectively diminish the roles of microstructural defects in Gilbert\ndamping. Future experimental studies of temperature-dependent damping in polycrystalline\nFe \flms may provide deeper insights. Regardless of the underlying mechanisms, the robust\nconsistency of 𝛼OOP (Fig. 3(b)) could be an indication that the intrinsic Gilbert damp-\ning parameter at a su\u000eciently high temperature is a local property of the ferromagnetic\nmetal, possibly averaged over the ferromagnetic exchange length of just a few nm48that is\ncomparable or smaller than the grain size. In this scenario, the impact on damping from\ngrain boundaries would be limited in comparison to the contributions to damping within\nthe grains.\n10Moreover, the misalignment of Fe grains evidently does not have much in\ruence on the\nintrinsic damping. This is reasonable considering that intrinsic Gilbert damping is predicted\nto be nearly isotropic in Fe at su\u000eciently high electronic scattering rates49{ e.g.,\u00181014s\u00001\nat room temperature where interband scattering is expected to be dominant15,18,47. It is\nalso worth emphasizing that 𝛼OOP remains unchanged for Fe \flms of various thicknesses\nwith di\u000berent magnitudes of strain (tetragonal distortion, as evidenced by the variation in\nthe out-of-plane lattice parameter in Fig. 1(e)). Strain in Fe grains is not expected to impact\nthe intrinsic damping, as Ref.18suggests that strain in bcc Fe does not signi\fcantly alter\nthe band structure near the Fermi level. Thus, polycrystalline Fe \flms exhibit essentially\nthe same magnitude of room-temperature intrinsic Gilbert damping as epitaxial Fe, as long\nas the grains retain the bcc crystal structure.\nThe observed invariance of intrinsic damping here is quite di\u000berent from the recent study\nof polycrystalline Co 25Fe75alloy \flms31, reporting a decrease in intrinsic damping with in-\ncreasing structural disorder. This inverse correlation between intrinsic damping and disorder\nin Ref.31is attributed to the dominance of intraband scattering, which is inversely propor-\ntional to the electronic scattering rate. It remains an open challenge to understand why the\nroom-temperature intrinsic Gilbert damping of some ferromagnetic metals might be more\nsensitive to structural disorder than others.\nIV. EXTRINSIC MAGNETIC RELAXATION PROBED BY IN-PLANE FMR\nAlthough we have shown via OOP FMR in Sec. III that intrinsic Gilbert damping is\nessentially independent of the structural properties of the Fe \flms, it might be expected\nthat microstructure has a pronounced impact on extrinsic magnetic relaxation driven by\ntwo-magnon scattering, which is generally present in IP FMR. IP magnetized \flms are more\ncommon in device applications than OOP magnetized \flms, since the shape anisotropy of\nthin \flms tends to keep the magnetization in the \flm plane. What governs the performance\nof such magnetic devices (e.g., quality factor50,51) may not be the intrinsic Gilbert damping\nparameter but the total FMR linewidth. Thus, for many magnetic device applications, it is\nessential to understand the contributions to the IP FMR linewidth.\nIP FMR measurements have been performed using a coplanar-waveguide-based spectrom-\neter, as detailed in Refs.18,37. Examples of the frequency dependence of IP FMR linewidth\n110501001502002500\n10203040506070050100150200250Cu/FeA\ng/Fe 2 nm \n6 nm \n8 nm \n10 nm \n15 nm \n25 nmΔHIP [Oe] \n12(\na) \nf\n [GHz](b)FIG. 4. (Color online) IP FMR half-width-at-half-maximum linewidth Δ𝐻IPas a function of\nresonance frequency 𝑓for (a) Cu/Fe and (b) Ag/Fe. The vertical dashed line at 12 GHz highlights\nthe hump in linewidth vs frequency seen for many of the samples.\nare shown in Fig. 4. In contrast to the linear frequency dependence that arises from in-\ntrinsic Gilbert damping in Fig. 3(a), a nonlinear hump is observed for most of the \flms\nin the vicinity of \u001912 GHz. In some \flms, e.g., 10 nm thick Cu/Fe \flm, the hump is so\nlarge that its peak even exceeds the linewidth at the highest measured frequency. Similar\nnonlinear IP FMR linewidth behavior has been observed in Fe alloy \flms52and epitaxial\nHeusler \flms53in previous studies, where two-magnon scattering has been identi\fed as a\nsigni\fcant contributor to the FMR linewidth. Therefore, in the following, we attribute the\nnonlinear behavior to two-magnon scattering.\nTo gain insight into the origin of two-magnon scattering, we plot the linewidth at 12\n122550751001251500\n5101520250255075100125150 Cu/Fe \nAg/Fe Cu/Fe \nAg/FeΔHIP @ 12 GHz [Oe](a)HC [Oe]T\nhickness [nm](b)FIG. 5. (Color online) (a) IP FMR half-width-at-half-maximum linewidth at 12 GHz { approxi-\nmately where the maximum (\\hump\") in linewidth vs frequency is seen (see Fig. 4) { as a function\nof \flm thickness for both Cu/Fe and Ag/Fe. (b) Coercivity 𝐻𝑐as a function of \flm thickness for\nboth Cu/Fe and Ag/Fe. The red shaded area highlights thickness region where the Cu/Fe sample\nseries show a peak behavior in both plots.\nGHz { approximately where the hump is seen in Fig. 4 { against the Fe \flm thickness in\nFig. 5(a). We do not observe a monotonic decay in the linewidth with increasing thickness\nthat would result from two-magnon scattering of interfacial origin54. Rather, we observe\na non-monotonic thickness dependence in Fig. 5(a), which indicates that the observed\ntwo-magnon scattering originates within the bulk of the \flms. We note that Ag/Fe with\ngreater interfacial disorder (see Sec. II) exhibits weaker two-magnon scattering than Cu/Fe,\nparticularly in the lower thickness regime ( .10 nm). This observation further corroborates\n13that the two-magnon scattering here is not governed by the interfacial roughness of Fe\n\flms. The contrast between Cu/Fe and Ag/Fe also might appear counterintuitive, since\ntwo-magnon scattering is induced by defects and hence might be expected to be stronger\nfor more \\defective\" \flms (i.e., Ag/Fe in this case). The counterintuitive nature of the\ntwo-magnon scattering here points to more subtle mechanisms at work.\nTo search for a possible correlation between static magnetic properties and two-magnon\nscattering, we have performed vibrating sample magnetometry (VSM) measurements with a\nMicrosense EZ9 VSM. Coercivity extracted from VSM measurements is plotted as a function\nof \flm thickness in Fig. 5(b), which shows a remarkably close correspondence with linewidth\nvs thickness (Fig. 5(a)). In particular, a pronounced peak in coercivity is observed for Cu/Fe\naround 10 nm, corresponding to the same thickness regime where the 12 GHz FMR linewidth\nfor Cu/Fe is maximized. Moreover, the 10 nm Cu/Fe sample (see Sec. II) exhibits a tall,\nnarrow bcc (110) di\u000braction peak, which suggests that its peculiar microstructure plays a\npossible role in the large two-magnon scattering and coercivity (e.g., via stronger domain\nwall pinning).\nWhile the trends shown in Fig. 5 provide some qualitative insights, we now attempt to\nquantitatively analyze the frequency dependence of FMR linewidth for the Cu/Fe and Ag/Fe\n\flms. We assume that the Gilbert damping parameter for IP FMR is equal to that for OOP\nFMR, i.e.,𝛼IP=𝛼OOP. This assumption is physically reasonable, considering that Gilbert\ndamping is theoretically expected to be isotropic in Fe \flms near room temperature49. While\na recent study has reported anisotropic Gilbert damping that scales quadratically with\nmagnetostriction55, this e\u000bect is likely negligible in elemental Fe whose magnetostriction is\nseveral times smaller56,57than that of the Fe 07Ga03alloy in Ref.55.\nThus, from the measured IP linewidth Δ𝐻IP, the extrinsic two-magnon scattering\nlinewidthΔ𝐻TMS can be obtained by\nΔ𝐻TMS=Δ𝐻IP\u00002𝜋\n𝛾𝛼IP (2)\nwhere2𝜋\n𝛾𝛼IPis the Gilbert damping contribution. Figure 6 shows the obtained Δ𝐻TMSand \ft\nattempts using the \\grain-to-grain\" two-magnon scattering model developed by McMicheal\nand Krivosik25. This model captures the inhomogeneity of the e\u000bective internal magnetic\n\feld in a \flm consisting of many magnetic grains. The magnetic inhomogeneity can arise\nfrom the distribution of magnetocrystalline anisotropy \feld directions associated with the\n14randomly oriented crystal grains52. In this model the two-magnon scattering linewidth\nΔ𝐻TMS is a function of the Gilbert damping parameter 𝛼IP, the e\u000bective anisotropy \feld\n𝐻𝑎of the randomly oriented grain, and the correlation length 𝜉within which the e\u000bective\ninternal magnetic \feld is correlated. Further details for computing Δ𝐻TMS are provided in\nthe Appendix and Refs.25,52,53. As we have speci\fed above, 𝛼IPis set to the value derived\nfrom OOP FMR results (i.e., 𝛼OOP in Fig. 3(b)). This leaves 𝜉and𝐻𝑎as the only free\nparameters in the \ftting process.\nThe modeling results are dependent on the choice of the correlation function 𝐶¹Rº, which\ncaptures how the e\u000bective internal magnetic \feld is correlated as a function of lateral distance\nRin the \flm plane. We \frst show results obtained with a simple exponentially decaying\ncorrelation function, as done in prior studies of two-magnon scattering25,52,53, i.e.,\n𝐶¹Rº=exp\u0012\n\u0000jRj\n𝜉\u0013\n (3)\nEquation 3 has the same form as the simplest correlation function used to model rough\ntopographical surfaces (when they are assumed to be \\self-a\u000ene\")58. Fit results with Eq. (3)\nare shown in dashed blue curves in Fig. 6. For most samples, the \ftted curve does not\nreproduce the experimental data quantitatively. Moreover, the \ftted values of 𝜉and𝐻𝑎\noften reach physically unrealistic values, e.g., with 𝐻𝑎¡104Oe and𝜉 1 nm (see Table I).\nThese results suggest that the model does not properly capture the underlying physics of\ntwo-magnon scattering in our samples.\nA possible cause for the failure to \ft the data is that the simple correlation function\n(Eq. 3) is inadequate. We therefore consider an alternative correlation function by again\ninvoking an analogy between the spatially varying height of a rough surface58and the spa-\ntially varying e\u000bective internal magnetic \feld in a \flm. Speci\fcally, we apply a correlation\nfunction (i.e., a special case of Eq. (4.3) in Ref.58where short-range roughness 𝛼=1) for\nthe so-called \\mounded surface,\" which incorporates the average distance 𝜆between peaks\nin topographical height (or, analogously, e\u000bective internal magnetic \feld):\n𝐶¹Rº=p\n2jRj\n𝜉𝐾1 p\n2jRj\n𝜉!\n𝐽0\u00122𝜋jRj\n𝜆\u0013\n (4)\nwhere𝐽0and𝐾1are the Bessel function of the \frst kind of order zero and the modi\fed Bessel\nfunction of the second kind of order one, respectively. This oscillatory decaying function is\nchosen because its Fourier transform (see Appendix) does not contain any transcendental\n15020406080100120 \nExperimental \nSelf-affine \nMoundedΔHTMS [Oe]Cu/FeA g/Fe6\n nm8\n nm1\n0 nm1\n5 nm2\n5 nm(a)( f)0\n50100150ΔHTMS [Oe](\nb)( g)0\n50100150ΔHTMS [Oe](\nc)( h)0\n255075100125ΔHTMS [Oe](\nd)( i)0\n2 04 06 0050100150200ΔHTMS [Oe]f\n [GHz](e)0\n2 04 06 0f\n [GHz](j)FIG. 6. (Color online) Extrinsic two-magnon scattering linewidth Δ𝐻TMSvs frequency 𝑓and \ftted\ncurves for 6, 8, 10, 15, and 25 nm Cu/Fe and Ag/Fe \flms. Black squares represent experimental\nFMR linewidth data. Dashed blue and solid red curves represent the \ftted curves using correlation\nfunctions proposed for modeling self-a\u000ene and mounded surfaces, respectively. In (d), (e), (h), (i),\ndashed blue curves overlap with solid red curves.\n16functions, which simpli\fes the numerical calculation. We also stress that while Eq. (4) in\nthe original context (Ref.58) was used to model topographical roughness, we are applying\nEq. (4) in an attempt to model the spatial \ructuations (\\roughness\") of the e\u000bective internal\nmagnetic \feld { rather than the roughness of the \flm topography.\nThe \ftted curves using the model with Eq. (4) are shown in solid red curves in Fig. 6. Fit\nresults for some samples show visible improvement, although this is perhaps not surprising\nwith the introduction of 𝜆as an additional free parameter. Nevertheless, the \ftted values\nof𝐻𝑎or𝜆still diverge to unrealistic values of ¡104Oe or¡104nm in some cases (see\nTable I), which means that the new correlation function (Eq. (4)) does not fully re\rect\nthe meaningful underlying physics of our samples either. More detailed characterization of\nthe microstructure and inhomogeneities, e.g., via synchrotron x-ray and neutron scattering,\ncould help determine the appropriate correlation function. It is also worth pointing out that\nfor some samples (e.g. 15 nm Cu/Fe and Ag/Fe \flms), essentially identical \ft curves are\nobtained regardless of the correlation function. This is because when 𝜆\u001d𝜉, the Fourier\ntransform of Eq. (4) has a very similar form as the Fourier transform of Eq. (3), as shown in\nthe Appendix. In such cases, the choice of the correlation function has almost no in\ruence\non the behavior of the two-magnon scattering model in the \ftting process.\nV. SUMMARY\nWe have examined room-temperature intrinsic and extrinsic damping in two series of\npolycrystalline Fe thin \flms with distinct structural properties. Out-of-plane FMR mea-\nsurements con\frm constant intrinsic Gilbert damping of \u00190.0024, essentially independent\nof \flm thickness and structural properties. This \fnding implies that intrinsic damping in\nFe at room temperature is predominantly governed by the crystalline and electronic band\nstructures within the grains, rather than scattering at grain boundaries or \flm surfaces. The\nresults from in-plane FMR, where extrinsic damping (i.e., two-magnon scattering) plays a\nsigni\fcant role, are far more nuanced. The conventional grain-to-grain two-magnon scatter-\ning model fails to reproduce the in-plane FMR linewidth data with physically reasonable\nparameters { pointing to the need to modify the model, along with more detailed character-\nization of the \flm microstructure. Our experimental \fndings advance the understanding of\nintrinsic Gilbert damping in polycrystalline Fe, while motivating further studies to uncover\n17TABLE I. Summary of IP FMR linewidth \ft results. Note the divergence to physically unreasonable\nvalues in many of the results. Standard error is calculated using equation√︁\nSSRDOF\u0002diag¹COVº,\nwhere SSR stands for the sum of squared residuals, DOF stands for degrees of freedom, and COV\nstands for the covariance matrix.\nSelf-a\u000ene Mounded\nSample\nSeriesThickness\n(nm)𝜉\n(nm)𝐻𝑎\n(Oe)𝜉\n(nm)𝐻𝑎\n(Oe)𝜆\n(nm)\nCu/Fe6 70\u000610 170\u000610 80\u000690 24\u00063 >1\u0002104\n8 200\u0006100 150\u000620 700\u00061000 25\u00062 900\u0006100\n10 140\u000640 200\u000620 160\u000650 33\u00061 800\u0006200\n15 9\u00062 800\u0006100 10\u000620 100\u000680 >1\u0002104\n25 0\u00065 >1\u000210460\u000630 >1\u000210410.41\u00060.01\nAg/Fe6 0\u000640 >1\u0002104150\u000640 >1\u000210411.7\u00060.7\n8 0\u000630 >1\u0002104170\u000650 >1\u000210412\u00064\n10 6\u00061 1500\u0006300 8\u000640 200\u0006500 >1\u0002104\n15 2\u00062 4000\u00063000 3\u00069 500\u0006900 >6\u0002103\n25 0\u00066 >1\u0002104140\u000650 >1\u000210415\u00066\nthe mechanisms of extrinsic damping in structurally disordered thin \flms.\nACKNOWLEDGMENTS\nS.W. acknowledges support by the ICTAS Junior Faculty Program. D.A.S. and S.E.\nacknowledge support by the National Science Foundation, Grant No. DMR-2003914. P.\nN. would like to acknowledge support through NASA Grant NASA CAN80NSSC18M0023.\nA. R. would like to acknowledge support through the Defense Advanced Research Project\nAgency (DARPA) program on Topological Excitations in Electronics (TEE) under Grant\nNo. D18AP00011. This work was supported by NanoEarth, a member of National Nan-\notechnology Coordinated Infrastructure (NNCI), supported by NSF (ECCS 1542100).\n18Appendix A: Details of the Two-Magnon Scattering Model\nIn the model developed by McMichael and Krivosik, the two-magnon scattering contri-\nbutionΔ𝐻TMS to the FMR linewidth is given by25,52,53\nΔ𝐻TMS=𝛾2𝐻2\n𝑎\n2𝜋𝑃𝐴¹𝜔º∫\nΛ0𝑘𝐶𝑘¹𝜉º𝛿𝛼¹𝜔\u0000𝜔𝑘ºd2𝑘 (A1)\nwhere𝜉is correlation length, 𝐻𝑎is the e\u000bective anisotropy \feld of the randomly oriented\ngrain.𝑃𝐴¹𝜔º=𝜕𝜔\n𝜕𝐻\f\f\n𝐻=𝐻FMR=√︃\n1¸¹4𝜋𝑀𝑠\n2𝜔𝛾º2accounts for the conversion between the fre-\nquency and \feld swept linewidth. Λ0𝑘represents the averaging of the anisotropy axis \ruc-\ntuations over the sample. It also takes into account the ellipticity of the precession for both\nthe uniform FMR mode and the spin wave mode52. The detailed expression of Λ0𝑘can\nbe found in the Appendix of Ref.52. The coe\u000ecients in the expression of Λ0𝑘depend on\nthe type of anisotropy of the system. Here, we used \frst-order cubic anisotropy for bcc Fe.\n𝛿𝛼¹𝜔\u0000𝜔𝑘ºselects all the degenerate modes, where 𝜔represents the FMR mode frequency\nand𝜔𝑘represents the spin wave mode frequency. The detailed expression of 𝜔𝑘can be found\nin Ref.25. In the ideal case where Gilbert damping is 0, 𝛿𝛼is the Dirac delta function. For a\n\fnite damping, 𝛿𝛼¹𝜔0\u0000𝜔𝑘ºis replaced by a Lorentzian function1\n𝜋¹𝛼IP𝜔𝑘𝛾º𝜕𝜔𝜕𝐻\n¹𝜔𝑘\u0000𝜔º2¸»¹𝛼IP𝜔𝑘𝛾º𝜕𝜔𝜕𝐻¼2,\nwhich is centered at 𝜔and has the width of ¹2𝛼IP𝜔𝑘𝛾º𝜕𝜔𝜕𝐻.\nFinally,𝐶𝑘¹𝜉º(or𝐶𝑘¹𝜉𝜆º) is the Fourier transform of the grain-to-grain internal \feld\ncorrelation function, Eq. (3) (or Eq. (4)). For the description of magnetic inhomogeneity\nanalogous to the simple self-a\u000ene topographical surface58, the Fourier transform of the\ncorrelation function, Eq. (3), is\n𝐶𝑘¹𝜉º=2𝜋𝜉2\n»1¸¹𝑘𝜉º2¼3\n2 (A2)\nas also used in Refs.25,52,53. For the description analogous to the mounded surface, the\nFourier transform of the correlation function, Eq. (4), is58\n𝐶𝑘¹𝜉𝜆º=8𝜋3𝜉2\u0010\n1¸2𝜋2𝜉2\n𝜆2¸𝜉2\n2𝑘2\u0011\n\u0014\u0010\n1¸2𝜋2𝜉2\n𝜆2¸𝜉2\n2𝑘2\u00112\n\u0000\u0010\n2𝜋𝜉2\n𝜆𝑘\u00112\u001532 (A3)\nWhen𝜆\u001d𝜉, Eq. (A3) becomes\n𝐶𝑘¹𝜉º\u00198𝜋3𝜉2\n\u0010\n1¸𝜉2\n2𝑘2\u00112 (A4)\n19100102104106108101010-2410-2210-2010-1810-1610-1410-1210-10 \nSelf-affine  \nMounded λ = 10 nm \nMounded λ = 100 nm \nMounded λ = 1000 nmCk [m2]k\n [m-1]ξ = 100 nmFIG. 7. Fourier transform of correlation function for mounded surfaces as a function of wavenumber\n𝑘for three di\u000berent 𝜆values. Fourier transform of correlation function for self-a\u000ene surfaces as a\nfunction of 𝑘is also included for comparison purpose. 𝜉is set as 100 nm for all curves.\nwhich has a similar form as Eq. (A2). This similarity can also be demonstrated graphically.\nFigure 7 plots a self-a\u000ene 𝐶𝑘curve (Eq. (A2)) at 𝜉=100 nm and three mounded 𝐶𝑘curves\n(Eq. (A3)) at 𝜆=10, 100, 1000 nm. 𝜉in mounded 𝐶𝑘curves is set as 100 nm as well. It\nis clearly shown in Fig. 7 that when 𝜆=1000 nm, the peak appearing in 𝜆=10 and 100\nnm mounded 𝐶𝑘curves disappears and the curve shape of mounded 𝐶𝑘resembles that of\nself-a\u000ene𝐶𝑘.\nThe hump feature in Fig. 4 is governed by both 𝛿𝛼and𝐶𝑘(see Eq. A1). 𝛿𝛼has the shape\nof1in reciprocal space ( 𝑘space), as shown in our videos in the Supplemental Material as\nwell as Fig. 5(b) of Ref.53and Fig 2 (b) of Ref.25. The size of the contour of the degenerated\nspin wave modes in 𝑘space increases as the microwave frequency 𝑓increases, which means\nthe number of available degenerate spin wave modes increases as 𝑓increases. As shown\nin Fig. 7, self-a\u000ene 𝐶𝑘is nearly constant with the wavenumber 𝑘until𝑘reaches\u00181𝜉.\nThis suggests that the system becomes e\u000bectively more uniform (i.e. weaker inhomogeneous\nperturbation) when the length scale falls below the characteristic correlation length 𝜉(i.e.,\n𝑘 ¡1𝜉). Because inhomogeneities serve as the scattering centers of two-magnon scattering\n20process, degenerate spin wave modes with 𝑘 ¡1𝜉are less likely to be scattered into.\nNow we consider the 𝑓dependence of the two-magnon scattering rate. When 𝑓is small,\nthe two-magnon scattering rate increases as 𝑓increases because more degenerate spin wave\nmodes become available as 𝑓increases. When 𝑓further increases, the wavenumber 𝑘of\nsome degenerate spin wave modes exceeds 1 𝜉. This will decrease the overall two-magnon\nscattering rate because the degenerate spin wave modes with 𝑘 ¡1𝜉are less likely to be\nscattered into, as discussed above. Furthermore, the portion of degenerate spin wave modes\nwith𝑘 ¡ 1𝜉increases as 𝑓continues to increase. When the impact of decreasing two-\nmagnon scattering rate for degenerate spin wave modes with high 𝑘surpasses the impact\nof increasing available degenerate spin wave modes, the overall two-magnon scattering rate\nwill start to decrease as 𝑓increases. Consequently, the nonlinear trend { i.e., a \\hump\" {\nin FMR linewidth Δ𝐻TMS vs𝑓appears in Fig. 4.\nHowever, the scenario discussed above can only happen when 𝜉is large enough, because\nthe wavenumber 𝑘of degenerate spin wave modes saturates (i.e., reaches a limit) as 𝑓\napproaches in\fnity. If the limit value of 𝑘is smaller than 1𝜉, the two-magnon scattering\nrate will increase monotonically as 𝑓increases. In that case the hump feature will not\nappear. See our videos in the Supplemental Material that display the 𝑓dependence of Λ0𝑘,\n𝛿𝛼¹𝜔\u0000𝜔𝑘º,𝐶𝑘¹𝜉º\n2𝜋𝜉2,Λ0𝑘𝐶𝑘¹𝜉º𝛿𝛼¹𝜔\u0000𝜔𝑘º\n2𝜋𝜉2 , andΔ𝐻TMS for various𝜉values.\nPrevious discussions of the hump feature are all based on the self-a\u000ene correlation func-\ntion (Eq. 3). The main di\u000berence between the mounded correlation function (Eq. 4) and the\nself-a\u000ene correlation function (Eq. 3) is that the mounded correlation function has a peak\nwhen𝜆is not much larger than 𝜉as shown in Fig. 7. This means when the wavenumber\n𝑘of degenerate spin wave modes enters (leaves) the peak region, two-magnon scattering\nrate will increase (decrease) much faster compared to the self-a\u000ene correlation function. In\nother words, the mounded correlation function can generate a narrower hump compared to\nthe self-a\u000ene correlation function in the two-magnon linewidth Δ𝐻TMS vs𝑓plot, which is\nshown in Fig. 6 (b, c).\n1Z. Diao, Z. Li, S. Wang, Y. Ding, A. Panchula, E. Chen, L.-C. 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Hals and Arne Brataas\nDepartment of Physics, Norwegian University of Science and Technology, NO-7491, Trondheim, Norway\nWe compute the Gilbert damping in (Ga,Mn)As based on the scattering theory of magnetization\nrelaxation. The disorder scattering is included non-perturbatively. In the clean limit, spin-pumping\nfrom the localized d-electrons to the itinerant holes dominates the relaxation processes. In the\ndi\u000busive regime, the breathing Fermi-surface e\u000bect is balanced by the e\u000bects of interband scattering,\nwhich cause the Gilbert damping constant to saturate at around 0.005. In small samples, the system\nshape induces a large anisotropy in the Gilbert damping.\nI. INTRODUCTION\nThe magnetization dynamics of a ferromagnet can be\ndescribed phenomenologically by the Landau-Lifshitz-\nGilbert (LLG) equation:1,2\n1\n\rdM\ndt=\u0000M\u0002He\u000b+M\u0002\"~G(M)\n\r2M2sdM\ndt#\n:(1)\nHere,\ris the gyromagnetic ratio, He\u000bis the e\u000bective\nmagnetic \feld (which is the functional derivative of the\nfree energy He\u000b=\u0000\u000eF[M]=\u000eM),Mis the magnetiza-\ntion andMsis its magnitude. The Gilbert damping con-\nstant ~G(M) parameterizes the dissipative friction process\nthat drives the magnetization towards an equilibrium\nstate.3In the most general case, ~G(M) is a symmetric\npositive de\fnite matrix that depends on the magnetiza-\ntion direction; however, it is often assumed to be inde-\npendent of Mand proportional to the unit matrix, as-\nsumptions which are valid for isotropic systems. Gilbert\ndamping is important in magnetization dynamics. It de-\ntermines the magnitudes of the external magnetic \felds4\nand the current densities1that are required to reorient\nthe magnetization direction of a ferromagnet. Therefore,\na thorough understanding of its properties is essential for\nmodeling ferromagnetic systems.\nThe main contribution to the Gilbert damping process\nin metallic ferromagnets is the generation of electron-\nhole pairs.1,2,5,6A model that captures this process was\ndeveloped by Kambersky.5In this model, the electrons\nare excited by a time-varying magnetization via electron-\nmagnon coupling. If the ferromagnet is in metallic con-\ntact with other materials, the spin-pumping into the ad-\njacent leads provides an additional contribution to the\nmagnetization relaxation.7A general theory that cap-\ntures both of these e\u000bects was recently developed.8The\nmodel expresses the ~G(M) tensor in terms of the scatter-\ning matrix Sof the ferromagnetic system ( m\u0011M=Ms):\n~Gij(m) =\r2\u0016h\n4\u0019Re\u001a\nTr\u0014@S\n@mi@Sy\n@mj\u0015\u001b\n: (2)\nThe expression is evaluated at the Fermi energy. Instead\nof~G(M), one often parameterizes the damping by the\ndimensionless Gilbert damping parameter ~ \u000b\u0011~G=\rM s.\nEq. (2) allows studying both the e\u000bects of the systemshape and the disorder dependency of the magnetization\ndamping beyond the relaxation time approximation.9\nIn anisotropic systems, the Gilbert damping is ex-\npected to be a symmetric tensor with non-vanishing o\u000b-\ndiagonal terms. We are interested in how this tensor\nstructure in\ruences the dynamics of the precessing mag-\nnetization in (Ga,Mn)As. Therefore, to brie\ry discuss\nthis issue, let us consider a homogenous ferromagnet in\nwhich the magnetization direction m=m0+\u000empre-\ncesses with a small angle around the equilibrium direc-\ntionm0that points along the external magnetic \feld\nHext.10For clarity, we neglect the anisotropy in the\nfree energy and choose the coordinate system such that\nm0= (0 0 1) and \u000em= (mxmy0). For the lowest order\nof Gilbert damping, the LLG equation can be rewritten\nas:_m=\u0000\rm\u0002Hext+\rm\u0002(~\u000b[Hext\u0002m]), where ~\u000b[:::]\nis the dimensionless Gilbert damping tensor that acts on\nthe vector Hext\u0002m. Linearizing the LLG equation re-\nsults in the following set of equations for mxandmy:\n\u0012\n_mx\n_my\u0013\n=\u0000\rHext \n\u000b(0)\nyy (1\u0000\u000b(0)\nxy)\n\u0000(1 +\u000b(0)\nxy)\u000b(0)\nxx!\u0012\nmx\nmy\u0013\n:\n(3)\nHere,\u000b(0)\nijare the matrix elements of ~ \u000bwhen the tensor\nis evaluated along the equilibrium magnetization direc-\ntionm0. For the lowest order of Gilbert damping, the\neigenvalues of (3) are \u0015\u0006=\u0006i\rH ext\u0000\rHext\u000b, and the\neigenvectors describe a precessing magnetization with a\ncharacteristic life time \u001c= (\u000b\rH ext)\u00001. The e\u000bective\ndamping coe\u000ecient \u000bis:10\n\u000b\u00111\n2\u0010\n\u000b(0)\nxx+\u000b(0)\nyy\u0011\n: (4)\nThe value of \u000bis generally anisotropic and depends on\nthe static magnetization direction m0. The magnetiza-\ntion damping is accessible via ferromagnetic resonance\n(FMR) experiments by measuring the linear relation-\nship between the FMR line width and the precession fre-\nquency. This linear relationship is proportional to \u001c\u00001\nand thus depends linearly on \u000b. Therefore, an FMR ex-\nperiment can be used to determine the e\u000bective damping\ncoe\u000ecient\u000b. In contrast, the o\u000b-diagonal terms, \u000b(0)\nxy\nand\u000b(0)\nyx, do not contribute to the lowest order in the\ndamping and are di\u000ecult to probe experimentally.\nIn this paper, we use Eq. (2) to study the anisotropy\nand disorder dependency of the Gilbert damping in thearXiv:1105.4148v2  [cond-mat.mtrl-sci]  2 Nov 20112\nferromagnetic semiconductor (Ga,Mn)As. Damping co-\ne\u000ecients of this material in the range of \u000b\u00180:004\u00000:04\nfor annealed samples have been reported.11{14The damp-\ning is anisotropically dependent on the magnetization\ndirection.11,12,14The few previous calculations of the\nGilbert damping constant in this material have indicated\nthat\u000b\u00180:003\u00000:04.11,15{17These theoretical works\nhave included the e\u000bects of disorder phenomenologically,\nfor instance, by applying the relaxation time approxima-\ntion. In contrast, Eq. (2) allows for studying the disor-\nder e\u000bects fully and non-perturbatively for the \frst time.\nIn agreement with Ref. 15, we show that spin-pumping\nfrom the localized d-electrons to the itinerant holes dom-\ninates the damping process in the clean limit. In the\ndi\u000busive regime, the breathing Fermi-surface e\u000bect is bal-\nanced by e\u000bects of the interband transitions, which cause\nthe damping to saturate. In determining the anisotropy\nof the Gilbert damping tensor, we \fnd that the shape of\nthe sample is typically more important than the e\u000bects\nof the strain and the cubic symmetry in the GaAs crys-\ntal.18This shape anisotropy of the Gilbert damping in\n(Ga,Mn)As has not been reported before and provides\na new direction for engineering the magnetization relax-\nation.\nII. MODEL\nThe kinetic-exchange e\u000bective Hamiltonian approach\ngives a reasonably good description of the electronic\nproperties of (Ga,Mn)As.19The model assumes that the\nelectronic states near the Fermi energy have the character\nof the host material GaAs and that the spins of the itin-\nerant quasiparticles interact with the localized magnetic\nMn impurities (with spin 5/2) via the isotropic Heisen-\nberg exchange interaction. If the s-d exchange interac-\ntion is modeled by a mean \feld, the e\u000bective Hamiltonian\ntakes the form:19,20\nH=HHoles +h(r)\u0001s; (5)\nwhereHHoles is the k\u0001pKohn-Luttinger Hamiltonian de-\nscribing the valence band structure of GaAs and h(r)\u0001s\nis a mean \feld description of the s-d exchange interac-\ntion between the itinerant holes and the local magnetic\nimpurities ( sis the spin operator). The exchange \feld\nhis antiparallel to the magnetization direction m. The\nexplicit form of HHoles that is needed for realistic model-\ning of the band structure of GaAs depends on the doping\nlevel of the system. Higher doping levels often require\nan eight-band model, but a six- or four-band model may\nbe su\u000ecient for lower doping levels. In the four-band\nmodel, the Hamiltonian is projected onto the subspace\nspanned by the four 3/2 spin states at the top of the\nGaAs valence band. The six-band model also includes\nthe spin-orbit split-o\u000b bands with spin 1/2. The spin-\norbit splitting of the spin 3/2 and 1/2 states in GaAs\nis 341 meV.21We consider a system with a Fermi level\nof 77 meV when measured from the lowest subband. Inthis limit, the following four-band model gives a su\u000ecient\ndescription:\nH=1\n2m\u0014\n(\r1+5\n2\r2)p2\u00002\r3(p\u0001J)2+h\u0001J\u0015\n+\n\r3\u0000\r2\nm(p2\nxJ2\nx+c:p:) +Hstrain +V(r): (6)\nHere, pis the momentum operator, Jiare the spin 3/2\nmatrices22and\r1,\r2and\r3are the Kohn-Luttinger pa-\nrameters.V(r) =P\niVi\u000e(r\u0000Ri) is the impurity scat-\ntering potential, where Riis the position of the impurity\niandViare the scattering strengths of the impurities23\nthat are randomly and uniformly distributed in the in-\nterval [\u0000V0=2;V0=2].Hstrain is a strain Hamiltonian and\narises because the (Ga,Mn)As system is grown on top\nof a substrate (such as GaAs).24The two \frst terms in\nEq. (6) have spherical symmetry, and the term propor-\ntional to\r3\u0000\r2represents the e\u000bects of the cubic sym-\nmetry of the GaAs crystal. Both this cubic symmetry\nterm25and the strain Hamiltonian24are small compared\nto the spherical portion of the Hamiltonian. A numeri-\ncal calculation shows that they give a correction to the\nGilbert damping on the order of 10%. However, the un-\ncertainty of the numerical results, due to issues such as\nthe sample-to-sample disorder \ructuations, is also about\n10%; therefore, we cannot conclude how these terms in-\n\ruence the anisotropy of the Gilbert damping. Instead,\nwe demonstrate that the shape of the system is the dom-\ninant factor in\ruencing the anisotropy of the damping.\nTherefore, we disregard the strain Hamiltonian Hstrain\nand the term proportional to \r3\u0000\r2in our investigation\nof the Gilbert damping.\nGaAs GaAs (Ga,Mn)As \nyx\nFIG. 1: We consider a (Ga,Mn)As system attached along the\n[010] direction to in\fnite ballistic GaAs leads. The scattering\nmatrix is calculated for the (Ga,Mn)As layer and one lattice\npoint into each of the leads. The magnetization is assumed to\nbe homogenous. In this paper, we denote the [100] direction\nas the x-axis, the [010] direction as the y-axis and the [001]\ndirection as the z-axis\nWe consider a discrete (Ga,Mn)As system with\ntransverse dimensions Lx2 f17;19;21gnm,Lz2\nf11;15;17gnm andLy= 50 nm and connected to in\fnite\nballistic GaAs leads, as illustrated in Fig. 1. The leads\nare modeled as being identical to the (Ga,Mn)As system,\nexcept for the magnetization and disorder. The lattice\nconstant is 1 nm, which is much less than the Fermi wave-\nlength\u0015F\u001810 nm. The Fermi energy is 0.077 eV when3\nmeasured from the lowest subband edge. The Kohn-\nLuttinger parameters are \r1= 7:0 and\r2=\r3= 2:5,\nimplying that we apply the spherical approximation for\nthe Luttinger Hamiltonian, as mentioned above.25We\nusejhj= 0:032 eV for the exchange-\feld strength. To\nestimate a typical saturation value of the magnetization,\nwe useMs= 10j\rj\u0016hx=a3\nGaAs withx= 0:05 as the doping\nlevel andaGaAs as the lattice constant for GaAs.26\nThe mean free path lfor the impurity strength V0is\ncalculated by \ftting the average transmission probability\nT=hGi=GshtoT(Ly) =l=(l+Ly),27whereGshis the\nSharvin conductance and hGiis the conductance for a\nsystem of length Ly.\nThe scattering matrix is calculated numerically us-\ning a stable transfer matrix method.28The disorder ef-\nfects are fully and non-perturbatively included by the\nensemble average h\u000bi=PNI\nn=1\u000bn=NI, whereNIis the\nnumber of di\u000berent impurity con\fgurations. All the\ncoe\u000ecients are averaged until an uncertainty \u000eh\u000bi=r\u0010\nh\u000b2i\u0000h\u000bi2\u0011\n=NIof less than 10% is achieved. The\nvertex corrections are exactly included in the scattering\nformalism.\nIII. RESULTS AND DISCUSSION\nWithout disorder, the Hamiltonian describing our sys-\ntem is rotationally symmetric around the axis parallel\ntoh. Let us brie\ry discuss how this in\ruences the\nparticular form of the Gilbert damping tensor ~ \u000b.29For\nclarity, we choose the coordinate axis such that the ex-\nchange \feld points along the z-axis. In this case, the\nHamiltonian is invariant under all rotations Rzaround\nthe z-axis. This symmetry requires the energy dissipa-\ntion _E/_mT~\u000b_m(_mTis the transposed of _m) of the\nmagnetic system to be invariant under the coordinate\ntransformations r0=Rzr(i.e., ( _m0)T~\u000b0_m0=_mT~\u000b_m\nwhere m0=Rzmand ~\u000b0is the Gilbert damping ten-\nsor in the rotated coordinate system). Because ~ \u000bonly\ndepends on the direction of m, which is unchanged\nunder the coordinate transformation Rz, ~\u000b= ~\u000b0and\nRT\nz~\u000bRz= ~\u000b. Thus, ~\u000bandRzhave common eigenvectors\b\nj\u0006i\u0011 (jxi\u0006ijyi)=p\n2;jzi\t\n, and the spectral decompo-\nsition of ~\u000bis ~\u000b=\u000b+j+ih+j+\u000b\u0000j\u0000ih\u0000j +\u000bzjzihzj.\nRepresenting the damping tensor in the fjxi;jyi;jzig\ncoordinate basis yields \u000b\u0011~\u000bxx= ~\u000byy= (\u000b++\u000b\u0000)=2,\n~\u000bzz=\u000bz, ~\u000byx= ~\u000bxy= 0, and\u000b+=\u000b\u0000. The last\nequality results from real tensor coe\u000ecients. However,\n\u000bzzcannot be determined uniquely from the energy dis-\nsipation formula _E/_mT~\u000b_mbecause _mis perpendicu-\nlar to the z-axis. Therefore, \u000bzzhas no physical signi\f-\ncance and the energy dissipation is governed by the single\nparameter\u000b. For an in\fnite system, this damping pa-\nrameter does not depend on the speci\fc direction of the\nmagnetization, i.e., it is isotropic because the symmetry\nof the Hamiltonian is not directly linked to the crystallo-\ngraphic axes of the underlying crystal lattice (when the\n0 0.5 1 1.5 200.2 0.4 0.6 0.8 1\nφ / πθ / π\n44.5 55.5 6x 10 −3 \n0 0.5 1 1.5 200.2 0.4 0.6 0.8 1\nφ / πθ / π\n4.5 55.5 6x 10 −3 a\nbFIG. 2: ( a) The dimensionless Gilbert damping parameter\n\u000bas a function of the magnetization direction for a system\nwhereLx= 17 nm,Ly= 50 nm and Lz= 17 nm. ( b) The\ndimensionless Gilbert damping parameter \u000bas a function of\nthe magnetization direction for a system where Lx= 21 nm,\nLy= 50 nm and Lz= 11 nm. Here, \u0012and\u001eare the polar and\nazimuth angles, respectively, that describe the local magne-\ntization direction m= (sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012). In both\nplots, the mean free path is l\u001822 nm\ncubic symmetry term in Eq. (6) is disregarded). For a\n\fnite system, the shape of the system induces anisotropy\nin the magnetization damping. This e\u000bect is illustrated\nin Fig. 2, which plots the e\u000bective damping in Eq. (4) as\na function of the magnetization directions for di\u000berent\nsystem shapes. When the cross-section of the conductor\nis deformed from a regular shape to the shape of a thin-\nner system, the anisotropy of the damping changes. The\nmagnetization damping varies from a minimum value of\naround 0.004 to a maximum value of 0.006, e.g., the\nanisotropy is around 50%. The relaxation process is\nlargest along the axis where the ballistic leads are con-\nnected, i.e., the y-axis. This shape anisotropy is about\nfour- to \fve-times stronger than the anisotropy induced\nby the strain and the cubic symmetry terms in the Hamil-\ntonian (6), which give corrections of about 10 percent.\nFor larger systems, we expect this shape e\u000bect to be-4\ncome less dominant. In these systems, the anisotropy\nof the bulk damping parameter, which is induced by the\nanisotropic terms in the Hamiltonian, should play a more\nsigni\fcant role. The determination of the system size\nwhen the strain and cubic anisotropy become comparable\nto the shape anisotropy e\u000bects is beyond the scope of this\npaper because the system size is restricted by the com-\nputing time. However, this question should be possible to\ninvestigate experimentally by measuring the anisotropy\nof the Gilbert damping as a function of the \flm thickness.\n0 1 2 3 4 52345678910 x 10 −3 \nLy/l ααmin\nαmax \nαmean \nFIG. 3: The e\u000bective dimensionless Gilbert damping (4) as a\nfunction of the disorder. Here, lis the mean free path and\nLyis the length of the ferromagnetic system in the transport\ndirection.\u000bminand\u000bmaxare the minimum and maximum val-\nues of the anisotropic Gilbert damping parameter and \u000bmean is\nthe e\u000bective damping parameter averaged over all the magne-\ntization directions. The system dimensions are Lx= 19 nm,\nLy= 50 nm and Lz= 15 nm.\nWe next investigate how the magnetization relaxation\nprocess depends on the disorder. Ref. 15 derives an ex-\npression that relates the Gilbert damping parameter to\nthe spin-\rip rate T2of the system: \u000b/T2(1 + (T2)2)\u00001.\nIn the low spin-\rip rate regime, this expression scales\nwithT2as\u000b/T\u00001\n2, while the damping parameter is\nproportional to \u000b/T2in the opposite limit . As ex-\nplained in Ref. 15, the low spin-\rip regime is dominated\nby the spin-pumping process in which angular momen-\ntum is transferred to the itinerant particles; the trans-\nferred spin is then relaxed with a rate proportional to\nT\u00001\n2. This process appears inside the ferromagnet itself,\ni.e., the spin is transferred from the magnetic system to\nthe itinerant particles in the ferromagnet, which are then\nrelaxed within the ferromagnet. Therefore, this relax-\nation mechanism is a bulk process and should not be\nconfused with the spin-pumping interface e\u000bect across\nthe normal metal jferromagnet interfaces reported in\nRef. 7. In (Ga, Mn)As, this bulk process corresponds to\nspin-pumping from the d-electrons of the magnetic Mn\nimpurities to the itinerant spin 3/2 holes in the valence\nband of the host compound GaAs. The transfer of spin\nto the holes is then relaxed by the impurity scatteringwithin the ferromagnet. By contrast, the opposite limit\nis dominated by the breathing Fermi-surface mechanism.\nIn this mechanism, the spins of the itinerant particles are\nnot able to follow the local magnetization direction adia-\nbatically and lag behind with a delay time of T2. In our\nsystem, which has a large spin-orbit coupling in the band\nstructure, we expect the spin-\rip rate to be proportional\nto the mean free path ( l/T2).30The e\u000bective dimen-\nsionless Gilbert damping (4) is plotted as a function of\ndisorder in Fig. 3. The damping ( \u000bmean) partly shows\nthe same behavior as that reported in Ref. 15. For clean\nsystems (i.e., those with a low spin-\rip rate regime), the\ndamping increases with disorder. In such a regime, the\ntransfer of angular momentum to the spin 3/2 holes is the\ndominant damping process, i.e., the bulk spin-pumping\nprocess dominates. \u000bmean starts to decrease for smaller\nmean free paths, implying that the main contribution\nto the damping comes from the breathing Fermi-surface\nprocess. Refs. 11,16,17 have reported that the Gilbert\ndamping may start to increase as a function of disorder in\ndirtier samples. The interband transitions become more\nimportant with decreasing quasi-particle life times and\nstart to dominate the intraband transitions (The intra-\nband transitions give rise to the breathing Fermi-surface\ne\u000bect). We do not observe an increasing behavior in the\nmore di\u000busive regime, but we \fnd that the damping sat-\nurates at a value of around 0.0046 (See Fig. 3). In this\nregime, we believe that the breathing Fermi-surface e\u000bect\nis balanced by the interband transitions. The damping\ndoes not vanish in the limit 1 =l= 0 due to scattering\nat the interface between the GaAs and (Ga,Mn)As lay-\ners in addition to spin-pumping into the adjacent leads\n(an interface spin-pumping e\u000bect, as explained above).\nFig. 3 shows that the shape anisotropy of the damp-\ning is reduced by disorder because the di\u000berence between\nthe maximum ( \u000bmax) and minimum ( \u000bmin) values of the\ndamping parameter decrease with disorder. We antici-\npate this result because disorder increases the bulk damp-\ning e\u000bect, which is expected to be isotropic for an in\fnite\nsystem.\nIV. SUMMARY\nIn this paper, we studied the magnetization damping\nin the ferromagnetic semiconductor (Ga,Mn)As. The\nGilbert damping was calculated numerically using a\nrecently developed scattering matrix theory of mag-\nnetization dissipation.8We conducted a detailed non-\nperturbative study of the e\u000bects of disorder and an inves-\ntigation of the damping anisotropy induced by the shape\nof the sample.\nOur analysis showed that the damping process is\nmainly governed by three relaxation mechanisms. In the\nclean limit with little disorder, we found that the magne-\ntization dissipation is dominated by spin-pumping from\nthe d-electrons to the itinerant holes. For shorter mean\nfree paths, the breathing Fermi-surface e\u000bect starts to5\ndominate, which causes the damping to decrease. In\nthe di\u000busive regime, the breathing Fermi-surface e\u000bect\nis balanced by the interband transitions and the e\u000bective\ndamping parameter saturates at a value on the order of\n0.005.\nFor the small samples considered in this study, we\nfound that the shape of the system was typically more\nimportant than the anisotropic terms in the Hamiltonian\nfor the directional dependency of the damping parame-\nter. This shape anisotropy has not been reported beforeand o\u000bers a new way of manipulating the magnetization\ndamping.\nV. ACKNOWLEDGMENTS\nThis work was partially supported by the European\nUnion FP7 Grant No. 251759 \\MACALO\".\n1For a review, see D. C. Ralph and M. Stiles, J. Magn.\nMagn. Mater. 320, 1190 (2008), and reference therein.\n2B. Heinrich, D. Fraitov\u0013 a, and V. Kambersky, Phys. Status\nSolidi 23, 501 (1967); V. Kambersky, Can. J. Phys. 48,\n2906 (1970); V. Korenman and R.E. Prange, Phys. Rev.\nB6, 2769 (1972); V.S. Lutovinov and M.Y. Reizer, Zh.\nEksp. Teor. Fiz. 77, 707 (1979) [Sov. Phys. JETP 50, 355\n(1979)]; V.L. Safonov and H.N. Bertram, Phys. Rev. B 61,\nR14893 (2000); J. Kunes and V. Kambersky, Phys. Rev. B\n65, 212411 (2002); V. Kambersky Phys. Rev. B 76, 134416\n(2007).\n3T.L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n4J.A.C. Bland and B. Heinrich, Ultrathin Magnetic Struc-\ntures III Fundamentals of Nanomagnetism (Springer Ver-\nlag, Heidelberg, 2004).\n5V. Kambersky, Czech. J. Phys. B 26, 1366 (1976).\n6K. Gilmore, Y.U. Idzerda, and M.D. Stiles, Phys. Rev.\nLett. 99, 027204 (2007).\n7Y. Tserkovnyak, A. Brataas and G.E.W. Bauer, Phys. Rev.\nLett. 88, 117601 (2002).\n8A. Brataas, Y. Tserkovnyak, and G.E.W. Bauer, Phys.\nRev. Lett. 101, 037207 (2008); Phys. Rev. B 84, 054416\n(2011).\n9A. A. Starikov, P. J. Kelly, A. Brataas, Y. Tserkovnyak,\nand G. E. W. Bauer, Phys. Rev. Lett. 105, 236601\n(2010); Y. Liu, Z. Yuan, A. A. Starikov, and P. J. Kelly,\narXiv:1102.5305.\n10A similar analysis is presented in J. Seib, D. Steiauf, and\nM. F ahnle, Physical Review B 79, 092418 (2009).\n11J. Sinova, T. Jungwirth, X. Liu, Y. Sasaki, J.K. Furdyna,\nW.A. Atkinson, and A.H. MacDonald, Phys. Rev. B 69,\n085209 (2004).\n12Y.H. Matsuda, A. Oiwa, K. Tanaka, and H. Munekata,\nPhysica B 376-377 , 668 (2006).\n13A. Wirthmann et al. , Appl. Phys. Lett. 92, 232106 (2008).\n14Kh. Khazen et al. , Phys. Rev. B 78, 195210 (2008).\n15Y. Tserkovnyak, G.A. Fiete, and B.I. Halperin, Appl.\nPhys. Lett. 84, 5234 (2004).\n16I. Garate and A.H. MacDonald, Phys. Rev. B 79, 064403(2009).\n17I. Garate and A.H. MacDonald, Phys. Rev. B 79, 064404\n(2009).\n18This shape anisotropy in the Gilbert damping should not\nbe confused with the shape anisotropy (in the anisotropy\n\feld) caused by surface dipoles in non-spherical systems.\n19T. Jungwirth, J. Sinova, J. Ma\u0014 sek, J. Ku\u0014 cera, and A. H.\nMacDonald, Rev. Mod. Phys. 78, 809864 (2006).\n20M. Abolfath, T. Jungwirth, J. Brum, and A. H. MacDon-\nald, Phys. Rev. B 63, 054418 (2001).\n21P.Y. Yu and M. Cardona, Fundamentals of Semicon-\nductors: Physics and Materials Properties , 3rd Edition\n(Springer Verlag, Berlin, 2005).\n22Note that in Eq. (6) the spin operator s(in the p-d ex-\nchange term) is represented in the basis consisting of the\nfour spin 3/2 states ( s=J=3). The factor 1 =3 is absorbed\nin the exchange \feld h.\n23In the discrete version of Eq. (6), as used in the numerical\ncalculation, we have one impurity at each lattice site.\n24A. Chernyshov, M. Overby, X. Liu, J.K. Furdyna, Y.\nLyanda-Geller, and L.P. Rokhinson, Nature Physics 5, 656\n(2009).\n25A. Baldereschi and N.O. Lipari, Phys. Rev. B 8, 2697\n(1973).\n26The prefactor of 10 comes from 4 Ga atoms per unit cell\ntimes spin 5/2 per substitutional Mn, which are assumed to\nbe fully polarized. The reduction of the net magnetization\ndue to the interstitial Mn ions and p holes are disregarded\nin our estimate.\n27S. Datta, Electronic Transport in Mesoscopic Systems\n(Cambridge University Press, Cambridge, England, 1995).\n28T. Usuki, M. Saito, M. Takatsu, R. A. Kiehl, and N.\nYokoyama, Phys. Rev. B 52, 8244 (1995).\n29D. Steiauf and M. F ahnle, Physical Review B 72, 064450\n(2005).\n30The spin relaxation time of holes in GaAs is on the scale of\nthe momentum relaxation time. See D.J. Hilton and C.L.\nTang, Phys. Rev. Lett. 89, 146601 (2002), and references\ntherein."    },    {        "title": "1405.4677v1.Comparison_of_micromagnetic_parameters_of_ferromagnetic_semiconductors__Ga_Mn__As_P__and__Ga_Mn_As.pdf",        "content": "1 \n Comparison of micromagnetic parameters of ferromagnetic \nsemiconductors (Ga,Mn)(As,P) and (Ga,Mn)As  \n \n \nN. Tesařová1, D. Butkovi čová1, R. P. Campion2, A.W. Rushforth2, K. W. Edmonds, \nP. Wadley2, B. L. Gallagher2, E. Schmoranzerová,1 F. Trojánek1, P. Malý1, P. Motloch4, \nV. Novák3, T. Jungwirth3, 2, and P. N ěmec1,* \n \n1 Faculty of Mathematics and Physics, Charles University in Prague, Ke Karlovu 3, 121 16 Prague 2,  \nCzech Republic \n2School of Physics and Astronomy, University of Nottingham, Nottingham NG72RD, United Kingdom \n3 Institute of Physics ASCR, v.v.i., Cukrovar nická 10, 16253 Prague  6, Czech Republic \n4 University of Chicago, Chicago, IL 60637, USA \n \nWe report on the determination of microm agnetic parameters of epilayers of the \nferromagnetic semiconductor (Ga,Mn)As, which has easy axis in the sample plane, and \n(Ga,Mn)(As,P) which has easy axis perpendicula r to the sample plane. We use an optical \nanalog of ferromagnetic resonance where the laser-pulse-induced precession of \nmagnetization is measured directly in the time domain. By the analysis of a single set of pump-and-probe magneto-optical data we determined the magnetic anisotropy fields, the \nspin stiffness and the Gilbert damping consta nt in these two materials. We show that \nincorporation of 10% of phosphorus in (Ga, Mn)As with 6% of manganese leads not only \nto the expected sign change of the perpendicu lar-to-plane anisotropy field but also to an \nincrease of the Gilbert damping and to a reduction of the spin stiffness. The observed changes in the micromagnetic parameters  upon incorporating P in (Ga,Mn)As are \nconsistent with the reduced hole density, conductivity, and Curie temperature of the (Ga,Mn)(As,P) material. We report that th e magnetization precession damping is stronger \nfor the n = 1 spin wave resonance mode than for the n = 0 uniform magnetization \nprecession mode.  \n \n PACS numbers: 75.50.Pp, 75.30.Gw, 75.70.-i, 78.20.Ls, 78.47.D- \n \n \nI. INTRODUCTION \n   \n(Ga,Mn)As is the most widely studied dilute d magnetic semiconductor (DMS) with a carrier-\nmediated ferromagnetism.1 Investigation of this material system can provide fundamental \ninsight into new physical phenomen a that are present also in ot her types of magnetic materials \n– like ferromagnetic metals – where they can  be exploited in spintronic applications.2-5 \nMoreover, the carrier concentration in DMSs is  several orders of ma gnitude lower than in \nconventional FM metals which enables manipul ation of magnetization by external stimuli – \ne.g. by electric6,7 and optical8,9 fields. Another remarkable propert y of this material is a strong \nsensitivity of the magnetic anisotropy to the ep itaxial strain. (Ga,Mn)A s epilayers are usually \nprepared on a GaAs substrate where the growth-i nduced compressive strain leads to in-plane \norientation of the easy axis (EA) for Mn concentrations ≥2%.10 However, for certain \nexperiments – e.g., for a visualization of magn etization orientation by the magneto-optical \npolar Kerr effect11-17 or the anomalous Hall effect12,18 – the EA orientation in the direction \nperpendicular to the sample plane is more suitable. To achieve this, (Ga,Mn)As layers have been grown on relaxed (In,Ga)As buffer laye rs that introduce a tensile strain in \n(Ga,Mn)As.\n11,12,14,16-18 However, the growth on (In,Ga)As la yers can result in a high density \nof line defects that can lead to high coerci vities and a strong pinning of domain walls 2 \n (DW).16,17 Alternatively, tensile strain and perpendicular-to-plane or ientation of the EA can be \nachieved by incorporation of  small amount s of phosphorus in (Ga,Mn)(As,P) layers.19,20 In \nthese epilayers, the EA can be in the sample plane for the as-grown material and perpendicular to the plane for fully annealed (Ga,Mn)(As,P).\n21 The possibility of magnetic \nanisotropy fine tuning by the thermal annealing tu rns out to be a very favorable property of \n(Ga,Mn)(As,P) because it enables the preparation of materials with extr emely low barriers for \nmagnetization switching.22,23 Compared to tensile-stained (Ga,Mn)As/(In,Ga)As films,  \n(Ga,Mn)(As,P)/GaAs epilayers show weaker DW pinning, which allows observation of the \nintrinsic flow regimes of DW propagation.13,15,24 \n Preparation of uniform (Ga,Mn)As epilayers  with minimized dens ity of unintentional \nextrinsic defects is a rather challenging task  which requires  optimized growth and post-\ngrowth annealing conditions.25 Moreover, the subsequent determination of material \nmicromagnetic parameters by the standard char acterization techniques, such as ferromagnetic \nresonance (FMR), is complicated by the fact th at these techniques require rather thick films, \nwhich may be magnetically inhomogeneous.25,26 Recently, we have reported the preparation \nof high-quality (Ga,Mn)As epila yers where the individually optimized synthesis protocols \nyielded systematic doping trends, whic h are microscopically well understood.25 \nSimultaneously with the optimization of the ma terial synthesis, we developed an optical \nanalog of FMR (optical-FMR)25, where all micromagnetic pa rameters of the in-plane \n(Ga,Mn)As were deduced from a single magneto -optical (MO) pump-and-probe experiment \nwhere a laser pulse induces precession of magnetization.27,28 In this method the anisotropy \nfields are determined from the dependence of the precession frequency on the magnitude and \nthe orientation of the external magnetic field, the Gilbert damping cons tant is deduced from \nthe damping of the precession signal, and the sp in stiffness is obtained from the mutual \nspacing of the spin wave resonance modes observe d in the measured MO signal. In this paper \nwe apply this all optical-FMR to (Ga,Mn)(As,P) . We demonstrate the applicability of this \nmethod also for the determination of microma gnetic parameters in DMS materials with a \nperpendicular-to-plane orientation of the EA. By  this method we show that the incorporation \nof P in (Ga,Mn)As leads not only to the expect ed sign change of the perpendicular-to-plane \nanisotropy field but also to a considerable  in crease of the Gilbert damping and to a reduction \nof the spin stiffness. Moreover, we illustrate that the all optical-FMR can be very effectively \nused not only for an investig ation of the uniform magnetizati on precession but also for a study \nof spin wave resonances.   \nII. EXPERIMENTAL \n  \nIn our previous work we reported in de tail on the preparation and micromagnetic \ncharacterization of (Ga,Mn)A s epilayers prepared in MBE laboratory in Prague.\n25 We also \npointed out that the preparati on of (Ga,Mn)As by this highly non-equilibrium synthesis in two \ndistinct MBE laboratories in Prague and in No ttingham led to a growth of epilayers with \nmicromagnetic parameters that showed the same doping trends.25 Nevertheless, the \npreparation of epilayers with identical paramete rs (e.g., thickness, nominal Mn content, etc.) \nin two distinct MBE machines is still a nontrivial task. Therefore, in this study of the role of \nthe phosphorus incorporation to (Ga,Mn)As we opted for a dire ct comparison of materials \nprepared in one MBE mach ine. The investigated Ga 1-xMn xAs and Ga 1-xMn xAs1-yPy epilayers \nwere prepared in Nottingham20 with the same nominal amount of Mn (x = 6%) and the same \ngrowth time on a GaAs substrate (with 50 nm  thick GaAsP buffer layer in the case of \n(Ga,Mn)(As,P)]. They differ only in the incorpor ation of P (y = 10%) in the latter epilayer. 3 \n The inferred epilayer thicknesses are (24.5 േ\t1.0) nm for both (Ga,Mn)As and \n(Ga,Mn)(As,P).29 The as-grown layers, wh ich both had the EA in th e epilayer plane, were \nthermally annealed (for 48 hours at 180°C). This led to an increase in Curie temperature and \nto a rotation of the EA to the perpendicular-to-plane orientation for (Ga,Mn)(As,P).20,21 \n The magnetic anisotropy of the samples was studied using a superconducting quantum \ninterference device (SQUID) magneto meter and by the all-optical FMR.25 The hole \nconcentration was determined by fitting to Hall  effect measurements at low temperatures \n(1.8 K) for external magnetic fields from 2 T to 6 T. In this range the magnetization is \nsaturated and one can obtain th e normal Hall coefficient af ter correction for the field \ndependence of the anomalous Hall du e to the weak magnetoresistance.30 The time-resolved \npump-and-probe MO experiments were performe d using a titanium sapphire pulsed laser \n(pulse width  200 fs) with a repetition rate of 82 MHz, which was tuned ( hυ = 1.64 eV) \nabove the GaAs band gap. The energy fl uence of the pump pulses was around 30 μJcm-2 and \nthe probe pulses were at least ten times weak er. The pump pulses were circularly polarized \n(with a helicity controlled by a quarter wave plate) and the probe pulses were linearly \npolarized (in a direction perpendicular to the external magnetic field). The time-resolved MO data reported here correspond to the polariz ation-independent part of the pump-induced \nrotation of probe polarization plane, which was computed from the measured data by \naveraging the signals obtained for the opposite  helicities of circularly polarized pump \npulses.\n27, 28 The experiment was performed close to the normal-incidence geometry, where the \nangles of incidence were 9° and 3° (measured from the sample normal) for the probe and the pump pulses, respectively.   The rotation of the probe polarization plane is caused by two MO effects – the polar \nKerr effect and the magnetic linear  dichroism, which are sensitiv e to perpendicular-to-plane \nand in-plane components of  magnetization, respectively.\n31-33 For all MO experiments, samples \nwere mounted in a cryostat and cooled down to ≈ 15 K. The cryostat was placed between the \npoles of an electromagnet and the external magnetic field Hext ranging from ≈ 0 to 585 mT \nwas applied in the sample plane,  either in the [010] or [110] crystallographic di rection of the \nsample (see inset in Fig. 1 for a definition of the coordinate system). Prior to all \nmeasurements, we always prepared the magnetiza tion in a well-defined state by first applying \na strong saturating magnetic field and then  reducing it to the desired magnitude of Hext. \n  \nIII. RESULTS AND DISCUSSION \n \nA. Sample characterization \n \n The hysteresis loops measured by SQUID magnetometry for external magnetic field \napplied along the in-plane [-110] and perpendicu lar-to-plane [001] crystallographic directions \nin (Ga,Mn)As and (Ga,Mn)(As,P) samples are s hown in Fig. 1(a) and Fig. 1(c), respectively. \nThese data confirm the expected in-plane and perp endicular-to-plane orient ations of the EA in \n(Ga,Mn)As and (Ga,Mn)(As,P), respectively. Moreover, they reveal that for the \n(Ga,Mn)(As,P) sample, an external magnetic field of  250 mT is needed to rotate the \nmagnetization into the sample plane. In Fig. 1(b) and Fig. 1(d) we show the temperature \ndependences of the remanent magnetization of the samples from which the Curie temperature \nT\nc of  130 K and  110 K can be deduced. The measur ed saturation ma gnetization also \nindicates very similar density of Mn moments contributing to the ferromagnetic state in the \ntwo samples. 4 \n  \nFig. 1 (Color online): Magnetic characterization of samples:  (a), (b) (Ga,Mn)As  and (c), (d) (Ga,Mn)(As,P). (a), \n(c) Hysteresis loops measured in at 2 K for the external magnetic field applied in the sample plane (along the \ncrystallographic direction [-110]) and perpendicular to sample plane (along the crystallographic direction [001]). (b), (d) Temperature dependence of the remanent magnetization. Inset: Definition of the coordinate system. \n \n The electrical characterization of the samp les is shown in Fig. 2. The measured data \nshow a sharp Curie point singula rity in the temperature derivative of the resistivity which \nconfirms the high quality of the samples.25 The hole densities inferred from Hall \nmeasurements are (1.3  0.2)  1021 cm-3 and (0.8  0.2)  1021 cm-3 for (Ga,Mn)As and \n(Ga,Mn)(As,P), respectively. The hole density obtained for (Ga, Mn)As is in agreement with \nour previous measurements for simila r films in magnetic  fields up 14 T.30 The reduction of \nthe density of itinerant holes quantitatively correlates with the observed increase of the resistivity of the (Ga,Mn)(As,P) film as compared to the (Ga,Mn)As sample. \n \n5 \n  \nFig. 2 (Color online): Electrical char acterization of samples. Temperature dependence of the resistivity (a) and \nits temperature derivative (b). \n \n  \nB. Time-resolved magnet o-optical experiment \n \n In Fig. 3(a) and 3(b) we show the measur ed MO signals that reflect the magnetization \ndynamics in (Ga,Mn)As and (Ga,Mn)(As,P) sa mples, respectively. Th ese signals can be \ndecomposed into the oscillatory parts [Figs. 3(c) and 3(d)] and the non-oscillatory pulse-like \nbackground [Fig. 3(e) and 3(f)].27, 28 The oscillatory part arises  from the precessional motion \nof magnetization around the quasi-equilibrium EA and the pulse-like function reflects the \nlaser-induced tilt of the EA and the laser-induced demagnetization.25,31 The pump \npolarization-independent MO data  reported here, which were measured at a relatively low \nexcitation intensity of 30 μJcm-2, can be attributed to the ma gnetization precession induced by \na transient heating of the sample due to the absorption of the laser pulse.8,9 Before absorption \nof the pump pulse the magnetization is along th e EA direction. Absorptio n of the laser pulse \nleads to a photo-injection of electron-hole pa irs. The subsequent fast non-radiative \nrecombination of photo-injected  electrons induces a transi ent increase of  the lattice \ntemperature (within tens of picoseconds afte r the impact of the pu mp pulse). The laser-\ninduced change of the lattice temperature then leads to a change of the EA position.34 As a \nresult, magnetization starts to follow th e EA shift by the precessional motion. Finally, \ndissipation of the heat leads to a return of  the EA to the equilibrium position and the \nprecession of magnetization is stopped by a Gilbert damping.25  It is apparent from Fig. 3 \nthat the measured MO signals are strongly dependent on a magnit ude of the external magnetic \nfield, which was applied in the epilayer plan e along the [010] crystall ographic direction in \nboth samples. In particular, absorption of the laser pulse does not induce precession of \nmagnetization in (Ga,Mn)(As,P) unless magnetic field stronger than 20 mT is applied [see \nFig. 3(d)].  \n  \n6 \n  \n \nFig. 3 (Color online): Time-resolved magneto-optical (MO) signals measured in (Ga,Mn)As (a) and \n(Ga,Mn)(As,P) (b) for two magnitudes of the external magnetic field applied along the [010] crystallographic \ndirection. The measured MO signals were decomposed in to oscillatory parts [(c) and (d]), which correspond to \nthe magnetization precession, and to non-oscillatory part s [(e) and (f)], which are connected with the quasi-\nequilibrium tilt of the easy axis and with the demagnetization. Note different x-scales in the left and in the right \ncolumns. \n \n The magnetization dynamics is describe d by the Landau-Lifshitz-Gilbert (LLG) \nequation that is usually expressed in the form35,36: \n \n ௗሺ௧ሻ\nௗ௧ൌെ ߛൣ ሺݐሻൈሺݐሻ൧ఈ\nெೞቂሺݐሻൈௗሺ௧ሻ\nௗ௧ቃ,      ( 1 )  \n  \nwhere   = (gμB)/ћ is the gyromagnetic ratio,  g is the Landé g-factor, μB is the Bohr magneton, \nħ is the reduced Planck constant,  is the Gilbert damping constant, and Heff is the effective \nmagnetic field. Nevertheless, it is more conve nient to express this equation in spherical \ncoordinates where the directi on of the magnetization vector M is given by the polar angle θ \nand azimuthal angle φ and where Heff can be directly connected w ith angular derivatives of the \nfree energy density functional F (see the Appendix).37 For small deviations δ and δ of \nmagnetization from its equilibrium position (given by 0 and 0), the solution of LLG \nequation can be written in the form (t) = 0 + δ(t) and (t) = 0 + δ(t) as \n \n ߠሺݐሻൌߠܣఏ݁ି௧ݏܿሺ2ݐ݂ߨΦ ఏሻ,        ( 2 )  \n ߮ሺݐሻൌ߮ܣఝ݁ି௧ݏܿ൫2ݐ݂ߨΦ ఝ൯,       ( 3 )  \n \nwhere the constants A (A) and  () represent the initial amplitude and phase of  (), \nrespectively, f is the magnetization precession frequency, and kd is the precession damping \nrate (see the Appendix). The pr ecession frequency reflects the in ternal magnetic anisotropy of \nthe sample that can be characterized by the cubic ( KC), in-plane uniaxial ( Ku) and out-of-plane \nuniaxial ( Kout) anisotropy fields (see Eq. (A4) in the Appendix).10 Moreover, f depends also on \nthe magnitude and on the orientation of Hext (see the Appendix) and, therefore, the magnetic \n7 \n field dependence of f can be used to evalua te the magnetic anisotropy fields in the sample. If \nthe applied in-plane magnetic field is strong e nough to align the magnetiz ation parallel with \nHext (i.e., for Hext exceeding the saturation field in the sa mple for a particular orientation of \nHext),   = H = π/2 and  = H and if the precession damping is relatively slow , i.e. α2 ≈ 0 f  \ncan be expressed as  \n \n ݂ൌఓಳ\nඩ൬ܪ௫௧െ2ܭ௨௧ሺଷା௦ସఝ ሻ\nଶ2ܭ௨݊݅ݏଶቀ߮ுെగ\nସቁ൰\nൈሺܪ௫௧2ܭݏܿ4߮ ுെ2ܭ௨݊݅ݏ2߮ுሻ,   (4) \n \n \n \nFig. 4 (Color online): Fourier spectrum of the oscillatory  part of the MO signal measured in (Ga,Mn)As for \nexternal magnetic fields applied alon g the [010] crystallo graphic direction. f0 and f1 indicate the frequencies of \nthe uniform magnetization precession and the fi rst spin wave resona nce, respectively. \n \n In Fig. 4 we show the fast Fourier transfor m (FFT) spectra of the oscillatory parts of \nthe MO signals measured in the (Ga,Mn )As sample for different values of Hext. This figure \nclearly reveals that for all external magnetic  fields there are two distinct oscillatory \nfrequencies present in the measured data . These precession modes are the spin wave \nresonances (SWRs) – i.e., spin waves (or magno ns) that are selectively amplified by fulfilling \nthe boundary conditions: In a homogeneous thin magnetic film  with a thickness L, only the \nperpendicular standing waves with a wave vector k fulfilling the resonant condition kL = n \n(where n is the mode number) are amplified.25,38-41 In our case – using the ferromagnetic films \nwith a thickness around 25 nm – we detect only42 the uniform magnetiza tion precession with \nzero k vector (i.e. the precession where at any instant of time all magnetic moments are \nparallel over the entire sample; n = 0 at frequency f0) and the first SWR (i.e.  n = 1 at \nfrequency f1). See the inset in Fig. 8 for a schematic de piction of the modes. In Fig. 5 we plot \nthe amplitudes of the uniform magnetization precession ( A0) and of the first SWR ( A1) as a \nfunction of the exte rnal magnetic field Hext. In the (Ga,Mn)As sample, the oscillations are \npresent even when no magnetic field is applied and the precession amplitude increases \nslightly with an increasing Hext (up to  20 mT for A0 and up to  60 mT for A1). Above this \nvalue, a further increase of Hext leads to a suppression of the oscillations, but the suppression \nof the first SWR is slower than that of the uniform magnetization precession [see Fig. 5(c)]. In \n8 \n (Ga,Mn)(As,P), the oscillatory signal starts to  appear at  50 mT, reaches its maximum for \nHext  175 mT, and a further increase of Hext leads to its monotonic d ecrease, like in the case \nof (Ga,Mn)As. The observed field dependence of the precession amplitude, which expresses \nthe sensitivity of the EA position on the laser- induced sample temperature change, can be \nqualitatively understood as follows. In (Ga,Mn)As,  the position of the EA in the sample plane \nis given by a competition between the cubic and the in-plane uniaxial magnetic \nanisotropies.10,25 The laser-induced heating of the sa mple leads to a reduction of the \nmagnetization magnitude M and, consequently, it enhances th e uniaxial anisotropy relative to \nthe cubic anisotropy.9 This is because the uniaxial anisotropy component scales with \nmagnetization as ~ M2 while the cubic component scales as ~ M4. The application of Hext \nalong the [010] crystallographic di rection deepens the minimum in  the [010] direction in the \nfree energy density functional F (due to the Zeeman term in F, see Eq. (A4) in the Appendix). \nMeasured data shown in Fig. 5 reve al that in the (Ga,Mn)As sample, Hext initially (for Hext up \nto  20 mT) destabilizes the posit ion of EA but stabilizes it for large values of Hext (where the \nposition of the energy minimum in F is dominated by the Zeeman term, which is not \ntemperature dependent). In the case of (Ga,Mn)( As,P), the position of the EA is determined \nby the strong perpendicular-to-p lane anisotropy. Therefore, w ithout an external magnetic \nfield, the laser-induced heating of the sample doe s not change significantl y the position of EA \nand, consequently, does not initiate the pr ecession of magnetization [see Fig. 5(b)]. The \napplication of an in-plane fi eld moves the energy minimum in F towards the sample plane \n[see Fig. 1(c)] which makes the EA position more  sensitive to the laser-induced temperature \nchange. Finally, for a sufficiently strong Hext, the sample magnetic anisotropy is dominated by \nthe temperature-independent Zeeman term, wh ich again suppresses the precession amplitude. \nThe markedly different ratio A1/A0 in the (Ga,Mn)As and (Ga,Mn )(As,P) samples is probably \nconnected with a different surface magnetic anis otropy and/or a slight difference in magnetic \nhomogeneity in these two samples.43,44 \n  \n \n \nFig. 5 (Color online): Dependence of the amplitude of the uniform magnetization precession ( A0) and the first \nspin wave resonance ( A1) on the magnitude of the external magnetic field ( Hext) applied along the [010] \ncrystallographic direction in (Ga,Mn )As (a) and (Ga,Mn)(As,P) (b). (c) and (d) Dependence of the ratio  A1 / A0 \non Hext.  \n9 \n C. Determination of magnetic anisotropy \n \n In Fig. 6 we plot the ma gnetic field dependences of f0 and f1 for two different \norientations of Hext. The frequency f0 of the spatially uniform precession of magnetization is \ngiven by Eq. (4). For the SWRs, where the local moments are no longer para llel (see the inset \nin Fig. 8), restoring torques due to exchange interaction and internal magnetic dipolar \ninteraction have to be included in the analysis.39-41,45 For Hext along the [010] crystallographic \ndirection (i.e., for φH = /2) Eq. (4) can be written as \n \n ݂ൌఓಳ\nඥሺܪ௫௧െ2ܭ௨௧ܭ∆ܪሻሺܪ௫௧െ2ܭെ2ܭ௨∆ܪሻ ,  (5) \n \nwhere Hn is the shift of the resonant field for the nth spin-wave mode with respect to the \nn = 0 uniform precession mode. Analogically, for Hext applied in the [110] crystallographic \ndirection (i.e., for φH = /4)  \n \n ݂ൌఓಳ\nඥሺܪ௫௧െ2ܭ௨௧2ܭܭ௨∆ܪሻሺܪ௫௧2ܭ∆ܪሻ.   (6) \n \nThe lines in Fig. 6 represent the fits  of all four measured dependencies fn = fn (Hext, H) [where \nn = 0; 1 and H = /4; /2] with a single set of anisotropy constants for each of the samples, \nwhich confirms the credibility of the fitting pr ocedure. The obtained an isotropy constants at \n≈ 15 K are: KC = (17 ± 3) mT, Ku = (11 ± 5) mT, Kout = (-200 ± 20) mT for (Ga,Mn)As and KC \n= (14 ± 3) mT, Ku = (11 ± 5) mT, Kout = (90 ± 10) mT for (Ga,Mn)(As,P), respectively (in \nboth cases we considered the Mn g-factor of 2). For (Ga,Mn)As, we can now compare these \nanisotropy constants with those obtained by the same fitting procedure for samples prepared \nin a different MBE laboratory (in Prague) – see Fig. 4 in Ref. 25. We see that the previously \nreported25 doping trends of KC and Kout predict for a sample with nominal Mn doping x = 6% \nthe anisotropy fields which are the same as thos e reported in this pape r for the sample grown \nin Nottingham. This observation is in accord with the current microscopic understanding of \ntheir origin – KC reflects the zinc-blende crystal st ructure of the host semiconductor and Kout \n  \n \n \nFig. 6 (Color online): Magnetic field dependence of the precession frequencies f0 and f1 for two different \norientations of the external magnetic field (points) measured in (Ga,Mn)As (a) and (Ga,Mn)(As,P) (b). Lines are the fits by Eqs. (5) and (6). ΔH\n1 indicates the shift of the resonant field for the first spin-wave mode with respect \nto the uniform precession mode. \n10 \n  \n is a sum of the anisotropy due to  the growth-induced lattice-ma tching strain and of the thin-\nfilm shape anisotropy, which should be the sa me for equally doped and optimally synthesized \nsamples, independent of the growth chamber. On  the other hand, the micr oscopic origin of in-\nplane uniaxial anisotropy field K\nu is still not established10,25 and our data reveal that it is \nconsiderably smaller in the sample grown in Nottingham. Th e incorporation of phosphorus \ndoes not change significa ntly the values of KC and Ku but it strongly modi fies the magnitude \nand changes the sign of Kout, which is in agreement with the previous results obtained by \nFMR experiment.22  \n  \nD. Determination of spin stiffness \n \n The observation of a higher-o rder SWR enables us to also determine the exchange \nspin stiffness constant D, which is a parameter that is rather difficult to extract from other \nexperiments in (Ga,Mn)As.25,46 In homogeneous thin films, Hn is given by the Kittel \nformula43 \n \n Δܪ\tܪെܪൌ݊ଶ\nఓಳగమ\nమ,          ( 7 )  \n \nwhere L is the thickness of the magnetic film. By fitting the data in Fig. 6, we obtained H1 = \n(363 ± 2) mT for (Ga,Mn)As and (271 ± 2) mT for (Ga,Mn)(As,P) which correspond to D = \n(2.5 ± 0.2) meVnm2 and (1.9 ± 0.2) meVnm2 for (Ga,Mn)As and (Ga,Mn)(As,P), respectively \n(note that the relatively large experimental error in D is given mainly by the uncertainty of the \nepilayer thickness).29 The value obtained for (Ga,Mn)As is again in agreement with that \nreported previously for samples grown in Prague,25 which also confirms the consistent \ndetermination of the epilayer thicknesses in both MBE laboratories.29 The incorporation of \nphosphorus leads to a  reduction of D which correlates with the decrease of the hole density,47 \nand the reduced Tc in (Ga,Mn)(As,P), as compared to its (Ga,Mn)As counterpart. \n   \nE. Determination of Gilbert damping \n \n The Gilbert damping constant α can be determined by fitting the measured dynamical \nMO signals by the LLG equation.\n35,36,48 For a relatively slow precession damping and a \nsufficiently strong external magnetic field, the analytical solution of the LLG equation gives \n(see the Appendix)  \n ݇\nௗൌߙఓಳ\nଶ൬2ܪ௫௧െ2ܭ௨௧ሺଷାହ௦ସఝ ಹሻ\nଶܭ௨ሺ1െ3݊݅ݏ2߮ ுሻ൰.  (8) \n \nEq. (8) shows not only that kd is proportional to  but also that for obtaining a correct value of \n from the measured MO precession signal damp ing it is necessary to take into account a \nrealistic magnetic anisotropy of the investigated sample. Nevert heless, the correct dependence \nof kd on magnetic anisotropy was not cons idered in the previous studies35,36,48 where only one \neffective magnetic field was used, which is probably one of the reasons why mutually \ninconsistent results were obtained for Ga 1-xMn xAs with a different Mn content x. An increase \nof  from  0.02 to  0.08 for an increase of x from 3.6% to 7.5% was reported in Ref. 36. On 11 \n the contrary, in Ref. 48 values of  from 0.06 to 0.19 – without any apparent doping trend – \nwere observed for x from 2% to 11%.  \n For numerical modeling of the measured MO data, we first computed from the LLG \nequation (Eqs. (A1) and (A2) in  the Appendix with th e measured magnetic anisotropy fields) \nthe time-dependent deviations of the spherical angles [ (t) and (t)] from the corresponding \nequilibrium values ( 0, 0). Then we calculated how such changes of  and  modify the \nstatic magneto-optical response of the samp le, which is the signal that we detect \nexperimentally31 \n \n         0\n00 2sin2 2cos2 ,MLD s MLD PKEPMt MPt Pt t MO .    (9) \n \nThe first two terms in Eq. (9) are connected wi th the out-of-plane and in-plane movement of \nmagnetization, and the last term describes a change of the sta tic magneto-optical response of \nthe sample due to the laser-induced demagnetization.31 PPKE and PMLD are MO coefficients \nthat describe the MO response of the sample which we measured independently in a static \nMO experiment,32,33 and β is the probe polarization orientation with respect to the \ncrystallographic direction [100].31 To further simplify the fitting procedure, we can extract the \noscillatory parts from the measured MO data (cf. Fig. 3), which effectively removes the MO \nsignals due to the laser-induced demagnetization [i .e., the last term in Eq. (9)] and due to the \nin-plane movement of the easy axis [i.e., a part  of the MO signal desc ribed by the second term \nin Eq. (9)].31 Examples of the fitting of the precessional MO optical data are shown in Fig. \n7(a) and (b) for (Ga,Mn)As and (Ga,Mn)(As,P), respectively. We stress that in our case the \nonly fitting parameters in the modeling are the damping coefficient  and the initial \ndeviations of the spherical angles from the corresponding equilibrium values. By this \nnumerical modeling we deduced a de pendence of the damping factor  on the external \nmagnetic field for two different orientations of Hext. At smaller fields, the dependences \nobtained show a strong anisotropy w ith respect to the field angle th at can be fully ascribed to \nthe field-angle dependence of  the precession frequency.25 However, when plotted as a \nfunction of the precession frequency, the de pendence on the field-an gle disappears – see \nFig. 7(c) and (d) for (Ga,Mn)As and (Ga,Mn )(As,P), respectively. For both materials,  \ninitially decreases monotonously with f and finally it saturates at a certain value for f ≥ \n10 GHz. A frequency-dependent (or magnetic field-dependent) damping parameter was \nreported in various magnetic materials and a va riety of underlying mechanisms responsible \nfor it were suggested  as an explanation.49-51 In our case, the most probable explanation seems \nto be the one that was used by Walowski et al.  to explain the experimental results obtained in \nthin films of nickel.49 They argued that in the low field range small magnetization \ninhomogeneities can be formed – the magnetizati on does not align parallel in an externally \napplied field, but forms ripples.49 Consequently, the measured MO signal which detects \nsample properties averaged over the laser spot size, which is in our case about 30 m wide \n(FWHM), experiences an apparent oscillation damping  because the magnetic properties \n(i.e., the precession frequencies) are slightly differing within the spot size (see Fig. 6 and 7 in \nRef. 49). On the other hand, for stronger external  fields the sample is fully homogeneous and, \ntherefore, the precession damping is not de pendent on the applied field (the precession \nfrequency), as expected for the in trinsic Gilbert damping coefficient.52,53 We note that the \nobserved monotonous frequency decrease of α is in fact a signature of a magnetic \nhomogeneity of the studied epilayers.25 The obtained frequency-independent values of α are \n(0.9 ± 0.2)  10-2 for (Ga,Mn)As and (1.9 ± 0.5)  10-2 for (Ga,Mn)(As,P), respectively. The 12 \n observed enhancement of the magnetization pre cession damping due to the incorporation of \nphosphorus is also clearly apparent  directly from Figs. 7(a) and 7(b) where the MO data with \nsimilar precession frequencies are shown for (G a,Mn)As and (Ga,Mn)(As,P), respectively. In \n(Ga,Mn)As the value of α obtained is again fully in accord with the reported Mn doping trend \nin α in this material.25  In (Ga,Mn)(As,P), the determined α is similar to the value 1.2  10-2 \nwhich was reported by Cubukcu et al. for (Ga,Mn)(As,P) with a si milar concentration of Mn \nand P.22 Comparing to the doping trends in the se ries of optimized (Ga,Mn)As materials,25 the \nvalue of α i n  o u r  ( G a , M n ) ( A s , P )  s a m p l e  i s  c o n s i s tent with the measured Gilbert damping \nconstant in  lower Mn-doped (Ga,Mn)As epilayers with similar hole densities and resistivities \nto those of the  (Ga,Mn)(As,P) film.  \n \nFig. 7 (Color online): Determination of the Gilbert da mping. (a) and (b) Oscillatory part of the MO signal \n(points) measured in (Ga,Mn)As for the external magnetic field 100 mT (a) and in (Ga,Mn)(As,P) for 350 mT \n(b); magnetic field applied along the [010] crystallographic direction leads to a similar frequency ( f0  7.5 GHz) \nin both cases. Lines are fits by the Landau-Lifshitz-G ilbert equation. (c) and (d) Dependence of the damping \nfactor () on the precession frequency for two different orienta tions of the external magnetic field in (Ga,Mn)As \n(c) and (Ga,Mn)(As,P) (d). \n \n The high quality of our MO data enables us  to evaluate not only the damping of the \nuniform magnetization precession, which is addresse d above, but also the damping of the first \nSWR. To illustrate this procedure, we show  in Fig. 8(a) the MO data measured for Hext = \n13 \n 250 mT applied along the [010] crystallographic direction in (Ga,Mn)As. The experimental \ndata (points) obtained can be fitted by a sum of two expone ntially damped cosine functions \n(line) which enables us to separate,  directly in a time domain, the contributions of the individual precession modes to the measured MO signal. In this particular case, the uniform \nmagnetization precession occurs at a frequency f\n0 = 12.2 GHz and this precession mode is \ndamped with a rate constant kd0 = 0.79 ns-1. Remarkably, the first SWR, which has a \nfrequency f1 = 23.0 GHz, has a considerably la rger damping rate constant kd1 = 1.7 ns-1 – see \nFig. 8(b) where the contribution of individual modes ar e directly compared and also Fig. 8(c) \nwhere Fourier spectra computed from the measured  MO data for two diffe rent ranges of time \ndelays are shown. To convert the damping rate constant kdn obtained to the damping constant \n \n \nFig. 8 (Color online): Comparison of the Gilbert damping of the uniform magnetization precession and of the \nfirst spin wave resonance. (a) Oscillatory part of the MO  signal (points) measured in (Ga,Mn)As for the external \nmagnetic field 250 mT applied along the [010] crystallographic direction. The solid line is a fit by a sum of two exponentially damped cosine functions that are shown in (b). Inset: Schematic illustration\n39 of the spin wave \nresonances with n = 0 (uniform magnetization precession with zero k vector) and n = 1 (perpendicular standing \nwave with a wave vector k fulfilling the resonant condition kL = ) in a magnetic film with a thickness L. (c) \nNormalized Fourier spectra computed fo r the depicted ranges of time delays from the measured MO data, which \nare shown in (a). (d) Dependence of the damping factor ( n) on the precession frequency for the uniform \nmagnetization precession ( n = 0) and the first spin wave resonance ( n = 1). \n14 \n  \nn for the n-th mode, we can use the ge neralized analytic al solution of the LLG equation. For \na sufficiently strong Hext along the [010] crystallographic direction (i.e., when φ  φH = /2), \nEq. (8) can be written as \n \n ݇ௗൌߙఓಳ\nଶሺ2ܪ௫௧2∆ܪ െ2ܭ௨௧2ܭܭ௨ሻ.    (10) \n \nFor the case of MO data measured at Hext = 250 mT, the damping constants obtained for \nmodes with n = 0 and 1 are 0 = 0.009 and 1 = 0.011, respectively. [We note that the value \nof 0 obtained from the analytical solution of LLG equation is identical to that determined by \nthe numerical fitting and shown in Fig. 7(c), which confirms the consistency of this \nprocedure.] In Fig. 8(d) we show the dependence of 0 and 1 on the precession frequency. \nThese data clearly show that  even if the modes with n = 0 and 1 were oscillating with the \nsame frequency, the SWR mode with n = 1 would have a larger  damping coefficient. \nHowever, for sufficiently high fr equencies (i.e., external magnetic  fields) the damping of the \ntwo modes is nearly equal [see Fig. 8(d)].  This feature can be ascribed to the presence of an \nextrinsic contribution to the damping coeffici ent for the SWR modes. The extrinsic damping \nprobably originates from small variations of the sample thickness (< 1 nm) within the laser \nspot size54 and/or from the presence of a weak bulk inhomogeneity,43 which is apparent as \nsmall variations of ΔHn. The frequency spacing and the (Ki ttel) character of the SWR modes \nis insensitive to such small variations of ΔHn but the resulting frequency variations (see Eq. 5) \ncan still strongly affect the observed damping of the oscillations. For high enough external \nmagnetic fields, the variations of ΔHn have a negligible role a nd the damping of the SWR \nmodes is governed solely by the intrinsic Gilbert damping parameter.   \nIV. CONCLUSIONS \n \n We used the optical analog of FMR, wh ich is based on a pump-and-probe magneto-\noptical technique, for the determination of micromagnetic parameters of (Ga,Mn)As and \n(Ga,Mn)(As,P) DMS materials. The main advantage of this technique is that it enables us to \ndetermine the anisotropy constants, the spin s tiffness and the Gilbert damping parameter from \na single set of the experimental magneto-optical  data measured in films with a thickness of \nonly several tens of nanometers. To addres s the role of phosphorus incorporation in \n(Ga,Mn)As, we measured simultaneously proper ties of (Ga,Mn)As and (Ga,Mn)(As,P) with \n6% Mn-doping which were grown under identical  conditions in the sa me MBE laboratory. \nWe have shown that the laser-i nduced precession of magnetization is closely connected with a \nmagnetic anisotropy of the samples. In partic ular, in (Ga,Mn)As with in-plane magnetic \nanisotropy the laser-pulse-induced precession of  magnetization was observed even when no \nexternal magnetic field was applied. On the cont rary, in (Ga,Mn)(As,P) with perpendicular-to-\nplane magnetic EA the precession of magnetizat ion was observed only when the EA  position \nwas destabilized by an external in-plane ma gnetic field. From the measured magneto-optical \ndata we deduced the anisotropy constants, spin  stiffness, and Gilber t damping parameter in \nboth materials. We have shown that the incorp oration of 10% of P in  (Ga,Mn)As leads not \nonly to the expected sign change of the perpendi cular-to-plane anisotropy field but also to a \nconsiderable increase of the G ilbert damping which correlates with the increased resistivity \nand reduced itinerant hole density in the (Ga,Mn)(As,P) material. We also observed a reduction of the spin stiffness consistent with the suppression of T\nc upon incorporating P in 15 \n (Ga,Mn)As. Finally, we found that in small exte rnal magnetic fields the damping of the first \nspin wave resonance is sizably stronger than  that of the uniform magnetization precession.  \n  \nACKNOWLEDGEMENTS \n \n This work was supported by the Grant Agency of the Czech Republic grant no. \nP204/12/0853 and 202/09/H041, by the Grant Agency of Charles University in Prague grant \nno. 1360313 and SVV-2013-267306, by EU grant ERC Advanced Grant 268066 - 0MSPIN, \nand by Praemium Academiae of the Academy of Sciences of the Czech Republic, from the \nMinistry of Education of the Czech Re public Grant No. LM2011026, and from the Czech \nScience Foundation Grant No. 14-37427G.   \nAPPENDIX \n \n Due to symmetry reasons, it is conveni ent to rewrite the LLG equation given by \nEq. (1) in spherical coordinates where M\nS describes the magnetiza tion magnitude and polar θ \nand azimuthal φ angles characterize its orientation. We define the perpendicular-to-plane \nangle θ (in-plane angle φ) in such a way that it is counted from the [001] ([100]) \ncrystallographic direction and it is positive wh en magnetization is tilted towards the [100] \n([010]) direction (see inset of Fig. 1 for the co ordinate system definition). The time evolution \nof magnetization is given by37 \n \n ௗெೞ\nௗ௧ൌ0,           ( A 1 )  \n ௗఏ\nௗ௧ൌെఊ\nሺଵାఈమሻெೞቀߙ ∙ܣ\n௦ఏቁൌΓఏሺߠ,߮ሻ ,       ( A 2 )  \n ௗఝ\nௗ௧ൌఊ\nሺଵାఈమሻெೞ௦ఏቀܣെఈ∙\n௦ఏቁൌΓఝሺߠ,߮ሻ ,       ( A 3 )  \n \nwhere A = dF/d and B = dF/d are the derivatives of the free energy density functional F \nwith respect to  and , respectively. We express F in a form10 \n \nܨൌܯ ௌܭ݊݅ݏଶߠቀଵ\nସ݊݅ݏଶ2݊݅ݏ߮ଶߠ ݏܿଶߠቁെܭ ௨௧ݏܿଶߠെೠ\nଶ݊݅ݏଶߠሺ1െ݊݅ݏ2߮ ሻെ\nെܪ௫௧൫ߠݏܿߠݏܿ ுߠ݊݅ݏߠ݊݅ݏ ுݏܿሺ߮െ߮ுሻ൯൩, (A4) \n \nwhere KC, Ku and Kout are the constants that characterize the cubic, uniaxial and out-of-plane \nmagnetic anisotropy fields in (Ga,Mn)As, respectively. Hext is the magnitude of the external \nmagnetic field whose orientati on is described by the angles θH and φH, which are again \ncounted from the [001] and [100] crystallographic direc tions, respectively. For small \ndeviations δθ and δφ from the equilibrium values θ0 and φ0, the Eqs. (A2) and (A3) can be \nwritten in a linear form as  \n ௗఏ\nௗ௧ൌܦଵሺߠെߠሻܦଶሺ߮െ߮ሻ,        ( A 5 )  \n ௗఝ\nௗ௧ൌܦଷሺߠെߠሻܦସሺ߮െ߮ሻ,        ( A 6 )  \n \nwhere    16 \n   ܦଵൌௗഇ\nௗୀబ,ୀబ ,          ( A 7 a )  \n  ܦଶൌௗഇ\nௗୀబ,ୀబ ,          ( A 7 b )  \nand analogically for D3, D4. The solution of Eqs. (A5) and (A6) is expressed by Eqs. (2) and \n(3) where the magnetizati on precession frequency f and the damping rate kd are given by \n \n ݂ൌඥସሺభరିమయሻିሺభାరሻమ\nସగ,          ( A 8 )  \n ݇ௗൌെభାర\nଶ.           ( A 9 )  \n \nEqs. (A8) and (A9) for F in the form (A4) can be simplified when the geometry of our \nexperiment – i.e., the in-plane orientation of the external magnetic field ( θH = π/2) – is taken \ninto account.  The equilibrium orientation of magnetization is in the sample plane for \n(Ga,Mn)As ( θ0 = π/2) and the same applies for (Ga,Mn)(As, P) if sufficiently strong external \nmagnetic field (see Fig. 1) is applied ( θ0 ≈ θH = π/2). In such conditi ons, the precession \nfrequency f and the damping rate kd are given by the following equations \n \n ݂ൌఓಳ\nଶగሺଵାఈమሻ\nۣളളളളളളളളളളളളളളളለ\n൬ܪ௫௧ݏܿሺ߮െ߮ுሻെ2ܭ௨௧ሺଷା௦ସఝ ሻ\nଶ2ܭ௨݊݅ݏଶቀ߮െగ\nସቁ൰ൈ\nൈሺܪ௫௧ݏܿሺ߮െ߮ுሻ2ܭݏܿ4߮െ2ܭ ௨݊݅ݏ2߮ሻ\nߙଶ\nەۖ۔ۖۓ൬ܪ௫௧ݏܿሺ߮െ߮ுሻെ2ܭ௨௧ሺଷା௦ସఝ ሻ\nଶ2ܭ௨݊݅ݏଶቀ߮െగ\nସቁ൰ൈ\nൈሺܪ௫௧ݏܿሺ߮െ߮ுሻ2ܭݏܿ4߮െ2ܭ ௨݊݅ݏ2߮ሻെ\nെቀܪ௫௧ݏܿሺ߮െ߮ுሻെܭ௨௧ሺଷାହ௦ସఝ ሻ\nସೠሺଵିଷ௦ଶఝ ሻ\nଶቁଶ\nۙۖۘۖۗ( A10)  \n݇ௗൌߙఓಳ\nଶሺଵାఈమሻ൬2ܪ௫௧ݏܿሺ߮െ߮ுሻെ2ܭ௨௧ሺଷାହ௦ସఝ ሻ\nଶܭ௨ሺ1െ3݊݅ݏ2߮ ሻ൰. (A11) \n  17 \n  \nREFERENCES \n \n* Corresponding author; nemec@karlov.mff.cuni.cz \n1 T. Jungwirth, J. Sinova, J. Mašek, J. Ku čera, and A. H. MacDonald, Rev. Mod. Phys. 78, \n809 (2006). \n2 Editorial , Nature Materials 9, 951 (2010). \n3 H. Ohno, Nature Materials 9, 952 (2010). \n4 T. 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The relative accu racy 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 ba sed on the following quantities: (i) the growth \ntime and the growth rate of the GaAs buffer layer measured by the RHEED oscillations (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 thic kness by the native oxidation (-1.5 nm ± 0.5 nm); \n(iv) reduction 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 (G a,Mn)As on AlAs, where \nan accurate X-ray reflectivity method to m easure the (Ga,Mn)As layer thickness could be \nused. 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"    },    {        "title": "1306.4680v1.Asymmetric_Ferromagnetic_Resonance__Universal_Walker_Breakdown__and_Counterflow_Domain_Wall_Motion_in_the_Presence_of_Multiple_Spin_Orbit_Torques.pdf",        "content": "Asymmetric Ferromagnetic Resonance, Universal Walker Breakdown, and Counterflow Domain\nWall Motion in the Presence of Multiple Spin-Orbit Torques\nJacob Linder\u0003and Mohammad Alidousty\nDepartment of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway\n(Dated: November 1, 2018)\nWe study the motion of several types of domain wall profiles in spin-orbit coupled magnetic nanowires and\nalso the influence of spin-orbit interaction on the ferromagnetic resonance of uniform magnetic films. Whereas\ndomain wall motion in systems without correlations between spin-space and real-space is not sensitive to the\nprecise magnetization texture of the domain wall, spin-orbit interactions break the equivalence between such\ntextures due to the coupling between the momentum and spin of the electrons. In particular, we extend previous\nstudies by fully considering not only the field-like contribution from the spin-orbit torque, but also the recently\nderived Slonczewski-like spin-orbit torque. We show that the latter interaction affects both the domain wall\nvelocity and the Walker breakdown threshold non-trivially, which suggests that it should be accounted in exper-\nimental data analysis. We find that the presence of multiple spin-orbit torques may render the Walker breakdown\nto be universal in the sense that the threshold is completely independent on the material-dependent Gilbert damp-\ning\u000b, non-adiabaticity \f, and the chirality \u001bof the domain wall. We also find that domain wall motion against\nthe current injection is sustained in the presence of multiple spin-orbit torques and that the wall profile will\ndetermine the qualitative influence of these different types of torques ( e.g.field-like and Slonczewski-like). In\naddition, we consider a uniform ferromagnetic layer under a current bias, and find that the resonance frequency\nbecomes asymmetric against the current direction in the presence of Slonczewski-like spin-orbit coupling. This\nis in contrast with those cases where such an interaction is absent, where the frequency is found to be symmetric\nwith respect to the current direction. This finding shows that spin-orbit interactions may offer additional con-\ntrol over pumped and absorbed energy in a ferromagnetic resonance setup by manipulating the injected current\ndirection.\nPACS numbers: 75.78.Fg,75.60.Jk,76.50.+g, 75.76.+j, 85.75.-d\nI. INTRODUCTION\nSpintronics has been a highly fertile research area espe-\ncially over the last two decades1, giving rise to practical devel-\nopments such as read-heads of harddrives, non-volatile mag-\nnetic memory, and other types of magnetic sensors2,3. The\nkey ingredient in this field is to utilize the spin-degree of free-\ndom in currents and materials to achieve the desired function-\nality, in particular with an eye to providing a feasible alterna-\ntive to semiconductor technology. One of the main obstacles\nto overcome in this regard is the high energy cost associated\nwith e.g.Joule heating when passing a spin-polarized current\nconsisting of electrons through a device: current-densities of\norder 106A/cm2are needed to perform magnetization switch-\ning via current-induced spin-transfer torque. As an alternative\nmechanism to spin-transfer torque which could circumvent\nthe Joule heating from electrons, magnon-induced magneti-\nzation dynamics has been investigated more recently4–7.\nCurrently, the topic of controllable domain wall motion is\nreceiving much attention (see e.g.Ref. 8 for a very recent re-\nview) due to its potential with regard to the storage and trans-\nfer of information. A domain wall is a topological defect in\na magnetic system where the local magnetic order parame-\nter typically rotates spatially in a fashion that reduces the net\nmagnetic moment of the domain wall area. Owing to their\nsmall size (\u001810 nm) and large velocities ( \u0018100 m/s)9,10,\ncontrollable domain wall motion represents holds real poten-\ntial for tailoring functional devices with fast writing speeds.\nIn addition, there has been several proposals11–14related to\nmagnetic memory functionality due to the non-volatile nature\nof magnetic domains. Walker breakdown15is nevertheless a\n-x+xz\ny\n-x+xz\ny\n +x-x\nxy(a): Bloch(z)\n(b): Bloch(y)(c): Head-to-head\nDomain wall ferromagn\netFIG. 1: (Color online) Schematic setup: a spin-polarized current\nis passed through a domain wall magnetic nanowire with spin-orbit\ncoupling. The spin-orbit interaction may be either intrinsic or in-\nduced via a heavy metal proximate host. We consider several types\nof domain wall configurations, since the presence of spin-orbit cou-\npling qualitatively distinguishes the domain wall motion with one\ntype of magnetization texture from another. More specifically, we\nconsider two types of Bloch-domain walls relevant for perpendicular\nmagnetic anisotropy systems in addition to a head-to-head domain\nwall with an in-plane magnetization easy anisotropy.\nlimiting factor in this regard.\nDomain walls can come in several different shapes depend-\ning on the anisotropy energies and dimensionality of the sys-\ntem at hand. In a low-dimensional system such as a magnetic\nnanowire, Bloch walls are one of the most frequent types en-arXiv:1306.4680v1  [cond-mat.mes-hall]  19 Jun 20132\ncountered. However, it is also possible to generate other sorts\nof magnetization textures such as head-to-head domain walls.\nBoth of these wall types are shown in Fig. 1. A key ques-\ntion is whether or not specific domain wall types are benefi-\ncial with regard to the objectives mentioned above ( e.g. fast\npropagation, low current densities to generate motion). The\nanswer to this question depends on if the spin and position\ndegrees of freedom are correlated in the system, for instance\nvia spin-orbit interaction. In the absence of such spin-orbit in-\nteractions, different types of domain walls behave in the same\nway - the exact magnetization texture has no effect and one\nobtains for instance the same terminal domain wall velocity in\nall cases. The fact changes when spin-orbit coupling is present\nsince the electron transport and spin torque now directly de-\npends on the precise magnetization texture, which warrants a\nspecific study for how domain wall motion is manifested for\ndifferent types of domain walls. A numerical investigation of\nthis issue was recently put forth in Ref. 16.\nThe influence of spin-orbit coupling on domain wall mo-\ntion has recently been considered extensively in several theo-\nretical works16–23. On the experimental stage24–27, it has been\ndemonstrated that the presence of spin-orbit coupling indeed\ninfluences the domain wall dynamics in a non-trivial way in-\ncluding anomalous behavior such as strongly enhanced do-\nmain wall velocities and induced wall motion in the opposite\ndirection of the electron flow. In order to explain these find-\nings, it was shown in Ref. 21 that the presence of spin-orbit\ncoupling would generate not only a field-like torque but also\na so-called Slonczewski-like torque28, named such due to its\nformal resemblence to standard current-induced torques in the\nabsence of spin-orbit coupling. Alternatively, these two types\nof spin-orbit torques may be characterized as out-of-plane and\nin-plane components of the total Rashba torque29.\nMotivated by this, we will in this paper derive exact ana-\nlytical expressions for the domain wall velocity and Walker\nbreakdown threshold for several types of domain wall config-\nurations when including both types of spin-orbit torques in or-\nder to investigate how the Slonczewski-like torque influences\nthe physics at hand. This way, we expand previous literature17\nwhich has only considered the field-like term and show that\nthe inclusion of the Slonczewski-like torque has profound im-\npact on the domain wall velocity and the threshold value of\nWalker breakdown. In fact, we will show that the existence\nof this torque renders the threshold value to be universal in\nthe sense that it is independent on both the Gilbert damping\n\u000b, the non-adiabiticity parameter \f, and the chirality \u001bof the\ndomain wall.\nWe will present a detailed derivation of the equations of\nmotion where possible and show precisely in which manner\nthe spin-orbit coupling influences both the domain wall ve-\nlocity and the Walker breakdown threshold value. Our analyt-\nical expressions show the precise conditions required to real-\nize domain wall motion against the current flow, as has been\nexperimentally observed recently27, and in particular how the\ndomain wall chirality affects this phenomenon.\nFinally, we investigate how the ferromagnetic resonance\nresponse of a material (or equivalently the dissipation and\npumping of energy) is altered due to the above men-tioned spin-orbit torques. The ferromagnetic resonance ex-\nperiment is an important technique for obtaining informa-\ntion about anisotropy, magnetic damping and magnetization\nreversal41–43,48,50–52. The influence of spin-polarized current\non Gilbert damping and ferromagnetic resonance have been\nextensively investigated in different situations30,31,44–47,49.\nConsidering a ferromagnetic resonance setup in the pres-\nence of a current bias, we analytically show that the spin-orbit\ninteractions render the resonance frequency to become asym-\nmetric with respect to the direction of current injection. This\nis different from previous works considering a ferromagnetic\nresonance setup in the presence of spin-transfer torques, albeit\nwithout spin-orbit coupling, where the frequency was found to\nbe symmetric with respect to the current direction30,49.\nThis paper is organized as follows. In Sec. II, we\noutline the theoretical framework to be used in our analy-\nsis, namely the Landau-Lifshitz-Gilbert (LLG) equation aug-\nmented to include the role of spin-orbit coupling combined\nwith a collective-coordinate description of the domain wall.\nWe then present our main findings in Sec. III, in four subsec-\ntions, where the LLG equation is solved in order to obtain both\nthe domain wall velocity, the Walker breakdown threshold and\nthe ferromagnetic resonance frequency. In Subsec. III A we\nconsider Block( z) wall profile, in Subsec III B Block( z) wall\nprofile is studied, in Subsec. III C a head-to-head domain wall\nstructure is investigated, and in Subsec. III D we present and\ndiscuss the results of absorbed power by a ferromagnetic film\nunder current injection in the presence of Slonczewski-like\nspin-orbit interaction. We finally summarize our results and\nfindings in Sec. IV.\nII. THEORY\nThe starting point of our analysis is the spatio-temporal\nLandau-Lifshitz-Gilbert equation32, augmented to include the\ncontribution from torque terms arising due to the presence of\nspin-orbit coupling. When a current-bias is applied along x\naxis, the full LLG equation takes the form21,33\n@tM =\u0000\rM\u0002(Heff+Hso\u0000\f\nM0M\u0002Hso)\n+\u000b\nM0M\u0002@tM+ \u0000@xM\u0000\f\u0000\nM0M\u0002@xM:(1)\nThe above equation describes the time-dynamics of the local\nmagnetic order parameter M(x;t). The effective field Heffis\nformally obtained by a functional derivative of the free energy\nwith respect to the magnetization and will vary depending on\ne.g.the anisotropy configuration of the wire35. The influence\nof spin-orbit interaction is captured as an effective field:\nHso=\u000bRmeS\n~eM0(1 +\f2)^z\u0002j; (2)\nwhere inversion symmetry is broken in the zdirection and\n\u000bRcharacterizes the strength of the spin-orbit coupling. S\nandjis the polarization and density of the injected current,\nwhereasmeandM0is the electron mass and magnitude of\nthe magnetization, respectively. The parameter \fis known as3\nthe non-adiabaticity parameter in the literature, a convention\nwe shall stick to although this terminology is not ideal34.\nThe terms in Eq. (1) have the following physical interpre-\ntation. The effective field causes a precession of the magneti-\nzation vector Mand has two extra contributions in terms of\nHsoandM\u0002Hsoin the presence of spin-orbit coupling. The\nformer of these has the exact form of an effective field-like\ntorque whereas the latter has the form of a Slonczewksi-like\ntorque. Interestingly, this term was conjectured to exist in the\nexperiment of Miron et al.27in order to explain the results, but\nit was only recently theoretically derived in Refs. 21, 29. A\nkey observation is that the Slonczewski like spin-orbit torque\ndepends on the non-adiabaticity parameter \fwhich also ap-\npears for the conventional non-adiabatic spin-transfer torque\n[last term in Eq. (1)] as is well-known. The term /@xM\nis the adiabatic spin-transfer torque originating from the as-\nsumption that the spin of the conduction electrons follow the\ndomain wall profile perfectly without any loss or spin scatter-\ning and \u0000 =\u0016BP=eM 0(1 +\f2).\nOne of the main goal in this work is to compute the domain\nwall velocity and analyze Walker breakdown for a domain\nwall nanowire with spin-orbit coupling, considering several\ntypes of experimentally relevant domain walls, both with in-\nplane and perpendicular magnetization relative the extension\nof the wire16,25–27. We will take into account both the field-\nlike and the Slonczewski-like spin-orbit induced torques. We\nunderline again that the various magnetization textures con-\nsidered in this paper will give qualitatively different behavior\nfor the wall velocity and Walker threshold values precisely\ndue to the spin-orbit interaction which correlates spin- and\nreal-space. For a Bloch( y) domain wall (see Fig. 1), an exact\nanalytical solution for the domain wall velocity vDWis permis-\nsible and we will derive this result in detail. For other types\nof domain walls, a general expression for vDWis not possi-\nble to obtain analytically, thus for completeness, we revert to\na numerical study for these cases. However, it is still possi-\nble to investigate analytically the Walker breakdown thresh-\nold for these domain walls and we show that the chirality of\nthe domain wall conspires with the presence of spin-orbit cou-\npling to qualitatively alter the behavior of Walker breakdown\nin spin-orbit coupled nanowires.\nIII. RESULTS AND DISCUSSION\nWe shall start by investigating domain wall motion in the\npresence of multiple spin-orbit torques and consider three\ntypes of domain wall structures as shown in Fig. 1. For\neach case, we will focus on the domain wall velocity and the\nWalker breakdown threshold value, giving exact analytical re-\nsults where possible. We note that such an exact solution for\nvDWconstitutes the most general analytical expression for the\ndomain wall velocity up to now, including fully the influence\nof spin-orbit coupling. We then study the ferromagnetic res-\nonance response of a magnetic layer with a Slonczewski-like\nspin-orbit interaction with an injected current into the plane of\nthe layer and using the absorbed power by the film, we drive\nthe ferromagnetic resonance expression analytically.A. Bloch(z) wall\nConsider first a domain wall profile relevant for magnetic\nnanowires with perpendicular anisotropy25–27(e.g. Co/Ni\nmultilayers), namely a so-called Bloch( z) wall which is\nparametrized as:\nm= (sin\u0012sin\u001e;sin\u0012cos\u001e;\u001bcos\u0012); (3)\nand a corresponding effective field:\nHeff=2Aex\nM2\n0r2m\u0000H?mx^x+Hkmz^z+Hext: (4)\nHere,H?andHkare the anisotropy fields along the hard and\neasy axes of magnetization, respectively, whereas Hextis an\nexternally applied magnetic field. The parameter \u001b=\u00061\ncharacterizes the chirality of the domain wall: both signs of\n\u001bgive allowed equilibrium solutions (\u001e= 0) of the LLG-\nequation and describes a spin texture changing from positive\nto negative depending on which direction one is moving in.\nNote that\u001bis also denoted the topological charge of the do-\nmain wall35: the winding direction of the local magnetization\ndictates the effective ”charge” since the sign of \u001bwill deter-\nmine the direction in which an external magnetic field moves\nthe domain wall. The components of the magnetization vector\ndepend on both space and time according to15\ncos\u0012= tanh\u0010x\u0000X(t)\n\u0015\u0011\n;\nsin\u0012=sech\u0010x\u0000X(t)\n\u0015\u0011\n: (5)\nEq. (5) is obtained by inserting the magnetization profile m\ninto the LLG equation and solving for \u0012and\u001eunder equilib-\nrium conditions (in which case X(t)is a constant and \u001e= 0).\nThe tilt angle \u001e=\u001e(t)is in general, however, time-dependent\nand causes the domain wall to acquire a finite component\nalong the hard magnetization axis in an non-equilibrium situ-\nation. A collective-coordinate description of the domain wall\nmotion is obtained if one may identify the time-dependence of\nthe domain-wall center position X(t)and the tilt angle \u001e(t).\nIn general, other modes of deformation can be allowed35.\nHowever, it can be shown that the domain wall may be treated\nas rigid [only depending on X(t)and\u001e(t)] in a collective-\ncoordinate framework when the easy axis anisotropy energy\nKis assumed larger than its hard axis equivalent K?36, i.e.\njKj\u001djK?j.\nIt is useful to write down an explicitly normalized form of\nthe LLG-equation which we will use for all the domain wall\nprofiles considered in this work. We normalize all quanti-\nties to a dimensionless form as defined by the following LLG\nequation:\n@\u001cm=\u0000m\u0002(Heff+Hso\u0000\fm\u0002Hso)\n+\u000bm\u0002@\u001cm+u@~xm\u0000\fum\u0002@~xm:(6)\nIn the specific case of a Bloch( z) wall, we then have the nor-\nmalized effective field:\nHeff= 2A~r2m\u0000H?mx^x+Hkmz^z;\nHso= ~\u000bRu^y: (7)4\nInserting Eq. (5) into Eq. (6) leads to one pair of equations of\nmotion for the collective coordinates Xand\u001e. These equa-\ntions may be simplified by using Thiele’s approach37where\none integrates over xand utilizesR1\n\u00001sin2\u0012dx= 2\u0015andR1\n\u00001sin\u0012dx=\u0015\u0019. We then find the following dimensionless\nequations:\n\u001a\n\u000b@\u001c\u001e\u0000\u001b@\u001cX=\u001bu\u00001\n2H?sin 2\u001e\u00001\n2~\u000bR\u0019usin\u001e;\n@\u001c\u001e+\u000b\u001b@\u001cX=1\n2\f~\u000bRu\u0019sin\u001e\u0000\fu\u001b:\nHere,X=X=\u0015 is the normalized spatial coordinate of the\ndomain-wall center and ~\u000bRis a dimensionless measure of the\nstrength of the spin-orbit interaction. In the limiting case of\nan absent Slonczewski-like spin-orbit torque where the terms\nproportional to \f\u0002~\u000bRare zero, our results are consistent\nwith Ref. 20. The sin\u001eterms in Eq. (8) makes an exact\nanalytical solution of the equations untractable. As we shall\nsee, a similar situation occurs for the head-to-head domain\nwall case. Nevertheless, it is possible to make further progress\nin the present case with regard to the appearance of so-called\nWalker breakdown15. This phenomenon refers to a threshold\nvalue of the current density for which the domain wall starts\nto rotate with a time-dependent \u001e=\u001e(\u001c)rather than simply\npropagating with a fixed magnetization texture, i.e. constant\n\u001e. In general, it is desirable with as large threshold value as\npossible for Walker breakdown. We note in passing here that\nthe presence of pinning potentials and defects in the sample\nmay also contribute to the threshold value of the current, but\nwe leave this issue for a future work.\nTo investigate the velocity at which breakdown occurs, we\ncombine the equations of motion into a single equation for the\ntilt angle\u001e:\n@\u001c\u001e=1\n1 +\u000b2h1\n2(\f\u0000\u000b)~\u000bRu\u0019sin\u001e\u0000\u001bu(\f\u0000\u000b)\n\u00001\n2\u000bH?sin 2\u001ei\n: (8)\nThere is no Walker breakdown as long as @\u001c\u001e= 0, which\nholds when the tilt angle \u001esatisfies the equation:\nsin 2\u001e=(\f\u0000\u000b)u\n\u000bH?(~\u000bR\u0019sin\u001e\u00002\u001b): (9)\nWalker breakdown will occur at a velocity ucsuch that for\nu > ucthere is no stable solution for this equation. Now,\nforj~\u000bR\u0019j<2the right hand side of Eq. (9) will have equal\nsign for its minimum and maximum value as \u001evaries from\n0 to2\u0019. Therefore, Walker breakdown will always occur by\nincreasingu: at some value uc, the minimum value of the\nright hand side of Eq. (9) will be larger than unity and thus\nrender the equation to be void of any solution. However, if\nj~\u000bR\u0019j>2, the minimum and maximum value of the right\nhand side have opposite signs. This means that there must\nbe a crossing of the 0 line at some values of \u001e, and thus an\nintersection with sin 2\u001e. In effect, we can always find a stable\nsolution and there will be no Walker breakdown regardless of\nthe velocity uwhen:\n\f\f\f~\u000bR\u0019\n2\f\f\f>1: (10)In other words, for a sufficiently large spin-orbit interaction,\nno Walker breakdown occurs. It is interesting to note that this\ncondition is universal in the sense that it is independent on the\ndamping parameter \u000b, the non-adiabiticity parameter \f, and\nthe chirality \u001bof the domain wall. This observation can be\nattributed directly to the presence of the new spin-orbit torque\nproportional to \f. To see this, consider a scenario where only\nthe field-like spin-orbit torque /M\u0002Hsois included. All\nterms proportional to \f\u0002~\u000bRare then zero, and we obtain the\nequation\nsin 2\u001e=2\u001bu\nH?\u0010\n1\u0000\f=\u000b\u0000\u001b~\u000bR\u0019\n2sin\u001e\u0001\n; (11)\nwhich must be satisfied to prevent Walker breakdown. As\nseen, whether or not the maximum and minimum value of the\nright hand side have equal sign depends on if\n\f\f\f~\u000bR\u0019\n2\f\f\f>j(1\u0000\f=\u000b)j: (12)\nIn this regime, we recover the results of Ref. 20. The effect\nof the Slonczewski-like spin-orbit torque is then to render the\nWalker breakdown universal (independent on \u000b;\f;\u001b ). Let us\nalso consider the implications this torque-term has with regard\nto the magnitude of the threshold value for Walker breakdown.\nComparing Eqs. (10) and (12), we see that the required spin-\norbit interaction ~\u000bRto completely remove the Walker thresh-\nold depends on the ratio \f=\u000b if one does not take into account\nthe Slonczewski-like spin-orbit torque. For \f=\u000b'1, the re-\nquired spin-orbit strength becomes very small. In the more\ngeneral case where the aforementioned torque is included,\nhowever, the required ~\u000bRhas a fixed value. This is shown\nin Fig. 2.\n0 1 2 3 400.511.522.53\nα/βThreshold ˜ αR\n  \nOnly field−like torque\nBoth FL− and SL−torque\nFIG. 2: (Color online) Threshold value for the magnitude of the spin-\norbit coupling above which there is no Walker breakdown. In the\nmore general scenario where both types of spin-orbit torques are ac-\ncounted for, the threshold value for ~\u000bRis constant. When only the\nfield-like torque is considered, the threshold is strongly increased in\nthe regime\u000b=\f < 0:5. In the limit \u000b=\f!1 , the asymptote is\n2=\u0019.5\nWe also give numerical results for the wall velocity for this\nBloch domain wall configuration, using a similar approach as\nin Ref. 38. Let us first note that it is possible to infer what\nthe qualitative effect is of the chirality \u001bdirectly from the\nequations of motion Eqs. (8). By making the transformation\n\u001e!\u001b\u001e, it is seen that the equations of motion become in-\ndependent on the chirality \u001b:This means that the domain wall\nvelocity will be the same regardless of the sign of \u001b, whereas\nthe tilt angle \u001eevolves in the opposite direction with time for\nopposite signs of \u001b. In Fig. 3, we therefore present results for\n\u001b= 1without loss of generality and consider two cases with\ndamping\u000blarger or smaller than the non-adiabaticity con-\nstant\fin (a) and (b), respectively. As seen, this qualitatively\naffects the domain wall velocity.\nA particular feature worth noting in (b) is that the abrupt\nchange in wall velocity at a given uis not necessarily syn-\nonymous with the occurrence of Walker breakdown. To see\nthis, consider Fig. 4 where we have plotted the left- and right-\nhand side of the Walker breakdown criterion Eq. (9) in addi-\ntion to the time-evolution of the tilt angle \u001eas an inset. We\nhave set\u000b= 0:005and\f= 0:01and consider two strengths\nof the spin-orbit coupling parameter ~\u000bRin (a) and (b). An\nintersection of the lines in the main panels means that there\nexists a solution to Eq. (9) and that Walker breakdown does\nnot occur. Considering Fig. 4(a) first, we see that increasing\nthe current density eventually causes Walker breakdown as the\ndashed and full lines no longer intersect. As a result, \u001eis no\nlonger a constant as seen in the inset and starts to grow with\ntime. We may therefore conclude that the abrupt change in\nwall velocity for ~\u000bR= 0:01seen in Fig. 3(b) does correspond\nto the occurrence of Walker breakdown. However, turning to\nFig. 4(b) it is seen that the dashed and full lines always in-\ntersect even when increasing the current density uabove the\nvalue at which the wall velocity abruptly changes in Fig. 3(b)\nfor~\u000bR= 1 (aroundu= 0:14). What is important to note\nis that their point of intersection changes discontinuously: the\ntilt angle\u001eremains constant so that there is no Walker break-\ndown in the sense of a continuously deforming domain wall.\nInstead, there is an abrupt change in the tilt angle where it\nchanges from one constant value to another.\nB. Bloch(y) wall\nAnother type of domain wall structure which may appear in\nsuch a systems with perpendicular magnetic anisotropy is the\nBloch (y)-wall, having the easy magnetization direction along\ntheyaxis whereas the hard axis remains along the wire direc-\ntion:\nm= (sin\u0012sin\u001e;\u001bcos\u0012;sin\u0012cos\u001e); (13)\nand a corresponding effective field:\nHeff=2Aex\nM2\n0r2m\u0000H?mx^x+Hkmy^y+Hext:(14)\nIn this case, the equations of motion for the collective coordi-\nnatesXand\u001etake a different form compared to the Bloch( z)\n0.10.20.30.40.50.60.7−0.6\n−0.5\n−0.4\n−0.3\n−0.2\n−0.1\n0/angbracketleftvDW/angbracketright(a)\n  \n00.10.20.30.40.50.60.7−0.6\n−0.5\n−0.4\n−0.3\n−0.2\n−0.1\n0\nu/angbracketleftvDW/angbracketright(b)\n  ˜αR= 0\n0.001\n0.01\n0.1\n1FIG. 3: (Color online) Domain wall velocity for a Bloch( z) wall\nplotted against the injected current. We have chosen \u001b= 1 without\nloss of generality (see text) and set \f= 0:01andH?=0.5. In (a)\n\u000b > \f (\u000b= 0:02) whereas in (b) \u000b < \f (\u000b= 0:005). Note the\ninverted sign of the yaxis, which simply corresponds to the direction\nof the wall motion.\ncase:\n\u001a\n\u001b@\u001cX+\u000b@\u001c\u001e=\f~\u000bRu\u00001\n2H?sin 2\u001e\u0000u\u001b;\n@\u001c\u001e\u0000\u000b\u001b@\u001cX=\fu\u001b+ ~\u000bRu:\nIn fact, these equations can now be solved analytically in an\nexact manner, using a similar approach as in Ref. 22. Com-\nbining the two above equations yields:\n@\u001c\u001e(1 +\u000b2) =\u0000\u000b\n2H?sin 2\u001e+u[\u001b(\f\u0000\u000b) + ~\u000bR(1 +\u000b\f)]:\n(15)\nConsider Eq. (15) with respect to \u001e=\u001e(\u001c). This is a separa-\nble equation and direct integration gives:\n\u001c=C0\u00001 +\u000b2\np\nA2\u0000\u000b2H2\n?=4atanh\u000bH?=2\u0000Atan\u001ep\nA2\u0000\u000b2H2\n?=4i\n;\n(16)\nwhereC0is an integration constant and we define:\nA\u0011u[\u001b(\f\u0000\u000b) + ~\u000bR(1 +\u000b\f)]: (17)\nFor brevity of notation, we also introduce B\u0011\u000bH?=2. The\nintegration constant depends on the initial conditions. At\n\u001c= 0, we assume that the domain wall is in its equilibrium\nconfiguration \u001e= 0, in which case we may write the solution6\n01 2 3 4 5 6 7−2−1.5−1−0.500.51\nφL.h.s and r.h.s of Eq. (10)(b)\n  \n01234567−1.5−1−0.500.51\nφL.h.s and r.h.s of Eq. (10)(a)\n  \n0 5 10\nx 105−400−2000\nτφ(τ)\n  \n0 5 10\nx 105−4−20\nτφ(τ)\n  \nsin(2 φ)\nFIG. 4: (Color online) Plot of left-hand side (dashed line) and right-hand side (full lines) of Eq. (9) in order to illustrate the intersection\npoints. When there is no intersection between the lines, Walker breakdown has occurred. We have set \f= 0:01,\u000b= 0:005and consider (a)\n~\u000bR= 0:01anduranging from 0.24 to 0.30 along the direction of the arrow, in addition to (b) ~\u000bR= 1anduranging from 0.10 to 0.16 along\nthe direction of the arrow. The black arrow between the circles in (b) highlights how the intersection point changes abruptly upon increasing\nu.Insets: Time-evolution of the tilt angle for the same choices of u.\nfor the tilt angle as:\ntan\u001e=B\nA\u0000p\nA2\u0000B2\nAtanh\natan(\u000bB=p\nA2\u0000B2)\n\u0000\u001cp\nA2\u0000B2=(1 +\u000b2)i\n: (18)\nHaving now obtained the full time-dependence of the tilt-\nangle, we insert this back into the original equation of motionin order to find the domain wall velocity _X=vDW. The gen-\neral expression for the domain wall velocity is rather large.\nHowever, by utilizing the fact that vDWwill display small-\nscale oscillations it is possible to find a simplified expression\nfor the average domain wall velocity hvDWi. The period of\noscillation is T= (1 +\u000b2)\u0019=p\nA2\u0000B2, which gives us:\nhvDWi=1\nTZT\n0d\u001c\u001b\n\u000b(\nA2\u0000B2\nA(1 +\u000b2)sec2\u0010\natan(\u000bB=p\nA2\u0000B2)\u0000\u001cp\nA2\u0000B2\n1 +\u000b2\u0011\n\u0002h\n1 +\u0010\u000bB\nA\u0000p\nA2\u0000B2Atan[atan(\u000bB=p\nA2\u0000B2)\u0000\u001cp\nA2\u0000B2=(1 +\u000b2)]\u00112i\u00001)\n\u0000u(~\u000bR\u001b+\f)=\u000b: (19)\nThe analytical solution to the above integral and the final result is:\nhvDWi=\u001b\n\u000b(1 +\u000b2)sgnfu\u001b(\f\u0000\u000b) +u~\u000bR(1 +\u000b\f)g\u0002Req\n[u\u001b(\f\u0000\u000b) +u~\u000bR(1 +\u000b\f)]2\u0000\u000b2H2\n?=4\u0000u(~\u000bR\u001b+\f)=\u000b;\n(20)\nwhere we have reinstated the original parameters contained in the quantities AandB.\nThe equation forhvDWishows the exact manner in which the\ndomain wall velocity depends on the various torque terms\nsuch as the non-adiabatic contribution \fand the spin-orbit\nterms ~\u000bR, and reveals several important features. It is seen\nthat for this particular domain wall configuration [Bloch (y)],\nthe effect of the Slonczewski-like spin-orbit torque is a smallquantitative correction of order O(\u000b\f), which thus can be ne-\nglected. However, the conventional field-like spin-orbit torque\nhas a strong qualitative influence on the wall dynamics. In\nfact, it is seen that the ~\u000bRterm plays the same role as the non-\nadiabatic conventional torque proportional to \f, but with one\nimportant difference: the spin-orbit torque contribution is chi-7\n0.1 0.2 0.3 0.4 0.5 0.6−0.7\n−0.6\n−0.5\n−0.4\n−0.3\n−0.2\n−0.1\n0/angbracketleftvDW/angbracketright(a)\n  \n0.1 0.2 0.3 0.4 0.5−0.5\n−0.4\n−0.3\n−0.2\n−0.1\n0\n0.1\n0.2(b)\n  \n00.1 0.2 0.3 0.4 0.5 0.6 0.7−0.7\n−0.6\n−0.5\n−0.4\n−0.3\n−0.2\n−0.1\n0\nu/angbracketleftvDW/angbracketright(c)\n  \n0 0.1 0.2 0.3 0.4 0.5−0.6\n−0.4\n−0.2\n0\n0.2\nu(d)\n  \n˜αR= 0\n0.001\n0.01\n0.1\n1\nFIG. 5: (Color on-\nline) The domain\nwall velocityhvDWi\nas a function of the\ncurrent density ufor\nvarious chiralities and\nspin-orbit coupling\nstrengths. (a): Positive\nchirality\u001b= +1 and\n\u000b > \f (\u000b= 0:02).\n(b): Negative chi-\nrality\u001b=\u00001and\n\u000b > \f (\u000b= 0:02).\n(c): Positive chirality\n\u001b= +1 and\u000b < \f\n(\u000b= 0:005). (d):\nNegative chirality\n\u001b=\u00001and\u000b < \f\n(\u000b= 0:005). For\nall plots, we have\nused\f= 0:01and\nH?= 0:5.\nrality dependent, i.e. changes sign with \u001b, whereas the \f-term\ndoes not. As a consequence, the wall may actually propagate\nin opposite direction of the applied current depending on the\nchirality\u001bof the domain wall, as was shown recently in Ref.\n22.\nIt is seen from Eq. (20) that there is either an enhancement\nof the domain wall velocity or a competition between the spin-\norbit induced torque and \f-torque depending on the sign of \u001b.\nWe show this in Fig. 5 where we consider the four possible\ncombinations of wall chirality \u001b(two values, \u001b=\u00061) com-\nbined with whether or not \u000bis larger than \f(two possibilities,\n\u000b > \f or\u000b < \f ). For a positive chirality \u001b= +1 displayed\nin Fig. 5 (a) and (c), the wall moves in the same direction for\nall current densities uas the torque terms in Eq. (20) have the\nsame sign. This is no longer the case for the opposite chirality\n\u001b=\u00001shown in Fig. 5(b) and (d) where the wall velocity\ncan actually change sign as uincreases. This is indicative of\ncounterflow domain wall motion where the wall moves in the\nopposite direction of the applied spin current.\nWalker breakdown for the domain wall occurs for veloci-\ntiesu\u0015ucwhere the root in Eq. (20) becomes imaginary,\nnamely:\nuc=\u000bH?\nj2\u001b(\f\u0000\u000b) + 2~\u000bR(1 +\u000b\f)j: (21)\nNote that this is the same as ucthat we would have found us-\ning the arguments in the previous section in order to identify\nthe Walker breakdown from the equations of motion (without\nactually solving them explicitly) and thus serves as a consis-\ntency check for the correctness of Eq. (20). This expression\nis quite generally valid, including the effects of both types of\nspin-orbit torques and both types of conventional spin-transfer\ntorques. As another consistency check, we observe that in\nthe absence of spin-orbit coupling ( ~\u000bR= 0), one finds that\njucj=\u000bH?=2j\f\u0000\u000bjwhich agrees with Ref. 33. The effectof the spin-orbit interaction is seen to depend explicitly on the\nchirality\u001bof the domain wall. Although Walker breakdown\nis inevitable for the present Bloch( y) domain wall, in contrast\nto the Bloch( z) one, the presence of spin-orbit interactions\n(~\u000bR6= 0) can strongly enhance the threshold velocity due to\nthe competition between the terms \u001b(\f\u0000\u000b)and~\u000bR(1 +\u000b\f)\nin the denominator. When these terms have different sign (ei-\nther for\u001b=\u00001and\f > \u000b or\u001b= 1 and\f < \u000b ), the\nspin-orbit coupling can very strongly enhance the threshold\ncurrent for Walker breakdown. This effect could be used to\ninfer information about the value of \u000band\fprecisely due\nto the non-monotonic behavior of the threshold current as a\nfunction of ~\u000bR.\nWe illustrate this behavior in Fig. 6 where we have cho-\nsen\u001b= +1 . As seen, the threshold velocity decreases in a\nmonotonic fashion with increasing ~\u000bRwhen the damping is\nlow,\u000b < \f . However, when the two terms in the denomina-\ntor differ in sign (which occurs precisely when \u000b > \f ), the\nthreshold velocity uchas a non-monotonic behavior and is in\nfact strongly increases near ~\u000bR=j\f\u0000\u000bj. In this way, one\nmay obtain information regarding the relative size of \u000band\f\nby measuring the threshold velocity.\nC. Head-to-head domain wall\nThe final type of domain wall structure we will consider\nappears for in-plane magnetized strips ( e.g.NiFe layer16) and\nis known as a so-called head-to-head domain wall. In this\ncase, the easy axis is parallell with the extension of the wire\nwhereas the hard axis is perpendicular to it:\nm= (\u0000\u001bcos\u0012;sin\u0012cos\u001e;sin\u0012sin\u001e); (22)8\n00.511.522.533.54uc/H⊥\n  \n0 0.005 0.01 0.015 0.02020406080\n˜αRuc/H⊥0.02\n0.015\n0.01\n0.005\n0.035 0.03 α= 0.025α\n(b)(a)\nFIG. 6: (Color online) Critical velocity uc=H?that triggers Walker\nbreakdown. We have chosen \f= 0:02as a representative value\nwhich demonstrates the fundamental behavior of uc. For sufficiently\nlow damping \u000b < \f shown in (a), the threshold velocity is lowered\nmonotonically as the spin-orbit interaction ~\u000bRis increased. When\nthe damping becomes stronger such that \u000b > \f ,ucis strongly en-\nhanced in a limited interval of ~\u000bR.\nand a corresponding effective field:\nHeff=2Aex\nM2\n0r2m\u0000H?mz^z+Hkmx^x+Hext:(23)\nUsing again Thiele’s approach as described in the previous\nsections, one arrives at exactly the same equations of motion\nas in the Bloch( z) case. The formal reason for this can be\ntraced back to the fact that the effective spin-orbit field Hso\nis directed along the yaxis. The magnetization textures of the\nBloch(z) and head-to-head domain walls may be transformed\ninto each other via an SO(3) rotation with an angle \u0019=2ofM\naround theyaxis. Such a rotation leaves Hsoinvariant and\none thus obtains the same equations of motion for both types\nof domain walls. Formally, one can see this by multiplying\nEq. (1) from the left side with:\nU=0\n@0 0\u00001\n0 1 0\n1 0 01\nA; (24)\nand using that\n(Ua)\u0002(Ub) =det(U)(U\u00001)T(a\u0002b): (25)\nSinceU2 SO(3), we have that (U\u00001)T=Uand det (U)=+1.\nBy direct multiplication, one observes that UHso=Hso,\nUMBloch(z)=Mhead-to-head andUHeff\nBloch(z)=Heff\nhead-to-head .\nNote that it is in drastic contrast with the Bloch( y) case whereHsoisnotinvariant under the matrix which rotates MBloch(z)\nintoMBloch(y). The same arguments and results related to the\ndomain wall velocity and Walker breakdown that were dis-\ncussed in Sec. III A then also hold for the present head-to-\nhead domain wall case.\nWe mention here that the equivalence of the Bloch( z) and\nhead-to-head domain wall case found here is contingent on\nthe specific setup we have considered in Fig. 1. Although\nthis model is the standard one and indeed the most frequently\nemployed setup experimentally, it was recently shown that\nsuch an equivalence does not hold when combining a mag-\nnetic strip/wire with a non-magnetic conductive layer with\nspin-orbit interaction in a non-parallell geometry16. Such a\nmethod actually provides a manner in which the direction of\nthe effective spin-orbit field can be changed which could then\nserve as a mean to distinguish between different types of do-\nmain walls, based on their response to an applied current.\nD. Ferromagnetic resonance (FMR) in the presence of\nspin-orbit torques\nWe now turn our attention to another setup where the aim\nis to identify the ferromagnetic resonance response of a ma-\nterial where spin-orbit interactions play a prominent role. To\ndo so, we consider the setup shown in Fig. 7 where a spin-\ncurrent with polarization magnitude and unit vector direction\nS2[0;1]and~S, respectively, is injected into the ferro-\nmagnetic layer where spin-orbit coupling is present. This di-\nrectly influences the susceptibility tensor and thus both the fer-\nromagnetic resonance frequency/linewidth and the absorbed\npower by the system39.\nTo facilitate the analytical calculations, we will operate\nwith two different coordinate systems. The laboratory (sta-\ntionary) framework xyzis shown in Fig. 7, where the xy\nplane spans the ferromagnetic layer, and xyz denotes a ro-\ntated coordinate system which we will specify the direction\nand purpose of below. A current is injected into the ferromag-\nnetic layer acting with a spin-transfer torque on the magneti-\nzation vector~M. This torque is modified due to the presence\nof spin-orbit coupling which is taken into account via a field\n~Hsoas in the domain-wall treatment. The time-dependent\nLLG motion equation describing the dynamic of ferromag-\nnetic layer magnetization vector then takes the following form\nin this new notation:\n@~M\n@t=\u0000\r~M\u0002~Ht+\u000b\nMS~M\u0002@~M\n@t\n+\r\nMS~M\u0002~M\u0002 (\f~Hso+Ps~S); (26)\n~Hso=\u000bRmeS\n~eMS(1 +\f2)(~n\u0002~Je); Ps=~SJe\n2eMSd:\nHere,\ris the electron gyromagnetic ratio and \u000bis the Gilbert\ndamping constant. Moreover, \fis the non-adiabaticity param-\neter discussed previously, Psis the spin-torque parameter, S\nis the polarization of injected current into the ferromagnetic9\nFIG. 7: (Color online) Schematic setup of the free ferromagnetic\n(FM) layer with a general saturation magnetization direction,~MS,\ndescribed by polar and azimuthal angles \u0012Mand'M, respectively.\nThe thickness of free ferromagnetic layer is denoted by d. The exter-\nnally applied static magnetic field~H0, polarization vector of injected\ncharge current~S, spin-orbit coupling torque vector~Hso, and finally\nnormal unity vector ~nare shown. The ferromagnetic film is located\nin the xyplane so that zaxis is normal to the ferromagnetic film. The\nspin-orbit coupling is assumed to be induced via a substrate layer into\nthe free ferromagnetic layer. The double dot represents the vector\nquantities in the non-rotated coordinate system (laboratory frame-\nwork).\nlayer, and a normal vector to the plane of ferromagnetic layer\nis represented by ~n(see Fig. 7).\nWe now introduce a rotated coordinate system xyz where\nthe saturation magnetization direction is parallel with the z\naxis. The orientation of the rotated system xyz compared to\nthe stationary one xyzis determined by calculating the equi-\nlibrium orientation of the magnetization order parameter and\nsetting thezaxis to be parallel with it. The details of the cal-\nculations will be discussed in what follows.\nWe define a transformation matrix which rotates the fixed\ncoordinate system so that its zaxis to be oriented along~MS.\nTherefore, all other vector quantities should be rotated via the\ndefined transformation to be described in this new rotated co-\nordinate system. If we describe~MSby polar and azimuthal\nangles i.e.\u0012Mand'M, in the fixed original coordinate sys-\ntem, a rotation around the zaxis equal to 'Mand then around\nthe rotated yaxis equal to \u0012Mare required for aligning zaxis\nand~MSorientations. Hence, the rotation matrices can be re-\nspectively given by (see Ref. 40 for more details):\nRz(\u0000'M) =0\n@cos'M\u0000sin'M0\nsin'Mcos'M0\n0 0 11\nA;\nRy(\u0012M) =0\n@cos\u0012M0\u0000sin\u0012M\n0 1 0\nsin\u0012M0 cos\u0012M1\nA:\nThe total rotation matrix is thus the multiplication of RyandRzi.e.\nRt=RyRz=0\n@cos\u0012Mcos'M\u0000cos\u0012Msin'M\u0000sin\u0012M\nsin'Mcos'M0\nsin\u0012Mcos'M\u0000sin\u0012Msin'Mcos\u0012M1\nA:(27)\nWe characterize each vector quantity by its polar and az-\nimuthal angle in the fixed original coordinate system shown\nin Fig. 7. Since we assume a homogeneous magnetization\ntexture (macrospin approximation), we have ~r2~M= 0. The\ntotal effective field entering the LLG-equation may now be\ndecomposed into the following terms:\n~Ht=~Hdip+~hdip(t) +~Ha+~ha(t) +~Hso\n+b~S+~H0+~hext(t)\n\u0011~H+~h(t): (28)\nAbove,f~Hdip;~hdip(t)gandf~Ha;~ha(t)gare the static and dy-\nnamic parts of the dipole and anisotropy fields respectively,\n~Hsois the spin-orbit field, b~Sis the spin-torque effective field\n(which is usually negligible), ~H0is the static externally ap-\nplied field, and finally ~hext(t)is a small rf field applied per-\npendicularly to the saturation magnetization direction zin or-\nder to probe the ferromagnetic resonance. To show an exam-\nple of how the quantities in the two coordinate systems are\nrelated, note that the x,y, andzcomponents of the externally\napplied static magnetic field ~H0in the rotated coordinate sys-\ntem are given by:\nH0x=H0\b\ncos\u0012Mcos'Msin\u0012H0cos'H0\u0000\ncos\u0012Msin'Msin\u0012H0cos'H0\u0000sin\u0012Mcos\u0012H0\t\n;(29)\nH0y=H0\b\nsin'Msin\u0012H0cos'H0+\n\b\ncos'Msin\u0012H0cos'H0\t\n; (30)\nH0z=H0\b\nsin\u0012Mcos'Msin\u0012H0cos'H0\u0000\nsin\u0012Msin'Msin\u0012H0cos'H0\u0000cos\u0012Mcos\u0012H0\t\n:(31)\nAs mentioned above, the dipole field can be divided into static\n~Hdipand dynamic ~hdip(t)parts. In the rotated coordinate\nsystem they may be obtained as31:\n~Hdip=Mcos\u0012M0\n@cos\u0012Msin'M\n\u0000cos'M\nsin\u0012Msin'M1\nA;\n~hdip(t) = 4\u0019my(t) sin\u0012M0\n@cos\u0012Msin'M\n\u0000cos'M\nsin\u0012Msin'M1\nA;\nwhereM\u0019 4\u0019MS\u0000Ha. Assuming a weak rf magnetic\nfield applied transverse to the ^z-direction, we may consider\nthe components of magnetization in the rotated coordinate\nsystem asMz=MS\u001dMx;My. In this case, the fol-\nlowing time-dependent coupled differential equations for the\nprecessing magnetization components are obtained;10\n@Mx\n@t=\u0000\rMyHt\nz+\rMx(\fHso\nz+PsSz)\n+\rMS(Ht\ny\u0000(\fHso\nx+PsSx))\u0000\u000b@My\n@t;\n@My\n@t=\rMxHt\nz+\rMy(\fHso\nz+PsSz)\n\u0000\rMS(Ht\nx+ (\fHso\ny+PsSy)) +\u000b@Mx\n@t;\n@Mz\n@t=@MS\n@t= 0 =\rMx(\u0000Ht\ny+ (\fHso\nx+PsSx))\n+\rMy(Ht\nx+ (\fHso\ny+PsSy)):\nSetting the transverse part of the magnetization and fields\nequal to zero in the above equations for @tMxand@tMy,\none obtains the equilibrium conditions which specify the ori-\nentation of the zaxis:\n\u001aHx+ (\fsoHso\ny+\fsSy) = 0\nHy\u0000(\fsoHso\nx+\fsSx) = 0: (32)\nThis is consistent with the equation for @tMzand our preas-\nsumption namely; Mz\u001dMx;My. In order to obtain the\nsolution for the transverse components MxandMyto lowest\norder, we now substitute these conditions back into the equa-\ntions of motion for the magnetization components above and\nobtain:\n@Mx\n@t=\u0000\rMyHz+\rMShy(t)\u0000\u000b@My\n@t\n+\rMx(\fHso\nz+PsSz);\n@My\n@t= +\rMxHz\u0000\rMShx(t) +\u000b@Mx\n@t(33)\n+\rMy(\fHso\nz+PsSz):\nIn our calculations we have set the time-dependent fields suf-\nficienty small so that those terms including higher orders of\ntime-dependent components are negligible. Assuming that\nthe the external time-dependent magnetic field induces the\nsame frequency in all time-dependent components of other\nvector quantities (including responses) as itself, \n, we get e.g.\n~hdip(t) =~hdipe\u0000i\nt. By substituting this time-dependency\ninto Eqs. (33) we arrive at ~M(t) =\u001f~hext(t)in which\n~M(t) = (Mx;My)T,~hext(t) = (hext\nx;hext\ny)T, and;\n\u001f=\u0012\n\u001fxx\u001fxy\n\u001fyx\u001fyy\u0013\n: (34)\n\u001fis known as the susceptibility tensor which determines the\nbehavior of magnetization in response to the external time-\ndependent magnetic field. The components of the obtained\nsusceptibility tensor in the presence of spin-orbit coupling\nread:\n\u001fxx= +\u0000f\rWy\u0004\u0000\u0001\u000b\n\u0000i(\r\u0001Wy+ \n\u000b\u0004)g;\n\u001fxy=\u0000\u0000f\u0006\u0004 + \u0001\n\u0000i(\u0001\u0006\u0000\n\u0004)g;\n\u001fyx= +\u0000f\u0006\u0004 + \u0001\n\u0000i(\u0001\u0006\u0000\n\u0004)g\n\u001fyy= +\u0000f\rWx\u0004\u0000\u0001\u000b\n\u0000i(\r\u0001Wx+ \n\u000b\u0004)g;where we have defined the following parameters;\n\u0000 =\rMS\n\u00042+ \u00012;\u0006 =\r(\fsoHso\nz+\fsSz);\n\u0004 = \u00072\u0000\n2(1 +\u000b2);\u0007 =q\n\r2WxWy+ \u00062;\n\u0001 = 2\u0006\n\u0000\r\u000b\n(Wx+Wy);\nWx=Hz+Msin\u0012Mcos\u0012Msin'M;\nWy=Hz+Msin\u0012Mcos'M:\nThe susceptibility tensor components may be used to com-\npute physical quantities of interest such as the absorbed\npower (which is experimentally relevant39) by the ferro-\nmagnetic sample with volume Vat frequency \n. In turn,\nthis gives a clear signal of ferromagnetic resonance in the\nabsorption spectrum. This energy dissipation is given by\nPabs\npower =ImfPpowergwherePpower is defined by:\nPpower =\u0000\n2Z\nVdV~hext\u0003\u0001~M=\u0000\n2Z\nVdV~hext\u0003\u0001\u001f~hext\n=\u0000\n2Z\nVdVn\njhext\nxj2\u001fxx+hext\nx\u0003hext\ny\u001fxy+\nhext\ny\u0003hext\nx\u001fyx+jhext\nyj2\u001fyyo\n:\nThis expression simplifies if the rf magnetic field only has one\ncomponent, e.g.~h(t) =hext\nx(t), in which case the power\nabsorbed at radio-frequency \ncan be expressed by:\nPabs\npower =\n2Z\nVdV\rMSjhext\nxj2\n\u00042+ \u00012(\r\u0001Wy+ \n\u000b\u0004):\nAlthough the above expressions may be numerically evalu-\nated in our system for a specific parameter choice, we focus\nbelow on analytical insights that may be gained. In partic-\nular, we are interested in the role played by spin-orbit inter-\nactions and the magnitude/direction of the injected current.\nSo far, our treatment has been general and accounted for sev-\neral terms contributing to the susceptibility tensor. In order\nto identify the role played by current-dependent spin-orbit\ncoupling in the ferromagnetic resonance, we need to derive\nan analytical expression for the ferromagnetic resonance fre-\nquency \nFMR. This is defined as the frequency where the\nPabs\npower has a maximum. In their general form shown above,\nthis cannot be done analytically in an exact manner. How-\never, progress can be made by considering the denominator\nofPabs\npower . This quantity has the following form when all the\nfrequency-dependence is written explicitly:\n\u00042+ \u00012= [\u00072\u0000\n2(1 +\u000b2)]2\n+ \n2[2\u0006\u0000\r\u000b(Wx+Wy)]2: (35)\nFollowing the standard procedure of neglecting the second\nterm above, one may identify the resonance frequency simi-\nlarly to Ref. 31 as \nFMR= \u0007. We have also verified that this\nholds numerically for a realistic parameter set.\nTo see how the spin-orbit coupling affects \nFMR, one should\nnote in particular its dependence on the current J. It is instruc-\ntive to consider first the scenario with zero spin-orbit coupling,11\nin which case the resonance frequency may be written as:\n\nFMR=p\nc1+c2J2; (36)\nwherec1andc2are determined by the quantities in Eq. (35)\nin the limit ~\u000bR!0. Importantly, they are independent on the\ncurrent bias J, which means that the resonance frequency is\ncompletely independent on the direction of the applied current\nas it is only the magnitude J2that enters. Therefore, the cur-\nrent direction cannot alter the \nFMR. Turning on the spin-orbit\ncoupling so that ~\u000bR6= 0, one may in a similar way show from\nthe above equations that the resonance frequency now can be\nwritten as:\n\nFMR=p\n(d1+DJ)(d2+DJ) +d3J2; (37)\nwhere again the coefficients diandDare determined from\nEq. (35). It then follows from Eq. (37) that the resonance\nfrequency will be asymmetric with respect to the applied cur-\nrent direction when spin-orbit coupling is present. In partic-\nular, one obtains different values for \nFMR by reversing the\ncurrentJ!(\u0000J)so that the Z2symmetry in Eq. (36) is\nlost. The main signature of spin-orbit coupling in the current-\nbiased ferromagnetic resonance setup under consideration is\nthen an asymmetric current dependence which should be dis-\ntinguishable from the scenario without spin-orbit interactions.\nIt is interesting to note that the current-dependence on the fer-\nromagnetic resonacne and the linewidth allows one to exert\nsome control over the magnetization dissipation/absorption\nin the system via J. The presence of spin-orbit interactions\nenhances this control since it introduces a directional depen-\ndence which is absent without such interactions.\nIV . SUMMARY\nIn summary, we have considered the influence of existence\nof spin-orbit interactions on both domain wall motion and fer-romagnetic resonance of a ferromagnetic film. Due to the cou-\npling between the momentum and spin of the electrons, the\ndegeneracy between domain wall textures is broken which in\nturn leads to qualitatively different behavior for various wall\nprofiles, e.g.Bloch vs. Neel domain walls. By taking into ac-\ncount both the field- and Slonczewski-like spin-orbit torque,\nwe have derived exact analytical expressions for the wall ve-\nlocity and the onset of Walker breakdown. One of the most\ninteresting consequences of the spin-orbit torques is that they\nrender Walker breakdown to be universal for some wall pro-\nfiles in the sense that the threshold is completely independent\non the material-dependent damping \u000b, non-adiabaticity \f, and\nthe chirality \u001bof the domain wall. We have also shown that\ndomain wall motion against the current flow is sustained in\nthe presence of multiple spin-orbit torques and that the wall\nprofile will determine the qualitative influence of these differ-\nent types of torques. 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Phys. 113, 17C732 (2013)."    },    {        "title": "1511.04802v1.Determination_of_intrinsic_damping_of_perpendicularly_magnetized_ultrathin_films_from_time_resolved_precessional_magnetization_measurements.pdf",        "content": "1 \n Determination of intrinsic damping of perpendicularly magnetized \nultrathin films from time resolved precessional magnetization \nmeasurements  \n \nAmir Capua1,*, See -hun Yang1, Timothy Phung1, Stuart S. P. Parkin1,2 \n \n1 IBM Research Division, Almaden Research Center, 650 Harry Rd., San Jose,  California \n95120, USA  \n2 Max Planck  Institute for Microstructure Physics, Halle (Saale), D -06120, Germany  \n \n*e-mail: acapua@us.ibm.com  \nPACS number(s) : 75.78. -n \n \n \nAbstract:  \nMagnetization dynamics are strongly influenced by damping, namely the loss of spin \nangular momentum from the magnetic system to the lattice. An “effective” damping \nconstant  αeff is often determined experimentally from the spectral linewidth of the free \ninduction decay of the magnetization after the system  is excited to its non -equilibrium state . \nSuch an  αeff, however, reflects both intrinsic damping as well as inhomogeneous \nbroadening that arises , for example,  from spatial variations of the anisotropy field.  In this \npaper we compare measurements of the m agnetization dynamics in ultrathin non -epitaxial \nfilms having perpendicular magnetic anisotropy using  two different techniques,  time-\nresolved magneto optical Kerr effect (TRMOKE ) and hybrid  optical -electrical  \nferromagnetic resonance  (OFMR) . By using a n external  magnetic field that is applied at \nvery small angles  to the film plane  in the TRMOKE studies , we develop  an explicit closed -\nform analytical expression  for the TRMOKE spectral linewidth and show how this can be \nused to reliably extract the intrinsic Gilbert damping constant. The damping constant \ndetermined in this way is in exc ellent agreement with that determined from the  OFMR \nmethod  on the same samples. Our studies indicate  that the asymptotic high -field approach \nthat is often used in the TRMOKE method to distinguish the  intrinsic damping  from the 2 \n effective damping may result in significant error , because such high external  magnetic \nfields are required  to make this approach valid that they  are out of reach . The error becomes \nlarger the lower is the intrinsic damping  constant, and thus may account for the \nanomalously high damping constants that are  often  reported  in TRMOKE studies . In \nconventional ferromagnetic resonance ( FMR ) studies , inhomogeneous contributions can \nbe readily distinguished from intrinsic damping contributions from the magnetic field \ndependence of the FMR linewidth. Using the analogous approach, w e show how reliable \nvalues of the intrinsic damping can be extracted from  TRMOKE in two distinct magnetic  \nsystems with significant perpendicular magnetic anisotropy: ultrathin CoFeB layers and \nCo/Ni/Co trilayers.  \n \n  3 \n I. Introduction  \nSpintronic nano -devices have been identified in recent years as one of the most \npromisin g emerging technologies for future low power microelectronic circuits1, 2. In the \nheart of the dynamical spin -state transi tion stands the energy loss  parameter of the Gilbert \ndamping . Its accurate dete rmination is of paramount importance  as it determines the \nperformance of key building blocks required for  spin manipulation  such as t he switching  \ncurrent threshold of the spin transfer torq ue magnetic tunnel junction  (MTJ)  used in \nmagnetic random access memory (MRAM) as w ell as the skyrmion velocities  and the \ndomain wall motion current threshold . Up-scaling for high logic and data capacities while \nobtaining stability with high retention energies  require in addition that large  magnetic  \nanisotropies be ind uced. T hese cannot be achieved simply by engineering the geometrical \nasymmetries in the nanometer -scale range , but rather  require harnessing the induced spin-\norbit interaction a t the interface of the  ferromagnet ic film to obtain perpendicular  magnetic  \nanisotropy  (PMA)2. Hence an increasing  effort is invested in the quest for perpendicular \nmagnetized materials  having large anisotropies with low Gilbert damping3-11. \nTwo distinct families of experimental methods  are typically used for measurement \nof Gilbert damping , namely,  time-resolved pump -probe and continuous microwave \nstimulated ferromagnetic resonance ( FMR ), either of which can be implemented using \noptical and/or electrical methods . While in some cases good agreement between these \ndistinct techniques have been reported12, 13, there is often significant disagreement between \nthe methods14, 15.  4 \n When the time resolved pump -probe  method  is implemented using the magneto \noptical Kerr effect ( TRMOKE), a clear advantage over the FMR  method  is gained  in the \nability to  operate at  very high fields and frequencies16, 17. On the other hand , the FMR \nmethod a llows operation over a wide r range of geometrical configurations . The \nfundamental geometrical restriction of the TRMOKE comes from the fact that  the \nmagnetization  precession s are initiated from  the perturbation of the  effective  anisotropy \nfield by the pump  pulse , by momentarily  increasing the lattice temperature18, 19. In cases \nwhere  the torque exerted by the  effective  anisotropy field is n egligible , the pump pulse \ncannot sufficiently perturb the magnetization . Such a case occurs  for example  whenever \nthe magnetization lays in the plane of the sample in uniaxial  thin films having \nperpendicular magneti c anisotropy . Similar limitations exist if the magnetic field i s applied  \nperpendicular to the film . Hence in TRMOKE experiments , the external field is usually \napplied at  angles  typically not smaller than about \n10\n  from  either the film plane or its \nnormal.  This fact has however the consequence that the steady state magnetization \norientation , determined by the bal ancing condition for  the torques , cannot be described  \nusing a n explicit -form algebraic expression , but rather a numerical approach  should be \ntaken5. Alternatively , the dynamics  can be  described using an effective damping  from \nwhich the intrinsic damping , or at least an upper bound o f its value , is estimated at the high \nmagnetic field  limit with the limit being  undetermined . These approaches are hence less \nintuitive while the latter does not indicate directly on the energy losses but rather on the  \ncombination of the energy loss rate , coherence  time of the spin ensemble  and geometry of \nthe measurement .  5 \n In this paper , we present an approach where the TRMOKE system is operated while \napplying the  magnetic field at  very sm all angles with respect to the sample plane. This \nenables  us to use explicit closed -form analytical expressions derived for a perfectly in -\nplane external magnetic field as an approximate  solution. Hence , extraction of the intrinsic \nGilbert damping using an analytical model becomes possible without the need to drive the \nsystem to the high magnetic field limit providing at the same time an intuitive \nunderstanding of the measured responses. The validity of t he method  is verified using a \nhighly sensitive hybrid optical -electrical FMR system (OFMR) capable of operating with \na perfectly in -plane magnetic field where the analytical expressions hold. In particular, we \nbring to test the high -field asymptotic approa ch used for evaluation of the intrinsic damping \nfrom the effective damping and show that in order for it to truly indicate the intrinsic \ndamping, extremely high fields need to be applied. Our analysis reveals the  resonance \nfrequency dispersion relation as well as the inhomogeneous broadening to be the source of \nthis requirement which becomes more difficult  to fulfill the smaller the intrinsic damping \nis. The presented method is applied  on two distinct families of technologically relevant \nperpendicularly mag netized systems; CoFeB4, 6 and Co/Ni/Co20-23. Interestingly, the results \nindicate  that the Ta seed layer  thickness used in  CoFeB  films  strongly affects the intrinsic \ndamping , while t he static characteristics of the  films remain intact . In the Co/Ni/Co trilayer \nsystem which has in contrast a large effective anisotropy field, unexpected ly large spectral \nlinewidth s are measured when  the external  magnetic field is comparable to the  effective  \nanisotropy field, which cannot be explained by the  conventional model  of no n-interacting \nspins  describing the inhomogeneous broadening . This  suggest s that under the low stiffness  6 \n conditions associated with such bias fi elds, cooperative  exchange  interactions, as two \nmagnon scattering, become relevant8, 24.  \nII. EXPERIMENT  \nThe experiments present ed were carried out on three PMA samples: two samples \nconsist ed of Co36Fe44B20 which differed by the thickness of the underlayer  and a third \nsample consisting of Co/Ni/Co trilayer . The CoFeB samples were characterized by low \neffective anisotropy  (Hkeff) values as well as by small distribution of its value  in contrast to \nthe Co /Ni/Co trilayer system . We define here Hkeff as 2Ku/Ms-4πMs where Ku is the \nanisotropy energy constant and Ms being the saturation magnetization.   \nThe structure s of the two CoFeB samples were 50Ta|11CoFeB |11MgO |30Ta, and \n100Ta|11CoFeB |11MgO |30Ta (units are in Å)  and had  similar Ms value s of 1200 emu/cc  \nand Hkeff of 1400 Oe and 1350 Oe respectively.  The t hird system studied was  \n100AlO x|20TaN |15Pt|8Pt 75Bi25|3Co|7Ni|1.5Co |50TaN  with Ms of 600 emu/cc  and Hkeff \nvalue  of about 4200 Oe . All samples were grown on oxidized Si substrates  using DC \nmagnetron sputtering  and exhibited sharp perpendicular switching characteristics . The \nsamples consisting of CoFeB were annealed for  30 min  at \n275\n  C in contrast to the \nCo/Ni/Co which was measured as deposited.  Since the resultant film has a polycrystalline  \ntexture , the in -plane anisotropy is averaged out and the films are regarded as uniaxial \ncrystals with the symmetry axis being perpendicular to the film plane.  7 \n The t wo configurations of the experimental setup were driven by a Ti:Sapphire  laser \nemitting 70 fs pulses at 800 nm  having energy of 6 nJ. In the first configuration a standard  \npolar  pump -probe TRMOKE was implemented with the probe  pulse being a ttenuated by \n15 dB compared to the  pump pulse. Both  beams were focused on the sample to an estimated \nspot size of 10.5 m defined by the full width at half maximum (FWHM) . In the hybrid \noptical -electrical  OFMR system , the Ti:Sapphire laser  served to pro be the magnetization \nstate via  the magneto -optical  Kerr effect after being attenuated  to pulse energies of about \n200 pJ  and was phase -locked with a microwave oscillator in a similar configuration to the \none reported in Ref. [ 25]. For this measurement , the film was patterned into a  20 m x 20 \nm square island with a Au wire  deposited in proximity  to it, which  was driven by the \nmicrowave signal. Prior to reaching the sample, the probing laser beam traversed the \noptical delay line that enabled mapping of the time axis and in particular the  out of plane -\nmz component of the magnetization  as in the  polar TRMOKE  experiment . With this \nconfiguration the OFMR realizes  a conventional FMR system where the magnetization \nstate is read in the time-domain using the magneto optical Kerr effect  and hence its high \nsensitivity . The OFMR  system therefore  enables operation  even when the external field is \napplied fully in the sample plane.  \nIII. RESULTS AND DISCUSSION  \nA. TRMOKE measurements on 50 Å-Ta CoFeB  film  \nThe first experiments we present  were performed  on the 50  Å-Ta CoFeB  system  \nwhich is similar to the one studied in Ref. [4]. The TRMOKE measurement was carried 8 \n out at two angles  of applied magnetic field,  \nH, of \n4\n and \n1\n  measured from the surface \nplane as indicated in Fig. 1. We de fine here in addition the comple mentary angle measured \nfrom the surface normal, \n2HH   . Having i ts origin in the effective anisotropy, the \ntorque  generated by the optical pump  is proportional to \n cos( )sins keffMH   with θ being \nthe angle of the magnetization relative to the normal of the sample plane.  Hence, f or \n1H\n, the angle θ becomes close to \n/2 , and the resultant torque generated by the optical pump \nis not strong enough to  initiate reasonable precessions . For the same reason, the maximum \nfield measureable for the \n1H\n  case is significantly lower than for the \n4H\n  case. This \nis clearly demonstrated in the m easured MOKE signals for the two \nH  angles in Fig. 2 (a). \nWhile for \n4H\n  the precessional motion  is clearly seen even at a bias field of 12 kOe, \nwith \n1H\n  the precessions are hardly observable already at a bias field of 5.5 kOe.  \nAdditionally, it is also possible that the  lower signal to noise ratio observed for  \n1H\n may \nbe due to a breakdown into domains with the almost in -plane applied magnetic field26. \nAfter reduction of the background signal, the measured data can be fitted to a decaying \nsinusoidal response from which t he frequency and decay time can be extracted in the usual \nmanner 6 (Fig. 2(b)) . The measured precession frequency as a function of the applied \nexternal field , H0, is plotted in Fig. 3(a). Significant differences near Hkeff are observed for \nmerely a change of three degrees in the angle of the applied magnetic field . In particular, \nthe trace for \n1H\n  exhibits a minimum point at approximately  Hkeff in contrast to the \nmonotonic behavior of the \n4H\n  case. The theoretical dependence of the resonance 9 \n frequency  on the magnetic  bias field expressed in normalized units, \n/keffH , with \n  \nbeing the resonance angular frequency and \n  the gyromagnetic ratio, is presented in Fig. \n3(b) for several  representative  angles of the applied  field. The resonance frequency at the \nvicinity of Hkeff is very sensitive to slight changes in the angle of the applied field  as \nobserved also in the experiment . Actually the derivative of the resonance frequency with \nrespect to the applied field  at the vicinity of Hkeff is even more sensitive where it diverges \nfor \n90\n  but reaches a value of zero for the slightest angle divergence.  A discrepancy \nbetween the measurement and the theoretical solution exists however.  At field values much \nhigher than Hkeff the precession frequenc y should be identical for all angles (Fig. 3(b)) but \nin practice the resonance frequency measured for \nH  of \n4\n  is consistently higher by nearly \n2 GHz  than at \n1\n . The t heory  also predicts that for the case of \n4\n , the resonance frequencies \nshould  exhibit a minimum point as well which is not observed  in the measurement . The \norigin of the difference is not clear and may be related to the inhomogeneities in the local \nfields  or to the higher orders of the  interface induced anisotropy  which were neglected in \nthe theoretical calculation .  \nIn Fig. 3(c), we plot the effective Lorentzian resonance linewidth in the frequency \ndomain , \neff , defined by \n2/eff eff  with \neff being the measured decay time extracted \nfrom the measured responses. Decompos ing the measured linewidth t o an  intrinsic \ncontribution that represent s the energy loss es upon precession  and an extrinsic contribution \nwhich represent s the inhomogeneities in the local fields and is not related  to energy loss of 10 \n the spin system , we express  the linewidth as : \nint eff IH     . \nint  is given by the \nSmit -Suhl formula27, 28 and equals \n2/  with \n  denoting  the intrinsic spin  precession  decay \ntime where as \nIH  represents the dispersion in the resonance frequencies due to the \ninhomogeneities. If the variations  in the resonance frequency  are assumed to be primarily \ncaused by variations in the local effective anisotrop y field \nkeffH , \nIH  may be given by : \n/IH keff keff d dH H  \n. For the case of \n/2H  or \n0H , \neff  has a closed  \nmathematical  form.  In PMA  films  with bias field applied in the sample plane , the \nexpression  for \neff  becomes : \n      \n0\n002\n00\n0\n0022\n002         for        H\n2\n2     for         Heff keff keff keff\nkeff\nkeff keff\neff keff keff\nkeffkeffHH H H H\nH H H\nHH HH H HHH HH \n      \n\n\n              ,    (1) \nwith \n denot ing the Gilbert damping . The first term s in Eq. (1) stem from  the intrinsic \ndamping , while the second term s stem  from the inhomogeneous broadening . Eq. (1)  shows \nthat while the contribution of the intrinsic part to the total spectral linewidth  is finite, as the \nexternal field approaches  Hkeff either from higher or lower field values, the inhomogeneous \ncontribution diverges. Equation  (1) further shows that for H0 >> Hkeff , the slope of \neff  \nbecomes \n2  with a constant offset given by \n/2keffH . Although Eq. (1) is valid only \nfor\n/2H , it is still  instructi ve to apply it on the measured linewidth for the  \n4H\n  case.  11 \n The theoretical intrinsic linewidth for \n/2H , inhomogeneous contribution and the sum \nof the two a fter fitting \n  and \nkeffH  in the range H 0 > 5000 Oe  are plotted in Fig. 3(c). The \nresul tant fitting values were 0.023 ±0.002 for the Gilbert damping and 175 Oe for \nkeffH . At \nexternal fields comparable to Hkeff the theoretical expression derived for the  \ninhomogeneous broadening for a perfectly in -plane field does not describe properly the \nexperiment . In the theoretical analysis , at fields comparable to Hkeff, the derivative \n0/d dH\ndiverges  and therefore also the derivative \n/keff d dH  as understood from Fig. 3(b).  In the \nexperiment however , \n/2H  and the actual derivative \n/keff d dH  approache s zero. \nHence any variation in Hkeff result s in minor  variation of the frequency . This mean s that the \ncontribution of the inhomogeneous broadening to the total linewidth is suppressed near \nHkeff in the experiment as opposed to being expanded in  the theoretical calculation which \nwas carried out for  \n/2H . The result  is an overestimate d theoretical linewidth  near \nHkeff. After reduction of the inhomogeneous broadening , the extracted  intrinsic measured \nlinewidth is  presented in Fig. 3(c) as well  showing the deviation from  the theor etical \nintrinsic contribution as the field approaches  Hkeff.  \nTo further investigate the e ffect of tilt ing the magnetic field , we study the TRMOKE \nresponses for the \n1H\n  case.  The measured linewidth for this  case is presented in Fig. \n3(d). In contrast to the \n4H\n  case, the measured linewidth now increases at fields near \nHkeff as expected theoretically . Furthermore,  the measured linewidth  for the \n1H\n  case is 12 \n well describe d by Eq. (1) even in the vicinity of Hkeff as well as for bias  fields smaller than \nHkeff. The fitting result s in the same damping value of 0.023 ±0.0015 as with the \n4H\n  \ncase,  and a variation in \nkeffH  of 155 Oe, which is 20 Oe smaller  than the value fitted for \nthe \n4H\n  case.  \nWe next turn to examine  the G ilbert damping. In the absence of the demagnetization \nand crystalline anisotrop y fields, the expression  for the intrinsic Gilbert damping is given \nby:  \n                                                     \n1 .                  (2) \nOnce the anisotropy and the demagnetization field s are included , the expression for the \nintrinsic Gilbert damping becomes :  \n          \n 0\n0\n0\n0\n001                                         for           \n21       for           \n2keff\nkeff\nkeff keffdHHHd\ndHHHd H H H H \n   \n   \n  ,          (3) \nand is valid  only for \n2H  and for  crystals having uniaxial symmetry. At oth er angles \na numerical method5 should be used  to relate the precession decay time to the Gilbert \ndamping. Eq. (3) is merely the intrinsic contribution in Eq. (1)  written in the form \nresembling Eq. (2) . At high fields both Eq s. (2) and (3) converge  to the same result since  13 \n \n1 0dH\nd. As seen in Fig. 3(b), at bias field s comparable to Hkeff the additional derivative \nterm of Eq. (3) becomes very significant . When substituting the measured  decay time,\neff\n, for \n , Eq. (2) gives what is often interpreted as  the “effective ” damping , αeff, from which \nthe intrinsic damping is measured by evaluating it at high fields when the damping becomes  \nasymptotically field independent. Additionally, t he asymptotic limit should be reached \nwith respect to  the inhomogeneous contribution of  Eq. (1). In Fig. 3(e), we plot the effective \ndamping  using \neff  and Eq. (2) . We further show the intrinsic damping value after  \nextracting the intrinsic linewidth  and using Eq. (3). Examining first the effective damping \nvalues, we see that for the two angles , the values are distinctively  different at low fields  \nbut converge at approximately 41 00 Oe  (Beyond  5500 Oe the data  for the \n1H\n  case \ncould not be measured).  In fact , the behavior of the effective damping seems to be related \nto the dependence of the resonance frequency on H0 (Fig. 3(a)) in which  for the \n1H\n  \ncase reaches an extremum while  the \n4H\n  case exhibits a monotonic behavior . Since Eq. \n(2) lacks  the derivative term \n0/dH d , near Hkeff the effective damping is related to the \nGilbert damping by the relation: \n01\neffd\ndH  for H0 > Hkeff. Furthermore, since  \n does not \ndepend on the magnetic field to the first order, the dependence  of the effective damping,\neff , on \nthe bias field stems from the derivative  term \n0 d dH which becomes larger and eventually \ndiverges to infinity  when the magnetic field reaches Hkeff as can be inferred from Fig. 3(b) \nfor the case of \n0H\n  for which Eq. (3) was derived . Hence the increase in \neff at bias 14 \n fields near Hkeff. The same considerations apply also for H0 < Hkeff. As the angle \nH  increases , \nthis analysis becomes  valid only for bias fields which are large enough or small enough \nrelative to Hkeff. When examined  separately, each effective damping trace may give the \nimpression that  at the higher fields  it has become bias field independent and reached its \nasymptotic value from which two very distinct  Gilbert  damping  values of ~0.027 and \n~0.039 are extracted at field values of 12 kOe and 5.5 kOe  for the \n4H\n and \n1H\nmeasurements , respectively . These values are also rather different from the intrinsic \ndamping value of 0.023  extracted using  the analytical model . In contrast  to the effective \ndamping , the intrinsic damping obtained from the analytical model  reveal s a constant and \ncontinuous behavior  which is field and angle independent.  The presumably negative values \nmeasure d for the \n4H\n case stem of course  from the fac t that the expressions in Eqs. (1) \nand (3)  are derived for the  \n2H  case. The error in using the effective damping in \nconjunction with the asymptotic approximation compared to using  the analytical model is \ntherefore 17% and 70% for the \n4H\n and \n1H\n  measurements respectively.  \nIt is important in addition to understand th e conse quence of using Eq. (2) rathe r than \nEq. (3) . In Fig. 3(f) we present the error  in the damping value  after accounting for the \ninhomogeneous broadening using Eq. (2) instead of the complete  expression of  Eq. (3) . As \nexpected , the error increases as the applied field approaches Hkeff. For the measurement \ntaken with \n4H\n  the error is significantly smaller due to the smaller value of the \nderivative\n0/d dH .  15 \n As mentioned previously, i n order to evaluate the intrinsic damping from the total \nmeasured linewidth , the asymptotic limit should be reached with respect to the \ninhomogeneous broadening as well  (Eq. (1) ). In Fig s. 3(c) and 3(d) we see that this is not \nthe case  where the contribution  of the inhomogeneous linewidth is still large compared to \nthe intrinsic l inewidth . Examining Figs. 3(d) and 3(f) for the case of \n1H\n , we see that \nthe overall error of 70% resulting in the asymptotic evaluation stems from both the  \ncontribution of  inhomogeneous broadening as well as from the use of Eq. ( 2) rather than \nEq. (3) while for \n4H\n  (Figs. 3(c) and 3(f)) the error of 17% is solely due to contribution \nof the inhomogeneous broadening which was not as negligible as conceived when applying \nthe asymptotic approximation .  \nB. Comparison of TRMOKE and OFMR measurements in 100  Å-Ta CoFeB  \nfilm \nWe next turn to  study the magnetization dynamics using the OFMR system where \nthe precession s are driven with the microwave signal . Hence, the external magnetic field \ncan be applied perfectly in the sample plane. The 100 Å-Ta CoFeB sample  was used for \nthis experiment. Before patterning the film for the OFMR measurement, a TRMOKE \nmeasurement was carried out at \n4H\n  which  exhibited a similar  behavior to that observed \nwith the sample having  50 Å Ta  as a seeding layer . The  dependence of the  resonance \nfrequency on the magnetic  field as well  as the measured linewidth and its  different \ncontributions  are presented in Figs. 4(a) and 4(b). Before reduction of the inhomogeneous 16 \n broadening the asymptotic effective damping was measured to be ~0.0168 while after \nextraction  of the intrinsic damping a value of 0.0109 ±0.0015 was measured marking a \ndifference of 54%  (Fig. 4(c)). The fitted \nkeffH  was 205 Oe. Fig. 4(b) shows that the origin \nof the error stems from significan t contribution of the inhomogeneous broadening \ncompared to the intrinsi c contribution which plays a mor e significant role when the \ndamping is low.  By us ing the criteria  for the  minimum  field that results in  \n10IH eff   \nto estimat e the point where the asymptotic approximation would be valid , we arrive to a \nvalue of at least 4.6 T which is rather impractic al. The threshold of this minimal f ield is \nhighly dependent on the damping so that for a lower damping an even higher field would \nbe required.  \nAn example of a measured trace using the OFMR system  at a low microwave \nfrequency of 2.5 GHz  is presented in Fig. 4(d). The square root of the magn etization \namplitude  (out of plane mz component)  while preserving its sign  is plotted to show detail . \nThe high sensitivity of the OFMR system enable s operation  at very low frequencies and \nbias fields. For every frequency  and DC  magnetic  field value , several  cycles of the \nmagnetization precession  were recorded  by scanning the optical delay  line. The magnetic \nfield was then swept  to fully capture the resonance . The trace should be examined  \nseparately in two sections, be low Hkeff and above Hkeff (marked in the figure by black dashed \nline). For frequencies of up to \nkeffH  two resonances are crossed as indicated by the guiding \nred dashed line which represents the out -of-phase component of the magnetization, namely \nthe imaginary part of the magnetic susceptibility . Hence the cross section along this line 17 \n gives  the field dependent absorption spectrum  from which the resonance frequency and \nlinewi dth can be identified. This spectrum is show n in Fig. 4( e) together with the fitted \nlorentzian lineshapes  for bias fields below and above Hkeff. The resultant resonance \nfrequencies  of all measurements are plotted in addition in Fig. 4(a).  \nThe resonance linewidth s extracted  for bias fields larger than Hkeff, are presented in \nFig. 4(f). Here the effective magnetic field linewidth , ΔHeff, that includes the contribution \nof the inhomogeneous broadening derived  from the same principles that led to Eq. (1) with \n/2H\n is given by: \n              \n02\n2\n0\n00\n2 2\n0211                     for       2\n4\n2\n                   \n        with       keff\neff keff keff\nkeff\nkeff keff\neff keff\nkeff\nkeffHH H H H\nH\nHH HHHH H H\nHH\n\n\n\n\n\n\n      \n      \n \n       \n 0      for       keff HH   (5).  \nThe second terms in Eq. (5) denote  the contribution of the inhomogeneous broadening , \nIHH\n, and are frequency dependent as opposed to the case where the field is applied out \nof the sample plane9. The dispersion in the effective anisotropy , \nkeffH , and the intrinsic \nGilbert damping  were found by fitting the linewidth in the  seemingly  linear range at \nfrequencies larger th an 7.5 GHz . The contribution s of the intrinsic  and inhomogeneous \nparts  and the ir sum are presented as well in Fig. 4(f).  18 \n It is apparent that the measured linewidth at the lower frequencies is much broader \nthan the theoretical one.  The reason for that lies in the fact that in practice the bias field is \nnot applied perfectly in the sample plane as well as in the fact that there migh t be locally  \ndifferent orientation s of the polycrystalline  grains  due to the natural imperfections of the \ninterfaces that further result  in angle distribution of \nH . Since  the measured  field linewidth \nis a projection of the spectral  linewidth into the  magnetic  field domain, the relation between \nthe frequency and the field  intrinsic  linewidth s is given by: \n1\nint int\n0dHdH   \n . The \nintrinsic linewidth , \nint , in the frequency domain near Hkeff is finite , as easily seen from \nEq. (1)  while the derivative term near Hkeff is zero for even the slightest angle misalignment  \nas already seen. H ence the field-domain  linewidth  diverg es to infinity as observed \nexperimentally. The inhomogeneous broadening component does not diverge in that \nmanner  but is rather suppressed . To show that the excessive linewidth at low field s is \nindeed related to the derivative of \n0/d dH  we empirically multiply the total theoretical \nlinewidth by the factor \n0 / ( )d d H  which  turns out to fit  the data surprisingly well (Fig. \n4(f)). This  is merely a phenomenological qualitative description, and a rigorous description \nshould still be derived.   \nThe fitted linewidth of Fig. 4(f) results in  the intrinsic damping value of \n0.011 ±0.0005  and is identical to the value obtained by the TRMOKE  method . Often \nconcerns regarding the differences between the TRMOKE and FMR measurements such \nas spin wave emission away from the pump laser spot in the TRMOKE29, increase of 19 \n damping due to thermal heating  by the pump pulse  as well as differences in the  nature of \nthe inhomogeneous broadening  are raised. Such effects do not seem to be significant here . \nAdditionally , it is worth noting that s ince the linewidth seems to reach a  linear  dependence \nwith respect to  the field  at high fields , it may be naively fitted using a constant frequency -\nindependent inhomogeneous broadening factor . In that case an underestimated value of \n~0.0096 would have been obtained . The origin of this misinterpretation is seen clearly by \nexamining the  inhomogeneous broadening contribu tion in Fig. 4(f) that show s it as well to \nexhibit a seemingly linear dependence at the high fields. Regarding the  inhomogeneous \nbroadening , the anisotropy field dispersion,  \neffKH , obtained with the TRMOKE was 205 \nOe while the value  obtained from the OFMR system was 169 Oe . Although these values  \nare of the same order of magnitude , the difference is rather significant. It is possible that \nthe discrepancy  is related to the differences in the measurement  techniques. For instance, \nthe fact that both  the pump and probe beams have the same spot size may cause an uneven \nexcitation across the probed region in the case of the TRMOKE measurement  while in the \ncase of the OFMR measurement the amplitude of the microwave field decays at increasin g \ndistances away from the microwire. These effects may  be reflect ed in the measurements as \ninhomogeneous broadening. Nevertheless , the measured intrinsic damping values are \nsimilar.  \nFinally , we compare the effective damping of the OFMR and the TRMOKE \nmeasu rements without correcting for the inhomogeneous broadening in Fig. 4(g). The 20 \n figure shows a deviation in the low field values which  is by now understood to  be unrelated \nto the energy losses  of the system .  \nFurthermore, we observe that the thickness of the Ta underlayer affects the \ndamping. The comparison  of the 50 Å -Ta CoFeB and the 100 Å -Ta CoFeB samples  shows \nthat the increase by merely 50 Å of Ta, reduced significantly the damping while leaving \nthe anisot ropy field unaffected.  \nC. TRMOKE and OFMR measurements in Co /Ni/Co film  \nIn the last set of measurements we study the Co /Ni/Co film which has distinctively \ndifferent static properties compared to the CoFeB samples . The sample was studied  using \nthe TRMOKE setup at two \nH  angles of \n1\n  and \n4\n  and using the OFMR system at\n0H\n. The resultant  resonance frequency traces are depicted in Fig. 5(a). The spectral linewidth \nmeasure d for \n4H\n using  the TRMOKE  setup  is presented in Fig . 5(b). A linear fit at the \nquasi linear high field range results in a large damping value of 0.081 ±0.015 and in a very \nlarge \neffKH  of 630 Oe . The large damping is attributed to the efficient spin pumping into \nPtBi30 layer  having large spin -orbit coupling . When the angle of the  applied  magnetic f ield \nis reduced to  \n1H\n  a clearer picture of the  contribution  of the inhomogeneous broadening \nto the total linewidth is obtained (Fig. 5(c)) revealing  that it cannot explain solely the \nmeasured  spectral  linewidth s. While the theoretical model predicts that  the increase in \nbandwidth spans  a relatively  narrow  field range around Hkeff, the measurement shows an \nincrease over a much larger range around Hkeff. The linewidth broadening originating from 21 \n the anisotropy dispersion was theoretically calculated under the assumption of a small \nperturbation of the resonance frequency. A large \nkeffH  value was measured however  from \nthe TRMOKE measurement taken at \n4H\n . Calculating numerically the exact variation \nof the resonance frequency improved slightly the fit but definitely  did not resolve the \ndiscrepancy  (not presented) . From this fact we understand that  there should be an  additional  \nsource  contributing  to the line broadening  at least near Hkeff. A possible explanation may \nbe related to the low stiffness27 associated with the \n0 keff HH  conditions . Under such \nconditions weaker torques which are usually neglected may become relevant24, 31. These  \ntorques could possibly originate from  dipolar or exchange coupling resulting in two \nmagnon scattering processes  or even in a breakdown into magnetic domains as described \nby Grolier et al.26. From the limited data range at this angle, the damping could not be \nmeasured.  \nThe OFMR system  enabl ed a wider range of fields and frequencies than the  ones \nmeasured with the TRMOKE  for \n1H\n  (Fig. 5(a)). Fig. 5(d) presents t he measured  \nOFMR linewidth . The quasi -linear regime of the linewidth seems to be reached  at \nfrequencies of 12 GHz corresponding to bias field values which are larger than 7500 Oe . \nThe resultant intrinsic damping after fitting to this range was 0.09±0.005  with a \nkeffH  of \n660 Oe  which differ  by approximately 10% from the  values obtained from the  TRMOKE \nmeasurement . The effective measured damping is plotted in Fig. 5(e). The asymptotic \ndamping value , though not fully reached  for this high damping sample , would be about 0.1.  22 \n This represents an error of about 10% which is smaller compared to the errors of 17% and \n54% encountered in the CoFeB samples because of the larger damping of the Co /Ni/Co \nsample.   \nD. Considerations of two -magnon scattering  \nIn general, two -magnon spin wave scattering by impurities may exist in our \nmeasurements  at all field ranges32, 33, not only near Hkeff as suggested in the discussion of \nthe previous section32, 33. The resultant additional linewidth broadening would then be \nregarded as an extr insic contribution to the damping34-36. While in isotropic films which \nexhibit low crystalline anisotropy or in films having in -plane  crystalline anisotropy, two -\nmagnon scattering is maximized when the external field is applied in the film plane, in \nPMA films this is not necessarily the case and the highest efficiency of two -magnon \nscattering may be obtained at some oblique angle35.  \nIn films where two -magnon scatt ering is significant , the measured linewidth should \nexhibit  an additional  nonlinear dependence on the external field which  cannot be accounted \nfor by the present model . In such case, a s trong dependence on the external  field would be  \nobserved  for fields below Hkeff  due to the variation in the  orientation of the magnet ization \nwith the external magnetic field. At higher fields the dependence on the external field is \nexpected to be moderate35.  \nWhile at bias field values below Hkeff our data is relatively limited, at external \nmagnetic fields that are larger than Hkeff, the observed linewidth seems to be described well 23 \n by our model  resulting in a field independent Gilbert damping coefficient . This seems to \nsupport our model that the scattering of spin waves does not have a prominent effect. It is \npossible however that a moderate dependence on the bias field, especially at high field \nvalues, may have been “linearized” and classified as intrinsic damping.  \n \nIV. CONCLUSION  \nIn conclusion, in this paper we studied the time domain magnetization dynamics  in \nnon-epitaxial  thin films  having perpendicular magnetic  anisotropy  using the TRMOKE  and \nOFMR  system s. The analytical model used to interpret the magnetization dynamics from \nthe TRMOKE  responses indicated that the asymptotic high -field approach often used to \ndistinguish the intrinsic damping from the effective damping may result in significant error  \nthat increases the lower the damping is . Two sources for the error  were  identified  while \nvalidity of th e asymptotic  approach was shown to require very high magnetic fields. \nAdditionally, the effective damping was shown to be highly affected by the derivative of \nthe resonance frequency with respect to  the magnetic field  \n0/d dH . The analytical \napproach  developed here was verified by use of the OFMR measurement showing excellent \nagreement whenever the intrinsic damping was compared and ruled out the possibility of \nthermal heating by the laser or  emission of spin waves away from  the probed  area.  \n  24 \n As to the systems studied, a  large impact of the seed layer  on the intrinsic damping \nwith minor effect on the static characteristics  of the CoFeB system was observed  and may \ngreatly aid in engineering the proper materials for the MTJ. Interestingly, the use of the \nanalytical model enabled identification of  an additional exchange torque when low stiffness \nconditions prevailed. While  effort still remains  to understand th e limits on the angle of the \napplied magnetic field to which the analytical solution  is valid , the approach  presented is \nbelieved to accelerate  the discovery of novel materials for new applications .  25 \n Acknowledgments:  \nA.C. thanks the Viterbi foundation and the Feder Family foundation for supporting this \nresearch.  \nReferences:  \n1 Z. Yue, Z. Weisheng, J. O. Klein, K. Wang, D. Querlioz, Z. Youguang, D. Ravelosona, and C. \nChappert, in Design, Automation and Test in Europe Conference and Exhibition (DATE), 2014 , \np. 1. \n2 A. D. Kent and D. C. Worledge, Nat Nano 10, 187 (2015).  \n3 M. Shigemi, Z. Xianmin, K. Takahide, N. Hiroshi, O. Mikihiko, A. Yasuo, and M. Terunobu, \nApplied Physics Express 4, 013005 (2011).  \n4 S. Iihama, S. Mizukami, H. Nagan uma, M. Oogane, Y. Ando, and T. Miyazaki, Physical Review \nB 89, 174416 (2014).  \n5 S. Mizukami, Journal of the Magnetics Society of Japan 39, 1 (2015).  \n6 G. 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Baberschke, D. Spoddig, R. Meckenstock, J. Pelzl, Z. Frait, \nand D. L. Mills, Physical Review B 68, 060102 (2003).  \n34 H. Suhl, Magnetics, IEEE Transactions on 34, 1834 (1998).  \n35 M. J. Hurben and C. E. Patton, Journal of Applied Physics 83, 4344 (1998).  \n36 D. L. Mills and S. M. Rezende, in Spin Dynamics in Confined Magnetic Structures II , edited by \nB. Hillebrands and K. Ounadjela (Springer Berlin Heidelberg, 2003).  \n \n \n  27 \n Figure 1  \n \nFIG 1. Illustration of the angles \nH , \nH and \n . M and H0 vectors denote the magnetization \nand external magnetic field , respectively.  \n  \n28 \n Figure 2 \n \nFIG. 2. Measured TRMOKE  responses  at \nH  angles of \n4\n and \n1\n . (a) TRMOKE signal at \nlow and high external magnetic field values. Traces are shifted for clarity. (b) Measured \nmagnetization responses after reduction of background signal (open circles) \n29 \n superimposed  with the fitted decaying sine  wave  (solid lines). Traces are shifted and \nnormalized to have the same peak  amplitude. Data presented for  low and high external \nmagnetic field values.  \n  30 \n Figure 3 \n  \nFIG. 3. TRMOKE measurements at \n4H\n  and \n1H\n . (a) Measured resonance \nfrequency versus magnetic field. (b) Theoretical dependence of resonance frequency on \nmagnetic field  presented in normalized units . (c) & (d)  Measured linewidth (blue) , fitted \ntheoretical con tributions to l inewidth (green, cyan, magenta) and extracted intrinsic \nlinewidth from measurement (red) for \n4H\n  and \n1H\n , respectively. (e) Intrinsic and \neffective damping. (f) Error in damping value when using Eq. (2) instead  of Eq. (3 ).  \n31 \n Figure 4 \n \nFIG. 4. TRMOKE and OFMR measurements at \n4H\n  and \n0H\n , respectively. (a) \nMeasured resonance frequency versus magnetic field. (b)  Measured linewidth (blue), fitted \ntheoretical contributions to linewidth (green, cyan, magenta) and extracted intrinsic \n32 \n linewidth from measurement (red) using the TRMOKE with  \n4H\n . (c) Intrinsic and \neffective damping  using TRMOKE . (d) Representative OFMR trace at 2.5 GHz.  The \nfunction sign( mz)(mz)1/2 is plotted.  (e) Field dependent absorption spectrum  (blue)  \nextracted from the cross section along the red dashed lined of (d) together with fitted \nlorentzian lineshapes (red).  (f) Measured linewidth (blue), fitted theoretical contributions \nto linewidth (green, cyan, black) and empirical fit that describes the angle misalignment \n(magenta) using the OFMR with \n0H\n . (g) Effective damping using the OFMR and \nTRMOKE .    33 \n Figure 5 \n \nFIG. 5. TRMOKE at \n4H\n  and \n1H\n  and OFMR measurement at \n0H\n  for Co/Ni/Co \nsample . (a) Measured resonance frequency versus magnetic field. (b ) Measured linewidth \n(blue), fitted theoretical contributions to linewidth (green, cyan, magenta) and extracted \nintrinsic linewidth from measurement (red) using the TRMOKE with \n4H\n . (c) \nMeasured linewidth (blue), fitted theoretical c ontributions to linewidth (green, cyan, \nmagenta) using the TRMOKE with \n1H\n . (d) Measured linewidth (blue), fitted \ntheoretical contributions to linewidth (green, cyan, black) using the OFMR with \n0H\n . \n34 \n (e) Effecti ve (blue)  and intrinsic  (black ) damping using the TRMOKE at \n4H\n  and \neffective damping measured with the OFMR at \n0H\n  (red).  "    },    {        "title": "1706.08488v1.Perpendicular_magnetic_anisotropy_in_insulating_ferrimagnetic_gadolinium_iron_garnet_thin_films.pdf",        "content": "Perpendicular magnetic anisotropy in insulating ferrimagnetic gadolinium iron garnet\nthin \flms\nH. Maier-Flaig,1, 2S. Gepr ags,1Z. Qiu,3, 4E. Saitoh,3, 4, 5, 6, 7R. Gross,1, 2, 8\nM. Weiler,1, 2H. Huebl,1, 2, 8and S. T. B. Goennenwein1, 2, 8, 9, 10\n1Walther-Mei\u0019ner-Institut, Bayerische Akademie der Wissenschaften, Garching, Germany\n2Physik-Department, Technische Universit at M unchen, Garching, Germany\n3WPI Advanced Institute for Materials Research, Tohoku University, Sendai, Japan\n4Spin Quantum Recti\fcation Project, ERATO, Japan Science and Technology Agency, Sendai, Japan\n5Institute for Materials Research, Tohoku University, Sendai, Japan\n6PRESTO, Japan Science and Technology Agency, Saitama, Japan\n7Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Japan\n8Nanosystems Initiative Munich, M unchen, Germany\n9Institut f ur Festk oper- und Materialphysik, Technische Universit at Dresden, Dresden, Germany\n10Center for Transport and Devices of Emergent Materials, Technische Universit at Dresden, 01062 Dresden\n(Dated: June 27, 2017)\nWe present experimental control of the magnetic anisotropy in a gadolinium iron garnet (GdIG)\nthin \flm from in-plane to perpendicular anisotropy by simply changing the sample temperature.\nThe magnetic hysteresis loops obtained by SQUID magnetometry measurements unambiguously\nreveal a change of the magnetically easy axis from out-of-plane to in-plane depending on the sam-\nple temperature. Additionally, we con\frm these \fndings by the use of temperature dependent\nbroadband ferromagnetic resonance spectroscopy (FMR). In order to determine the e\u000bective mag-\nnetization, we utilize the intrinsic advantage of FMR spectroscopy which allows to determine the\nmagnetic anisotropy independent of the paramagnetic substrate, while magnetometry determines\nthe combined magnetic moment from \flm and substrate. This enables us to quantitatively evalu-\nate the anisotropy and the smooth transition from in-plane to perpendicular magnetic anisotropy.\nFurthermore, we derive the temperature dependent g-factor and the Gilbert damping of the GdIG\nthin \flm.\nControlling the magnetization direction of magnetic\nsystems without the need to switch an external static\nmagnetic \feld is a challenge that has seen tremendous\nprogress in the past years. It is of considerable interest\nfor applications as it is a key prerequisite to store infor-\nmation in magnetic media in a fast, reliable and energy\ne\u000ecient way. Two notable approaches to achieve this in\nthin magnetic \flms are switching the magnetization by\nshort laser pulses[1, 2] and switching the magnetization\nvia spin orbit torques[3{5]. For both methods, materials\nwith an easy magnetic anisotropy axis oriented perpen-\ndicular to the \flm plane are of particular interest. While\nall-optical switching requires a magnetization component\nperpendicular to the \flm plane in order to transfer angu-\nlar momentum[2], spin orbit torque switching with per-\npendicularly polarized materials allows fast and reliable\noperation at low current densities[3]. Therefore great ef-\nforts have been undertaken to achieve magnetic thin \flms\nwith perpendicular magnetic anisotropy.[6] However, re-\nsearch has mainly been focused on conducting ferromag-\nnets that are subject to eddy current losses and thus of-\nten feature large magnetization damping. Magnetic gar-\nnets are a class of highly tailorable magnetic insulators\nthat have been under investigation and in use in appli-\ncations for the past six decades.[7{9] The deposition of\ngarnet thin \flms using sputtering, pulsed laser deposition\nor liquid phase epitaxy, and their properties are very well\nunderstood. In particular, doping the parent compound\n(yttrium iron garnet, YIG) with rare earth elements is apowerful means to tune the static and dynamic magnetic\nproperties of these materials.[7, 10, 11]\nHere, we study the magnetic properties of a gadolin-\nium iron garnet thin \flm sample using broadband fer-\nromagnetic resonance (FMR) and SQUID magnetome-\ntry. By changing the temperature, we achieve a transi-\ntion from the typical in-plane magnetic anisotropy (IPA),\ndominated by the magnetic shape anisotropy, to a per-\npendicular magnetic anisotropy (PMA) at about 190 K.\nWe furthermore report the magnetodynamic properties\nof GdIG con\frming and extending previous results.[7]\nI. MATERIAL AND SAMPLE DETAILS\nWe investigate a 2 :6µm thick gadolinium iron gar-\nnet (Gd 3Fe5O3, GdIG) \flm grown by liquid phase epi-\ntaxy (LPE) on a (111)-oriented gadolinium gallium gar-\nnet substrate (GGG). The sample is identical to the\none used in Ref. 12 and is described there in detail.\nGdIG is a compensating ferrimagnet composed of two\ne\u000bective magnetic sublattices: The magnetic sublattice\nof the Gd ions and an e\u000bective sublattice of the two\nstrongly antiferromagnetically coupled Fe sublattices.\nThe magnetization of the coupled Fe sublattices shows a\nweak temperature dependence below room temperature\nand decreases from approximately 190 kA m\u00001at 5 K to\n140 kA m\u00001at 300 K.[8] The Gd sublattice magnetization\nfollows a Brillouin-like function and decreases drasticallyarXiv:1706.08488v1  [cond-mat.mtrl-sci]  26 Jun 20172\nfrom approximately 800 kA m\u00001at 5 K to 120 kA m\u00001at\n300 K.[8] As the Gd and the net Fe sublattice magneti-\nzations are aligned anti-parallel, the remanent magneti-\nzations cancel each other at the so-called compensation\ntemperature Tcomp = 285 K of the material.[13] Hence,\nthe remanent net magnetization Mof GdIG vanishes at\nTcomp.\nThe typical magnetic anisotropies in thin garnet \flms\nare the shape anisotropy and the cubic magnetocrys-\ntalline anisotropy, but also growth induced anisotropies\nand magnetoelastic e\u000bects due to epitaxial strain have\nbeen reported in literature.[14, 15] We \fnd that our ex-\nperimental data can be understood by taking into ac-\ncount only shape anisotropy and an additional anisotropy\n\feld perpendicular to the \flm plane. A full determina-\ntion of the anisotropy contributions is in principle pos-\nsible with FMR. Angle dependent FMR measurements\n(not shown) indicate an anisotropy of cubic symmetry\nwith the easy axis along the crystal [111] direction in\nagreement with literature.[16] The measurements sug-\ngest that the origin of the additional anisotropy \feld\nperpendicular to the \flm plane is the cubic magnetocrys-\ntalline anisotropy. However, the low signal amplitude and\nthe large FMR linewidth towards Tcomp in combination\nwith a small misalignment of the sample, render a com-\nplete, temperature dependent anisotropy analysis impos-\nsible. In the following, we therefore focus only on shape\nanisotropy and the additional out-of-plane anisotropy\n\feld.\nII. SQUID MAGNETOMETRY\nSQUID magnetometry measures the projection of the\nmagnetic moment of a sample on the applied magnetic\n\feld direction. For thin magnetic \flms, however, the\nbackground signal from the comparatively thick sub-\nstrate can be on the order of or even exceed the magnetic\nmomentmof the thin \flm and hereby impede the quan-\ntitative determination of m. Our 2:6µm thick GdIG \flm\nis grown on a 500 µm thick GGG substrate warranting a\ncareful subtraction of the paramagnetic background sig-\nnal of the substrate. In our experiments, H0is applied\nperpendicular to the \flm plane and thus, the projection\nof the net magnetization M=m=Vto the out-of-plane\naxis is recorded as M?. Fig. 1 shows M?of the GdIG \flm\nas function of the externally applied magnetic \feld H0.\nIn the investigated small region of H0, the magnetization\nof the paramagnetic substrate can be approximated by\na linear background that has been subtracted from the\ndata. The two magnetic hysteresis loops shown in Fig. 1\nare typical for low temperatures ( T.170 K) and for\ntemperatures close to Tcomp. The hysteresis loops unam-\nbiguously evidence hard and easy axis behavior, respec-\ntively. Towards low temperatures ( T= 170 K, Fig. 1 (a))\nthe net magnetization M=jMjincreases and hence, the\nanisotropy energy associated with the demagnetization\n\feldHshape =\u0000M?[17] dominates and forces the mag-\n−150 −100 −50 0 50 100 150\nµ0H0(mT)-40-2002040M⊥(kA/m)a\n170 K (1)(2)(3)\nMH0z\n−100 mT 10 mTz\n−150 −100 −50 0 50 100 150\nµ0H0(mT)-4-2024M⊥(kA/m)b\n250 K(2)(3)\n10 mTz − HC+ HC\n−100 mTH0M z\n(1)FIG. 1. Out-of-plane magnetization component M?mea-\nsured by SQUID magnetometry. For di\u000berent temperatures,\nmagnetically hard (170K, (a)) and easy (250K, (b)) axis loops\nare observed. The arrows on the data indicate the sweep di-\nrection ofH0. The insets schematically show the magnetiza-\ntion direction MandH0=H0zwith the \flm normal zat\nthe indicated values of H0.\nnetization to stay in-plane. At these low temperatures,\nthe anisotropy \feld perpendicular to the \flm plane, Hk,\ncaused by the additional anisotropy contribution has a\nconstant, comparatively small magnitude. We therefore\nobserve a hard axis loop in the out-of-plane direction:\nUpon increasing H0from\u0000150 mT to +150 mT, Mcon-\ntinuously rotates from the out-of-plane (oop) direction to\nthe in-plane (ip) direction and back to the oop direction\nagain. The same continuous rotation happens for the op-\nposite sweep direction of H0with very little hysteresis.\nFor temperatures close to Tcomp (T= 250 K, Fig. 1 (b)),\nHshape becomes negligible due to the decreasing Mwhile\nHkincreases as shown below. Hence, the out-of-plane di-\nrection becomes the magnetically easy axis and, in turn,\nan easy-axis hysteresis loop is observed: After applying\na large negative H0[(1) in Fig. 1 (a)] MandH0are\n\frst parallel. Sweeping to a positive H0,M\frst stays\nparallel to the \flm normal and thus M?remains con-\nstant [(2) in Fig. 1 (a)] until it suddenly \rips to being\naligned anti-parallel to the \flm normal at H0>+Hc\n[(3) in Fig. 1 (a)]. These loops clearly demonstrate that\nthe nature of the anisotropy changes from IPA to PMA\non varying temperature.3\n0.0 0.5 1.0\nµ0H0(T)0102030ωres/2π (GHz)110K 190K 240Ka\n0.70 0.75µ0H0(T)\n−40040∆S21×103\n110 K0.30.40.5\nµ0H0(T)−0.50.00.5\n∆S21×103\n240 K\n0 50 100 150 200 250 300\nT(K)0.00.5−µ 0Hi(T)b\nHani=−Meff\nHshape =−M⊥\nHk=M⊥−MeffIPA PMA\nRe ImReIm\nFIG. 2. Broadband FMR spectroscopy data reveiling a smooth transition from in-plane to perpendicular anisotropy. (a)\nFMR resonance frequency plotted against H0taken for three di\u000berent temperatures (symbols) and \ft to Eq. (3) (solid lines).\nFor an IPA, a positive e\u000bective magnetization Me\u000b(positivex-axis intercept) is extracted, whereas Me\u000bis negative for a PMA.\n(inset) Exemplary resonance spectra (symbols) at 14 :5 GHz recorded at 110 K and 240 K as well as the \fts to Eq. (1) used\nto determine !res(solid lines). A complex o\u000bset S0\n21has been subtracted for visual clarity, plotted is \u0001 S21=S21\u0000S0\n21.(b)\nAnisotropy \feld Hani=\u0000Me\u000bas a function of temperature (open squares). Prediction for shape anisotropy Hshape based on\nSQUID magnetometry data (solid line) from Ref. 18. The additional perpendicular anisotropy \feld Hk=M?\u0000Me\u000b(red dots)\nincreases to approximately 0 :18 T at 250 K where its value is essentially identical to Hanidue to the vanishing M?.\nIII. BROADBAND FERROMAGNETIC\nRESONANCE\nIn order to quantify the transition from in-plane to per-\npendicular anisotropy found in the SQUID magnetome-\ntry data, broadband FMR is performed as a function of\ntemperature with the external magnetic \feld H0applied\nalong the \flm normal.[19] For this, H0is swept while\nthe complex microwave transmission S21of a coplanar\nwaveguide loaded with the sample is recorded at vari-\nous \fxed frequencies between 10 GHz and 25 GHz. We\nperform \fts of S21to[20]\nS21(H0)j!=\u0000iZ\u001f(H0) +A+B\u0001H0 (1)\nwith the complex parameters AandBaccounting for a\nlinear \feld-dependent background signal of S21, the com-\nplex FMR amplitude Z, and the Polder susceptibility[21,\n22]\n\u001f(H0) =Me\u000b(H\u0000Me\u000b)\n(H\u0000Me\u000b)2\u0000H2\ne\u000b+i\u0001H\n2(H\u0000Me\u000b):(2)\nHere,\ris the gyromagnetic ratio, He\u000b=!=(\r\u00160), and\n!is the microwave frequency and the e\u000bective magneti-\nzationMe\u000b=Hres\u0000!res=(\r\u00160). From the \ft, the res-\nonance \feld Hresand the full width at half-maximum\n(FWHM) linewidth \u0001 His extracted. Exemplary data\nforS21(data points) and the \fts to Eq. (1) (solid lines)\nat two distinct temperatures are shown in the two in-\nsets of Fig. 2 (a). We obtain excellent agreement of the\n\fts and the data. The insets furthermore show that the\nsignal amplitude is signi\fcantly smaller for T= 240 K\nthan for 110 K. This is expected as the signal amplitude\nis proportional to the net magnetization Mof the sam-\nple which decreases considerably with increasing temper-\nature (cf. Fig. 2 (b)). At the same time, the linewidthdrastically increases as discussed in the following section.\nThese two aspects prevent a reliable analysis of the FMR\nsignal in the temperature region 250 K <T < 300 K (i.e.\naround the compensation temperature). Therefore we do\nnot report data in this temperature region. Nevertheless,\nFMR is ideally suited to investigate the magnetic prop-\nerties of the GdIG \flm selectively, i.e. independent of\nthe substrate, for temperatures below Tcomp.\nAs all measurements are performed in the high \feld\nlimit of FMR, the dispersions shown in Fig. 2 (a) are\nlinear and we can use the Kittel equation\n!res=\r\u00160(Hres\u0000Me\u000b) (3)\nto extract\randMe\u000b. It is customary to describe the\nmagnetic anisotropy using Me\u000bwhich can be related to\nan anisotropy \feld Hanialong + zasMe\u000b=\u0000Hani=\nM?\u0000Hkfor positive H0. Here,Haniis given by the\ndemagnetization \feld Hshape =\u0000M?(along\u0000z) and\nthe anisotropy \feld Hkof the additional perpendicular\nanisotropy (along + z). Evidently, Me\u000bcan be deter-\nmined by linearly extrapolating the data to !res= 0.\nThe FMR dispersion and the \ft to Eq. (3) (solid lines) are\nshown for three selected temperatures in Fig. 2 (a). At\n110 K (blue curve) Me\u000bis positive. Therefore, M >H k\nindicating that shape anisotropy dominates, and the \flm\nplane is a magnetically easy plane while the oop direction\nis a magnetically hard axis. At 240 K (red curve) Me\u000bis\nnegative and hence, the oop direction is a magnetically\neasy axis. Figure 2 (b) shows the extracted Me\u000b(T).\nAt 190 K, Me\u000bchanges sign. Above this tempera-\nture (marked in red), the oop axis is magnetically easy\n(PMA) and below this temperature (marked in blue),\nthe oop axis is magnetically hard (IPA). The knowledge\nofM?(T) obtained from SQUID measurements allows\nto separate the additional anisotropy \feld HkfromMe\u000b4\n1.61.82.02.2g\na\n10−310−210−1αb\n0 50 100 150 200 250\nT(K)100030005000∆ω0/2π(MHz)c\nFIG. 3. Key parameters characterizing the magnetiza-\ntion dynamics of GdIG as a function of T:(a)g-factor\ng=\r~=\u0016B,(b)Gilbert damping constant \u000band(c)inhomo-\ngeneous linewidth \u0001 !0=(2\u0019)\n(red dots in Fig. 2 (b)). Hk=M?\u0000Me\u000bincreases\nconsiderably for temperatures close to Tcomp while at\nthe same time the contribution of the shape anisotropy,\nHshape =\u0000M?trends to zero. For T'180 K,Hkex-\nceedsHshape which is indicated by the sign change of\nMe\u000b. Above this temperature, we thus observe PMA. We\nuse the magnetization Mdetermined using SQUID mag-\nnetometry from Ref. 18 normalized to the here recorded\nMe\u000bat 10 K in order to quantify Hk. The maximal value\n\u00160Hk= 0:18 T is obtained at 250 K which is the highest\nmeasured temperature due to the decreasing signal-to-\nnoise ratio towards Tcomp.\nWe can furthermore extract the g-factor and damp-\ning parameters from FMR. The evolution of the g-factor\ng=\r~\n\u0016Bwith temperature is shown in Fig. 3 (a). We ob-\nserve a substantial decrease of gtowardsTcomp. This is\nconsistent with reports in literature for bulk GIG and can\nbe explained considering that the g-factors of Gd and Fe\nions are slightly di\u000berent such that the angular momen-\ntum compensation temperature is larger than the mag-\nnetization compensation temperature.[23] The linewidth\n\u0001!=\r\u0001Hcan be separated into a inhomogeneous con-\ntribution \u0001 !0= \u0001!(H0= 0) and a damping contribu-\ntion varying linear with frequency with the slope \u000b:\n\u0001!= 2\u000b\u0001!res+ \u0001!0: (4)\nClose toTcomp = 285 K, the dominant contribution to\nthe linewidth is \u0001 !0which increases by more than an\norder of magnitude from 390 MHz at 10 K to 6350 MHz\nat 250 K [Fig. 3 (c)]. This temperature dependence of the\nlinewidth has been described theoretically by Clogston\net al.[24, 25] in terms of a dipole narrowing of the in-homogeneous broadening and was reported experimen-\ntally before[7, 16]. As opposed to these single frequency\nexperiments, our broadband experiments allow to sepa-\nrate inhomogeneous and intrinsic damping contributions\nto the linewidth. We \fnd that in addition to the in-\nhomogeneous broadening of the line, also the Gilbert-\nlike (linearly frequency dependent) contribution to the\nlinewidth changes signi\fcantly: Upon approaching Tcomp\n[Fig. 3 (b)], the Gilbert damping parameter \u000bincreases\nby an order of magnitude. Note, however, that due to\nthe large linewidth and the small magnetic moment of\nthe \flm, the determination of \u000bhas a relatively large\nuncertainty.[26] A more reliable determination of the\ntemperature evolution of \u000busing a single crystal GdIG\nsample that gives access to the intrinsic bulk damping\nparameters remains an important task.\nIV. CONCLUSIONS\nWe investigate the temperature evolution of the mag-\nnetic anisotropy of a GdIG thin \flm using SQUID mag-\nnetometry as well as broadband ferromagnetic resonance\nspectroscopy. At temperatures far away from the com-\npensation temperature Tcomp, the SQUID magnetome-\ntry reveals hard axis hysteresis loops in the out-of-plane\ndirection due to shape anisotropy dominating the mag-\nnetic con\fguration. In contrast, at temperatures close to\nthe compensation point, we observe easy axis hysteresis\nloops. Broadband ferromagnetic resonance spectroscopy\nreveals a sign change of the e\u000bective magnetization (the\nmagnetic anisotropy \feld) which is in line with the mag-\nnetometry measurements and allows a quantitative anal-\nysis of the anisotropy \felds. We explain the qualitative\nanisotropy modi\fcations as a function of temperature by\nthe fact that the magnetic shape anisotropy contribu-\ntion is reduced considerably close to Tcomp due to the re-\nduced net magnetization, while the additional perpendic-\nular anisotropy \feld increases considerably. We conclude\nthat by changing the temperature the nature of the mag-\nnetic anisotropy can be changed from an in-plane mag-\nnetic anisotropy to a perpendicular magnetic anisotropy.\nThis perpendicular anisotropy close to Tcomp in combi-\nnation with the small magnetization of the material may\nenable optical switching experiments in insulating fer-\nromagnetic garnet materials. Furthermore, we analyze\nthe temperature dependence of the FMR linewidth and\ntheg-factor of the GdIG thin \flm where we \fnd values\ncompatible with bulk GdIG[7, 25]. The linewidth can\nbe separated into a Gilbert-like and an inhomogeneous\ncontribution. We show that in addition to the previously\nreported increase of the inhomogeneous broadening, also\nthe Gilbert-like damping increases signi\fcantly when ap-\nproachingTcomp5\nV. ACKNOWLEDGMENTS\nWe gratefully acknowledge funding via the priority pro-\ngram Spin Caloric Transport (spinCAT), (Projects GO\n944/4 and GR 1132/18), the priority program SPP 1601(HU 1896/2-1) and the collaborative research center SFB\n631 of the Deutsche Forschungsgemeinschaft.\nVI. BIBLIOGRAPHY\n[1] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk,\nA. Tsukamoto, A. Itoh, and T. Rasing, Physical Review\nLetters 99, 1 (2007).\n[2] C.-H. Lambert, S. Mangin, B. S. D. C. S. Varaprasad,\nY. K. Takahashi, M. Hehn, M. Cinchetti, G. Malinowski,\nK. Hono, Y. Fainman, M. Aeschlimann, and E. E. Fuller-\nton, Science 345, 1337 (2014).\n[3] K. Garello, C. O. Avci, I. M. Miron, M. Baumgartner,\nA. Ghosh, S. Au\u000bret, O. Boulle, G. Gaudin, and P. Gam-\nbardella, Applied Physics Letters 105, 212402 (2014).\n[4] I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten,\nM. V. Costache, S. Au\u000bret, S. Bandiera, B. Rodmacq,\nA. Schuhl, and P. Gambardella, Nature 476, 189 (2011).\n[5] A. Brataas, A. D. Kent, and H. Ohno, Nature Materials\n11, 372 (2012).\n[6] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D.\nGan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura,\nand H. Ohno, Nature Materials 9, 721 (2010).\n[7] B. Calhoun, J. Overmeyer, and W. Smith, Physical Re-\nview107(1957).\n[8] G. F. Dionne, Journal of Applied Physics 42, 2142\n(1971).\n[9] J. D. Adam, L. E. Davis, G. F. Dionne, E. F. Schloemann,\nand S. N. Stitzer, IEEE Transactions on Microwave The-\nory and Techniques 50, 721 (2002).\n[10] K. P. Belov, L. A. Malevskaya, and V. I. Sokoldv, Soviet\nPhysics JETP 12, 1074 (1961).\n[11] P. R oschmann and W. Tolksdorf, Materials Research\nBulletin 18, 449 (1983).\n[12] H. Maier-Flaig, M. Harder, S. Klingler, Z. Qiu, E. Saitoh,\nM. Weiler, S. Gepr ags, R. Gross, S. T. B. Goennen-\nwein, and H. Huebl, Applied Physics Letters 110, 132401\n(2017).\n[13] G. F. Dionne, Magnetic Oxides (Springer US, Boston,\nMA, 2009).\n[14] S. A. Manuilov, S. I. Khartsev, and A. M. Grishin, Jour-\nnal of Applied Physics 106, 123917 (2009).\n[15] S. A. Manuilov and A. M. Grishin, Journal of Applied\nPhysics 108, 013902 (2010).\n[16] G. P. Rodrigue, H. Meyer, and R. V. Jones, Journal of\nApplied Physics 31, S376 (1960).\n[17] We use the demagnetization factors of a in\fnite thin \flm:\nNx;y;z= (0;0;1).\n[18] S. Gepr ags, A. Kehlberger, F. D. Coletta, Z. Qiu, E.-J.\nGuo, T. Schulz, C. Mix, S. Meyer, A. Kamra, M. Al-\nthammer, H. Huebl, G. Jakob, Y. Ohnuma, H. Adachi,\nJ. Barker, S. Maekawa, G. E. W. Bauer, E. Saitoh,\nR. Gross, S. T. B. Goennenwein, and M. Kl aui, Nature\nCommunications 7, 10452 (2016).\n[19] The alignment of the sample is con\frmed at low temper-\natures by performing rotations of the magnetic \feld di-\nrection at \fxed magnetic \feld magnitude while recording\nthe frequency of resonance !res. As the shape anisotropydominates at low temperatures, !resgoes through an\neasy-to-identify minimum when the sample is aligned\noop.\n[20] H. Maier-Flaig, S. T. B. Goennenwein, R. Ohshima,\nM. Shiraishi, R. Gross, H. Huebl, and M. Weiler, arXiv\npreprint arXiv:1705.05694 .\n[21] J. M. Shaw, H. T. Nembach, and T. J. Silva, Physical\nReview B 87, 054416 (2013).\n[22] H. T. Nembach, T. J. Silva, J. M. Shaw, M. L. Schneider,\nM. J. Carey, S. Maat, and J. R. Childress, Physical\nReview B 84, 054424 (2011).\n[23] R. K. Wangsness, American Journal of Physics 24, 60\n(1956).\n[24] A. M. Clogston, Journal of Applied Physics 29, 334\n(1958).\n[25] S. Geschwind and A. M. Clogston, Physical Review 108,\n49 (1957).\n[26] For the given signal-to-noise ratio and the large\nlinewidth, \u000band \u0001!0are correlated to a non-\nnegligible degree with a correlation coe\u000ecient of\nC(intercept;slope) = \u00000:967."    },    {        "title": "1410.0439v1.Investigation_of_the_temperature_dependence_of_ferromagnetic_resonance_and_spin_waves_in_Co2FeAl0_5Si0_5.pdf",        "content": "1\n \nInvestigation of the temp erature-dependence of fe rromagnetic resonance and \nspin waves in Co 2FeAl 0.5Si0.5 \n \nLi Ming Loong1, Jae Hyun Kwon1, Praveen Deorani1, Chris Nga Tung Yu2, Atsufumi \nHirohata3,a), and Hyunsoo Yang1,b) \n \n1Department of Electrical and Computer Engine ering, National University of Singapore, 117576 \nSingapore \n2Department of Physics, The Univers ity of York, York, YO10 5DD, UK \n3Department of Electronics, The Unive rsity of York, York, YO10 5DD, UK \n \n \n  Co\n2FeAl 0.5Si0.5 (CFAS) is a Heusler compound th at is of interest for sp intronics applications, due \nto its high spin polarization a nd relatively low Gilbert dampi ng constant. In this study, the \nbehavior of ferromagnetic resonance as a functi on of temperature was investigated in CFAS, \nyielding a decreasing trend of damping constant as  the temperature was increased from 13 to 300 \nK. Furthermore, we studied spin waves in CF AS using both frequency domain and time domain \ntechniques, obtaining group velocities and atte nuation lengths as high as 26 km/s and 23.3 m, \nrespectively, at room temperature. \n  \na) Electronic mail: atsufumi.hirohata@york.ac.uk \nb) Electronic mail: eleyang@nus.edu.sg 2\nHalf-metallic Heusler compounds with  low Gilbert damping constant ( ) are promising \ncandidates for spin transfer torque-based (STT) spintronic devices,1-3 spin-based logic systems,4 \nas well as spin wave-based data comm unication in microelectronic circuits.5 Hence, a deeper \nfundamental understanding of the magnetiza tion dynamics, such as the behavior of \nferromagnetic resonance (FMR) and spin waves in Heusler compounds, could enable better \nengineering and utilization of these compounds fo r the aforementioned applications. In previous \nwork, FMR has been investigated in several Heusler compounds, such as Co 2FeAl (CFA),6 \nCo2MnSi (CMS),7 and Co 2FeAl 0.5Si0.5 (CFAS).8 In addition, the variation of  with temperature \nhas been studied for other material s, such as Co, Fe, Ni, and CoFeB.9-11 However, the \ntemperature-dependence of  in Heusler compounds has not be en reported yet. Furthermore, \nwhile there have been some studies of spin  waves in Heusler compounds, such as CMS and \nCo2Mn 0.6Fe0.4Si (CMFS),7,12 these studies have focused on frequency domain measurements. \nThus, time domain measurements remain scarce, and mainly consist of time-resolved magneto-optic Kerr effect (TR-MOKE) experiments.\n13 In this work, we investigate the temperature-\ndependence of  in CFAS, a half-metallic Heusler compound.14,15 Moreover, we utilize both \nfrequency domain and pulsed inductive micr owave magnetometry (PIMM) time domain \nmeasurements to study the magnetiza tion dynamics in CFAS. We obtain of 0.0025 at room \ntemperature, which is 6 times lower than the va lue at 13 K. In addition, we evaluate the group \nvelocity ( vg) and the attenuation length ( ) in CFAS, leading to values as high as 26 km/s and \n23.3 m respectively, at room temperature. \n CFAS (30 nm thick) was grown by ultrah igh vacuum (UHV) molecular beam epitaxy \n(MBE) on single crystal MgO (001) substrates and capped with 5 nm  of Au. The base pressure \nwas 1.210-8 Pa and the pressure during deposition was typically 1.6 10-7 Pa. The substrates 3\nwere cleaned with acetone, IPA and deionised wate r in an ultrasonic bath before being loaded \ninto the chamber. After the film growth, the samples were in-situ  annealed at 600 °C for 1 hour. \nCFAS alloy and Au pellets were used as targ ets for electron-beam bombardment. Figure 1(a) \nshows the vibrating sample magnetometry (V SM) results, from which the saturation \nmagnetization ( Ms) was extracted. The measurement was also  repeated at different temperatures \nto extract the corresponding values of Ms for subsequent data fitting. The Ms value increases \nfrom 1100 emu/cc at 300 K, to 1160 emu/cc at 13 K.  From the VSM data, we verify a hard axis \nalong [100] and an easy axis along [110] , consistent with earlier reports.3,8 In addition, the -2 \nXRD data shown in Fig. 1(b) verified the presen ce of the characteristic (004) peak, indicating \nthat the CFAS film wa s at least B2-ordered.1,14 As shown in Fig. 1(c), the film was patterned into \nmesas, which were integrated with asymme tric coplanar waveguides (ACPW). The ACPWs \nwere electrically isolated from the mesa by 50 nm of Al 2O3, which was deposited by RF \nsputtering. Vector network analy zer (VNA) and PIMM techniques were used to excite and detect \nferromagnetic resonance (FMR) as well as spin waves in CFAS. The former technique allows \nfrequency domain measurements, while the la tter technique was us ed for time domain \nmeasurements. The experimental setup enable d the excitation of Damon-Eshbach-type (DE) \nmodes, as the external magnetic field was applied along the ACPWs, shown in Fig. 1(c).16  \n A VNA was connected to the AC PWs, and reflection as well as transmission signals were \nmeasured to study the FMR and spin wave pr opagation, respectively. Background subtraction \nwas performed to obtain the resonance peaks.  Figure 2(a) shows th e FMR frequency as a \nfunction of applied magnetic fiel d at different temperatures, with  the corresponding fits using the \nKittel formula,17 \n݂ൌఊ\nଶగඥሺܪܪሻሺܪܪ4ܯߨ ௦ሻ,      (1)4\nwhere f is the resonance frequency,  is the gyromagnetic ratio,  H is the applied magnetic field,  \nand Ha is the anisotropy field. The g factor, which was extr acted using the equation \t\tߛ ൌ\n2ߤ݃ߨ/݄ ,where B is the Bohr magneton and h is Planck’s constant, was found to be 2.03 0.02, \nwhile Ha generally decreased from 130 Oe at  13 K to 70 Oe at 300 K. The ( g – 2) value is lower \nthan those of Co and Ni, but comparable to th ose of other Heusler comp ounds, such as CMS and \nCo2MnAl (CMA).18 The deviation of the g factor from the free electron  value of 2 is correlated \nwith the spin-orbit interaction in a material, where a smaller deviation indicates weaker spin-\norbit interaction, and lower .18 The inset of Fig. 2(a) show s the resonance frequency at H = \n1040 Oe as a function of temperature, with a Bloch fitting, indicating a Curie temperature of \napproximately 1000 K. The Bloch fitting was perf ormed by substituting the following equation19 \ninto Eq. (1): \nܯ௦ൌܽ൫1െܽ ଵܶଷ/ଶെܽଶܶହ/ଶെܽଷܶ/ଶ൯,      ( 2 )  \nwhere T is temperature, and a0, a1, a2, and  a3 are positive coefficients. \n As shown in Fig. 2(b), the extracted FMR field linewidths were fitted with the linear \nequation20 ΔH\tൌ\tΔH0\t\t4αf/, where H is the field linewidth and H0is the extrinsic field \nlinewidth . This enabled the extraction of the intrinsic Gilbert damping ( ) from the fit line slopes. \nFigure 2(c) shows that  increases as the temperature decreases. The value of  at room \ntemperature was found to be 0.0025, which is comp arable with the previously reported room \ntemperature value for CFAS.8 The trend of  with temperature is consistent with previous first-\nprinciple calculations,9 and could be attributed to longe r electron scattering time at lower \ntemperatures, due to a reduction in phonon-elec tron scattering. Consequently, the angular \nmomentum transfer at low temperatures occu rs predominantly by direct damping through \nintraband transitions.11 Similar temperature-dependence of  has also been observed 5\nexperimentally. For example, the of Co 20Fe60B20 has been found to increase by a factor of 3 \nfrom 0.007 at 300 K, to 0.023 at 5 K.11 This is comparable to our results, where  increases by a \nfactor of almost 6 from 0.0025 at 300 K to 0.014 at 13 K. It shoul d be noted that spin pumping \ninto the Au cap layer could have contributed to  the measured resonance linewidth, thus causing \nthe extracted  to be higher than its actual value ( CFAS). Thus,  = CFAS + sp, where sp \ndenotes the spin pumping c ontribution to the damping.21 While an investigation of sp in the \nCFAS/Au system would exceed the scope of this work, sp values for a Fe/Au system21,22 have \nnonetheless been included in Fig. 2(c) to prov ide a gauge of the temperature dependence of sp, \nas well as a rough estimation of the magnitude of sp in the CFAS/Au system. Figure 2(d) shows \nan increase in H0 as temperature increases. This could be due to the effect of temperature on the \ninteraction between magnetic precession and sample inhomogeneities, or on magnon-magnon \nscattering, as these f actors contribute to H0.20,23 In both Fig. 2(c) and 2(d), room temperature \nvalues of  and H0 for sputter-deposited CFAS were in cluded, for comparison with the MBE \nsample. It can be seen that the  is higher for the sputter-deposite d sample, consistent with lower \nhalf-metallic character due to greater structural disorder.1,6 \n We have also measured the time domain PI MM data at 300 K as shown in Fig. 3(a), \nwhere SW15 and SW30 denote edge-to-edge signal line separations of 15 and 30 m, \nrespectively. The width of all the signal lines was fixed at 10 m. Using the temporal positions \nof the centers of the Gaussian wavepackets ( t15 and t30, respectively), the group velocity ( vg) was \ncalculated with the equation5,24 vg\tൌ \t1 5 \tm/ሺt30\t– \tt15ሻ. Fast Fourier transform (FFT) was \nperformed on the PIMM data, as shown in Fig. 3(b), verifying the presence of multiple modes, \nwhere each mode manifested as a dark-light-dark oscillation. The vg decreases from 26 km/s at \n50 Oe to 11 km/s at 370 Oe, as shown in Fig. 3( c). Moreover, from the VNA transmission data, 6\nwhich is another measure of spin wave propagation, attenuation length ( ) and were extracted \nas a function of magnetic fiel d at room temperature, using the method reported elsewhere.24,25 \nThe spin wave amplitude was extracted from  Lorentzian fittings of the VNA transmission \nresonance peaks, which were measured using wa veguides with different center-to-center signal \nline-signal line (S-S) spacings. Then,  was extracted using the equation24 A1expሺ x1/ሻ\t ൌ\t\nA2expሺ x2/ሻ, where A1 and A2 denote the measured spin wave amplitudes, while  x1 and x2 denote \nthe different S-S spacings for the corresponding waveguides. The  decreases from 23.3 m at \n460 Oe, to 12.1 m at 1430 Oe, as shown in Fig. 3(c). Using the following equation,25  was \ncalculated at different magnetic fields, as shown in Fig. 3(d) \n ߙൌఊሺଶగெೞሻమௗషమೖ\nଶగሺுାଶగெ ೞሻ       ( 3 )   \nwhere d is the film thickness and k is a spin wave vector, which can be estimated by 2 /(signal \nline width).5 The  values (0.0026 – 0.0031) are consistent with the room temperature value \n(0.0025) obtained from the FMR measurements.  \n As shown in Fig. 3(c),  and vg decreased as the applie d magnetic field increased, \nconsistent with previous experimental11 and theoretical5 results. This trend can be understood in \nterms of the following equation,5,26 \n ݒൌఊమఓబమெೞమௗ\n଼గ݁ିଶௗ,        ( 4 )  \nwhere μ0 is the permeability of free space. As the a pplied magnetic field increases, the resonance \nfrequency increases, thus vg decreases. In addition,  for a given value of , the magnetic \nprecession will decay within a ce rtain amount of time. Hence, the distance travelled by the \nprecessional disturbance within that amount of time depends on its propagation velocity, vg. \nConsequently, the higher the vg, the longer the distance travelled, and thus, the higher the .The 7\nobtained values of  and vg are comparable to those of other ferromagnetic materials for the \nDamon-Eshbach surface spin wave mode.5,11,12 For example,  of 18.95 m was extracted for \nCFA by micromagnetic simulations,5 while  and vg values as high as 23.9 m and 25 km/s, \nrespectively, were experimentally observed in CoFeB.11 Furthermore,  as high as 16.7 m was \nexperimentally observed in CMFS.12 \n In conclusion, we have found a decreasing trend of  with increasing temperature for \nMBE-grown Co 2FeAl 0.5Si0.5, in the temperature range of 13 – 300 K. The room temperature \nvalue of  was found to be 0.0025, which was approximately 6 times lower than that at 13 K. \nWe have also investigated vg and  in CFAS, obtaining values as high as 26 km/s and 23.3 m \nrespectively, at room temperature. \n \nThis work was supported by the Singa pore NRF CRP Award No. NRF-CRP 4-2008-06. \n  8\nReferences \n1 T. Graf, C. Felser, and S. S. P. Parkin,  Prog. Solid State Chem. 39, 1 (2011). \n2 S.  Ikeda, H. Sato, M. Yamanouchi, H. Ga n, K. Miura, K. Mizunuma, S. Kanai, S. \nFukami, F. Matsukura, N. Kasai, and H. Ohno,  SPIN 2, 1240003 (2012). \n3 H. Sukegawa, Z. C. Wen, K. Kondou, S. Kasai,  S. Mitani, and K. Inomata,  Appl. Phys. \nLett. 100, 042508 (2012). \n4 M. Jamali, J. H. Kwon, S. M. Seo, K. J. Lee, and H. Yang,  Scientific Reports 3, 3160 \n(2013). \n5 J. H. Kwon, S. S. Mukherjee, P. Deorani,  M. Hayashi, and H. Yang,  Appl. Phys. A: \nMater. Sci. Process. 111, 369 (2013). \n6 S. Mizukami, D. Watanabe, M. Oogane, Y. A ndo, Y. Miura, M. Shirai, and T. Miyazaki,  \nJ. Appl. Phys. 105, 07D306 (2009). \n7 H. Pandey, P. C. Joshi, R. P. Pant, R. Pras ad, S. Auluck, and R. C. Budhani,  J. Appl. \nPhys. 111, 023912 (2012). \n8 L. H. Bai, N. Tezuka, M. Kohda, and J. Nitta,  Jpn. J. Appl. Phys. 51, 083001 (2012). \n9 K. Gilmore, Y. U. Idzerda, and M. D. Stiles,  Phys. Rev. Lett. 99, 027204 (2007). \n10 B. Heinrich, D. J. Meredith, and J. F. Cochran,  J. Appl. Phys. 50, 7726 (1979). \n11 H. M. Yu, R. Huber, T. Schwarze, F. Bra ndl, T. Rapp, P. Berberich, G. Duerr, and D. \nGrundler,  Appl. Phys. Lett. 100, 262412 (2012). \n12 T. Sebastian, Y. Ohdaira, T. Kubota, P. Pi rro, T. Bracher, K. Vogt , A. A. Serga, H. \nNaganuma, M. Oogane, Y. Ando, and B. Hillebrands,  Appl. Phys. Lett. 100, 112402 \n(2012). 9\n13 Y. Liu, L. R. Shelford, V. V. Kruglyak, R. J. Hicken, Y. Sakuraba, M. Oogane, and Y. \nAndo,  Phys. Rev. B 81, 094402 (2010). \n14 B. Balke, G. H. Fecher, and C. Felser,  Appl. Phys. Lett. 90, 242503 (2007). \n15 R. Shan, H. Sukegawa, W. H.  Wang, M. Kodzuka, T. Furubayashi, T. Ohkubo, S. Mitani, \nK. Inomata, and K. Hono,  Phys. Rev. Lett. 102, 246601 (2009). \n16 R. W. Damon and J. R. Eshbach,  J. Appl. Phys. 31, S104 (1960). \n17 C. Kittel,  Physical Review 73, 155 (1948). \n18 B. Aktas and F. Mikailov, Advances in Nanoscale Magnetism: Proceedings of the \nInternational Conference on Nanoscale Magnetism ICNM-2 007, June 25 -29, Istanbul, \nTurkey . (Springer, 2008). p. 63. \n19 S. T. Lin and R. E. Ogilvie,  J. Appl. Phys. 34, 1372 (1963). \n20 P. Krivosik, N. Mo, S. Kalarickal , and C. E. Patton,  J. Appl. Phys. 101, 083901 (2007). \n21 E. Montoya, B. Kardasz, C. Burrowes, W. Hu ttema, E. Girt, and B. Heinrich,  J. Appl. \nPhys. 111, 07C512 (2012). \n22 M. Zwierzycki, Y. Tserkovnyak, P. J. Kelly, A. Brataas, and G. E. W. Bauer,  Phys. Rev. \nB 71, 064420 (2005). \n23 M. Farle,  Reports on Progress in Physics 61, 755 (1998). \n24 K. Sekiguchi, K. Yamada, S. M. Seo, K. J. Le e, D. Chiba, K. Kobayashi, and T. Ono,  \nAppl. Phys. Lett. 97, 022508 (2010). \n25 K. Sekiguchi, K. Yamada, S. M. Seo, K. J. Le e, D. Chiba, K. Kobayashi, and T. Ono,  \nPhys. Rev. Lett. 108, 017203 (2012). \n26 D. D. Stancil, Theory of Magnetostatic Waves . (Springer, Berlin, 1993). \n 10\nFigure captions  \n \nFIG. 1. (a) Normalized magnetic hysteresis data along the crystallographic hard [100] and easy \n[110] axes of MBE-grown CFAS. Ms is the saturation magnetization. (b) -2 XRD data of the \nMBE-grown CFAS sample. (c) Optical microscopy image of the CFAS me sa integrated with \nasymmetric coplanar waveguides (ACPW). The or ientation of the in-plane magnetic field ( H) is \nindicated.  \nFIG. 2. (a) FMR frequency at different magnetic fields. Inset: FMR frequency for a fixed field \n(1040 Oe) at different temperat ures. (b) Resonance linewidth as a function of frequency at \ndifferent temperatures (symbols), with correspo nding fit lines. (c) Gilbert damping parameter ( ) \nat different temperatures. The spin  pumping contribution to damping ( sp) for a Fe/Au system \nhas been included, where all sp values were obtained from literature, except those at 13 K and \nroom temperature, which were obtained by extra polating the literature valu es. (d) Extrinsic field \nlinewidth (H0) at different temperatures. \nFIG. 3. (a) PIMM data from two differ ent signal line-signal line spacings at H = 50 Oe for 300 K. \n(b) Fast Fourier transform (FFT) of room temp erature PIMM data. (c) Room temperature group \nvelocity ( vg, axis: left and bottom) and attenuation length ( , axis: top and right) at different \nmagnetic fields. (d) Room temperature Gilbert damping parameter ( ) at different magnetic \nfields.  \n  11\n \n\n \nFIG. 1\n\n\n12\n \nFIG. 2 800 1000 1200 140091011121314\n(d) (c)(b) (a)\n  \n 13 K\n 90 K\n 210 K\n 294 KResonance frequency (GHz)\nMagnetic field (Oe)10 11 12 13 1450100150200250300\n 90 K\n 210 K\n 294 K H (Oe)\nResonance frequency (GHz)\n0 100 200 3000.0000.0050.0100.0150.020\n  \nTemperature (K) MBE\n Sputtered\n sp Au-Fe\n0 100 200 300050100150200 H0 (Oe)\nTemperature (K) MBE\n Sputtered0 200 800 100012000412 fR (GHz)\nTemperature (K)13\n\nFIG. 3 \n\n"    },    {        "title": "2010.07694v1.Spin_injection_characteristics_of_Py_graphene_Pt_by_gigahertz_and_terahertz_magnetization_dynamics_driven_by_femtosecond_laser_pulse.pdf",        "content": " 1Spin injection characteristics  of Py/graphene/Pt by gigahertz a nd terahertz magnetization \ndynamics driven by femtosecond laser pulse  \nH. Idzuchi1-3*, S. Iihama4,5#, M. Shimura6, A. Kumatani1,2,6,7, S. Mizukami1,2,5, Y . P. Chen3,8,9,1,2,5 \n \n1 WPI Advanced Institute for Material s Research (AIMR), Tohoku University \nSendai 980-8577, Japan \n2 Center for Science and Innovation in Spintronics (CSIS), Tohoku University \nSendai 980-8577, Japan \n3 Purdue Quantum Science and Engineering Ins titute and Department of Physics and Astronomy, \nPurdue University, West Lafayette, Indiana 47907, USA \n4 Frontier Research Institute for Interdiscipli nary Sciences (FRIS), Tohoku University, Sendai \n980-8578, Japan \n5 Center for Spintronics Research Network (CSRN), Tohoku University, Sendai 980-8577, Japan \n6 Graduate School of Environmental Studies, Tohoku University, Sendai 980-8579, Japan \n7 WPI-International Center fo r Materials Nanoarchitectonics (MANA), National Institute for \nMaterial Science, Tsukuba 305-0044, Japan \n8 School of Electrical and Computer Engi neering and Birck N anotechnology Center \nPurdue University, West Lafayette, Indiana 47907, USA \n9Institute of Physics and Astronomy and Villum Center for Hybrid Quantum Materials and \nDevices, Aarhus University, 8000, Aarhus-C, Denmark \n \n           \n \n*idzuchi@tohoku.ac.jp \n#) H. Idzuchi and S. Iihama cont ributed equally to this work.  2Abstract \nSpin transport characteristics of graphene has been extensively  studied so far. The spin transport along c-\naxis is however reported by rather limited number of papers. We  have studied spin transport characteristics \nthrough graphene along c-axis with permalloy(Py)/graphene(Gr)/Pt by gigahertz (GHz) and  terahertz (THz) \nmagnetization dynamics driven by femtosecond laser pulses. The relatively simple sample structure does \nnot require electrodes on the sample. The graphene layer was pr epared by chemical vapor deposition and \ntransferred on Pt film. The quality of graphene layer was chara cterized by Raman microscopy. Time \nresolved magneto-optical Kerr effect is used to characterize gi gahertz magnetization dynamics. \nMagnetization precession is clearly observed both for Pt/Py and  Pt/Gr/Py. The Gilbert damping constant of \nPt/Py was 0.015, indicates spin pumping effect from Py to Pt. T he Gilbert damping constant of Pt/Gr/Py is \nfound to be 0.011, indicates spin injection is blocked by graph ene layer. We have also performed the \nmeasurement of THz emission for Pt/Py and Pt/Gr/Py. While the T Hz emission is clearly observed for Pt/Py, \na strong reduction of THz emission is observed for Pt/Gr/Py. Wi th these two different experiments, and \nhighly anisotropic resistivity of graphite, we conclude that th e vertical spin transport is strongly suppressed \nby the graphene layer.       3Recently, two-dimensional (2D) materials have attracted conside rable attention. Two-dimensional \nmaterials provide handful access on highly crystalline samples,  offering new spintronics research directions. \nSince spin currents can flow in nonmagnetic materials, so far s uch spin transport is widely studied in in-\nplane direction of nonmagnetic 2D materials [1,2,3]. In three-d imensional materials such as Pt, spin \ntransport in out-of-plane direction is often studied with spin Hall effect. The spin current is converted to \ncharge transport with certain geometry: the spin polarization, the flow direction of spin current and the \ndirection of detecti on voltage need to be a ll perpendicular to each other. This geometrical constraint makes \nit difficult to study out-of-plane spin transport through c-axis of 2D material while the spin transport in-\nplane has been relatively well studied. Here, we optically inve stigated spin transport characteristics in c-axis \nof graphene by using gigahertz ( GHz) and terahertz (THz) magnet ization dynamics excited by a \nfemtosecond pulse laser. This makes it more easily to satisfy s uch geometrical conditions as the injector and \ndetector are not required to be electrically connected.  \nFigure 1 represents our sample structure as well as a brief mea surement set up. Here, we employed \nspin pumping and THz emission, both induced by magnetization dy namics excited by a pulsed laser as \ndescribed below. Previously, vertical spin transport in multila yers of graphene have been studied by ferro \nmagnetic resonance. Patra et al fabricated the sample on a co-p laner wave guide and used broad band \nfrequency to characterize spin transport. They found the Gilber t damping is significantly enhanced for Py/Gr \ncompared to Py/Pt where Py stands for permalloy (Ni 80Fe20) [4]. Later Gannett et al studied series of the \nsamples with different thickness of Py to characterize transpor t properties, which shows no detectable \nenhancement for Py/Gr/Cu [5]. While the interface of graphene c an be complicated, in our experiment the \nsample structure is simple (just a multilayer film) and complim entary characteristics are obtained by two \nmethods, which should help reveal t he intrinsic interface spin transport properties.  \n In this study, spin transport was studied on Pt/graphene/Py an d Pt/Py . Pt, graphene, and Py were \nchosen for representative materials for spin Hall effect, 2D ma terial, and soft ferromagnet. Pt film was \nprepared by sputtering with the thickness of 3 nm on silicon su bstrate and glass substrate. The graphene \nfilm was transferred onto Pt in ambient condition. Graphene fil m was prepared on thin copper foil by a \nstandard chemical vapor deposition (CVD) method and transferred  onto the Pt film. Raman microscopy \nwas used to characterize the number of layers in graphene where  the laser wavelength is 532 nm. We have \nobserved clear peaks of D, G, and 2D bands from left to right a s shown in Fig.2a. Particularly from the 2D \npeak, we confirmed the crystallinity of the graphene film and t he film was not folded [6,7]. The Py film and \nMgO capping layer was sputtere d on graphene film with the base pressure of ~ 10-7 T o r r .  T h e  s t a t i c  \nmagnetization process of the film was examined by magneto optic al Kerr effect. For measuring GHz \nmagnetization dynamics induced by femtosecond laser pulse (Fig.  1a), time-resolved magneto-optical Kerr \neffect (TRMOKE) was employed [8]. The wavelength, pulse duratio n, and repetition rate for both the pump \nand probe laser pulses were 800 nm , 120 fs, and 1 kHz, respecti vely. The pump laser beam was irradiated \non the sample from the film normal and the incident angle of th e probe laser beam was ~ 5 degree measured \nfrom the film normal. Kerr rotation angle of the reflected prob e beam was detected via balanced photo- 4detector. The pump laser pulse was modulated by the mechanical chopper with the frequency of 360 Hz \nand then the pump-laser induced change in Kerr rotation angle w as detected by a Lock-in amplifier. A \nmagnetic field was applied with a  10 degree angle measured from  the film normal. The magnetization \nprecession can be excited by the reduction of demagnetizing fie ld due to laser heating. The damping of \nmagnetization precession reflects the transfer of spin into adj acent normal metal layer, referred as spin-\npumping effect [9, 10]. To study spin-transport induced by THz magnetization dynamics, THz time-domain \nspectroscopy was employed [11] (Fig. 1b), in which THz spin-cur rent can be generated by ultrafast \ndemagnetization of Py layer and its angular momentum can be tra nsferred to Pt layer [12,13]. Then, THz \nelectric field can be generated through spin-to-charge conversi on (inverse spin Hall effect) in Pt layer. \nWavelength, pulse duration, repetition rate for the laser pulse  were 800 nm, 120 fs, and 80 MHz, respectively. \nThe femtosecond laser was irradiated from substrate side and th en THz electric field emitted from the film \nside was measured. The THz electric field was detected by elect ro-optic sampling method using a ZnTe \n(110) crystal. All measurements  were conducted at room temperat ure. \n   The spin transport of graphene in vertical direction can be accessed by spin pumping with \nadditional layers. We compare Pt/P y bilayer with Pt/graphene/Py  trilayers to characterize spin transport \nproperties across graphene layer. Figure 2b shows typical TRMOK E signal for Pt/Gr/Py(10nm)/MgO on Si \nsubstrate under the external magnetic field of 10.7 kOe in a di rection tilted by 10 degrees from perpendicular \nto the substrate. We have clearly observed spin precession slow ly decaying over long period right after \ninitial ultrafast dynamics, for the samples of both with and wi thout graphene layer. Those oscillations are \nfitted to the following equations \n𝐴𝐵⋅e x p ሺെ𝑣𝑡 ሻ𝐶⋅e x p ሺെ𝑡 𝜏⁄ሻsinሺ2𝜋𝑓𝑡  𝜙 ሻ \nwhere, A, B, , C, f, , and 0 are signal offset, magnitude of exponential background signal due to recovery \nof magnetization, decay rate, oscillation amplitude, oscillatio n frequency, oscillation life-time, and initial \nphase, respectively. The TRMOKE signals are well fitted by the above equation shown as solid curve in \nFig. 2(b). The f and 1/ values evaluated by fitting with different applied magnetic fi elds are shown in Fig. \n3. f and 1/ can be calculated theoretically  using Landau-Lifshitz Gilbert (LLG) equation as [8,14], \n𝑓ୋ ൌఊ\nଶగඥ𝐻ଵ𝐻ଶ,    ( 1 )  \nଵ\nఛైైృൌଵଶ𝛼𝛾ሺ𝐻ଵ𝐻 ଶሻ,     ( 2 )  \n𝐻ଵൌ𝐻c o s ሺ𝜃െ𝜃 ுሻെ4 𝜋 𝑀 ୣcosଶ𝜃,  (3) \n𝐻ଶൌ𝐻c o s ሺ𝜃െ𝜃 ுሻെ4 𝜋 𝑀 ୣcos 2𝜃 ,  (4) \nwhere H, H (=10 degree in this study), , 4Meff, , and  are external magnetic f ield, field angle, \nmagnetization angle, effective de magnetizing field, gyromagneti c ratio and Gilbert damping constant \nrespectively.  is given by the relation,  =gB/ℏ. The  is determined by the energy minimum condition as,  5𝐻s i n ሺ 𝜃 ுെ𝜃 ሻ2 𝜋 𝑀 ୣsin 2𝜃 ൌ 0 ,  (5) \nThe measured f is well fitted by Eq. (1) with the parameters g = 2.09 (2.06) and 4 Meff = -9.8 (-8.8) kOe for \nPt / Gr / Py (Pt / Py) film. The 1/  calculated using Eq. (2) are sh own in Fig. 3(b) and 3(c). 1/  for Pt / Gr / \nPy / MgO sample (Fig. 3(c)) can be well explained by Eq. (2) wi th  = 0.011. However, 1/  for Pt / Py / \nMgO cannot be explained by Eq. (2), which is due to inhomogeneo us linewidth broadening. Therefore, 1/  \nenhancement due to inhomogeneous linewidth broadening is consid ered as follows, \nଵ\nఛ౪౪ൌଵ\nఛైైృଵଶቚௗሺଶగ ైైృ ሻ\nௗఏಹቚΔ𝜃ு ,    ( 6 )  \nwhere, the first term is identi cal to Eq. (2) and the second te rm is 1/ enhancement due to distribution of H \nwhich may be related to surface roughness of the film [14]. The  black solid and blue dashed curves in Fig. \n3(b) are the calculated results of the first and second terms o f Eq. (6). Hext dependence of 1/  for Pt / Py / \nMgO filmis well explained by the summation of two contributions  in Eq. (6) with the parameters,  = 0.015 \nand H = 0.05 rad [green broken curv e in Fig. 3(b)]. The enhancement of  is due to spin-pumping effect \nat Pt / Py interface associated with dissipation of angular mom entum. This indicates strong suppression of \nspin current with graphene, cons istent with Gannett et al [5]. Previously, it was shown that graphene has \nlong spin diffusion length by means of lateral spin transport w here spin current is flowing in-plane with \nlong spin lifetime probed by Hanle effect [1,15]. Transport alo ng c-direction can be rather different from \nthe one in ab plane. In early studies, graphite crystal shows rather anisotr opic charge transport properties \nand the resistivity of c-axis is reported to larger than the one for ab plane by a factor of 102 to 103 [16]. The \nresistive nature of the graphene along c axis may prevent spin transport.      \nIn the THz method, by irradiating femtosecond laser pulse on th is kind of multilayers, THz electric \nfield can be generated [12, 13]. Ultrafast spin current in nonm agnetic layer can be generated by the ultrafast \ndemagnetization in Py layer, and spin-charge conversion via inv erse spin-Hall effect in Pt layer create \nterahertz charge current and electric field. In our bilayer Pt/ Py, we observed clear THz emission and its \nsignal is inverted with reversed bias magnetic field, as shown in the top panel (a) of Fig. 4. Interestingly, on \ntwo Pt/graphene/Py samples, the THz signal was largely suppress ed [Fig. 4(b) and 4(c)]. This implies strong \nsuppression of spin injection from Py to Pt by graphene monolay er in the terahertz frequency region. The \ninterpretation is qualitatively c onsistent with spin pumping st udy (using ultrafast laser heating and GHz \nmagnetization dynamics). The strong reduction of THz signal is attributed to the strong suppression of spin-\ntransport by inserting graphene monolayer with high resistivity  along the c-axis. Strong reduction of THz \nemission was also reported for Co/ZnO/Pt junctions[17]. With th ese two different characterization methods, \nwe conclude graphene monolayer effectively blocks vertical spin  current. For the second sample of \nPt/graphene/Py (Fig.4c), a small peak appeared around 1 ps. We notice this may or may not be a delayed \nTHz emission, whose precise mechanism (e.g., how it may be rela ted to the graphene barrier, whether it \nmay also be generated by Py itself etc.) is not clear yet at th is stage and open for future study.  \nIn conclusion, we have investigat ed spin injection characterist ics of Py/graphene/Pt by means of  6gigahertz and terahertz magneti zation dynamics driven by a femt osecond laser pulse. Graphene layer was \ngrown by CVD method and the Raman characteristics on Pt showed the characteristics of single layer \ngraphene film. We have clearly observed GHz magnetization prece ssion induced by the laser pulse for the \nsamples of both with and without graphene (Py/Pt). Graphene is observed to give an apparent suppression \nof the damping enhancement due to spin-pumping effect at Py / P t interface, indicating reduction of angular \nmomentum dissipation by graphene monolayer. THz emission induce d by femto-second laser pulse was \nobserved for Py/Pt bilayer, while the THz emission was strongly  suppressed for Py/graphene/Pt, which clear \nindicates graphene blocks spin current in transport along c-axis. Both experiments on spin pumping and \nTHz method can be understood by th e strongly suppressed spin tr ansport across the graphene layer.    \n Data Availability Statements The data that support the findings of this study are available from the corresponding author upon reasonable \nrequest.  \nAcknowledgments We acknowledge the support from AIMR common equipment unit. Thi s work was supported in part by \nAdvanced Institute for Materials R esearch (AIMR) under World Pr emier International Research Center \nInitiative (WPI) of MEXT, Japan, and by AIMR fusion research pr ogram, by the Mazda Foundation, and \nby a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and \nTechnology (MEXT), JSPS KAKENHI (G rant Number 18H 03858, 18H0447 3, 20H04623, and 20K14399). \n  7References  \n[1] N. Tombros et al., Nature  448, 571 (2007). \n[2] Y. P. Liu et al.,  Appl. Phys. Lett.  102, 033105 (2013). \n[3] H. Idzuchi, A. Fert, Y. Otani, Phys. Rev. B  91, 241407 (2015). \n[4] A. K. Patra et al., Appl. Phys. Lett.  101 162407 (2012). \n[5] W. Gannett et al., J. Appl. Phys.  117 213907 (2015). \n[6] L. M. Malard et al., Phys. Rep.  473 51 (2009). \n[7] A. C. Ferrari, D. M. Basko, Nat. Nanotech.  8, 235 (2013). \n[8] S. Iihama et al.,  Phys. Rev. B  89, 174416 (2014). \n[9] Y. Tserkovnyak et al., Phys. Rev. Lett.  88, 117601 (2002). \n[10] S. Mizukami et al., Phys. Rev. B  66, 104413 (2002). \n[11] Y. Sasaki et al., Appl. Phys. Lett.  111, 102401 (2017). \n[12] T. Kampfrath et al., Nat. Nanotech.  8 256 (2013). \n[13] T. Seifert et al., Nat. Photon.  10 485 (2016). \n[14] S. Mizukami et al., Jpn. J. Appl. Phys.  40, 580 (2001). \n[15] D. Khokhriakov et al., Carbon  161, 892 (2020). \n[16] W. Primak and L. H. Fuchs, Phys. Rev.  95 22 (1954). \n[17] G. Li et al., J. Phys. D: Appl. Phys.  51, 134001 (2018). \n    8Figures \n \nFig. 1.  Concept and schematic image of experimental set up of this stu dy. Graphene, spin injector (Py) and detector \n(Pt) are depicted in black hexagon, purple box, and gray box, r espectively. (a) The set up for pump-probe \nmeasurement for magnetization dynamics. The probe beam is sligh tly (~ 5 deg) tilted from the film normal. The \nmagnetic field is applied with a 10 degree angle measured from the film normal. (b) The set up for THz emission. \nThe magnetic field is applied in plane.  \n \n  \nFig.1   9 \nFig. 2. (a) Raman spectroscopy of a graphene layer transferred on to a glass/Pt substrate (glass with Pt sputtered). \nThree main Raman peaks (D, G, and 2D) are clearly observed. Ins et shows the 2D peak, clearly different from bilayer \nor multilayers graphene [6].  (b) Magneto-optical Kerr effect (MOKE ) signal plotted as a function  of pump-probe \ndelay time for Pt/Gr/Py(10nm)/MgO under an external magnetic fi eld of 10.7 kOe. The field is applied with a \n10degree angle measured from the film normal. The measurement s et up is schematically shown in Fig.1a. \n \n \n  \n  \nFig.2   10 \n \nFig. 3. Characterization of GHz magnetization dynamics for multilayers with and without graphene. The \nmeasurement set up is schemati cally shown in Fig.1a. (a) preces sion frequency as a function of the magnetic field. \nThe field is applied with a 10 degree angle measured from the f ilm normal. Closed red squares and open green circles \nindicate data from the Pt/Gr/Py/MgO and Pt/Py/MgO respectively where the thickness of Py is 10 nm for both case. \nThe solid curves are obtained by Eq. (1) with the parameters in  the main text. Inverse lifetime as a function of the \nmagnetic field for (b) Pt / Py / MgO and (c) Pt / Gr / Py / MgO  films. The black solid curves shown in (b) and (c) \ncorrespond to the calculated valu e using LLG eq. (Eq. (2)). The  green dotted and blue broken curves shown in (b) \nare the left hand side and the second term in the right hand si de in Eq. (6), respectively, with the parameters in the \nmain text.      \nFig.3    11  \nFig. 4. Detection of THz electric field generated by femto-second laser  pulse on (a) Pt(3) / Py(2) /MgO , (b) Pt(3) / \nGr / Py(2) / MgO and (c) Pt(3) / Gr / Py(5) / MgO made on glass  substrates. The measurement set up is schematically \nshown in Fig.1b. The numbers in bracket indicate the thickness of the layers in the unit of nanometers. The magnetic \nfield was applied in in-plane direction. Blue solid and red ope n symbols are the signal obtained with opposite \nmagnetic field direction. \n \nFig.4   "    },    {        "title": "1706.00777v2.Power_Loss_for_a_Periodically_Driven_Ferromagnetic_Nanoparticle_in_a_Viscous_Fluid__the_Finite_Anisotropy_Aspects.pdf",        "content": "Power Loss for a Periodically Driven Ferromagnetic Nanoparticle in a Viscous Fluid:\nthe Finite Anisotropy Aspects\nT. V. Lyutyy\u0003, O. M. Hryshko, A. A. Kovner\nSumy State University, 2 Rimsky-Korsakov Street, UA-40007 Sumy, Ukraine\nAbstract\nThe coupled magnetic and mechanical motion of a ferromagnetic nanoparticle in a viscous \ruid is considered within the\ndynamical approach. The equation based on the total momentum conservation law is used for the description of the\nmechanical rotation, while the modi\fed Landau-Lifshitz-Gilbert equation is utilized for the description of the internal\nmagnetic dynamics. The exact expressions for the particles trajectories and the power loss are obtained in the linear\napproximation. The comparison with the results of other widespread approaches, such as the model of \fxed particle\nand the model of rigid dipole, is performed. It is established that in the small oscillations mode the damping precession\nof the nanopartile magnetic moment is the main channel of energy dissipation, but the motion of the nanoparticle easy\naxis can signi\fcantly in\ruence the value of the resulting power loss.\nKeywords: Ferro\ruid, \fnite anisotropy, spherical motion, damping precession, Landau-Lifshitz-Gilbert equation\n1. Introduction\nThe correct description of a ferromagnetic nanoparti-\ncle dynamics in a viscous carrier \ruid is a key to under-\nstanding the ferro\ruid dynamics for all possible applica-\ntions. Up to now, for ferro\ruids composed of small enough\nnanoparticles, the response to a time-periodic magnetic\n\feld was considered \frstly within the concept of com-\nplex magnetic susceptibility, which is well described in [1].\nHowever, when the nanoparticle magnetic energy is com-\nparable with the thermal one, the response of a nanopar-\nticle will be based mainly on the individual trajectories of\neach nanoparticle. For example, the regular viscous rota-\ntion is considered as the main energy dissipation channel\nfor large enough nanoparticles driven by an external al-\nternating \feld [2]. This gives reason to believe that the\nanalytical description of the single nanoparticle motion is\ndemanded.\nTwo components of the nanoparticle dynamics should\nbe considered simultaneously for the trajectory descrip-\ntion: 1) the mechanical rotation (or the so-called spher-\nical motion) of a nanoparticle with respect to a viscous\n\ruid, 2) the internal motion of the nanoparticle magne-\ntization in the framework of the crystal lattice. Since\nthe simultaneous description is faced with some di\u000ecul-\nties, two approximations are utilized instead: 1) the rigid\ndipole approach [3], when the nanoparticle magnetic mo-\nment is supposed to be locked in the nanoparticle crys-\ntal lattice, 2) the \fxed nanoparticle approach [4], when\n\u0003Corresponding author\nEmail address: lyutyy@oeph.sumdu.edu.ua (T. V. Lyutyy)a nanoparticle is assumed to be immobilized because of\nthe rigid bound with a media carrier. Despite the restric-\ntions, both approaches are widely used for the description\nof the response to an alternating \feld of a ferromagnetic\nparticle in a viscous \ruid, including the power loss calcula-\ntion problem, which is closely related to the magnetic \ruid\nhyperthermia for cancer therapy [5, 6]. Thus, the model\nof rigid dipole was applied successfully for the dynamical\nand stochastic approximations: the power loss was found\nfor a circularly-polarized [7, 8, 9] and a linearly-polarized\n[10, 9] \felds. The e\u000bective Langevin equation and the\nkey characteristics of the rotational dynamics were estab-\nlished in [11]. The power loss calculation within the \fxed\nnanoparticle model, where only a damping precession of\nthe magnetic moment is taken into account, was performed\nin [12, 13, 14]. Finally, this problem was investigated in\n[15, 16] for the nanoparticles ensemble.\nThe coupled dynamics of a nanoparticle cannot be de-\nscribed by a simple superposition of these two types of\nmotion because of the essential changes in the equations\nof motion. The coupled motion of the particle magnetic\nmoment and the whole particle was \frstly described in\n[17]. Despite this, the discussion about the basic equations\nof motion in the case of the coupled dynamics is contin-\nued till now [18, 19, 20, 21]. It is especially important in\nthe context of a ferro\ruid heating by an alternating \feld,\nwhen both these types of motion can produce heat. One of\nthe \frst successful attempts concerning the energy absorp-\ntion description was reported in [22]. There the power loss\nwas obtained in the dynamical approximation by lineariza-\ntion of the Lagrangian equation in some speci\fc cases.\nBut within this approach the equations of motion were\nPreprint submitted to Journal of Magnetism and Magnetic Materials October 1, 2018 arXiv:1706.00777v2  [cond-mat.other]  20 Jul 2017not used. The study of the forced coupled dynamics in\na circularly-polarized magnetic \feld using the simpli\fed\nequations of motion was presented in [23], but the energy\nabsorption problem was not considered. The power loss\nwas calculated in recent studies [18, 19]. Unfortunately,\nthe correct explicit form of the equations of motion was\nnot applied there that facilitates the discussion about the\nbasic model equations [20, 21]. And only recently the es-\nsential progress in the description of energy absorption by\na viscously coupled nanoparticle with a \fnite anisotropy\nwas achieved [24]. Here the microwave absorption spec-\ntra was investigated using the linear response approach.\nBut the viscous term as well as in [17] was not taken into\naccount that motivates further research.\nTherefore, we use the correct equations set, presented\nin [20] to investigate the nanoparticle response to an ex-\nternal alternating \feld. Absorption of the \feld energy,\nwhich further is transformed into heating, is in our main\nfocus. In particular, the in\ruence of the easy axis mo-\nbility on the resonance dependencies of the power loss on\nthe \feld frequency is examined. Then, we consider the\nresults obtained for the same conditions within the \fxed\nnanoparticle and the rigid dipole approximations. In this\nway we reveal the role of both the viscous rotation of a\nwhole particle and the damping precession of its magnetic\nmoment inside in the energy dissipation process. We make\na conclusion about the complex character of the coupled\ndynamics and indistinguishability of the contribution of\neach type of motion into the mutual heating in the dy-\nnamical approximation.\n2. Description of the Model\nLet us consider a uniform spherical single-domain fer-\nromagnetic nanoparticle of radius R, magnetization ( M,\njMj=M= const) and density \u001a. This particle performs\nthe spherical motion (or motion with the \fxed center of\nmass) with respect to a \ruid of viscosity \u0011. Then, we as-\nsume that the nanoparticle is driven by the external time-\nperiodic \feld of the following type:\nH(t) =exHcos(\nt) +ey\u001bHsin(\nt); (1)\nwhere ex,eyare the unit vectors of the Cartesian coor-\ndinates,His the \feld amplitude, \n is the \feld frequency,\ntis the time, and \u001bis the factor which determines the\npolarization type ( \u001b=\u00061 corresponds to the circularly\npolarized \feld, 0 <j\u001bj<1 corresponds to the elliptically\npolarized \feld, and \u001b= 0 corresponds to the linearly po-\nlarized \feld). Both the interaction with the viscous media\nand the damping precession of Minside the particle lead to\ndissipation of the nanoparticle energy and further heating\nof the surrounding environment. These losses are compen-\nsated by the energy absorption of the external \feld of type\n(1) and can be characterized by the dimensionless powerloss per period calculated as [14]\nq=\n2\u0019MH aZ2\u0019\n\n0dtH@M\n@t: (2)\n2.1. Equations of Motion\nThree approaches will be considered: 1) the model\nof viscously coupled nanoparticle with a \fnite anisotropy\n(FA-model), 2) the model of \fxed particle (FP-model), 3)\nthe model of rigid dipole (RD-model). Let us start with\nthe \frst one as more important and novel. As follows\nfrom [20], the coupled magnetic dynamics and mechanical\nmotion in the deterministic case obey a pair of coupled\nequations\n_n=!\u0002n;\nJ_!=\r\u00001V_M+VM\u0002H\u00006\u0011V!; (3)\nwhere nis the unit vector which determines the anisotropy\naxis direction, !is the angular velocity of the particle,\nJ(= 8\u0019\u001aR5=15) is the moment of inertia of the particle, \r\nis the gyromagnetic ratio, Vis the nanoparticle volume,\nand dots over symbols represent derivatives with respect\nto time.\nIn fact, the \frst equation in the set (3) is the condi-\ntion of spherical motion for a rigid body, and the second\none is the classical torque equation, where the \frst term\nconstitutes the main di\u000berence from the same equation for\nthe FP-model. This term is originated from the motion\nof magnetization inside the nanoparticle with respect to\nits crystal lattice. In turn, the magnetization dynamics is\ndescribed by the modi\fed Landau-Lifshitz-Gilbert (LLG)\nequation\n_M=\u0000\rM\u0002Heff+\u000bM\u00001\u0010\nM\u0002_M\u0000!\u0002M\u0011\n;(4)\nwhere\u000bis the damping parameter, Heffis the e\u000bective\nmagnetic \feld, which accounts the uniaxial anisotropy \feld\nof magnitude Ha\nHeff=H+HaM\u00001(Mn)n: (5)\nHere, the di\u000berence from the original LLG equation\nconsists in the term which is proportional to M\u0002!\u0002M\nthat excludes the component of Mrotating together with\nthe crystal lattice. As a rule, the inertia term in (3) can be\nneglected even for large enough nanoparticles ( R> 20 nm)\nin a wide frequency range. Therefore, for further analysis\nwe transform the equations of motion (3) and (4) into the\nconvenient form\n_ n=MH a[_ m\u0002n=\nr+ (m\u0002h)\u0002n]=6\u0011;\n_ m(1 +\f) =\u0000\nrm\u0002h1\neff+\u000bm\u0002_ m;(6)\n2where \n r=\rHais the ferromagnetic resonance fre-\nquency,\f=\u000bM= 6\r\u0011,\nh1\neff= (exhcos \nt+eyh\u001bsin \nt) (1 +\f) + (mn)n;(7)\nand, \fnally, m=M=M,h=H=H aare the dimension-\nless magnetic moment and \fled amplitude, respectively.\nThe FP-model is described by the well-known LLG\nequation\n_M=\u0000\rM\u0002Heff+\u000bM\u00001M\u0002_M (8)\nor in the dimensionless form\n_ m=\u0000\nrm\u0002heff+\u000bm\u0002_ m; (9)\nwhere heff=Heff=Ha.\nFinally, the RD-model is described by the set of equa-\ntions similar to the Eqs. (3), but without the term propor-\ntional to _M\n_n=!\u0002n;\nJ_!=VMn\u0002H\u00006\u0011V!: (10)\nWhen the inertia momentum is neglected, Eqs. (10) are\ntransformed into a simple form\n_n=\u0000\ncrn\u0002(n\u0002h); (11)\nwhere \n cr=MH a=6\u0011is the characteristic frequency\nof the uniform mechanical rotation.\n2.2. Validation of the Dynamical Approximation\nThe used systems of equations are valid if thermal \ruc-\ntuations do not signi\fcantly in\ruence the obtained trajec-\ntories. There are two principal issues in this regard needed\nto be considered. Firstly, the magnetic energy should be\nmuch larger than the thermal energy, or \u0000 \u001d1, where\n\u0000 =MHV= (kBT),Tis the thermodynamic temperature,\nkBis the Boltzmann constant. In this case, primarily small\ndeviations from the dynamical trajectories take place. Sec-\nondly, the requirement to the relaxation time \u001cNexists.\nHere, relaxation time is the time, during which the rare,\nbut large \ructuations can occur. When the period of an\nexternal \feld is much smaller than the relaxation time,\nor \n\u00001\u001c\u001cN, the probability of such \ructuation is neg-\nligible, and the dynamical approach remains valid. Fol-\nlowing Brown [4], the relaxation time \u001cNcan be found\nas\u001cN= (\u0000=\u0019)\u00001=2exp(\u0000)(2\u000b\rH )\u00001. Both these factors\ntogether impose the requirements to the nanoparticle size\nand values of the \feld frequency and amplitude. For exam-\nple, \u0000\u001911:9 for the real nanoparticles of maghemite [25]\nwith the following parameters: average radius R= 20 nm,\nHa= 910 Oe,M= 338 G, temperature T= 315 K and ex-\nternal \feld amplitude H= 0:05Ha. Then, the frequency\nshould to be larger than \u001c\u00001\nN, which for the parameters\nstated above and \u000b= 0:05 is equal to \n N\u00191:11\u0001103Hz.These conditions are su\u000ecient for the FP-model. But\nwhen we consider the mechanical rotation in addition to\nthe magnetic dynamics within the FA-model, one needs\nto take into account the conditions of stable spherical mo-\ntion. The signi\fcant changes in the angular coordinates\ncan occur due to thermal excitation, when the observa-\ntion time is much more than the Brownian relaxation time\n\u001cB= 3\u0011V=(kBT) [26]. It imposes the existence of an-\nother characteristic frequency \n B=\u001c\u00001\nB=kBT=(3\u0011V).\nFor the above-mentioned maghemite nanoparticles of ra-\ndiusR= 20 nm and water at temperature of T= 315 K\nand viscosity of \u0011= 0:006 P this frequency is equal to\n\nB\u00192:26\u0001105Hz. One more requirement to the fre-\nquency arises from the condition which represents the va-\nlidity of the Stokes approximation for the friction momen-\ntum [27]: Re = \u001al\nSR2=\u0011\u001810. Here Re is the so-called\nReynolds number, \u001alis the liquid density, \n Sis the corre-\nsponding characteristic frequency, which de\fnes the upper\nlimit of the \feld frequency. Straightforward calculations\ngive \n S\u00181012Hz in our case. Summarizing, one can\nobtain that max[\n B;\nN]\u001c\n\u001c\nS. Therefore, the fre-\nquency interval, where the dynamical approach is valid for\nthe calculation, is \n = (105\u00001012) Hz which includes the\nfrequencies acceptable for the magnetic \ruid hyperthermia\nmethod.\nFinally, the conditions of using the RD-model include\nall the stated above for the FA-model and contain ad-\nditionally the requirement to the \feld amplitudes, which\nshould be much smaller than the e\u000bective anisotropy \feld\n(H\u001cHa). The last inequality satis\fes the above cal-\nculations and corresponds to the limitations of the linear\napproximation utilized for the processing of the equations\nof motion.\nThe importance of the dynamical approximation is not\nrestricted by its validity in a certain interval of the sys-\ntem parameters. The dynamical approximation reveals\nthe main microscopic mechanisms of the ferro\ruid sensi-\ntivity to external \felds. In this way we can estimate the\nupper limits of such important performance criteria as the\nmagnetic susceptibility or the power loss. It is very impor-\ntant in a light of \fctionalization of ferro\ruids and creation\nof the properties demanded in the applications.\n3. Results\nThe solution of the set of equations (6), (11) and (9)\ncan be found in the linear approximation for the small\noscillations mode. In this mode, vectors mandnare\nrotated in a small vicinity around the initial position of\nthe easy axis which, in turn, is de\fned by the angles \u00120\nand'0(see Fig. 1). This takes place for small enough\n\feld amplitudes ( h\u001c1). The linearization procedure\nused here is similar to the reported in [14] and consists in\nthe following. Let us introduce a new coordinate system\nx0y0z0in the way depicted in Fig. 1. Actually, it is rotated\nwith respect to the laboratory system xyzby the angles\n3Figure 1: (Color online) Schematic representation of the model and\nthe coordinate systems.\n\u00120and'0. In this new coordinate system, vectors mand\nncan be represented in the linear approximation as\nm=ex0mx0+ey0my0+ez0; (12)\nn=ex0nx0+ey0ny0+ez0; (13)\nwhere ex0;ey0;ez0are the unit vectors of the coordinate\nsystemx0y0z0. In this system, the external \feld (1) can be\nwritten using the rotation matrix as\nh0=C\u00010\n@hcos \nt\n\u001bhsin \nt\n01\nA; (14)\nC=0\n@cos\u00120cos'0cos\u00120sin'0\u0000sin\u00120\n\u0000sin'0 cos'0 0\nsin\u00120cos'0sin\u00120sin'0 cos\u001201\nA;(15)\nh0=0\n@hcos\u00120cos'0cos \nt+\u001bhcos\u00120sin'0sin \nt\n\u0000hsin'0cos \nt+\u001bhcos'0sin \nt\nhsin\u00120cos'0cos \nt+\u001bhsin\u00120sin'0sin \nt1\nA:\n(16)\nAll the above allows to analyze the features of the\nnanoparticle response to the external \feld (1) for three\napproximations in an uniform manner. The analytical so-\nlutions obtained below describe the principal di\u000berence be-\ntween the coupled and separated motion of the magnetic\nmoment and the whole particle that constitutes our main\nresults.\n3.1. Coupled Oscillations of the Easy Axis and the Mag-\nnetic Moment\nWe start from the most complicated, however, the most\ninteresting case: the case when both the mechanical rota-\ntion and internal magnetic dynamics occur simultaneously,\nor the FA-model. Using (16), assuming nx0;ny0;mx0;my0\u0018\nh, and neglecting all the nonlinear terms with respect to h,we, \fnally, derive from (6) the linearized system of equa-\ntions for mandnin the following form:\n_nx0=MH a( _my0=\nr+hx0)=6\u0011;\n_ny0=\u0000MH a( _mx0=\nr\u0000hy0)=6\u0011;\n(1 +\f) _mx0=\u0000\nr(my0\u0000hy0\u0000ny0)\u0000\u000b_my0;\n(1 +\f) _my0= \nr(mx0\u0000hx0\u0000nx0)\u0000\u000b_mx0:(17)\nSolution of this set of linear equations can be written\nin the standard form\nnx0=ancos \nt+bnsin \nt;\nny0=cncos \nt+dnsin \nt;\nmx0=amcos \nt+bmsin \nt;\nmy0=cmcos \nt+dmsin \nt;(18)\nwherean,bn,cn,dn,am,bm,cm, anddmare the con-\nstant coe\u000ecients which should be de\fned. Substituting\n(18) into (17) and using the linear independence of the\ntrigonometric functions, we obtain the system of linear al-\ngebraic equations for the coe\u000ecients corresponding to m\n(1 +\f)~\nam=dm+\u000ebm\u0000\u000b~\ncm\u0000Am;\n(1 +\f)~\nbm=\u0000cm\u0000\u000eam\u0000\u000b~\ndm+Bm;\n(1 +\f)~\ncm=\u0000bm+\u000edm+\u000b~\nam\u0000Cm;\n(1 +\f)~\ndm=am\u0000\u000ecm+\u000b~\nbm+Dm(19)\nand the explicit expressions for the coe\u000ecients corre-\nsponding to n\nan=\u000ecm\u0000\u001b~\n\u00001hcos\u00120sin'0;\nbn=\u000edm+~\n\u00001hcos\u00120cos'0;\ncn=\u0000\u000eam\u0000\u001b~\n\u00001hcos'0;\ndn=\u0000\u000ebm\u0000~\n\u00001hsin'0:(20)\nHere ~\n = \n=\nr,\u000e=\f=\u000b and\nAm=\u001bh(1 +\f) cos'0\u0000~\n\u00001hsin'0;\nBm=\u0000h(1 +\f) sin'0\u0000\u001b~\n\u00001hcos'0;\nCm=\u0000\u001bh(1 +\f) cos\u00120sin'0\u0000~\n\u00001hcos\u00120cos'0;\nDm=\u0000h(1 +\f) cos\u00120cos'0+\u001b~\n\u00001hcos\u00120sin'0:\nFrom (19) one straightforwardly obtains the unknown\nconstantsam,bm,cm, anddmas follows\nam=Z\u00001h\n~\n1Dm+~\n2Bm+~\n3Cm+~\n4Ami\n;\nbm=Z\u00001h\n~\n1Cm+~\n2Am\u0000~\n3Dm\u0000~\n4Bmi\n;\ncm=Z\u00001h\n\u0000~\n1Bm+~\n2Dm\u0000~\n3Am+~\n4Cmi\n;\ndm=Z\u00001h\n\u0000~\n1Am+~\n2Cm+~\n3Bm\u0000~\n4Dmi\n;\n(21)\n4where\nZ=~\n4\u000b4+ 2~\n4\u000b2\f2+~\n4\f4+ 4~\n4\u000b2\f+\n+ 4~\n4\f3+ 2~\n4\u000b2+ 6~\n4\f2\u00002~\n2\u000b2\u000e2+\n+ 2~\n2\f2\u000e2+ 4~\n4\f+ 8~\n2\u000b\f\u000e+\n+ 4~\n2\f\u000e2+~\n4+ 2~\n2\u000b2+ 8~\n2\u000b\u000e\u0000\n\u00002~\n2\f2+ 2~\n2\u000e2+\u000e4\u00004~\n2\f\u00002~\n2+\n+ 2\u000e2+ 1;(22)\n~\n1=\u0000~\n2\u000b2\u00002~\n2\u000b\f\u000e\u00002~\n2\u000b\u000e+\n+~\n2\f2+ 2~\n2\f+~\n2\u0000\u000e2\u00001;\n~\n2=\u0000~\n2\u000b2\u000e+ 2~\n2\u000b\f+ 2~\n2\u000b+\n+~\n2\f2\u000e+ 2~\n2\f\u000e+~\n2\u000e+\u000e3+\u000e;\n~\n3=~\n3\u000b3+~\n3\u000b\f2+ 2~\n3\u000b\f+\n+~\n3\u000b\u0000~\n\u000b\u000e2+~\n\u000b+ 2~\n\f\u000e+\n+ 2~\n\u000e;\n~\n4=\u0000~\n3\u000b2\f\u0000~\n3\u000b2\u0000~\n3\f3\u0000\n\u00003~\n3\f2\u00003~\n3\f\u0000~\n3\u00002~\n\u000b\u000e\u0000\n\u0000~\n\f\u000e2+~\n\f\u0000~\n\u000e2+~\n:\nUsing (21), one can easily derive the set of constants\n(20), which de\fne the dynamics of the whole particle.\nThe obtained expressions for the nanoparticle trajec-\ntories let us to write the analytical relation for the power\nlossq. Direct integration of (2) with substitution of (18),\n(21) and (20) yields the following formula:\nq= 0:5~\n\nr(bmhcos\u00120cos'0\u0000am\u001ahcos\u00120sin'0\u0000\n\u0000dmhsin'0\u0000cm\u001ahcos'0+bman\u0000ambn+dmcn\u0000\n\u0000cmdn):\n(23)\nThe dependence of qon the system parameters, espe-\ncially on the external \feld frequency, is of great interest\nand will be analyzed below. But, the comparison of this\nresult with similar one obtained within other approxima-\ntions, such as the FP-model and the RD-model, is no less\ninteresting.\n3.2. Oscillations of the Magnetic Moment in a Fixed Nanopar-\nticle\nAs the next stage, let us consider the magnetic dynam-\nics only within the FP-model. As in the previous case,\nthe linearized equations are written under the assumption\nmx0;my0\u0018h, and all the nonlinear terms with respect to\nhare dropped. Using (16), we, \fnally, obtain from (9) the\nlinearized system of equations for mas follows\n_mx0=\u0000\nr( _my0\u0000hy0)\u0000\u000b_my0;\n_my0= \nr( _mx0\u0000hx0)\u0000\u000b_mx0:(24)\nThen, the general form of the solution of (24) can be\neasily written in the standard form\nmx0=afpcos \nt+bfpsin \nt;\nmy0=cfpcos \nt+dfpsin \nt;(25)whereafp,bfp,cfp, anddfpare the oscillation am-\nplitudes for magnetic the moment inside the immobilized\nnanoparticle. Substitution of (25) into (24) yields the sys-\ntem of linear algebraic equations for the desired amplitudes\n~\nafp= (dfp\u0000\u001bhcos'0)\u0000\u000b~\ncfp;\n~\nbfp=\u0000(cfp+hsin'0)\u0000\u000b~\ndfp;\n~\ncfp=\u0000(bfp\u0000\u001bhcos\u00120sin'0) +\u000b~\nafp;\n~\ndfp= (afp\u0000hcos\u00120cos'0) +\u000b~\nbfp:(26)\nAfter the calculations, we \fnd the solution of (26)\nafp=\u0000Z\u00001\nfph\n~\nfp\n1Bfp+~\nfp\n2Afpi\n;\nbfp=Z\u00001\nfph\n~\nfp\n1Cfp+~\nfp\n2Dfpi\n;\nafp=Z\u00001\nfph\n~\nfp\n1Afp\u0000~\nfp\n2Bfpi\n;\ndfp=Z\u00001\nfph\n\u0000~\nfp\n1Dfp+~\nfp\n2Cfpi\n;(27)\nwhere\nZfp= 4\u000b2~\n4+\u0010\n(\u000b2\u00001)~\n\u00002+ 1\u00112\n; (28)\n\nfp\n1= 2\u000b~\n2;\n\nfp\n2= (\u000b2\u00001)~\n2+ 1;\nAfp=\u001bh~\n (\u000bcos\u00120sin'0\u0000cos'0)\u0000hcos\u00120cos'0;\nBfp=\u001bh~\n (cos'0sin'0+\u000bcos'0) +hsin'0;\nCfp=h~\n (cos\u00120cos'0\u0000\u000bsin'0) +\u001bhcos'0;\nDfp=h~\n (\u000bcos\u00120cos'0+ sin'0) +\u001bhcos\u00120sin'0:\nThe power loss in this case can be also found by direct\nintegration of (2) with substitution of (25) and (27)\nq= 0:5h~\n\nrZ\u00001\nfpf~\nfp\n1[2\u001bhcos\u00120+\n+h~\nD] +~\nfp\n2\u000bh~\nDg;(29)\nwhere\nD= cos2\u00120(cos2'0+\u001b2sin2'0) +\u001b2cos2'0+ sin2'0:\n(30)\nThe obtained expression (29) is similar to the reported\nin [14], but accounts an arbitrary orientation of the nanopar-\nticle easy axis. Despite the quantitative di\u000berence caused\nby the turn of the easy axis, the qualitative character of\nthe frequency behavior of qremains.\n3.3. Oscillations of a Nanoparticle with the Locked Mag-\nnetic Moment\nAnd \fnally, we consider within the same framework\nthe widely used approach when the nanoparticle magnetic\nmoment is rigidly bound with the nanoparticle crystal lat-\ntice. In this so-called RD-model, the linearized equations\nhave the simplest form. Expanding the vector equation\n5(11) and taking into account (16), we write the linearized\nsystem of equations for nas follows\n_nx0= \ncrhx0;\n_ny0= \ncrhy0:(31)\nAs in the previous case, we use the trigonometric rep-\nresentation of the solution of (31)\nnx0=ardcos \nt+brdsin \nt;\nny0=crdcos \nt+drdsin \nt:(32)\nAfter direct substitution of (32) into (31), one can eas-\nily obtain the unknown constants, which are the ampli-\ntudes of the vector n\nard=h\ncrsin'0=\n;\nbrd=h\ncrcos\u00120cos'0=\n;\ncrd=\u0000h\ncrcos'0=\n;\ndrd=h\ncrcos\u00120sin'0=\n:(33)\nAnd at last, we can directly \fnd the power loss from\n(2) substituting (32) and (33)\nq= 0:5\ncrh2D: (34)\nIt is remarkable that qdoes not depend on the fre-\nquency, because while \n increases, the coe\u000ecients (33) de-\ncrease proportionally that compensates the possible growth\nof the power loss.\n4. Discussion and Conclusions\nWe have considered the response of a uniaxial ferro-\nmagnetic nanoparticle placed into a viscous \ruid to an al-\nternating \feld in the linear approximation for three mod-\nels, namely, the FA-model (viscously coupled nanoparticle\nwith a \fnite anisotropy), the FP-model (\fxed particle),\nand the RD-model (rigid dipole). As a result, we have\nobtained the expressions for the nanoparticle trajectories\nand for the power loss produced by both the rotation of\na nanoparticle in a viscous media and the internal damp-\ning precession of the nanoparticles magnetic moment. Our\nmain aims were the understanding of 1) the power loss be-\nhavior depending on di\u000berent parameters; 2) the role of\ndissipation mechanisms when they both are present; and\n3) the correlations between the mechanical rotation of a\nnanoparticle and the internal motion of its magnetic mo-\nment. The analysis of three approximations simultane-\nously helps us comprehend the restrictions and applicabil-\nity limits that, in turn, allows to systematize the results\nobtained by other authors. The relevance of our \fndings is\nclosely bounded with the application issues, such as heat-\ning rate during magnetic \ruid hyperthermia or absorption\nfrequency range of the microwave absorbing materials.\nThe comparison of the expressions for the power loss\nderived in the previous section allows a number of con-\nclusions, and some of them are rather unexpected at \frstsight. Firstly, the role of the internal magnetic motion is\nprimary. As follows from (23) and (29), the dependencies\nof the dimensionless power loss on the reduced frequency\nq(~\n) for the model of \fxed particle and for the model of\nviscously coupled nanoparticle with a \fnite anisotropy are\nsimilar: they both demonstrate a resonant behavior. At\nthe same time, for the model of rigid dipole it remains\nconstant (see (34)). Therefore, the dynamics of the mag-\nnetic moment represented by unit vector mdetermines\nthe resulting power loss in a wide range of realistic pa-\nrameters. But the quantitative comparison of these de-\npendencies lets us assume that the easy axis oscillations\ncan considerably modify the power loss induced by damp-\ning precession. The reasons for that are in the character\nof the collective motion of easy axis, which is represented\nby vector n, and magnetic moment m. Although the har-\nmonic motion takes place, the ratio of their phases and\namplitudes can lead to quite di\u000berent values of the en-\nergy dissipation in the system. Further we consider the\nbehavior of q(~\n) in context of the features of the mandn\nmotion.\nThe behavior of q(~\n) is caused by the features of co-\ne\u000ecientsam(~\n),bm(~\n),cm(~\n), anddm(~\n), which de-\ntermine the mdynamics, and an(~\n),bn(~\n),cn(~\n), and\ndn(~\n) de\fning the dynamics of n(see Fig. 2). As seen,\nfor frequencies far from the resonance one, vectors nand\nmalmost coincide and are rotated synchronously. Here,\nthe model of viscously coupled nanoparticle with a \fnite\nanisotropy and the model of \fxed particle yield very close\nvalues of the power loss. But near the resonance, in the\nvicinity of ~\n = 1, coe\u000ecients am(~\n),bm(~\n),cm(~\n), and\ndm(~\n) have the pronounced maxima and change the signs,\nwhile coe\u000ecients an(~\n),bn(~\n),cn(~\n), anddn(~\n) remain\nthe same. Therefore, vectors nandmare rotated in an\nasynchronous way now that leads to a larger angle between\nthe magnetic moment and the resulting or e\u000bective \feld\nheff. Together with increasing precession angle of m, this\ncauses the growth of the power loss compared with the\ncase of \fxed particle (see Fig. 3).\nFor small viscosity, vector nbecomes more suscepti-\nble to the external \feld, and the rotating magnetic mo-\nment can easily involve a whole nanoparticle into rotation.\nBut this does not induce a more intense motion in result.\nFirstly, a considerable decrease in the coe\u000ecients am(~\n),\nbm(~\n),cm(~\n), anddm(~\n) near the resonance takes place\nin comparison with the case of larger viscosity. Then, only\nbm(~\n) anddm(~\n) change the signs now (see Fig. 4). Fi-\nnally, the dependencies an(~\n),bn(~\n),cn(~\n), anddn(~\n)\nget the local maxima (Fig. 4) and slightly decrease in ab-\nsolute values in the vicinity of ~\n = 1. Therefore, the ef-\nfect of the pronounced asynchronous rotation of nandm,\nwhich is actual for the foregoing case, eliminates now, and\nthey become almost parallel for a whole range of frequen-\ncies. Since the angle between the magnetic moment and\nthe resulting \feld is reduced, the model of viscously cou-\npled nanoparticle with a \fnite anisotropy predicts lower\nvalues of the power loss than the model of \fxed particle\n6Figure 2: (Color online) The dependencies of the amplitudes of\ncoupled oscillations of the magnetic moment (21) and the easy axis\n(20) on the \feld frequency. The parameters used are M= 338 G,\nHa= 910 Oe,\u0011= 0:006 P,\u000b= 0:05 that corresponds to maghemite\nnanoparticles ( \r\u0000Fe2O3) in water at the temperature of 42\u000eC,\n\u001b=\u00001,h= 0:01,\u00120= 0:4\u0019,'0= 0:125\u0019.\nFigure 3: (Color online) The frequency dependencies of the power\nloss for the cases of rigid dipole (RD-model), \fxed particle (FP-\nmodel), and viscously coupled nanoparticle with a \fnite anisotropy\n(FA-model). The parameters used are the same as in the caption to\nFig. 2.\nFigure 4: (Color online) The dependencies of the amplitudes of\ncoupled oscillations of the magnetic moment (21) and the easy axis\n(20) on the \feld frequency. The parameters used are the same as in\nthe caption to Fig. 2, but \u0011= 4:0\u00005P.\nnear the resonance (Fig. 5).\nThe situation described above is an origin for extreme\nsensitivity of the power loss to the system parameters,\nwhich may be useful in the applications and can be uti-\nlized in a number of cases. In contrary, in other cases such\nsensitivity can be very undesirable, and we have to take\nmeasures to prevent it. Independently of the further pur-\nposes, one needs to investigate the in\ruence of the main\nparameters in detail. It is especially important for the de-\nsign of the nanoparticle ensembles with the speci\fed prop-\nerties for key applications, such as microwave absorbing\nor magnetic \ruid hyperthermia, where the heating or/and\nabsorbing rates are the primary characteristics.\nIn this regard, the similar parameters \u000band\u0011are the\nmost interesting. In Fig. 6a, the comparison of the power\nloss for two values of \u000bare plotted using the \fxed particle\napproximation and the approximation of viscously cou-\npled nanoparticle with a \fnite anisotropy. As expected,\nthe decrease in \u000bleads to the proportional increase in the\npower loss for both approximations. At the same time, the\nchange in\u0011results in di\u000berent behavior of the power loss\nobtained using the rigid dipole approximation and the ap-\nproximation of viscously coupled nanoparticle with a \fnite\nanisotropy (see Fig. 6b). For the \frst case, the propor-\ntional growth of q(~\n) with decreasing \u0011takes place. But\nfor the second case, account of the \fnite anisotropy leads\nto the opposite results. Here we report a nonlinear growth\ninq(~\n) with increasing viscosity \u0011. As it was explained\n7Figure 5: (Color online) The frequency dependencies of the power\nloss for the cases of rigid dipole (RD-model), \fxed particle (FP-\nmodel), and viscously coupled nanoparticle with a \fnite anisotropy\n(FA-model). The parameters used are the same as in the caption to\nFig. 2, but \u0011= 4:0\u00005P.\nabove, the origin of this e\u000bect lies in the relative motion\nof vectors nandm. Then, to estimate the applicability of\nthe model of rigid dipole, one needs to compare the power\nloss values for these two cases. As seen from Fig. 6b, vari-\nous situations are possible because there are two di\u000berent\nbehavior types when mis unlocked. The \frst type is the\nasynchronous oscillations of mandn, wherein the values\nofq(~\n) for the model of viscously coupled nanoparticle\nwith a \fnite anisotropy can be considerably larger than\nthe values predicted by the model of rigid dipole. The\nsecond type is the synchronous motion of the magnetic\nmoment and the easy axis. Here, both dissipation mecha-\nnisms are suppressed because the amplitudes of nandm\noscillations become smaller. As a result, the power loss for\nthe \fnite anisotropy case can be substantially lower than\nthe value obtained for the model of rigid dipole. This al-\nlows us to conclude about a low applicability of the model\nof rigid dipole in a high frequency limit.\nAnother important issue which needs to be accounted\nis the in\ruence of the external \feld orientation with re-\nspect to the nanoparticle position. As follows from (23),\n(29), (34), this orientation is de\fned by the polarization\ntype and the initial position of the easy axis. The model\nof rigid dipole predicts the di\u000berence of the power loss not\nmore than two times when \u001bvaries in the range of [ \u00001:::1].\nIn accordance with two other models, the dependence of\nthe power loss on \u001bis more strong. As seen from Fig. 7a,\nq(~\n) can be at least 10 times di\u000berent depending on \u001bfor\nthe model of viscously coupled nanoparticle with a \fnite\nanisotropy. Here we need to note that this dependence is\nnot linear and the lowest curve q(~\n) does not correspond\nto\u001b= 0 or\u001b=\u00061. The initial position of the easy\naxis given by angle \u00120essentially in\ruences the power loss\nas well. As seen from Fig. 7b, this di\u000berence may be at\nleast 20 times. Since nanoparticles in real ferro\ruids are\nnon-uniformly distributed, one can highlight the following.\nFirstly, the dipole interaction, which tries to arrange the\nensemble, can considerably in\ruence the power loss. Sec-\nondly, an external magnetic \feld gradient, which is used\nFigure 6: (Color online) The sensitivity of the power loss to the\nattenuation parameters. Plot (a): \fxed particle (FP-model) and\nviscously coupled nanoparticle with a \fnite anisotropy (FA-model)\nand di\u000berent values of the magnetic damping parameter \u000b. Plot (b):\nrigid dipole (RD-model) and viscously coupled nanoparticle with a\n\fnite anisotropy (FA-model) and di\u000berent values of the \u0011and results\nobtained for the cases rigid dipole (RD-model) and viscously coupled\nnanoparticle with a \fnite anisotropy (FA-model). The parameters\nused here and not stated in the \fgure legend are the same as in the\ncaption to Fig. 2, but \u00120= 0:25\u0019.\n8Figure 7: (Color online) The sensitivity of the power loss to the\norientation of the nanoparticle with respect to the external \feld for\nthe case of viscously coupled nanoparticle with a \fnite anisotropy\n(FA-model). The parameters used here and not stated in the \fgure\nlegend are the same as in the caption to Fig. 2, but \u00120= 0:25\u0019for\nthe plot (a) and \u001b= 1 for the plot (b).\nfor the ferro\ruid control during hyperthermia, also de\fnes\nthe power loss. And, thirdly, we can easily control the\npower loss in a wide range of values by a permanent exter-\nnal \feld, which speci\fes the direction of the nanoparticle\neasy axis.\nWe summarize our \fndings as follows. 1) The small os-\ncillations mode is considered for the coupled magnetic and\nmechanical motion for the viscously coupled nanoparticle\nwith a \fnite anisotropy. This mode takes place when the\namplitude of the external alternating \feld is much smaller\nthan the value of the nanoparticle uniaxial anisotropy \feld\n(H\u001cHa). 2) The damping precession of the magnetic\nmoment inside the nanoparticle primarily determines the\nvalue of the power loss and the resonance character of its\nfrequency dependence. 3) The power loss can be signi\f-\ncantly changed by the nanoparticle easy axis motion. For\nthe realistic system parameters, the power loss obtained\nfor the model of viscously coupled nanoparticle with a \f-\nnite anisotropy is larger than the value obtained for the\n\fxed particle model. 4) The decrease in the \ruid car-\nrier viscosity leads to the nonproportional decrease in the\npower loss, which near the resonance can be much smaller\nthan the value obtained for the \fxed particle model. 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Xue, Precessional dynamics of\nsingle-domain magnetic nanoparticles driven by small ac mag-\nnetic \felds, Journal of Physics D: Applied Physics 39 (22) (2006)\n4746.\nURL http://stacks.iop.org/0022-3727/39/i=22/a=002\n[23] J. C\u0016 \u0010murs, A. C\u0016 ebers, Dynamics of anisotropic superparamag-\nnetic particles in a precessing magnetic \feld, Phys. Rev. E 87\n(2013) 062318. doi:10.1103/PhysRevE.87.062318 .\nURL https://link.aps.org/doi/10.1103/PhysRevE.87.\n062318\n[24] H. Keshtgar, S. Streib, A. Kamra, Y. M. Blanter, G. E. W.\nBauer, Magnetomechanical coupling and ferromagnetic res-\nonance in magnetic nanoparticles, Phys. Rev. B 95 (2017)\n134447. doi:10.1103/PhysRevB.95.134447 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.95.134447\n[25] M. I. Dar, S. A. Shivashankar, Single crystalline magnetite,\nmaghemite, and hematite nanoparticles with rich coercivity,\nRSC Adv. 4 (2014) 4105{4113. doi:10.1039/C3RA45457F .\nURL http://dx.doi.org/10.1039/C3RA45457F\n[26] Y. L. Raikher, M. I. Shliomis, The e\u000bective \feld method in\nthe orientational kinetics of magnetic \ruids and liquid crystals,\nAdvances in Chemical Physics 87 (1994) 595{751. doi:http:\n//dx.doi.org/10.1002/9780470141465.ch8 .\nURL http://dx.doi.org/10.1002/9780470141465.ch8\n[27] J. Frenkel, Kinetic Theory of Liquids, Dover Publications,\nDover, 1955.\nURL https://books.google.com.ua/books?id=ORdSQwAACAAJ\n10"    },    {        "title": "1404.1488v2.Gilbert_damping_in_noncollinear_ferromagnets.pdf",        "content": "arXiv:1404.1488v2  [cond-mat.mtrl-sci]  27 Nov 2014Gilbert damping in noncollinear ferromagnets\nZhe Yuan,1,∗Kjetil M. D. Hals,2,3Yi Liu,1Anton A. Starikov,1Arne Brataas,2and 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\n2Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\n3Niels Bohr International Academy and the Center for Quantum Devices,\nNiels Bohr Institute, University of Copenhagen, 2100 Copen hagen, Denmark\nThe precession and damping of a collinear magnetization dis placed from its equilibrium are well\ndescribed by the Landau-Lifshitz-Gilbert equation. The th eoretical and experimental complexity\nof noncollinear magnetizations is such that it is not known h ow the damping is modified by the\nnoncollinearity. We use first-principles scattering theor y to investigate transverse domain walls\n(DWs) of the important ferromagnetic alloy Ni 80Fe20and show that the damping depends not only\non the magnetization texture but also on the specific dynamic modes of Bloch and N´ eel DWs in ways\nthat were not theoretically predicted. Even in the highly di sordered Ni 80Fe20alloy, the damping is\nfound to be remarkably nonlocal.\nPACS numbers: 72.25.Rb, 75.60.Ch, 75.78.-n, 75.60.Jk\nIntroduction. —The key common ingredient in various\nproposed nanoscale spintronics devices involving mag-\nnetic droplet solitons [ 1], skyrmions [ 2,3], or magnetic\ndomain walls (DWs) [ 4,5], is a noncollinear magneti-\nzation that can be manipulated using current-induced\ntorques (CITs) [ 6]. Different microscopic mechanisms\nhave been proposed for the CIT including spin trans-\nfer [7,8], spin-orbit interaction with broken inversion\nsymmetry in the bulk or at interfaces [ 9–11], the spin-\nHalleffect[ 12]orproximity-inducedanisotropicmagnetic\nproperties in adjacent normal metals [ 13]. Their contri-\nbutions are hotly debated but can only be disentangled\nif the Gilbert damping torque is accurately known. This\nis not the case [ 14]. Theoretical work [ 15–19] suggest-\ning that noncollinearity can modify the Gilbert damping\ndue to the absorption of the pumped spin current by the\nadjacent precessing magnetization has stimulated exper-\nimental efforts to confirm this quantitatively [ 14,20]. In\nthis Letter, we use first-principles scattering calculations\nto show that the Gilbert damping in a noncollinear alloy\ncan be significantly enhanced depending on the partic-\nular precession modes and surprisingly, that even in a\nhighly disordered alloy like Ni 80Fe20, the nonlocal char-\nacterofthe dampingis verysubstantial. Ourfindingsare\nimportant for understanding field- and/or current-driven\nnoncollinear magnetization dynamics and for designing\nnew spintronics devices.\nGilbert damping in Ni 80Fe20DWs.—Gilbert damping\nis in general described by a symmetric 3 ×3 tensor.\nFor a substitutional, cubic binary alloy like Permalloy,\nNi80Fe20, this tensor is essentially diagonal and isotropic\nand reduces to scalar form when the magnetization is\ncollinear. A value of this dimensionless scalar calculated\nfrom first-principles, αcoll= 0.0046, is in good agree-\nment with values extracted from room temperature ex-\nperiments that range between 0.004 and 0.009 [ 21]. In a\none-dimensional (1D) transverse DW, the Gilbert damp-ing tensor is still diagonal but, as a consequence of the\nlowered symmetry [ 22], it contains two unequal compo-\nnents. The magnetization in static N´ eel or Bloch DWs\n(a) \n(b) \n(c) \nφ\nθ\n x\n y z\nφ\nθ\n0 0.1 0.2 0.3 0.4 \n1/( /h w) (nm -1 )00.01 0.02 0.03 _eff Néel \nBloch \njSO =0 50 20 10 5 3 /h w (nm) \n_oeff _ieff \nFIG. 1. (color online). Sketch of N´ eel (a) and Bloch (b)\nDWs. (c) Calculated effective Gilbert damping parameters\nfor Permalloy DWs (N´ eel, black lines; Bloch, red lines) as a\nfunction of the inverse of the DW width λw. Without spin-\norbit coupling, calculations for the two DW types yield the\nsame results (blue lines). The green dot represents the valu e\nof Gilbert damping calculated for collinear Permalloy. For\neach value of λw, we typically consider 8 different disorder\nconfigurations and the error bars are a measure of the spread\nof the results.2\nliesinsidewelldefinedplanesthatareillustratedinFig. 1.\nAn angle θrepresents the in-plane rotation with respect\nto the magnetizationin the left domainand it variesfrom\n0 toπthrough a 180◦DW. If the plane changes in time,\nas it does when the magnetization precesses, an angle φ\ncan be used to describe its rotation. We define an out-\nof-plane damping component αocorresponding to varia-\ntion inφ, and an in-plane component αicorresponding\nto time-dependent θ. Rigid translation of the DW, i.e.\nmaking the DW center rwvary in time, is a specific ex-\nample of the latter.\nFor Walker-profile DWs [ 23], an effective (dimension-\nless) in-plane ( αeff\ni) and out-of-plane damping ( αeff\no) can\nbe calculated in terms of the scattering matrix Sof the\nsystem using the scattering theory of magnetization dis-\nsipation [ 24,25]. Both calculated values are plotted in\nFig.1(c) as a function of the inverse DW width 1 /λwfor\nN´ eel and Bloch DWs. Results with the spin-orbit cou-\npling (SOC) artificially switched off are shown for com-\nparison; because spin space is then decoupled from real\nspace, the results for the two DW profiles are identical\nand both αeff\niandαeff\novanish in the large λwlimit con-\nfirming that SOC is the origin of intrinsic Gilbert damp-\ning for collinear magnetization. With SOC switched on,\nN´ eel and Bloch DWs have identical values within the\nnumerical accuracy, reflecting the negligibly small mag-\nnetocrystalline anisotropy in Permalloy. Both αeff\niand\nαeff\noapproach the collinear value αcoll[21], shown as a\ngreen dot in the figure, in the wide DW limit. For finite\nwidths, theyexhibit aquadraticandapredominantlylin-\near dependence on 1 /(πλw), respectively, both with and\nwithoutSOC;forlargevaluesof λw, thereisahintofnon-\nlinearity in αeff\no(λw). However, phenomenological theo-\nries [15–17] predict that αeff\nishould be independent of λw\nand equal to αcollwhileαeff\noshould be a quadratic func-\ntion of the magnetization gradient. Neither of these pre-\ndicted behaviours is observed in Fig. 1(c) indicating that\nexisting theoretical models of texture-enhanced Gilbert\ndamping need to be reexamined.\nTheαeffshown in Fig. 1(c) is an effective damping\nconstant because the magnetization gradient dθ/dzof a\nWalker profile DW is inhomogeneous. Our aim in the\nfollowing is to understand the physical mechanisms of\ntexture-enhanced Gilbert damping with a view to deter-\nmining how the local damping depends on the magneti-\nzation gradient, as well as the corresponding parameters\nfor Permalloy, and finally expressing these in a form suit-\nable for use in micromagnetic simulations.\nIn-plane damping αi.—To get a clearer picture of how\nthe in-plane damping depends on the gradient, we calcu-\nlate the energy pumping Er≡Tr/parenleftBig\n∂S\n∂rs∂S†\n∂rs/parenrightBig\nfor a finite\nlengthLof a Bloch-DW-type spin spiral (SS) centered\natrs. In this SS segment (SSS), dθ/dzis constant ex-\ncept at the ends. Figure 2(b) showsthe resultscalculated\nwithout SOC for a single PermalloySSS with dθ/dz= 6◦0 10 20 30 40 \nL (nm) 020 40 Er (nm -2 )Without smearing \nWith smearing 0 4 2 6Winding angle ( /)\n0 1 2 3 4 \nNumber of SSSs z0n//L de/dz L L \n(c) (a) \n(b) \nFIG. 2. (color online). (a) Sketch of the magnetization gra-\ndient for two SSSs separated by collinear magnetization wit h\n(green, dashed) and without (red, solid) a broadening of the\nmagnetization gradient at the ends of the SSSs. The length\nof each segment is L. (b) Calculated energy pumping Eras a\nfunction of Lfor asingle Permalloy Bloch-DW-typeSSSwith-\nout SOC. The upper horizontal axis shows the total winding\nangle of the SSS. (c) Calculated energy pumping Erwithout\nSOC as a function of the number of SSSs that are separated\nby a stretch of collinear magnetization.\nper atomic layer; Fig. 1(c) shows that SOC does not in-\nfluence the quadratic behaviour essentially. Eris seen\nto be independent of Lindicating there is no dissipation\nwhendθ/dzis constant in the absence of SOC. In this\ncase, the only contribution arises from the ends of the\nSSS where dθ/dzchanges abruptly; see Fig. 2(a). If we\nreplace the step function of dθ/dzby a Fermi-like func-\ntion with a smearing width equal to one atomic layer, Er\ndecreasessignificantly(greensquares). Formultiple SSSs\nseparated by collinear magnetization, we find that Eris\nproportional to the number of segments; see Fig. 2(c).\nWhat remains is to understand the physical origin of\nthe damping at the ends of the SSSs. Rigid translation\nof a SSS or of a DW allows for a dissipative spin cur-\nrentj′′\ns∼ −m×∂z∂tmthat breaks time-reversal sym-\nmetry [19]. The divergence of j′′\nsgives rise to a local\ndissipative torque, whose transverse component is the\nenhancement of the in-plane Gilbert damping from the\nmagnetizationtexture. After straightforwardalgebra, we\nobtain the texture-enhanced in-plane damping torque\nα′′/bracketleftbig\n(m·∂z∂tm)m×∂zm−m×∂2\nz∂tm/bracketrightbig\n,(1)\nwhereα′′is a material parameter with dimensions\nof length squared. In 1D SSs or DWs, Eq. ( 1)\nleads to the local energy dissipation rate ˙E(r) =\n(α′′Ms/γ)∂tθ∂t(d2θ/dz2) [25], where Msis the satura-\ntion magnetization and γ=gµB//planckover2pi1is the gyromagnetic\nratio expressed in terms of the Land´ e g-factor and the\nBohr magneton µB. This results shows explicitly that\nthe in-plane damping enhancement is related to finite\nd2θ/dz2. Using the calculated data in Fig. 1(c), we ex-3\ntract a value for the coefficient α′′= 0.016 nm2that is\nindependent of specific textures m(r) [25].\nOut-of-plane damping αo.—We begin our analysis of\nthe out-of-plane damping with a simple two-band free-\nelectron DW model [ 25]. Because the linearity of the\ndamping enhancement does not depend on SOC, we ex-\namine the SOC free case for which there is no differ-\nence between N´ eel and Bloch DW profiles and we use\nN´ eel DWs in the following. Without disorder, we can\nuse the known φ-dependence of the scattering matrix for\nthis model [ 31] to obtain αeff\noanalytically,\nαeff\no=gµB\n4πAMsλw/summationdisplay\nk/bardbl/parenleftbigg/vextendsingle/vextendsingle/vextendsinglerk/bardbl\n↑↓/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsinglerk/bardbl\n↓↑/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingletk/bardbl\n↑↓/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingletk/bardbl\n↓↑/vextendsingle/vextendsingle/vextendsingle2/parenrightbigg\n≈gµB\n4πAMsλwh\ne2GSh,. (2)\nwhereAis the cross sectional area and the convention\nused for the reflection ( r) and transmission ( t) probabil-\nity amplitudes is shown in Fig. 3(a). Note that |tk/bardbl\n↑↓|2and\n|tk/bardbl\n↓↑|2are of the order of unity and much larger than the\nothertwotermsbetweenthebracketsunlesstheexchange\nsplitting is very large and the DW width very small. It\nis then a good approximation to replace the quantities in\nbracketsbythenumberofpropagatingmodesat k/bardbltoob-\ntain the second line of Eq. ( 2), where GShis the Sharvin\nconductance that only depends on the free-electron den-\nsity. Equation ( 2) shows analytically that αeff\nois pro-\nportional to 1 /λwin the ballistic regime. This is repro-\nduced by the results of numerical calculations for ideal\nfree-electron DWs shown as black circles in Fig. 3(b).\nIntroducing site disorder [ 32] into the free-electron\nmodel results in a finite resistivity. The out-of-plane\ndamping calculated for disordered free-electron DWs ex-\nhibits a transition as a function of its width. For narrow\nDWs (ballistic limit), αeff\nois inversely proportional to λw\nand the green, red and blue circles in Fig. 3(b) tend to\nbecomeparalleltothevioletlineforsmallvaluesof λw. If\nλwis sufficiently large, αeff\nobecomes proportional to λ−2\nw\nin agreement with phenomenological predictions [ 15–17]\nwhere the diffusive limit is assumed. This demonstrates\nthe different behaviour of αeff\noin these two regimes.\nWe can construct an expression that describes both\nthe ballistic and diffusive regimes by introducing an ex-\nplicit spatial correlation in the nonlocal form of the out-\nof-plane Gilbert damping tensor that was derived using\nthe fluctuation-dissipation theorem [ 15]\n[αo]ij(r,r′) =αcollδijδ(r−r′)+α′D(r,r′;l0)\n×[m(r)×∂zm(r)]i[m(r′)×∂z′m(r′)]j.(3)\nHereα′isamaterialparameterwithdimensionsoflength\nsquared and Dis a correlation function with an effective\ncorrelation length l0. In practice, we use D(r,r′;l0) =\n1√πAl0e−(z−z′)2/l2\n0, which reduces to δ(r−r′) in the dif-\nfusive limit ( l0≪λw) and reproduces earlier results [ 15–\n17]. In the ballistic limit, both α′andl0are infinite,0.01 0.05 0.1 0.5 \n1/( /h w) (nm -1 )10 -4 10 -3 10 -2 10 -1 _oeff \nBallistic \nl=2.7 !1 cm \nl=25 !1 cm \nl=94 !1 cm 100 50 30 20 10 5 2/h w (nm) \n~1/ hw\n~1/ hw2(a) \n(b) \nand . By definition, for weak splitting 1, but for all commonplace \ns the Fermi wavelength 2 is orders of magnitude smaller than . This \nimplies a wall resistance that is vanishingly small, because of the exponential depen- \ndence. For the example of iron, 2 is only 1 or 2 A , depending on which band \nis in question, whilst the wall thickness is some thousands of A . This leads to a \n10 . The physical reason for this is that waves are only scattered very much \nby potential steps that are abrupt on the scale of the wavelength of that wave, as \nsketched in figure 13. \nFor strong splitting ( it was found to be necessary to restrict the \nculation to a very narrow wall, viz. me 1. In practice this means \nmic abruptness. In this case a variable ¼ ð ÞÞ , trivially \nconnected to the definitions of in equations (2) and (3), determines the DW \nce. The obvious relationship with the Stearns definition of polarisation, \nequation (3), emphasises that the theory is essentially one of tunnelling between \none domain and the next. The DW resistance vanishes as 1, as might be \nd. As !1 uivalent to unity), the material becomes half-metallic \nand the wall resistance also !1 . A multi-domain half-metal, with no opportunity \nfor spin relaxation, is an insulator, no matter how high is. \nCabrera and Falicov satisfied themselves that, once the diamagnetic Lorentz \nforce e that give rise to additional resistance at the wall were properly treated \n[178], their theory could account for the results found in the Fe whiskers. However, \nit does not describe most cases encountered experimentally because the condition Abrupt \nFigure 13. Spin-resolved potential profiles and resulting wavefunctions at abrupt \nand wide (adiabatic) domain walls. The wavefunctions are travelling from left to right. In the \nadiabatic case, the wavelengths of the two wavefunctions are exchanged, but the change in \npotential energy is slow enough that there is no change in the amplitude of the transmitted \nwave. When the wall is abrupt the wavelength change is accompanied by substantial reflection, \nlting in a much lower transmitted amplitude (the reflected part of the wavefunction is not \nshown). This gives rise to domain wall resistance. C. H. Marrows Downloaded By: [University of California, Berkeley] At: 14:36 9 June 2010  V↑ V↓\n↓\n↑  e±ik↑z e±ik↓z\n e±ik↓z e±ik↑z\n. By definition, for weak splitting 1, but for all commonplace \nmi wavelength 2 is orders of magnitude smaller than . This \na wall resistance that is vanishingly small, because of the exponential depen- \ne of iron, 2 is only 1 or 2 A , depending on which band \nis in question, whilst the wall thickness is some thousands of A . This leads to a \n10 . The physical reason for this is that waves are only scattered very much \nby potential steps that are abrupt on the scale of the wavelength of that wave, as \nd in figure 13. \nit was found to be necessary to restrict the \nto a very narrow wall, 1. In practice this means \nabruptness. In this case a variable ¼ ð ÞÞ , trivially \nto the definitions of in equations (2) and (3), determines the DW \n. The obvious relationship with the Stearns definition of polarisation, \non (3), emphasises that the theory is essentially one of tunnelling between \nDW resistance vanishes as 1, as might be \nd. As !1 to , the material becomes half-metallic \n!1 . A multi-domain half-metal, with no opportunity \nis an insulator, no matter how high \nto additional resistance at the wall were properly treated \nld account for the results found in the Fe whiskers. However, \nit does not describe most cases encountered experimentally because the condition at abrupt \nto right. In the \nof the two wavefunctions are exchanged, but the change in \nis slow enough that there is no change in the amplitude of the transmitted \nis abrupt the wavelength change is accompanied by substantial reflection, \nin a much lower transmitted amplitude (the reflected part of the wavefunction is not \nto domain wall resistance. C. H. Marrows Downloaded By: [University of California, Berkeley] At: 14:36 9 June 2010 \n t↑↑ t↓↓\n↓↓\n t↑↓  t↓↑\nFIG. 3. (color online). (a) Cartoon of electronic transport\nin a two-band, free-electron DW. The global quantization\naxis of the system is defined by the majority and minority\nspin states in the left domain. (b) Calculated αeff\nofor two-\nband free-electron DWs as a function of 1 /(πλw) on a log-log\nscale. The black circles show the calculated results for the\nclean DWs, whichare in perfect agreement with theanalytica l\nmodel Eq. ( 2), shown as a dashed violet line. When disorder\n(characterized by the resistivity ρcalculated for the corre-\nsponding collinear magnetization) is introduced, αeff\noshows a\ntransition from a linear dependence on 1 /λwfor narrow DWs\ntoaquadraticbehaviourfor wideDWs. The solid lines arefits\nusing Eq. ( S24). The dashed orange lines illustrate quadratic\nbehaviour.\nbut the product α′D(r,r′;l0) =α′/(√πAl0) is finite and\nrelated to the Sharvin conductance of the system [ 33],\nconsistent with Eq. ( 2). We then fit the calculated val-\nues ofαeff\noshown in Fig. 3(b) using Eq. ( S24) [25]. With\nthe parameters α′andl0listed in Table I, the fit is seen\nto be excellent over the whole range of λw. The out-\nof-plane damping enhancement arises from the pumped\nspin current j′\ns∼∂tm×∂zmin a magnetization tex-\nture [15,17], where the magnitude of j′\nsis related to the\nTABLEI. Fitparameters usedtodescribe thedampingshown\nin Fig.1for Permalloy DWs and in Fig. 3for free-electron\nDWs with Eq. ( S24). The resistivity is determined for the\ncorresponding collinear magnetization.\nSystem ρ(µΩ cm) α′(nm2)l0(nm)\nFree electron 2 .69 45 .0 13 .8\nFree electron 24 .8 1 .96 4 .50\nFree electron 94 .3 0 .324 2 .78\nPy (ξSO= 0) 0 .504 23 .1 28 .3\nPy (ξSO/negationslash= 0) 3 .45 5 .91 13 .14\nconductivity [ 15]. This is the reason why α′is larger in\na system with a lower resistivity in Table I.l0is a mea-\nsure of how far the pumped transverse spin current can\npropagate before being absorbed by the local magnetiza-\ntion. It is worth distinguishing the relevant characteris-\ntic lengths in microscopic spin transport that define the\ndiffusive regimes for different transport processes. The\nmean free path lmis the length scale for diffusive charge\ntransport. The spin-flip diffusion length lsfcharacterizes\nthe length scale for diffusive transport of a longitudinal\nspin current, and l0is the corresponding length scale for\ntransverse spin currents. Only when the system size is\nlarger than the corresponding characteristic length can\ntransport be described in a local approximation.\nWe can use Eq. ( S24) to fit the calculated αeff\noshown\nin Fig.1for Permalloy DWs. The results are shown in\nFig.S4. Since the values of αeff\nowe calculate for N´ eel\nand Bloch DWs are nearly identical, we take their aver-\nage for the SOC case. Intuitively, we would expect the\nout-of-plane damping for a highly disordered alloy like\nPermalloy to be in the diffusive regime corresponding to\na shortl0. But the fitted values of l0are remarkably\nlarge, as long as 28.3 nm without SOC. With SOC, l0\nis reduced to 13.1 nm implying that nonlocal damping\ncan play an important role in nanoscale magnetization\ntextures in Permalloy, whose length scale in experiment\nis usually about 100 nm and can be reduced to be even\nsmaller than l0by manipulating the shape anisotropy of\nexperimental samples [ 34,35].\nAs shown in Table I,l0is positively correlated with\nthe conductivity. The large value of l0and the low re-\nsistivity of Permalloy can be qualitatively understood in\nterms of its electronic structure and spin-dependent scat-\ntering. The Ni and Fe potentials seen by majority-spin\nelectrons around the Fermi level in Permalloy are almost\nidentical [ 25] so that they are only very weakly scattered.\nThe Ni and Fe potentials seen by minority-spin electrons\nare howeverquite different leading to strongscattering in\ntransport. The strong asymmetric spin-dependent scat-\ntering can also be seen in the resistivity of Permalloy\ncalculated without SOC, where ρ↓/ρ↑>200 [21,36]. As\na result, conduction in Permalloy is dominated by the\nweakly scattered majority-spin electrons resulting in a\nlow total resistivity and a large value of l0. This short-\ncircuit effect is only slightly reduced by SOC-induced\nspin-flipscatteringbecausetheSOCin3 dtransitionmet-\nals is in energy terms small compared to the bandwidth\nand exchange splitting. Indeed, αeff\no−αcollcalculated\nwith SOC (the red curve in Fig. S4) shows a greater cur-\nvature at large widths than without SOC, but is still\nquite different from the quadratic function characteristic\nofdiffusive behaviourforthe widest DWs wecould study.\nBothαeff\niandαeff\nooriginate from locally pumped spin\ncurrents proportional to m×∂tm. Because of the spa-\ntially varying magnetization, the spin currents pumped\ntotheleftandrightdonotcancelexactlyandthenetspin0.02 0.05 0.1 0.2 0.5\n1/(πλw) (nm-1)0.0010.010.05αoeff-αcoll\nξSO≠0\nξSO=040 30 20 15 10 5 3 2πλw (nm)\n~1/λw\n~1/λw2\nFIG. 4. (color online). Calculated out-of-plane damping\nαeff\no−αcollfrom Fig. 1plotted as a function of 1 /(πλw) on a\nlog-log scale. The solid lines are fitted using Eq. ( S24). The\ndashed violet and orange lines illustrate linear and quadra tic\nbehaviour, respectively.\ncurrent contains two components, j′′\ns∼ −m×∂z∂tm[19]\nandj′\ns∼∂tm×∂zm[15,17]. For out-of-plane damping,\n∂zmis perpendicular to ∂tmso there is large enhance-\nment due to the lowest order derivative. For the rigid\nmotion of a 1D DW, ∂zmis parallel to ∂tmso thatj′\ns\nvanishes. The enhancement of in-plane damping arising\nfromj′′\nsdue to the higher-orderspatial derivative of mag-\nnetization is then smaller.\nConclusions.— We have discovered an anisotropic\ntexture-enhanced Gilbert damping in Permalloy DWs\nusing first-principles calculations. The findings are ex-\npressed in a form [Eqs. ( 1) and (S24)] suitable for ap-\nplication to micromagnetic simulations of the dynamics\nof magnetization textures. The nonlocal character of the\nmagnetization dissipation suggests that field and/or cur-\nrentdrivenDW motion, whichis alwaysassumedto be in\nthe diffusive limit, needs to be reexamined. The more ac-\ncurate form of the damping that we propose can be used\nto deduce the CITs in magnetization textures where the\nusual way to study them quantitatively is by comparing\nexperimental observations with simulations.\nCurrent-drivenDWs movewith velocities that arepro-\nportional to β/αwhereβis the nonadiabatic spin trans-\nfer torque parameter. The order of magnitude spread in\nvalues of βdeduced for Permalloy from measurements of\nthe velocities of vortex DWs [ 37–40] may be a result of\nassumingthat αis a scalarconstant. Ourpredictions can\nbe tested by reexamining these studies using the expres-\nsions for αgiven in this paper as input to micromagnetic\ncalculations.\nWe would like to thank Geert Brocks and Taher Am-\nlaki for useful discussions. This work was financially\nsupported by the “Nederlandse Organisatie voor Weten-\nschappelijk Onderzoek” (NWO) through the research\nprogramme of “Stichting voor Fundamenteel Onderzoek\nder Materie” (FOM) and the supercomputer facilities5\nof NWO “Exacte Wetenschappen (Physical Sciences)”.\nIt was also partly supported by the Royal Netherlands\nAcademy of Arts and Sciences (KNAW). 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Box 217, 7500 AE Enschede, The Net herlands\n2Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\n3Niels Bohr International Academy and the Center for Quantum Devices,\nNiels Bohr Institute, University of Copenhagen, 2100 Copen hagen, Denmark\nI. COMPUTATIONAL DETAILS.\nTaking the concrete example of Walker profile domain\nwalls (DWs), the effective (dimensionless) in-plane and\nout-of-plane damping parameters can be expressed in\nterms of the scattering matrix Sof the system as, re-\nspectively,\nαeff\ni=gµBλw\n8πAMsTr/parenleftbigg∂S\n∂rw∂S†\n∂rw/parenrightbigg\n, (S1)\nαeff\no=gµB\n8πAMsλwTr/parenleftbigg∂S\n∂φ∂S†\n∂φ/parenrightbigg\n,(S2)\nusing the scattering theory of magnetization dissipation\n[S1,S2]. Heregis the Land´ e g-factor,µBis the Bohr\nmagneton, λwdenotes the DW width, Ais the cross sec-\ntional area, and Msis the saturation magnetization.\nIt is interesting to compare the scheme for calculat-\ning the Gilbert damping of DWs using Eqs. ( S1) and\n(S2) [S1,S2] with that used for collinear magnetiza-\ntion [S3,S4]. Both of them are based upon the energy\npumping theory [ S2,S3]. To calculate the damping αcoll\nfor the collinear case, the magnetization is made to pre-\ncessuniformlyandthelocalenergydissipationishomoge-\nneous throughout the ferromagnet. The total energy loss\ndue to Gilbert damping is then proportional to the vol-\nume of the ferromagnetic material and the homogenous\nlocal damping αcollcan be determined from the damp-\ning per unit volume. When the magnetization of a DW is\nmade to change either by moving its center rwor varying\nits orientation φ, this results in a relatively large preces-\nsion at the center of the DW; the further from the center,\nthe less the magnetization changes. The local contribu-\ntion to the total energydissipationofthe DWis weighted\nby the magnitude of the magnetization precession when\nrworφvaries. For a fixed DW width, the total damping\nis not proportional to the volume of the scattering region\nbut converges to a constant once the scattering region is\nlarge compared to the DW. In practice, αeff\niandαeff\nocal-\nculated using Eqs. ( S1) and (S2) are well converged for\na scattering region 10 times longer than λw. Effectively,\nαeffcan be regardedas a weighted averageof the (dimen-\nsionless) damping constant in the region of a DW. In the\nwide DW limit, αeff\niandαeff\noboth approach αcollwith\nspin-orbit coupling (SOC) and vanish in its absence.\nTo evaluate the effective Gilbert damping of a DWusing Eqs. ( S1) and (S2), we attached semiinfinite (cop-\nper) leads to a finite length of Ni 80Fe20alloy (Permal-\nloy, Py) and rotated the local magnetization to make\na 180◦DW using the Walker profile. Specifically, we\nusedm= (sechz−rw\nλw,0,tanhz−rw\nλw) for N´ eel DWs and\nm= (−tanhz−rw\nλw,−sechz−rw\nλw,0) for Bloch DWs. The\nscatteringpropertiesofthedisorderedregionwereprobed\nby studying how Bloch waves in the Cu leads incident\nfrom the left or right sides weretransmitted and reflected\n[S4,S5]. Thescatteringmatrixwasobtainedusingafirst-\nprinciples “wave-function matching” scheme [ S6] imple-\nmented with tight-binding linearized muffin-tin orbitals\n(TB-LMTOs) [ S7]. SOC was included using a Pauli\nHamiltonian. The calculations were rendered tractable\nby imposing periodic boundary conditions transverse to\nthe transport direction. It turned out that good results\ncould be achieved even when these so-called “lateral su-\npercells” were quite modest in size. In practice, we used\n5×5 lateral supercells and the longest DW we consid-\nered was more than 500 atomic monolayers thick. After\nembedding the DW between collinear Py and Cu leads,\nthe largest scattering region contained 13300 atoms. For\nevery DW width, we averaged over about 8 random dis-\norder configurations.\nA potential profile for the scattering region was con-\nstructed within the framework of the local spin den-\nsity approximation of density functional theory as fol-\nlows. For a slab of collinear Py binary alloy sandwiched\nbetween Cu leads, atomic-sphere-approximation (ASA)\npotentials [ S7] were calculated self-consistently without\nSOC using a surface Green’s function (SGF) method im-\nplemented [ S8] with TB-LMTOs. Chargeand spin densi-\nties for binary alloy AandBsites were calculated using\nthe coherent potential approximation [ S9] generalized to\nlayer structures [ S8]. For the scattering matrix calcu-\nlation, the resulting ASA potentials were assigned ran-\ndomly to sites in the lateral supercells subject to mainte-\nnance of the appropriate concentration of the alloy [ S6]\nand SOC was included. The exchange potentials are ro-\ntated in spin space [ S10] so that the local quantization\naxis for each atomic sphere follows the DW profile. The\nDW width is determined in reality by a competition be-\ntween interatomic exchange interactions and magnetic\nanisotropy. For a nanowire composed of a soft mag-\nnetic material like Py, the latter is dominated by the2\nshape anisotropy that arises from long range magnetic\ndipole-dipole interactions and depends on the nanowire\nprofile. Experimentallyitcanbetailoredbychangingthe\nnanowire dimensions leading to the considerable spread\nof reported DW widths [ S11]. In electronic structure cal-\nculations, that do not contain magnetic dipole-dipole in-\nteractions, we simulate a change of demagnetization en-\nergy by varying the DW width. In this way we can study\nthe dependence of Gilbert damping on the magnetization\ngradient by performing a series of calculations for DWs\nwith different widths.\nFor the self-consistent SGF calculations (without\nSOC), the two-dimensional(2D) Brillouin zone (BZ) cor-\nresponding to the 1 ×1 interface unit cell was sampled\nwith a 120 ×120 grid. The transport calculations includ-\ning SOC were performed with a 32 ×32 2D BZ grid for a\n5×5 lateral supercell, which is equivalent to a 160 ×160\ngrid in the 1 ×1 2D BZ.\nII. EXTRACTING α′′\nWe first briefly derive the form of the in-plane damp-\ning. It has been argued phenomenologically [ S12] that\nfor a noncollinear magnetization texture varying slowly\nin time the lowest order term in an expansion of the\ntransverse component of the spin current in spatial and\ntime derivatives that breaks time-reversal symmetry and\nis therefore dissipative is\nj′′\ns=−ηm×∂z∂tm, (S3)\nwhereηis a coefficient depending on the material and\nmis a unit vector in the direction of the magnetization.\nThe divergence of the spin current,\n∂zj′′\ns=−η/parenleftbig\n∂zm×∂z∂tm+m×∂2\nz∂tm/parenrightbig\n,(S4)\ngives the corresponding dissipative torque exerted on the\nlocal magnetization. While the second term in brackets\nin Eq. (S4) is perpendicular to m, the first term contains\nboth perpendicular and parallel components. Since we\nare only interested in the transverse component of the\ntorque, we subtract the parallel component to find the\ndamping torque\nτ′′=−η/braceleftbig\n(1−mm)·(∂zm×∂z∂tm)+m×∂2\nz∂tm/bracerightbig\n=−η/braceleftbig\n[m×(∂zm×∂z∂tm)]×m+m×∂2\nz∂tm/bracerightbig\n=η/bracketleftbig\n(m·∂z∂tm)m×∂zm−m×∂2\nz∂tm/bracketrightbig\n.(S5)\nThe Landau-Lifshitz-Gilbert equation including the\ndamping torque τ′′reads\n∂tm=−γm×Heff+αcollm×∂tm+γτ′′\nMs\n=−γm×Heff+αcollm×∂tm\n+α′′/bracketleftbig\n(m·∂z∂tm)m×∂zm−m×∂2\nz∂tm/bracketrightbig\n,(S6)where the in-plane damping parameter α′′≡γη/Mshas\nthe dimension of length squared.\nIn the following, we explain how α′′can be extracted\nfrom calculations on Walker DWs and show that it is ap-\nplicable to other profiles. The formulation is essentially\nindependent of the DW type (Bloch or N´ eel) and we use\na Bloch DW in the following derivation for which\nm(z) = [cosθ(z),sinθ(z),0], (S7)\nwhereθ(z) represents the in-plane rotation (see Fig. 1 in\nthe paper). The local energy dissipation associated with\na time-dependent θis given by [ S2]\nγ\nMs˙E(z) =αcoll∂tm·∂tm\n+α′′/bracketleftbig\n(m·∂z∂tm)∂tm·∂zm−∂tm·∂2\nz∂tm/bracketrightbig\n.(S8)\nFor the one-dimensional profile Eq. ( S7), this can be sim-\nplified as\nγ\nMs˙E(z) =αcoll/parenleftbiggdθ\ndt/parenrightbigg2\n−α′′dθ\ndtd\ndt/parenleftbiggd2θ\ndz2/parenrightbigg\n.(S9)\nSubstituting into Eq. ( S9) the Walker profile\nθ(z) =−π\n2−arcsin/parenleftbigg\ntanhz−rw\nλw/parenrightbigg\n,(S10)\nthat we used in the calculations, we obtain for the total\nenergy dissipation associated with the motion of a rigid\nDW for which ˙θ= ˙rwdθ/drw,\n˙E=/integraldisplay\nd3r˙E(z) =2MsA\nγλw/parenleftbigg\nαcoll+α′′\n3λ2w/parenrightbigg\n˙r2\nw.(S11)\nComparing this to the energy dissipation expressed in\nterms of the effective in-plane damping αeff\ni[S2]\n˙E=2MsA\nγλwαeff\ni˙r2\nw, (S12)\nwe arrive at\nαeff\ni(λw) =αcoll+α′′\n3λ2w. (S13)\nUsing Eq. ( S13), we perform a least squares linear fitting\nofαeff\nias a function of λ−2\nwto obtain αcollandα′′. The\nfitting is shown in Fig. S1and the parameters are listed\nin Table SI. Note that αcollis in perfect agreement with\nindependent calculations for collinear Py [ S4].\nTo confirm that α′′is independent of texture, we con-\nsider another analytical DW profile in which the in-plane\nrotation is described by a Fermi-like function,\nθ(z) =−π+π\n1+ez−rF\nλF. (S14)\nHererFandλFdenote the DW center and width, re-\nspectively. Substituting Eq. ( S14) into Eq. ( S9), we find\nthe energy dissipation for “Fermi” DWs to be\n˙E=π2MsA\n6γλF/parenleftbigg\nαcoll+α′′\n5λ2\nF/parenrightbigg\n˙r2\nF,(S15)3\n0 0.5 1.0 1.5\n1/λw2 (nm-2)00.0050.0100.015αieff\nBloch\nNéel\nξSO=0Walker\nFIG. S1. Calculated αeff\nifor Walker-profile Permalloy DWs.\nN´ eel DWs: black circles, Bloch DWs: red circles. Without\nSOC, calculations for the twoDWtypesyield thesame results\n(blue circles). The dashed lines are linear fits using Eq. ( S13).\nwhich suggests the effective in-plane damping\nαeff\ni(λF) =αcoll+α′′\n5λ2\nF. (S16)\nEq. (S16) is plotted as solid lines in Fig. S2with the\nvalues of αcollandα′′taken from Table SI.\nSince the energy pumping can be expressed in terms\nof the scattering matrix Sas\n˙E=/planckover2pi1\n4πTr/parenleftbigg∂S\n∂t∂S†\n∂t/parenrightbigg\n=/planckover2pi1\n4πTr/parenleftbigg∂S\n∂rF∂S†\n∂rF/parenrightbigg\n˙r2\nF,(S17)\nwe can calculate the effective in-plane damping for a\nFermi DW from the Smatrix to be\nαeff\ni=3/planckover2pi1γλF\n2π3MsATr/parenleftbigg∂S\n∂rF∂S†\n∂rF/parenrightbigg\n.(S18)\nWe plot the values of αeff\nicalculated using the derivative\nof the scattering matrix Eq. ( S18) as circles in Fig. S2.\nThe good agreement between the circles and the solid\nlines demonstratesthe validity ofthe form ofthe in-plane\ndamping torque in Eq. ( S6) and that the parameter α′′\ndoes not depend on a specific magnetization texture.\nTABLE SI. Fit parameters to describe the in-plane Gilbert\ndamping in Permalloy DWs.\nDW type αcoll α′′(nm2)\nBloch (4.6 ±0.1)×10−30.016±0.001\nN´ eel (4.5 ±0.1)×10−30.016±0.001\nξSO=0 (2.0 ±1.0)×10−60.017±0.0010 0.5 1.0 1.5 2.0 2.5 3.0\n1/λF2 (nm-2)00.0050.0100.015αieff\nBloch\nξSO=0Fermi\nFIG. S2. Calculated αeff\nifor Permalloy Bloch DWs (red cir-\ncles) with the Fermi profile Eq. ( S14). The blue circles are\nresults calculated without SOC. The solid lines are the an-\nalytical expression Eq. ( S16) using the parameters listed in\nTableSI.\nIII. THE FREE-ELECTRON MODEL USING\nMUFFIN-TIN ORBITALS\nWe take constant potentials, V↑=−0.2 Ry,V↓=\n−0.1Ry inside atomic sphereswith an exchangesplitting\n∆V= 0.1 Ry between majority and minority spins and a\nFermi level EF= 0. The atomic spheres are placed on a\nface-centered cubic (fcc) lattice with the lattice constant\nof nickel, 3.52 ˚A. The magnetic moment on each atom is\nthen 0.072µB. The transport direction is along the fcc\n[111]. In the scattering calculation, we use a 300 ×300\n01020 30 4050 60\nL (nm)306090102030AR (fΩ m2)456(a)\n(b)\n(c)ρ=2.69±0.06 µΩ cm\nρ=24.8±0.5 µΩ cm\nρ=94.3±4.4 µΩ cm\nFIG.S3. Resistancecalculatedforthedisorderedfree-ele ctron\nmodel as a function of the length of the scattering region for\nthree values of V0, the disorder strength: 0.05 Ry (a), 0.15\nRy (b) and 0.25 Ry (c). The lines are the linear fitting used\nto determine the resistivity.4\nk-point mesh in the 2D BZ. The calculated Sharvin con-\nductances for majority and minority channels are 0.306\nand 0.153 e2/hper unit cell, respectively, compared with\nanalytical values of 0.305 and 0.153.\nTo mimic disordered free-electron systems, we intro-\nduce a 5 ×5 lateral supercell and distribute constant\npotentials uniformly in the energy range [ −V0/2,V0/2]\nwhereV0is some given strength [ S13] and spatially at\nrandom on every atomic sphere in the scattering re-gion. The calculated total resistance as a function of the\nlengthLof the (disordered) scattering region is shown in\nFig.S3withV0= 0.05 Ry (a), 0.15 Ry (b) and 0.25 Ry\n(c). The resistivity increases with the impurity strength\nas expected and can be extracted with a linear fitting\nAR(L) =AR0+ρL. For each system, we calculate about\n10randomconfigurationsand takethe averageofthe cal-\nculatedresults. Wellconvergedresultsareobtainedusing\na 32×32k-point mesh for the 5 ×5 supercell.\nIV. FITTING α′ANDl0\nWith a nonlocal Gilbert damping, α(r,r′), the energy dissipation rate is given by [ S2]\n˙E=Ms\nγ/integraldisplay\nd3r˙m(r)·/integraldisplay\nd3r′α(r,r′)·˙m(r′). (S19)\nIf we consider the out-of-plane damping of a N´ eel DW, i.e. for which the angle φvaries in time (see Fig. 1 in the\npaper), we have\n˙m(r) =˙φsechz−rw\nλwˆy. (S20)\nConsidering again a Walker profile, we find the explicit form of the out- of-plane damping matrix element\nαo(z,z′) =αcollδ(z−z′)+α′\nλ2wsechz−rw\nλwsechz′−rw\nλw1√πAl0e−(z−z′\nl0)2. (S21)\nSubstituting Eq. ( S21) and Eq. ( S20) into Eq. ( S19), we obtain explicitly the energy dissipation rate\n˙E=2MsAλw\nγαcoll˙φ2+MsAα′˙φ2\n√πγl0λ2w/integraldisplay\ndzsech2z−rw\nλw/integraldisplay\ndz′sech2z′−rw\nλwe−(z−z′\nl0)2\n. (S22)\nThe calculated effective out-of-plane Gilbert damping for a DW with th e Walker profile is related to the energy\ndissipation rate as [ S2]\n˙E=2MsAλw\nγαeff\no˙φ2. (S23)\nComparing Eqs. ( S22) and (S23), we arrive at\nαeff\no=αcoll+α′\n2√πλ3wl0/integraldisplay\ndzsech2z−rw\nλw/integraldisplay\ndz′sech2z′−rw\nλwe−(z−z′\nl0)2\n. (S24)\nThe last equation is used to fit α′andl0toαeff\nocalculated for different λw. For Bloch DWs, it is straightforward to\nrepeat the above derivation and find the same result, Eq. ( S24).\nV. BAND STRUCTURES OF NI AND FE IN\nPERMALLOY\nIn the coherent potential approximation (CPA) [ S8,\nS9], the single-site approximation involves calculating\nauxiliary (spin-dependent) potentials for Ni and Fe self-\nconsistently. In our transport calculations, these auxil-\niary potentials are distributed randomly in the scattering\nregion. It is instructive to place the Ni potentials (for\nmajority- and minority-spin electrons) on an fcc latticeand to calculate the band structure non-self-consistently.\nThen we do the same using the Fe potentials. The cor-\nresponding band structures are plotted in Fig. S4. At\nthe Fermi level, where electron transport takes place,\nthe majority-spin bands for Ni and Fe are almost identi-\ncal, including their angular momentum character. This\nmeans that majority-spin electrons in a disordered al-\nloy see essentially the same potentials on all lattice sites\nand are only very weakly scattered in transport by the\nrandomly distributed Ni and Fe potentials. In contrast,5\n-9-6-303E-EF (eV)Majority Spin Minority Spin\nX Γ L-9-6-303E-EF (eV)\nX Γ LNi Ni\nFe Fe\nFIG. S4. Band structures calculated with the auxiliary Ni\nand Fe atomic sphere potentials and Fermi energy that were\ncalculated self-consistently forNi 80Fe20usingthecoherentpo-\ntential approximation. The red bars indicates the amount of\nscharacter in each band.\nthe minority-spin bands are quite different for Ni and Fe.\nThiscanbeunderstoodintermsofthe differentexchange\nsplitting between majority- and minority-spin bands; the\ncalculated magnetic moments of Ni and Fe in Permalloy\nin the CPA are 0.63 and 2.61 µB, respectively. The ran-\ndom distribution of Ni and Fe potentials in Permalloy\nthen leads to strong scattering of minority-spin electrons\nin transport.∗Present address: Institut f¨ ur Physik, Johannes\nGutenberg–Universit¨ at Mainz, Staudingerweg 7, 55128\nMainz, Germany; zyuan@uni-mainz.de\n[S1] K. M. D. Hals, A. K. Nguyen, and A. Brataas,\nPhys. Rev. Lett. 102, 256601 (2009) .\n[S2] A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer,\nPhys. Rev. B 84, 054416 (2011) .\n[S3] A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer,\nPhys. Rev. Lett. 101, 037207 (2008) .\n[S4] A. A. Starikov, P. J. Kelly, A. Brataas,\nY. Tserkovnyak, and G. E. W. Bauer,\nPhys. Rev. Lett. 105, 236601 (2010) .\n[S5] Z. Yuan, Y. Liu, A. A. Starikov, P. J. Kelly, and\nA. Brataas, Phys. Rev. Lett. 109, 267201 (2012) .\n[S6] K. Xia, M. Zwierzycki, M. Talanana, P. J. Kelly, and\nG. E. W. Bauer, Phys. Rev. B 73, 064420 (2006) .\n[S7] O. K. Andersen, Z. Pawlowska, and O. Jepsen,\nPhys. Rev. B 34, 5253 (1986) .\n[S8] I. Turek, V. Drchal, J. Kudrnovsk´ y, M. ˇSob, and\nP. Weinberger, Electronic Structure of Disordered Al-\nloys, Surfaces and Interfaces (Kluwer, Boston-London-\nDordrecht, 1997).\n[S9] P. Soven, Phys. Rev. 156, 809 (1967) .\n[S10] S. Wang, Y. Xu, and K. Xia,\nPhys. Rev. B 77, 184430 (2008) .\n[S11] O. Boulle, G. Malinowski, and M. Kl¨ aui,\nMat. Science and Eng. R 72, 159 (2011) .\n[S12] Y. Tserkovnyak, E. M. Hankiewicz, and G. Vignale,\nPhys. Rev. B 79, 094415 (2009) .\n[S13] A. K. Nguyen and A. Brataas,\nPhys. Rev. Lett. 101, 016801 (2008) ."    },    {        "title": "1802.05548v1.Damping_s_effect_on_the_magnetodynamics_of_spin_Hall_nano_oscillators.pdf",        "content": "Damping's e\u000bect on the magnetodynamics of spin Hall nano-oscillators\nYuli Yin,1, 2,\u0003Philipp D urrenfeld,2Mykola Dvornik,2Martina\nAhlberg,2Afshin Houshang,2Ya Zhai,1and Johan \u0017Akerman2, 3\n1Department of Physics, Southeast University, 211189 Nanjing, China\n2Department of Physics, University of Gothenburg, 412 96 Gothenburg, Sweden\n3Department of Materials and Nano Physics, School of Information and Communication Technology,\nKTH Royal Institute of Technology, Electrum 229, 164 40 Kista, Sweden\n(Dated: July 27, 2021)\nWe study the impact of spin wave damping ( \u000b) on the auto-oscillation properties of nano-\nconstriction based spin Hall nano-oscillators (SHNOs). The SHNOs are based on a 5 nm Pt layer\ninterfaced to a 5 nm Py 100\u0000x\u0000yPtxAgymagnetic layer, where the Pt and Ag contents are co-varied\nto keep the saturation magnetization constant (within 10 %), while \u000bvaries close to a factor of\nthree. We systematically investigate the in\ruence of the Gilbert damping on the magnetodynamics\nof these SHNOs by means of electrical microwave measurements. Under the condition of a constant\n\feld, the threshold current scales with the damping in the magnetic layer. The threshold current as\na function of \feld shows a parabolic-like behavior, which we attribute to the evolution of the spatial\npro\fle of the auto-oscillation mode. The signal linewidth is smaller for the high-damping materials\nin low magnetic \felds, although the lowest observed linewidth was measured for the alloy with least\ndamping.\nPACS numbers: 75.70.-i, 76.50.+g, 75.78.-n\nINTRODUCTION\nSpin Hall nano-oscillators (SHNO) are spintronic de-\nvices in which magnetization oscillations are induced by\npure spin currents [1]. These pure spin currents can be\nexperimentally realized via the spin Hall e\u000bect (SHE)\nin an adjacent heavy metal layer [2{4] or by non-local\nspin injection [5, 6]. SHNOs, which use the SHE in\na heavy metal layer, have been fabricated in a vari-\nety of device layouts, which all utilize the focusing of\ncharge current into a region with a lateral size of tens\nto hundreds of nanometers. This focusing is commonly\ndone via a nano-gap between two highly conductive elec-\ntrodes [3, 7, 8], with a nanoconstriction [9{13], or with\na nanowire [14, 15]. Most recently, nanoconstriction-\nSHNOs have attracted large interest, due to their rel-\native ease of fabrication, their direct optical access to\nthe magnetization oscillation area, and their potential for\nlarge scale and large distance synchronization of multiple\nSHNOs [16, 17].\nNanoconstriction-SHNOs consist of a bilayer of a fer-\nromagnetic free layer and a SHE inducing heavy metal\nlayer. Since the SHE and the concomitant spin accumu-\nlation at the bilayer interface are only in\ruenced by the\ncurrent density in the heavy metal layer, magnetization\noscillations of the device under a constant current can\nbe directly linked to the magnetodynamic properties of\nthe magnetic free layer. Until now, the variety of materi-\nals from which SHNOs has been fabricated is limited to a\nfew standards like permalloy (Py, Ni 80Fe20), (Co,Fe)B, or\nyttrium iron garnet (YIG). However, these materials are\ndi\u000berent from each other in every one of the key magneto-\ndynamic parameters, such as magnetization ( M), Gilbertdamping (\u000b), or exchange constant ( A).\nIn a recent study, we have shown how the magneto-\ndynamic properties of Py can be engineered by alloying\nwith the noble metals Pt, Au, and Ag [18]. While alloy-\ning with Pt leads to a large increase in damping but only\na small decrease in magnetization, alloying with Ag has\nonly a weak e\u000bect on the damping but reduces the mag-\nnetization relatively strongly. Co-alloying with both ele-\nments Pt and Ag thus results in Py 100\u0000x\u0000yPtxAgy\flms,\nwhoseMand\u000bcan be tuned independently, e.g. the\nmagnetization can be kept constant, while the damping\nis strongly increased with increasing Pt concentration.\nHere, we employ a series of alloyed Py 100\u0000x\u0000yPtxAgy\nthin \flms in nanoconstriction-SHNOs, where we vary the\ne\u000bective damping of the free layer by a factor of three,\nwhile we keep the magnetization of the \flms constant.\nBased on these \flms, we fabricate geometrically identical\nnanoconstriction-SHNOs and compare their microwave\nauto-oscillation characteristics. This allows us to directly\nanalyze the in\ruence of one single magnetodynamic prop-\nerty, namely the Gilbert damping, on the spectral char-\nacteristics, i.e. the onset current ( Ith\nDC), the output power\n(P), and the linewidth (\u0001 f).\nSPIN HALL NANO-OSCILLATOR DEVICES\nBilayers of 5 nm Py 100\u0000x\u0000yPtxAgyand 5 nm Pt were\nsputter-deposited onto sapphire substrates in a high-\nvacuum chamber with a base pressure of less than\n3\u000210\u00008Torr. The deposition was carried out with\n3 mTorr argon gas at a \row rate of 30 sccm. The alloyed\nlayers were co-sputtered from up to 3 targets, and the\nPy target power was kept constant at 350 W, while thearXiv:1802.05548v1  [cond-mat.mes-hall]  15 Feb 20182\n150 nm\n200 nm\n(a) (c)\n(b)\nHIPCurrent\nFIG. 1. (a) Schematic representation of the sputtered bilayer\nstructure. (b) SEM micrograph of a nanoconstriction-SHNO\nshowing the relative orientations of current and \feld. (c) Op-\ntical micrograph showing the microwave wave guide used for\ncontacting the SHNOs.\nnoble metal sputtering powers and the sputtering time\nwas adjusted for composition and thickness, respectively.\nThe top Pt layer was magnetron sputtered with a dc\npower of 50 W. The alloy compositions are Py 84Ag16\n(S01), Py 77:5Pt10Ag12:5(S02), Py 75Pt15Ag10(S03), and\nPy73Pt19Ag8(S04), chosen to result in a constant satura-\ntion magnetization throughout the series of SHNOs [18].\nDevices for electrical measurements were fabricated\nfrom these bilayers by electron beam lithography and ar-\ngon ion beam etching, using the negative resist as an etch-\ning mask. Nanoconstrictions were formed by two sym-\nmetrical indentations with a 50 nm tip radius into 4 µm\nwide stripes, see Fig. 1(b). The width of the nanocon-\nstrictions is 150 nm. Finally, 1 µm thick copper waveg-\nuides with a 150 µm pitch were fabricated by optical\nlithography and lift-o\u000b, see Fig. 1(c).\nFILM CHARACTERIZATION\nCharacterization of the extended bilayer samples was\nperformed by ferromagnetic resonance (FMR), and two-\npoint anisotropic magneotresistance (AMR) measure-\nments. The FMR was carried out with in-plane applied\n\felds using a NanOsc Instruments PhaseFMR with a\n200µm wide coplanar waveguide (CPW). An asymmet-\nric Lorentzian was \ft to the absorption peaks. The fre-\nquency dependence of the determined resonance \felds\nand linewidths was subsequently used to extract the ef-\nfective magnetization ( \u00160Me\u000b) and the damping param-\n0.0\n0.2\n0.4\n0.6\n0.8\nμ0M eff\nα\n0.02\n0.03\n0.04\n0.05\n0.06\nα  \n0\n5\n10\n15\n20\n0.3\n0.4\n0.5\nAMR (%)\nxPt(%)φ (°)\n0\n90\n180\n270\n360\n0.0\n0.1\n0.2\n0.3\n0.4\nMR (%)\nS01μ0Meff(T) μ0Meff = 0.617 TFIG. 2. (a) Magnetization and damping of the alloyed \flms\nin the bilayer as measured by CPW-based FMR. (b) AMR of\nthe extended layer structure. The inset shows the angular-\ndependent relative resistance of the Py 84Ag16/Pt (S01) bi-\nlayer, together with a \ft to a cos2-function.\neter (\u000b), respectively [18]. Figure 2(a) shows the two\nparameters, \u00160Me\u000band\u000b, as a function of Pt concen-\ntration. The magnetization is constant throughout the\nsample series ( \u00160Me\u000b= 0.617(34) T), while the damp-\ning increases linearly from 0 :023(1) to 0 :058(3) as the\nPt concentration increases from 0 (Py 84Ag16) to 19 %\n(Py73Pt19Ag8). The small layer thickness compared to\nthe \flms in Ref. 18 results in a slightly lower magnetiza-\ntion, whereas the damping is enhanced as a consequence\nof spin pumping into the adjacent Pt layer [19{21].\nThe AMR was determined by probing the resistance\nof 4µm wide stripes in a rotating 90 mT in-plane mag-\nnetic \feld. A representative AMR measurement is pre-\nsented in the inset of Fig. 2(b), together with a \ft of a\ncos2-function to the data. The angle '= 0\u000edenotes a\nperpendicular orientation between current and \feld, and\nthe AMR (Fig. 2(b)) is calculated by the di\u000berence in\nresistance at perpendicular and parallel alignments via\nAMR =Rk\u0000R?\nR?. The AMR is below 1 %, which is a re-\nsult of the majority of the current \rowing through the\nnonmagnetic platinum layer, which has a higher conduc-\ntivity than the Py alloys. The AMR reduces by \u001930 %\nacross the samples series, but the absolute resistance of\nthe bilayers decreases by less than 5 %. The AMR magni-\ntude is therefore most likely governed by the alloy compo-\nsition, since the amount of current in the magnetic layer\ndoes not change signi\fcantly.3\nf(GHz)\nCurrent (mA)\n2.2 2.4 2.6 2.8 3.0 3.2 3.45.96.06.1\n2468S01, H = 500 mT\n5.85 5.95 6.05 6.150246\nf(GHz)PSD (nV2/Hz)\nFIG. 3. Power spectral density (PSD) of the Py 84Ag16/Pt\n(S01) SHNO as a function of current in an external \feld of\n\u00160Hext= 0:5 T, tilted 80\u000eOOP. The inset shows the PSD at\nIDC= 3:26 mA and the solid line is a Lorentzian \ft resulting\nin \u0001f= 5:98 MHz and P= 1:02 pW.\nMICROWAVE EMISSION MEASUREMENTS\nAND DISCUSSION\nThe microwave measurements were conducted with the\ndevices placed in a magnetic \feld oriented at an out-of-\nplane (OOP) angle of 80\u000efrom the \flm plane, and an\nin-plane angle of '= 0\u000e. The in-plane component of the\nmagnetic \feld ( HIP\next) was thus perpendicular to the cur-\nrent \row direction, as sketched in Fig. 1(b). The relative\norientation of the current and HIP\nextyields a spin-torque\ncaused by the spin current from the Pt layer, which re-\nduces the damping in the Py layer and leads to auto-\noscillations for su\u000eciently large positive applied dc cur-\nrents (IDC) [22]. The current was applied to the samples\nvia the dc port of a bias-tee and the resulting microwave\nsignals from the devices were extracted from the rf port of\nthe bias-tee. The microwave signals were then ampli\fed\nby a broadband (0 :1 to 40 GHz) low-noise ampli\fer with\na gain of +32 dB before being recorded by a spectrum an-\nalyzer (Rohde&Schwarz FSV-40) with a resolution band-\nwidth of 500 kHz. All measurements were carried out at\nroom temperature.\nA typical microwave measurement of a Py 84Ag16/Pt\n(S01) device in a constant \feld of \u00160Hext= 0:5 T and\na varying current is displayed in Fig. 3. The peak fre-\nquency \frst decreases slightly after the oscillation onset\natIth\nDC= 2:26 mA, then reaches a minimum at \u00182:6 mA,\nand \fnally increases up to the maximum applied current\nof 3:4 mA. A Lorentzian peak function \fts well to the\nauto-oscillation signal, see inset of Fig. 3, allowing for de-\ntermination of the full-width at half-maximum linewidth\n(\u0001f) and the integrated output power ( P). Besides the\n10\n100\nΔf(MHz)\n5.9\n6.0\n6.1\nS01\nS02\nS03\nS04\nf(GHz)\n2.2\n2.4\n2.6\n2.8\n3.0\n3.2\n3.4\n0.1\n1\nPower (pW)\nI (mA)(a)\n(b)\n(c)FIG. 4. (a) Frequency, (b) linewidth, and (c) integrated power\nof the microwave auto-oscillations as a function of current for\nfour di\u000berent SHNOs with increasing damping. The applied\n\feld is\u00160Hext= 0:5 T, tilted 80\u000eOOP.\nhighly coherent auto-oscillation mode, no other modes\nare excited under these \feld conditions.\nFigure 4 shows the determined auto-oscillation char-\nacteristics of SHNOs with di\u000berent alloy composition\nand damping. The measurements were again made in\na constant \feld of 0 :5 T. The oscillation frequencies in\nFig. 4(a) lie around 6.0 \u00060.1 GHz for all samples, and the\ncurrent-frequency dependence is virtually identical above\nthe individual threshold currents. However, the current\nrange where fdecreases with IDCis missing for the S04\nsample, which suggests that the threshold current is un-\nderestimated for this device. The comparable frequencies\nof all samples con\frm that the saturation magnetization\nis constant throughout the alloy series. Furthermore, the\nquantitatively similar current tunability implies that the\nincreased damping does not change the fundamental na-\nture of the excited auto-oscillation mode.\nThe linewidth of the SHNOs decreases rapidly after the\nauto-oscillation onset and then levels o\u000b for higher IDC\nvalues, as shown in Fig. 4(b). This behavior is consistent\nwith previous studies on nanoconstriction-SHNOs made4\n300\n400\n500\n600\n700\n800\n2.0\n2.4\n2.8\n3.2\nS02\nS01\nS03\nS04\nIth\nDC(mA)\nField (mT)\nFIG. 5. Threshold current ( Ith\nDC) as a function of external\nmagnetic \feld for the four devices of this study.\nof permalloy \flms [11, 17]. The low-damping device S01\nreaches its minimum level at \u0001 f\u001811 MHz, while the\nSHNOs with higher damping materials all have a simi-\nlar minimum linewidth of \u0001 f\u00185 MHz. The linewidth\nis inversely proportional to the mode volume [23], and\nthe decrease in \u0001 fcan therefore be attributed to a spa-\ntial growth of the auto-oscillation region as the damping\nincreases. Nevertheless, the active area of the device is\ncon\fned to the nanoconstriction, which limits the reduc-\ntion in linewidth.\nThe output power of the four nanoconstriction-SHNOs\nis shown as a function of IDCin Fig. 4(c). The power\ngrows almost exponentially with increasing current for\nall samples. However, Pdrops dramatically as the Pt\nconcentration increases. The AMR also decreases in the\nhigher damping samples, but the reduction is too small\nto fully account for the drop in power. Together with the\ntrend in linewidth, the evolution of the power contradicts\nthe general assumption \u0001 f/\u000b=P [23{25]. This equa-\ntion is only valid in the vicinity of the threshold current\nand a direct comparison to the data is problematic, due\nto the experimental di\u000eculties of determining Ith\nDC. Still,\nthe direct relation between the intrinsic oscillator power\nand the electrically measured power is put into question\ndue to the remarkable decrease in the measured P. A\nnumber of factors could in\ruence the signal strength, e.g.\nrecti\fcation, spin-pumping, and the inverse spin-Hall ef-\nfect.\nThe onset current for auto-oscillations was determined\nby current scans for external \felds ranging between 0 :3 T\nand 0:8 T, and the results are shown in Fig. 5. The\n\feld dependence of Ith\nDCis parabola-like for all samples.\nThis kind of behavior has been predicted in a numerical\nstudy by Dvornik et al. [13]. The non-monotonic behav-\nior of threshold current as a function of applied \feld is\na result of a re-localization of the auto-oscillation mode\nand a corresponding change in the spin-transfer-torque\n(STT) e\u000eciency. In weak oblique magnetic \felds, the\nmode is of edge type and samples a signi\fcant portion of\nthe pure spin current, which is largest at the nanocon-striction edges due to the inhomogeneous current den-\nsity. When the \feld strength increases, the mode shows\nan even stronger localization towards the region of the\nhigher current density. Thereby, the STT e\u000eciency in-\ncreases and the threshold current drops. When the \feld\nstrength increases further, the mode detaches from the\nedges and eventually transforms to the bulk type. As\nthis transformation gradually reduces the spatial corre-\nlation between the spin current density and the location\nof the mode, the STT e\u000eciency drops and the threshold\ncurrent increases. The lower \feld tunability of Ith\nDCof\nthe high damping samples imply an initially larger mode\nvolume, which also was suggested by the evolution of the\nlinewidth.\nThe \feld and current range with detectable auto-\noscillations is strongly dependent on \u000b. The threshold\ncurrent should increase linearly with damping [13] and\nthe minimum Ith\nDCindeed scales with \u000b. The enhance-\nment is smaller than predicted (a factor of three), which\nindicates that the increase in damping is accompanied\nwith a higher STT e\u000eciency. A possible reason for the\nimproved e\u000eciency is a larger SHE through a more trans-\nparent interface for alloyed \flms. The origin of the ob-\nserved damping dependence of the threshold \feld is un-\nclear at this stage, calling for a closer inspection of the\nimpact of the applied \feld on the spectral characteristics.\nThus, a further investigation of our devices is targeted\ntowards the microwave emission as a function of \feld\nwith a constant IDC= 3.2 mA, i.e. above or at the pre-\nviously measured auto-oscillation threshold for all \felds.\nWhile the peak frequencies are virtually identical for all\nthe samples, see Fig. 6(a), the varied damping manifests\nin a clear pattern in Pand \u0001f. The microwave power,\nshown in Fig. 6(c), \frstly increases for all samples with\nincreasing \feld, peaks for an intermediate \feld, and \f-\nnally drops relatively sharp until a point where no more\noscillations are detectable. An opposite behavior can be\nseen for \u0001f, which shows a minimum for intermediate\n\felds. The \feld at which the SHNOs emit their maxi-\nmum output power decreases monotonically from 0.64 T\nto 0.4 T with increasing damping. The same trend is\nvisible for the point of minimum linewidth, which de-\ncreases with increased damping from 0.71 T to 0.49 T,\nand is therefore at a typically \u00180.1 T larger \feld than the\nrespective maximum power. The lowest overall linewidth\ncan be achieved for the lowest damping SHNO (S01) at\nhigh \felds, where only this device still shows a detectable\nsignal, i.e., \u0001 f= 1:2 MHz at\u00160Hext= 0:71 T. How-\never, at low applied \felds \u00160Hext\u00140:48 T a clear trend\nis noticeable towards smaller linewidths for the alloyed\npermalloy \flms with larger damping.\nIn light of this inverse trend, we can argue that auto-\noscillations in nanoconstriction-SHNO should also be de-\nscribed in the framework of non-linear auto-oscillators,\nalthough the study in Ref. 13 has shown that oscilla-\ntions in nanoconstriction-SHNOs emerge from a local-5\nΔf(MHz)\n(b)\n(c)\n4\n6\n8\n10\nS01\nS02S03\nS04\n1\n10\n100\n300\n400\n500\n600\n700\n800\n0.1\n1\nPower (pW)\nField (mT)(a)f(GHz)\nFIG. 6. (a) Frequency, (b) linewidth, and (c) integrated power\nof the auto-oscillations as a function of the applied external\nmagnetic \feld at a constant drive current IDC= 3:2 mA.\nized linear mode. The generation linewidth of nanocon-\ntact spin torque oscillators, which are a prime example\nof non-linear auto-oscillators, has been studied analyti-\ncally [23, 26] and experimentally [27]. The linewidth as\na function of current and magnetic \feld angle was shown\nto follow the expression:\n\u0001f=\u00000\n2\u0019\u0012kBT\nE0\u0013\"\n1 +\u0012N\n\u0000e\u000b\u00132#\n; (1)\nwherekB,T, andE0(IDC=Ith\nDC) are the Boltzmann con-\nstant, temperature and the average oscillator energy, re-\nspectively. Nis the nonlinear frequency shift, a material\nproperty that is determined by the internal magnetic \feld\nand the magnetization [28]. \u0000 e\u000bis the e\u000bective nonlinear\ndamping rate and \u0000 0is the positive damping rate, and\nboth have an explicit linear dependence on the Gilbert\ndamping\u000b[23]. Assuming everything else equal amongst\nour devices, a decrease of the linewidth with \u000bcan be\nthus expected, when the second term in the brackets in\nEq. 1 dominates. This is likely for low to intermediate\felds, since Ncan be calculated to take up the largest\nvalues under these conditions [28], which are thus in ac-\ncordance with the range of \felds, where we observe the\ndiscussed linewidth vs. damping behavior in our devices.\nCONCLUSIONS\nIn conclusion, we have fabricated a series of sam-\nples where the magnetization is constant, while the\nspin wave damping is varied by a factor of three. We\nhave shown that the damping of the magnetic layer in\nnanoconstriction-SHNOs has an important in\ruence on\nall the spectral characteristics of the devices. The re-\nsults of our study will encourage the application of tai-\nlored materials in SHNOs and can be used for a further\nunderstanding of the magnetodynamics in nanodevices,\ne.g. the coupling mechanisms in mutually synchronized\nSHNOs.\nACKNOWLEDGMENTS\nWe acknowledge \fnancial support from the China\nScholarship Council (CSC), the G oran Gustafsson\nFoundation, the Swedish Research Council (VR), the\nKnut and Alice Wallenberg Foundation (KAW), and\nthe Swedish Foundation for Strategic Research (SSF).\nThis work was also supported by the European Re-\nsearch Council (ERC) under the European Communitys\nSeventh Framework Programme (FP/2007-2013)/ERC\nGrant 307144 \\MUSTANG\".\n\u0003yuri@seu.edu.cn\n[1] T. Chen, R. K. Dumas, A. Eklund, P. K. Muduli,\nA. Houshang, A. A. Awad, P. D urrenfeld, B. G. Malm,\nA. Rusu, and J. \u0017Akerman, \\Spin-Torque and Spin-Hall\nNano-Oscillators,\" Proc. IEEE 104, 1919 (2016).\n[2] L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and\nR. A. 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B 76, 024437 (2007)."    },    {        "title": "1711.06759v2.Shot_noise_of_charge_and_spin_transport_in_a_junction_with_a_precessing_molecular_spin.pdf",        "content": "Shot noise of charge and spin transport in a junction with a precessing molecular spin\nMilena Filipović and Wolfgang Belzig\nFachbereich Physik, Universität Konstanz, D-78457 Konstanz, Germany\n(Dated: April 6, 2018)\nMagnetic molecules and nanomagnets can be used to influence the electronic transport in meso-\nscopic junction. In a magnetic field the precessional motion leads to resonances in the dc- and\nac-transport properties of a nanocontact, in which the electrons are coupled to the precession.\nQuantities such as the dc conductance or the ac response provide valuable information, such as\nthe level structure and the coupling parameters. Here, we address the current-noise properties\nof such contacts. This encompasses the charge current and spin-torque shot noise, which both\nshow a steplike behavior as functions of bias voltage and magnetic field. The charge-current noise\nshows pronounced dips around the steps, which we trace back to interference effects of electrons\nin quasienergy levels coupled by the molecular spin precession. We show that some components\nof the noise of the spin-torque currents are directly related to the Gilbert damping and hence are\nexperimentally accessible. Our results show that the noise characteristics allow us to investigate in\nmore detail the coherence of spin transport in contacts containing magnetic molecules.\nI. INTRODUCTION\nShot noise of charge current has become an active re-\nsearchtopicinrecentdecades, sinceitenablestheinvesti-\ngation of microscopic transport properties, which cannot\nbe obtained from the charge current or conductance.1\nIt has been demonstrated that spin-flip induced fluctua-\ntions in diffusive conductors connected to ferromagnetic\nleads enhance the noise power, approaching the Pois-\nsonian value.2,3Accordingly, the Fano factor defined as\nF=S(0)=ejIj, which describes the deviation of the shot\nnoise from the average charge current, equals 1 in this\ncase. On the other hand, it has been shown that shot\nnoise in a ferromagnet-quantum-dot-ferromagnet system\nwith antiparallel magnetization alignments can be sup-\npressed due to spin flip, with F < 1=2.4\nThe quantum-interference phenomenon, which is a\nmanifestation of the wave nature of electrons, has at-\ntracted a lot of attention. The quantum-interference ef-\nfects occur between coherent electron waves in nanoscale\njunctions.5Quantum interference in molecular junc-\ntions influences their electronic properties.6–10The Fano\neffect11due to the interference between a discrete state\nand the continuum has an important role in investigation\nof the interference effects in nanojunctions, which behave\nin an analogous way, and are manifested in the conduc-\ntance or noise spectra.5,12,13Particularly interesting ex-\namples involve spin-flip processes, such as in the presence\nof Rashba spin-orbit interaction,14,15a rotating magnetic\nfield,16or in the case of the magnetotransport.17–19\nIn the domain of spin transport it is interesting to\ninvestigate the noise properties, as the discrete na-\nture of electron spin leads to the correlations between\nspin-carrying particles. The spin current is usually\na nonconserved quantity that is difficult to measure,\nand its shot noise depends on spin-flip processes lead-\ning to spin-current correlations with opposite spins.20–22\nThe investigation of the spin-dependent scattering, spin\naccumulation,23and attractive or repulsive interactions\nin mesoscopic systems can be obtained using the shotnoise of spin current,24as well as measuring the spin re-\nlaxation time.20,24Even in the absence of charge cur-\nrent, a nonzero spin current and its noise can still\nemerge.22,25,26Several works have studied the shot noise\nof a spin current using, e.g., the nonequilibrium Green’s\nfunctions method and scattering matrix theory.22,27–29\nIt was demonstrated that the magnetization noise\noriginates from transferred spin current noise via\na fluctuating spin-transfer torque in ferromagnetic-\nnormal-ferromagnetic systems,30and magnetic tunnel\njunctions.31Experimentally, Spin Hall noise measur-\nments have been demonstrated,32and in a similar fash-\nion the spin-current shot noise due to magnon currents\ncan be related to the nonquantized spin of interact-\ning magnons in ferri-, ferro-, and antiferromagnets.33,34\nQuantum noise generated from the scatterings between\nthe magnetization of a nanomagnet and spin-polarized\nelectrons has been studied theoretically as well.35,36The\nshot noise of spin-transfer torque was studied recently\nusing a magnetic quantum dot connected to two non-\ncollinear magnetic contacts.29According to the defini-\ntion of the spin-transfer torque,37,38both autocorrela-\ntions and cross-correlations of the spin-current compo-\nnents contribute to the spin-torque noise.\nIn this article, we study theoretically the noise of\ncharge and spin currents and spin-transfer torque in a\njunction connected to two normal metallic leads. The\ntransport occurs via a single electron energy level inter-\nacting with a molecular magnet in a constant magnetic\nfield. The spin of the molecular magnet precesses around\nthe magnetic field with the Larmor frequency, which is\nkept undamped, e.g., due to external driving. The elec-\ntronic level may belong to a neighboring quantum dot or\nit may be an orbital of the molecular magnet itself. The\nelectroniclevelandthemolecularspinarecoupledviaex-\nchange interaction. We derive expressions for the noise\ncomponents using the Keldysh nonequilibrium Green’s\nfunctions formalism.39–41The noise of charge current is\ncontributed by both elastic processes driven by the bias\nvoltage, and inelastic tunneling processes driven by thearXiv:1711.06759v2  [cond-mat.mes-hall]  5 Apr 20182\nµLΓLΓRBS(t)µRJgµBevacRcos(Ωt+φR)∼eVµR0s(t)\nFIG. 1. (Color online) Tunneling through a single molecular\nlevel with energy \u000f0in the presence of a precessing molecular\nspin~S(t)in a constant magnetic field ~B, connected to two\nmetallic leads with chemical potentials \u0016\u0018,\u0018=L;R. The\nmolecular level is coupled to the spin of the molecule via ex-\nchange interaction with the coupling constant J. The applied\ndc-bias voltage eV=\u0016L\u0000\u0016R, and the tunnel rates are \u0000\u0018.\nmolecular spin precession. We observe diplike features\nin the shot noise due to inelastic tunneling processes and\ndestructive quantum interference between electron trans-\nport channels involved in the spin-flip processes. The\ndriving mechanism of the correlations of the spin-torque\ncomponents in the same spatial direction involves both\nprecession of the molecular spin and the bias voltage.\nHence, they are contributed by elastic and inelastic pro-\ncesses, with the change of energy equal to one or two Lar-\nmor frequencies. The nonzero correlations of the perpen-\ndicular spin-torque components are driven by the molec-\nular spin precession, with contributions of spin-flip tun-\nneling processes only. These components are related to\nthe previously obtained Gilbert damping coefficient,42,43\nwhich characterize the Gilbert damping term of the spin-\ntransfer torque,44–46at arbitrary temperature.\nThe article is organized as follows. The model and\ntheoretical framework based on the Keldysh nonequi-\nlibrium Green’s functions formalism39–41are given in\nSec. II. Here we derive expressions for the noise of spin\nand charge currents. In Sec. III we investigate and an-\nalyze the properties of the charge-current shot noise. In\nSec. IV we derive and analyze the noise of spin-transfer\ntorque. The conclusions are given in Sec. V.\nII. MODEL AND THEORETICAL\nFRAMEWORK\nThe junction under consideration consists of a nonin-\nteracting single-level quantum dot in the presence of a\nprecessing molecular spin in a magnetic field along the\nz-axis,~B=B~ ez, coupled to two noninteracting leads\n(Fig. 1). The junction is described by the Hamiltonian\n^H(t) =X\n\u00182fL;Rg^H\u0018+^HT+^HD(t) +^HS;(1)where\n^H\u0018=X\nk;\u001b\u000fk\u0018^cy\nk\u001b\u0018^ck\u001b\u0018 (2)\nis the Hamiltonian of contact \u0018=L;R. The spin-(up or\ndown) state of the electrons is denoted by the subscript\n\u001b=\";#= 1;2 =\u00061. The tunnel coupling between the\nquantum dot and the leads reads\n^HT=X\nk;\u001b;\u0018[Vk\u0018^cy\nk\u001b\u0018^d\u001b+V\u0003\nk\u0018^dy\n\u001b^ck\u001b\u0018];(3)\nwith spin-independent matrix element Vk\u0018. The creation\n(annihilation) operators of the electrons in the leads and\nthequantumdotaregivenby ^cy\nk\u001b\u0018(^ck\u001b\u0018)and^dy\n\u001b(^d\u001b). The\nHamiltonian of the electronic level equals\n^HD(t) =X\n\u001b\u000f0^dy\n\u001b^d\u001b+g\u0016B^~ s~B+J^~ s~S(t):(4)\nThe first term in Eq. (4) is the Hamiltonian of the non-\ninteracting single-level quantum dot with energy \u000f0. The\nsecond term describes the electronic spin in the dot,\n^~ s= (~=2)P\n\u001b\u001b0(~ \u001b)\u001b\u001b0^dy\n\u001b^d\u001b0, in the presence of a constant\nmagnetic field ~B, and the third term represents the ex-\nchange interaction between the electronic spin and the\nmolecular spin ~S(t). The vector of the Pauli matrices is\ngiven by ^~ \u001b= (^\u001bx;^\u001by;^\u001bz)T. The g-factor of the electron\nand the Bohr magneton are gand\u0016B, whereasJis the\nexchange coupling constant between the electronic and\nmolecular spins.\nThe last term of Eq. (1) can be written as\n^HS=g\u0016B~S~B; (5)\nand represents the energy of the molecular spin ~Sin\nthe magnetic field ~B. We assume that j~Sj\u001d~and ne-\nglecting quantum fluctuations treat ~Sas a classical vari-\nable. The magnetic field ~Bgenerates a torque on the\nspin~Sthat causes the spin to precess around the field\naxis with Larmor frequency !L=g\u0016BB=~. The dy-\nnamics of the molecular spin is kept constant, which\ncan be realized, e.g., by external rf fields47to cancel\nthe loss of magnetic energy due to the interaction with\nthe itinerant electrons. Thus, the precessing spin ~S(t)\npumps spin currents into the leads, but its dynamics\nremains unaffected by the spin currents, i.e., the spin-\ntransfer torque exerted on the molecular spin is compen-\nsated by the above mentioned external means. The un-\ndamped precessional motion of the molecular spin, sup-\nported by the external sources, is then given by ~S(t) =\nS?cos(!Lt)~ ex+S?sin(!Lt)~ ey+Sz~ ez, with\u0012the tilt an-\ngle between ~Band~S, andS?=Ssin(\u0012)the magnitude\nof the instantaneous projection of ~S(t)onto thexyplane.\nThe component of the molecular spin along the field axis\nequalsSz=Scos(\u0012).3\nThe charge- and spin-current operators of the lead \u0018\nare given by the Heisenberg equation39,40\n^I\u0018\u0017(t) =q\u0017d^N\u0018\u0017\ndt=q\u0017i\n~[^H;^N\u0018\u0017]; (6)\nwhere [;]denotes the commutator, while ^NL\u0017=P\nk;\u001b;\u001b0^cy\nk\u001bL(\u001b\u0017)\u001b\u001b0^ck\u001b0Lis the charge ( \u0017= 0andq0=\n\u0000e) and spin ( \u0017=x;y;zandq\u00176=0=~=2) occupation\nnumber operator of the contact \u0018. Here ^\u001b0=^1is the\nidentity matrix. Taking into account that only the tun-\nneling Hamiltonian ^HTgenerates a nonzero commutator\nin Eq. (6), the current operator ^I\u0018\u0017(t)can be expressed\nas\n^I\u0018\u0017(t) =\u0000q\u0017i\n~X\n\u001b;\u001b0(\u001b\u0017)\u001b\u001b0^I\u0018;\u001b\u001b0(t); (7)\nwhere the operator component ^I\u0018;\u001b\u001b0(t)equals\n^I\u0018;\u001b\u001b0(t) =X\nk[Vk\u0018^cy\nk\u001b\u0018(t)^d\u001b0(t)\u0000V\u0003\nk\u0018^dy\n\u001b(t)^ck\u001b0\u0018(t)]:(8)\nThe nonsymmetrized noise of charge and spin current\nis defined as the correlation between fluctuations of cur-\nrentsI\u0018\u0017andI\u0010\u0016,1,40\nS\u0017\u0016\n\u0018\u0010(t;t0) =h\u000e^I\u0018\u0017(t)\u000e^I\u0010\u0016(t0)i; (9)\nwith\u0017=\u0016= 0for the charge-current noise. The fluctu-\nation operator of the charge and spin current in lead \u0018is\ngiven by\n\u000e^I\u0018\u0017(t) =^I\u0018\u0017(t)\u0000h^I\u0018\u0017(t)i: (10)\nUsing Eqs. (7) and (10), the noise becomes\nS\u0017\u0016\n\u0018\u0010(t;t0) =\u0000q\u0017q\u0016\n~2X\n\u001b\u001b0X\n\u0015\u0011(\u001b\u0017)\u001b\u001b0(\u001b\u0016)\u0015\u0011S\u001b\u001b0;\u0015\u0011\n\u0018\u0010(t;t0);\n(11)\nwhereS\u001b\u001b0;\u0015\u0011\n\u0018\u0010(t;t0) =h\u000e^I\u0018;\u001b\u001b0(t)\u000e^I\u0010;\u0015\u0011(t0)i. The for-\nmal expression for S\u0017\u0016\n\u0018\u0010(t;t0)is given by Eq. (A10) in\nthe Appendix, where it is obtained using Eq. (11) and\nEqs. (A1)–(A9).\nUsing Fourier transformations of the central-region\nGreen’s functions given by Eqs. (A6)–(A8) and self-\nenergies in the wide-band limit, the correlations given\nby Eq. (A9) can be further simplified. Some correla-\ntion functions are not just functions of time difference\nt\u0000t0. Thus, as in Ref. 48, we used Wigner representa-\ntion assuming that in experiments fluctuations are mea-\nsured on timescales much larger than the driving pe-\nriodT= 2\u0019=!L, which is the period of one molecular\nspin precession. The Wigner coordinates are given by\nT0= (t+t0)=2and\u001c=t\u0000t0, while the correlation func-\ntions are defined as\nS\u001b\u001b0;\u0015\u0011\n\u0018\u0010(\u001c) =1\nTZT\n0dth\u000e^I\u0018;\u001b\u001b0(t+\u001c)\u000e^I\u0010;\u0015\u0011(t)i:(12)The Fourier transform of S\u001b\u001b0;\u0015\u0011\n\u0018\u0010(\u001c)is given by\nS\u001b\u001b0;\u0015\u0011\n\u0018\u0010(\n;\n0) = 2\u0019\u000e(\n\u0000\n0)S\u001b\u001b0;\u0015\u0011\n\u0018\u0010(\n);(13)\nwhere\nS\u001b\u001b0;\u0015\u0011\n\u0018\u0010(\n) =Z\nd\u001cei\n\u001cS\u001b\u001b0;\u0015\u0011\n\u0018\u0010(\u001c):(14)\nFor the correlations which depend only on t\u0000t0, the\nWigner representation is identical to the standard repre-\nsentation.\nThe symmetrized noise of charge and spin currents\nreads1,40\nS\u0017\u0016\n\u0018\u0010S(t;t0) =1\n2hf\u000e^I\u0018\u0017(t);\u000e^I\u0010\u0016(t0)gi;(15)\nwheref;gdenotes the anticommutator. According to\nEqs. (11), (12), (14), and (15), in the Wigner represen-\ntation the nonsymmetrized noise spectrum reads\nS\u0017\u0016\n\u0018\u0010(\n) =Z\nd\u001cei\n\u001cS\u0017\u0016\n\u0018\u0010(\u001c)\n=Z\nd\u001cei\n\u001c1\nTZT\n0dth\u000e^I\u0018\u0017(t+\u001c)\u000e^I\u0010\u0016(t)i\n=\u0000q\u0017q\u0016\n~2X\n\u001b\u001b0X\n\u0015\u0011(\u001b\u0017)\u001b\u001b0(\u001b\u0016)\u0015\u0011S\u001b\u001b0;\u0015\u0011\n\u0018\u0010(\n);(16)\nwhile the symmetrized noise spectrum equals\nS\u0017\u0016\n\u0018\u0010S(\n) =1\n2[S\u0017\u0016\n\u0018\u0010(\n) +S\u0016\u0017\n\u0010\u0018(\u0000\n)]\n=\u0000q\u0017q\u0016\n2~2X\n\u001b\u001b0X\n\u0015\u0011(\u001b\u0017)\u001b\u001b0(\u001b\u0016)\u0015\u0011S\u001b\u001b0;\u0015\u0011\n\u0018\u0010S(\n);\n(17)\nwhereS\u001b\u001b0;\u0015\u0011\n\u0018\u0010S(\n) = [S\u001b\u001b0;\u0015\u0011\n\u0018\u0010(\n) +S\u0015\u0011;\u001b\u001b0\n\u0010\u0018(\u0000\n)]=2. The\nexperimentallymosteasilyaccessiblequantityisthezero-\nfrequency noise power.\nIII. SHOT NOISE OF CHARGE CURRENT\nFor the charge-current noise, it is convenient to drop\nthe superscripts \u0017=\u0016= 0. The charge-current noise\nspectrum can be obtained as24\nS\u0018\u0010(\n) =\u0000e2\n~2[S11;11\n\u0018\u0010+S11;22\n\u0018\u0010+S22;11\n\u0018\u0010+S22;22\n\u0018\u0010](\n):(18)\nInthissection, weanalyzethezero-frequencynoisepower\nof the charge current S\u0018\u0010=S\u0018\u0010(0)at zero temperature.\nTakingintoaccountthatthermalnoisedisappearsatzero\ntemperature, the only contribution to the charge-current\nnoise comes from the shot noise. The tunnel couplings\nbetween the molecular orbital and the leads, \u0000\u0018(\u000f) =\n2\u0019P\nkjVk\u0018j2\u000e(\u000f\u0000\u000fk\u0018), are considered symmetric and in\nthe wide-band limit \u0000L= \u0000R= \u0000=2.4\n-4-2024-2-1012\neV@e0DIL@10-2e0DHaL\nmL,R=e1±eV2mL,R=e0±eV2mL,R=±eV2mL,R=eV,0\n4202401\neVΕ0102Ε0b\nΜL,RΕ1eV2ΜL,RΕ0eV2ΜL,ReV2ΜL,ReV, 0SLL\nFIG. 2. (Color online) (a) Charge current ILand (b) auto-correlation shot noise SLLas functions of bias-voltage eV. All\nplots are obtained at zero temperature, with ~B=B~ ez. The other parameters are \u0000L= \u0000R= \u0000=2,\u0000 = 0:05\u000f0,!L= 0:5\u000f0,\nJ= 0:01\u000f0,S= 100, and\u0012=\u0019=2. The molecular quasienergy levels are located at \u000f1= 0:25\u000f0,\u000f2= 0:75\u000f0,\u000f3= 1:25\u000f0, and\n\u000f4= 1:75\u000f0.\nThe average charge current from lead \u0018can be ex-\npressed as\nI\u0018=e\u0000\u0018\u0000\u0010\n~Zd\u000f\n2\u0019[f\u0018(\u000f)\u0000f\u0010(\u000f)]\n\u0002X\n\u001b\u001b0\n\u001b6=\u001b0jG0r\n\u001b\u001b(\u000f)j2[1 +\r2jG0r\n\u001b0\u001b0(\u000f+\u001b0!L)j2]\nj1\u0000\r2G0r\u001b\u001b(\u000f)G0r\n\u001b0\u001b0(\u000f+\u001b0!L)j2;(19)\nwhere\u00186=\u0010, whileG0r\n\u001b\u001b(\u000f)are matrix elements of\n^G0r(\u000f) = [\u000f\u0000\u000f0+iP\n\u0018\u0000\u0018=2\u0000^\u001bz(g\u0016BB+JSz)=2]\u00001.49,50\nIn the above expression, f\u0018(\u000f) = [e(\u000f\u0000\u0016\u0018)=kBT+ 1]\u00001is\nthe Fermi-Dirac distribution of the electrons in lead \u0018,\nwithkBthe Boltzmann constant and Tthe tempera-\nture. The conservation of the charge current implies that\nSLL(0) +SLR(0) = 0. Thus, it is sufficient to study only\none correlation function.\nTunning the parameters in the system such as the bias\nvoltageeV=\u0016L\u0000\u0016R(where\u0016Land\u0016Rare the chemical\npotentials of the leads), ~B, and the tilt angle \u0012, the shot\nnoise can be controlled and minimized. The shot noise\nin the small precession frequency limit !L\u001ckBTis in\nagreement with Ref. 22 for eV= 0.\nIn Fig. 2(a) we present the average charge current as\na staircase function of bias voltage, where the bias is\nvaried in four different ways. In the presence of the ex-\nternal magnetic field and the precessing molecular spin,\nthe initially degenerate electronic level with energy \u000f0\nresults in four nondegenerate transport channels, which\nhas an important influence on the noise. Each step corre-\nsponds to a new available transport channel. The trans-\nport channels are located at the Floquet quasienergies43\n\u000f1=\u000f0\u0000(!L=2)\u0000(JS=2),\u000f2=\u000f0+ (!L=2)\u0000(JS=2),\n\u000f3=\u000f0\u0000(!L=2)+(JS=2), and\u000f4=\u000f0+(!L=2)+(JS=2),\nwhich are calculated using the Floquet theorem.16,51–54\nThe correlated current fluctuations give nonzero noise\npower, which is presented in Fig. 2(b). The noise power\nshowsthemolecularquasienergyspectrum, andeachstepor diplike feature in the noise denotes the energy of a new\navailable transport channel. The noise has two steps and\ntwo diplike features that correspond to these resonances.\nCharge current and noise power are saturated for large\nbias voltages. If the Fermi levels of the leads lie below the\nresonances, the shot noise approaches zero for eV!0\n[red and dashed pink lines in Fig. 2(b)]. This is due to\nthe fact that a small number of electron states can par-\nticipate in transport inside this small bias window and\nboth current and noise are close to 0. If the bias voltage\nis varied with respect to the resonant energy \u000f1such that\n\u0016L;R=\u000f1\u0006eV=2[dot-dashed blue line in Fig. 2(b)], or\nwith respect to \u000f0such that\u0016L;R=\u000f0\u0006eV=2[green line\nin Fig. 2(b)], we observe a valley at zero bias eV= 0,\nwhich corresponds to \u0016L=\u0016R=\u000f1in the first case, and\nnonzero noise in the second case. For eV= 0, the charge\ncurrent is zero, but the precession-assisted inelastic pro-\ncesses involving the absorption of an energy quantum !L\ngive rise to the noise here.\nAt small bias voltage, the Fano factor F=SLL=ejILj\nis inversely proportional to eVand hence diverges as\neV!0, indicating that the noise is super-Poissonian,\nas depicted in Fig. 3. Due to absorption (emission)\nprocesses16and quantum interference effects, the Fano\nfactor is a deformed steplike function, where each step\ncorresponds to a resonance. As the bias voltage is in-\ncreased, the noise is enhanced since the number of the\ncorrelated electron pairs increases with the increase of\nthe Fermi level. For larger bias, due to the absorption\nand emission of an energy quantum !L, electrons can\njump to a level with higher energy or lower level during\nthe transport, and the Fano factor F < 1indicates the\nsub-Poissonian noise. Around the resonances \u0016L;R=\u000fi,\ni= 1;2;3;4, the probability of transmission is very high,\nresulting in the small Fano factor. Elastic tunneling con-\ntributes to the sub-Poissonian Fano factor around the\nresonances and competes with the spin-flip events caused\nby the molecular spin precession. However, if the reso-5\n4 2 0 2 40.40.60.81.01.21.4\neV Ε0ΜL,RΕ1eV2ΜL,RΕ0eV2ΜL,ReV2ΜL,ReV,0F\nFIG. 3. (Color online) Fano factor Fas a function of bias-\nvoltageeV. All plots are obtained at zero temperature, with\n~B=B~ ez. The other parameters are set to \u0000 = 0:05\u000f0,\u0000L=\n\u0000R= \u0000=2,!L= 0:5\u000f0,J= 0:01\u000f0,S= 100, and\u0012=\u0019=2.\nThe positions of the molecular quasienergy levels are \u000f1=\n0:25\u000f0,\u000f2= 0:75\u000f0,\u000f3= 1:25\u000f0, and\u000f4= 1:75\u000f0.\n0.0 0.5 1.0 1.5 2.0 2.5 3.01\nΩLΕ0102Ε0\nΘ 2Θ 4Θ 6Θ 15SLL\n0\nFIG. 4. (Color online) Shot noise of charge current SLLas a\nfunction of the Larmor frequency !Lfor different tilt angles\n\u0012, with~B=B~ ez, at zero temperature. The other parameters\nare\u0000 = 0:05\u000f0,\u0000L= \u0000R= \u0000=2,\u0016L= 0:75\u000f0,\u0016R= 0:25\u000f0,\nJ= 0:01\u000f0, andS= 100. For!L=\u0016L\u0000\u0016R, we observe a\ndip due to destructive quantum interference.\nnant quasienergy levels are much higher than the Fermi\nenergyoftheleads, theprobabilityoftransmissionisvery\nlow and the Fano factor is close to 1, as shown in Fig. 3\n(red line). This means that the stochastic processes are\nuncorrelated. If the two levels connected with the inelas-\ntic photon emission (absorption) tunnel processes, or all\nfour levels, lie between the Fermi levels of the leads, the\nFano factor approaches 1/2, which is in agreement with\nRef. 55. For eV=\u000f3[see Fig. 3 (red line)] a spin-down\nelectron can tunnel elastically, or inelastically in a spin-\nflip process, leading to the increase of the Fano factor.\nSpin-flip processes increase the electron traveling time,\nleading to sub-Poissonian noise. Similarly, the Pauli ex-\nclusionprincipleisknowntoleadtosub-Poissoniannoise,\nsince it prevents the double occupancy of a level.\n0.0 0.5 1.0 1.5 2.012\nΜΕ0102Ε0ΜΜ LΜR\n 3Ε0 0.25Ε0 0.05Ε0SLL\n0FIG. 5. (Color online) Shot noise of charge current SLLas\na function of the chemical potential of the leads \u0016=\u0016L=\n\u0016R, with~B=B~ ez, for three different couplings \u0000, where\n\u0000L= \u0000R= \u0000=2, at zero temperature. The other parameters\nare!L= 0:5\u000f0,J= 0:01\u000f0,S= 100, and\u0012=\u0019=2. The\nmolecular quasienergy levels are positioned at \u000f1= 0:25\u000f0,\n\u000f2= 0:75\u000f0,\u000f3= 1:25\u000f0, and\u000f4= 1:75\u000f0.\nThe precessing molecular spin induces quantum inter-\nference between the transport channels connected with\nspin-flip events and the change of energy by one energy\nquantum!L, i.e., between levels with energies \u000f1and\n\u000f2=\u000f1+!L, or\u000f3and\u000f4=\u000f3+!L. The destruc-\ntive quantum-interference effects manifest themselves in\nthe form of diplike features in Fig. 2(b). When one or\nboth pairs of the levels connected with spin-flip events\nenter the bias-voltage window, then an electron from the\nleft lead can tunnel through both levels via elastic or in-\nelastic spin-flip processes. Different tunneling pathways\nending in the final state with the same energy destruc-\ntively interfere, similarly as in the Fano effect.11Namely,\nthe state with lower energy \u000f1(or\u000f3) mimics the discrete\nstate in the Fano effect. An electron tunnels into the\nstate\u000f1(or\u000f3), undergoes a spin flip and absorbs an en-\nergy quantum !L. The other state with energy \u000f2(or\u000f4)\nis an analog of the continuum in the Fano effect, and the\nelectron tunnels elastically through this level. These two\ntunneling processes (one elastic and the other inelastic)\ninterfere, leading to a diplike feature in the noise power.\nIf we vary, for instance, the bias-voltage as eV=\u0016L,\nwhere\u0016R= 0[Fig. 2(b), red line], we observe diplike\nfeatures for eV=\u000f2andeV=\u000f4.\nThe destructive interference effect is also presented in\nFig. 4, where noise power SLLis depicted as a function\nof!L. Here, we observe a dip due to the quantum-\ninterference effect around !L= 0:5\u000f0, which corresponds\nto\u0016L=\u000f2and\u0016R=\u000f1. The other two steps in Fig. 4\noccur when the Fermi energy of the right or left lead\nis in resonance with one of the quasienergy levels. The\nmagnitude of the precessing component of the molecular\nspin, which induces spin-flip processes between molecu-\nlar quasienergy levels, equals JSsin(\u0012)=2. Therefore, the\ndip increases with the increase of the tilt angle \u0012, and it6\nis maximal and distinct for \u0012=\u0019=2.\nFinally, in Fig. 5 we plotted the noise power of charge\ncurrentSLLas a function of \u0016=\u0016L=\u0016Rat zero tem-\nperature. It shows a nonmonotonic dependence on the\ntunneling rates \u0000. For small \u0000(Fig. 5, red line) the noise\nis increased if \u0016is positioned between levels connected\nwith spin-flip events, and is contributed only by absorp-\ntion processes of an energy quantum !Las we vary the\nchemical potentials. For larger \u0000(Fig. 5, green line), the\ncharge-current noise is increased since levels broaden and\noverlap, and more electrons can tunnel. With further in-\ncrease of \u0000(Fig. 5, dotted blue line) the noise starts to\ndecrease, and it is finally suppressed for \u0000\u001d!Lsince\na current-carrying electron sees the molecular spin as\nnearly static in this case, leading to a reduction of the\ninelastic spin-flip processes.\nIV. SHOT NOISE OF SPIN CURRENT AND\nSPIN-TRANSFER TORQUE\nIn this section we present the spin-current noise spec-\ntrum components and relations between them. Later we\nintroduce the noise of spin-transfer torque and we inves-\ntigate the zero-frequency spin-torque shot noise at zero\ntemperature. The components of the nonsymmetrized\nspin-current noise spectrum read\nSxx\n\u0018\u0010(\n) =\u00001\n4[S12;21\n\u0018\u0010+S21;12\n\u0018\u0010](\n); (20)\nSxy\n\u0018\u0010(\n) =\u0000i\n4[S12;21\n\u0018\u0010\u0000S21;12\n\u0018\u0010](\n); (21)\nSzz\n\u0018\u0010(\n) =\u00001\n4[S11;11\n\u0018\u0010\u0000S11;22\n\u0018\u0010\u0000S22;11\n\u0018\u0010+S22;22\n\u0018\u0010](\n);\n(22)\nwhere Eq. (22) denotes the noise of the zcomponent of\nthe spin current.22,24Since the polarization of the spin\ncurrent precesses in the xyplane, the remaining com-\nponents of the spin-current noise spectrum satisfy the\nfollowing relations:\nSyy\n\u0018\u0010(\n) =Sxx\n\u0018\u0010(\n); (23)\nSyx\n\u0018\u0010(\n) =\u0000Sxy\n\u0018\u0010(\n); (24)\nSxz\n\u0018\u0010(\n) =Szx\n\u0018\u0010(\n) =Syz\n\u0018\u0010(\n) =Szy\n\u0018\u0010(\n) = 0:(25)\nTaking into account that the spin current is not a\nconserved quantity, it is important to notice that the\ncomplete information from the noise spectrum can be\nobtained by studying both the autocorrelation noise\nspectrumSjk\n\u0018\u0018(\n)and cross-correlation noise spectrum\nSjk\n\u0018\u0010(\n),\u00106=\u0018. Therefore, it is more convenient to in-\nvestigate the spin-torque noise spectrum, where both au-\ntocorrelation and cross-correlation noise components of\nspin currents are included. The spin-transfer torque op-\nerator can be defined as\n^Tj=\u0000(^ILj+^IRj); j =x;y;z ;(26)while its fluctuation reads\n\u000e^Tj(t) =\u0000[\u000e^ILj(t) +\u000e^IRj(t)]: (27)\nAccordingly, the nonsymmetrized and symmetrized spin-\ntorque noise can be obtained using the spin-current noise\ncomponents as\nSjk\nT(t;t0) =h\u000e^Tj(t)\u000e^Tk(t0)i\n=X\n\u0018\u0010Sjk\n\u0018\u0010(t;t0); j;k =x;y;z ;(28)\nSjk\nTS(t;t0) =1\n2[Sjk\nT(t;t0) +Skj\nT(t0;t)]; (29)\nwith the corresponding noise spectrums given by\nSjk\nT(\n) =X\n\u0018\u0010Sjk\n\u0018\u0010(\n); (30)\nSjk\nTS(\n) =X\n\u0018\u0010Sjk\n\u0018\u0010S(\n): (31)\nAccording to Eqs.(23),(24), and (30), Sxx\nT(\n) =Syy\nT(\n)\nandSyx\nT(\n) =\u0000Sxy\nT(\n).\nIn the remainder of the section we investigate the zero-\nfrequency spin-torque shot noise Sjk\nT=Sjk\nT(0)at zero\ntemperature, where Sxx\nT(0) =Sxx\nTS(0),Syy\nT(0) =Syy\nTS(0),\nSzz\nT(0) =Szz\nTS(0), whileSxy\nT(0)is a complex imaginary\nfunction, and Sxy\nTS(0) = 0 according to Eqs. (24) and\n(31). Since Sxx\nT(0) =Syy\nT(0), all results and discussions\nrelated toSxx\nT(0)also refer to Syy\nT(0).\nSpincurrents I\u0018xandI\u0018yareperiodicfunctionsoftime,\nwith periodT= 2\u0019=!L, whileI\u0018zis time-independent.\nIt has already been demonstrated that spin-flip pro-\ncesses contribute to the noise of spin current.22The pres-\nence of the precessing molecular spin affects the spin-\ncurrent noise. Since the number of particles with differ-\nent spins changes due to spin-flip processes, additional\nspin-current fluctuations are generated. Currents with\nthe same and with different spin orientations are corre-\nlated during transport. Due to the precessional motion\nof the molecular spin, inelastic spin currents with spin-\nflip events induce noise of spin currents and spin-torque\nnoise, which can be nonzero even for eV= 0. The noise\ncomponent Sxy\nTis induced by the molecular spin preces-\nsion and vanishes for a static molecular spin. The noises\nof spin currents and spin-transfer torque are driven by\nthe bias voltage and by the molecular spin precession.\nHence, in the case when both the molecular spin is static\n(absence of inelastic spin-flip processes) and eV= 0(no\ncontribution of elastic tunneling processes), they are all\nequal to zero. The noise of spin-transfer torque can be\nmodified by adjusting system parameters such as the bias\nvoltageeV, the magnetic field ~B, or the tilt angle \u0012.\nIn Fig. 6 we present the zero-frequency spin-torque\nnoise components Sxx\nT=Syy\nT,ImfSxy\nTg, andSzz\nTas func-\ntions of the bias voltage eV=\u0016L\u0000\u0016R, for\u0016R= 0\nand different tilt angles \u0012between~Band~Sat zero tem-\nperature. They give information on available transport7\n0.0 0.5 1.0 1.51\neV Ε0ST102Ε0\nST,Θ0ST,Θ2ST,Θ ΜLeV\nΜR0\nImST,Θ2ST,Θ2zz xx\nxyxx\nxxjk\n0\nFIG.6. (Coloronline)Spin-torqueshot-noisecomponents Sjk\nT\nas functions of the bias voltage eVfor\u0016R= 0,\u0016L=eV. All\nplots are obtained at zero temperature, with ~B=B~ ez, and\n\u0000L= \u0000R= \u0000=2, for \u0000 = 0:05\u000f0. The other parameters are\nset to!L= 0:5\u000f0,J= 0:01\u000f0, andS= 100. The molecular\nquasienergylevelslieat \u000f1= 0:25\u000f0,\u000f2= 0:75\u000f0,\u000f3= 1:25\u000f0,\nand\u000f4= 1:75\u000f0.\nchannelsandinelasticspin-flipprocesses. Themagnitude\nof the torque noise at resonance energies \u000fi,i= 1;2;3;4,\nis determined by \u0012. In cases \u0012= 0and\u0012=\u0019, there\nare only two transport channels of opposite spins deter-\nmined by the resulting Zeeman field B\u0006JS=g\u0016B. The\ncomponent Sxx\nTshows two steps with equal heights lo-\ncated at these resonances, where the only contribution to\nthe spin-torque noise comes from elastic tunneling events\n(dotted purple and red lines in Fig. 6). For \u0012=\u0019=2, the\nelastic tunneling contributes with four steps with equal\nheights located at resonances \u000fi, but due to the contri-\nbutions of the inelastic precession-assisted processes be-\ntween quasienergy levels \u000f1(\u000f3) and\u000f2(\u000f4), the heights of\nthe steps in Sxx\nTare not equal anymore (dot-dashed pink\nline in Fig. 6). Here, we observed that the contribution\nof the inelastic tunneling processes to Sxx\nT, involving ab-\nsorption of an energy quantum !Land a spin-flip, shows\nsteps at spin-down quasienergy levels \u000f1and\u000f3, while it\nis constant between and after the bias has passed these\nlevels. The component Szz\nTshows similar behavior (green\nline in Fig. 6). As in the case of the inelastic tunneling\ninvolving the absorption of one energy quantum !L, in\nSxx\nT=Syy\nTwe observed inelastic spin-flip processes in-\nvolving the absorption of two energy quanta 2!Lin the\nform of steps at spin-down levels \u000f1,\u000f3,\u000f2\u00002!L, and\n\u000f4\u00002!L, which have negligible contribution compared\nto the other terms. These processes are a result of cor-\nrelations of two oscillating spin-currents. For large bias\nvoltage, the spin-torque noise components Sxx\nTandSzz\nT\nsaturate.\nThe behavior of the component ImfSxy\nTgis completely\ndifferent in nature. It is contributed only by one energy\nquantum!Labsorption (emission) spin-flip processes.\nInterestingly, we obtained the following relation between\nthe Gilbert damping parameter \u000b,42,43andImfSxy\nTgat\n3210123101\nΩLΕ0102Ε0STΘ2ImSTSTxxzzxySTjkFIG.7. (Coloronline)Spin-torqueshot-noisecomponents Sjk\nT\nas functions of the Larmor frequency !Lfor\u0012=\u0019=2,\u0016R= 0,\nand\u0016L= 1:5\u000f0. All plots are obtained for ~B=B~ ezat zero\ntemperature. The other parameters are \u0000L= \u0000R= \u0000=2,\n\u0000 = 0:05\u000f0,J= 0:01\u000f0, andS= 100.\narbitrary temperature\nImfSxy\nTg=!LSsin2(\u0012)\n2\u000b: (32)\nHence, the component ImfSxy\nTgis increased for Fermi\nlevelsoftheleadspositionedintheregionswhereinelastic\ntunneling processes occur (blue line in Fig. 6).\nThe spin-torque noise is influenced by the magnetic\nfield~Bsince it determines the spin-up and spin-down\nmolecular quasienergy levels. The dependence of Sxx\nT,\nImfSxy\nTg, andSzz\nTon the Larmor frequency !Lis de-\npicted in Fig. 7. The steps, dips, or peaks in the\nplots are located at resonant tunneling frequencies !L=\n\u0006j2\u0016L;R\u00002\u000f0\u0006JSj. For!L= 0there are only two\ntransport channels, one at energy \u000f0+JS=2, which is\nequal to the Fermi energy of the left lead, and the other\nat\u000f0\u0000JS=2located between \u0016Land\u0016R. The contribu-\ntions of the elastic spin transport processes through these\nlevelsresultindipsinthecomponents Sxx\nTandSzz\nT, while\nImfSxy\nTg= 0. For!=\u000f0corresponding to \u0016R=\u000f1and\n\u0016R=\u000f4\u00002!L, both the elastic and spin-flip tunneling\nevents involving the absorption of energy of one quantum\n!Lcontribute with a dip, while the spin-flip processes\ninvolving the absorption of an energy equal to 2!Lcon-\ntribute with a peak to the component Sxx\nT. For!L= 2\u000f0\nand!L= 3\u000f0corresponding to \u0016L=\u000f2and\u0016R=\u000f3,\nboth elastic and spin-flip processes with the absorption\nof an energy equal to !Lcontribute with a step, while the\ninelastic processes involving the absorption of an energy\n2!Lgive negligible contribution to Sxx\nT. The component\nSzz\nTshows dips at these two points, since here the domi-\nnantcontributioncomesfrominelastictunnelingspin-flip\nevents. The component Szz\nTis an even function of !L,\nwhile ImfSxy\nTgis an odd function of !L. The spin-torque\nnoiseSxx\nTis an even function of !Lfor\u0012=\u0019=2.\nThe spin-torque noise components as functions of \u0012for\n\u0016L=\u000f3and\u0016R= 0at zero temperature are shown in8\n0 1 2 3 4 5 62468\nΘ103Ε0b\nST\nΜLΕ3\nΜR0\nImSTSTxx\nxyzzSTjk\n0\nFIG. 8. (Color online) Spin-torque shot-noise components as\nfunctions of the tilt angle \u0012for\u0016L=\u000f3,\u0016R= 0. All plots\nare obtained at zero temperature, with ~B=B~ ez,\u0000 = 0:05\u000f0,\nand\u0000L= \u0000R= \u0000=2. The other parameters are !L= 0:5\u000f0,\nJ= 0:01\u000f0, andS= 100.\nFig. 8. The magnitudes and the appearance of the spin-\ntorque noise components at resonance energies \u000fican be\ncontrolled by \u0012, since it influences the polarization of the\nspin current. Here we see that both Szz\nTandImfSxy\nTg\nare zero for \u0012= 0and\u0012=\u0019, as the molecular spin is\nstatic and its magnitude is constant along zdirection in\nboth cases. These torque-noise components take their\nmaximum values for \u0012=\u0019=2, where both elastic and\ninelastic tunneling contributions are maximal. The com-\nponentSxx\nTtakes its minimum value for \u0012= 0and its\nmaximum value for \u0012=\u0019, with only elastic tunneling\ncontributions in both cases. For \u0012=\u0019=2, the inelastic\ntunneling events make a maximal contribution while en-\nergy conserving processes give minimal contribution to\nSxx\nT.\nV. CONCLUSIONS\nIn this article, we studied theoretically the noise of\ncharge and spin transport through a small junction, con-sisting of a single molecular orbital in the presence of a\nmolecular spin precessing with Larmor frequency !Lin\na constant magnetic field. The orbital is connected to\ntwo Fermi leads. We used the Keldysh nonequilibrium\nGreen’s functions method to derive the noise components\nof charge and spin currents and spin-transfer torque.\nThen, we analyzed the shot noise of charge current\nand observed characteristics that differ from the ones in\nthe current. In the noise power, we observed diplike fea-\ntures which we attribute to inelastic processes, due to\nthe molecular spin precession, leading to the quantum-\ninterference effect between correlated transport channels.\nSince the inelastic tunneling processes lead to a spin-\ntransfer torque acting on the molecular spin, we have\nalso investigated the spin-torque noise components con-\ntributed by these processes, involving the change of en-\nergy by an energy quantum !L. The spin-torque noise\ncomponents are driven by both the bias voltage and the\nmolecular spin precession. The in-plane noise compo-\nnentsSxx\nTandSyy\nTare also contributed by the processes\ninvolving the absorption of an energy equal to 2!L. We\nobtained the relation between ImfSxy\nTgand the Gilbert\ndamping coefficient \u000bat arbitrary temperature.\nTaking into account that the noise of charge and spin\ntransport can be controlled by the parameters such as\nbias voltage and external magnetic field, our results\nmight be useful in molecular electronics and spintron-\nics. The experimental observation of the predicted noise\npropertiesmightbeachallengingtaskduetocomplicated\ntunnelling processes through molecular magnets. Find-\ning a way to control the spin states of single-molecule\nmagnets in tunnel junctions could be one of the future\ntasks.\nACKNOWLEDGMENTS\nWe would like to thank Fei Xu for useful discussions.\nWe gratefully acknowledge the financial support from the\nDeutsche Forschungsgemeinschaft through the SFB 767\nControlled Nanosystems , the Center of Applied Photon-\nics, the DAAD through a STIBET scholarship, and an\nERCAdvancedGrant UltraPhase ofAlfredLeitenstorfer.9\nAPPENDIX: FORMAL EXPRESSION FOR THE NONSYMMETRIZED NOISE\nHere, we present the derivation of the formal expression for the nonsymmetrized noise S\u0017\u0016\n\u0018\u0010(t;t0). The correlation\nfunctionsS\u001b\u001b0;\u0015\u0011\n\u0018\u0010(t;t0), introduced in Eq. (11), can be expressed by means of the Wick’s theorem56as\nS\u001b\u001b0;\u0015\u0011\n\u0018\u0010(t;t0) =X\nkk0[Vk\u0018Vk0\u0010G>\n\u001b0;k0\u0015\u0010(t;t0)G<\n\u0011;k\u001b\u0018(t0;t)\n\u0000Vk\u0018V\u0003\nk0\u0010G>\n\u001b0\u0015(t;t0)G<\nk0\u0011\u0010;k\u001b\u0018 (t0;t)\n\u0000V\u0003\nk\u0018Vk0\u0010G>\nk\u001b0\u0018;k0\u0015\u0010(t;t0)G<\n\u0011\u001b(t0;t)\n+V\u0003\nk\u0018V\u0003\nk0\u0010G>\nk\u001b0\u0018;\u0015(t;t0)G<\nk0\u0011\u0010;\u001b(t0;t)]; (A1)\nwith the mixed Green’s functions defined, using units in which ~=e= 1, as\nG<\n\u0011;k\u001b\u0018(t;t0) =ih^cy\nk\u001b\u0018(t0)^d\u0011(t)i; (A2)\nG>\n\u001b0;k0\u0015\u0010(t;t0) =\u0000ih^d\u001b0(t)^cy\nk0\u0015\u0010(t0)i; (A3)\nwhile Green’s functions G<\nk\u001b\u0018;\u0011(t;t0) =\u0000[G<\n\u0011;k\u001b\u0018(t0;t)]\u0003andG>\nk0\u0015\u0010;\u001b0(t;t0) =\u0000[G>\n\u001b0;k0\u0015\u0010(t0;t)]\u0003. The Green’s functions\nof the leads and the central region are defined as\nG<\nk\u001b\u0018;k0\u001b0\u0010(t;t0) =ih^cy\nk0\u001b0\u0010(t0)^ck\u001b\u0018(t)i; (A4)\nG>\nk\u001b\u0018;k0\u001b0\u0010(t;t0) =\u0000ih^ck\u001b\u0018(t)^cy\nk0\u001b0\u0010(t0)i; (A5)\nG<\n\u001b\u001b0(t;t0) =ih^dy\n\u001b0(t0)^d\u001b(t)i; (A6)\nG>\n\u001b\u001b0(t;t0) =\u0000ih^d\u001b(t)^dy\n\u001b0(t0)i; (A7)\nGr;a\n\u001b\u001b0(t;t0) =\u0007i\u0012(\u0006t\u0007t0)hf^d\u001b(t);^dy\n\u001b0(t0)gi: (A8)\nSince the self-energies originating from the coupling between the electronic level and the lead \u0018are diagonal in the\nelectron spin space, their entries can be written as \u0006<;>;r;a\n\u0018(t;t0) =P\nkVk\u0018g<;>;r;a\nk\u0018(t;t0)V\u0003\nk\u0018, whereg<;>;r;a(t;t0)are\nthe Green’s functions of the free electrons in lead \u0018. Applying Langreth analytical continuation rules,57Eq. (A1)\ntransforms into\nS\u001b\u001b0;\u0015\u0011\n\u0018\u0010(t;t0) =Z\ndt1Z\ndt2\n\u0002\b\n[Gr\n\u001b0\u0015(t;t1)\u0006>\n\u0010(t1;t0) +G>\n\u001b0\u0015(t;t1)\u0006a\n\u0010(t1;t0)][Gr\n\u0011\u001b(t0;t2)\u0006<\n\u0018(t2;t) +G<\n\u0011\u001b(t0;t2)\u0006a\n\u0018(t2;t)]\n+[\u0006>\n\u0018(t;t1)Ga\n\u001b0\u0015(t1;t0) + \u0006r\n\u0018(t;t1)G>\n\u001b0\u0015(t1;t0)][\u0006<\n\u0010(t0;t2)Ga\n\u0011\u001b(t2;t) + \u0006r\n\u0010(t0;t2)G<\n\u0011\u001b(t2;t)]\n\u0000G>\n\u001b0\u0015(t;t0)[\u0006r\n\u0010(t0;t1)Gr\n\u0011\u001b(t1;t2)\u0006<\n\u0018(t2;t) + \u0006<\n\u0010(t0;t1)Ga\n\u0011\u001b(t1;t2)\u0006a\n\u0018(t2;t)\n+\u0006r\n\u0010(t0;t1)G<\n\u0011\u001b(t1;t2)\u0006a\n\u0018(t2;t)]\u0000[\u0006r\n\u0018(t;t1)Gr\n\u001b0\u0015(t1;t2)\u0006>\n\u0010(t2;t0)\n+\u0006>\n\u0018(t;t1)Ga\n\u001b0\u0015(t1;t2)\u0006a\n\u0010(t2;t0) + \u0006r\n\u0018(t;t1)G>\n\u001b0\u0015(t1;t2)\u0006a\n\u0010(t2;t0)]G<\n\u0011\u001b(t0;t)\t\n\u0000\u000e\u0018\u0010[\u000e\u0011\u001bG>\n\u001b0\u0015(t;t0)\u0006<\n\u0018(t0;t) +\u000e\u001b0\u0015\u0006>\n\u0018(t;t0)G<\n\u0011\u001b(t0;t)]: (A9)\nFinally, using Eqs. (11) and (A9), the obtained formal expression for the nonsymmetrized noise of charge current40,58\nand spin currents in standard coordinates tandt0can be written as\nS\u0017\u0016\n\u0018\u0010(t;t0) =\u0000q\u0017q\u0016\n~2TrnZ\ndt1Z\ndt2\n\u0002\b\n^\u001b\u0017[^Gr(t;t1)^\u0006>\n\u0010(t1;t0) +^G>(t;t1)^\u0006a\n\u0010(t1;t0)]^\u001b\u0016[^Gr(t0;t2)^\u0006<\n\u0018(t2;t) +^G<(t0;t2)^\u0006a\n\u0018(t2;t)]\n+ ^\u001b\u0017[^\u0006>\n\u0018(t;t1)^Ga(t1;t0) +^\u0006r\n\u0018(t;t1)^G>(t1;t0)]^\u001b\u0016[^\u0006<\n\u0010(t0;t2)^Ga(t2;t) +^\u0006r\n\u0010(t0;t2)^G<(t2;t)]\n\u0000^\u001b\u0017^G>(t;t0)^\u001b\u0016[^\u0006r\n\u0010(t0;t1)^Gr(t1;t2)^\u0006<\n\u0018(t2;t) +^\u0006<\n\u0010(t0;t1)^Ga(t1;t2)^\u0006a\n\u0018(t2;t) +^\u0006r\n\u0010(t0;t1)^G<(t1;t2)^\u0006a\n\u0018(t2;t)]\n\u0000^\u001b\u0017[^\u0006r\n\u0018(t;t1)^Gr(t1;t2)^\u0006>\n\u0010(t2;t0) +^\u0006>\n\u0018(t;t1)^Ga(t1;t2)^\u0006a\n\u0010(t2;t0) +^\u0006r\n\u0018(t;t1)^G>(t1;t2)^\u0006a\n\u0010(t2;t0)]^\u001b\u0016^G<(t0;t)\t\n\u0000\u000e\u0018\u0010^\u001b\u0017[^G>(t;t0)^\u001b\u0016^\u0006<\n\u0018(t0;t) +^\u0006>\n\u0018(t;t0)^\u001b\u0016^G<(t0;t)]o\n; (A10)10\nwhere Trdenotes the trace in the electronic spin space.\n1Y. 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B\n77, 075302 (2008)."    },    {        "title": "1704.07006v1.Spin_injection_into_silicon_detected_by_broadband_ferromagnetic_resonance_spectroscopy.pdf",        "content": "1 \n Spin injection into silicon  \ndetected by broadband ferromagnetic resonance spectroscopy  \n \nRyo Ohshima,1 Stefan Klingler,2,3, Sergey Dushenko,1 Yuichiro Ando,1 Mathias Weiler,2,3 \nHans Huebl,2,3,4 Teruya Shinjo,1 Sebastian T. B. Goennenwein,2,3,4 and Masashi Shiraishi1* \n \n1Department of Electronic Science and Engineering, Kyoto Univ., 615 -8510 Kyoto, Japan.  \n2Walther -Meißner -Institut, Bayerische Akademie der Wissenschaften, 85748 Garching,  \nGermany.  \n3Physik -Department, Technische Universität München, 85748 Garching, Germany  \n4Nanosystems Initiative Munich , 80799 München, Germany  \n \n \nWe studied the spin injection in a NiFe(Py)/Si system using broadband \nferromagnetic resonance spectroscopy. The Gilbert damping parameter  of the Py layer \non top of the Si channel was determined as a function of the Si doping concentration  and \nPy layer thickness . For fixed Py thickness w e observe d an increase of the Gilbert damping \nparameter with decreasing resistivity of the Si channel . For a fixed Si doping \nconcentration we measured an increasing Gilbert damping parameter for decreasing Py \nlayer thickness.  No increase of the Gilbert damping param eter was found Py/Si samples \nwith an insulating interlayer . We attribute our observations to an enhanced spin injection \ninto the low -resistivity Si  by spin pumping . \n  2 \n Spin injection into semiconductors was relentlessly studied in recent years in hope to \nharness their long spin relaxation time, and gate tunability to realize spin metal -oxide -\nsemiconductor field -effect -transistors (MOSFET s). A central obstacle for a spin injection into \nsemiconductors was the conductance mismatch1 between ferromagnetic metals (used for the \nspin injection) and semiconductor channels. In an electrical spin injection method —widely \nused from the early years of the non -local spin transport studies —tunnel barriers between the \nsemiconductor and ferromagne t were formed to avoid the conductance mismatch problem2–5. \nUnfortunately, it complicated the production process of the devices , as high quality tunnel \nbarriers are not easy to grow , and presence of impurities, defects and pinholes take s a heavy \ntoll on th e spin injection efficiency and/or induces spurious effects. Meanwhile, the fabrication \nof electrical Si spin devices, like spin MOSFETs, with different resistivities is a time -\nconsuming process, which so far prevented systematic studies of the spin inject ion properties \n(such as spin lifetime, spin injection efficiency etc.). However,  such a systematic study is \nnecessary for further progress towards practical applications of spin MOSFETs.  \nIn 2002, a dynamical spin injection method, known as spin pumping, was introduced \nto the scene of spintronics research6,7. While the method was initially used in the metallic \nmultilayer sy stems, it was later  implemented to inject spin current s into semiconductors. In \ncontrast to electrical spin injection, spin pumping doe s not require the application of an electric \ncurrent across the ferromagnet/semiconductor interface. Devices that operate using spin \ncurrent s instead of charge current s can potentially reduce heat generation and power \nconsumption problems of modern electro nics. From a technological point of view, spin \npumping is also appealing because it does not require a tunnel barrier.  Spin injection —using \nspin pumping —into semiconductors from an adjacent ferromagnetic metal was achieved \ndespite the existence of conducti vity mismatch8-12. However, so far there was no systematic \nstudy of the spin pumping based spin injection in dependence on the resistivity of the Si channel.  3 \n In this letter, we focus on the study of spin injection  by spin pumping  in the NiFe(Py)/Si system  \nwith different resistivities  of the Si channel  using broadband ferromagnetic resonance (FMR) . \nThe broadband FMR method allows for a  precise determination of  the Gilbert damping \nparameter 𝛼, which increases in the presence of spin pumping, and  thus, spin injection . By \ntracking the change of the Gilbert damping parameter in various  Py/Si system s, we determine d \nthe spin pumping  efficiency  in the broad range of resistivities of the Si channel.   \nFor a first set of sample s 7nm-thick Py film s were  deposited  by electron beam \nevaporation  on top of various Si substrates  (1×1 cm2 in size) with resistivit ies in the range from \n10-3 to 103 ･cm (see Table 1 for the list of the prepared samples ). The oxidized surface of the \nSi substrates  was removed using 10% hydrofluoric acid (HF) prior to the Py evaporation. For \na second set of samples  Py films with  thickness es 𝑑Py between 5nm and 80nm  were deposited \non P -doped SOI  (silicon on insulator)  with the same technique . As a control  experiment , \nPy/AlO x and Py/ TiO x films were grown  on Si, P -doped SOI and SiO 2 substrates , as spin \npumping should be suppressed  in systems with an insulati ng barrier13 (see Table 2) . Both Al (3 \nnm, thermal deposition) and Ti  (2 nm, electron beam evaporation) were evaporated on the non -\ntreated substrates and left in the air for one day for oxidation of the surface (for the Al layer , \nthe process was repeated 3 times, with 1  nm of Al evaporated and oxidized at each step). After \noxidation, we evaporated 7nm thick  Py film s on the top of the tunnel barrier s. The properties \nof the prepared samples  are summarized in Table s 1 and 2 . \nA sketch of the broadband ferromagnetic res onance setup is shown in Fig. 1 (a). T he \nsamples were placed  face down on the center conductor  of a coplanar waveguide  (CPW), which \nwas located between the pole shoes of an electromagnet . A static magnetic field  |𝜇0𝐻|≤2.5 \nT was applied perpendicular to the surface of the samples to avoid extra damping due to two -\nmagnon scattering14. One end of the CPW was connected to a microwave source , where \nmicrowaves with frequenc y f < 40 GHz were generated . The other end of the CPW was 4 \n connected to a microwave diode and a lock -in amplifier to measure the rectified microwave \nvoltage  as a function of the applied magnetic field . All measurements were carried out at room \ntemperature.  \nThe microwave current in the  CPW generates an oscillating magnetic field around the \ncenter conductor  which results  in an oscillating torque on the s ample ’s magnetization. For \n𝜇0𝐻=𝜇0𝐻FMR  this torque results in a n absorption of microwave power . The resonance \ncondition i s given by the out -of-plane Kittel equation15,16: \n ℎ𝑓\n𝑔𝜇B=𝜇0𝐻FMR −𝜇0𝑀eff. (1)  \nHere,  ℎ is the Plan ck constant,  𝑔 is the Landé g-factor, 𝜇B is the Bohr magneton, 𝜇0 is \nthe vacuum permeability, and 𝑀eff is the effective saturation magnetization of Py.   \nWe use  the Gilbert damping model, which phenomenologically models the viscous \ndamping of the magnetic resonance. The linear relat ion between the full width at half maximum \n𝛥𝐻 of the resonance and the applied microwave frequency f is given by the Gilbert damping \nequation 17: \n 𝜇0𝛥𝐻=𝜇0𝛥𝐻0+2𝛼ℎ𝑓\n𝑔𝜇B. (2)  \nHere,  𝛥𝐻0 corresponds to  frequency independent scattering processes  and 𝛼 is the Gilbert  \ndamping parameter18,19: \n 𝛼=𝛼0+𝛼SP+𝛼EC. (3)  \nHere, 𝛼0 is the intrinsic Gilbert damping , 𝛼SP=𝑔𝜇𝐵𝑔r↑↓4𝜋𝑀S𝑑Py ⁄  is the damping due to \nspin pumping20, 𝑔r↑↓ is the real part of the spin mixing conductance,  and 𝛼EC=𝐶EC𝑑Py2 is the \neddy -current damping . The parameter  𝐶EC describes efficiency of  the eddy -current damping . \nTo realize a net spin injection via spin pumping, the following conditions should be \nfulfilled in the system: (i ) carriers should be present in the underlying channel, (ii) the spin \nrelaxation time in the channel should be small enough. The available carriers in the channel \ntransfer spin angular momentum away from the spin injection interface, allowing propagation 5 \n of the spin current. On the other hand, the long spin relaxation time in the channel leads to a \nlarge spin accumulation at the interface and generates a diffusive spin backflow in the direction \nopposite to the spin pumping current20 (see Figs. 1(b) and 1(c )). Thus, the spin backflow \neffectively cancels out the spin pumping current for long spin relaxation time, and the spin \npumping contribution to the Gilbert damping parameter should no longer be present in the \nsystem.  \nIn addition to spin pumping,  charge  currents can be induced in the Si channel  and Py  \nlayer due to  the Faraday ’s law  and Py magnetization precession, which results  in a Gilbert -like \ndamping contribution . These processes are  refer red to as radiative damping  and eddy -current \ndamping18. An e nhancement of the Gilbert damping parameter due to these processes  is \nexpected to be especially large for the Si channels with low resistivit ies and thick Py films , \nsince energy dissipation through eddy currents scales linearly  with the conductivity  of the  Si \nlayer  and quadratically with the Py layer thickness . Hence, both spin pumping and eddy current \ndamping are expected to be most efficien t for low -resistivity Si. I n contrast  to spin pumping , \nthe radiative damping  contribution doe s not  require a direct electrical contact between Py and \nSi, and is hence unaffected by the tunnel barrie r. \nFigure 1(d ) show s a typical FMR spectr um of a 7nm thick Py film on a phosphorous \nP-doped Si on insulator (SOI) substrate , where t he microwave frequency was fixed at 30 GHz  \nduring the sweep of the magnetic field. A single FMR  signal  was observed  (Fig. 1(d) red filled \ncircles) , from which 𝜇0𝐻FMR  and 𝜇0𝛥𝐻  were extracted  by a fit of the magnetic ac \nsusceptibility  (Fig. 1(d) black line) 16,21 (see Supplemental Material for additional fitting \nexamples).  An excellent agreement of the fit with the measurement is achieved.  \nFigures 2(a) and (b) show 𝐻FMR and 𝛥𝐻 versus the applied microwave frequency f \nfor the Py/P-doped SOI, Py/SOI and Py/SiO 2 samples. From the f itting of the frequency \ndependence of HFMR with Eq.(1) , 𝑔  and 𝜇0𝑀eff  of the Py /P-doped SOI ( Py/SOI) were 6 \n estimated to be 2.049 (2.051) and 0.732 T (0.724 T), and those of the Py /SiO 2 were estimated \nto be 2.038 and 0.935 T, respectively  (Supplemental Material  for fitting  and data from other \nsamples ). The difference in the 𝑔  and 𝑀eff  between the Py /Si and the Py/ SiO 2 sample  is \nattributed to the inter -diffusion of the Fe/Ni and Si at the  interface , which is  always present to \nsome extent during the growth at room temperature22,23. The Gilbert damping  𝛼 of the Py /P-\ndoped SOI , Py/SOI and Py/SiO 2 were estimated to be 1.25 ×10-2, 9.02 ×10-3 and 8.49 ×10-3, \nrespectively , from the linewidth vs. frequency evolution . The intrinsic Gilbert damping \nparameter 𝛼0  is determined from the linewidth evolution of the Py/SiO 2 and Py/quartz \nsamples to be 8.5×10−3 and 8.6×10−3, respectively, since no spin pumping contribution is \nexpected in these insulating materials ( 𝛼=𝛼0). From this we can see an increasing Gilbert \ndamping with decreasing resistivity. We additionally measured the samples with a n insulating \ntunnel barrier between the Si channel and the Py film. We found Gilbert damping parameters \nof 𝛼 = 8.8×10-3 for the Py/ AlO x/Si samples and 𝛼 = 7.5×10-3 for the Py/TiO x/Si samples, \nindependent of the Si resistivity.  The damping  values are in agreement with the intrinsic \ndamping extracted from the Py/SiO 2 sample , indicating that radiative damping is negligible in \nour samples.  \nFigure 3 (a) summarizes  the dependence of the Gilbert damping parameter 𝛼 on the \nresistivity of the Si channel (see Supplemental Material D for the g -factor, the effective \nsaturation magnetization and the frequency independent term ), including the measured control \nsamples.  The dashed lines  show  the intrinsic contributions 𝛼0  to the Gilbert damping \nparameter 𝛼 measured from the Py/SiO 2 and Py/quartz samples (red dashed), Py/AlO x/SiO 2 \n(blue dashed) and Py/TiO x/SiO 2 (green dashed). All  samples with Py on top of the conductive \nsubstrates without an additional tunnel barrier exhibited the Gilbert damping parameter 𝛼 \nlarger than the intrinsic contribution 𝛼0.  \nThe experimentally measured Gilbert damping parameter decreases logarithmically 7 \n with the resistivity. This result is in agreement with condition (i) for the spin pumping. In the \nSi channels with a small resistivity more carriers were available to transfer the injected angular \nmomentum, leading to an effective spin pumping. Additionally, both electron spin resonance25–\n28 and non -local 4 -terminal Hanle precession29,30 experiments showed, that the spin lifetime in \nSi is increas ing with increasing resistivity . Our samples with low resistivities have a large \ndoping concentration (see Table 1), leading to shorter spin relaxation time. In accordance with \nthe spin pumping condition (ii), the decrease of the spin relaxation time should lead to the \nincrease of the spi n pumping contribution, as now observe d experimentally. While the spin \npumping shows a logarithmic dependence on the resistivity of the channel, we note that an \nincreased Gilbert damping parameter  is observed even for the Si channel with high resistivit ies. \nWe comment on the Sb -doped sample, where the experimentally measured 𝛼SP was lower \nthan one expected from the logarithmic trend of the other samples. We speculate that this might \noriginate from the different doping profile, compared to the other samples . We note, that further \nstudies are necessary to separate the influence of the number of carriers in the channel and the \nspin relaxation time on the spin pumping process.  \nFinally, we show that the Py damping is increased in a broad range of Si resistivitie s \nand attribute this effect to the enhanced spin injection via spin pumping ( a discussion  of the \nspin mixing conductance  for various Si  resistivities is given in the Supplemental M aterial A). \nFigure 3(b) shows the Py thickness dependence of the 𝛼 for Py/P -doped SOI samples. The \nsolid line shows a fit of Eq.(3) to the measured data and a very good agreement of the spin \npumping theory with our measurements is achieved. From the fit we estimate 𝛼0=6.1×10−3, \n𝑔r↑↓=1.2×1019 m-2 and 𝐶EC=2.9×1011 m-2. Both the intrinsic damping and the real part \nof the spin mixing conductance are in good agreement with previous measurements30. The blue \ndashed line indicates 𝛼0+𝛼SP  and the green dashed line shows 𝛼0+𝛼EC . Dominant \ninfluence of spin pumping to the total damping is observed in samples with  small Py thickness , 8 \n while  eddy current contribution is dominant in samples with  thick Py  layer . For the 7 nm -thick \nPy sample, we find 𝛼SP = 3.6 ×10-3 and 𝛼EC = 1.4 ×10-5. Thus, the eddy -current damping in \nour 7 nm Py samples is negligibly small and cannot explain the increase of the damping with \ndecreasing resistivity. The Py thickness dependence of the Gilbert damping indicates  spin \npumping into the Si substrates.  \nIn conclusion , we studied spin pumping based spin injection from a Py layer into Si \nchannels with various resistivit ies using broadband ferromagnetic resonance . We determine d \nthe spin pumping contribution from the  change of the Gilbert damping parameter. The observed \nlogarithmic decrease of the Gilbert damping parameter with  increasing resistivity of the Si \nchannel is attribute  to the decrease in the number of carriers in the channel, and the increase in \nthe spin lifetime. De spite  the reduction of the spin pumping contribution to the Gilbert damping \nparameter with the increasing  resistivity  of the Si channel , we observe  spin pumping even for \nthe channels with high resistivity . We furthermore observe an increase of the Gilbert damping \nparameter for decreasing Py thickness which is in agreement with the spin pumping theory. \nOur results show that spin pumping can be potentially used in  a spin transistors, where low \ndoping concentration in the channel is necessary for the gate control of the device.  \n \nSupplement al Material  \n See Supplementary M aterial for a discussion  of the spin mixing conductance  for \nvarious Si resistivities and additional  fitting examples.  \n \nACKNOWLEGEMENTS  \nThis research was supported in part by a Gran t-in-Aid for Scientific Research from \nthe Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan, \nInnovative Area “Nano Spin Conversion Science ” (No. 2 6103003), Scientific Research (S) 9 \n “Semico nductor Spincurrentronics ” (No. 16H0633) and JSPS KAKENHI Grant  (No. \n16J00485).  R.O. acknowledges JSPS Research Fellowship. S.D. acknowledges support by \nJSPS Postdoctoral Fellowship and JSPS KAKENHI Grant No. 16F16064.  \n  10 \n References  \n1 A. Fert and H. Jaffrès, Phys. Rev. B 64, 184420 (2001).  \n2 E.I. Rashba, Phys. Rev. B 62, R16267 (2000).  \n3 I. Appelbaum, B. Huang, and D.J. Monsma, Nature 447, 295 (2007).  \n4 O.M.J. van ’t Erve, A.T. Hanbicki, M. Holub, C.H. Li, C. Aw o-Affouda, P.E. Thompson, \nand B.T. Jonker, Appl. Phys. Lett. 91, 212109 (2007).  \n5 T. Sasaki, T. Oikawa, T. Suzuki, M. Shiraishi, Y. Suzuki, and K. Tagami, Appl. Phys. \nExpress 2, 53003 (2009).  \n6 Y. Tserkovnyak, A. Brataas, and G.E.W. Bauer, Phys. Rev. B 66, 224403 (2002).  \n7 S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B 66, 104413 (2002).  \n8 K. Ando, S. Takahashi, J. Ieda, H. Kurebayashi, T. Trypiniotis, C.H.W. Barnes, S. \nMaekawa, and E. Saitoh, Nat. Mater. 10, 655 (2011).  \n9 K. Ando and E. Saitoh, Nat. Commun. 3, 629 (2012).  \n10 E. Shikoh, K. Ando, K. Kubo, E. Saitoh, T. Shinjo, and M. 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Shiraishi, Y. Suzuki, and K. Noguchi, Appl. Phys. \nExpress 4, 23003 (2011).  \n29 T. Tahara, Y. Ando, M. Kameno, H. Koike, K. Tanaka, S. Miwa, Y. Suzuki, T. Sasaki, T. \nOikawa, and M. Shiraishi, Phys. Rev. B 93, 214406 (2016).  \n30 Note, that the sample set with various Py thicknesses was grown in a different batch than \nthe samples with v arious Si doping. Hence, small deviations in the damping and the spin 12 \n mixing conductance are due to slightly different growth conditions . \n \nFigure 1: (a) Experimental setup for the broadband FMR measurement. The samples were \nplaced with the Py layer facing down on a coplanar waveguide. External magnetic field and \nmicrowave field from the waveguide induce the FMR of the Py and spins are injec ted into Si \nvia spin pumping. Schematic images of spin injection and dephasing in Si that have (b) long  \nand (c) short spin lifetimes. 𝜏1 and 𝜏2 are the spin lifetim e of Si in the case of (b) and (c), \nrespectively. Spin injection efficiency becomes large in the case of (c) because of a reduction \nof the backflow of spins. (d ) The derivative of the FMR signal of Py at 30 GHz microwave \nfrequency.  I is the microwave absorption intensity.  \n \nFigure 2: Frequency dependence of the (a) resonance field 𝐻FMR and (b) full width at half \nmaximum 𝛥𝐻 of the FMR spectra obtained from Py on top of P -doped SOI, SOI and  SiO 2. \nThe solid lines show fitting using  Eqs. (1) and (2) of 𝐻FMR and 𝛥𝐻, respectively.  \n \nFigure  3: (a) Si resistivity dependence of the Gilbert damping parameter  𝛼. The damping of \nthe samples with an insulating layer represents the intrinsic damping of the Py layer and is \nshown by the dashed line s. Red, blue and green coloration represents Py, Py/AlO x and Py/TiO x \nsamples, respectively. The damping of the Py/ P-doped SOI is an averaged value extracted from \nthe two Py/P-doped SOI samples fabricated at different times.  (b) Py thickness dependence of  \n𝛼. The solid line shows  a fit of Eq. (3) to the data . The b lue line shows 𝛼0+𝛼SP, whereas the \ngreen line shows 𝛼0+𝛼EC. \n \n \n 13 \n Fig. 1 R. Ohshima et al . \n \n \nFig. 2  R. Ohshima et al.  \n \n14 \n Fig. 3  R. Ohshima et al.  \n \n \nTable 1: Sample summary  \nName  Dopant  Doping \ndensity (cm-3) Structure  Resistivity \n(･cm) Gilbert damping \nparameter  Mixing \nconductance (m-2) \nPy/P -\ndoped SOI  P 6.5×1019 Py(7 nm)/Si(100 nm)/  \nSiO 2(200 nm)/Si  1.3×10−3 1.1×10−2 5.7×1018 \nPy/Sb -\ndoped Si  Sb 1×1019 Py(7 nm)/Si  5.0×10−3 9.3×10−3 2.3×1018 \nPy/N -\ndoped Si  N 1×1019 Py(7 nm)/Si  1.0×10−1 9.5×10−3 2.6×1018 \nPy/SOI  N/A 1×1015 Py(7 nm)/Si(100 nm)/  \nSiO 2(200 nm)/Si  4.5 9.0×10−3 1.3×1018 \nPy/P -\ndoped Si  P 1×1013 Py(7 nm)/Si  1.0×103 8.7×10−3 5.1×1017 \nPy/SiO 2 - - Py(7 nm)  \n/SiO 2(500 nm)/Si  - 8.5×10−3 - \nPy/Quartz  - - Py(7 nm) /Quartz  - 8.6×10−3 - \n \n \n \n \n \n15 \n Table 2: List of the samples for the control experiment  \nName  Dopant  Doping \ndensity (cm-3) Structure  Resistivity \n(･cm) Gilbert damping \nparameter  \nPy/AlO x/P-\ndoped SOI  P 6.5×1019 Py(7 nm)/AlO x(3 \nnm)/Si(100 nm)/  \nSiO 2(200 nm)/Si  1.3×10−3 8.6×10−3 \nPy/AlO x/P-\ndoped Si  P 1×1013 Py(7 nm)/AlO x(3 nm)/Si  1.0×103 8.5×10−3 \nPy/AlO x/ \nSiO 2 - - Py(7 nm)/ AlO x(3 nm)/  \nSiO 2(500 nm)/Si  - 8.8×10−3 \nPy/TiO x/P-\ndoped SOI  P 6.5×1019 Py(7 nm)/TiO x(2 \nnm)/Si(100 nm)/  \nSiO 2(200 nm)/Si  1.3×10−3 7.5×10−3 \nPy/TiO x/P-\ndoped Si  P 1×1013 Py(7 nm)/TiO x(2 nm)/Si  1.0×103 7.9×10−3 \nPy/TiO x/ \nSiO 2 - - Py(7 nm)/TiO x(2 nm)/  \nSiO 2(500 nm)/Si  - 7.8×10−3 \n \n "    },    {        "title": "1712.07323v1.Unifying_ultrafast_demagnetization_and_intrinsic_Gilbert_damping_in_Co_Ni_bilayers_with_electronic_relaxation_near_the_Fermi_surface.pdf",        "content": " 1 Unifying ultrafast demagnetization and intrinsic Gilbert damping in Co/Ni \nbilayers with electronic relaxation near the Fermi surface  \nWei Zhang, Wei He*, Xiang -Qun Zhang, and Zhao -Hua Cheng*  \nState Key Laboratory of Magnetism and Beijing National Laboratory for \nCondensed Matter Physics, Institute of Physics, Chinese Academy of \nSciences, Beijing 100190, P. R. China  \nJiao Teng  \nDepartment of Materials Physics and Chemistry, University of Sci ence and \nTechnology Beijing, Beijing 100083, P. R. China  \nManfred Fä hnle  \nMax Planck Institute for Intelligent Systems, Heisenbergstra e 3, 70569 \nStuttgart, Germany  \nAbstract  \nThe ability to controllably manipulate the laser -induced ultrafast magnetic \ndynamics  is a prerequisite for future high speed spintronic devices. The optimization \nof devices requires the controllability of the ultrafast demagnetization time, , and \nintrinsic Gilbert damping, . In previous attempts to establish the relationship \nbetween \nM  and \nrint  , the rare -earth doping of a permalloy film with two different \ndemagnetization mechanism is not a suitable candidate. Here, we  choose Co/Ni \nbilayers to  investigate the relations between and  by means of ti me-resolved \nmagneto -optical Kerr effect (TRMOKE) via adjusting the thickness of the Ni layers, \nand obtain an approximately  proportional  relation between these two parameters.   \nM\nintr\nM\nintr 2 The remarkable agreement between TRMOKE experiment and the prediction of \nbreathi ng Fermi -surface model confirms that a large Elliott -Yafet spin -mixing \nparameter \n2b  is relevant to the strong spin -orbital coupling at the Co/Ni interface. \nMore importantly, a proportional relation between \nM  and \nintr  in such metallic \nfilms or heterostructures  with electronic relaxation near Fermi surface suggests the \nlocal spin -flip scattering domains the mechanism of ultrafast demagnetization, \notherwise the spin -current mechanism domains. It is a n effective method to \ndistinguish the dominant contributions to ultrafast magnetic quenching in metallic \nheterostructures by investigating both the ultrafast demagnetization time and Gilbert \ndamping simultaneously. Our work can open a novel avenue to manip ulate the  \nmagnitude and efficiency of Terahertz  emission in metallic heterostructures such as \nthe perpendicular magnetic anisotropic Ta/Pt/ Co/Ni /Pt/Ta  multi layers,  and then it has \nan immediate implication of the design of high frequency spintronic devices . \n \nPACS numbers: 75.78.Jp, 75.40.Gb, 76.50.+g, 78.47.+p  \n*Correspondence and requests for materials  should be addressed to Z.H.C \n(zhcheng@iphy.ac.cn ) or W.H. ( hewei@iphy.ac.cn ) \n   3 Since the pioneering work on ultrafast demagnetization of Ni thin film after \nfemtosecond laser irradiation was demonstrated in 1996 by Beaurepaire et al1, the \nquest for ultrafast modification of the magnetic moments has triggered a new field of \nresearch : Femtomagnetism . It leads to the dawn of a new ear for breaking the ultimate \nphysical limit for the speed of magnetic switching and manipulation, which are \nrelevant to current and future information storage. In the past two decades, the \nultrafast dynamics  in hundreds of femtoseconds have been probed with the \nfemtosecond laser pulse using magneto -optical Kerr1 or Faraday effect2, or other \ntime-resolved techniques such as the high -harmonic generation (HHG) of extreme \nultraviolet(XUV) radiation3, magnetic cir cular dichroism4, or spin resolved two -photo \nphotoemission5.  \nNevertheless, the microscopic mechanism underlying ultrafast quenching of \nmagnetization remains elusive. Various mechanisms including electron -phonon \nmediated spin -flip scattering6-9, electron -electron scattering10,11, electron -magnon \nscattering12,13, direct angular momentum transfer from photon to electron mediated by \nspin-orbit coupling14,15, coherent interaction among spins electrons and photons16, \nwere proposed to explain the ultraf ast spin dynamics.  In addition, since Malinowski17 \net al first proposed that the laser excited spin current transport could increase and \nspeed up the magnetic quenching in metallic heterostructures, the laser -induced \nsuper -diffusive spin current was raise d to play an important role in determining the \nultrafast demagnetization in metallic films or heterostructures18-22. However, the \nrecent demonstration23 shows that the unpolarized hot electrons transport can  4 demagnetize a ferromagnet, indicating the local spin angular momentum dissipation is \nunavoidable even when super -diffusive spin transport domains in the metallic \nheterostructures. Moreover, even in th e similar samples, the local spin -flip scattering \nand nonlocal spin transport mechanism were proposed respectively by different \nexperimental tools19, 24 to explain the ultrafast demagnetization . It is harmful for \nclarifying the underlying ultrafast demagne tization mechanism in such metallic \nheterostructures. Therefore, an effective method to distinguish  the two dominant \ncontribution s to ultrafast  demagnetization in metallic heterostructures is highly \ndesirable19,23,24. Here, we propose that investigating bo th the ultrafast demagnetization \ntime and Gilbert damping25 simultaneously is a candidate method, although the \nrelationship between the two parameters has never been unified successfully so far \nbetween the experiments and theoretical predictions.    \nAn inv erse relation between and was first derived by Koopmans et al. \nfrom a quantum -mechanical calculation on the basis of the Elliot t-Yalfet (EY) \nspin-flip scattering model6. Later, the attempted experiments have ever been carried \nout to demonstrate the predict ion in rare -earth -doped permalloys26,27 and amorphous \nTbFeCo films28. In this case, t he localized 4f electrons  rather than itinerant 5 d6s \nelectrons domain most of the large magnetic moment  in rare -earth elements. Because \nthe 4 f electrons are far from the Fermi level, their ultrafast demagnetization processes \nare medicated by 5 d6s electrons after laser pulse excitation7. The indirect excitation \nleads to the so called type_II ultrafast demagnetization behavior in rare -earth elemen ts, \nwhich is much slower than that of itinerant electrons. Therefore, it is not unexpected \nM\nintr 5 that the ultrafast demagnetization time \nM  of permalloys increases with the doping \ncontents of rare -earth elements increasing. Meanwhile , it happ ened  that the Gilbert \ndamping constant of permalloys is  also increased by doping 4 f elements, which \nmainly comes from the so called “slow relaxing impurities mechanism”29. Therefore, \nby introducing the extra mechanism unavoidablely ，a trivial consequence wa s \nobtained that the ultrafast demagnetization time increases as the Gilbert damping  \n\n increases in rare -earth -doped permalloys26. In hindsight, from this experiment, one \ncan not confirm the relation between ultrafast demagnetization time  and Gilbert \ndamping  \ndue to the defects of the experimental design. A genuine relation between \nultrafast demagnetization time and Gilbert damping should be explored in a clean \nsystem without extra demagnetization mechanism. So far, the explicit relationship \nbetween the two parameters has never been unified successfully between the \nexperiments and theoretical predictions. Our work in Co/Ni bilayers with the electrons \nrelaxing at the Fermi surface can fill in the blank.   \nIn the cas e of pure 3 d itinerant electrons  relaxing near the Fermi surface after the \nlaser excitation , both ultrafast demagnetization  and Gilbert damping are determined \nby spin -flip scattering of itinerant electrons at quasi -particles or impurities . Based on \nthe breathing Fermi -surface model of Gilbert damping and on the EY relation for the \nspin-relaxation time, a proportional relation between and was derived by \nFä hnle et al30,31 for the materials with conductivity -like damping. And an inverse \nrelation was also d erived which is similar with that proposed by B. Koopmans et al \nwhen the resistivity -type damping domains in the materials.  Although the predicted \nM\nM\nM\nintr 6 single numerical values of intr/M are in good agreement with the experimental ones \nfor Fe, Ni, or Co, for a confirmation of the explicit relation between and  one \nhas to vary the values on the two parameters systematically for one system, as we do \nit in our paper by changing the thickness of the films.    \nCo/Ni bilayers with a stack of Ta (3 nm)/Pt (2 nm)/Co ( 0.8 nm)/Ni ( dNi nm)/Pt (1 \nnm)/Ta (3 nm) were grown on glass substrates by DC magnetron sputtering32, 33. The \nthickness of Ni layer changes from dNi = 0.4 nm to dNi = 2.0 nm. T heir static \nproperties have been shown in the Part Ⅰof the Supplementary Materials34. Both\nand for Co/Ni bilayer systems have been achieved by using time -resolved \nmagneto -optical Kerr effect (TRMOKE) technique21, 35. The reasons for selecting the \nCo/Ni bilayers are three -fold. First, Co/Ni bilayers with perpendicular magnetic \nanisotropy  (PMA) are one of candidates for perpendicular magnetic recording (PMR) \nmedia and spintronic devices36-39. Second, the electrons in both Co and Ni are \nitinerant near the Fermi surface and they have the same order of magnitude of \ndemagnetization time7,10. Without rare earth element doping in 3 d metals, one can \nexclude the possibility of an extra slow demagnetization accompanied by doping with \n4f rare-earth metals. Third, both and in Co/Ni bilayers can be tuned by \nchanging the Ni thickness. Therefore, Co/Ni bilayers provide an ideal system to \ninvestigate the relation between and . A nearly p roportional  relationship \nbetween and was evident  in Co/Ni bilayers, suggesting that the \nconductivity -like damping30, 31 plays a dominant role. It is distinct i n physics with \nprevious experiments26 where  the seemingly similar results have been obtained via \nM\nintr\nM\nintr\nM\nintr\nM\nintr\nM\nintr 7 introducing extra slow demagnetization mechanism. Moreover, we discussed the \norigin of Gilbert damping, analyzed its influence on the relation between \nM  and \nintr\n and proposed a new approach to distinguish the intrinsic spin -flip and extrinsic \nspin current mechanism for ultrafast demagnetization in metallic heterostructures. The \nfinding for this unification can provid e the possibility for manipulating the \nlaser -induced ultrafast demagnetization via Gilbert damping in high frequency or \nultrafast spintronic devices  such as the Terahertz emitters .  \nFig. 1(a) shows time -resolved MOKE signals40 for films with various Ni lay er \nthickness measured with an external field  Oe. The quantitative values of \nintrinsic Gilbert damping constant41-44 in Fig. 1(b) can be obtained by eliminating the \nextrinsic contributions (See the Supplementary Materials [34],  PartⅡ for details). It \nwas observed that intr decreases with increasing Ni layer thickness. On the one hand, \nprevious investigations39, 45 have been reported that the large PMA origins from the \nstrong spin -orbit coupling effect at Co/Ni interface. A thickness modification in Co/Ni \nbilayer can change the competition between interface and volume effect, and \nconsequently the PMA. When we plot the intrinsic Gilbert damping constant as a \nfunction of effective anisotropy field  in Fig. 4 in the Part Ⅱ in Supplementary \nMaterial (See the Supplementary Materials [34],  PartⅡ for details), a proportional \nrelation was confirmed in our Co/Ni bilayer system, which demonstrates that \nspin-orbit coupling contributes to both Gilbert damping and PMA  (Also, for the \nachievement of effec tive anisotropy field, please see the Supplementary Materials [34]  \nPartⅡfor details ). On the other hand, the interface between Ni and Pt maybe also \n4000H 8 modified via changing Ni layer thickness. Because the Gilbert damping increases \nlinearly when the Ni layer b ecomes thinner, it seems that the spin current dissipation \nis involved partly. A similar trend was observed in a Pt/CoFeB/Pt system46, in which a \npure non -local spin pumping effect domains the Gilbert damping. Therefore, the total \nGilbert damping equals to  α=𝛼𝑖𝑛𝑡𝑟 +𝛼𝑠𝑝 , in which 𝛼𝑠𝑝  represents the \ncontributions from spin current. Due to the low spin diffusion length of Pt, the \nmagnetization precession in Ni layer entering the Pt layer would be absorbed \ncompletely like in the system of Py/Pt and Py/Pd47 and so on. H owever，we have to \naddress that, i n the case of the variation of ferromagnetic layer thickness, the amount \nof spin current pumped out of ferromagnet is determined entirely by the parameter of \ninterfacial mixing conductance  𝐺𝑒𝑓𝑓𝑚𝑖𝑥 48,49.  It is a constant value once the normal \nmetal thickness is fixed , although the Gilbert damping in thinner magnetic layer is \nenhanced. Therefore, given the spin current contributes partly to the Gilbert damping \nat present, the spin angular momentum transferring from Ni layer to Pt layer would be  \nthe same  for various Ni lay er thickness.       \n  The central strategy  of our study is to establish a direct correlation between \nultrafast demagnetization time and the intrinsic Gilbert damping constant. The \nintrinsic Gilbert damping constant was extracted from magnetization precessi on in \nhundreds of ps timescale. The laser -induced ultrafast demagnetization dynamics has \nbeen measured carefully within time delay of 2.5  ps at a step of 15  fs and low laser \nfluence of 1 was used. Fig. 2 (a) shows the TRMOKE signals of the ultrafast \ndemagn etization evolution after optical excitation. A rapid decrease of magnetization \n2/cmmJ 9 takes place on the sub -picosecond timescale followed by a pronounced recovery. As \ncan be seen in this figure, the ultrafast demagnetization rate is different by changing \nthe Ni  thickness.  \nTo identify the effect of the heat transport across the film thickness on \ndemagnetization time, a numerical simulation50 was carried out to demonstrate that \nthe demagnetization time  variation induced with the thicknesses ranged from 1.2 nm \nto 2.8 nm  is so small that can be ignored  (See the Supplementary Materials [34], Part \nⅢ for details), although a relatively large error of  could be resulted in when the \nsample thickness spans very large. According to the simulation results, the heat \ntransport not only affects  the rate of ultrafast magnetization loss but also the \nmaximum magnetic quenching. So, in experiment we obtain the ultrafast \ndemagnetization time for various samples with almost the sa me maximum quenching \nof 9% to suppress the influence of heat transport7, 21, 51 -54 as well as the non local spin \ncurrent effect17. The temporal evolution of magnetization in sub -picosecond time \nscale was fitted by the analytic solution based on the  phenome nological  three \ntemperature model (3TM)1, 17:  \n \n(1)  \nwhere  presents the convolution product with the Gaussian laser pulse \nprofile, whose full width at half maximum (FWHM) is . A temporal stretching of \nthe laser pulse was introduced by the excited hot ele ctrons55, which is the trigger for \nthe observed ultrafast demagnetization. In the fitting procedure, the demagnetization \nM\n),()()()(\n1)(\n321 1 2\n5.0\n01\nGt\nM EEt\nM EM EtGtAteAAeAA\ntA\nMtMM M \n\n \n\n\n\n\n\n\n\n\n \n),(GtG\nG 10 time  we cared can be influenced by the value of , which is inter -dependence \nwith   within the three temperature model. As is shown in  Table 1  in the \nSupplementary Material34 Part Ⅳ,  was fixed at 330 fs for various samples to \neliminate its relevance with . The time variable in eq. (5) corresponds to  \n, with the free fit parameter characterizing the onset of the \ndemagnetization dynamic s of the actual data trace, which is fixed as 100 fs for various \nsamples.  is a step function,  is the Dirac delta function and  are \nthe fitting constants. The two critical time parameters  are the ultrafast \ndemagnetization time and magnetization recov ery time, respectively. The well fitted \ncurves by 3TM are also shown as the solid lines in Fig. 3(a) from which the ultrafast \ndemagnetization time  and the magnetization recovery time  were evaluated. \nWithin 3TM model, the magnetization recovery process is affected by , \ncharactering the electron -phonon relaxation, and , representing heat transport \ntimescale through the substrates as well as demagnetization time . In the fitting \nprocedure by 3TM model, we assigned a fixed value to  and varies slightly to \nexclude the heat transport effect through thickness. Via changing the single \nparameter ， , we can accurately reproduce the experimental results for various \nsamples. And the heat transport across the thickness domains within 3TM model \ncharacterized by the  parameter of , which is shown in  Table. 1 in  Part Ⅳ of \nSupplementary Material34 as around 2 ps. It is about three times bigger than  \nindicating that we are not mixing the heat transport and the electron -phonon \nrelaxation56. Only in this case, are both th e values of and  genuine. The value \nM\nG\nM\nG\nM\n0 expt tt\n0t\n)(t\n)(t\n3 2 1,,AAA\nE M,\nM\nE\nE\n0\nM\nE\n0\nM\n0\nE\nE\nM 11 of  indicates that the heat was transferred through the substrate in less than 3 ps in \nthis paper, rather than what was observed by F. Busse et al57 where the heat was \ntrapped laterally in the Gaussian profile up to 1  ns. Therefore, the lateral heat \ntransport effect can be ignored, and hencely the modification of precessional \ndynamics here. As illustrated in Fig. 2(b), it can be clearly seen that decreases with \nincreasing dNi.  \n      \nBy replotting Fig. 1(b) and Fig. 2(b), an approximately proportional \nrelationship between and intr was confirmed by our experimental results \n(Fig. 2(c)). This relationship between intr and is consistent well with the \ntheoretical prediction  based on the breathing Fermi -surface model30,31,58 \nfor materials with conductivity -like damping contributions. On the basis of the \nbreathing Fermi -surface model, the Elliott -Yafet spin -mixing parameter 𝑏2 in Co/Ni \nbilayers can be estimated from the theoretical equation30, 31 shown as the red solid l ine \nin Fig. 2(c): \n                                              (2) \nwhere the quantity contains the derivatives of the single -electron energies with respect \nto the orientation e of the magnetization M=Me. p is a material -specific parameter \nwhich should be close to 4. If we use =   from ab initio density \nfunctional electron theory calculation for fcc bulk Ni31, the experimental value of \nElliott -Yafet spin -mixing parameter 𝑏2 = 0.28 can be estimated in Co/Ni bilayers, \nwhich is far larger than that of Co or Ni. The significant enhancement of spin -mixing \n0\nM\nM\nM\nint Mr\n2pbFM\nelM\nelF\nJ231087.1 12 parameters is related to the strong spin -orbital coupling at the Co/Ni interface since b2 \nis proportional to 2 in first -order  perturbation theory, where  is the coefficient of the \nspin-orbit coupling. A detailed ab initio calculation  for Elliott -Yafet spin -mixing \nparameter in Co/Ni bilayers is highly desirable.  For a derivation of eq. (2) it must be \nassumed that the same types of spin-flip scattering processes are relevant for the \nultrafast  demagnetization and for the damping. The assumption does not say anything  \nabout these  detailed  types. It has been shown in Ref. 9 that mere electron -phonon  \nscatterings cannot explain the expe rimentally observed demagnetization  quantitatively. \nIn reality there are also contributions from  electron -electron scatterings11, \nelectron -magnon scatterings12 and from a combination of electron -phonon and \nelectron -magnon scatterings13. Because both for de magnetization and for damping ，\nthe spin  angular momentum  has to be transferred from the electronic spin system to \nthe lattice,  there is no reason why different types of theses spin -flip scatterings  should \nbe relevant for the two situations. Therefore , the Elliott -Yafet  relation, eq. (2) should \nbe applicable for our system. It would not be  valid if non -local spin -diffusion \nprocesses would contribute a lot to  demagnetization. Examples are a superdiffusive \nspin current in the  direction perpendicular to th e film plane, or a lateral diffusion out \nof the spot irradiated by the laser pulse and investigated by the TRMOKE.  However, \nwe definitely found the validity of the Elliott -Yafet relation,  and this shows that \nnonlocal spin -diffusion processes are so small t hat can be neglected in our \nexperiment.  \nDespite this , previous demonstrations17,19-21 show that the ultrafast spin current  13 caused by the transport of spin -majority and spin -minority electrons in the antiparallel \n(AP) state of magnetic multilayers  after the  laser pulse  accelerates the ultrafast \ndemagnetization. Similarly, as is indicated in Fig. 1(b), with the assistance of interface \nbetween FM (Ni) and NM (Pt), the spin current induced by the flow of spin -up and \nspin-down electrons in opposite directions59 may contribute partly to the Gilbert \ndamping in Pt/Co/Ni/Pt mulitilayers.  The femtosecond laser induced spin current lives \nvery shortly which is in sub -picosecond timescale, while the duration of spin current \ntriggered by spin precession is in the timescal e of nanosecond. The difference of the \nduration of the spin current is just related to the timescale of the perturbation of the \nsystem. One has to note that spin currents at the femtosecond time scale gives rise to a \nlowering of the demanetization time17, while spin pumping induced spin current gives \nrise to the enhancement of Gilbert damping and thus a lowering of the relaxation time. \nTherefore, when spin current contributes largely to both ultrafast demagnetization and \nspin precession dynamics, an inverse  relationship between ultrafast demagnetization \ntime and Gilbert damping could be expected. That is, t he more spin current \ntransferred from ferromagnetic layer to normal metal, the faster ultrafast \ndemagnetization should be. Therefore, a t present paper, to  explain the experimental \nresults the local Ellio tt-Yafet scattering theory suffices. And , the non -local spin \ncurrent effect can be ignored, although it contributes partly to the fitted value of \nspin-mixing parameter  𝑏2 . The discussions here inspire us t o continuously  clarify \nthe various relationships between ultrafast demagnetization time and Gilbert dam ping \ncoming from different microscopic mechanisms, which is helpful for understanding  14 the underlying physics of ultrafast spin dynamics as well as the ap plication of ultrafast \nspin current triggered by ultrashort laser60, 61. For instance, recently, the researchers \nare seeking for the potential candidates as the Terahertz waves emitters including the \nmetallic heterostructures. Previous demonstrations show that the magnitude and \nefficiency of Terahertz signals in these multilayers are determined by Gilbert \ndamping60. The investigations of the relationship between Gilbert damping and \nultrafast demagnetization time will open up a new avenue to tailor the Terah ertz \nemission.  \nMeanwhile , the dominant contribution to ultrafast demagnetization in metallic \nheterostructures, either from the localized spin -flip scattering or non -local spin \ntransport, has been a controversial issue for a long time23. Here, a new approa ch, by \nestablishing the relation between the demagnetization time and Gilbert damping, is \nproposed to distinguish the two mechanisms . The proportional relationship indicates \nthe localized spin -flip scattering mechanism domains, otherwise the nonlocal spin \ncurrent domains.          \nIn conclusion, the fast and ultrafast dynamic properties of Ta(3 nm)/Pt(2 \nnm)/Co(0.8 nm)/Ni( dNi nm)/Pt(1 nm)/Ta(3 nm) bilayers with  the electrons  relaxing \nnear the Fermi surface have been investigated by using TRMOKE pump -probe \ntechnique. An genuine  proportional relationship , contrast to previous trivial \nconsequence induced by impurities mechanism,  between ultrafast demagnetization \ntime and Gilbert damping constant is confirmed fr om experimental results. The \nestimated value of spin -mixing parameter on the basis of breathing Fermi -surface  15 model is far larger than that of Co or Ni , which is  originated from the strong \nspin-orbital coupling at the interface. More importantly, distingui shing the dominant \nmechanism underlying ultrafast demagnetization in metallic heterstructures has been a \ntough task for a long time. Here, an effective method by unification of the ultrafast \ndemagnetization time and Gilbert damping is proposed to solve thi s task, namely that, \na proportional relation  between the two parameters indicates the local spin flip \nscattering mechanism domains, otherwise the non local spin current effect domains.   \n   16 Acknowledgments  \nThis work was supported by the National Basic Research Program of China (973 \nprogram, Grant Nos. 2015CB921403 and 2016YFA0300701), the National Natural \nSciences Foundation of China (51427801, 11374350, and 11274361). The authors \nthank Hai -Feng Du, Da -Li Su n and Qing -feng Zhan for critical reading and \nconstructive suggestions for the manuscript.  The authors are indebted to B. Koopmans \nand M. Haag for helpful discussions.  \nAuthor Contributions  \nZ.H.C. supervised project. Z.H.C. and W.Z conceived and designed th e \nexperiments. W.Z. and W.H. performed the polar Kerr loops and TRMOKE \nmeasurement. 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Macizo, Modeling of ultrafast laser -induced magnetization dynamics within \nthe Landau -Lifshitz -Bloch approach , PhD Thesis 2012. P.78  \n55. B. Vodungb o, B. Tudu, J. Perron, R. Delaunay, L. Mü ller, M. H. Berntsen, G. \nGrü bel, G. Malinowski, C. Weier, J. Gautier, G. Lambert, P. Zeitoun, C. Gutt, E. \nJal, A. H. Reid, P. W. Granitzka, N. Jaouen, G. L. Dakovski, S. Moeller, M. P. \nMinitti, A. Mitra, S. Carron, B. Pfau, C. von Korff Schmising, M. Schneider, S. \nEisebitt, and J. Lü ning, Sci. Rep, 6, 18970 (2016).  \n56. F. Dalla Longa, J. T. Kohlhepp, W. J. M. de Jonge, and B. Koopmans, Phys. Rev. \nB, 75, 224431 (2007).  \n57. F. Busse, M. Mansurova, B. Lenk, M. von der Ehe and M. Mü nzenberg, Sci. Rep. \n5, 12824 (2015).  \n58. K. Gilmore, Y. -U. Idzerda, and M. -D. Stiles, Phys. Rev. Lett. 99, 027204 (2007).  \n59. I. Zutic and H. Dery, Nature Mater. 10, 647 (2011).  \n60. J. Shen, X. Fan, Z. Y. Chen, M. F. DeCamp, H. W. Zhang, and J. Q. Xiao, Appl. \nPhys. Lett. 101, 072401 (2012).  \n61. T. Seifert, S. Jaiswal, U. Martens, J. Hannegan, L. Braun, P. Maldonado, F. \nFreimuth, A. Kronenberg, J. Henrizi, I. Radu, E. Beaurepaire, Y. Mokrousov , P. \nM. Oppeneer , M. Jourdan , G. Jakob , D. Turchinovich , L. M. M. Hayde n , M. \nWolf , M. Mü nzenberg , M. Klä ui , and T. Kampfrath ,  Nat. Photonics. 10, 483 \n(2016).   23  \n \n \nFigure caption:  \nFIG. 1 Spin precession.  (a)TRMOKE signals of Co/Ni bilayers with dNi=0.4-2.0 nm \nin applied field H = 4000 Oe. (b) Intrinsic Gilbert damping constant as a function of \ndNi. \nFIG. 2 Ultrafast demagnetization.  (a) Ultrafast demagnetization curves with various \nNi layer thickness. (b) Ultrafast demagnetization time as a function of Ni layer \nthickness. (c) Ult rafast demagnetization time as a function of Gilbert damping \nconstant. The red full line indicates theoretical fitting .  \n   24  \n \n \n \n \n                  \n \n \n \n \n \n \n \nFig. 1 (Color Online)  Spin precession.  (a)TRMOKE signals of Co/Ni bilayers \nwith dNi=0.4-2.0 nm in applied field H = 4000 Oe. (b) Intrinsic Gilbert damping \nconstant as a function of dNi. \n  \n 25  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig.2 (Color Online)  Ultrafast demagnetization.  (a) Ultrafast demagnetization \ncurves with various Ni layer thickness. (b) Ultrafast demagnetization time as a \nfunction of Ni layer thickness. (c) Ultrafast demagnetization time as a function of \nGilbert damping constant. The red full line indicates theoret ical fitting.  \n \n \n 26  \nSupplementary Information  \n \nUnifying ultrafast demagnetization and intrinsic Gilbert damping in Co/Ni \nbilayers with electronic relaxation near the Fermi surface  \n \n \n \nPartⅠ \n \n \nThe measurements of static properties for Ta (3 nm)/Pt (2 nm)/Co (0.8 nm)/Ni \n(dNi nm)/Pt (1 nm)/Ta (3 nm) . \n \nFig. 1(a) shows the polar magneto -optical Kerr signal measured at room \ntemperature with maximum applied field of 300 Oe. The static polar Kerr loops of \nCo/Ni bilayers were acquired using a laser diode with a wavel ength of 650 nm.  All \nsamples show very square loops  with a remanence ratio of about 100% , indicating t he \nwell-established perpendicular magnetization anisotropy ( PMA) of the samples. The \nmeasured coercivity Hc decreases with dNi from 103Oe for dNi = 0.4 nm to 37Oe for \ndNi =2.0 nm (Fig. 1(b)). The decrease of coercivity implies that the PMA decreases \nwith the thickness of Ni.  \n \n  27  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig.1  Static magnetic properties of of Ta (3 nm)/Pt (2 nm)/Co (0.8 nm)/Ni (d Ni \nnm)/Pt (1 nm)/Ta (3 nm) bilayers.  (a) Polar -MOKE loops with various thickness of \nNi layer d Ni. (b) Coercivity Hc and effective anisotropy field as a function of Ni \nlayer thickness d Ni. \n \neff\nKH\n 28  \n \n \n \n \nPartⅡ \n \nThe measurements of spin dynamics for Co/Ni bilayers in ns timescales and the \nanalysis of extrinsic contributions to spin precession  \n \nIn this part, we show the details of spin precession experiment.  For example, Fig. \n2(a) illustrates the scheme for laser -induced magnetization precession. The direction \nof applied field is fixed at .  \nThe typical time -resolved magnetization dynamics with various applied fields for \nTa(3 nm)/Pt(2 nm)/Co(0.8 nm)/Ni(0.8 nm)/Pt(1 nm)/Ta(3 nm) shown in Fig. 2(b) can \nbe best fitted by using the damped harmonic function added to an \nexponential -decaying background1:  \n                (1) \nwhere A and B are the background magnitudes, and  is the background recovery rate. \nC, , f and  are magnetization precession amplitude, relaxation time, frequency and \nphase, respectively. From the f itting curves shown in Fig. 2(b) as the solid lines, the \nvalues of precession frequency f and relaxation time  are extracted. Since the applied \nfields are large enough, we can obtain the Gilbert damping constant using the \nfollowing relationship2  \n                                                (2).  \n80H\n( ) exp( ) exp( )sin(2 )tM t A B t C ft         \n\n1)2( f 29 In the case of films with a relatively low Gilbert damping3-7 as well as thickness \nlarger than the optical penetration depth8, ultrafast laser may generate non -uniform \nspin waves and  affect the relation ship between demagnetization and Gilbert damping \nas extrinsic contributions . In order to check the contribution of non -uniform modes, \nwe performed a fast Fourier transform shown in Fig. 2(c). Only the uniform \nprecession mode was excited at present Co/Ni bi layers with perpendicular magnetic \nanisotropy.     \nBoth  and f as a function of H are plotted in Fig.3. Since the overall damping \nconstant consists of intrinsic damping and extrinsic damping whereby the second one \narises from inhomogeneities in the sample , the Gilbert damping constant decreases \nmonotonously to a constant value as the applied field increases (Fig. 3(a)).  In the low \nexternal fields range, the inhomogeneously distributed anisotropy may lead to higher \n values. Fortunately, the sufficient high  field we used can suppress the extrinsic \ncontributions to the magnetization precession, because for high fields the \nmagnetization dynamics is mainly determined by the external field9. In addition, \nbecause of the interaction between femtosecond laser sourc e and the thin films, the \nlateral heat distribution across the film plane has to be considered as another  \ncandidate contributions to affect the processional dynamics. As is shown by F. Busse \net al6, the heat was trapped as the Gaussian distribution across the film plane of \nCoFeB up to 1 ns due to the use of regenerative amplifier. It can enhance the laser \npower largely while the pump laser spot kept as large as around 90  μm. This \nfacilitates the occurrence of the temperature profile, and consequently the sp in-waves  30 in the range of laser spot size. However, in the absent of regenerative amplifier at \npresent, the laser spot is so small as less than 10 1,10 that one can excite the \nnonequilibrium state of the samples. And the laser fluence used here is around 1\n, which is far weaker than that used in previous report6. Although smaller \nlaser spot seems easier to trigger the nonuniform spin waves, the very low laser power \nwe used here can suppress the influence of lateral heat distribution on the relaxation \ntime o f spin dynamics at present.  M oreover, the absence of non -uniform spin wave \ndemonstrated in Fig. 2(c) in the pump laser spot confirms that the lateral heat \ntransport can be neglected here. In fact, it is found in the main text, within the three \ntemperature  model  (3TM model) describing the ultrafast demagnetization dynamics, \nthat the heat induced by laser pulse mainly transports along the thickness direction to \nsubstrate in less than a few picoseconds. The observation of pronounced \nmagnetization recovery aft er ultrafast demagnetization can exclude the possibility of \nlateral heat trap.     \n   In order to avoid the effect of extrinsic damping constant, the intrinsic \ndamping constants were obtained by fitting the overall damping factor as the function \nof applied  fields with the expression  shown as the red line in Fig. 3(a) : \n                                            (3) \nwhere  and are the intrinsic and extrinsic parts of the damping factor, \nrespectively. The intrinsic part is independent of the external field  or precession \nfrequency, while the extrinsic part is field -dependent.  \nm\n2/cmmJ\n0/\nint 1HH\nrae \nintr\n0/\n1HHe 31 The experimental f-H relation in Fig. 3(b) can be fitted by analytic Kittel formula \nderived from LLG equation2: \n                                                (4) \nwhere , . The \nequilibrium angle of magnetization  was calculated from the relationship\n. The direction of applied field is fixed at . In the \nabove equations, and are the effective perpendicular magnetization \nanisotropy and gyromagnetic ratio, respectively, wher e , . In \nour calculation, the Lande factor  was set to 2.2 as the bulk Co value2. is the \nonly adjustable parameter. The variation of effective field with the thickness of Ni \nlayer was also plotted in Fig. 1(b).  When we plot the intrinsic Gilbert damping \nconstant as a function of effective anisotropy field  in Fig. 4, a proportional relation \nwas confirmed in our Co/Ni bilayer system, which demonstrates that spin -orbit \ncoupling contributes to both Gilbert damping and PMA .   \n \n \n \n \n \n \n \n2 12HH f\n 2\n1 cos ) cos(eff\nK H H H H  \n  2cos ) cos(2eff\nK H H H H  \n\n) sin(22sin    H eff\nKHH\n80H\neff\nKH\n\nseff eff\nKMKH2\n2Bg\nh\ng\neff\nKH 32  \n \n \n \n \n \n \n \n \n \n \nThe numerical simulation for ultrafast demagnetization  \n \n \n \n \n \nFig. 2   (a) Scheme of TRMOKE. (b): TRMOKE signals with various applied \nfield for Ta (3 nm)/Pt (2 nm)/Co (0.8 nm)/Ni (0.8 nm)/Pt (1 nm)/Ta (3 nm) \nbilayers. (c): Fast Fourier transform ation s ignals.  \n \n \n \n \n \n \n 33  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 3  Gilbert damping and precession frequency. Field dependence of overall \ndamping constant (a) and precession frequency (b) of Co/Ni bilayers with\n \n \nnm dnm dNi Co 8.0 ,8.0  \n 34  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 4  Dependence of intrinsic Gilbert damping constant on the effective \nanisotropy field.  \n \n \n \n \n \n \n \n 35  \n \n \n \nPartⅢ \n \nNumerical simulation for the effect of heat transport across the film thickness \non the ultrafast demagnetization time  \n \n \nTo estimate the evolution of heat  transport profile in time, w e carried out a \nnumerical simulation based on M3TM11 model, in which the heat transport12 was \ndominated by electrons and a temperature gradient across the film thickness was \nintroduced.  It is divided in thin slabs in the direct ion normal to the film plane, and the \nslabs is 0.1 nm thick. For each slab, the evolution of the electron and phonon \ntemperatures \neT  and \npT  are determined by a  set of coupled differential equations :13  \n)),)(coth()( 1()()()()),( )(()()),( )(( ))( ()()(\nzTmTzmTzTzRmdtzdmzTzTgdtz dTCzTzTg zTdtzdTzT\nec\ncpp e epp\npe p ep ez ze\ne\n    \n                (5) \n \nWhere \nsMMm ,\n)()(\n0zTzT\npe 4, \n228\nD atB atcBep sf\nEVTkgaR ,with \nat the atomic \nmagnetic moment in units of Bohr magneton \nB , \natV  the atomic volume, and \nDE is \nthe Debye energy.  \neC and \npC  are the heat capacities of the e and p systems \nrespectively. \n)(zTez is the electron temperature gradient normal to the film . \nBk is  36 the Boltzmann constant. \n0k is the material dependent electronic thermal conductivity. \nepg\nis the e -p coupling constant and determines  the decay of the electronic \ntemperature until equilibrium is  reached14. \nsfa represents the spin -flip probability11. \nThe equations of motion for each slab thus describe heating of the electron system by \na Gaussian laser pulse, heat diffusi on by electrons to neighboring slabs, e -p \nequilibration, and finally the evolution of the magnetization due to e -p spin -flip \nscattering. In the simulation, the total magneto -optical signal was obtained by the \ncalculation of \ndzztzm t ) exp(),( )(  .  \nThe electronic system after the action of the laser pulse is in a strongly  \nnon-equilibrium situation. Nevertheless, one can describe the electron  system by use \nof an electron temperature. The reason is that the laser  photons excite electrons, but \nthese excite d electrons thermalize more or  less instantly due to very rapid and \nfrequent electron -electron  scatterings via their Coulomb interactions. This is the \nassumption of the  accepted Elliott -Yafet scenario  which describes the  effect of the \nlaser pulse directly after the action of the laser pulse.   \nFig.4(a) shows the simulated ultrafast demagnetization curves for various film \nthicknesses. We can clearly observe that the evolution of magnetization curves looks \nalmost identical for various film thicknesses , indicating that the effect of heat \ntransport on the demagnetization time can be neglected. Despite this, for the \nremagnetization part, a deviation from the experimental curves occurs. This is mainly \nbecause that the heat diffusion can almost be neglected d uring the ultrafast \ndemagnetization timescale, but starts playing an increasing role from ps timescale  37 onwards. The similar phenomenon was reported previously by B. Koopmans et al. \nFortunately, what we should be focused on here is in the ultrafast demagnet ization \ntimescale, in which the effect of heat transport can be neglected.  In fact, as is shown \nin Fig. 4(b), less than 10 fs variation was induced with the thicknesses ranged from \n1.2 nm to 2.8 nm . The parameters used in the simulation is given in Table.1 .   \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  38  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 4(a) Dependence of demagnetization as a function of delay time after pulsed laser \nheating at \n0t  (b) Maximum demagnetization and demagnetization time  \nversus the sample thickness.  \n \n \n \n 39  \n \n \n \n \n \nTable 1: Parameters used in the M3TM12,13,15. \n \nParameters  Value  Units  \n\n 5400  \n) /(23KmJ \npC\n \n61033.2  \n) /(3KmJ  \nepg\n \n181005.4  \n) /(3sKmJ  \nDE\n 0.036  \neV  \nat\n 0.62  \ncT\n 630 \nK  \n0\n 90.7 \n) /(smKJ \nsfa\n 0.185  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  40  \n \n \n \n \n \n \nPart  Ⅳ \n \nTable . 1 Values of the main fit parameters of ultrafast demagnetizations curves \nfor various thicknesses of the samples.  \n \n \n \n \n \n \n \n \nReferences :  dNi (nm)       \n0.4 200 860 2.3 330 100 \n0.8 170 860 2.1 330 100 \n1.0 150 860 2.0 330 100 \n1.5 120 860 2.3 330 100 \n2.0 90 860 2.0 330 100 \n)(fsM\n)fsE（\n)(0ps\n)(fsG\n)(0fst 41 1、W. He, B. Hu, Q. F. Zhan, X. Q. Zhang, and Z. H. Cheng, Appl. Phys. Lett. 104, \n142405 (2014).  \n2、H. S. Song, K. D. Lee, J. W. Sohn, S. H. Yang, Stuart S. P. Parkin, C. Y. You, and \nS. C. Shin, Appl. Phys. Lett. 103, 022406 (2013).  \n3、Y. Au, M. Dvornik, T. Davison, E. Ahmad, P. S. Keatley, A. Vansteenkiste, B. Van \nWaeyenberge, and V.V. Kruglyak, Phys. Rev. Lett. 110, 097201 (2013).  \n4、C.Y. Cheng, K. K. Meng, S. F. Li, J. H. Zhao, and T. S. Lai, Appl. Phys. Lett. 103, \n232406 (2013).  \n5、Y. Au, T. Davison, E. Ahmad, P. S. Keatley, R. J. Hicken, and V. V. Kruglyak, \nAppl. Phys. Lett. 98, 122506 (2011).  \n6、F. Busse, M. Mansurova, B. Lenk, M. von der Ehe and M. Mü nzenberg, Sci. Rep. 5, \n12824 (2015).  \n7、B. Lenk, G. Eilers, J. Hamrle, and M. Mü nzenberg, Phys. Rev. B 82,134443 \n(2010).  \n8、M. van Kampen, C. Jozsa, J.T. Kohlhepp, P. LeClair, L. Lagae, W. J.M. de Jonge, \nand B. Koopmans, Phys. Rev. Lett.  88, 227201 (2002).  \n9、S. Mizukami, F. Wu, A. Sakuma, J. Walowski, D. Watanabe, T. Kubota, X. Zhang, \nH. Naganuma, M. Oogane, Y. Ando, and T. Miyazaki, Phys. Rev. Lett. 106, 117201 \n(2011)  \n10、W. He, T. Zhu, X. -Q. Zhang, H. -T. Yang, and Z. -H. Cheng, Sci. Rep. 3, 2883 \n(2013).   42 11、B. Koopmans,  G. Malinowski, F. Dalla Longa, D. Steiauf, M. Fä hnle, T. Roth, M. \nCinchetti and M. Aeschlimann, Nat. Mater . 9, 259265 (2010).  \n12、K. C. Kuiper, G. Malinowski, F. Dalla Longa, and B. Koopmans, J. Appl. Phys \n109, 07D316 (2011).  \n13、K. C. Kuiper, T. Roth, A. J. Schellekens, O. Schmitt, B. Koopmans,  M. Cinchetti, \nand M. Aeschlimann, Appl. Phys. Lett. 105, 202402 (2014).  \n14、L.I. Berger, Optical properties of selected inorganica and organic solids, in \nHandbook of Chemistry and Physic s, 88th Edition, p. 12 -144 - 12-159. 3.1.5, 3.4  \n15、U. A. Macizo, Modeling of ultrafast laser -induced magnetization dynamics within \nthe Landau -Lifshitz -Bloch approach , PhD Thesis 2012. P.78  \n \n \n "    },    {        "title": "1601.02210v3.Interfacial_Dzyaloshinskii_Moriya_interaction__surface_anisotropy_energy_and_spin_pumping_at_spin_orbit_coupled_Ir_Co_interface.pdf",        "content": "1 Interfacial Dzyaloshinskii -Moriya interaction, surface anisotropy  energy,  \nand spin pumping at spin orbit  coupled Ir/Co interface  \n \nNam -Hui Kim,1 Jinyong Jung,1 Jaehun Cho,1 Dong -Soo Han,3 Yuxiang Yin,3 June-Seo \nKim,3,a) Henk J. M. Swagten,3 and Chun -Yeol You1,2,b) \n1Department of Physics, Inha University, Incheon , 22212 , South Korea  \n2Department of Emerging Materials Science, DGIST, Daegu , 42988 , South  Korea   \n3Department of Applied Physics,  Center for Nano materials, Eindhoven University of \nTechnology, PO Box 513,  5600 MB Eindhoven, The Netherlands  \n \nThe interfacial Dzyaloshinskii -Moriya interaction  (iDMI) , surface anisotropy  energy,  and \nspin pumping at the Ir/Co interface  are experimentally investigated  by performing Brillouin \nlight scattering.  Contrary to previous reports, we suggest that the sign of  the iDMI at the \nIr/Co interface is the same as in  the case of the Pt/Co  interface . We also find  that the \nmagnitude of the iDM I energy density is relatively smaller than in the case  of the Pt/Co \ninterface , despite the large strong  spin-orbit coupling (SOC)  of Ir . The saturation \nmagnetization and the perpendicular magnetic anisotropy (PMA) energy are significantly \nimproved due to a strong  SOC . Our finding s suggest  that a n SOC in an Ir/Co system behaves \nin different ways for iDMI and PMA. Finally , we determine the spin pumping effect at the \nIr/Co interface , and it increases  the Gilbert damping constant from 0.01 2 to 0.024  for 1.5 nm-\nthick Co.   \n  \n                                          \na) E-mail: spin2mtj@gmail.com   \nb) E-mail: cyyou@inha.ac.kr  2 Spin-orbit coupling (SOC)  plays a crucial role  in various basic magnetic phenomena such as  \nmagnetic crystalline anisotrop ies, magnetostriction  effect s, magneto -optical Kerr effect s, \nanomalous Hall effects, anisotropic magnetoresistance s, and magnetic damping processes .1 \nRecently the  SOC has been of growing interest due to  novel physics and  emerging \ntechnolog ies of “spin  orbitronics” such as  the spin Hall effect,2,3,4 the Rashba effect,5,6 the \nDzyaloshinskii -Moriya interaction (DMI) ,7,8,9 and spin pumping effects .10,11 Therefore, an \nunderstanding  the SOC , in particular  at the interface between ferromagnetic (FM) material \nand heavy metals (HM) , is of great importance for an investigation of the underlying physics \nof this exotic phenomena . Among these SOC -related phenomena, a n interfacial DMI (iDMI) \nhas drawn recent interest . The iDMI  is known to arise at the interface due to inversion \nsymmetry breaking and large SOC in heavy metals (HM). The iDMI  manifest s itself by \nforming spiral spin configurations with a preferred chirality . Therefore, it plays  an important \nrole in the dynamics of the chiral domain wall (DW)12,13\n and the skyrmion formation.14,15 \nExperimental  determination of the iDMI energy density is needed. H owever, general \napproach es for the iDMI measurements (i.e., asymmetric domain wall motion or bubble \nexpansion ) have been difficul t to obtain  the exact value of  the DMI energy density.16,17,18  \nRecently, direct  measurement s of the iDMI via Brillouin light scattering (BLS)  have been \ndemonstrated in Pt/Co/AlO x, Pt/CoFeB/AlO x,19 and Ta/Pt/Co/AlO x20 systems . Based on the \nnon-reciprocal spin wave (SW) theorem under the iDMI,21 BLS can provide a direct value of \nthe iDMI energy density from a measurement of the frequency difference between negative \n(Stokes) and positive (anti -Stokes) SW frequencies . Therefore, BLS is considered to be a \nsuperior approach for the study of iDMI.22,23,24 Since BLS is able to  measure intrinsic \nmaterial parameters such as the  saturation magnetization  (MS), the effective magnetic \nanisotropy energy, and the Gilbert damping constant simultaneously, it is suitable for the \nmeasurement of complex SOC -related phenomena accompanying perpendicul ar magnetic \nanisotropy (PMA), the enhancement of Gilbert damping by  spin pumping, and the \nenhancement of saturat ion magnetization by the proximity effect. For example, it is well \nknown that PMA is proportional to the square of SOC strength ,25,26 whereas spin pumping is \nproportional to the SOC strength.10,11 Furthermore, the enhancement of proximity -induced \nmagnetization  (PIM) is also related to SOC.27  \n  3 In this Letter , we experimentally investigate the interfac ial magnetic properties of an \nIr/Co( tCo)/AlO x system by BLS .19,20 Iridium, which is a  heavy metal , was chosen to compare \nthe SOC at the interface between Cobalt and HM. From systematic BLS measurements, a \nlarge  value of  MS due to  a proximity  effect  and an enhanced  value of  KS due to  SOC at  the \nIr/Co interface were  observed. We measure d iDM I energy densities  from the magnetic field-\ndependent frequency difference measurements , and f ound  that they are  smaller than in the \ncase of the Pt/Co interface. Contrary to  previous reports,28,29,30 the measured sign of the \niDMI is the same with Pt/Co /AlO x systems . Furthermore, a noticeable enhancement of the \nGilbert  damping constant  due to a strong  spin pumping  effect  was observed . This fact implies \nthat the role s of SOC for iDMI, PMA, proximity -induced magnetization, and  a spin pumping  \neffect  are not the same based on our experimental observations.   \nThe inset in Fig. 1(a) shows the sample geometry . A wedge -shaped Ta(4 nm)/Ir(4 \nnm)/Co( tCo)/AlO x(2 nm) sample was deposited o n a thermally -oxidized Si  wafer . A \nmagnetron sputtering system was used to deposit all the layers. Especially, t he Co layer was \ndeposited wedge d shape in the range of 1 to 3 nm. In order to break the inversion symmetry, a \n2-nm AlO x capping was used on the top of the Co layer. In our BLS measurement s, a p-\npolarized laser  with 300 mW of power and a 532 nm wavelength was used to create and \nannihilate magnons at the surfaces of the Co layer, which is called the Damon -Eshbach \nsurface mode.31 More  detailed descriptions about BLS measurements are given  in the \nreferences .19,32,33  \nFigure 1(a) indicates the frequency differences ( f) between Stokes and anti -Stokes peak \npositions in BLS spectra as a function of the Co thickness. Due to the non -reciprocal SW \npropagation properties with finite iDMI, f is directly proportional to the iDM I energy \ndensity  (D). The correlation between f and the D is given by21 \n∆𝑓=2𝛾𝐷\n𝜋𝑀s𝑘x                                                    (1) \nwhere  𝑘x and D are the propagating spin wave  (SW)  wave  vector along the x-direction and \nthe iDM I energy density, respectively.  In this study, the in-plane k-vector is fixed at 𝑘x = \n0.0167 nm-1, corresponding to the back -scattered light by thermal excitation with an angle of \nincidence of 45o. 4  In Fig. 1(b), the deduced iDM I energy densities based on  Eq. (1) are plotted as a function \nof 𝑡Co−1. The figure  shows inverse proportionality, which is direct evidence that the iDMI at \nthe Ir/Co bilayer is generated  at the interface .19,20 We note  that the iDM I energy densities are \nmuch smaller than in the case of the Pt/Co interface , as shown in Fig. 1(b) . \nNext, we consider ed the magnitude and sign of iDMI for  an Ir/Co /AlO x system. Recently, \nusing  the Vienna ab initio  simulation package (V ASP), Yang et al. 30 reported that the iDMI \nof an Ir/Co system is much smaller than those of Pt/Co systems . They also found that the \nmagnitude and sign of  the iDMI at Ir/Co is very small and has a direction  opposite to the  case \nof the  Pt/Co  interface , respectively. This is consistent with reported experimental \nobservation s of a Pt(111)/Ir( tIr)/[Ni/Co] N and Ta/Pt/Co/Ir( tIr)/Pt system.17,28 Chen et al.28 \nobserved real space images by spin -polarized low -energy electron microscopy (SPLEEM), \nand they found  the transition of DW chirality to be a function of  an Ir inserting layer between \nPt and [N i/Co] N. This implies that the iDMI of the interface of Ir/[Ni/Co] N is opposite to that \nof Pt/[Ni/Co] N. Hrabec et al.17 used a magneto -optical Kerr effect (MOKE) micr oscope to \nobserve asymmetric magnetic domain expansion due to the iDMI , and they inserted  an Ir \nlayer in to the  Ta/Pt/Co/Ir( tIr)/Pt system.  Based on the asymmetric DW velocities , they \nconclude d that the sign of Co/Ir is opposite that of the Co/Pt interfaces.  \nContrary to previous theoretical and experimental result s,17,28,30 our experimental data \nindicate s that the sign of iDMI at the Ir/Co /AlO x interface  is the same with the Pt/Co/AlO x, \nPt/CoFeB/AlO x, and Ta/Pt/Co/AlO x system s.19,20 In order to cross -check  our field dependence \nf measurements  (see Fig. 1(b) ), SW propagation angle  dependence measurements  were \nperformed . The relation between f and 𝜃α, the angle b etween the direction of the \npropagating SWs and the direction of the applied field , is given by34\n∆𝑓(𝜃α)=∆𝑓0sin𝜃α,          (2) \nwhere f0 is the maximum  f when𝜃α equals  ±π/2; this  is a typical measurement geometry  \nfor our magnetic field and SW wave  vector dependence measurements . The angle 𝜃α-\ndependent f measurements from  -90o to +90o with a 22.5o interval  for the Ir/Co ( tCo = 1.75 \nnm and 2.0 nm) and Pt/Co ( tCo = 1.8 nm) cases are shown  in Fig. 2.  All sinusoidal fitting \ncurves are well matched with the measured f as a function of 𝜃α. The green area  showing  5 that f is less than about 0. 3 GHz i ndicates the limit of our BLS experiment s. Consequently , \nwe suggest  that the signs of iDM I are the same at Ir/Co and Pt/Co interfaces.  \nCurrent ly, the physical origin of  the sign of iDMI at Ir/Co /AlO x is not fully unde rstood . \nHowever, there are several possible reasons for the opposite sign reported in previous \nexperimental results .17, 28 Our measurements were conducted on Ir(4 nm) /Co/AlO x, whereas \nthe previous results had different sample stacks ; i.e., Ta/Pt/Co/Ir/Pt17 and \nPt(111)/Ir/[Ni/Co] N.28 For example, Hrabec et al. 17 inserted an Ir layer on the top of a Co \nlayer and covered it with  Pt capping . Chen et al. investigate d the Ir/Ni and Co/Ir interface,28,35 \nnot the Ir/Co interface. Based on our experimental results, we suggest that the iDM I energy \ndensity is quite sensitive to the detail s of the multilayer structures. Moreover, the iDMI is not  \nonly determined by the nearest interfaces , it may have long range characteristics , as predicted \nby Fert and Levy.36  \nWe investigat e other important and SOC -related quantities in  a Ta/Ir/Co/AlO x system by \nusing BLS measurement s. We ignored the effect of iDMI on the SW excitation frequency, \nand took the  average of the Stokes and anti -Stokes SW frequencies without  the iDMI \ncontribution . This can be expressed as37 \n𝑓SW=𝛾\n2𝜋√𝐻ex(𝐻ex−2𝐾eff\n𝜇0𝑀S)            (3) \nHere, we ignored the exchange energy and the bulk PMA contribution . 𝐾eff=2𝐾s\n𝑡Co−1\n2𝜇0𝑀S2, \nKS is the surface anisotropy energy , is the gyromagnetic ratio , and Hex is the applied \nmagnetic field.  We can extract the effective anisotropy energy Keff from the external field -\ndependent  fsw using Eq. (3), and 𝐾eff×𝑡Co versus tCo is shown in Fig. 3. From the slope and \ny-intercept, the saturation magnetization MS (=1.68±0.02×106 A/m) and the surface \nanisotropy energy  KS (=1.36 ±0.01 mJ/m2) were  determined , respectively. For tCo > 1.5 nm, \nthe effective anisotropy becomes negative such that the easy axis of the sample is in -plane.  \nSurprisingly, the obtained  values of  KS and MS at the Ir/Co interface are significantly \nenhanced in comparison with the case of  the Pt/Co  (MS= 1.42±0.02×106 A/m)  interface .20 \nEspecially, the measured MS (=1.68 ×106 A/m) is 20% greater than the bulk magnetization of \nCo (MS = 1.4×106 A/m). Experimental evidence that a large PIM exists at  the Ir/Co interfa ce 6 has recently been reported .27 The reported PIM in an  Ir/Co/Ni/Co system is 19%, which is \nquite similar to our observed value.  \nThe KS value (1.36 mJ/m2) of the Ir/Co system is noticeably enhanced compared to the \nvalues for  Pt/Co/AlO x (KS = 0.54 mJ/m2) and Ta/Pt/Co/AlO x (KS = 1.1 mJ/m2) in our previous \nreports.19,20 On a theoretical basis, it  has been reported that Ir monolayer capping induces the \nstrongest surface PMA of an Fe(001) layer .38 They found that the PIM and the PMA of Ir is \neven larger than  that of  Pt. This gives a clue regarding the observed enhancement s of the \nvalues of  MS and KS in our Ir/Co/AlO x system. Regardless , their study is about 5 d transition \nmetals with Fe , and  not Co . Furthermore, Broeder et al.  reported that KS for Ir/Co (~  0.8 \nmJ/m2) is larger than  that of  Pt/Co (0.5  ~ 0.58 mJ/m2).39 \nAnother important effect of a strong SOC is the enhancement of Gilbert damping due to a \nstrong  spin pumping  effect .10,11 The precess ion of spins  in a ferromagnetic layer induces a \nspin current in the adjacent layer and a loss of angular momentum , and causes additional \ndamping. The amount of spin pumping is closely related to the SOC through the spin flip \nrelaxation time and the interface mixing conductance. As a result, spin pumping is an \nimportant  path for the magnetic damping of HM/FM structure s. It has been reported that spin \npumping can be suppressed by int erface engineering , or by introducing a nano -oxide layer \nbetween HM and FM  by using vector -network analyzer ferromagnetic resonance (VNA -\nFMR) .11 \nWe obtain a full -width at half maximum (FWHM) from each resonance frequency spectrum  \nfrom BLS SW spectra , similar to  the VNA -FMR experiment . To extract the Gilbert damping \nconstant, we applied a modified equation which used from  FMR system  as the condition of \nthe applied large  in-plain magnetic field in a PMA system . The FWHM (∆𝑓res) has the \nfollowing relation  with the Gilbert damping constant 𝛼:  \n∆𝑓res= 2𝛼𝜇0𝛾\n𝜋𝐻ex+∆𝑓resextrinsic      (4) \nwhere ∆𝑓resextrinsic is the additional linewidth at the resonance frequency by an extrinsic \nsource . Figure 4(a) clearly shows the linear relation s between linewidths  (∆𝑓res) and the  \napplied mag netic fields  (𝐻ex) for tCo = 1.5 and 3.0 nm.  7  We examined another possible mechanism of FWHM broadening: the two -magnon \nscattering (TMS) process. It is well known that TMS occurs when 𝜃α(𝑘⃗ ∥), the angle between \nthe SW propagation direction and the in -plane field (or magnetization direction), is smaller \nthan the critical angle 𝜃c.40 Here, the critical angle 𝜃c=sin−1[𝐻ex\n𝐵0+𝐻𝑆]1\n2 = 41.2o when the in -\nplane external field Hex is 0.2 T. B0 = Hex+4πMS; HS = 2KS/Msd; and MS, KS, and d are the \nsaturation magnetization, the surface anisotropy, and the thickness of the cobalt, respectively.  \nThe necessary condition of TMS is 𝜃α(k||) < 𝜃c for k|| > 0, which corresponds to the anti -\nStokes case in our experimental geometry. Since we extracted the FHWM from the anti -\nStokes peaks and  𝜃α(𝑘⃗ ∥)=90o configuration, there is no contribution from TMS. Furthermore, \nif TMS contributes to the linewidth, we require a non -zero f for the 𝜃α(𝑘⃗ ∥)=0 case. \nHowever, we already show a negligible f for 𝜃α(𝑘⃗ ∥)=0 in Fig. 2. Therefore, we can exclude \nthe TMS contribution in the observed FWHM broadening.  \nFrom the slope s of the linear fit tings, the magnetic damping constant s 𝛼 for each Co \nthickness  were  deduced  and are shown  in Fig. 4(b). Normally, the energy dissipation  in a \nmagnetic system depends on the imaginary part of the susceptibility  of the magn etic system. \nIt has been claimed that the imaginary part of the eigenfrequency is modified by a factor of \n(1+𝑓DM(𝑘)/𝑓0(𝑘)) due to the iDMI .23 Here, 𝑓DM  and 𝑓0 are the resonance frequency with \niDMI and without iDMI, respectively. For general cases  (𝑓DM\n𝑓0≪1), the enhancement of \ndamping due to the iDMI  is not significant. However, since  the observed damping constant \nenhancement in our experiment is about double  at tCo = 1.2 nm compared with Bulk, the \nenhancement due to the iDMI is about 10%. Therefore, we can rule out a contribution due to \niDMI in the enhancement of the damping constant.  \n Figure 4(b) shows that  the 𝑡Co−1 dependen ce of the damping constant  is mainly due to  the \nspin pumping effect . Consequently, d ue to spin pumping at the Ir/Co interface, an \nenhancement of the damping constant  (= 0.02) is observed at tCo = 1.5 nm. With increasing \ntCo, the measured  decreases as shown in Fig. 4( b), since spin pumping is a kind of interface \neffect . Therefore, the spin pumping  is being s meared away when the thickness of the FM \nlayer  increases . In Fig. 4(b), t he measured versus 𝑡Co−1 and a linear dependency with a \nfinite y-intercept is seen. The physical meaning of the damping at 𝑡Co−1=0 (tCo→∞) is the 8 damping constant ( bulk) of bulk c obalt . In these  measurement s, we determined that bulk ~ \n0.012, which is in good agreement  with the magnetic damping constant for bulk Co ( 𝛼Cobulk= \n0.011).41  \nIn conclusion, we used BLS to observe  SOC -related physical quantities such as  the \ninterfacial Dzyaloshins kii-Moriya interaction , surface magnetic  anisotropy , and the magnetic \ndamping constant  accompanying the spin pumping effect  at the Ir/Co interface . From \nsystematic BLS measurement s, we suggest that the measured iDM I energy density is \nrelatively smaller than in the case of the Pt/Co interface . On the other hand, the saturation \nmagnetization and the surface magnetic anisotropy are significantly improved due to a higher \nproximity -induced magnetization. From this result, we  believe  that the iDMI and PMA \nbehave  in different way s at the Ir/Co interface . Based on the results in  previous reports , the \nsign of the iDMI at  the Ir/Co interface is the same as that of the Pt/Co interface . 9 Acknowledgement  \nThis work is supported by the Research Programme  of the Foundation for Fundamental \nResearch on Matter (FOM), which is part of the Netherlands Organisation for Scientific \nResearch (NWO) and the National Research Foundation of Korea (Grant nos. \n2015M3D1A1070467, 2013R1A1A2011936, and 2015M2A2A6021171 ). 10 Figure Captions  \nFig. 1. (a) iDM I induced spin -wave frequency differences ( △f) as fun ction of tCo. (b) iDM I \nenergy density as a function of 𝑡Co−1 for the external magnetic field dependence ( DH) in Ir/Co  \n(black squares) , Pt/Co/AlO x (blue circles) , and Ta/Pt/Co/AlO x (red triangles), respectively.  \nThe iDM I energy density  of the Ir/Co  interface has a maximum value  (0.7 mJ/m2) that is \nmuch smaller than that of Pt/Co/AlO x (1.3 mJ/m2) and Ta/Pt/Co/AlO x (1.7 mJ/m2). \n \nFig. 2. The frequency differences ( △f) between Stokes and anti -Stokes fr om 𝜃α= -90∘ to +90∘ \nin Ta/Ir/Co/AlO x (tCo=1.75 nm and  2.0 nm) and Ta/Pt/Co/AlO x (tCo=1.8 nm). The measured f \nas a function of the  𝜃α. The solid line is the fitting curve from  Eq. (2).  \n \nFig. 3. Keff × tCo versus  tCo plot with a linear fitting. KS and MS were extracted from the slope \nand y-intercept . For tCo > 1.54 nm, the effective uniaxial anisotropy becomes negative , which \nmeans the direction of the easy axis changes  perpendicular to  the in-plane.  \n \nFig. 4. (a) The linewidths as a function of Hex for Ta  (4 nm)/Ir  (4 nm)/Co( tCo)/AlO x (2 nm). \nThe black and red open symbols  are experimental values from each of spectr um, and the solid \nlines are results of linear  fitting (tCo = 1.5, 3.0 nm ). The error bars were obtained from \nLorentzian fitting of SW peaks . (b) The Gilbert damping parameters  as function of inverse Co \nthickness. 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Grains\nconsisting of FePt like hard magnetic material with two di\u000berent grain sizes d= 5 nm and\nd= 7 nm are considered and simulations of heat-assisted magnetic recording (HAMR) are\nperformed with the atomistic simulation program VAMPIRE. The simulations display that\nthe SNR saturates for damping constants larger or equal than 0.1. Additionally, we can\nshow that the Gilbert damping together with the bit length have a major e\u000bect on the SNR\nwhereas other write head and material parameters only have a minor relevance on the SNR.\nI. INTRODUCTION\nThe next generation recording technology to increase\nthe areal storage density of hard drives beyond 1.5 Tb/in2\nis heat-assisted magnetic recording (HAMR)1{6. Higher\nareal storage densities (ADs) require smaller recording\ngrains. These grains need to have high anisotropy to be\nthermally stable. HAMR uses a heat pulse to locally\nenhance the temperature of the high anisotropy record-\ning medium beyond the Curie temperature. Due to the\nheating, the coercivity of the grain drops and it can be\nwritten with the available head \felds. After the grain\nis written, the medium is cooled and the information is\nsafely stored. A good indicator for the quality of the\nwritten bits is the so-called signal-to-noise ratio (SNR)\nwhich gives the power of the signal over the power of the\nnoise7. To achieve high areal storage densities, record-\ning materials that show good magnetic properties even\nat small grain sizes and thus yield high SNR values are\nneeded. However, a priori it is not clear which parame-\nters have the strongest in\ruence on the SNR.\nIn this work, we investigate the e\u000bect of a varying damp-\ning constant on the SNR. HAMR simulations with the\natomistic simulation program VAMPIRE8are performed\nfor cylindrical recording grains with two di\u000berent diame-\ntersd= 5 nm and d= 7 nm and a height h= 8 nm. The\nmaterial parameters of FePt like hard magnetic record-\ning media according to the Advanced Storage Technol-\nogy Consortium (ASTC)9are used. Damping constants\nbetween\u000b= 0:01 and\u000b= 0:5 are considered. Addition-\nally, we present an equation to include the in\ruence of\nthe bit length to the SNR. With this we can explain a\nSNR decrease of about 8.25 dB for 5- nm grains, which\nresults when changing the material and writing parame-\nters in the HAMR simulations from those used in former\nsimulations10{12to those according to the Advanced Stor-\nage Technology Consortium9, with the damping constant\nand the bit length only.\nThe structure of this paper is as follows: In Section II,\nthe HAMR model is introduced and it is explained how\na)Electronic mail: olivia.muthsam@univie.ac.atthe SNR is determined. In Section III, the results are\npresented and in Section IV they are discussed.\nII. HAMR MODEL\nCylindrical recording grains with height h= 8 nm and\ndiametersd= 5 nm and d= 7 nm are considered. One\ngrain can be interpreted as one grain of a state-of-the-\nart granular recording medium. A simple cubic crystal\nstructure is used. The exchange interaction Jijand the\ne\u000bective lattice parameter aare adjusted so that the sim-\nulations lead to the experimentally obtained saturation\nmagnetization and Curie temperature13,14. In the simu-\nlations, only nearest neighbor exchange interactions be-\ntween the atoms are included. A continuous laser pulse\nwith Gaussian shape and the full width at half maximum\n(FWHM) of 60 nm is assumed in the simulations. The\ntemperature pro\fle of the heat pulse is given by\nT(x;y;t ) = (Twrite\u0000Tmin)e\u0000x2+y2\n2\u001b2+Tmin (1)\n=Tpeak(y)\u0001e\u0000x2\n2\u001b2+Tmin (2)\nwith\n\u001b=FWHMp\n8 ln(2)(3)\nand\nTpeak(y) = (Twrite\u0000Tmin)e\u0000y2\n2\u001b2: (4)\nv= 15 m/s is the speed of the write head. xandylabel\nthe down-track and the o\u000b-track position of the grain,\nrespectively. In our simulations both the down-track po-\nsitionxand the o\u000b-track position yare variable. The\nambient and thus minimum temperature of all simula-\ntions isTmin= 300 K. The applied \feld is modeled as\na trapezoidal \feld with a \feld duration of 0.57 ns and a\n\feld rise and decay time of 0.1 ns, resulting in a bit length\nof 10.2 nm. The \feld strength is assumed to be +0 :8 T\nand\u00000:8 T inz-direction. Initially, the magnetization of\neach grain points in + z-direction. The trapezoidal \feldarXiv:1907.04577v2  [physics.app-ph]  24 Sep 20192\ntries to switch the magnetization of the grain from + z-\ndirection to\u0000z-direction. At the end of every simulation,\nit is evaluated if the bit has switched or not.\nThe material and write head parameters according to the\nAdvanced Storage Technology Consortium9are shown\nTable I.\nA. Determination of SNR\nTo calculate the signal-to-noise ratio, the read-back\nsignal of a written bit pattern has to be determined. To\nwrite the bit pattern and get the read-back signal from it,\nthe following procedure is used. First, a switching prob-\nability phase diagram is needed for the writing process of\nthe bit pattern. Since it is very time consuming to com-\npute a switching probability phase diagram with atom-\nistic or micromagnetic simulations, an analytical model\ndeveloped by Slanovc et al15is used in this work. The\nmodel uses eight input parameters (the maximum switch-\ning probability Pmax, the down-track jitter \u001bdown;the o\u000b-\ntrack jitter \u001bo\u000b;the transition curvature c, the bit length\nb, the half maximum temperature F50, the position p2of\nthe phase diagram in Tpeakdirection and the position p3\nof the phase diagram in down-track direction) to deter-\nmine a switching probability phase diagram. Slanovc et\nalshowed that the maximum switching probability Pmax\nand the down-track jitter \u001bdown are the input parameters\nwith the strongest in\ruence on the SNR. Note, that the\nbit lengthbalso has a strong in\ruence on the SNR. In\nthe further course of this work, an equation to include\nthe bit length to the SNR calculations is shown. Thus,\nthe bit length can be assumed constant during the SNR\ndetermination. The transition curvature cdid not show\nstrong in\ruence on the SNR for the used reader model\nand the o\u000b-track jitter \u001bo\u000bis neglectable since the reader\nwidth is with 30 :13 nm smaller than the track width with\n44:34 nm and thus does not sense the o\u000b-track jitter. p2\nandp3only shift the bit pattern and can thus be \fxed\nfor comparability. For this reason, it is reasonable to \fx\nthe input parameters, except for the maximum switch-\ning probability Pmaxand the down-track jitter \u001bdown.\nThe \fxed input parameters are determined by a least\nsquare \ft from a switching probability phase diagram\ncomputed with a coarse-grained Landau-Lifshitz-Bloch\n(LLB) model16for pure hard magnetic grains with mate-\nrial parameters given in Table I. The \ftting parameters\nare summarized in Table II for grain diameters d= 5 nm\nandd= 7 nm.\nFurther, it is necessary to compute the down-track jitter\n\u001bdown and the maximal switching probability Pmaxfor\nthe considered set of material and write head parameters,\nsee Table I. In the simulations, the switching probability\nof a recording grain at various down-track positions xat\na peak temperature Tpeak=Tc+ 60 K is calculated with\nthe atomistic simulations program VAMPIRE8, yielding\na down-track probability function P(x). To get the down-\ntrack jitter and the maximum switching probability, the\nswitching probability curve is \ftted with a Gaussian cu-mulative distribution function\n\b\u0016;\u001b2=1\n2(1 + erf(x\u0000\u0016p\n2\u001b2))\u0001Pmax (5)\nwith\nerf(x) =2p\u0019Zx\n0e\u0000\u001c2d\u001c; (6)\nwhere the mean value \u0016, the standard deviation \u001band\nthe mean maximum switching probability Pmax2[0;1]\nare the \ftting parameters. The standard deviation \u001b,\nwhich determines the steepness of the transition function,\nis a measure for the transition jitter and thus for the\nachievable maximum areal grain density of a recording\nmedium. The \ftting parameter Pmaxis a measure for\nthe average switching probability at the bit center. Note,\nthat the calculated jitter values \u001bdown only consider the\ndown-track contribution of the write jitter. The so-called\na\u0000parameter is given by\na=q\n\u001b2\ndown+\u001b2g (7)\nwhere\u001bgis a grain-size-dependent jitter contribution17.\nThe write jitter can then be calculated by\n\u001bwrite\u0019ar\nS\nW(8)\nwhereWis the reader width and S=D+Bis the grain\nsize, i.e. the sum of the grain diameter Dand the non-\nmagnetic boundary B15,18.\nFor each\u001bdown andPmaxcombination a switching prob-\nability phase diagram is computed with the analytical\nmodel. With the resulting phase diagram, the writing\nprocess of a certain bit pattern is simulated on granu-\nlar recording medium15. Here, the switching probabil-\nity of the grain is set according to its position in the\nphase diagram. The writing process is repeated for 50\ndi\u000berent randomly initialized granular media. Finally,\nthe read-back signal is determined with a reader model\nwhere the reader width is 30.13 nm and the reader res-\nolution in down-track direction is 13 :26 nm. The SNR\ncan then be computed from the read-back signal with\nthe help of a SNR calculator provided by SEAGATE19.\nThe resulting SNR value is given in dB (SNR dB). In the\nfollowing, the SNR dBis simply called SNR unless it is\nexplicitly noted di\u000berent.\nIII. RESULTS\nA. SNR Dependency on Damping\nFirst, the in\ruence of the damping constant on the\nSNR is investigated in more detail. The damping con-\nstant is varied from \u000b= 0:01 to\u000b= 0:5 for two di\u000berent\ngrain sizesd= 5 nm and d= 7 nm. All other parameters\nare taken from Table I. The bit length in the simula-\ntions is 10:2 nm and the track width is 44 :34 nm. The\ndown-track jitter curves are computed at Tpeak= 760 K\nand \ftted with eq. (5). In Figure 1, the SNR over the3\nCurie temp.\nTC[K]Damping\u000bUniaxial anisotropy\nku[J/link]Jij[J/link] \u0016s[\u0016B]v[m/s]\feld duration\n(fd) [ns]FWHM [nm]\n693.5 0.02 9:124\u000210\u0000236:72\u000210\u0000211.6 15 0.57 60\nTABLE I. Material and write head parameters of a FePt like hard magnetic granular recording medium accoring to the Advanced\nStorage Technology Consortium.\ngrain size\n5 nm 7 nm\n\u001bo\u000b[K] 22.5 14.4\nPmax 0.995 0.997\nF50[K] 602 628\nb[nm] 10.2 10.2\nc[10\u00004nm/K2]3.88 4.89\np2[K] 839 830\np3[nm] 27.5 25.8\nTABLE II. Reference parameters that are evaluated via least\nsquare \ft of the simulated phase diagrams for grain sizes 5 nm\nand 7 nm. Details of the parameters can be found in15.\ndamping constants for both grain sizes is visible. Note\nthat the SNR is proportional to the number of grains,\nmeaning that the number of grains per bit has to be kept\nconstant to determine a nearly constant SNR20. How-\never, since the dimensions of the granular media used for\nthe writing and reading process are \fxed, less grains form\none bit ford= 7 nm. Thus, the SNR values for the larger\ngrain size are smaller than for the small grains.\nThe results show that changing the damping constant\nfrom\u000b= 0:01 to\u000b= 0:02 already increases the SNR by\n3.66 dB for 5 nm-grains. For d= 7 nm, the SNR gain is\n1.65 dB. For 5 nm-grains, damping constants \u000b\u00150:1 lead\nto the best results with a total improvement of 6 dB com-\npared to\u000b= 0:01. Surprisingly, enhancing the damping\nconstant beyond 0 :1 does not show any further improve-\nment, the SNR saturates. This behavior is the same for\nthe 7 nm-grains. However, the total betterment of the\nSNR is only 2.24 dB for the larger grains. The SNR sat-\nuration results from the fact that Pmax= 1 for\u000b\u00150:1.\nSimultaneously, the down-track jitter \u001bdown varies only\nmarginally for \u000b\u00150:1 (see Table III) such that it does\nnot alter the SNR. The correlation between the SNR and\nthe maximum switching probability Pmaxis shown in Fig-\nure 2. It shows that the \ftted SNR curve reproduces the\ndata very well.\nBy further studying the switching dynamics of a 5 nm-\ngrain, one can show that the assumed pulse duration of\nthe heat pulse and the applied \feld strength are crucial\nfor the saturation of the SNR. In Figure 3, it is displayed\nhow the duration of the heat pulse in\ruences the maxi-\nmum switching probability and with it the SNR. During\nthe duration of the heat pulse the \feld is considered to\nconstantly point in \u0000z\u0000direction. The results demon-\nstrate that Pmaxdoes not saturate for small pulse du-\nrations. If longer pulse durations \u00150:5 ns are assumed,\naPmaxsaturation can be seen. A similar e\u000bect can be\nseen for a change of the \feld strength when the pulse\nduration is assumed to be 0 :5 ns (see Figure 4). For a\nsmall head \feld with a strength of 0 :5 T,Pmaxshows no\nsaturation whereas it does for larger head \felds. From\nFIG. 1. Resulting SNR for various damping constants \u000bfor\ngrains with two di\u000berent diameters d= 5 nm and d= 7 nm.\nFIG. 2. SNR and Pmaxdepending on the damping constant\n\u000bfor grain size with a diameter of 5 nm.\nthe simulations with varying duration of the heat pulse\nand \feld strength, it can also be seen that the SNR can\nbe improved for smaller damping constants if the dura-\ntion of the heat pulse is increased due to a smaller head\nvelocity or the \feld strength are enhanced.\nB. SNR Dependency on bit length\nThe in\ruence of the bit length on the SNR was al-\nready studied by Slanovc et al15. In this work, the fol-4\n5 nm 7 nm\n\u000b\u001bdown[nm]Pmax\u001bdown[nm]Pmax\n0.01 2.0 0.917 1.13 0.955\n0.02 0.9475 0.974 0.83 0.99\n0.05 0.7 0.989 0.549 1.0\n0.1 0.688 1.0 0.442 1.0\n0.3 0.495 1.0 0.48 1.0\n0.5 0.64 1.0 0.636 1.0\nTABLE III. Resulting down-track jitter parameters and mean maximum switching probability values for pure hard magnetic\nmaterial with di\u000berent damping constants \u000b.\nFIG. 3. Maximum switching probability Pmaxover damping\n\u000bfor di\u000berent pulse lengths of the heat pulse. A \feld strength\nof\u00000:8 T for grains with diameter d= 5 nm is assumed.\nlowing calculation is important. For the SNR calcula-\ntions a bit length b1= 10:2 nm is assumed since this is\nthe bit length resulting from the ASTC parameters. The\ntrack width in the simulations is again 44 :34 nm. How-\never, the bit length can change due to a variation of the\nwrite head parameters (\feld duration and head velocity).\nTherefore, the bit length for the former parameters10{12\nis 22 nm. To write a bit pattern with larger bit lengths\n(b >12 nm) the simulations of new granular media are\nrequired. This is computationally very expensive. Thus,\na di\u000berent approach is needed to qualitatively investi-\ngate the in\ruence of the bit length. For the SNR with\nSNR dB= 10 log10(SNR), there holds18\nSNR/\u0012b\na\u00132\u0012T50\nb\u0013\u0012W\nS\u0013\n(9)\nwith the bit length band the read-back pulse width T50\nwhich is proportional to the reader resolution in down-\ntrack direction. The ratio T50=bis called user bit density\nand is usually kept constant18. Further, the reader width\nWand the grain size Sare constant. Since the aim is to\nqualitatively describe the SNR for a bit length b2from\nSNR calculations with a bit length b1;the a-parameter a\nis also assumed to be constant. The SNR dBfor a di\u000berent\nbit lengthb2can then be calculated by\nFIG. 4. Maximum switching probability Pmaxover damping\n\u000bfor di\u000berent \feld strengths. The durations of the heat pulse\nof 0:5 ns for grains with diameter d= 5 nm is assumed.\nSNR dB(b2)\u0000SNR dB(b1)\n= 10 log10(SNR(b2))\u000010 log10(SNR(b1))\n= 10 log10(b2\n2)\u000010 log10(b2\n1) = 20 log10(b2\nb1) (10)\nsince all other parameters are the same for both bit\nlengths. Thus, one can compute the SNR dBvalue for\na varied bit length b2via the SNR dBof the bit length b1\nby\nSNR dB(b2) = SNR dB(b1) + 20 log10(b2\nb1): (11)\nThe curve achieved by eq. (11) with b1= 10:2 nm agrees\nqualitatively very well with the SNR(bit length) data\nfrom Slanovc et al15. It is thus reasonable to use this\nequation to include the bit length to the SNR.\nC. Combination of damping and bit length5\nCurie temp.\nTC[K]Damping\u000bUniaxial anisotropy\nku[J/link]Jij[J/link] \u0016s[\u0016B]v[m/s]\feld duration\n(fd) [ns]FWHM [nm]\n536.6 0.1 9:12\u000210\u0000235:17\u000210\u0000211.7 20 1.0 20\nTABLE IV. Material and write head parameters of a FePt like hard magnetic granular recording medium that were used in\nformer works10{12.\nParameter set diameter [nm] Tpeak[K]bit length [nm] Pmax\u001bdown [nm] SNR [dB]\nASTC 5 760 10.2 0.974 0.95 17.51\nParameters of former\nworks10{12 5 600 22 0.984 0.384 25.76\nASTC 7 760 10.2 0.99 0.83 15.35\nParameters of former\nworks10{12 7 600 22 1.0 0.44 22.75\nTABLE V. Resulting Pmax; \u001bdown and SNR values for the simulations with ASTC parameters and those used in former\nsimulations.\nThe simulations with write head and material parameters\naccording to the ASTC are compared to simulations with\nparameters used in former works10{12. Main di\u000berences\nto the currently used parameters are the bit length, the\ndamping constant, the height of the grain, the exchange\ninteraction, the atomistic spin moment, the full width at\nhalf maximum, the head velocity and the \feld duration.\nThese former parameters are summarized in Table IV.\nComparing the SNR values of both parameter sets shows\nthat ford= 5 nm the SNR is about 8.25 dB larger for the\nformer used parameters than for the ASTC parameters\nand ford= 7 nm it is\u00187:4 dB larger. The question is\nif the damping and bit length variation can fully explain\nthis deviation.\nIncreasing the damping constant from \u000b= 0:02 to\u000b=\n0:1, yields about +2 :25 dB ford= 5 nm and +0 :72 dB for\nd= 7 nm. Additionally, with the calculations from Sec-\ntion III B, one can show that by changing the bit length\nfromb1= 10:2 nm tob2= 22 nm gives\nSNR dB(b2) = SNR dB(b1) + 6:85 dB: (12)\nCombined, this shows that the di\u000berence in the SNR\ncan be attributed entirely to the damping and the bit\nlength enhancement. Moreover, simulations where the\nother material and write head parameters are changed\none by one con\frm this \fndings. The other write head\nand material parameters that are changed in the simu-\nlations have only minor relevance on the SNR compared\nto the damping constant and the bit length.\nIV. CONCLUSION\nTo conclude, we investigated how the damping con-\nstant a\u000bects the SNR. The damping constant was varied\nbetween\u000b= 0:01 and\u000b= 0:5 for two di\u000berent grain sizes\nd= 5 nm and d= 7 nm and the SNR was determined.\nIn practice, the damping constant of FePt might be in-\ncreased by enhancing the Pt concentration21,22. Another\noption would be to use a high/low Tcbilayer structure23\nand increase the damping of the soft magnetic layer by\ndoping with transition metals24{28. An interesting \fnd-\ning of the study is the enormous SNR improvement of6 dB that can be achieved for 5 nm-grains when enhanc-\ning the damping constant from \u000b= 0:01 to\u000b= 0:1\nand beyond. It is reasonable that the SNR improves\nwith larger damping. This results from the oscillatory\nbehavior of the magnetization for small damping dur-\ning switching. In fact, smaller damping facilitates the\n\frst switching but with larger damping it is more likely\nthat the grain will switch stably during the cooling of the\nthermal pulse29. This leads to a smaller switching time\ndistribution for larger damping constants and in the fur-\nther course to higher SNR values. However, an increase\nof the duration of the heat pulse due to a smaller head\nvelocity or an increase of the \feld strength can improve\nthe SNR even for smaller damping constant.\nFurthermore, the results display a SNR saturation for\ndamping constants \u000b\u00150:1. This SNR saturation can be\nexplained with the saturation of the maximum switching\nprobability and the only marginal change of the down-\ntrack jitter for \u000b\u00150:1. Indeed, one can check that for\nshorter pulse widths and smaller \feld strength, the be-\nhavior is di\u000berent and the SNR does not saturate. In\nthis case, the SNR rises for increasing damping constants.\nSummarizing, the SNR saturation for a varying damping\nconstant depends strongly on the used \feld strength and\nthe duration of the heat pulse.\nThe qualitative behavior for 7 nm-grains is the same. In-\nterestingly, the SNR change for a varying damping con-\nstant is not as signi\fcant as for grains with d= 5 nm.\nThis results from the higher maximum switching proba-\nbility and the smaller down-track jitter \u001bdown for 7 nm-\ngrains even for small damping constants. This is as\nexpected since larger grain sizes lead to an elevated\nmaximum switching probability11and smaller transition\njitter7compared to smaller grain sizes. This limits the\npossible increase of the recording performance in terms\nofPmaxand\u001bdown and thus the possible SNR gain. Ad-\nditionally, the SNR saturation value is smaller for 7 nm-\ngrains since one bit consists of fewer grains.\nThe overall goal was to explain the decrease of the SNR\nby about 8:25 dB and 7 :4 dB ford= 5 nm and d= 7 nm,\nrespectively, when changing from recording parameters\nused in former simulations10{12to the new ASTC pa-\nrameter. Indeed, together with the bit length variation,\nthe SNR variation could be fully attributed to the damp-6\ning enhancement. The other changed parameters like the\natomistic spin moment, the system height, the exchange\ninteraction and the full width at half maximum have only\na minor relevance compared to the in\ruence of the damp-\ning\u000band the bit length.\nIn fact, the variation of the bit length gave the largest\nSNR change. However, since an increase of the bit length\nis not realistic in recording devices, the variation of the\nmaterial parameters, especially the increase of the damp-\ning constant, is a more promising way to improve the\nSNR.\nV. ACKNOWLEDGEMENTS\nThe authors would like to thank the Vienna Sci-\nence and Technology Fund (WWTF) under grant No.\nMA14-044, the Advanced Storage Technology Consor-\ntium (ASTC), and the Austrian Science Fund (FWF)\nunder grant No. I2214-N20 for \fnancial support. 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Applied Physics Letters ,\n100(10):102402, 2012."    },    {        "title": "1910.11200v1.Spin_waves_in_ferromagnetic_thin_films.pdf",        "content": "arXiv:1910.11200v1  [cond-mat.mes-hall]  24 Oct 2019Spin waves in ferromagnetic thin films\nZhiwei Sun\nSchool of Mathematical Sciences, Soochow University, Suzh ou, China\nJingrun Chen∗\nSchool of Mathematical Sciences, Soochow University, Suzh ou, China and\nMathematical Center for Interdisciplinary Research, Sooc how University, Suzhou, China\n(Dated: October 25, 2019)\nA spin wave is the disturbance of intrinsic spin order in magn etic materials. In this paper, a spin\nwave in the Landau-Lifshitz-Gilbert equation is obtained b ased on the assumption that the spin\nwave maintains its shape while it propagates at a constant ve locity. Our main findings include:\n(1) in the absence of Gilbert damping, the spin wave propagat es at a constant velocity with the\nincrement proportional tothe strength of the magnetic field ; (2) in the absence of magnetic field, at a\ngiven time the spin wave converges exponentially fast to its initial profile as the damping parameter\ngoes to zero and in the long time the relaxation dynamics of th e spin wave converges exponentially\nfast to the easy-axis direction with the exponent proportio nal to the damping parameter; (3) in\nthe presence of both Gilbert damping and magnetic field, the s pin wave converges to the easy-axis\ndirection exponentially fast at a small timescale while pro pagates at a constant velocity beyond\nthat. These provides a comprehensive understanding of spin waves in ferromagnetic materials.\nPACS numbers: 05.45.Yv, 75.70.-i, 75.78.-n\nI. INTRODUCTION\nA spin wave is the disturbance of intrinsic spin order in magnetic mater ials. It is usually excited using magnetic\nfields and offers unique properties such as charge-less propagatio n and high group velocities, which are important for\nsignal transformations and magnetic logic applications [1–6].\nThepropagationofspinwavesisdescribedbytheLandau-Lifshitz- Gilbert(LLG) equation[7,8]in thedimensionless\nform\nmt=−m×h−αm×(m×h), (1)\nwhere the magnetization m= (m1,m2,m3)Tis a three dimensional vector with unit length, αis the Gilbert damping\nparameter. The effective field hincludes the exchange term, the anisotropy term with easy axis alon g the x-axis and\nthe anisotropy constant q, and the external field\nh= ∆m+qm1e1+hexte1. (2)\nhextis the strength of the external field applied along the x-axis with e1the unit vector. This model is often used to\ndescribe the magnetization dynamics in ferromagnetic thin films.\nFrom a theoretical perspective, a spin wave is known as a solitotary wave, which appears as the solution of a weakly\nnonlinear dispersive partial differential equation. In LLG equation ( 1)-(2), a soliton is caused by the cancellation of\nnonlinear and dispersive effects in the magnetic material. Solitons are of interests for quite a long time [9–13]. Most\nof works consider the one dimensional case and drop the damping te rm [9, 10, 13]. In [11], using the stereographic\nprojection, the authors found that the presence of Gilbert damp ing was merely a rescaling of time by a complex\nconstant. However, this was found to be valid only for a single spin in a constant magnetic field [12].\nIn this work, we give a comprehensive study of an explicit spin wave in t he LLG equation. Our starting point is\nthat the spin wave maintains its shape while it propagates at a consta nt velocity and the derivation is based on the\ngeneralization of the method of characteristics. The main findings a re: (1) in the absence of Gilbert damping, the\nspin wave propagates at a constant velocity with the increment pro portional to the strength of the magnetic field; (2)\nin the absence of magnetic field, at a given time the spin wave converg es exponentially fast to its initial profile as the\ndamping parameter goesto zeroand in the long time the relaxationdy namics of the spin waveconvergesexponentially\n∗Electronic address: jingrunchen@suda.edu.cn2\nfast to the easy-axis direction with the exponent proportional to the damping parameter; (3) in the presence of both\nGilbert damping and magnetic field, the spin wave converges to the ea sy-axis direction exponentially fast at a small\ntimescale while propagates at a constant velocity beyond that.\nII. DERIVATION AND RESULTS\nAs mentioned above, we start with the assumption that a spin wave m aintains its shape while it propagates at a\nconstant velocity. This can be seen from the method characterist ics in simple situations.\nIn 1D when α=q=hext= 0, one can check that\nm(x,t) =\ncosθ0\nsinθ0cos/parenleftig\nc\ncosθ0(x+ct)/parenrightig\nsinθ0sin/parenleftig\nc\ncosθ0(x+ct)/parenrightig\n(3)\nsolvesmt=−m×mxx. Hereθ0is determined by the initial condition and u=x+ctis the characteristic line. (3)\nprovides a solitary solution with the traveling speed c. A detailed derivation of (3) can be found in Chapter 2 of [13].\nA generalization of the method of characteristics yields a spin wave t omt=−m×∆m\nm(x,t) =\ncosθ0\nsinθ0cosv\ncosθ0\nsinθ0sinv\ncosθ0\n, (4)\nwherev=c1x+c2y+c3z+(c2\n1+c2\n2+c2\n3)t=c·x+(c·c)twithc= (c1,c2,c3)T. The speed field is cwith magnitude\n|c|. Actually, both (3) and (4) can be rewritten as\nm(x,t) =\ncosθ0\nsinθ0cos(w0·x+ϕ(t))\nsinθ0sin(w0·x+ϕ(t))\n, (5)\nwherew0=c/cosθ0andϕ(t) =/parenleftbig\n|c|2/cosθ0/parenrightbig\nt.\n(3)-(5) are obtained in the absence of Gilbert damping. In order to take the Gilbert damping and the other terms\nin (2) into account, we make an ansatz for the spin wave profile in the following form\nm(x,t) =\ncosθ(t)\nsinθ(t)cos(w0·x+ϕ(t))\nsinθ(t)sin(w0·x+ϕ(t))\n, (6)\nwhereθandϕare independent of xand only depend on t.\nSubstituting (6) into (2) and (1) and denoting w0·x+ϕ(t) byu(x,t), we have\nh=\n0\n−|w0|2sinθcosu(x,t)\n−|w0|2sinθsinu(x,t)\n+q\ncosθ\n0\n0\n+hext\n1\n0\n0\n,\nm×h=\n0\n|w0|2sinθcosθsinu(x,t)\n−|w0|2sinθcosθcosu(x,t)\n+q\n0\nsinθcosθsinu(x,t)\n−sinθcosθcosu(x,t)\n+hext\n0\nsinθsinu(x,t)\n−sinθcosu(x,t)\n,\nm×(m×h) =\n−|w0|2sin2θcosθ\n|w0|2sinθcos2θcosu(x,t)\n|w0|2sinθcos2θsinu(x,t)\n+q\n−sin2θcosθ\nsinθcos2θcosu(x,t)\nsinθcos2θsinu(x,t)\n+hext\n−sin2θ\nsinθcosθcosu(x,t)\nsinθcosθsinu(x,t)\n,\nand\nmt=\n−θtsinθ\nθtcosθcosu(x,t)−ϕtsinθsinu(x,t)\nθtcosθsinu(x,t)+ϕtsinθcosu(x,t)\n.\nAfter algebraic simplifications, we arrive at\n/braceleftbigg\nθt=−α(|w0|2+q)sinθcosθ−αhextsinθ\nϕt= (|w0|2+q)cosθ+hext. (7)3\nA. The absence of Gilbert damping\nWhenα= 0, we have θ=θ0andϕ=/parenleftbig\n(|w0|2+q)cosθ0+hext/parenrightbig\nt. Therefore we have the solution\nm=\ncosθ0\nsinθ0cos/parenleftbig\nw0·x+t/parenleftbig\n|w0|2cosθ0+qcosθ0+hext/parenrightbig/parenrightbig\nsinθ0sin/parenleftbig\nw0·x+t/parenleftbig\n|w0|2cosθ0+qcosθ0+hext/parenrightbig/parenrightbig\n. (8)\nNote that this recovers (5) when q= 0 and hext= 0. It is easy to see that the spin wave (8) propagates at a consta nt\nvelocity. The increment of the velocity field is qcosθ0w0\n|w0|2with magnitude|qcosθ0|\n|w0|, due to the magnetic anisotropy.\nThe increment of the velocity field is hextw0\n|w0|2with magnitude|hext|\n|w0|, due to the magnetic field.\nB. The absence of magnetic field\nWhenhext= 0, (7) reduces to\n/braceleftbigg\nθt=−α(|w0|2+q)sinθcosθ\nϕt= (|w0|2+q)cosθ. (9)\nFor the first equation in (9), assuming 0 ≤θ0< π/2, by separation of variables, we have\nα(|w0|2+q)t+C1= lncotθ,\nwhereC1is a constant determined by the initial condition.\nDenote˜t=α(|w0|2+q)t+C1. It follows that\ntanθ=e−˜t, (10)\nfrom which one has\ncosθ=1/radicalbig\n1+e−2˜t, (11)\nsinθ=1/radicalbig\n1+e2˜t. (12)\nWhent= 0, (11) turns to\ncosθ0=1√\n1+e−2C1, (13)\nfrom which we can determine C1by the initial condition θ0.\nAs forϕ, from the second equation in (9), one has that\ndϕ\ndθ=dϕ\ndt·dt\ndθ=−1\nαsinθ.\nTherefore\nαϕ=−/integraldisplaydθ\nsinθ=1\n2ln/parenleftbigg1+cosθ\n1−cosθ/parenrightbigg\n+C2= lncot1\n2θ+C2, (14)\nwhere\nC2=−1\n2ln/parenleftbigg1+cosθ0\n1−cosθ0/parenrightbigg\n.\nSubstituting (11) and (13) into (14) yields\nϕ=1\nαln/parenleftigg\ne˜t+/radicalbig\ne2˜t+1\neC1+√\ne2C1+1/parenrightigg\n=1\nα/parenleftbigg\nlncot1\n2θ−lncot1\n2θ0/parenrightbigg\n. (15)4\nIn short summary, the spin wave when α/negationslash= 0 takes the form\nm=1/radicalbig\n1+e2˜t\ne˜t\ncos(w0·x+ϕ)\nsin(w0·x+ϕ)\n. (16)\nThe above derivation is valid when 0 ≤θ0< π/2. Ifπ/2< θ0≤π, we choose the other solution of (11)\ncosθ=−1/radicalbig\n1+e−2˜t, (17)\nand\nϕ=−1\nαln/parenleftigg\ne˜t+/radicalbig\ne2˜t+1\neC1+√\ne2C1+1/parenrightigg\n.\n(16) remains unchanged.\nWhenα→0, we have ˜t→C1and\nlim\nα→0m=1√\n1+e2C1\neC1\ncos/parenleftig\nw0·x+ lim\nα→0ϕ/parenrightig\nsin/parenleftig\nw0·x+ lim\nα→0ϕ/parenrightig\n.\nBy L’Hospital’s rule, one has that\nlim\nα→0ϕ= lim\nα→0d\ndα/parenleftigg\nln/parenleftigg\ne˜t+/radicalbig\ne2˜t+1\neC1+√\ne2C1+1/parenrightigg/parenrightigg\n= lim\nα→0e˜t\n/radicalbig\ne2˜t+1(|w0|2+q)t=eC1\n√\neC1+1(|w0|2+q)t.(18)\nTherefore it follows that\nlim\nα→0m=\ncosθ0\nsinθ0cos/parenleftbig\nw·x+cosθ0/parenleftbig\n|w0|2+q/parenrightbig\nt/parenrightbig\nsinθ0sin/parenleftbig\nw·x+cosθ0/parenleftbig\n|w0|2+q/parenrightbig\nt/parenrightbig\n. (19)\nThis is exactly the solution (8) when hext= 0.\nIn addition, when α→0, bothθandϕconverges exponentially fast to initial conditions; see equations (1 1), (12),\n(13), and (18). Therefore, at a giventime t, (19) convergesexponentiallyfast to the initial spin wave(8) when hext= 0\nwith the exponent proportionalto the damping parameter α. Moreover,in the long time, i.e., when t→+∞,˜t→+∞\nandθ→0, (16) converges to (1 ,0,0)T(the easy-axis direction) exponentially fast with the rate proport ional to the\ndamping parameter α. Whenπ/2< θ0≤π, from (17), we have that (16) converges to ( −1,0,0)T(again the easy-axis\ndirection) exponentially fast with the rate proportional to the dam ping parameter α.\nIt is easy to check that the right-hand side of (19) is the solution of (1) when hext= 0 with the initial condition\nθ0=π/2. Therefore, Gilbert damping does not have any influence on magne tization dynamics in this case.\nIn [11], the authors used the stereographic projection and obser ved that the effect of Gilbert damping was only a\nrescaling of time by a complex constant. However, this was latter fo und to be valid only for a single spin in a constant\nmagnetic field [12]. Our result provides an explicit characterization of magnetization dynamics in the presence of\nGilbert damping.\nC. The presence of both Gilbert damping and magnetic field\nIt is difficult to get the explicit solution of (7) in general. To understan d the magnetization dynamics, we use the\nmethod of asymptotic expansion. For small external magnetic field ,θandϕadmit the following expansions\nθ(t,hext) =θ0(t)+θ1(t)hext+θ2(t)h2\next+···,\nϕ(t,hext) =ϕ0(t)+ϕ1(t)hext+ϕ2(t)h2\next+···.5\nTherefore one has that\nθt(t,hext) =θ0\nt(t)+θ1\nt(t)hext+θ2\nt(t)h2\next+···, (20)\nϕt(t,hext) =ϕ0\nt(t)+ϕ1\nt(t)hext+ϕ2\nt(t)h2\next+···. (21)\nOn the other hand, from (7), it follows that\nθt=−α(|w0|2+q)sinθ0cosθ0−/parenleftbig\nα(|w0|2+q)θ1cos2θ0+αsinθ0/parenrightbig\nhext+···, (22)\nϕt= (|w0|2+q)cosθ0+/parenleftbig\n−(|w0|2+q)θ1sinθ0+1/parenrightbig\nhext+···. (23)\nCombining (20) and (21) with (22) and (23), for the zero-order te rm, one has\n/braceleftbigg\nθ0\nt=−α(|w0|2+q)sinθ0cosθ0\nϕ0\nt= (|w0|2+q)cosθ0 , (24)\nwhich recovers (9) with solution (10) and (15).\nAs for the first-order term, one has that\n/braceleftbigg\nθ1\nt=−α(|w0|2+q)θ1cos2θ0−αsinθ0\nϕ1\nt=−(|w0|2+q)θ1sinθ0+1. (25)\nUsing variation of parameters, one can assume θ1=C(t)\ne˜t+e−˜tand it follows that\nC′(t) =−α(e˜t+e−˜t)sinθ0=−α(tanθ0+tan−1θ0)sinθ0.\nSince\n/integraldisplay\n−αtanθ0sinθ0dt=/integraldisplay\n−α(|w0|2+q)sinθ0cosθ0∗(|w0|2+q)−1sinθ0\ncos2θ0dt\n=/integraldisplay\n(|w0|2+q)−1sinθ0\ncos2θ0dθ0\n=(|w0|2+q)−11\ncosθ0,\nand\n/integraldisplay\n−αtan−1θ0sinθ0dt=/integraldisplay\n−αcosθ0dt\n=−α(|w0|2+q)−1ϕ0,\none can get C(t) = (|w0|2+q)−1(1\ncosθ0−αϕ0), and it follows that\nθ1= (|w0|2+q)−1(sinθ0−αsinθ0cosθ0ϕ0). (26)\nSubstituting the first equation in (24) into the second equation in (2 5), one has\n−/integraldisplay\n(|w0|2+q)θ1sinθ0dt\n=1\nα/integraldisplayθ1\ncosθ0dθ0\n=1\nα(|w0|2+q)/integraldisplay\ntanθ0−sinθ0(lncot1\n2θ0+C2)dθ0(using (26))\n=−t+1\n|w0|2+q(ϕ0cosθ0+α−1C1),\nand thus\nϕ1=1\n|w0|2+q(ϕ0cosθ0+α−1C1). (27)6\nTherefore, when hextis small, it has the approximate solution\n/braceleftbigg\nθ∗=θ0+(|w0|2+q)−1(sinθ0−αsinθ0cosθ0ϕ0)hext\nϕ∗=ϕ0+(|w0|2+q)−1(ϕ0cosθ0+α−1C1)hext(28)\nwithθ0andϕ0satisfying (9).\nFrom (10) and (15), θ0converges exponentially fast to the easy-axis direction, while ϕ0grows linearly. Therefore,\nfrom (26), θ1converges exponentially fast to 0 as well with a larger exponent. Th is relaxation dynamics happens at\na small timescale.\nMeanwhile, from (25), the difference between ϕ∗andϕ0satisfies\nϕ∗\nt−ϕ0\nt=ϕ1\nthext=−(|w0|2+q)θ1sinθ0hext+hext. (29)\nSinceθ1sinθ0convergesto 0 at a small timescale, the dynamics of ϕ∗−ϕ0is determined by the external field at longer\ntimescales. As a consequence, the increment of the velocity field is hextw0\n|w0|2with magnitude|hext|\n|w0|. This validates the\nWalker’s ansatz [14] for a spin wave.\nWhenπ/2< θ0≤πand the magnetic field is applied along the negative x-axis, and if ( |w|2+q)|cosθ0| ≤hext, the\nresult above will be correct.\nNote that θ0=π/2 does not fall into the above two cases since the magnetization dyn amics will change the spin\nwave profile. In fact, as t→+∞,θ→0 if the magnetic field is applied along the positive x-axis direction and θ→π\nif the magnetic field is applied along the negative x-axis direction.\nIII. CONCLUDING REMARKS\nIn this work, we study the magnetization dynamics in Landau-Lifshit z-Gilbert equation. By generalizing the\nmethod of characteristics, we are able to have an explicit characte rization of spin wave dynamics in the presence of\nboth Gilbert damping and magnetic field. Gilbert damping drives the spin wave converge exponentially fast to the\neasy-axis direction with the exponent proportional to the damping parameter at a small timescale and the magnetic\nfield drives the spin wave propagate at a constant velocity at longer timescales.\nIt will be of interests whether the technique developed here applies to the antiferromagnetic case [15, 16] and how\nrigorous the results obtained here can be proved from a mathemat ical perspective.\nIV. ACKNOWLEDGEMENTS\nWe thank Professor Yun Wang for helpful discussions. This work wa s partially supported by National Natural\nScience Foundation of China via grant 21602149 and 11971021.\n[1] M. P. Kostylev, A. A. Serga, T. Schneider, B. Leven, and B. Hillebrands, Applied Physics Letters 87, 153501 (2005).\n[2] Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi,\net al., Nature 464, 262 (2010).\n[3] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebra nds, Nature Physics 11, 453 (2015).\n[4] S. Woo, T. Delaney, and G. S. D. Beach, Nature Physics (201 7).\n[5] A. V. Chumak and H. Schultheiss, Journal of Physics D: App lied Physics 50, 300201 (2017).\n[6] M. Langer, R. A. Gallardo, T. Schneider, S. Stienen, A. Ro ld´ an-Molina, Y. Yuan, K. Lenz, J. Lindner, P. Landeros, and\nJ. Fassbender, Physical Review B 99(2019).\n[7] L. Landau and E. Lifshitz, Physikalische Zeitschrift de r Sowjetunion 8, 153 (1935).\n[8] T. Gilbert, Physical Review 100, 1243 (1955).\n[9] K. Nakamura and T. Sasada, Physics Letters A 48, 321 (1974).\n[10] H. J. Mikeska, Journal of Physics C: Solid State Physics 11, L29 (1977).\n[11] M. Lakshmanan and K. Nakamura, Physical Review Letters 53, 2497 (1984).\n[12] E. Magyari, H. Thomas, and R. Weber, Physical Review Let ters56, 1756 (1986).\n[13] B. Guo and S. Ding, Landau-Lifshitz Equation (World Scientific, 2007).\n[14] N. L. Schryer and L. R. Walker, Journal of Applied Physic s45, 5406 (1974).\n[15] H. J. Mikeska, Journal of Physics C: Solid State Physics 13, 2913 (1980).\n[16] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak, Reviews of Modern Physics 90, 015005 (2018)."    },    {        "title": "1709.03775v1.Green_s_function_formalism_for_spin_transport_in_metal_insulator_metal_heterostructures.pdf",        "content": "Green’s function formalism for spin transport in metal-insulator-metal\nheterostructures\nJiansen Zheng,1Scott Bender,1Jogundas Armaitis,2Roberto E. Troncoso,3,4and Rembert A. Duine1,5\n1Institute for Theoretical Physics and Center for Extreme Matter and Emergent Phenomena,\nUtrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands\n2Institute of Theoretical Physics and Astronomy,\nVilnius University, Saul˙ etekio Ave. 3, LT-10222 Vilnius, Lithuania\n3Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\n4Departamento de Física, Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso, Chile\n5Department of Applied Physics, Eindhoven University of Technology,\nPO Box 513, 5600 MB Eindhoven, The Netherlands\n(Dated: September 13, 2017)\nWe develop a Green’s function formalism for spin transport through heterostructures that contain\nmetallic leads and insulating ferromagnets. While this formalism in principle allows for the inclusion\nof various magnonic interactions, we focus on Gilbert damping. As an application, we consider\nballistic spin transport by exchange magnons in a metal-insulator-metal heterostructure with and\nwithout disorder. For the former case, we show that the interplay between disorder and Gilbert\ndamping leads to spin current fluctuations. For the case without disorder, we obtain the dependence\nof the transmitted spin current on the thickness of the ferromagnet. Moreover, we show that the\nresults of the Green’s function formalism agree in the clean and continuum limit with those obtained\nfrom the linearized stochastic Landau-Lifshitz-Gilbert equation. The developed Green’s function\nformalism is a natural starting point for numerical studies of magnon transport in heterostructures\nthat contain normal metals and magnetic insulators.\nPACS numbers: 05.30.Jp, 03.75.-b, 67.10.Jn, 64.60.Ht\nI. INTRODUCTION\nMagnons are the bosonic quanta of spin waves, oscil-\nlations in the magnetization orientation in magnets1,2.\nInterest in magnons has recently revived as enhanced ex-\nperimental control has made them attractive as potential\ndata carriers of spin information over long distances and\nwithoutOhmicdissipation3. Ingeneral, magnonsexistin\ntwo regimes. One is the dipolar magnon with long wave-\nlengths that is dominated by long-range dipolar interac-\ntions and which can be generated e.g. by ferromagnetic\nresonance4,5. The other type is the exchange magnon6,\ndominated by exchange interactions and which generally\nhas higher frequency and therefore perhaps more poten-\ntial for applications in magnon based devices3. In this\npaper, we focus on transport of exchange magnons.\nThermally driven magnon transport has been widely\ninvestigated, and is closely related to spin pumping of\nspin currents across the interface between insulating fer-\nromagnets (FMs) and normal metals (NM)7–9and de-\ntection of spin current by the inverse spin Hall Effect10.\nThe most-often studied thermal effect in this context is\nthe spin Seebeck effect, which is the generation of a spin\ncurrent by a temperature gradient applied to a magnetic\ninsulator that is detected in an adjacent normal metal\nvia the inverse spin Hall effect11,12. Here, thermal fluc-\ntuations in the NM contacts drive spin transport into the\nFM, while the dissipation of spin back into the NM by\nmagnetic dynamics is facilitated by the above mentioned\nspin-pumping mechanism.\nThe injection of spin into a FM can also be accom-plished electrically, via the interaction of spin polarized\nelectrons in the NM and the localized magnetic mo-\nments of the FM. Reciprocal to spin-pumping is the spin-\ntransfer torque, which, in the presence of a spin accu-\nmulation (typically generated by the spin Hall effect)\nin the NM, drives magnetic dynamics in the FM13,14.\nSpin pumping likewise underlies the flow of spin back\ninto the NM contacts, which serve as magnon reservoirs.\nIn two-terminal set-ups based on YIG and Pt, the char-\nacteristic length scales and device-specific parameter de-\npendence of magnon transport has attracted enormous\nattention, both in experiments and theory. Cornelis-\nsenet al.15studied the excitation and detection of high-\nfrequency magnons in YIG and measured the propagat-\ning length of magnons, which reaches up to 10\u0016m in\na thin YIG film at room temperature. Other experi-\nments have shown that the polarity reversal of detected\nspins of thermal magnons in non-local devices of YIG\nare strongly dependent on temperature, YIG film thick-\nness, and injector-detector separation distance16. That\nthe interfaces are crucial can e.g. be seen by changing the\ninterface electron-magnon coupling, which was found to\nsignificantly alter the longitudinal spin Seebeck effect17.\nA linear-response transport theory was developed for\ndiffusive spin and heat transport by magnons in mag-\nnetic insulators with metallic contacts. Among other\nquantities, this theory is parameterized by relaxation\nlengths for the magnon chemical potential and magnon-\nphonon energy relaxation18,19. In a different but closely-\nrelated development, Onsager relations for the magnon\nspin and heat currents driven by magnetic field andarXiv:1709.03775v1  [cond-mat.mes-hall]  12 Sep 20172\ntemperature differences were established for insulating\nferromagnet junctions, and a magnon analogue of the\nWiedemann-Franz law was is also predicted20,21. Wang\net al.22consider ballistic transport of magnons through\nmagnetic insulators with magnonic reservoirs — rather\nthanthemoreexperimentallyrelevantsituationofmetal-\nlic reservoirs considered here — and use a nonequilib-\nrium Green’s function formalism (NEGF) to arrive at\nLanduaer-Bütikker-type expressions for the magnon cur-\nrent. Theabove-mentionedworksareeitherinthelinear-\nresponse regime or do not consider Gilbert damping\nand/or metallic reservoirs. So far, a complete quantum\nmechanical framework to study exchange magnon trans-\nport through heterostructures containing metallic reser-\nvoirs that can access different regimes, ranging from bal-\nlistic to diffusive, large or small Gilbert damping, and/or\nsmall or large interfacial magnon-electron coupling, and\nthat can incorporate Gilbert damping, is lacking.\nFigure 1: Illustration of the system where magnon transport\nin a ferromagnet (orange region) is driven by a spin accu-\nmulation difference \u0001\u0016L\u0000\u0001\u0016Rand temperature difference\nTL\u0000TRbetween two normal-metal leads (blue regions). Spin-\nflip scattering at the interface converts electronic to magnonic\nspin current. Here, Sis the local spin density in equilibrium.\nIn this paper we develop the non-equilibrium Green’s\nfunctionformalism23forspintransportthroughNM-FM-\nNM heterostructures (see Fig. 1). In principle, this for-\nmalism straightforwardly allows for adding arbitrary in-\nteractions, such as scattering of magnons with impuri-\nties and phonons, Gilbert damping, and magnon-magnon\ninteractions, and provides a suitable platform to study\nmagnon spin transport numerically, in particular beyond\nlinear response. Here, we apply the formalism to ballistic\nmagnon transport through a low-dimensional channel in\nthe presence of Gilbert damping. For that case, we com-\npute the magnon spin current as a function of channel\nlength both numerically and analytically. For the clean\ncase in the continuum limit we show how to recover our\nresults from the linearized stochastic Landau-Lifshitz-\nGilbert (LLG) equation24used previously to study ther-\nmal magnon transport in the ballistic regime25that ap-\nplies to to clean systems at low temperatures such that\nGilbert damping is the only relaxation mechanism. Us-\ning this formalism we also consider the interplay between\nGilbert damping and disorder and show that it leads to\nspin-current fluctuations.\nThis paper is organized as follows. In Sec. II, we\ndiscuss the non-equilibrium Green’s function approachto magnon transport and derive an expression for the\nmagnon spin current. Additionally a Landauer-Büttiker\nformula for the magnon spin current is derived. In\nSec. III, we illustrate the formalism by numerically con-\nsidering ballistic magnon transport and determine the\ndependence of the spin current on thickness of the ferro-\nmagnet. To further understand these numerical results,\nwe consider the formalism analytically in the continuum\nlimit in Sec. IV, and also show that in that limit we ob-\ntain the same results using the stochastic LLG equation.\nWe give a further discussion and outlook in section V.\nII. NON-EQUILIBRIUM GREEN’S FUNCTION\nFORMALISM\nIn this section we describe our model and, using\nKeldysh theory, arrive at an expression for the density\nmatrix of the magnons from which all observables can be\ncalculated. The reader interested in applying the final re-\nsult of our formalism may skip ahead to Sec. IIE where\nwe give a summary on how to implement it.\nA. Model\nj j/primej j/primej j/prime∆µL\nTL∆µR\nTR\nTFMNNM FM NM\nJLJRρj,j/prime G(±)\nj,j/prime(t,t/prime)\nGk,k/prime(t,t/prime) Gk,k/prime(t,t/prime)\nΣFM,(±)ΣL,(±)ΣR,(±)\nSelf-energy\nFigure 2: Schematic for the NM-FM-NM heterostructure and\nnotationfortheGreen’sfunctionsandself-energies. Thearray\nof circles denotes the localized magnetic moments, while the\ntwo regions outside the parabolic lines denote the leads, i.e.,\nreservoirs of polarized electrons. Moreover, JL=R\nj;kk0denotes the\ninterface coupling, and TL=Rand\u0001\u0016L=Rdenote the temper-\nature and spin accumulation for the leads. The properties of\nthe magnons are encoded in G(+)\nj;j0(t;t0), the retarded magnon\nGreen’s function, and the magnon density matrix \u001aj;j0. The\nnumber of sites in the spin-current direction is N. The self-\nenergies \u0006FM; (\u0006),\u0006L;(\u0006),\u0006R;(\u0006)are due to Gilbert damping,\nand the left and right lead, respectively.\nWe consider a magnetic insulator connected to two\nnonmagnetic metallic leads, as shown in Fig. 2. For3\nour formalism it is most convenient to consider both the\nmagnons and the electrons as hopping on the lattice for\nthe ferromagnet. Here, we consider the simplest versions\nof such cubic lattice models; extensions, e.g. to multi-\nple magnon and/or electron bands, and multiple leads\nare straightforward. The leads have a temperature TL=R\nand a spin accumulation \u0001\u0016L=Rthat injects spin cur-\nrent from the non-magnetic metal into the magnetic in-\nsulator. This nonzero spin accumulation could, e.g., be\nestablished by the spin Hall effect.\nThe total Hamiltonian is a sum of the uncoupled\nmagnon and lead Hamiltonians together with a coupling\nterm:\n^Htot=^HFM+^HNM+^HC: (1)\nHere, ^HFMdenotesthefreeHamiltonianforthemagnons,\n^HFM=\u0000X\n<j;j0>Jj;j0by\nj0bj+X\nj\u0001jby\njbj\u0011X\n<j;j0>hj0;jby\nj0bj:\n(2)\nwherebj(by\nj)is a magnon annihilation (creation) opera-\ntor. This hamiltonian can be derived from a spin hamil-\ntonian using the Holstein-Primakoff transformation26,27\nand expanding up to second order in the bosonic fields.\nEq. (2) describes hopping of the magnons with amplitude\nJj;j0between sites labeled by jandj0on the lattice, with\nan on-site potential energy \u0001jthat, if taken to be homo-\ngeneous, would correspond to the magnon gap induced\nby a magnetic field and anisotropy. We have taken the\nexternal field in the \u0000zdirection, so that one magnon,\ncreated at site jby the operator ^by\nj, corresponds to spin\n+~.\nThe Hamiltonian for the electrons in the leads is\n^HNM=\u0000X\nr2fL;RgX\n<k;k0>X\n\u001b2\";#tr^ y\nk\u001br^ k0\u001br+h:c:(3)\nwhere the electron creation (  y\nk\u001br) and annihilation\n( k\u001br) operators are labelled by the lattice position k,\nspin\u001b, and an index rdistinguishing (L)eft and (R)ight\nleads. The hopping amplitude for the electrons is de-\nnotedbytrandcouldinprinciplebedifferentfordifferent\nleads. Moreover, terms to describe hopping beyond near-\nestneighborcanbestraightforwardlyincluded. Belowwe\nwill show that microscopic details will be incorporated in\na single parameter per lead that describes the coupling\nbetween electrons and magnons.\nFinally, the Hamiltonian that describes the coupling\nbetween metal and insulator, ^HC, is given by28\n^HC=X\nr;j;kk0\u0010\nJr\nj;kk0^by\nj^ y\nk#r^ k0\"r+ h:c:\u0011\n;(4)\nwith the matrix elements Jr\nj;kk0that depend on the mi-\ncroscopic details of the interface. An electron spin that\nflips from up to down at the interface creates one magnon\nwithspin +~inthemagneticinsulator. Thisformofcou-\npling between electrons and magnons derives from inter-\nface exchange coupling between spins in the insulators\nwith electronic spins in the metal28.\nGk/prime,k/prime/prime;↑\nGk/prime/prime/prime,k;↓t/prime, j/primet, jFigure 3: Feynman diagram for the spin-flip processes emit-\nting and absorbing magnons that are represented by the wavy\nlines. The two vertices indicate the exchange coupling at one\nof the interfaces of the magnetic insulator (sites j;j0) and nor-\nmal metal (sites k;k0;k00;k000).Gk0k00;\"andGk000k;#denotes\nthe electron Keldysh Green’s function of one of the leads.\nB. Magnon density matrix and current\nOur objective is to calculate the steady-state magnon\nGreen’s function iG<\nj;j0(t;t0) =h^by\nj0(t0)^bj(t)i, from which\nall observables are calculated (note that time-dependent\noperators refer to the Heisenberg picture). This “lesser”\nGreen’s function follows from the Keldysh Green’s func-\ntion\niGj;j0(t;t0)\u0011Trh\n^\u001a(t0)TC1\u0010\n^bj(t)^by\nj0(t0)\u0011i\n;(5)\nwith ^\u001a(t0)the initial (at time t0) density matrix, and\nC1the Keldysh contour, and Tr[:::]stands for perform-\ning a trace average. The time-ordering operator on this\ncontour is defined by\nTC1\u0010\n^O(t)^O0(t0)\u0011\n\u0011\u0012(t;t0)^O(t)^O0(t0)\u0006\u0012(t0;t)^O0(t0)^O(t);\n(6)\nwith\u0012(t;t0)the corresponding Heaviside step function\nand the +(\u0000)sign applies when the operators have\nbosonic (fermionic) commutation relations. In Fig. 2\nwe schematically indicate the relevant quantities enter-\ning our theory.\nAtt= 0, the spin accumulation in the two leads is\nWe compute the magnon self energy due the coupling\nbetween magnons and electrons to second order in the\ncoupling matrix elements Jj;kk0. This implies that the\nmagnons acquire a Keldysh self-energy due to lead r\ngiven by\n~\u0006r\nj;j0(t;t0) =\u0000i\n~X\nkk0k00k000Jr\nj;kk0(Jr)\u0003\nj0;k00k000\n\u0002Gk0k00;r\"(t;t0)Gk000k;r#(t0;t);(7)\nwhereGk0k00;r\u001b(t;t0)denotes the electron Keldysh\nGreen’s function of lead r, that reads\nGkk0;r\u001b(t;t0) =\u0000ihTC1^ kr\u001b(t)^ y\nk0r\u001b(t0)i:(8)4\nThe Feynman diagram for this self-energy is shown in\nFig. 3. While this self-energy is computed to second\norder inJr\nj;kk0, the magnon Green’s function and the\nmagnon spin current, both of which we evaluate below,\ncontain all orders in Jr\nj;kk0, which therefore does not need\nto be small. In this respect, our approach is different\nfrom the work of Ohnuma et al.[29], who evaluate the\ninterfacial spin current to second order in the electron-\nmagnon coupling. Irreducible diagrams other than that\nin Fig. 3 involve one or more magnon propagators as in-\nternal lines and therefore correspond to magnon-magnon\ninteractions at the interface induced by electrons in the\nnormal metal. For the small magnon densities of interest\nto use here these can be safely neglected and the self-\nenergy in Eq. (7) thus takes into account the dominant\nprocess of spin transfer between metal and insulator.\nThe lesser and greater component of the electronic\nGreen’s functions can be expressed in terms of the spec-\ntral functions Akk0;r(\u000f)via\n\u0000iG<\nkk0;r\u001b=Akk0;r(\u000f)NF\u0012\u000f\u0000\u0016r\u001b\nkBTr\u0013\n;\niG>\nkk0;r\u001b=Akk0;r(\u000f)\u0014\n1\u0000NF\u0012\u000f\u0000\u0016r\u001b\nkBTr\u0013\u0015\n;(9)\nwithNF(x) = [ex+ 1]\u00001the Fermi distribution function,\nTrthe temperature of lead r(kBbeing Boltzmann’s con-\nstant) and\u0016\u001b;rthe chemical potential of spin projection\n\u001bin leadr. As we will see later on, the lead chemical\npotential are taken spin-dependent to be able to imple-\nment nonzero spin accumulation. The spectral function\nis related to the retarded Green’s function via\nAkk0;r(\u000f) =\u00002Imh\nG(+)\nkk0;r(\u000f)i\n; (10)\nwhich does not depend on spin as the leads are taken to\nbe normal metals. While the retarded Green’s function\nof the leads can be determined explicitly for the model\nthat we consider here, we will show below that such a\nlevel of detail is not needed but that, instead, we can pa-\nrameterize the electron-magnon coupling by an effective\ninterface parameter.\nAs mentioned before, all steady-state properties of the\nmagnon system are determined by the magnon lesser\nGreen’s function leading to the magnon density matrix.\nIt is ultimately given by the kinetic equation23,30\n\u001aj;j0\u0011h^by\nj0(t)^bj(t)i=Zd\u000f\n(2\u0019)h\nG(+)(\u000f)i~\u0006<(\u000f)G(\u0000)(\u000f)i\nj;j0;\n(11)\nwhere ~\u0006j;j0(t;t0)is the total magnon self-energy dis-\ncussed in detail below, of which the \"lesser\" component\nenters in the above equation. In the above and what\nfollows, quantities with suppressed site indexes are in-\nterpreted as matrices, and matrix multiplication applies\nfor products of these quantities. The retarded (+)and\nadvanced (\u0000)magnon Green’s functions satisfy\nh\n\u000f\u0006\u0000h\u0000~\u0006(\u0006)(\u000f)i\nG(\u0006)(\u000f) = 1;(12)where\u000f\u0006=\u000f\u0006i0. The magnon self-energies have con-\ntributions from the leads, as well as a contribution from\nthe bulk denoted by ~\u0006FM:\n~\u0006(\u000f) =~\u0006FM(\u000f) +X\nr2fL;Rg~\u0006r(\u000f):(13)\nFrom Eq. (7) we find that for the retarded and advanced\ncomponent, the contribution due to the leads is given by\n~\u0006r;(\u0006)\nj;j0(\u000f) =X\nkk0k00k000Jr\nj;kk0(Jr)\u0003\nj0;k00k000Zd\u000f0\n(2\u0019)Zd\u000f00\n(2\u0019)\n\u0002Ak0k00;r(\u000f0)Ak000k;r(\u000f00)NF\u0010\n\u000f0\u0000\u0016r\"\nkBTr\u0011\n\u0000NF\u0010\n\u000f00\u0000\u0016r#\nkBTr\u0011\n\u0000\u000f\u0006+\u000f0\u0000\u000f00\n;\n(14)\nwhereas the \"lesser\" self-energy can be shown to be of\nthe form:\n~\u0006r;<\nj;j0(\u000f) = 2iNB\u0012\u000f\u0000\u0001\u0016r\nkBTr\u0013\nImh\n~\u0006r;(+)\nj;j0(\u000f)i\n;(15)\nwithNB(x) = [ex\u00001]\u00001the Bose-Einstein distribution\nfunction and \u0001\u0016r=\u0016r\"\u0000\u0016r#the spin accumulation in\nleadr.\nHaving established the contributions due to the leads,\nwe consider the bulk self-energy ~\u0006FM, which in princi-\nple could include various contributions, such as magnon\nconserving and nonconsering magnon-phonon interac-\ntions, or magnon-magnon interactions. Here, we consider\nmagnon non-conserving magnon-phonon coupling as the\nsource of the bulk self-energy and use the Gilbert damp-\ning phenomenology to parameterize it by the constant\n\u000bwhich for the magnetic insulator YIG is of the order\nof10\u00004. Gilbert damping corresponds to a decay of the\nmagnons into phonons with a rate proportional to their\nenergy. This thus leads to the contributions\n~\u0006FM;<\nj;j0(\u000f) = 2NB\u0012\u000f\nkBTFM\u0013\n~\u0006FM;(+)\nj;j0(\u000f) ;\n~\u0006FM;(+)\nj;j0(\u000f) =\u0000i\u000b\u000f\u000ej;j0; (16)\nwhereTFMis the temperature of phonon bath. We note\nthat in principle the temperature could be taken position\ndependent to implement a temperature gradient, but we\ndo not consider this situation here.\nWith the results above, the density-matrix elements\n\u001aj;j0can be explicitly computed from the magnon re-\ntarded and advanced Green’s function and the “lesser”\ncomponent of the total magnon self-energy using\nEq. (11). The magnon self-energy is evaluated using the\nexplicitexpressionfortheretardedandadvancedmagnon\nself-energies due to leads and Gilbert damping ~\u0006FM, see\nEq. (16).\nWe are interested in the computation of the magnon\nspin currenthjm;jj0iin the bulk of the FM from site j5\nto sitej0, which in terms of the magnon density matrix\nreads,\nhjm;jj0i=\u0000i(hj;j0\u001aj0;j\u0000c:c:); (17)\nand follows from evaluating the change in time of the lo-\ncal spin density, ~dh^by\nj^bji=dt, using the Heisenberg equa-\ntions of motion. The magnon spin current in the bulk\nthus follows straightforwardly from the magnon density\nmatrix.\nWhile the formalism presented so far provides a com-\nplete description of the magnon spin transport driven by\nmetallic reservoirs, we discuss two simplifying develop-\nments below. First, we derive a Landauer-Bütikker-like\nformula for the spin current from metallic reservoirs to\nthe magnon system. Second, we discuss how to replace\nthe matrix elements Jr\nj;k;k0by a single phenomenologi-\ncal parameter that characterizes the interface between\nmetallic reservoirs and the magnetic insulator.\nC. Landauer-Büttiker formula\nIn this section we derive a Landauer-Büttiker formula\nfor the magnon transport. Using the Heisenberg equa-\ntions of motion for the local spin density, we find that\nthe spin current from the left reservoir into the magnon\nsystem is given by\njL\ns\u0011\u0000~\n2*\nd\ndtX\nk\u0010\n^ y\nk\"L k\"L\u0000 y\nk#L k#L\u0011+\n=\u00002\n~X\nj;kk0Re[\u0000\nJL\u0001\u0003\nj;kk0g<\nj;kk0(t;t0)];(18)\nin terms of the Green’s function\ng<\nj;kk0(t;t0)\u0011ih^ y\nk0\"L(t0)^ k#L(t0)^bj(t)i:(19)\nThis “lesser” coupling Green’s function g<\nj;kk0(t;t0)is cal-\nculated using Wick’s theorem and standard Keldysh\nmethods as described below.\nWe introduce the spin-flip operator for lead r\n^dy\nkk0;r(t) =^ y\nk0\"r(t)^ k#r(t); (20)\nso that the coupling Green’s function becomes\ng<\nj;kk0(t;t0)\u0011ih^dy\nkk0;L(t0)^bj(t)i: (21)\nThe Keldysh Green’s function for the spin-flip operator\nis given by\n\u0005r\nkk0k00k000(t;t0) =\u0000ihTC1^dkk0;r(t)^dy\nk00k000;r(t0)i(22)\nand using Wick’s theorem we find that\n\u0005r;>\nkk0k00k000(t;t0) =\u0000iG>\nkk000;r#(t;t0)G<\nk0k00;r\"(t0;t) ;\n\u0005r;<\nkk0k00k000(t;t0) =\u0000iG>\nk0k00;r\"(t0;t)G<\nkk000;r#(t;t0) ;\n\u0005r;(+)\nkk0k00k000(t;t0)\n=\u0000i\u0012(t\u0000t0)h\nG>\nkk000;r#(t;t0)G<\nk0k00;r\"(t0;t)\n\u0000G>\nk0k00;r\"(t0;t)G<\nkk000;r#(t;t0)i\n;(23)where we used the definition for the electron Green’s\nfunction in Eq. (8).\nApplying the Langreth theorem30and Fourier trans-\nforming, we write down the lesser coupling Green’s\nfunction in terms of the spin-flip Green’s function and\nmagnon Green’s function\ng<\nj;kk0(\u000f) =X\nj0;k00k000JL\nj0;k00k000\u0010\nG(+)\nj;j0(\u000f)\u0005L;<\nkk0k00k000(\u000f)\n+G<\nj0;j(\u000f)\u0005L;(\u0000)\nkk0k00k000(\u000f)\u0011\n; (24)\nwhere the retarded and “lesser\" magnon Green’s function\nare given by Eq. (11) and Eq. (12). Using these results,\nwe ultimately find that\njL\ns=Zd\u000f\n2\u0019\u0014\nNB\u0012\u000f\u0000\u0001\u0016L\nkBTL\u0013\n\u0000NB\u0012\u000f\u0000\u0001\u0016R\nkBTR\u0013\u0015\nT(\u000f)\n+Zd\u000f\n2\u0019\u0014\nNB\u0012\u000f\u0000\u0001\u0016L\nkBTL\u0013\n\u0000NB\u0012\u000f\nkBTFM\u0013\u0015\n\u0002Trh\n~\u0000L(\u000f)G(+)(\u000f)~\u0000FM(\u000f)G(\u0000)(\u000f)i\n; (25)\nwith the transmission function\nT(\u000f)\u0011Trh\n~\u0000L(\u000f)G(+)(\u000f)~\u0000R(\u000f)G(\u0000)(\u000f)i\n:(26)\nIn the above, the rates ~\u0000L=R(\u000f)are defined by\n~\u0000r(\u000f)\u0011\u00002Imh\n~\u0006r;(+)(\u000f)i\n; (27)\nand\n~\u0000FM(\u000f)\u0011\u00002Imh\n~\u0006FM;(+)(\u000f)i\n;(28)\nand correspond to the decay rates of magnons with en-\nergy\u000fdue to interactions with electrons in the normal\nmetal at the interfaces, and phonons in the bulk, respec-\ntively. This result is similar to the Laudauer-Büttiker\nformalism23for electronic transport using single-particle\nscattering theory. In the present context, a Landauer-\nBüttiker-like for spin transport was first derived by Ben-\nderet al.[28] for a single NM-FM interface. In the\nabsence of Gilbert damping, the spin current would cor-\nrespond to the expected result from Landauer-Bütikker\ntheory, i.e., the spin current from left to the right lead is\nthen given by the first line of Eq. (25). The presence of\ndamping gives leakage of spin current due to the coupling\nwith the phononic reservoir, as the second term shows.\nFinally,wenotethatthespincurrentfromtherightreser-\nvoir into the system is obtained by interchanging labels L\nand R in the first term, and the label L replaced by R in\nthe second one. Due to the presence of Gilbert damping,\nhowever, we have in general that jL\ns6=\u0000jR\ns.\nD. Determining the interface coupling\nWe now proceed to express the magnon spin current\n(Eq.(25))intermsofamacroscopic,measurablequantity6\nratherthantheinterfacialexchangeconstants Jr\nj;k;k0. For\n\u0001\u0016r\u001c\u000fF(with\u000fFthe Fermi energy of the metallic\nleads), which is in practice always obeyed, we have for\nlow energies and temperatures that\n~\u0006r;(\u0006)\nj;j0(\u000f)'\u0007i1\n4\u0019X\nkk0k00k000Jr\nj;kk0(Jr)\u0003\nj0;k00k000\nAk0;k00;r(\u000fF)Ak000;k;r(\u000fF)(\u000f\u0000\u0001\u0016r):(29)\nHere, we also neglected the real part of this self-energy\nwhich provides a small renormalization of the magnon\nenergies but is otherwise unimportant. The expansion\nfor small energies in Eq. (29) is valid as long as \u000f\u001c\u000fF,\nwhich applies since \u000fis a magnon energy, and therefore\nat most on the order of the thermal energy. Typically,\nthe above self-energy is strongly peaked for j;j0at the\ninterface because the magnon-electron interactions occur\nat the interface. For j;j0at the interface we have that\nthe self-energy depends weakly on varying j;j0along the\ninterface provided that the properties of the interface do\nnot vary substantially from position to position. We can\nthus make the identification:\n~\u0006r;(\u0006)\nj;j0(\u000f)'\u0007i\u0011r(\u000f\u0000\u0001\u0016r)\u000ej;j0\u000ej;jr;(30)\nwithjrthe positions of the sites at the r-th interface,\nand\u0011rparametrizing the coupling between electrons and\nmagnons at the interface. Note that \u0011rcan be read off\nfrom Eq. (29). Rather than evaluating this parameter\nin terms of the matrix elements Jr\nj;kk0and the electronic\nspectral functions of the leads Ak;k0;r(\u000f), we determine it\nin terms of the real part of the spin-mixing conductance\ng\"#;r, a phenomenological parameter that characterizes\nthespin-transferefficiencyattheinterface31. Thiscanbe\ndonebynotingthatintheclassicallimittheself-energyin\nEq. (30) leads to an interfacial contribution, determined\nby the damping constant \u0011r=N, to the Gilbert damping\nofthehomogeneousmode, where Nisthenumberofsites\nof the system perpendicular to the leads, as indicated in\nFig. 2. In terms of the spin-mixing conductance, we have\nthat this contribution is given by32g\"#;r=4\u0019srN, withsr\nthe saturation spin density per area of the ferromagnet\nat the interface with the r-th lead. Hence, we find that\n\u0011r=g\"#;r\n4\u0019sr; (31)\nwhich is used to express the reservoir contributions to the\nmagnon self-energies in terms of measurable quantities.\nThe spin-mixing conductance can be up to 5~nm\u00002for\nYIG-Pt interfaces33, leading to the conclusion that \u0011can\nbe of the order 1\u000010for that case.\nE. Summary on implementation\nWe end this section with some summarizing remarks\non implementation that may facilitate the reader who is\ninterested in applying the formalism presented here.Table I: Parameters chosen for numerical calculations based\non the NEGF formalism (unless otherwise noted).\nQuantity Value\nJ 0:05eV\n\u0001\u0016L=J 2:0\u000210\u00005\n\u0001\u0016R=J 0:0\n\u0011 8\n\u0001=J 2:0\u000210\u00003\nkBTFM=J0:60\nFirst, one determines the retarded and advanced\nmagnon Green’s functions. This can be done given\na magnon hamiltonian characterized by matrix ele-\nmentshj;j0in Eq. (2), mixing conductances for the\nmetal-insulator interfaces g\"#;r, and a value for the\nGilbert damping constant \u000b, from which one computes\nthe retarded self-energies at the interfaces in Eq. (30)\nwith Eq. (31), and Eq. (16). The retarded and ad-\nvanced magnon Green’s functions are then computed via\nEq. (12), which amounts to a matrix inversion. The next\nstepistocalculatethedensitymatrixforthemagnonsus-\ning Eq. (11), with as input the expressions for the “lesser”\nself-energies in Eqs. (15) and (16). Finally, the spin cur-\nrent is evaluated using Eq. (17) in the bulk of the FM or\nEq. (25) at the NM-FM interface. In the next sections,\nwe discuss some applications of our formalism.\nIII. NUMERICAL RESULTS\nIn this section, we present results of numerical calcula-\ntions using the formalism presented in the previous sec-\ntion.\nA. Clean system\nFor simplicity, we consider now the situation where\nthe leads and magnetic insulators are one dimensional.\nThe values of various parameters are displayed in Ta-\nble I, where we take the hopping amplitudes Jj;j0=\nJ(\u000ej;j0+1+\u000ej;j0\u00001), i.e.,Jj;j0is equal toJbetween near-\nest neighbours, and zero otherwise. We focus on trans-\nport driven by spin accumulation in the leads and set\nall temperatures equal, i.e., TL=TR=TFM\u0011T.\nWe also assume both interfaces to have equal proper-\nties, i.e., for the magnon-electron coupling parameters to\nobey\u0011L=\u0011R\u0011\u0011. First we consider the case without\ndisorder and take \u0001j= \u0001.\nWe are interested in how the Gilbert damping affects\nthe magnon spin current. In particular, we calculate the\nspincurrentinjectedintherightreservoirasafunctionof\nsystem size. The results of this calculation are shown in\nFig. 4 for various temperatures, which indicates that for7\na certain fixed spin accumulation, the injected spin cur-\nrent decays with the thickness of the system for N > 25,\nfor the parameters we have chosen. We come back to the\nvarious regimes of thickness dependence when we present\nanalyticalresultsforcleansystemsinthecontinuumlimit\nin Sec. IV. From these results we define a magnon relax-\n0 20 40 60 80 100 120 140 160\nsystem/uni00A0size/uni00A0(d/a)10/uni00AD410/uni00AD310/uni00AD210/uni00AD1100magnon/uni00A0spin/uni00A0current/uni00A0(J)kBT/J=0.012\nkBT/J=0.024\nkBT/J=0.048\nkBT/J=0.108\nkBT/J=0.192\nFigure 4: System-size dependence of spin current ejected in\nthe right reservoir for \u000b= 6:9\u000210\u00002;\u0011= 8:0and various\ntemperatures.\nation length drelaxusing the definition\njm(d)/exp(\u0000d=drelax); (32)\napplied to the region N > 25and where d= Nawith\nathe lattice constant. The magnon relaxation length\ndepends on system temperature and is shown in Fig. 5.\nWe attempt to fit the temperature dependence with\ndrelax(T\u0003) =a(\r0+\r1p\nT\u0003+\r2\nT\u0003);(33)\nwith\r0;\r1;\r2constantsand T\u0003definedasthedimension-\nless temperature T\u0003\u0011kBT=J. The term proportional to\n\r1is expected for quadratically dispersing magnons with\nGilbert damping as the only relaxation mechanism15,25.\nThe terms proportional to \r0and\r2are added to charac-\nterize the deviation from this expected form. Our results\nshow that the relaxation length has not only \u00181=p\nT\nbehaviour. This is due to the finite system size, the con-\ntact resistance that the spin current experiences at the\ninterface between metal and magnetic insulator, and the\ndeviation of the magnon dispersion from a quadratic one\ndue to the presence of the lattice.\nB. Disordered system\nWe now consider the effects of disorder on the spin\ncurrent as a function of the thickness of the FM. We con-\nsider a one-dimensional system with a disorder potential\ndrelax/a=γ0+γ1\nT*+γ2\nT*\nNumerical result\nFitted curve\n0.0 0.2 0.4 0.6 0.8115120125130135140145150\nT*=kBT/Jdrelax/aFigure 5: Magnon relaxation length as a function of dimen-\nsionlesstemperature T\u0003for\u000b= 6:9\u000210\u00002;\u0011= 8:0. Thefitted\nparameters are obtained as \r0= 114:33;\r1= 0:96;\r2= 0:32.\nimplemented by taking \u0001j= \u0001(1+\u000ej), where\u000ejis a ran-\ndom number evenly distributed between \u0000\u000eand\u000e(with\n\u000e\u001c1andpositive)thatisuncorrelatedbetweendifferent\nsites. In one dimension, all magnon states are Anderson\nlocalized34. Since this is an interference phenomenon, it\nis expected that Gilbert damping diminishes such local-\nization effects. The effect of disorder on spin waves was\ninvestigated using a classical model in Ref. [35], whereas\nRef. [36] presents a general discussion of the effect of\ndissipation on Anderson localization. Very recently, the\neffect of Dzyaloshinskii-Moriya interactions on magnon\nlocalization was studied37. Here we consider how the in-\nterplay between Gilbert damping and the disorder affects\nthe magnon transport.\nFor a system without Gilbert damping the spin current\ncarried by magnons is conserved and therefore indepen-\ndent of position regardless of the presence or absence of\ndisorder. DuetothepresenceofGilbertdampingthespin\ncurrent decays as a function of position. Adding disorder\non top of the dissipation due to Gilbert damping causes\nthe spin current to fluctuate from position to position.\nFor large Gilbert damping, however, the effects of dis-\norder are suppressed as the Gilbert damping suppresses\nthe localization of magnon states. In Fig. 6 we show nu-\nmerical results of the position dependence of the magnon\ncurrent for different combinations of disorder and Gilbert\ndamping constants. The plots clearly show that the spin\ncurrent fluctuates in position due to the combined ef-\nfect of disorder and Gilbert damping, whereas it is con-\nstant without Gilbert damping, and decays in the case\nwith damping but without disorder. Note that for the\ntwo cases without Gilbert damping the magnitude of the\nspin current is different because the disorder alters the\nconductance of the system and each curve in Fig. 6 cor-\nresponds to a different realization of disorder.\nTo characterize the fluctuations in the spin current, we8\n0 20 40 60 80\nsite j1.01.52.02.53.03.54.04.5magnon spin current (J)1e7\n=0.0,=0.0\n=0.0,=0.0015\n=0.0069,=0.0\n=0.0069,=0.0015\nFigure 6: Spatial dependence of local magnon current for the\ncase without Gilbert damping and disorder ( \u000b= 0;\u000e= 0),\nwithout disorder ( \u000b= 6:9\u000210\u00003;\u000e= 0), without Gilbert\ndamping (\u000b= 0;\u000e= 1:5\u000210\u00003), and both disorder and\nGilbert damping ( \u000b= 6:9\u000210\u00003;\u000e= 1:5\u000210\u00003). The inter-\nface coupling parameter is taken equal to \u0011= 0:8.\ndefine the correlation function\nCj=vuut\u0000\njm;j;j+1\u0000jm;j;j+1\u00012\n\u0000\njm;j;j+1\u00012; (34)\nwhere the bar stands for performing averaging over the\nrealizations of disorder. Fig. (7) shows this correlation\nfunction for j=N\u00001as a function of Gilbert damp-\ning for various strengths of the disorder. As we expect,\nbasedonthepreviousdiscussion, thefluctuationsbecome\nsmall as the Gilbert damping becomes very large or zero,\nleaving an intermediate range where there are sizeable\nfluctuations in the spin current.\nIV. ANALYTICAL RESULTS\nIn this section we analytically compute the magnon\ntransmission function in the continuum limit a!0\nfor a clean system. We consider again the situation\nof a magnon hopping amplitude Jj;j0that is equal to\nJand nonzero only for nearest neighbors, and a con-\nstant magnon gap \u0001j= \u0001. We compute the magnon\ndensity matrix, denoted by \u001a(x;x0), and retarded and\nadvanced Green’s functions, denoted by G(\u0006)(x;x00;\u000f).\nHere, the spatial coordinates in the continuum are de-\nnoted byx;x0;x00;\u0001\u0001\u0001. We take the system to be trans-\nlationally invariant in the y\u0000z-plane and the current\ndirection as shown in Fig. 1 to be x.\nIn the continuum limit, the imaginary part of the vari-\n0.000 0.005 0.010 0.015 0.020\nα012345CN−1\nδ=0.0005\nδ=0.001\nδ=0.0015Figure 7: Correlation function Cjthat characterizes the fluc-\ntuations in the spin curent for j=N\u00001as a function of the\nGilbert damping constant, for three strengths of the disorder\npotential. The curves are obtained by performing averaging\nover 100 realizations of the disorder. The interface coupling\nparameter is taken equal to \u0011= 0:8.\nous self-energies acquired by the magnons have the form:\nImh\n~\u0006r;(+)(x;x0;\u000f)i\n=\n\u0000~\u0011r(\u000f\u0000\u0016r)\u000e(x\u0000xr)\u000e(x\u0000x0) ;\nImh\n~\u0006FM;(+)(x;x0;\u000f)i\n=\u0000\u000b\u000f\u000e(x\u0000x0);(35)\nwherexris the position of the r-th lead, and where ~\u0011ris\nthe parameter that characterizes the interfacial coupling\nbetween magnons and electrons. We use a different nota-\ntion for this parameter as in the continuum situation its\ndimension is different with respect to the discrete case.\nTo express ~\u0011rin terms of the spin-mixing conductance\nwe have that ~\u0011r=g\"#=4\u0019~srwhere ~sris now the three-\ndimensional saturated spin density of the ferromagnet.\nWe proceed by evaluating the magnon transmission\nfunction from Eq. (26). We compute the rates in Eq. (27)\nfrom the self-energies Eqs. (35) and find for the transmis-\nsion function in the first instance that\nT(\u000f) = 4~\u0011L~\u0011R(\u000f\u0000\u0001\u0016L)(\u000f\u0000\u0001\u0016R)\n\u0002Zdq\n(2\u0019)2g(+)(xL;xR;q;\u000f)g(\u0000)(xR;xL;q;\u000f);(36)\nwhere qis the two-dimensional momentum that results\nfromFouriertransforminginthe y\u0000z-plane. TheGreen’s\nfunctionsg(\u0006)(x;x0;q;\u000f)obey [compare Eq. (12)]\n\u0014\n(1\u0006i\u000b)\u000f+Ad2\ndx2\u0000Aq2\u0000\u0001\n\u0006iX\nr2fL;Rg~\u0011r(\u000f\u0000\u0001\u0016r)\u000e(x\u0000xr)3\n5g(\u0006)(x;x0;q;\u000f)\n=\u000e(x\u0000x0); (37)9\nwhereA=Ja2. This Green’s function is evaluated using\nstandard techniques for inhomogeneous boundary value\nproblems (see Appendix A) to ultimately yield\nT(\u000f) = 4~\u00112(\u000f\u0000\u0001\u0016L)(\u000f\u0000\u0001\u0016R)Zdq\n(2\u0019)2jt(q;\u000f)j2;(38)\nwith\nt(q;\u000f) =A\u0014\u0002\u0000\nA2\u00142\u0000~\u00112(\u000f\u0000\u0001\u0016L)(\u000f\u0000\u0001\u0016R)\u0001\nsinh(\u0014d)\n\u0000iA~\u0011\u0014(2\u000f\u0000\u0001\u0016L\u0000\u0001\u0016R) cosh(\u0014d)]\u00001; (39)\nwith\u0014=p\n(Aq2+ \u0001\u0000\u000f\u0000i\u000b\u000f)=Aand whered=xR\u0000\nxL. Note that we have at this point taken both interfaces\nequal for simplicity, so that ~\u0011L= ~\u0011R\u0011~\u0011. In terms of an\ninterfacial Gilbert damping parameter \u000b0we have that\n~\u0011=d\u000b0.\nLet us identify the magnon decay length\nl\u0011\u0015\n\u000b;\nwhere\u0015=p\nA=kBTis proportional to the thermal de\nBroglie wavelength. Equipped with a closed, analytic\nexpression, we may now, in an analogous way as Hoffman\net al.[25], investigate the behavior of Eq. (38) in the thin\nFM (d\u001cl) and thick FM ( d\u001dl) regimes. In order\nto do so, we take \u0016L= 0so that the second term in\nEq. (25) vanishes and the spin current is fully determined\nby the transmission coefficient T(\u000f). Before analyzing\nthe result for the spin current more closely, we remark\nthat the result for the transmission function may also be\nobtained from the linearized stochastic Landau-Lifshitz-\nGilbert equation, as shown in Appendix B.\nA. Thin film regime ( d\u001cl)\nIn the thin film regime, the transmission coefficient\nT(\u000f)exhibits scattering resonances near \u000f=\u000fnqfor given\nq, where\n\u000fnq\nA=q2+1\n\u00182+n2\u00192\nd2\nandnisanintegerandwhere \u0018=p\nA=\u0001isthecoherence\nlength of the ferromagnet. When the ferromagnet is suf-\nficiently thin ( d\u001c\u0015=\u000b1=2=p\u000bl), one finds that these\npeaks are well separated, and the transmission coefficient\nis approximated as a sum of Lorentzians: T =P1\nn=0Tn,\nwhere:\nTn(\u000f)\u0019Anq\u0000L\nn\u0000R\nn\n\u0000Ln+ \u0000Rn+ \u0000FMn(40)\nwith\nAnq(\u000f) =\u0000n\n(\u000f\u0000\u000fnq)2+ (\u0000n=2)2(41)as the spin wave spectral density. The broadening rates\nare given by \u0000FM\nn= 2\u000b\u000f,\u0000L\n0= 2\u000b0\u000f,\u0000R\n0= 2\u000b0(\u000f\u0000\u0016R),\n\u0000L\nn6=0= 4\u000b0\u000f,\u0000R\nn6=0= 4\u000b0(\u000f\u0000\u0016R)and\u0000n= \u0000FM\nn+ \u0000L\nn+\n\u0000R\nn. Intheextremesmalldissipationlimit(i.e. neglecting\nspectral broadening by the Gilbert damping), one has:\nAnq(\u000f)!2\u0019\u000e(\u000f\u0000\u000fnq); (42)\nand the current has the simple form, jL\ns=P1\nn=0jn,\nwhere\njn=a2Zd2q\n(2\u0019)2\u0000L\nn\u0000R\nn\n\u0000n\u0014\nNB\u0012\u000fnq\nkBT\u0013\n\u0000NB\u0012\u000fnq\u0000\u0001\u0016R\nkBT\u0013\u0015\n(43)\nwhere \u0000L\nn,\u0000R\nnand \u0000FM\nnare all evaluated at \u000f=\u000fnq.\nEq. (43) allows one to estimate the thickness dependence\nof the signal. Supposing \u0016R.\u000fnq, whend\u001cg\"#=s\u000b,\nthen\u000b0\u001d\u000b, and \u0000L\nn\u0000R\nn=\u0000FM\nn\u0018jL\ns;cl\u00181=d; whend\u001d\ng\"#=s\u000b, then\u000b0\u001c\u000b, andjL\ns;cl\u00181=d2. The enhancement\nof the spin current for small dis in rough agreement with\nour numerical results in the previous section as shown in\nFig. 4.\nB. Thick film regime ( d\u001dl)\nIn the thick film regime, the transmission function be-\ncomes\nT(\u000f)\u0019(4Ad)2\u0000L\nx\u0000R\nxp\n(\u000f\u0000\u000f0q)2+ (\u0000FMx=2)2e\u00002\u0014rd\nj(4A\u0014)2\u0000(d)2\u0000Lx\u0000Rx\u0000i4dA\u0000Rx\u0014S(\u0014r)j2\nwhere \u0000L=R= FM\nx = \u0000L=R= FM\nn6=0,\u0014r= Re[\u0014], andS(\u0014r)isthe\nsign of\u0014r. For\u000b\u001c1, we have\u0014=ikx\u0000\n1 +i\u000b\u000f=2Ak2\nx\u0001\n,\nwherekx=p\nq2+\u0018\u00002\u0000\u000f=A. For energies \u000f > A (q2+\n\u0018\u00002),kxis imaginary, and the contribution to the spin\ncurrent decays rapidly with d. When, however, \u000f <\nA(q2+\u0018\u00002),kxis real, and \u0014r=\u0000\u000b\u0000\nq2+\u0018\u00002\u0001\n=2kx\u0018\n\u000b=\u0015(for thermal magnons), so that the signal decays\nover a length scale l/1=p\nT, in agreement with our\nnumerical results as shown in Fig. 5.\nC. Comparison with numerical results\nIn order to compare the numerical with the analyti-\ncal results we plot in Fig. 8 the transmission function\nas a function of energy. Here, the numerical result is\nevaluated for a clean system using Eq. (26) while the an-\nalytical result is that of Eq. (38). While they agree in\nthe appropriate limit ( N!1;a!0), for finite Nthere\nare substantial deviations that are due to the increased\nimportance of interfacing coupling relative to the Gilbert\ndamping for small systems and the deviations of the dis-\npersion from a quadratic one.10\nΔ/J=0.2N=20\nAnalytic\nNumerical\n0 1 2 3 4 50.000.020.040.060.08\nϵ/JT(ϵ)\nFigure 8: Magnon transmission function as a function of en-\nergy. The parameters are chosen to be \u0001=J= 0:2;\u000b=\n0:069;\u0011= 8:0.\nV. DISCUSSION AND OUTLOOK\nWe have developed a NEGF formalism for exchange\nmagnon transport in a NM-FM-NM heterostructure. We\nhave illustrated the formalism with numerical and ana-\nlytical calculations and determined the thickness depen-\ndence of the magnon spin current. We have also con-\nsidered magnon disorder scattering and shown that the\ninterplay between disorder and Gilbert damping leads to\nspin-current fluctuations.\nWehavealsodemonstratedthatforacleansystem,i.e.,\nwithout disorder, in the continuum limit the results ob-\ntained from the NEGF formalism agree with those fromthe stochastic LLG formalism. The latter is suitable for\na clean system in the continuum limit where the vari-\nous boundary conditions on the solutions of the stochas-\ntic equations are easily imposed. The NEGF formal-\nism is geared towards real-space implementation, such\nthat, e.g., disorder scattering due to impurities are more\nstraightforwardly included as illustrated by our example\napplication. The NEGF formalism is also more flexi-\nble for systematically including self-energies due to ad-\nditional physical processes, such as magnon-conserving\nmagnon-phonon scattering and magnon-magnon scatter-\ning, or, for example, for treating strong-coupling regimes\nintowhichthestochasticLandau-Lifshitz-Gilbertformal-\nism has no natural extension.\nUsing our formalism, a variety of mesoscopic transport\nfeatures of magnon transport can be investigated includ-\ning, e.g., magnon shot noise38. The generalization of our\nformalism to elliptical magnons and magnons in antifer-\nromagnets is an attractive direction for future research.\nAcknowledgments\nThis work was supported by the Stichting voor Funda-\nmenteel Onderzoek der Materie (FOM), the Netherlands\nOrganization for Scientific Research (NWO), and by the\nEuropean Research Council (ERC) under the Seventh\nFramework Program (FP7). J. Z. would like to thank\nthe China Scholarship Council. J. A. has received fund-\ning from the European Union’s Horizon 2020 research\nand innovation programme under the Marie Skłodowska-\nCurie grant agreement No 706839 (SPINSOCS).\nAppendix A: Evaluation of magnon Green’s function in the continuum limit\nIn this appendix we evaluate the magnon Green’s function in the continuum limit that is determined by Eq. (37).\nFor simplicity we take the momentum qequal to zero and suppress it in the notation, as it can be trivially restored\nafterwards. The Green’s function is then determined by\n2\n4\u000f\u0006i\u000b\u000f\u0006iX\nr2fL;Rg~\u0011r(\u000f\u0000\u0001\u0016r)\u000e(x\u0000xr) +Ad2\ndx2\u0000H3\n5g(\u0006)(x;x0;\u000f) =\u000e(x\u0000x0): (A1)\nTo determine this Green’s function we first solve for the states \u001f\u0006(x)that obey:\n2\n4\u000f\u0006i\u000b\u000f\u0006iX\nr2fL;Rg~\u0011r(\u000f\u0000\u0001\u0016r)\u000e(x\u0000xr) +Ad2\ndx2\u0000H3\n5\u001f\u0006(x) = 0: (A2)\nIntegrating this equation across x=xLandx=xRleads to the boundary conditions:\nx=xL:\u0006i~\u0011L(\u000f\u0000\u0001\u0016L)\u001f\u0006(xL) +Ad\u001f\u0006(x)\ndxjx=xL= 0; (A3)\nx=xR:\u0006i~\u0011R(\u000f\u0000\u0001\u0016R)\u001f\u0006(xR)\u0000Ad\u001f\u0006(x)\ndxjx=xR= 0: (A4)11\nForxL<x<xR, the general solution is:\n\u001f\u0006(x) =Beik\u0006x+Ce\u0000ik\u0006x; (A5)\nwithk\u0006=p\n(\u000f\u0006i\u000b\u000f\u0000H)=A. We write the solution obeying the boundary condition at x=xLas\n\u001f\u0006\nL(x) =eik\u0006x+C\u0006e\u0000ik\u0006x; (A6)\nWith the boundary condition at x=xL( Eq.A4), we find that\nC\u0006=\u0014Ak\u0006\u0006~\u0011L(\u000f\u0000\u0001\u0016L)\nAk\u0006\u0007~\u0011L(\u000f\u0000\u0001\u0016L)\u0015\ne2ik\u0006xL:\nFor the solution obeying the boundary condition at x=xR, we write\n\u001f\u0006\nR(x) =B\u0006eik\u0006x+e\u0000ik\u0006x: (A7)\nWith the boundary condition at x=xR( Eq.A4), we have:\nA(iB\u0006k\u0006eik\u0006xR\u0000ik\u0006e\u0000ik\u0006xR) =\u0006i~\u0011R(\u000f\u0000\u0016R)(B\u0006eik\u0006xR+e\u0000ik\u0006xR);\nso that\nB\u0006=\u0014Ak\u0006\u0006~\u0011R(\u000f\u0000\u0001\u0016R)\nAk\u0006\u0007~\u0011R(\u000f\u0000\u0001\u0016R)\u0015\ne\u00002ik\u0006xR:\nThe Green’s function is now given by39\ng(\u0006)(x;x0;\u000f) =8\n<\n:\u001f(\u0006)\nL(x0)\u001f(\u0006)\nR(x)\nAW(\u0006)(x0)forx>x0;\n\u001f(\u0006)\nL(x)\u001f(\u0006)\nR(x0)\nAW(\u0006)(x0)forx<x0:(A8)\nwith the Wronskian\nW\u0006(x0) =\u001f\u0006\nL(x0)d\u001f\u0006\nR(x0)\ndx0\u0000\u001f\u0006\nR(x0)d\u001f\u0006\nL(x0)\ndx0:\nInserting the result for the Green’s function in Eq. (36) and using that k+=i\u0014andk\u0000= (k+)\u0003, we obtain Eqs. (38)\nand (39) after restoring the q-dependence and taking ~\u0011R= ~\u0011L= ~\u0011.\nAppendix B: Stochastic Formalism for Spin Transport in a Ferromagnet\nHere, we show how to recover our analytical results from the stochastic Landau-Lifshitz-Gilbert equation, general-\nizing the results of Ref. [25] to the case of nonzero spin accumulation in the metallic reservoirs. The dynamics of the\nspin density unit vector nis governed by:\n(1 +\u000bn\u0002)~_n+n\u0002(H+h)\u0000An\u0002r2n= 0; (B1)\nwhere H= \u0001^zis the effective applied magnetic field (in units of energy) and his the bulk stochastic field24. We\nassume a spin accumulation \u00160= \u0001\u0016Rzin the right normal metal, while the spin accumulation in the left lead is\ntaken zero. The boundary condition at x= 0reads\njs(x= 0) =\u0000A~sn\u0002@xnjx=0\n=\u0014g\"#\n4\u0019(n\u0002(n\u0002\u00160) +n\u0002~_n) +n\u0002h0\nL\u0015\nx=0(B2)\nand atx=d:\njs(x=d) =\u0000A~sn\u0002@xnjx=d\n=\u0000\u0014g\"#\n4\u0019(n\u0002~_n) +n\u0002h0\nR\u0015\nx=d: (B3)12\nDefining (x;t) =n(x;t)p\n~s=2, wheren\u0011nx\u0000iny, we linearize the dynamics around the equilibrium orientation\nn=\u0000z. Fourier transforming:\n (x;q;\u000f) =Zdt\n2\u0019~Zd2r?\n2\u0019ei\u000ft=~e\u0000ir?\u0001q (x;r?;t);\nthe bulk equation of motion reads:\nA\u0000\n@2\nx\u0000\u00142\u0001\n =hp\n~s : (B4)\nThe bulk transformed stochastic field h=hx\u0000ihyobeys the fluctuation dissipation theorem:\nhh\u0003(x;q;\u000f)h(x0;q0;\u000f0)i= 2 (2\u0019)3\u000b(~2=~s)\u000f\n\u0002\u000e(x\u0000x0)\u000e(q\u0000q0)\u000e(\u000f\u0000\u000f0)\ntanh [\u000f=2kBT]: (B5)\nThe boundary conditions, Eqs. (B2) and (B3), become respectively:\nA@x \u0000ig\"#\n4\u0019~s(\u000f\u0000\u0001\u0016R) =hRp\n2~s(B6)\natx= 0and\nA@x +ig\"#\n4\u0019~s\u000f =hLp\n2~s(B7)\natx=d, where we have taken the coupling at both interfaces equal. Similarly, the interfacial stochastic fields obey:\nD\nh0\u0003\nR(q;\u000f)h0\nR(q0;\u000f0)E\n=2 (2\u0019)3\u000b0d~2~s(\u000f\u0000\u0001\u0016R)\u000e(q\u0000q0)\u000e(\u000f\u0000\u000f0)\ntanh [(\u000f\u0000\u0001\u0016R)=2kBT](B8)\nand\nD\nh0\u0003\nL(q;\u000f0)h0\nL(q0;\u000f)E\n=2 (2\u0019)3\u000b0d~2~s\u000f\u000e(q\u0000q0)\u000e(\u000f\u0000\u000f0)\ntanh [\u000f=2kBT]: (B9)\nUsing Eqs. (B4)-(B9), one finds the current on the left side of the structure: jL\ns\u0011z\u0001js(x= 0)to be of the form:\njL\ns=Zd\u000f\n2\u0019\u0014\nNB\u0012\u000f\nkBT\u0013\n\u0000NB\u0012\u000f\u0000\u0001\u0016R\nkBT\u0013\u0015\nT(\u000f) (B10)\nwhere T(\u000f)is the transmission coefficient in Eq. (38). 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Introduction to partial differential\nequations. 2015."    },    {        "title": "0708.0463v1.Strong_spin_orbit_induced_Gilbert_damping_and_g_shift_in_iron_platinum_nanoparticles.pdf",        "content": "arXiv:0708.0463v1  [cond-mat.other]  3 Aug 2007Strong spin-orbit induced Gilbert damping and g-shift in ir on-platinum nanoparticles\nJ¨ urgen K¨ otzler, Detlef G¨ orlitz, and Frank Wiekhorst\nInstitut f¨ ur Angewandte Physik und Zentrum f¨ ur Mikrostruk turforschung,\nUniversit¨ at Hamburg, Jungiusstrasse 11, D-20355 Hamburg , Germany\n(Dated: October 30, 2018)\nThe shape of ferromagnetic resonance spectra of highly disp ersed, chemically disordered\nFe0.2Pt0.8nanospheres is perfectly described by the solution of the La ndau-Lifshitz-Gilbert (LLG)\nequation excluding effects by crystalline anisotropy and su perparamagnetic fluctuations. Upon\ndecreasing temperature , the LLG damping α(T) and a negative g-shift,g(T)−g0, increase pro-\nportional to the particle magnetic moments determined from the Langevin analysis of the magneti-\nzation isotherms. These novel features are explained by the scattering of the q→0 magnon from\nan electron-hole (e/h) pair mediated by the spin-orbit coup ling, while the sd-exchange can be ruled\nout. The large saturation values, α(0) = 0.76 andg(0)/g0−1 =−0.37, indicate the dominance of\nan overdamped 1 meV e/h-pair which seems to originate from th e discrete levels of the itinerant\nelectrons in the dp= 3nmnanoparticles.\nPACS numbers: 76.50.+g, 78.67.Bf, 76.30.-v, 76.60.Es\nI. INTRODUCTION\nTheongoingdownscalingofmagneto-electronicdevices\nmaintainstheyetintenseresearchofspindynamicsinfer-\nromagnetic structures with restricted dimensions. The\neffect of surfaces, interfaces, and disorder in ultrathin\nfilms1, multilayers, and nanowires2has been examined\nand discussed in great detail. On structures confined to\nthe nm-scale in all three dimensions, like ferromagnetic\nnanoparticles, the impact of anisotropy3and particle-\nparticle interactions4on the Ne´ el-Brown type dynamics,\nwhich controls the switching of the longitudinal magneti-\nzation, is now also well understood. On the other hand,\nthe dynamics of the transverse magnetization, which e.g.\ndetermines the externally induced, ultrafast magnetic\nswitching in ferromagnetic nanoparticles, is still a top-\nical issue. Such fast switching requires a large, i.e. a\ncritical value of the LLG damping parameter α5. This\ndamping has been studied by conventional6,7and, more\nrecently, by advanced8ferromagnetic resonance (FMR)\ntechniques, revealing enhanced values of αup to the or-\nder of one.\nBy now, the LLG damping of bulkferromagnets is al-\nmost quantitatively explained by the scattering of the\nq= 0 magnon by conduction electron-hole (e/h) pairs\ndue to the spin-orbit coupling Ω so9. According to recent\nab initio bandstructure calculations10, the rather small\nvalues for α≈Ω2\nsoτresult from the small (Drude) re-\nlaxation time τof the electrons. For nanoparticles, the\nDrude scattering and also the wave-vector conservation\nare ill-defined, and ab initio many-body approaches to\nthe spin dynamics should be more appropriate. Numeri-\ncal workby Cehovin et al.11considersthe modification of\nthe FMR spectrum by the discrete level structure of the\nitinerant electrons in the particle. However, the effect of\nthe resulting electron-hole excitation, ǫp∼v−1\np, wherevp\nis the nanoparticle volume on the intrinsic damping has\nnot yet been considered.\nHere we present FMR-spectra recorded atω/2π=9.1 GHz on Fe0.2Pt0.8nanospheres, the struc-\ntural and magnetic properties of which are summarized\nin Sect. II. In Sect. III the measured FMR-shapes will be\nexamined by solutions of the LLG-equation of motion for\nthe particle moments. Several effects, in particular those\npredicted for crystalline anisotropy12and superparam-\nagnetic (SPM) fluctuations of the particle moments13\nwill be considered. In Sect. IV, the central results of this\nstudy, i.e. the LLG-damping α(T) reaching values of\nalmost one and a large g-shift,g(T)−g0, are presented.\nSince both α(T) and g(T) increase proportional to the\nparticle magnetization, they can be related to spin-orbit\ndamping torques, which, due to the large values of αand\n∆gare rather strongly correlated. It will be discussed\nwhich features of the e/h-excitations are responsible for\nthese correlations in a nanoparticle. A summary and the\nconclusions are given in the final section.\nII. NANOPARTICLE CHARACTERIZATION\nThe nanoparticleassemblyhasbeen prepared14follow-\ning the wet-chemical route by Sun et al.15. In order\nto minimize the effect of particle-particle interactions,\nthe nanoparticles were highly dispersed14. Transmis-\nsion electron microscopy (TEM) revealed nearly spher-\nical shapes with mean diameter dp= 3.1nmand a\nrather small width of the log-normal size distribution,\nσd= 0.17(3). Wide angle X-ray diffraction provided the\nchemically disordered fccstructure with a lattice con-\nstant a 0=0.3861 nm14.\nThemeanmagneticmomentsofthenanospheres µp(T)\nhave been extracted from fits of the magnetization\nisotherms M(H,T), measured by a SQUID magnetome-\nter (QUANTUM DESIGN, MPMS2) in units emu/g=\n1.1·1020µB/g, to\nML(H,T) =Npµp(T)L(µpH\nkBT). (1)2\nHere are L(y) = coth( y)−y−1withy=µp(T)H/kBT\nthe Langevin function and Npthe number of nanoparti-\ncles per gram. The fits shown in Fig.1(a) demanded for\na small paramagnetic background, M−ML=χb(T)H,\nwith a strong Curie-liketemperature variationof the sus-\nceptibility χb, signalizing the presence of paramagnetic\nimpurities. According to the inset of Fig. 1(b) this 1 /T-\nlaw turns out to agree with the temperature dependence\nof the intensity of a weak, narrow magnetic resonance\nwithgi= 4.3 depicted in Figs. 2 and 3. Such narrow\nline with the same g-factor has been observed by Berger\net al.16on partially crystallized iron-containing borate\nglass and could be traced to isolated Fe3+ions.\nThe results for µp(T) depicted in Fig.1(b) show the\nmoments to saturate at µp(T→0) = (910 ±30)µB.\nThis yields a mean moment per atom in the fccunit\ncell ofµ(0) =µpa3\n0/4vp= 0.7µBcorresponding to a\nspontaneous magnetization Ms(0) = 5.5kOe. Accord-\ning to previous work by Menshikov et al.17on chemi-\ncally disordered FexPt1−xthis corresponds to an iron-\nconcentration of x=0.20. Upon rising temperature the\nmoments decrease rapidly, which above 40 K can be\nrather well parameterized by the empirical power law,\nµp(T≥40K)∼(1−T/TC)βrevealing β= 2 and for\nthe Curie temperature TC= (320±20)K. This is con-\nsistent with TC= (310±10)KforFe0.2Pt0.8emerging\nfrom a slight extrapolation of results for TC(x≥0.25)\nofFexPt1−x18. No quantitative argument is at hand\nfor the exponent β= 2, which is much larger than the\nmean field value βMF= 1/2. We believe that β= 2\nmay arise from a reduced thermal stability of the magne-\ntization due to strong fluctuations of the ferromagnetic\nexchange between FeandPtin the disordered struc-\nture and also to additional effects of the antiferromag-\nneticFe−FeandPt−Ptexchange interactions. In\nthis context, it may be interesting to note that for low\nFeconcentrations, x≤0.3, bulkFexPt1−xexhibits fer-\nromagnetism only in the disordered structure17, while\nstructural ordering leads to para- or antiferromagnetism.\nRecent first-principle calculations of the electronic struc-\nture produced clear evidence for the stabilizing effect of\ndisorder on the ferromagnetism in FePt19.\nFrom the Langevin fits in Fig.1(a) we obtain for the\nparticle density Np= 3.5·1017g−1. Basing on the well\nknown mass densities of Fe0.2Pt0.8and the organic ma-\ntrix, we find by a little calculation20for the volume con-\ncentration of the particles cp= 0.013 and, hence, for the\nmean inter-particle distance, dpp=dp/c1/3\np= 13.5nm.\nThis implies forthe maximum (i.e. T=0) dipolar interac-\ntion between nearest particles, µ2\np(0)/4πµ0d3\npp= 0.20K,\nso that at the present temperatures, T≥20K, the sam-\nple should act as an ensemble of independent ferromag-\nnetic nanospheres. Since also the blocking temperature,\nTb= 9K, as determined from the maximum of the ac-\nsusceptibility at 0.1 Hz in zero magnetic field20, turned\nout to be low, the Langevin-analysis in Fig.1(a) is valid.FIG. 1: Fig. 1. (a) Magnetization isotherms of the\nnanospheres fitted to the Langevin model plus a small para-\nmagnetic background χb·H; (b) temperature dependences\nof the magnetic moments of the nanoparticles µpand of the\nbackground susceptibility χb( inset units are emu/g kOe )\nfitted to the indicated relations with TC= (320±20)K. The\ninset shows also data from the intensity of the paramagnetic\nresonance at 1.45 kOe, see Figs. 2 and 3.\nIII. RESONANCE SHAPE\nMagnetic resonances at a fixed X-band frequency of\n9.095 GHz have been recorded by a home-made mi-\ncrowave reflectometer equipped with field modulation to\nenhancethesensitivity. Adouble-walledquartztubecon-\ntaining the sample powder has been inserted to a mul-\ntipurpose, gold-plated VARIAN cavity (model V-4531).\nKeeping the cavity at room temperature, the sample\ncouldbeeithercooledbymeansofacontinuousflowcryo-\nstat (Oxford Instruments, model ESR 900) down to 15 K\norheatedupto500Kbyanexternal Pt-resistancewire20.\nAt all temperatures, the incident microwave power was\nvaried in order to assure the linear response.\nSomeexamplesofthespectrarecordedbelowthe Curie3\nFIG. 2: (a) Derivative of the microwave (f=9.095 GHz) ab-\nsorption spectrum recorded at T=52 K i.e. close to magnetic\nsaturation of the nanospheres. The solid and dashed curves\nare based on fits to Eqs.(4) and (8), respectively, which both\nignore SPM fluctuations and assume either a g-shift and zero\nanisotropy field HA(’∆g-FM’) or ∆ g= 0 and a randomly\ndistributed HA=0.5 kOe (’a-FM’), Eq. (9). Also shown are\nfits to predictions by Eq.(11), which account for SPM fluctu-\nations, with HA=0.5 kOe and ∆ g= 0 (’a-SPM’) and, using\nEq.(11), for ∆ g/negationslash= 0 and HA=0 (’∆g-SPM’). The weak, nar-\nrow resonance at 1.45 kOe is attributed to the paramagnetic\nbackground with g= 4.3±0.1 indicating Fe3+ 16impurities.\ntemperature are shown in Figs.2 and 3. The spectra have\nbeen measured from -9.5 kOe to +9.5 kOe and proved to\nbe independent of the sign of H and free of any hystere-\nsis. Thiscanbeexpectedduetothecompletelyreversible\nbehavior of the magnetization isotherms above 20 K and\nthe low blocking temperature of the particles. Lower-\ning the temperature, we observe a downward shift of the\nmainresonanceaccompaniedbyastrongbroadening. On\nthe other hand, the position and width of the weak nar-\nrow line at (1 .50±0.05)kOecorresponding to gi∼=4.3\nremain independent of temperature. This can be at-\ntributed to the previously detected paramagnetic Fe3+-\nimpurities16and is supported by the integrated intensity\nof this impurity resonance Ii(T) evaluated from the am-\nplitudedifferenceofthederivativepeaks. Sincetheinten-\nsity of a paramagnetic resonance is given by the param-\nagnetic susceptibility, Ii(T)∼/integraltext\ndH χ′′\nxx(H,T)∼χi(T)\ncan be compared directly to the background suscepti-\nbilityχb(T), see inset to Fig.1(b). The good agree-\nment between both temperature dependencies suggests\nto attribute χbto these Fe3+-impurities. An analysis of\nthe fitted Curie-constant, Ci= 5emuK/g kOe , yields\nNi= 164.1017g−1for theFe3+- concentration, which\ncorresponds to 50 Fe3+per 1150 atoms of a nanosphere.\nWith regard to the main intensity, we want to extract\na maximum possible information, in particular, on theFIG. 3: Fig. 3. Derivative spectra at some representative\ntemperatures and fits to Eq.(4). The LLG-damping, g-shift,\nand intensities are depicted in Fig. 4.\nintrinsic magnetic damping in nanoparticles. Unlike the\nconventional analysis of resonance fields and linewidths,\nas applied e.g. to Ni6and Co7nanoparticles, our ob-\njective is a complete shape analysis in order to disen-\ntangle effects by the crystalline anisotropy12, by SPM\nfluctuations13, by an electronic g-shift21, and by differ-\nent forms ofthe damping torque /vectorR22. Additional difficul-\nties may enter the analysis due to non-spherical particle\nshapes, size distributions and particle interaction, all of\nwhich, however, can be safely excluded for the present\nnanoparticle assembly.\nThe starting point of most FMR analyses is the phe-\nnomenological equation of motion for a particle moment\n( see e.g. Ref. 13 )\nd\ndt/vector µp=γ/vectorHeff×/vector µp−/vectorR , (2)\nusing either the original Landau-Lifshitz (LL) damping\nwith damping frequency λL\n/vectorRL=λL\nMs(/vectorHeff×/vector µp)×/vector sp ,(3)\northeGilbert-dampingwiththe Gilbertdampingparam-\neterαG,\n/vectorRG=αGd/vector µp\ndt×/vector sp , (4)\nwhere/vector sp=/vector µp/µpdenotes the direction of the particle\nmoment. In Eq.(2), the gyromagneticratio, γ=g0µB/¯h,\nis determined by the regular g-factorg0of the precessing\nmoments. It shouldperhapsbe notedthat the validityof\nthe micromagnetic approximationunderlying Eq. (2) has\nbeen questioned23for volumes smaller than (2 λsw(T))3,4\nwhereλsw= 2a0TC/Tis the smallest wavelengthof ther-\nmally excited spin waves. For the present particles, this\nestimate leads to a fairly large temperature of ∼0.7TC\nup to which micromagnetics should hold.\nAt first, we ignore the anisotropy being small in cubic\nFexPt1−x24,25, so that for the present nanospheres the\neffective field is identical to the applied field, /vectorHeff=\n/vectorH. Then, the solutions of Eq. (2) for the susceptibility\nof the two normal, i.e. circularly polarized modes, of\nNpindependent nanoparticles per gram take the simple\nforms\nχL\n±(H) =Npµpγ1∓iα\nγH(1∓iα)∓ω(5)\nfor/vectorR=/vectorRLwithα=λL/γMsand for the Gilbert torque\n/vectorRG\nχG\n±(H) =Npµpγ1\nγH∓ω(1+iαG).(6)\nFor the LL damping, the experimental, transverse sus-\nceptibility, χxx=1\n2(χ++χ−), takes the form\nχL\nxx(H) =NpµpγγH(1+α2)−iαω\n(γH)2(1+α2)−ω2−2iαωγH.\n(7)\nAs thesameshape isobtainedforthe Gilbert torquewith\nα=αG, the damping is frequently denoted as LLG pa-\nrameter. However, the gyromagnetic ratio in Eq.(7) has\ntobereplacedby γ/(1+α2), whichonlyfor α≪1implies\nalso the same resonance field Hr. Upon increasing the\ndamping up to α≈0.7 (the regime of interest here), the\nresonance field HrofχG\nxx(H) , determined by dχ′′/dH=\n0,remains constant, HG\nr≈ω/γ, whileHL\nrdecreases\nrapidly. After renormalization γ/(1+α2) the resonance\nfields and also the shapes of χL(H) andχG(H) become\nidentical. This effect should be observed when determin-\ning theg-factor from the resonance fields of broad lines.\nIt becomes even more important if the downward shift of\nHris attributed to anisotropy, as done recently for the\nratherbroadFMR absorptionof FexPt1−xnanoparticles\nwith larger Fe-content, x≥0.326.\nIn order to check here for both damping torques,\nwe selected the shape measured at a low temperature,\nT=52 K, where the linewidth proved to be large (see\nFig. 2) and the magnetic moment µp(T) was close to sat-\nuration (Fig. 1(b)). None of both damping terms could\nexplain both, the observed resonance field Hrand the\nlinewidth ∆ H=αω/γ, and, hence, the lineshape. By\nusing/vectorRG, the shift of Hrfromω/γ= 3.00kOewas not\nreproduced by HG\nr=ω/γ, while for /vectorRLthe resonance\nfieldHL\nr, demanded by the line width, became signifi-\ncantly smaller than Hr.\nThis result suggested to consider as next the effect of a\ncrystallineanisotropyfield /vectorHAonthetransversesuscepti-\nbility, which has been calculated by Netzelmann from thefree energy of a ferromagnetic grain12. Specializing his\ngeneral ansatz to a uniaxial /vectorHAoriented at angles ( θ,φ)\nwithrespecttothedc-field /vectorH||/vector ezandthemicrowavefield,\none obtains by minimizing\nF(θ,φ,ϑ,ϕ) =−µp[Hcosϑ+\n1\n2HA(sinϑsinθ−cos(ϕ−φ)+cosθcosϑ)2] (8)\nthe equilibrium orientation ( ϑ0,ϕ0) of the moment /vector µpof\na spherical grain. After performing the trivial average\noverφ, one finds for the transverse susceptibility of a\nparticle with orientation θ\nχL\nxx(θ,H) =γµp\n2×\n(Fϑ0ϑ0+Fϕ0ϕ0/tan2ϑ0)(1+α2)−iαµpω(1+cos2ϑ0)\n(1+α2)(γHeff)2−ω2−iαωγ∆H.\n(9)\nHereH2\neff= (Fϑ0ϑ0Fϕ0ϕ0−F2\nϑ0ϕ0/(µpsinϑ0)2) and\n∆H= (Fϑ0ϑ0+Fϕ0ϕ0/sin2ϑ0)/µpare given by the sec-\nond derivatives of F at the equilibrium orientation of /vector µp.\nFor the randomly distributed /vectorHAofNpindependent par-\nticles per gram one has\nχL\nxx(H) =/integraldisplayπ/2\n0d(cosθ)χL\nxx(θ,H).(10)\nIn a strict sense, this result should be valid at fields\nlarger than the so called thermal fluctuation field HT=\nkBT/µp(T), see e.g. Ref. 13, which for the present case\namounts to HT= 1.0kOe. Hence, in Fig. 2 we fit-\nted the data starting at high fields, reaching there an\nalmost perfect agreement with the curve a-FM. The fit\nyields a rather small HA= 0.5kOewhich implies a small\nanisotropyenergyper atom, EA=1\n2µp(0)HA= 1.0µeV.\nThis number is smaller than the calculated value for bulk\nfcc FePt ,EA= 4.0µeV25, most probably due to the\nlowerFe-concentration (x=0.20) and the strong struc-\ntural disorder in our nanospheres. We emphasize, that\nthe main defect of this a-FM fit curve arises from the\nfinite value of dχ′′\nxx/dHatH= 0. By means of Eq. (9)\none finds χ′′\nxx(H→0,θ)∼HAH/ω2, which remains fi-\nnite even after averaging over all orientations according\ntoθ(Eq. (10)).\nThefinite value ofthe derivativeof χ′′\nxx(H→0)should\ndisappear if superparamagnetic (SPM) fluctuations of\nthe particles are taken into account. Classical work27\npredicted the anisotropy field to be reduced by SPM,\nHA(y) =HA·(1/L(y)−3/y), which for y=H/HT≪1\nimpliesHA(y) =HA·y/5 and, therefore, χ′′\nxx(H→0)∼\nH2. A statistical theory for χL\nxx(H,T) which considers\nthe effect of SPM fluctuations exists only to first order in\nHA/H21. The result of this linear model (LM) in HA/H\nwhich generalizes Eq. (4), can be cast in the form\nχLM\n±(θ,H) =NpµpL(y)γ(1+A∓iαA)\nγ(1+B∓iαB)H∓ω.\n(11)5\nThe additional parameters are given in Ref. 21 and con-\ntain, depending on the symmetry of HA, higher-order\nLangevin functions Lj(y) and their derivatives. Observ-\ning the validity of the LM for H≫HA= 0.5kOe, we\nfitted the data in Fig. 2 to Eq. 11 with χLM\nxx(θ,H) =\n1\n2(χLM\n++χLM\n−) at larger fields. There one has also H≫\nHT= 1.0kOeand the fit, denoted as a-SPM, agrees\nwith the ferromagnetic result (a-FM). However, increas-\ning deviations appear below fields of 4 kOe. By varying\nHAandα, we tried to improve the fit near the resonance\nHr= 2.3kOeand obtained unsatisfying results. For low\nanisotropy, HA≤3kOe, the resonance field could be\nreproduced only by significantly lower values of α, which\nare inconsistent with the measured width and shape. For\nHA>3kOe, a small shift of Hroccurs, but at the\nsame time the lineshape became distorted, tending to a\ntwo-peak structure also found in previous simulations13.\nEven at the lowest temperature, T= 22K, where the\nthermal field drops to HT= 0.4kOe, no signatures of\nsuch inhomogeneous broadening appear (see Fig. 3). Fi-\nnally, it should be mentioned that all above attempts to\nincorporate the anisotropy in the discussion of the line-\nshape were based on the simplest non-trivial, i.e. uni-\naxial symmetry, which for FePtwas also considered by\nthe theory25. For cubic anisotropy, the same qualitative\ndiscrepancies were found in our simulations20. This in-\nsensitivitywith respecttothe symmetry of HAoriginates\nfrom the orientational averaging in the range of the HA-\nvalues of relevance here.\nAs a finite anisotropy failed to reproduce Hr,∆H, and\nalso the shape, we tried a novel ansatz for the magnetic\nresonanceofnanoparticlesbyintroducingacomplexLLG\nparameter,\nˆα(T) =α(T)−i β(T). (12)\nAccording to Eq. (4) this is equivalent to a negative g-\nshift,g(T)−g0=−β(T)g0, which is intended to com-\npensate the too large downward shift of HL\nrdemanded\nbyχL\nxx(H) due to the large linewidth. In fact, insert-\ning this ansatz in Eq. (5), the fit, denoted as ∆ g-FM\nin Fig. 2, provides a convincing description of the line-\nshape down to zero magnetic field. It may be interest-\ning to note that the resulting parameters, α= 0.56 and\nβ= 0.27, revealed the same shape as obtained by using\nthe Gilbert-susceptibilities, Eq. (6).\nIn spite of the agreement of the ∆ g-FM model with\nthe data, we also tried to include here SPM fluctuations\nby using ˆ α(T,H) = ˆα(T)(1/L(y)−1/y)21forHA= 0.\nThe result, designated as ∆ g-SPM in Fig. 2 agrees with\nthe ∆g-FM curve for H≫HTwhere ˆα(T,H) = ˆα(T),\nbut again significant deviations occur at lower fields.\nThey indicate that SPM fluctuations do not play any\nrole here, and this conclusion is also confirmed by the\nresults at higher temperatures. There, the thermal fluc-\ntuation field, HT=kBT/µp(T), increases to values\nlarger than the maximum measuring field, H= 10kOe,\nso that SPM fluctuations should cause a strong ther-/s48 /s51/s48/s48/s32/s75/s48/s55/s46/s53/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s56/s49/s46/s48/s32\n/s32\n/s32/s97/s41\n/s48/s46/s49/s56/s32/s43/s32/s48/s46/s53/s56/s32/s40/s32/s49/s45/s84/s47/s84\n/s67/s32/s41/s32/s50\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s53/s48/s48/s48/s46/s48/s48/s46/s50\n/s48/s46/s51/s57/s32/s40/s32/s49/s45/s84/s47/s84\n/s67/s32/s41/s32/s50\n/s32/s84/s32/s40/s32/s75/s32/s41/s103/s32/s47/s32/s103\n/s48\n/s32/s32\n/s98/s41/s32\n/s32/s73/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s126/s32/s40/s32/s49/s45/s84/s47/s84\n/s67/s32/s41/s32/s50\nFIG. 4: Temperature variation (a) of the LLG-damping αand\n(b) of the relative g-shifts with g0= 2.16 (following from the\nresonance fields at T > T C). Within the error margins, α(T)\nand ∆g(T) and also the fitted intensity of the LLG-shape\n(see inset) display the same temperature dependence as the\nparticle moments in Fig. 1(b).\nmal, homogeneous broadening of the resonance due to\nˆα(H≫HT) = ˆα·2HT/H. However, upon increasing\ntemperature, the fitted linewidths, (Fig. 3) and damping\nparameters (Fig. 4) display the reverse behavior.\nIV. COMPLEX DAMPING\nIn order to shed more light on the magnetization dy-\nnamics of the nanospheres we examined the tempera-\nture variation of the FMR spectra. Figure 3 shows\nsome examples recorded below the Curie temperature,\nTC= 320K, together with fits to the a−FM model\noutlined in the last section. Above TC, the resonance\nfields and the linewidths are temperature independent\nrevealing a mean g−factor,g0= 2.16±0.02, and a\ndamping parameter ∆ H/Hr=α0= 0.18±0.01. Since\ng0is consistent with a recent report on g-values of\nFexPt1−xforx≥0.4321, we suspect that this resonance\narises from small FexPt1−x-clusters in the inhomoge-\nneousFe0.2Pt0.8structure. Fluctuations of g0and of\nlocal fields may be responsible for the rather large width.6\nThis interpretation is supported by the observation that\naboveTCthe lineshape is closer to a Gaussian than to\nthe Lorentzian following from Eq.(7) for small α.\nThe temperature variation for both components of the\ncomplex damping, obtained from the fits below TCto\nEq. (7), are shown in Fig. 4. Clearly, they obey the same\npowerlaw as the moments, µp(T), displayed in Fig. 1(b),\nwhich implies\nˆα(T) = (α−i β)ms(T)+α0.(13)\nHerems(T)=µp(T)/µp(0),α=0.58, and β=-∆g(0)/g0=\n0.39 denote the reduced spontaneous magnetization and\nthe saturation values for the complex damping, respec-\ntively. It should be emphasized that the fitted inten-\nsityI(T) of the spectra, shown by the inset to Fig. 4(b),\nexhibits the same temperature variation I(T)∼µp(T).\nThis behavior is predicted by the ferromagnetic model,\nEq. (7), and is a further indication for the absence of\nSPM effects on the magnetic resonance. If the reso-\nnance were dominated by SPM fluctuations, the inten-\nsity should decrease like the SPM Curie-susceptibility,\nISPM(T)∼µ2\np(T)/T, following from Eq. (11), being\nmuch stronger than the observed I(T).\nAt the beginning of a physical discussion of ˆ α(T), we\nshould point out that the almost perfect fits of the line-\nshape to Eq. (7) indicate that the complex damping is\nrelated to an intrinsic mechanism and that eventual in-\nhomogeneous effects by distributions of particle sizes and\nshapes in the assembly, as well as by structural disorder\nareratherunlikely. Sinceageneraltheoryofthemagneti-\nzation dynamics in nanoparticles is not yet available, we\nstart with the current knowledge on the LLG-damping\ninbulkand thin film ferromagnets, as recently reviewed\nby B. Heinrich5. Based on experimental work on the\narchetypal metallic ferromagnets and on recent ab initio\nband structure calculations10there is now ratherfirm ev-\nidence that the damping ofthe q=0-magnonis associated\nwith the torques /vectorTso=/vector ms×/summationtext\nj(ξj/vectorLj×/vectorS) on the spin /vectorS\ndue to the spin-orbit interaction ξjat the lattice sites j.\nThe action of the torque is limited by the finite lifetime\nτof an e/h excitation, the finite energy ǫof which may\ncause a phase, i.e. a g-shift. As a result of this magnon\n- e/h-pair scattering, the temperature dependent part of\nthe LLG damping parameter becomes\nˆα(T)−α0=λL(T)\nγMs(T)\n=(Ωso·ms(T))2\nτ−1+i ǫ/¯h·1\nγMs(T). (14)\nForintraband scattering, ǫ≪¯h/τ, the aforementioned\nnumerical work10revealed Ω so= 0.8·1011s−1and 0.3·\n1011s−1as effective spin-orbit coupling in fcc Niand\nbcc Fe, respectively. Hence, the narrow unshifted (∆ g=\n0)bulkFMR lines in pure crystals, where α≤10−2,\nare related to intraband scattering with ǫ≪¯h/τand toelectronic (momentum) relaxation times τsmaller than\n10−13s.\nBasing on Eq. (14), we discuss at first the temperature\nvariation,whichimpliesalineardependence, ˆ α(T)−α0∼\nms(T). Obviously, both, the real and imaginary part of\nˆα(T)−α0, agree perfectly with the fits to the data in\nFig. 4, if the relaxation time τremains constant. It may\nbe interestingtonote herethat the observedtemperature\nvariation of the complex damping λL(T) is not predicted\nby the classical model28incorporating the sd-exchange\ncoupling Jsd. According to this model, which has been\nadvanced recently to ferromagnets with small spin-orbit\ninteraction29and ferromagnetic multilayers30,Jsdtrans-\nfers spin from the localized 3d-moments to the delocal-\nized s-electron spins within their spin-flip time τsf. From\nthe mean field treatment of their equations of motion by\nTurov31, we find a form analogous to Eq. (14)\nαsd(T) =Ω2\nsdχs\nτ−1\nsf+i/tildewideΩsd·1\nγMs(T)(15)\nwhere Ω sd=Jsd/¯his the exchange frequency , χsthe\nPauli-susceptibility of the s-electrons and /tildewideΩsd/Ωsd=\n(1 + Ω sdχs/γMd). The same form follows from more\ndetailed considerations of the involved scattering process\n(see e.g. Ref. 5). As a matter of fact, the LLG-damping\nαsd=λsd/γMdcannot account for the observed tem-\nperature dependence, because Ω sdandχsare constants.\nThe variation of the spin-torques with the spontaneous\nmagnetization ms(T) drops out in this model, since the\nsd-scattering involves transitions between the 3d spin-up\nand -down bands due to the splitting by the exchange\nfieldJsdms(T).\nBy passing from the bulk to the nanoparticle ferro-\nmagnet, we use Eq. (14) to discuss our results for the\ncomplex ˆ α(T), Eq. (13). Recently, for Conanoparticles\nwith diameters 1-4 nm, the existence of a discrete level\nstructure near ǫFhas been evidenced32, which suggests\nto associate the e/h-energy ǫwith the level difference ǫp\nat the Fermi energy. From Eqs. (13),(14) we obtain rela-\ntions between ǫand the lifetime of the e/h-pair and the\nexperimental parameters αandβ:\nτ−1=α\nβǫ\n¯h, (16a)\nǫ\n¯h=β\nα2+β2Ω2\nso\nγMs(0). (16b)\nDue toα/β= 1.5, Eq. (16a) reveals a strongly over-\ndamped excitation, which is a rather well-founded con-\nclusion. The evaluation of ǫ, on the other hand, de-\npends on an estimate for the effective spin-orbit cou-\npling, Ω so=ηLχ1/2\neξso/¯hwhereηLrepresents the ma-\ntrix element of the orbital angular momentum between7\nthe e/h states5. The spin-orbit coupling of the minor-\nityFe-spins in FePthas been calculated by Sakuma24,\nξso= 45meV, while the density of states D(ǫF)≈1/(eV\natom)24,33yields a rather high susceptibility of the elec-\ntrons,ηLχe=µ2\nBD(ǫF) = 4.5·10−5. Assuming ηL=1,\nboth results lead to Ω so≈3.5·1011s−1, which is by\none order of magnitude larger than the values for Fe\nandNimentioned above. One reason for this enhance-\nment and for a large matrix element, ηL=1, may be the\nstrong hybridization between 3 dand 4d−Ptorbitals24in\nFexPt1−x. By inserting this result into Eq. (16b) we find\nǫ= 0.8 meV. In fact, this value is comparable to an esti-\nmate for the level difference at ǫF32,ǫp= (D(ǫF)·Np)−1\nwhich for ourparticles with Np= (2π/3)(dp/a0)3= 1060\natoms yields ǫp= 0.9 meV. Regarding the several in-\nvolved approximations, we believe that this good agree-\nment between the two results on the energy of the e/h\nexcitation, ǫ≈ǫp, maybe accidental. However,wethink,\nthat this analysisprovidesa fairlystrongevidence for the\nmagnon-scattering by this excitation, i.e. for the gap in\nthe electronic states due to confinement of the itinerant\nelectrons to the nanoparticle.\nV. SUMMARY AND CONCLUSIONS\nThe analysis of magnetization isotherms explored\nthe mean magnetic moments of Fe0.2Pt0.8nanospheres\n(dp= 3.1nm) suspended in an organic matrix, their\ntemperature variation up to the Curie temperature TC,\nthe large mean particle-particle distance Dpp≫dpand\nthe presence of Fe3+impurities. Above TC, the res-\nonance field Hrof the 9.1GHzmicrowave absorption\nyielded a temperature independent mean g-factor,g0=\n2.16, consistent with a previous report21for paramag-\nneticFexPt1−xclusters. There, the lineshape proved\nto be closer to a gaussian with rather large linewidth,\n∆H/Hr= 0.18, which may be associated with fluctua-\ntions of g0and local fields both due to the chemically\ndisordered fccstructure of the nanospheres.\nBelow the Curie temperature, a detailed discussion of\nthe shape of the magnetic resonance spectra revealed a\nnumber of novel and unexpected features.\n(i) Starting at zero magnetic field, the shapes could be\ndescribed almost perfectly up to highest field of 10 kOe\nby the solution of the LLG equation of motion for inde-\npendentferromagneticsphereswithnegligibleanisotropy.\nSignatures of SPM fluctuations on the damping, which\nhave been predicted to occur below the thermal field\nHT=kBT/µp(T), could not be realized.\n(ii) Upon decreasing temperature, the LLG damping in-\ncreases proportional to µp(T), i.e. to the spontaneous\nmagnetization of the particles, reaching a rather largevalueα= 0.7 forT≪TC. We suspect that this high\nintrinsic damping may be responsible for the absence of\nthe predicted SPM effects on the FMR, since the under-\nlying statistical theory13has been developed for α≪1.\nThis conjecturemayfurther be based onthe fact that the\nlarge intrinsic damping field ∆ H=α·ω/γ= 2.1kOe\ncauses a rapid relaxation of the transverse magnetization\n(q= 0 magnon) as compared to the effect of statistical\nfluctuations of HTadded to Heffin the equation of mo-\ntion, Eq.(2)13.\n(iii) Along with the strong damping, the lineshape analy-\nsis revealed a significant reduction of the g-factor, which\nalso proved to be proportional to µp(T). Any attempts\nto account for this shift by introducing uniaxial or cubic\nanisotropy fields failed, since low values of /vectorHAhad no\neffects on the resonance field due to the orientational av-\neraging. On the other hand, larger /vectorHA’s, by which some\nsmall shifts of Hrcould be obtained, produced severe\ndistortions of the calculated lineshape.\nThe central results of this work are the temperature\nvariation and the large magnitudes of both α(T) and\n∆g(T). They were discussed by using the model of the\nspin-orbit induced scattering of the q= 0 magnon by\nan e/h excitation ǫ, well established for bulk ferromag-\nnets, where strong intraband scattering with ǫ≪¯h/τ\nproved to dominate5. In nanoparticles, the continuous\nǫ(/vectork)-spectrum of a bulk ferromagnet is expected to be\nsplit into discrete levels due to the finite number of lat-\ntice sites creating an e/h excitation ǫp. According to\nthe measured ratio between damping and g-shift, this\ne/h pair proved to be overdamped, ¯ h/τp= 1.5ǫp. Based\non the free electron approximation for ǫp32and the den-\nsity of states D(ǫF) from band-structure calculations\nforFexPt1−x24,33, one obtains a rough estimate ǫp≈\n0.9 meV for the present nanoparticles. Using a reason-\nable estimate of the effective spin-orbit coupling to the\nminority Fe-spins, this value could be well reproduced\nby the measured LLG damping, α= 0.59. Therefore we\nconclude that the noveland unexpected results of the dy-\nnamics of the transverse magnetization reported here are\ndue to the presence of a broade/h excitation with energy\nǫp≈1meV. Deeper quantitative conclusions, however,\nmust await more detailed information on the real elec-\ntronic structure of nanoparticles near ǫF, which are also\nrequired to explain the overdamping of the e/h-pairs, as\nit is inferred from our data.\nThe authors are indebted to E. Shevchenko and H.\nWeller (Hamburg) for the synthesis and the structural\ncharacterizationof the nanoparticles. One of the authors\n(J. 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Tsnerkovnyak, G. A. Fiete, B. I. Halperin,\nAppl. Phys. Lett. 84, 5234 (2004).\n30A. Costa , R. B. Muniz, D. L. Mills, Phys. Rev. B 73,\n054426 (2006).\n31E. A. Turov in Ferromagnetic Resonance ed. by\nS. N. Vonsovski, Chapt. V, 7, Moscow 1961.\n32S. Gu´ eron, M. M. Deshmukh, E. B. Myers, and\nD. C. Ralph, Phys. Rev. Lett. 83, 4148 (1999); S. Kleff,\nJ. von Delft, M. M. Deshmukh, D. C. Ralph, Phys. Rev.\nB64, 220401(R) (2004).\n33E. T. Kulatov, Y.-A. Uspenskii, S.-V. Halilov,\nJ. Magn. Magn. Mat. 163, 331 (1996)."    },    {        "title": "1509.01807v1.Study_of_spin_dynamics_and_damping_on_the_magnetic_nanowire_arrays_with_various_nanowire_widths.pdf",        "content": "1 \n Study of spin dynamics  and damping  on the magnetic nanowire  arrays   \nwith various nanowire widths  \n \nJaehun Cho a, Yuya  Fujii b, Katsunori  Konioshi b, Jungbum Yoon c, Nam -Hui Kim a, Jinyong Jung a, \nShinji Miwa b, Myung -Hwa Jung d, Yoshishige Suzuki b, and Chun -Yeol You  a,* \n \na Department of Physics, Inha University , Inch eon, 402-751, South Korea  \nb Graduate School of Engineering Science,  \nOsaka University,  Toyonaka, Osaka 560 -8531,  Japan  \nc Department of Electrical and Computer  Engineering ,  \nNational University of Singapore , Singapore  117576  \nd Department of Physics, Sogang University, Seoul, 121 -742, South Korea  \n \nAbstract  \nWe investigate the spin dynamics including Gilbert damping in the ferromagnetic  nanowire \narray s. We have measured  the ferromagnetic resonance of ferromagnetic nanowire arrays \nusing vector -network analyzer ferromagnetic resonance  (VNA -FMR)  and analyzed the results \nwith the micromagnetic si mulations . We find excellent agree ment  between the experimental \nVNA -FMR spectra and micromagnetic simulations result  for various applied magnetic fields . \nWe find that  the demagnetization factor for longitudinal conditions, Nz (Ny) increases  \n(decreas es) as decreasing the nanowire width  in the micromagnetic simulation s. For the \ntransverse magnetic field , Nz (Ny) increas es (decreas es) as increasing the nanowire width . We \nalso find that t he Gilbert damping constant increases  from 0.018 to 0.051 as the incr easing \nnanowire width  for the transverse case , while it is almost constant as 0.021  for the \nlongitudinal case . \n 2 \n * Corresponding author.  FAX: +82 32 872 7562.  \nE-mail address:  cyyou@inha.ac.kr  \nKeywords : Nanowires , Ferromagnetic Resonance , Micromagnetic  Simulations , Gilbert \ndamping  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 3 \n  \nFerromagnetic nanostructures have recently attracted much interest for the wide potential \napplications in high density spintronic information storage , logic devices and various spin \norbit torque phenomena .1,2,3,4,5 It is well known that the detail spin dynamics of nanostructure \nis far from the one of the bulk’s because of many reasons, different boundary conditions, \nchanges of the magnetic properties including the saturat ion magnetization, anisotropy energy, \nand exchange stiffness constant, etc. Since the magnetic properties are usually sensitive \nfunctions of the sample fabrication conditions, it has been widely accepted that the detail \nsample fabrications are also importa nt in the study of spin dynamics. However, the relatively \nless caution has been made for the boundary conditions of the spin dynamics in the \nnanostructure.  \nIn the spin transfer torque magnetic random access memory  (STT -MRAM), the magnetic \ndamping constant is important because the switching current density  is proportional to the \ndamping constant .6 In the nanowire, damping constant also plays crucial role in the spin \ndynamics including domain wall motion with magnetic field7 and spin transfer torque .8 \nFurthermore, it is the most important material parameter in spin wave (SW) dynamics .9 \nDespite of the importance of the damping constant, many studies about spin dynamics in \nferromagnetic nanowires  have not taken into account the damping constant  properly .10,11,12 \nOnly a few studies paid attention to the magnetic  damping in the nanowires  spin 4 \n dynamics .13,14 \nIn this study, arrays of CoFeB nanowire s are prepared by e -beam lithography , and they are  \ncovered coplanar  wave -guide  for the ferromagnetic resonance  (FMR) measurement  as shown \nin Fig. 1 . We measured FMR signal with longitudinal (wire direction) and transverse \nmagnetic field s in order to investigate the spin dynamics with different boundary conditions. \nAlso w e extract Gilbert damping constant  using micromagnetic  simulations  with the different \napplied magnetic field directions in various nanowire arrays . We find the damping constant \ndecreas es with increasing the nanowire width for the transverse magnetic field  with constant \ninput damping consta nt in micromagnetic simulations, while we obtain almost constant \ndamping constant for the longitudinal field.  \nThe film s were  prepared using DC magnetron sputtering. The stack s consist of Ta (5  \nnm)/Co 16Fe64B20 (30 nm)/Ta (5  nm) on single crystal MgO (001) substrate s. The film s are \npatterned as 100 -nm-width wire array s with 200 -nm-space each wires using e -beam \nlithography and an Ar ion milling technique  as shown in Fig. 1. The width is determined with \na scanning electron microscope (SEM).  These nanowire arrays are covered by coplanar wave \nguide in order to characterized with the Vector Network Analyzer ( VNA )-FMR technique \ndescribed elsewhere .15 We prepare nanowire arrays  as shown in Fig. 1 , and external DC \nmagnetic field direction  for FMR measurement  is also depicted.  \nWe use  VNA -FMR spectra to measure  imaginary parts of the susceptibility of the samples.16 5 \n The measured imaginary parts of the susceptibility raw data are calibrated with the careful \ncalibration procedures .16 The calibrated  imaginary parts of the susceptibility are shown in Fig \n2(a) and (b)  for an applied magnetic field at 0.194  T for the nanowire arrays . The un -\npatterned thin film is also examined for the reference. We find two resonance frequencies, \n17.2 and 26 .4 GHz, as shown in Fig. 2(a) for the nanowire array, while  there is only one peak \nat 16.8 GHz  for the  un-patterned  thin film  as shown in Fig 2(b). We believe that the smaller \npeak  (17.2 GHz) in Fig. 2(a) is originated  from the un-patterned part of the nanowire array, \nbecause the frequency is closed to the un -patterned thin film’s peak (16.8 GHz). Probably, the \nun-patterned part of the nanowire is formed due to poor e -beam lithography processes. On the \nother hand, t he resonance f requency  near 26.0 GHz is calculated from micromagnetic \nsimulation  at an applied  magnetic  field at 0.200 T , as shown in Fig. 2 (c).  We clarif y the \nsource of the main  peak (26.4 GHz) is nanowire  arrays  by using micromagnetic simulation.  \nThese two peaks name d as the uniform FMR  mode (smaller peak position) and nanowire \nmode (higher peak position).  \nIn order to determine the saturation magnetization, the resonance frequencies are measured \nas a function of the applied magnetic field, and the results are fitted with the Kittel ’s \nequation .17 This equation employs the corresponding demagnetization factors of Nx = 0, Ny = \n0 and Nz = 1 for un -patterned film, when applied magnetic field H is x- direction  with \nfollowing equations , 6 \n  \n    2y x s z x s f H N N M H N N M\n    \n.   (1) \n \n Here,  is the gyromagnetic ratio, H is the applied magnetic field, Ms is saturated \nmagnetization, Nx, Ny, and Nz are the demagnetization factor s applying the cyclic permutation \nfor the  applied magnetic field direction .  \nThe micromagnetic simulations are performed by using the Objective -Oriented -\nMicroMagnetic Framework (OOMMF)18 with 2-dimensional  periodic boundary  condition \n(PBC ).19 We select a square slat of 100 nm × 100 nm × 30 nm  nanowire separated 200 nm in \ny- direction with a cell size of 5 nm × 5 nm ×  30 nm. The material parameters of CoFeB used \nin our simulation are summarized as follows: Ms = 15.79 × 105 A/m, the exchange stiffness \n1.5 × 1011 J/m, the gyromagnetic ratio 2.32  × 1011 m/(A ∙s) and we ignore the magneto -\ncrystalline anisotropy. In this simulation, the Gilbert damping constant  of 0.0 27 is fixed. The \nsaturation magnetization and Gilbert damping constant are determined by using VNA -FMR  \nmeasurement for un -patterned thin film . For t he exchange  stiffness constant, experimentally \ndetermined values are range of 0.98 to 2.84 × 1011 J/m which value  has dependence on the \nfabrication processes20 and composition of ferromagnetic materials ,21 while we have picked \n1.5 × 1011 J/m as the exchange  stiffness  constant . The determination  method of Gilbert \ndamping constant will be described later.  7 \n In order to mimic FMR experiments in the micromagnetic simulations , a “sinc”  function\n0 0 0 ( ) sin 2 / 2y H HH t H f t t f t t    \n, with H0 = 10 mT, and field frequency fH = 45 \nGHz, is applied the whole nanowire  area.22 We obtain the FMR spectra in the corresponding \nfrequency range from 0 to 45 GHz . The FMR spectra due to the RF -magnetic field are \nobtained by the fast Fourier  transform  (FFT) of stored My(x) (x, y, t ) in longitudinal  (transverse)  \nH0 field. More details can be described elsewhere .23 \nThe closed blue circles  in Fig. 3 is the calculated values with the fitting parameter using Eq. \n(1) which are fitted with the experimental data of un -patterned thin film. The obt ained Ms is \n15.79 × 105 A/m while gyromagnetic ratio is fixed as 2.32 × 1011 m/(A∙s) . The obtained Ms \nvalue is similar  with vibrating sample magnetometer  method24 which CoFeB structure has Ta \nbuffer layer. The resonance frequencies of uniform FMR mode in nanowire arrays are plotted \nas open red circles in Fig. 3. The resonance frequencies of uniform FMR mode is measured \nby VNA -FMR are agreed well with resonance freq uency of un-patterned thin film measured \nby VNA -FMR. In Fig. 3, the applied field dependences of  the resonance frequencies \nMeasured by VNA -FAM for the nanowire are plotted as open black  rectangular , along with \nthe result of micromagnetics calculated with E q. (1)  as depicted  closed black rectangular . It is \nalso well agreed with the experimental result in nanowire mode and micromagnetic \nsimulation result  in the nanowire arrays.  \nIn order to reveal the effect s of spin dynamics properties with various  nanowire width s, we 8 \n perform micromagnetic simulat ions. The nanowire width s are varied from 50 to 150 nm in \n25-nm step for fixed 200 -nm-space with PBC , it causes changes of the demagnetization \nfactor  of the nanowire . In Fig. 4 (a) shows  the nanowire width  dependences of the resonance \nfrequencies for the longitudinal magnetic field (open symbols) along with the resonance \nfrequencies calculated with Eq. (1)  (solid lines) . The demagnetization factors can be \ndetermined by fitting Eq. (1) while Nx is fixed as 0  to represent infinitely long wire . The \nagreements between the results of micromagnetic simulations  (open circles)  and Eq. (1)  \n(solid lines)  are excellent.  \nFor the transverse  magnetic  field, the direction of applied magnetic field is y - axis, Eq. (1) \ncan be rewritten as follows:  \n \n    2x y s z y s f H N N M H N N M\n    \n.    (2) \n \nIn this equation, we use the relation of demagnetization factors , \n1x y zN N N   , in \norder to remove uncertainty in the fitting procedure . In the transverse field, the \ndemagnetization factors are determined by Eq. (2). The resonance frequencies  for transverse \nmagnetic field  which are obtained by micromagnetic simulation  (open circles)  and \ncalculated  by Eq. (2)  (solid lines)  as a function of the appl ied magnetic field with various \nnanowire width  are displayed in Fig. 4(b). The longitudinal case, when the field direction is 9 \n easy axis, they are saturated with small field. However, the transverse case, when the field \ndirection is hard axis, certain amoun t of field is necessary to saturate along the transverse \ndirections. The narrower nanowire, the larger field is required as shown in Fig. 4 (b).  \nFig. 5(a) and (b) show the changes of demagnetization factors in longitudinal and transverse \nmagnetic  fields as a functi on of  the nanowire width , respectively.  The demagnetization \nfactors play important role in the domain wall dynamics, for example the Walker breakdown \nis determined by the demagnetization factors .25 Furthermore, they are essential physical \nquantities to analyze the details of the spin dynamics. It is clear ly shown  that the Nz (Ny) \nincrease s (decrease s) with increasing the nanowire width  in longitudinal magnetic  field. For \nthe transverse magnetic  field, Nz (Ny) increase s (decrease s) with increasing the nanowire \nwidth , during Nx is almost zero value.  The demagnetization factors both longitudinal and \ntransverse have similar tendency with the effective demagnetization factors of dynamic \norigin26 and the static demagnetization factors for the prism geometry.27  \nNow, let us discuss about the Gilbert damping constant . The relation of the full width and \nhalf maxim a (f) of a resonance peak s as a function of applied field are shown in Fig. 6 for \nlongitudinal (a) and transverse (b). The f is given by15: \n \n,\n,2\n22xy\ns z ex yxN\nf H M N N f\n         \n. (3)  \n 10 \n where, fex is the extrinsic line width  contributions , when  the applied magnetic field  is x-(y-\n)axis for longitudinal (transverse) case . The symbol s are the result s of the micromagnetic \nsimulations and the solid lines are the fitting result of Eq. (3) . We use pre -determined \ndemagnetization factors (Fig. 5) during fitting procedures, and the agree ments are excellent.  \nWe have plotted the Gilbert damping constant as a function of the wavevector in nanowire \nwidth  (\n/ qa , a is the nanowire width ) in Fig. 7. The black open rectangles are data \nextracted from the transverse field and the red open circles are longitudinal field data. We \nfind that the Gilbert damping constant varied from 0.051 to 0.018 by changing  the \nwavevector  in nanowire width  in transverse field. On the other hand, lon gitudinal field case \nthe damping constant is almost constant as 0.021. Let us discuss about the un -expected \nbehavior of the damping constant of transverse case. T he wire width acts as a kind of cut -off \nwavelength of the SW excitations in the confined geome try. SWs whose wavelength are \nlarger than 2 a are not allowed in the nanowire. Therefore, only limited SW can be excited for \nthe narrower wire, while more SW can be existed in the wider wire. For example, we show \ntransverse standing SW as profiled  in the inset of Fig. 6 for 150 -nm width nanowire in our \nmicromagnetic simulations. More possible SW excitations imply more energy dissipation \npaths, it causes larger damping constant. For narrower nanowire (50 -nm), only limited SWs \ncan be excited, so that the damping constant is smaller. However, for the limit case of infinite \na case, it is the same with un -patterned thin films, there is no boundar y so that only uniform 11 \n mode can be excited, the obtained damping constant must be the input  value.  \nIn summary, the VNA -FMR experiments is employed  to investigate the magnetic properties \nof CoFeB nanowire  arrays  and the micromagnetic simulations is proposed to understand the \nmagnetic properties including Gilbert damping constant of various CoFeB nanowire  arrays  \nwidth. We f ind that the demagnetization factors are similar with the dynamic origin and static \nfor the prism geometry. The wire width or SW wavevector dependent damping constants can \nbe explained with number of SW excitation modes.  \n \nACKNOWLEDGMENTS  \nThis work  was supported by the National Research Foundation of Korea  (NRF) Grants  (Nos. \n616-2011 -C00017  and 2013R1A12011936 ). \nReferences  \n                                            \n1 S. S. P. Parkin,  M. Hayashi, and L. Thomas, Science 320, 190 (2008) . \n2 D. A. Allwood,  G. Xiong, C. C. Faulkner, D. Atkinson, D. Petit, and R. P. Cowburn,  \nScience 309, 1688 (2005) . \n3 I. M. Miron,  G. Gaudin, S. Aufftet, B. Rodmacq, A. Schuhl, S. Pizzini, J. V ogel and P. \nGambardella, Nature Materials 9, 230 (2010) . \n4 H-R Lee , K. Lee, J. Cho, Y . -H. Choi, C. -Y . You, M. -H. Jung, F. Bonell, Y. Shiota, S. Miwa \nand Y . Suzuki, Sci. Rep.  4, 6548 (2014).  \n5 J. Cho, et al.  Nat. Commun. 6, 7635 (2015).  \n6 S. Ikeda , K. 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Sharma, B. K. Kuanr, M. Sharma and Ananjan Nasu, IEEE Trans. Magn 50, 4300110 \n(2014).  \n15 D.-H. Kim, H .-H. Kim, and Chun -Yeol You, Appl. Phys. Lett. 99, 072502 (2011).  \n16 D.-H. Kim, H. -H. Kim, Chun -Yeol You and Hyungsuk Kim, J. Magn etics 16, 206 (2011).  \n17 C. Kittel, Introduction to Solid State Physics, 7th ed., pp. 504, (1996) . \n18 M. J. Donahue and D. G. Porter: OOMMF User’ s Guide : Ver. 1.0, NISTIR 6376  (National \nInstitute of Standards and Technology, Gaithersburg, Maryland, United States, 1999 ). \n19 W. Wang, C. Mu, B. Zhang, Q. Liu, J. Wang, D. Xue, Comput. Mater. Sci. 49, 84 (2010) . \nSee: http://oommf -2dpbc.sourceforge.net.  \n20 J. Cho, J . Jung, K .-E. Kim, S .-I. Kim, S .-Y. Park, M .-H. Jung, C .-Y. You, J. Magn. Magn. \nMater. 339, 36 (2013).  \n21 C. Bilzer, T. Devolder, J -V . Kim, G. Counil, C. Chappert, S. Cardoso and P. P. Feitas , J. \nAppl. Phys. 100, 053903 (2006).  \n22 K.-S. Lee, D. -S. Han , S.-K. Kim, Phys. Rev. Let t. 102, 127202 (2009).  \n23 J. Yoon, C. -Y . You, Y . Jo, S. -Y. Park, M. H. Jung , J. Korean Phys. Soc. 57, 1594 (2010) .  \n24 Y . Shiota, F. Bonell, S. Miwa, N. Muzuochi, T. Shinjo and Y . Suzuki , Appl. Phys. Le tt. 103. \n082410 (2013),  \n25 I. Purnama, I. S. Kerk, G. J. Lim  and W. S. Lew , Sci. Rep. 5, 8754 (2015).  \n26 J. Ding, M. Kostylev, and A. O. Adeyeye, Phys. Rev. B. 84, 054425 (2011) . \n27 A. Aharoni, J. Appl. Phys. 83, 3432 (2011) . 13 \n Figure  Captions  \n \nFig. 1. Measurement geometry with SEM image s of the 100 -nm-width nanowires with a gap \nof 200 nm between nanowires . The longitudinal nanowire arrays are shown. After the \nnanowire patterns have been  defined by e -beam lithography, they are covered by co -planar \nwave guides.  \n \nFig. 2. (a) The measured FMR  spectrum of the CoFeB nanowire with H =0.194 T. The red  (lower \npeak)  and blue  (higher peak)  arrows indicate t he resonance frequencies of the  uniform FMR  mode and \nthe nanowire mode, respectively. (b) The measured FMR spectrum of the CoFeB thin  film with H \n=0.194 T. (c) Simulated FMR spectrum of the CoFeB nanowire with H= 0.200 T.  \n \nFig. 3. Measured and calculated FMR frequencies with the applied magnetic field for 100 -\nnm-width nanowire. The open black rectangles  are nanowire mode and open red circles are \nthe uniform FMR mode for CoFeB thin film. The closed black rectangles  are calculated by \nOOMMF and the closed blue circles  are theoretical ly calculated by Eq. (1)  using fitted \nparameters form un -patterned film . \n \nFig. 4. Variation of resonanc e frequencies with the applied magnetic field for the different \nPBC wire width for (a) longitudinal field and (b) transverse field.  Inset: The geometry of 2 -\ndimensional PBC micromagnetic simulation with nanowire width a and a gap of 200 nm \nbetween nanowire s. The black open rectangles, red open circles, green open upper triangles, \nblue open down triangles, cyan open diamonds represent as nanowire width as 50 nm, 75nm, \n100 nm, 125nm, and 150 nm, respectively.  \n \nFig. 5. Demagnetization factor with PBC wire widt h for (a) longitudinal and (b) transverse \nfield.  The black open circles, red open rectangles, blue open upper triangles represent as \ndemagnetization factors, Ny, Nz, and Nx, respectively.  \n \nFig. 6. Full width and half maxim a with the applied magnetic field for (a) longitudinal and (b) \ntransverse field.  The black open rectangles, red open circles, green open upper triangles, blue \nopen down triangles, cyan open diamonds represent as nanowire width as 50 nm, 75nm, 100 \nnm, 125nm, and 150 nm, respectively.  \n \nFig. 7. Damping constants with wavevector for transverse ( the black open rectangles ) and \nlongitudinal ( the red open circles) field with errors. The black line is the input value which is \ndetermined from un -patterned film. Inset presents the profile of the trans verse spin density as SWs.  \n Fig. 1 \n \n \n    \n \n      Fig. 2. \n \n \n \nFig. 3. \n \n \n \n  \nFig. 4 \n \n` \n \n \n \nFig. 5 \n \n \n \nFig. 6 \n \n \n \nFig. 7 \n \n \n"    },    {        "title": "2005.05011v1.Manipulating_1_dimensinal_skyrmion_motion_by_external_magnetic_field_gradient.pdf",        "content": "Manipulating 1-dimensinal skyrmion  motion by external  magnetic field gradient  \nJaehun  Cho1, 2, Eiiti Tamura2, 3, 4, Chaozhe  Liu5, Soma Miki2, 3, Chun-Yeol You6, June-Seo Kim1, \nHikaru  Nomura2, 3, 5, Minori  Goto2, 3, Ryoichi Nakatani5, Yoshishige  Suzuki2, 3 \n \n1Division of Nanotechnology, Daegu Gyeongbuk Institute of Science and Technology (DGIST), \nDaegu, Republic of Korea  \n2Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka, Japan  \n3Center for Spintronics Research Network, Osaka University, Toyonaka, Osaka, Japan  \n4Department of Electronic Science and Engineering, Kyoto University, Kyoto, Kyoto, Japan  \n5Graduate School of Engineering, Osaka University, Suita, Osaka, Japan  \n6Department of Emerging Materials Science, Daegu Gyeongbuk Institute of Science and \nTechnology (DGIST ), Daegu, South Korea  \n \nAbstract  \nWe have investigated an analytic formula of the 1 -dimensional magnetic skyrmion dynamics \nunder external magnetic field gradient. We find excellent agreement between the analytical \nmodel and micromagnetic simulation results for various magnetic parameters su ch as the \nmagnetic field gradient, Gilbert damping constant. We also observe much faster velocity of the \nchiral domain wall  (DW)  motion . The chiral DW  is exist with smaller interfacial \nDzyaloshinskii -Moriya interaction energy density cases. These results p rovide to develop \nefficient control of skyrmion for spintronic devices.   \n \n \n \n \n \n \n \n \n  \nIntroduction  \nIn magnetic multilayer systems, the strong competition among Heisenberg exchange \ninteraction, Dzyaloshinskii -Moriya (DM) interaction, and magnetocrystalline  anisotropy can \nexhibit complex spin textures such as Skyrmions [ 1, 2], chiral magnetic domain walls (DWs) \n[3 - 5], Bloch lines [6, 7], and so on. A crucial contribution of interfacial DM interaction is \ndirectly related to a strong spin -orbit coupling at the interfaces between heavy metals and \nferromagnets combined with the broken inversion symmetry at the interfaces [8 - 12]. These \nexotic spin textures based on DM interaction are topologically stable, which are appropriate \nfor various applications as an inform ation carrier and manipulator [13, 14]. Indeed, numerous \ntheoretical and num erical studies have shown for the positive possibilities that magnetic \nskyrmions and chiral DWs could be essential ingredients for the next -generation spintronic \ndevices for storage devices and logic application [8, 14]. A single or bunch of these topological \nobjects are manipulated by laterally applied electrical currents due to  spin transfer torque  [15, \n16] or spin -orbit torque  [17, 18]. For the case below a critical current density, the magnetic \ntextures are fastened due to a  large pinning potential. When the electric current is larger than a \ncritical current density, the non -negligible displacements of topological objects occur. While \nthe electrical current injection technique is a promising method to drive multiple skrymions  \nand chiral DWs synchronously, a large critical current density caused by poor resistivities of \nmagnetic materials and the extremely narrow and long nanoscale wire architectures makes an \ninsurmountable obstacle which is so -called “Joule heating problem ” owing to Ohmic losses \n[19, 20]. Furthermore, the extra contributions such as Rashba effect and spin Hall effects lead \nto even more complex magnetization dynamics.  \nThe magnetic field driven chiral DWs and magnetic solitons  are received attentions because \nthe system is totally governed by the the Landau –Lifshitz (LL) equation , which is equivalent \nto the the Landau –Lifshitz –Gilbert  (LLG)  equation  when the magnetic damping constant of \nthe system is small enough. Moreover, vari ous manipulation idea such as DC or AC magnetic \nfield driven magnetic solitons, the transverse magnetic field pulse induced DW and skyrmion \nracetrack are demonstrated recently  [21, 22]. For realistic applications for information storage \ndevices or logic applications, the magnetic skyrmion or DW racetrack should be compatible \nfor the complementary metal -oxide -semiconductor (CMOS) architectures and the continuous \nminiaturization of CMOS architectures is essential for increasing the data capacity of the \ndevices. For the magnetic field driven skyrmion or DW motions in real spintronic d evices, the \nexternal magnetic field is applied from the ultrashort electrical current pulses passing through \nthe conduction lines adjacent skyrmion or DW racetracks to minimize the energy consumptions. \nNaturally, the applied magnetic field to the magnetic racetracks are not uniform due to Oersted \nlaw.  \nIn this work, the magnetic skyrmion and DW dynamics by applying gradient magnetic fields \nare systematically investigated by performing LLG simulations and Thiele approach . We described analytical and micromagnetic simulation studies of magnetic skyrmion dynamics in \na 1-dimensional nanowire, force by magnetic field gradient  along  the z-direction  while field \ngradient applied x-direction. According to the analytic model, the skyrm ion dynamics in the \nnanowire with magnetic field gradient is proportionality to the skyrmion width and radius, and \nits dynamics in good agreement with the analytic model and micromagnetic simulation results. \nIn micromagnetic simulations of DW dynamics, we observed much higher DW velocities than \nskyrmion one.  \n \nAnalytical model for magnetic field gradient driven skyrmions  \nWe briefly describe a simple theory for the skyrmion motion in our system. The motion of \nskyrmion in a two -dimensional film can be expressed by a Thiele ’s equations [ 23] for \nsufficiently slow varying and not too strong forces is  fellow:  \n𝑮×𝑹̇+𝛼𝒟𝑹̇=𝑭 (1) \nHere, R is the center coordinat e, G is the gyromagnetic coupling vector  with the winding \nnumber of the skyrmion q,  is the Gilbert damping constant, 𝒟 the dissipation dyadic and F \nthe external force e.g. by electric currents, magnetic field gradients, and thermal fluctuations.   \n The gyromagnetic coupling vector G is given by,  \n𝑮=∫𝑔̂𝑖𝑗𝑑𝑉𝑉. Here, 𝑔̂𝑖𝑗=𝑀𝑠\n|𝛾|𝒎⋅(𝜕𝒎\n𝜕𝑥𝑖×𝜕𝒎\n𝜕𝑥𝑗), Ms is the saturation magnetization and  is the \ngyromagnetic ratio . The components of the dissipative force, which is second term of Eq. (1), \n𝛼𝒟 describes the friction of the skyrmion, 𝒟=∫𝑑𝑖𝑗𝑑𝑉𝑉. Here, 𝑑𝑖𝑗=∫𝑀𝑠\n|𝛾|𝜕𝒎\n𝜕𝑥𝑖⋅𝜕𝒎\n𝜕𝑥𝑗𝑑𝑉𝑉.  \nWe study the effects of a non -uniform perpendicular magnetic field with a longitudinal \ndirection of nanowire . We neglect thermal fluctuation. As the skyrmion has a large magnetic \nmoment relative to the ferromagnetic nanowire, the field gradient leads to a force acting on the \nskyrmion. The force is given by  \n𝑭𝑔=−∫𝐻(𝑟)⋅𝜕𝑀\n𝜕𝑥𝑖𝑑2𝑟  \n=−𝑀𝑠∫𝐻(𝑟)⋅𝜕𝑛\n𝜕𝑥𝑖𝑑2𝑟. (2) \nBecause we apply the gradient magnetic field along  the z-direction  while field gradient \napplied x-direction,  force by magnetic field gradient can be expressed as  \n \n(𝑭𝑔(ℎ𝑔𝑧))\n𝑥=−𝑀𝑠ℎ𝑔𝑧∫𝑥⋅𝜕𝑛𝑧\n𝜕𝑥𝑑2𝑟 =𝑀𝑠ℎ𝑔𝑧∫𝑑𝑟𝑟2𝑠𝑖𝑛𝛩𝜕𝛩\n𝜕𝑟∞\n0∫𝑑𝜑𝑐𝑜𝑠2𝜑2𝜋\n0 \n=𝜋𝑀𝑠ℎ𝑔𝑧∫𝑑𝑟𝑟2𝑠𝑖𝑛𝛩𝜕𝛩\n𝜕𝑟∞\n0 \n=2𝜋𝑀𝑠ℎ𝑔𝑧𝑤2∫𝑑𝑡2𝑡2𝑠𝑖𝑛ℎ2(𝑥)𝑠𝑖𝑛ℎ(𝑡)𝑐𝑜𝑠ℎ(𝑡)\n[𝑠𝑖𝑛ℎ2(𝑥)+𝑠𝑖𝑛ℎ2(𝑡)]2∞\n0. (3) \nhere, ℎ𝑔𝑧  is perpendicular  magnetic field gradient , 𝑥=𝑅\n𝑤,  𝑡=𝑟\n𝑤 , where, R is the radius of \nskyrmion,  w is width of the skyrmion, and r is the position of magnetization.  \nThen, we consider the 1 -dimensional equation of skyrmion motion in Eq. (1). Using magnetic \nfield gradient force and dissipation above, we can calculate how t he velocity of skyrmion \ndepends  on the magnetic field gradient as below  \n(𝑅̇)𝑥=(𝑭𝑔)𝑥\n𝛼𝐷=𝑀𝑠ℎ𝑔𝑧∫𝑥𝜕𝑛𝑧\n𝜕𝑥𝑑𝑥𝑑𝑦\n𝛼𝑀𝑠\n|𝛾|∫𝜕𝑛⃗⃗ \n𝜕𝑥⋅𝜕𝑛⃗⃗ \n𝜕𝑥𝑑𝑥𝑑𝑦≅|𝛾|\n𝛼ℎ𝑔𝑧𝑤2𝑥2\n𝑥+𝑞2\n𝑥. (4) \nFor q = ± 1 and x ≫ 1, the velocity of skyrmion is simply calculated by  \n𝑣𝑥≈|𝛾|\n𝛼ℎ𝑔𝑧𝑤𝑅. (5) \nThe value of 𝑣𝑥 increase with increasing skyrmion radius . The magnetic field gradient force \nhas a large effect to large skyrmion because of large magnetic field differences from left and \nright side of skyrmion.  Surprisingly, the skyrmion velocity is proportional to not only radius of \nskyrmion but also width of skyrmion. The skyrmion width  and radius  are determined as \nskyrmion shape which is correlated with magnetic parameters, thereby, the skyrmion width is \nnot an independent variable in Eq. (5). Therefore, we find the relationship between the \nskyrmion velocities divided by skyrmion width ( 𝑣𝑥𝑤⁄) and the skyrmion radius as follows:  \n \n(𝑣𝑥\n𝑤)≈|𝛾|\n𝛼ℎ𝑔𝑧𝑅 (6) \n \nMicromagnetic simulations  \nThe micromagnetic simulations were performed by using the MuMax3 which is numerically \nsolve d the Landau -Lifshitz -Gilbert equation [ 24]. Here, we consider a nanowire  shaped with \n1000 -nm-length, 100 -nm-width, and 1.2 -nm-thick . The discretized cell for simulations is set to \nbe 2 × 2 × 1.2 nm3. In the simulation, a magnetic skyrmion is nucleated at the center of nanowire \nas shown in Fig. 1(a). As an example, here the mag netic parameters are  saturation \nmagnetization,  Ms = 560 kA/m , exchange stiffness constant, Aex = 12 pJ/m , perpendicular \nmagnetic anisotropy energy K = 1.1 MJ/m3, interfacial Dzyaloshinskii -Moriya interaction energy density D = 4.0 mJ/m2. To characterize the size and the shape of skyrmion, we take mz \nprofiles across the center of skyrmion. We approximate the line profile across a skyrmion  along \nthe longitudinal direction of nanowire  using a standard 360  o domain wall profile [ 25, 26] as \n𝑚𝑧=𝑐𝑜𝑠(2𝑎𝑟𝑐𝑡𝑎𝑛(𝑠𝑖𝑛ℎ((𝑟−𝑐\n𝑤))\n(𝑠𝑖𝑛ℎ(𝑅\n𝑤)))). (7) \nWhere r is the position of magnetization, c is the skyrmion center position, w is width of the \nskyrmion and R is the radius of skyrmion. The open black circles in Fig. 1(b) calculated values \nwith the fitting parameter using Eq. 7. The obtained R and w are 17.56  nm and 3.54 nm, \nrespectively . The material parameters in our simulations are chosen as table 1. The magnetic \nparameters selected for stable skyrmion conditions. In order to investigate skyrmion dynamics \nunder applied magnetic field gradient, ℎ𝑔𝑧=(𝐻final−𝐻initial)𝐿⁄ , is applied the whole \nnano wire along x - direction . We calculated the difference between the skyrmion center position \nat the initial time and the position at the final time as displacement . The final time is determined \nas the moment when the skyrmion stopped by nanowire edge. The determined skyrmion \nvelocity is defined as the skyrmion displacement with respect to time.  \n \nMagnetic field gradient driven skyrmions  in nanowire  \nThe skyrmion width a nd radius are determined by magnetic parameters such as Ms, Aex, K and \nD, the skyrmion width and radius cannot consider separate . Since Eq. (5) implies the skyrmion \nvelocity proportional to the skyrmion radius, however the skyrmion velocity are not matched \nwell with skyrmion radius as shown in inset of Fig. 2.  Because the values of skyrmion width \nare different with each skyrmions , the obtained skyrmion width are  from 3.57 to 5.78 nm  in the \nmicromagnetic simulation  results . The represented skyrmion radius dependence of the \nskyrmion velocities divided by skyrmion width ( 𝑣𝑥𝑤⁄) is plotted in Fig. 2. The black open \ncircles indicate the  simulation result for each skyrmion shown in table 1. The saturation \nmagnetization of a whole skyrmion is 560 kA/m. The red line is the theoretical calculation with \n = 176 GHz/T ,  = 0.3 , and ℎ𝑔𝑧 = 10.0 mT/ m. The value of 𝑣𝑥𝑤⁄ increase with increasing \nskyrmion radius . The micromagnetic results are  well matched as theoretical expectation results \nas shown in Eq. (6).   \nBased on the theoretical calculation, o ther magnetic parameters such as the damping constant \nand field gradient can also affect the skyrmion motion. In order to reveal the effects of skyrmion \ndynamics with various damping constant  () and field gradient  (ℎ𝑔𝑧 ), we perform \nmicromagnetic simulation s. Figure 3(a) and (b) indicate the 𝑣𝑥𝑤⁄  with various  and ℎ𝑔𝑧 , \nrespectively. In Fig. 3 (a) shows simulation results about the damping dependences of the \nskyrmion 𝑣𝑥𝑤⁄  for the skyrmion radius (open symbols) along with the 𝑣𝑥𝑤⁄  calculated \nwith Eq. (6) (solid lines).  The damping constant varied from 0.3 to 0.05  with  = 176 GHz/T, \nand ℎ𝑔𝑧  = 10.0 mT/ m. We find that the 𝑣𝑥𝑤⁄  value increase by decreasing the damping constant. The 𝑣𝑥𝑤⁄ for various ℎ𝑔𝑧 obtained by micromagnetic simulations (open symbols) \nand calculated  by Eq. (6) (solid lines)  as a function of skyrmion radius are displayed in Fig. 3  \n(b). The field gradient varied from 10. 0 to 2.5 mT/ m with  = 176 GHz/T,  = 0.05. As the \nfield gradient decrease, the value of 𝑣𝑥𝑤⁄ decreases. The agreements between the results of \nmicromagneti c simulation and Eq. (6) are excellent.  \nIn Fig. 4, we have plotted the 𝑣𝑥𝑤⁄  as a function of . Magnetic field gradient driven \nskyrmions  dependence  on the damping parameter is investigated at ℎ𝑔𝑧  = 10.0 mT/ m and \nskyrmion radius, R = 23.30 nm . Here the magnetic parameters are Ms = 560 kA/m, Aex = 12 \npJ/m, K = 1.1 MJ/m3, and D = 4.0 mJ/m2. The black open circles are micromagnetic simulation \nresults and red solid line is calculated values using Eq. (6). The results  show the inverse \nproportionality between 𝑣𝑥𝑤⁄ and  which is consistent with Eq. (6).  The decrease in alpha \nfrom 0.5 to 0.0125 resulted in an increase in 𝑣𝑥𝑤⁄ from 0.07 to 1.79 GHz.  \n \nSkyrmion dynamics for small interfacial DMI  \n \nWe set the DW center magnetization initially along the y direction and relax it with magnetic \nfield gradient , ℎ𝑔𝑧  as 10.0 mT/ 𝜇m. The iDMI energy density values coincides with the \nexperimentally determined values [refs.]. The snapshot of the magnetizations with Ms = 560 \nkA/m, Aex = 12 pJ/m, K = 1.1 MJ/m3,  = 0.1, and D = 0.1 mJ/m2, depicted in Fig . 5(a) and the \nchiral DW displacement s are shown in Fig. 5(b). As shown in colored lines in Fig. 5(b), the \nDW displacements are saturated after t = 35 ns for iDMI energy density is 0.1 mJ/m2 and t = \n33 ns for larger than 0.5 mJ/m2.  \n \nSummary  \nIn summary, we have investigated the skyrmion dynamics forced by a magnetic field gradient. \nThe velocity of skyrmion is predicted analytically  through the Thiele approach, which agrees \nwell with micromagnetic simulation results. The skyrmion dynamics is related with skyrmion \nshape, Gilbert damping, magnetic field gradient. Interestingly, skyrmion velocities divided by \nskyrmion width is proportional to th e skyrmion radius, magnetic field gradient and inverse \nGilbert damping constant. For the DW dynamics case which is small iDMI energy density, DW \nvelocity is much faster than the skyrmion velocities.  \n \n \n References  \n1. T. H. R. Skyrme, Nucl. Phys. 31, 556 (1962).,  \n2. U. K. Rössler, A. N. Bogdanov, and C. Pfleiderer, Nature 442 (7104), 797 (2006).  \n3. 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Yin, D. -S. Han, N. J. J. \nvan Hoof, H. J. M. Swagten, B. Koopmans, and C. -Y . You, Nat. Commun. 6, 7635 , (2015).  \n10. N.-H. Kim, D. -S. Han, J. Jung, J. Cho, J. -S. Kim, H. J. M. Swagten, and C. -Y . You, Appl. \nPhys. Lett. 107, 142408 , (2015). \n11. M. Belmeguenai, J. -P. Adam, Y . Roussigné, S. Eimer, T. Devolder, J. -V . Kim, S. M. Cherif, \nA. Stashkevich, and A. Thiaville, Phys. Rev. B 91, 180405(R) , (2015).  \n12. H. T. Nembach, J. M. Shaw, M. Weiler, E. Jué, and T. J. Silva, Nat. Phys. 11, 825, (2015).  \n13. J. Sampaio, V . Cros, S. Rohart, A. Thiaville, and A. Fert, Nat. Nanotech., 8, 839 (2013).  \n14. X. Zhang, M. Ezawa, and Y . Zhou, Sci. Rep. 5, 9400 (2015).  \n15. F. Jonietz, S. Muhlbauer, C. Pfleiderer, A. Neubauer, W. Munzer, A. Bauer, T. Adams, R. \nGeorgii, P. Boni , R. A. Duine, K. Everschor, M. Garst, and A. Rosch, Science 330 (6011), \n1648 (2010).  \n16. X. Z. Yu, N. Kanazawa, W. Z. Zhang, T. Nagai, T. Hara, K. Kimoto, Y . Matsui, Y . Onose, \nand Y . Tokura, Nat. Commun. 3, 988 (2012) . \n17. W. Jiang, P. Upadhyaya, W. Zhang, G. Yu,  M. B. Jungfleisch, F. Y . Fradin, J. E. Pearson, \nY . Tserkovnyak, K. L. Wang, O. Heinonen, S. G. E. Te Velthuis, and A. Hoffmann, Science \n349 (6245), 283 (2015).  \n18. S. Woo, K. Litzius, B. Kruger, M. Y . Im, L. Caretta, K. Richter, M. Mann, A. Krone, R. M. \nReeve , M. Weigand, P. Agrawal, I. Lemesh, M. A. Mawass, P. Fischer, M. Klaui, and G. S. Beach, Nat. Mater. 15 (5), 501 (2016).  \n19. Chun -Yeol You, In Mo Sung, and Byung -Kyu Joe , Appl. Phys. Lett. 89, 222513 (2006) . \n20. Chun -Yeol You and Seung -Seok Ha , Appl.  Phys. Lett. 91, 022507 (2007) . \n21. S. S. Parkin, M. Hayashi, L. Tomas, Science 320, 190 -194 (2008).,  \n22. R. Tomasello, E. Martinez, R. Zivieri, L. Torres, M. Carpentieri and G. Finocchio , Sci. Rep. \n4, 6784 (2015).  \n23. Thiele A. A, Phys. Rev. Lett, 30, 230 (1973).  \n24. Vansteenkiste , Arne, Leliaert, Jonathan, Dvornik, Mykola, Helsen, Mathias, Garcia -\nSanchez, Felipe, and Waeyenberge, Bartel Van, The Design and verification of Mumax3, \nAIP Advances 4, 107133 (2014).  \n25. H.-B. Braun, Phys. Rev. B 50, 16485 (1994).,  \n26. A. Kubetzka, O. Pietzsch , M. Bode, and R. Wiesendanger,  Phys. Rev. B 67, 020401 (2003).  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Figure captions  \nFig. 1 . (a) Spin structure of a hedgehog ( Néel-type) skyrmion texture in a nanowire film. (b)  \nLine profile across a skyrmion along the longitudinal direction of nanowire.  \n \nFig. 2 . The skyrmion radius dependence of skyrmion velocity divided by skyrmion width (𝑣𝑠∕𝑤) \nwith 𝛾 = 176 GHz/T, 𝛼 = 0.3, and magnetic field gradient ℎ𝑔𝑧  = 1.00 mT/ 𝜇m. inset: The \nskyrmion radius dependence of skyrmion velocity  \n \nFig. 3. The skyrmion radius dependent of skyrmion motion velocity divided by skyrmion width \n(𝑣𝑠∕𝑤) on the (a) damping constant, 𝛼 and (b)magnetic field gradient ℎ𝑔𝑧 dependence.  \n \nFig. 4 . The damping constant, 𝛼 dependent of skyrmion motion velocity divid ed by skyrmion \nwidth (𝑣𝑠 ∕𝑤) with 𝛾 = 176 GHz/T, 𝑅 = 23.39 nm, and magnetic field gradient ℎ𝑔𝑧 = 10.0 mT/ 𝜇m. \n \nFig. 5. (a) The snapshots of the magnetization with various time. (b) DW displacements as a \nfunction of time with various iDMI energy densities. All lines are the DW displacements \nanalyzed by Mz/Ms values.  \n \nTable caption  \nTable 1. magnetic parameters for chosen stable skyrmion. The saturation magnetization of a \nwhole skyrmions are 560 kA/m.  \n  \nFigure 1.  \n \nFigure 2.  \n \nFigure 3.  \n \nFigure 4.  \n \nFigure 5.  \n \n \n \n \n \nAex (pJ/m)  K (MJ/m3) D (mJ/m2) \n12 1.1 4 \n15 0.7 3 \n15 0.8 3.5 \n15 0.9 4 \n20 1.1 5 \n15 0.4 2 \nTable 1.  "    },    {        "title": "1810.06875v1.Spin_wave_induced_lateral_temperature_gradient_in_a_YIG_thin_film_GGG_system_excited_in_an_ESR_cavity.pdf",        "content": "1  Spin-wave-induced lateral temperature gradient in a YIG thin film/GGG system excited in an ESR cavity Ei Shigematsu, Yuichiro Ando, Sergey Dushenko, Teruya Shinjo, and Masashi Shiraishi Department of Electronic Science and Engineering, Kyoto University, 615-8510, Kyoto, Japan  Lateral thermal gradient of an yttrium iron garnet (YIG) film under the microwave application in the cavity of the electron spin resonance system (ESR) was measured at room temperature by fabricating a Cu/Sb thermocouple onto it. To date, thermal transport in YIG films caused by the Damon-Eshbach mode (DEM)—the unidirectional spin-wave heat conveyer effect—was demonstrated only by the excitation using coplanar waveguides. Here we show that effect exists even under YIG excitation using the ESR cavity—tool often employed to realize spin pumping. The temperature difference observed around the ferromagnetic resonance (FMR) field under the 4 mW microwave power peaked at 13 mK. The observed thermoelectric signal indicates the imbalance of the population between the DEMs that propagate near the top and bottom surfaces of the YIG film. We attribute the DEM population imbalance to the different magnetic damping near the top and bottom YIG surfaces. Additionally, the spin wave dynamics of the system were investigated using the micromagnetic simulations. The micromagnetic simulations confirmed the existence of the DEM imbalance in the system with the increased Gilbert damping at one of the YIG interfaces. The reported results are indispensable for the quantitative estimation of the electromotive force in the spin-charge conversion experiments using ESR cavities.   2  Spin caloritronics1—young but quickly developing spintronics field—is in pursuit of the comprehensive understanding of the connection between heat and spin currents. Following the discovery of the spin Seebeck effect2, plenty of experimental demonstrations of spin caloritronic phenomena have been reported, such as the spin-dependent Seebeck effect3 and the spin Peltier effect4. Especially, the heat transport via spin waves and spin-phonon interaction has attracted attention after the unidirectional spin-wave heat conveyer effect was reported5,6. In contrast to the conventional case of the heat transport against the temperature gradient, Damon-Eshbach mode (DEM) spin wave7,8 induces heat transport in the direction of the thermal gradient. Apart from this surprising achievement, the unidirectional spin-wave heat conveyer effect has important implications in the field of the dynamical spin injection, also known as spin pumping, since they often occur simultaneously in a studied system. Spin pumping9,10 is a method of generation of the pure spin current in the material due to the coupling of the interface spins to the precession magnetization of the adjacent ferromagnet layer. It quickly gained popularity as a spin injection method that can easily be used in any bilayer system consisting of nonmagnetic/ferromagnetic material11,12 whereas the electrical spin injection needs more elaborate surface treatment, formation of tunnel barriers and nanofabrication13–15. Using the spin pumping technique, the spin-to-charge conversion-related properties of various heavy metals11,16, semimetals17–19, semiconductors20–22 and even two-dimensional materials23–25 were unveiled, along with the spin transport properties of the materials 24,26,27. The DEM is the surface spin wave that is excited under the conditions close to the ferromagnetic resonance (FMR) and propagates in the opposite directions on the top and bottom surfaces28,29. When DEM reaches the end of the sample, its energy is damped as the heat, raising the temperature near the sample edge. In the case of the uniform excitation across the ferromagnet, the population of the DEM on top and bottom surfaces is the same, 3  and the net quantity of the transported heat cancels out. However, when the equivalence of the population of the two DEM propagating in the opposite directions is broken, the unidirectional thermal transport takes place. In the previous studies of the unidirectional spin-wave heat conveyer effect5,30, such inequivalence was shown to be present in case of the DEM excitation using the microstrip lines waveguides. In that case, the bottom surface of the ferromagnet is located closer to the microstrip line than the top surface, thus difference in the intensity of the microwave AC magnetic field causes population difference of the two DEM spin waves. Induced unidirectional heat transport happens in the direction of the propagation of the dominant DEM. Importantly, direction of the propagation of the dominant DEM (wave vector k) can be reversed by reversing the direction of the external magnetic field. Thus, voltage generated due to thermal effects (for example, the Seebeck effect) also reversed with the direction of the magnetic field. Incidentally, the spin pumping and spin-charge conversion experiments rely heavily on the reversal of the magnetic field to exclude non-magnetic spurious effects, including the thermal ones: sign reversal of the generated electromotive force with the external magnetic field usually taken as a proof of its spin-charge conversion origin. Thus, to confirm the origin of the electromotive force in the spin-charge conversion experiments, it is crucial to precisely estimate the unidirectional heat transfer induced by the DEM. While there were a few experimental5,30 and theoretical31 studies on the unidirectional heat transfer effect under the microwave excitation using wave guides, there were no such reports in the microwave cavities. In contrast, broad variety of the spin pumping and spin-charge conversion experiments are carried out using the TE011 cavity of the electron spin resonance (ESR) systems11,32,33. Our study fills the experimental gap, and reports the observation of the heat transfer by the DEM in the TE011 ESR cavity. We also performed the micromagnetic simulations, and discuss the origin of the DEM imbalance 4  observed experimentally. We now proceed to the experimental details and results. The 1.2-µm-thick YIG film was grown by liquid phase epitaxy on top of the GGG substrate and is available commercially (Granopt, Japan). We fabricated thermocouple on top of the YIG surface to measure temperature difference generated due to the heat transport by the DEM. While there are many types of thermocouples available commercially, the most common ones (types E, J, K, T) use ferromagnetic metals nickel (Ni) and iron (Fe), or their alloys, which may exhibit ferromagnetism due to the insufficient uniformity of the alloy. In the spin pumping experiments the lateral static magnetic field is applied in plane of the samples under the ferromagnetic resonance condition. In this geometry, the anomalous Hall effect in the thermocouple may be induced by the heating of the YIG film, which would add up to the electromotive force generated by the lateral thermal gradient of YIG film and prevent its quantitative estimation. To realize a thermocouple comprised of nonmagnetic metals, we focus on the combination of copper (Cu) and antimony (Sb), and use Cu wiring to make an electrical contact to the sample. First, we formed a 50-nm-thick SiO2 insulating layer on top of YIG to exclude the influence of spin pumping, which was shown to decrease exponentially with the thickness of the tunnel barrier34 . On top of it, 50-nm-thick Sb layer was deposited by resistance heating deposition. Finally, the third layer consisting of two Cu pads separated by a 1 mm gap was deposited. The sample with the formed thermocouple was set in the Seebeck effect measurement system (Fig.1(a)). Room temperature acted as a baseline level, while the hot and cold heat sinks—the temperature of which was controlled by the Peltier elements—were attached to the opposite sides of the sample. Lateral temperature difference and thermoelectric electromotive force were monitored simultaneously. For the ferromagnetic resonance measurements, the sample was mounted into the TE011 cavity of the ESR system (JEOL JES-FA200) at room temperature. The DC and AC magnetic fields were applied in 5  plane of the sample in DEM geometry as shown in Fig.1(b). The frequency of the AC magnetic field was set to 9.12 GHz, and applied microwave power was set to 4 mW. An estimated value of AC magnetic field applied to the sample was 2.2 µT. The DC magnetic field was swept through the FMR field of the YIG film, while the microwave absorption spectrum and the electromotive force between Cu electrodes were measured simultaneously. Since we used a bipolar electromagnet, measurements in 0° and 180° DC magnetic field are carried out without rotating the sample position Figure 1(a) and Figure 2 show the schematic layout and the detected thermoelectric electromotive force in the Seebeck effect measurement of the Cu/Sb thermocouple fabricated on top of the the YIG/GGG sample. We follow the conventional definition of the Seebeck coefficient S:  Δ𝑉=−𝑆Δ𝑇. (1) where ΔV and ΔT are the thermoelectric electromotive force and the temperature difference, respectively. From the linear fitting (black solid line in Fig. 2), the Seebeck coefficient of the fabricated sample was determined to be +15 nV/mK. This result is comparable to the Seebeck coefficient of amorphous Sb film reported in the literature35.  Next, the sample was placed in the cavity of the ESR system for the measurement of the magnetic-field-dependent heat transport induced by the DEM. The ferromagnetic resonance measurements with simultaneous detection of the electromotive force and FMR spectra were carried for the opposite directions of the DC magnetic field 0° and 180°. The wave vector k of the DEM is parallel to the cross product of the DC component of the magnetization of the YIG film M and the normal vector to the surface n. Direction of the k determines the direction ζΔT of the generated temperature difference ΔT on the propagation surface5,7,  k // ζΔT // M × n (2) 6  Therefore, we extracted the magnetization-dependent component of the observed thermoelectric signal by subtracting the signals measured at 0° and 180° direction of external magnetic field (V0° and V180°, correspondingly). Figure 3 shows the thermoelectric signal generated by the DEM, which is given by Vm = (V0° - V180°)/2, and the FMR spectra at the DC magnetic fields of 0° and 180°. The coincidence of the two FMR spectra confirms the identical resonance conditions for the opposite directions of the DC magnetic field. Interestingly, the DEM thermoelectric signal shows reversal of the polarity when approaching the FMR condition. Following the results of the Seebeck effect measurement of the sample, positive Vm signal corresponds to +y direction of the thermal gradient ζΔT, which is due to the DEM at the GGG/YIG interface, and negative Vm to -y direction, which is due to the DEM at the SiO2/YIG interface, respectively. The amplitude of the negative peak of the thermoelectric signal was measured to be -190 nV. Using the Seebeck coefficient of the sample, the estimated temperature difference between the Cu pads separated by the 1 mm gap (-y direction) is 13 mK. Figure 4 shows the schematic layout of the DEM excitation in our measuring geometry for 0° direction of the external magnetic field. The thermal gradient direction ζΔT of -y (+y) suggests the contribution of the DEM from the YIG interface with the SiO2 (the GGG) film. Note that the uniformly excited DEMs in the thin ferromagnetic film has the same population of the +k and -k modes, thus they transfer the equal amount of heat in the opposite directions and the induced temperature differences by the two modes cancel each other out. Therefore, the negative peak of the thermoelectric electromotive force at the DC magnetic field close to the FMR condition signifies that the magnitude of the DEM at the SiO2/YIG interface is superior to that on the GGG/YIG interface. The previous analytical magnetostatic studies of the DEM assumed that the ferromagnetic film was placed in the vacuum and did not treat the symmetry breaking of the top and bottom sides of the film7. Furthermore, the influence of the Gilbert damping on the 7  DEM propagation and damping was not considered. We carried out numerical micromagnetic simulations that evaluate effect of the symmetry breaking in our SiO2/YIG/GGG system on the DEM population using program MuMax3 36. GGG is known as a paramagnetic material with substantially large magnetization. Influence of the GGG layer attached to the YIG interface on the Gilbert damping of the surface YIG layer was already pointed out30. Thus, in the micromagnetic simulations, we set the Gilbert damping parameter α of one marginal layer next to the YIG/GGG interface (we refer to it as the bottom layer) larger than the other layers. The Gilbert damping parameter of the bottom layer was 0.02 and that of the other layers was 0.002 (Fig. 5(a)). As for the other magnetic parameters, we use those of permalloy presented in the specification paper of MuMax3 36, as a simple magnetic thin film model. The saturation magnetization and the exchange stiffness were set to be 8.6×105 A/m and 1.3×10-11 J/m, respectively. At the beginning of the simulation, the DC magnetic field was set, and the magnetization of the whole system was relaxed. Following that, the AC excitation of the magnetic field was applied, and—after the magnetization precession reached the steady state—we extracted the z component of the magnetization of each spin cell in the slice of x = 25 (where coordinate represents layer number in that direction). The magnetization motion in the slice consists of a non-time-dependent bias, standing waves, and traveling waves along the y direction. We can evaluate the DEM by extracting a portion of the traveling waves. For this purpose, the Fourier transform was implemented:  𝑚!𝑓,𝑘!2𝜋=W(𝑡)∙W(𝑦)∙𝑚!𝑡,𝑦𝑒!!!!!\"!!!!!!d𝑡d𝑦!!\"#!!\"#!!\"#!!\"# (3) where mz, W, f, and ky denote the z component of the normalized magnetization, the hamming window function, the excitation frequency, and the wavenumber of the magnetization in y direction, respectively. As we extracted the data in the finite range of [[ymin, ymax],[tmin, tmax]], we applied the hamming function to reduce obstructive sub lobes in the resulting Fourier spectra. We focus on the Fourier spectra in the frequency of the excitation f0, which was set 8  to 15 GHz. When 𝑚!𝑓!,!!!! is deconvoluted into 𝑚!𝑓!,!!!!=𝐴+𝐵j and 𝑚!−𝑓!,!!!!!=𝐶+𝐷j, the absolute amplitude with wavenumber k leads to 𝑚!!\"#!!!!=𝐴+𝐶!+−𝐵+𝐷!. Then we obtain the wave distribution as a function of the wavenumber ky/2π. A plot of the amplitude mzabs vs. wavenumber at different DC magnetic fields is shown in Fig. 5(b). The extracted plot at the magnetic field of 325.6 mT is also shown in Fig. 5(c). The clear break of the symmetry can be seen between the wavenumber spectra in the top (red line) and bottom layers (green line). The peak height in the top layer was superior to that in the bottom layer. Figure 5(d) shows the absolute amplitude mzabs in the central layer (top box), which is analogous to the FMR spectrum detected experimentally; the subtraction of the mzabs between the top and bottom layers (middle box), which characterizes the imbalance of the DEM between them; and the wave number (bottom box) corresponding to the maximum mzabs in the top (red filled circles) and bottom (green filled circles) layers. The peak of the DEM in simulated spectra appeared at the lower magnetic field than the FMR, in agreement with theoretical and experimental results in the literature37. The DEM modes in the top and bottom layers had opposite sign of the wave number (Fig. 4(d) bottom), i.e. the propagation direction of the DEM, and were consistent with DEM propagation direction in the previous studies5,30. Thus, the numerical simulation showed the imbalance between the DEM in the top and bottom layers due to the difference in damping constant. Finally, we performed MuMax3 simulations using parameters close to the experimental values. The AC magnetic field excitation frequency was set to f0 = 9.12 GHz. The saturation magnetization of the YIG layer was set to be 1.275×105 A/m, and the exchange stiffness to 3.7×10-12 J/m38. The calculated geometry is illustrated in Fig. 6(a). The Gilbert damping was 0.01 and 0.001 in the bottom layer and bulk, respectively. As in the case of permalloy, the DEM amplitudes in the top and bottom layers showed a clear difference (Fig. 6(b)). The difference in the amplitude between the two DEM propagating in the opposite directions peaked around the 9  FMR resonance field (Fig. 6(c)). Thus, numerical simulation of YIG also indicated that the top-layer DEM is dominant over the bottom-layer DEM due to the break of the reflection symmetry because of the different magnetic damping at the surfaces. This result explains the dominance of the heat generated by the top DEM observed in the experiment. We note that micromagnetic calculations for the full-size sample are necessary for the precise quantitative simulation of the experimental results, which was limited by the computational power in this study. Additionally, the quantitative determination of the heat drift velocity—a key parameter in the thermal distribution induced by the DEM imbalance—needs a more elaborate analysis of the spin-phonon interaction, which is left for the further study. However, our experimental and numerical results clearly show that reflection symmetry breaking between the two YIG surfaces by the magnetic damping at the interfaces induces the imbalanced DEM population and the unidirectional heat transfer.  In conclusion, we observed the unidirectional spin-wave heat conveyer effect in a 1.2-µm-thick YIG film under the uniform microwave excitation in the ESR cavity. The origin of the DEM imbalance that led to the heat transport is explained by the increased Gilbert damping at one of the YIG interfaces. The micromagnetic simulations confirmed the existence of the DEM imbalance in such system. Our study fills the experimental gap that existed in the literature on the unidirectional spin-wave heat conveyer effect generated in ESR cavity. The reported results are indispensable for the quantitative estimation of the electromotive force in the spin-charge conversion experiments using ESR cavities. Supplementary Material The supplementary material includes the following information, microwave power dependence of the detected electromotive force, dependence of the detected electromotive force on the direction and speed of the DC magnetic field sweep, reproducibility of the results, and details of the MuMax3 calculation. 10  Acknowledgements E.S. acknowledges the financial support from the JSPS Research Fellowship for Young Researchers and JSPS KAKENHI Grant No. 17J09520. This work was supported in part by MEXT (Innovative Area “Nano Spin Conversion Science” KAKENHI No. 26103003), Grant-in-Aid for Scientific Research (S) No. 16H06330, and Grant-in-Aid for Young Scientists (A) No. 16H06089. S.D. acknowledges support by JSPS Postdoctoral Fellowship and JSPS KAKENHI Grant No. 16F16064. The authors thank T. Takenobu and K. Kanahashi for the informative advice regarding the Seebeck coefficient measurement.    11  References 1 G.E. W Bauer, E. Saitoh, and B.J. van Wees, Nat. Mater. 11, 391 (2012). 2 K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778 (2008). 3 A. Slachter, F.L. Bakker, J.-P. Adam, and B.J. van Wees, Nat. Phys. 6, 879 (2010). 4 J. Flipse, F.K. Dejene, D. Wagenaar, G.E.W. Bauer, J. Ben Youssef, and B.J. Van Wees, Phys. Rev. Lett. 113, 027601 (2014). 5 T. An, V.I. Vasyuchka, K. Uchida, A. V Chumak, K. Yamaguchi, K. Harii, J. Ohe, M.B. Jungfleisch, Y. Kajiwara, H. Adachi, B. 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The fabricated sample was attached to the two Peltier elements that controlled the temperature difference between the edges of the sample. (b) A schematic image of the measurement of the DEM heat transfer under the FMR excitation in the ESR TE011 cavity.     \n Fig. 2. The thermoelectric electromotive force dependence on the applied temperature gradient for the Sb/Cu thermocouple fabricated on top of the YIG film. The black solid line is a linear fitting.   \n0100200300400500600-8000-6000-4000-20000ΔV\t\t[nV]ΔT\t\t[m K ]15      \n Fig. 3. Two lines are the electromotive forces observed under the microwave excitation of 4 mW at the DC magnetic fields of 0° and 180° A line in the middle box is the halved subtraction of the electromotive force (Vm). Two overlapped lines in the lower box are FMR spectra measured for 0° (red) and 180° (green) direction of the DC magnetic field.     SiO2faceGGGface-5. 0x10-70. 0\tV0°\tV180°EMF\t [V]200 nV\n-2. 0x10-70. 02. 0x10-7\t(V0°-V180°)/2Vm\t[V ]200 nV\n246. 0246. 5247. 0247. 5\t0°\t180°FMR\tspectrum\t[arb. \tuni t]Magneti c\tFi el d\t[mT]16   Fig. 4. A cross-sectional illustration of the thermal gradient generation by the DEM under the application of the external magnetic field in 0° direction. Direction of the k wave vector of the DEM is locked to the direction of the cross product of the YIG magnetization M and surface normal vector n.   \n  Fig. 5. (a) A schematic illustration of the structure used in micromagnetic simulation. The magnetic film had 31 layers in the z direction, and each layer consisted of 51 × 51 unit cells. Gilbert damping parameter was set to be 0.02 in the bottom layer, and 0.002 in the other layers. The directions of the DC and AC magnetic fields are indicated by the arrows. (b) The waterfall plot of the absolute amplitude of mz component in the x = 25 slice with respect to ky/2π and the DC magnetic field. The red and green lines indicate the top and bottom layer, respectively. (c) The absolute amplitude of mz component in the x = 25 slice in the top and bottom layer at the DC magnetic field 325.6 mT. The wave form of the top (bottom) layer is biased to the -ky/2π  (+ky/2π) direction, indicating the propagation direction of the DEM. The \n(a)(c)(d)\n(b)\nAbsolute amplitude [arb. unit]0. 00. 20. 40. 6\n-0. 020-0. 015-0. 010-0. 0050. 000\n300310320330340350-0. 004-0. 0020. 0000. 0020. 004FMR\t[arb. \tuni t]DEM\t[arb. \tuni t]\tz\t=\t30\t(Top)\tz\t=\t0\t(Bottom)ky/2π\t[1/51cel l ]Magneti c\tFi l ed\t[mT]-0. 10-0. 050. 000. 050. 100. 000. 020. 040. 060. 080. 10Absol ute\tampl i tude\t[arb. \tuni t]ky/2π  [1/51cel l ]\tz\t=\t30\t\t\t\t\t\t\t\t\t(T op)\tz\t=\t0\t\t\t\t\t\t\t\t\t(B ottom )325. 6\tmT17  amplitude of the main lobe in the top layer is higher than that in the bottom layer. (d) The upper box: the FMR intensity represented by the absolute amplitude of mz in the z = 15 layer). The middle box: difference in the absolute amplitude of mz between the bottom and top layers, indicating the DEM imbalance and heat transport amplitude. The lower box: the wave numbers corresponding to the maximum of the absolute amplitude of mz in top (red) and bottom (green) layers.   18    \n Fig. 6. (a) A schematic illustration of the structure used in micromagnetic simulation. The magnetic film had 21 layers in the z direction, and each layer consisted of 101 × 301 unit cells. Gilbert damping was set to be 0.01 in the bottom layer, and 0.001 in the others. (b) The absolute amplitude of mz component of the magnetization in the x = 50 slice in the top (red) and bottom (green) layers. The DC magnetic field was set to 251 (left box) and 253 mT (right box). (c) The upper box: The FMR intensity (the absolute amplitude of the z = 15 layer). The lower box: the DEM imbalance between bottom and top layers calculated from the micromagnetic simulation. Experimental measurements indicated the similar dominance of the top DEM from the observed heat transport.  (a)(c)(b)\n-1E-0401E-048. 4308. 4328. 434\n-1E-0401E-043. 1163. 1183. 120\n0.001Absol ute\tampl i tude\t[arb. \tuni t]ky/2π\t[1/301\tcel l ]\tz\t=\t20\t(Top)\tz\t=\t0\t(Bottom)251\tmT0246810\n246248250252254-8x10-4-6x10-4-4x10-4-2x10-40FMR\t[arb. \tuni t]\nMagneti c\tFi el d\t[mT]DEM\t[arb. \tuni t]253\tmTky/2π\t[1/301\tcel l ]\tz\t=\t20\t(Top)\tz\t=\t0\t(Bottom)0.001-1x10-40   1x10-4-1x10-40   1x10-4"    },    {        "title": "1510.06793v1.Laser_induced_THz_magnetization_precession_for_a_tetragonal_Heusler_like_nearly_compensated_ferrimagnet.pdf",        "content": "arXiv:1510.06793v1  [cond-mat.mtrl-sci]  23 Oct 2015Laser-induced THz magnetization precession for a tetragon al Heusler-like nearly\ncompensated ferrimagnet\nS. Mizukami,1,a)A. Sugihara,1S. Iihama,2Y. Sasaki,2K. Z. Suzuki,1and T. Miyazaki1\n1)WPI Advanced Institute for Materials Research, Tohoku Univ ersity,\nSendai 980-8577, Japan\n2)Department of Applied Physics, Tohoku University, Sendai 9 80-8579,\nJapan\n(Dated: 23 July 2018)\nLaser-inducedmagnetizationprecessional dynamicswasinvestiga tedinepitaxialfilms\nof Mn 3Ge, which is a tetragonal Heusler-like nearly compensated ferrimag net. The\nferromagnetic resonance (FMR) mode was observed, the preces sion frequency for\nwhich exceeded 0.5 THz and originated from the large magnetic anisot ropy field of\napproximately 200 kOe for this ferrimagnet. The effective damping c onstant was\napproximately 0.03. The corresponding effective Landau-Lifshitz c onstant of approx-\nimately 60 Mrad/s and is comparable to those of the similar Mn-Ga mate rials. The\nphysical mechanisms for the Gilbert damping and for the laser-induc ed excitation of\nthe FMR mode were also discussed in terms of the spin-orbit-induced damping and\nthe laser-induced ultrafast modulation of the magnetic anisotropy , respectively.\na)Electronic mail: mizukami@wpi-aimr.tohoku.ac.jp\n1Among the various types of magnetization dynamics, coherent mag netization precession,\ni.e.,ferromagneticresonance(FMR),isthemostfundamentaltype, andplaysamajorrolein\nrf spintronics applications based on spin pumping1–5and the spin-transfer-torque (STT).6,7\nSpin pumping is a phenomenon through which magnetization precessio n generates dc and rf\nspin currents in conductors that are in contact with magnetic films. The spin current can be\nconverted into anelectric voltage throughthe inverse spin-Hall eff ect.8The magnitude of the\nspin current generatedvia spinpumping is proportionaltothe FMRf requency fFMR;4,5thus,\nthe output electric voltage is enhanced with increased fFMR. In the case of STT oscillators\nand diodes, the fFMRvalue for the free layer of a given magnetoresistive devices primarily\ndetermines the frequency range for those devices.9,10An STT oscillator and diode detector\nat a frequency of approximately 40 GHz have already been demonst rated;11–13therefore, one\nof the issues for consideration as regards practical applications is the possibility of increasing\nfFMRto hundreds of GHz or to the THz wave range (0.1-3 THz).11,14\nOnesimple methodthroughwhich fFMRcanbeincreased utilizes magneticmaterials with\nlarge perpendicular magnetic anisotropy fields Heff\nkand small Gilbert damping constants\nα.13,15,16This is because fFMRis proportional to Heff\nkand, also, because the FMR quality\nfactor and critical current of an STT-oscillator are inversely and d irectly proportional to α,\nrespectively. The Heff\nkvalue is determined by the relation Heff\nk= 2Ku/Ms−4πMsfor thin\nfilms, where KuandMsare the perpendicular magnetic anisotropy constant and saturat ion\nmagnetization, respectively. Thus, materials with a small Ms, largeKu, and low αare\nvery favorable; these characteristics are similar to those of mate rials used in the free layers\nof magnetic tunnel junctions integrated in gigabit STT memory applic ation.17We have\npreviously reported that the Mn-Ga metallic compound satisfies the above requirements,\nand that magnetization precession at fFMRof up to 0.28 THz was observed in this case.18\nA couple of research groups have studied magnetization precessio n dynamics in the THz\nwave range for the FePt films with a large Heff\nk, and reported an αvalue that is a factor of\nabout 10 larger than that of Mn-Ga.19–21Thus, it is important to examine whether there are\nmaterialsexhibiting properties similartothoseofMn-Gaexist, inorde r tobetter understand\nthe physics behind this behavior.\nIn this letter, we report on observed magnetization precession at fFMRof more than 0.5\nTHz for an epitaxial film of a Mn 3Ge metallic compound. Also, we discuss the relatively\nsmall observed Gilbert damping. Such THz-wave-range dynamics ca n be investigated by\n2means of a THz wave22or pulse laser. Here, we use the all-optical technique proposed\npreviously;23therefore, the mechanism of laser-induced magnetization preces sion is also dis-\ncussed, because this is not very clearly understood.\nMn3Ge has a tetragonal D0 22structure, and the lattice constants are a= 3.816 and\nc= 7.261˚A in bulk materials [Fig. 1(a)].24,25The Mn atoms occupy at two non-equivalent\nsites in the unit-cell. The magnetic moment of Mn I(∼3.0µB) is anti-parallel to that of\nMnII(∼1.9µB), because of anti-ferromagnetic exchange coupling, and the net magnetic\nmoment is ∼0.8µB/f.u. In other words, this material is a nearly compensated ferrima gnet\nwith a Curie temperature Tcover 800 K.26The tetragonal structure induces a uniaxial\nmagnetic anisotropy, where the c-axis is the easy axis.24The D0 22structure is identical to\nthat of tetragonally-distorted D0 3, which is a class similar to the L2 1Heusler structure;\nthus, D0 22Mn3Ge is also known as a tetragonal Heusler-like compound, as is Mn 3Ga.27\nThe growth of epitaxial films of D0 22Mn3Ge has been reported quite recently, with these\nfilms exhibiting a large Kuand small Ms, similar to Mn-Ga.28–30Note that Mn 3Ge films\nwith a single D0 22phase can be grown for near stoichiometric compositions.29,30Further, an\nextremely large tunnel magnetoresistance is expected in the magn etic tunnel junction with\nMn3Ge electrodes, owing to the fully spin-polarized energy band with ∆ 1symmetry and the\nBloch wave vector parallel to the c-axis at the Fermi level.29,31These properties constitute\nthe qualitative differences between the Mn 3Ge and Mn 3Ga compounds from the material\nperspective.\nAll-optical measurement for the time-resolved magneto-optical K err effect was employed\nusing a standard optical pump-probe setup with a Ti: sapphire laser and a regenerative\namplifier. The wavelength and duration of the laser pulse were appro ximately 800 nm and\n150 fs, respectively, while the pulse repetition rate was 1 kHz. The p ulse laser beam was\ndivided into an intense pump beam and a weaker probe beam; both bea ms weres-polarized.\nThe pump beam was almost perpendicularly incident to the film surface , whereas the angle\nof incidence of the probe beam was ∼6◦with respect to the film normal [Fig. 1(b)].\nBoth laser beams were focused on the film surface and the beam spo ts were overlapped\nspatially. The probe and pump beams had spot sizes with 0.6 and 1.3 mm, respectively.\nThe Kerr rotation angle of the probe beam reflected at the film surf ace was analyzed using\na Wollaston prism and balanced photodiodes. The pump beam intensity was modulated\nby a mechanical chopper at a frequency of 360 Hz. Then, the volta ge output from the\n3FIG. 1. (a) Illustration of D0 22crystal structure unit cell for Mn 3Ge. (b) Diagram showing\ncoordinate system used for optical measurement and ferroma gnetic resonance mode of magnetiza-\ntion precession. The net magnetization (= MII−MI) precesses about the equilibrium angle of\nmagnetization θ, whereMI(MII) is the magnetization vector for the Mn I(MnII) sub-lattice. (b)\nOut-of-plane normalized hysteresis loop of the Kerr rotati on angle φkmeasured for the sample.\nphotodiodes was detected using a lock-in amplifier, as a function of d elay time of the pump-\nprobe laser pulses. The pump pulse fluence was ∼0.6 mJ/cm2. Note that the weakest\npossible fluence was used in order to reduce the temperature incre ase while maintaining the\nsignal-to-noise ratio. A magnetic field Hof 1.95 T with variable direction θHwas applied\nusing an electromagnet [Fig. 1 (b)].\nThec-axis-oriented Mn 3Ge epitaxial films were grown on a single-crystalline (001) MgO\nsubstrate with a Cr seed layer, and were capped with thin MgO/Al lay ers at room tempera-\nture using a sputtering method with a base pressure below 1 ×10−7Pa. The characteristics\nof a 130-nm-thick film with slightly off-stoichiometric composition (74 a t% Mn) deposited\nat 500◦C are reported here, because this sample showed the smallest coer civity (less than\n1 T) and the largest saturation magnetization (117 emu/cm3) of a number of films grown\nwith various thicknesses, compositions, and temperatures. Thes e properties are important\nto obtaining the data of time-resolved Kerr rotation angle φkwith a higher signal-to-noise\nratio, because, as noted above, Mn 3Ge films have a large perpendicular magnetic anisotropy\n4field and a small Kerr rotation angle.30Figure 1(c) displays an out-of-plane hysteresis loop\nofφkobtained for a sample without pump-beam irradiation. The loop is norm alized by the\nsaturation value φk,sat 1.95 T. The light skin depth is considered to be about 30 nm for the\nemployed laser wavelength, so that the φkvalue measured using the setup described above\nwas almost proportional to the out-of-plane component of the ma gnetization Mzwithin the\nlight skin depth depth. The loop shape is consistent with that measur ed using a vibrating\nsample magnetometer, indicating that the film is magnetically homogen eous along the film\nthickness and that value of φk/φk,sapproximates to the Mz/Msvalue.\nFigure 2(a) shows the pump-pulse-induced change in the normalized Kerr rotation angle\n∆φk/φk,s(∆φk=φk−φk,s) as a function of the pump-probe delay time ∆ twith an applied\nmagnetic field Hperpendicular to the film plane. ∆ φk/φk,sdecreases quickly immediately\nafter the pump-laser pulse irradiation, but it rapidly recovers within ∼2.0 ps. This change\nis attributed to the ultrafast reduction and ps restoration of Mswithin the light skin depth\nregion, and is involved in the process of thermal equilibration among t he internal degrees of\nfreedom, i.e., the electron, spin, and lattice systems.32. After the electron system absorbs\nlight energy, the spin temperature increases in the sub-ps timesca le because of the heat\nflow from the electron system, which corresponds to a reduction in Ms. Subsequently, the\nelectron and spin systems are cooled by the dissipation of heat into t he lattices, which have\na high heat capacity. Then, all of the systems reach thermal equilib rium. This process is\nreflected in the ps restoration of Ms. Even after thermal equilibrium among these systems is\nreached, the heat energy remains within the light skin depth region a nd the temperature is\nslightly higher than the initial value. However, this region gradually co ols via the diffusion\nof this heat deeper into the film and substrate over a longer timesca le. Thus, the remaining\nheat causing the increased temperature corresponds to the sma ll reduction of ∆ φk/φk,safter\n∼2.0 ps.\nWith increasing θHfrom out-of-plane to in-plane, a damped oscillation becomes visible\nin the ∆φk/φk,sdata in the 2-12 ps range [Fig. 2(b)]. Additionally, a fast Fourier tran sform\nof this data clearly indicates a single spectrum at a frequency of 0.5- 0.6 THz [Fig. 2(c)].\nThese damped oscillations are attributed to the temporal oscillation ofMz, which reflects\nthe damped magnetization precession,23because the zcomponent of the magnetization\nprecession vector increases with increasing θH. Further, the single spectrum apparent in\nFig. 2(c) indicates that there are no excited standing spin-waves ( such as those observed in\n5thick Ni films), even though the film is thicker than the optical skin de pth.23\nFerrimagnets generally have two magnetization precession modes, i.e., the FMR and\nexchange modes, because of the presence of sub-lattices.33In the FMR mode, sub-lattice\nmagnetization vectors precess while maintaining an anti-parallel dire ction, as illustrated in\nFig. 1(b), such that their frequency is independent of the exchan ge coupling energy between\nthe sub-lattice magnetizations. On the other hand, the sub-lattic e magnetization vectors\nare canted in the exchange mode; therefore, the precession fre quency is proportional to the\nexchange coupling energy between them and is much higher than tha t of the FMR mode.\nAs observed in the case of amorphous ferrimagnets, the FMR mode is preferentially excited\nwhen the pump laser intensity is so weak that the increase in tempera ture is lower than the\nferrimagnet compensation temperature.34No compensation temperature is observed in the\nbulk Mn 3Ge.25,26Also, the temperature increase in this experiment is significantly sma ller\nthanTcbecause the reduction of Msis up to 4 %, as can be seen in Fig. 2(a). Therefore,\nthe observed magnetization precession is attributed to the FMR mo de. Further, as the\nmode excitation is limited to the light skin depth, the amplitude, freque ncy, and etc., for\nthe excited mode are dependent on the film thickness with respect t o the light skin depth.\nThis is because the locally excited magnetization precession propaga tes more deeply into\nthe film as a spin wave in cases where fFMRis in the GHz range.23Note that it is reasonably\nassumed that such a non-local effect is negligible in this study, becau se the timescale of the\ndamped precession discussed here ( ∼1-10 ps) is significantly shorter than that relevant to a\nspin wave with wavelength comparable to the light skin depth ( ∼100 ps).\nThe FMR mode in the THz-wave range is quantitatively examined below. When the ex-\nchangecouplingbetween thesub-latticemagnetizationsissufficient ly strongandthetemper-\nature is well below both Tcand the compensation temperature, the magnetization dynamics\nfor a ferrimagnet can be described using the effective Landau-Lifs hitz-Gilbert equation35\ndm\ndt=−γeffm×/bracketleftbig\nH+Heff\nk(m·z)z/bracketrightbig\n+αeffm×dm\ndt, (1)\nwheremis the unit vector of the net magnetization parallel (anti-parallel) to the magnetiza-\ntion vector MII(MI) for the Mn II(MnI) sub-lattice [Fig. 1(b)]. Here, the spatial change of\nmis negligible, as mentioned above. Heff\nkis the effective value of the perpendicular magnetic\nanisotropyfieldincluding thedemagnetizationfield, even thoughthe demagnetizationfieldis\nnegligibly small for thisferrimagnet (4 πMs= 1.5 kOe). Further, γeffandαeffaretheeffective\n6FIG. 2. Change in Kerr rotation angle ∆ φknormalized by the saturation value φk,sas a function\nof pump-probe delay time ∆ t: (a) for a short time-frame at θH= 0◦and (b) for a relatively long\ntime-frame and different values of θH. The solid curves in (a) and (b) are a visual guide and values\nfitted to the data, respectively. The data in (b) are plotted w ith offsets for clarity. (c) Power\nspectral density as a function of frequency fand magnetic field angle θH.\n7values of the gyromagnetic ratio and the damping constant, respe ctively, which are defined\nasγeff= (MII−MI)/(MII/γII−MI/γI) andαeff= (αIIMII/γII−αIMI/γI)/(MII/γII−MI/γI),\nrespectively, using the gyromagnetic ratio γI(II)and damping constant αI(II)for the sub-\nlattice magnetization of Mn I(II). In the case of Heff\nk≫H,fFMRand the relaxation time of\nthe FMR mode τFMRare derived from Eq. (1) as\nfFMR=γeff/2π/parenleftbig\nHeff\nk+Hz/parenrightbig\n, (2)\n1/τFMR= 2παefffFMR. (3)\nHere,Hzis the normal component of H. Figure 3(a) shows the Hzdependence of the\nprecession frequency fp. This is obtained using the experimental data on the oscillatory\npart of the change in ∆ φk/φk,svia least-square fitting to the damped sinusoidal func-\ntion, ∆φk,p/φk,sexp(−t/τp)sin(2πfp+φp), with an offset approximating the slow change\nof ∆φk/φk,s[solid curves, Fig. 2(b)]. Here, ∆ φk,p/φk,s,τp, andφpare the normalized am-\nplitude, relaxation time, and phase for the oscillatory part of ∆ φk/φk,s, respectively. The\nleast-square fitting of Eq. (2) to the fpvs.Hzdata yields γeff/2π= 2.83 GHz/kOe and\nHeff\nk= 183 kOe [solid line, Fig. 3(a)]. The γeffvalue is close to 2.80 GHz/kOe for the free\nelectron. The value of Heff\nkis equal to the value determined via static measurement (198\nkOe)30within the accepted range of experimental error. Thus, the analy sis confirms that\nthe THz-wave range FMR mode primarily results from the large magne tic anisotropy field in\nthe Mn 3Ge material. The αeffvalues, which are estimated using the relation αeff= 1/2πfpτp\nfollowing Eq. (3), are also plotted in Fig. 3(a). The experimental αeffvalues are indepen-\ndent ofHzwithin the accepted range of experimental error, being in accorda nce with Eq.\n(3); the mean value is 0.03. This value of αefffor D0 22Mn3Ge is slightly larger than the\npreviously reported values for for D0 22Mn2.12Ga (∼0.015) and L1 0Mn1.54Ga (∼0.008).18\nIn the case of metallic magnets, the Gilbert damping at ambient tempe rature is primarily\ncaused by phonon and atomic-disorder scattering for electrons a t the Fermi level in the\nBloch states that are perturbed by the spin-orbit interaction. Th is mechanism, the so-\ncalled Kambersky mechanism,36,37predicts α∝M−1\ns, so that it is more preferable to use\nthe Landau-Lifshitz constants λ(≡αγMs) for discussion of the experimental values of α\nfor different materials. Interestingly, λeff(≡αeffγeffMs) for Mn 3Ge was estimated to be 61\nMrad/s, which is almost identical to the values for D0 22Mn2.12Ga (∼81 Mrad/s) and L1 0\nMn1.54Ga(∼66Mrad/s). The λfortheKamberkymechanism isapproximatelyproportional\n8FIG. 3. (a) Normal component of magnetic field Hdependence on precession frequency fpand\neffective dampingconstant αeffforMn 3Gefilm. (b)Oscillation amplitudeoftheKerrrotation angle\n∆φk,p/φk,scorresponding to the magnetization precession as a functio n of the in-plane component\nofH. The solid line and curve are fit to the data. The dashed line de notes the mean value of αeff.\ntoλ2\nSOD(EF), whereλSOisthespin-orbitinteractionconstant and D(EF)isthetotaldensity\nof states at the Fermi level.37The theoretical values of D(EF) for the above materials are\nroughly identical, because of the similar crystal structures and co nstituent elements, even\nthough the band structures around at the Fermi level differ slight ly, as mentioned at the\nbeginning.18,29Furthermore, the spin-orbit interactions for Ga or Ge, depending on the\natomic number, may not differ significantly. Thus, the difference in αefffor these materials\ncan be understood qualitatively in terms of the Kambersky mechanis m. Further discussion\nbased on additional experiments is required in order to obtain more p recise values for αeff\nand to examine whether other relaxation mechanisms, such as extr insic mechanisms (related\nto the magnetic inhomogeneities), must also be considered.\nFinally, the excitation mechanism of magnetization precession in this s tudy is discussed\nbelow, in the context of a previously proposed scenario for laser-in duced magnetization\n9precession in Ni films.23The initial equilibrium direction of magnetization θis determined\nby thebalance between HandHeff\nk[Fig. 1(b)]. Duringtheperiodinwhich thethree internal\nsystems are not in thermal equilibrium for ∆ t <∼2.0 ps after the pump-laser irradiation\n[Fig. 2(a)], not only the value of Ms, but also the value of the uniaxial magnetic anisotropy,\ni.e.,Heff\nk, is altered. Thus, the equilibrium direction deviates slightly from θand is restored,\nwhich causes magnetization precession. This mechanism may be exam ined by considering\nthe angular dependence of the magnetization precession amplitude . Because the precession\namplitudemaybeproportionaltoanimpulsive torquegeneratedfro mthemodulationof Heff\nk\nin Eq. (1), the torque has the angular dependence |m0×(m0·z)z|, wherem0is the initial\ndirection of the magnetization. Consequently, the z-component of the precession amplitude,\ni.e., ∆φk,p/φk,s, is expressed as ∆ φk,p/φk,s=ζcosθsin2θ∼ζ/parenleftbig\nHx/Heff\nk/parenrightbig2, whereζis the\nproportionalityconstant and Hxisthe in-plane component of H. The experimental values of\n∆φk,p/φk,sare plotted as a function of Hxin Fig. 3(b). The measured data match the above\nrelation, which supports the above-described scenario. Although ζcould be determined via\nthe magnitude and the period of modulation of Heff\nk, it is necessary to consider the ultrafast\ndynamics of the electron, spin, and lattice in the non-equilibrium stat e in order to obtain a\nmore quantitative evaluation;38,39this is beyond the scope of this report.\nIn summary, magnetization precessional dynamics was studied in a D 022Mn3Ge epitaxial\nfilm using an all-optical pump-probe technique. The FMR mode at fFMRup to 0.56 THz\nwas observed, which was caused by the extremely large Heff\nk. A relatively small damping\nconstant of approximately 0.03 was also obtained, and the corresp onding Landau-Lifshitz\nconstant for Mn 3Ge were shown to be almost identical to that for Mn-Ga, being in quali-\ntatively accordance with the prediction of the Kambersky spin-orb it mechanism. 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Here we point out the possibility of nondissipative electric\ndriving of magnetization dynamics, if the valence electronic states have nontrivial topology in the\ncombined space of crystal momentum and magnetization con\fguration. We provide a hybrid in-\nsulator system to demonstrate that the topology-based nonconservative electrical generalized force\nis capable of supporting sustained magnetization motion in the presence of Gilbert damping, with\nquantized and steady energy pumping into magnetization motion from the electric \feld. We also\ngeneralize our results to magnetic textures, and discuss electric \feld induced Dzyaloshinskii-Moriya\ninteraction which can be nonconservative.\nI. INTRODUCTION\nThe study of electrical control of magnetization dy-\nnamics has occupied a large part of solid state research\nfor many decades, which generally falls into two separate\ncategories known as multiferroics [1{3] and spintronics [4]\ndepending on the conductive behavior of the hosting ma-\nterials. The former deals with insulators where electrical\ne\u000bects on magnetization is characterized through the free\nenergy [5], and the resulting torque would be naturally\nconsidered as conservative and unable to drive sustained\nmotion of the magnetization for a static electric \feld. In\nthe latter, one \fnds various current induced magnetic\ntorques in metals and semiconductors [6{8], which can\nprovide a persistent source of energy for sustained mo-\ntion of magnetization, but one has to deal with wasteful\nand prohibitive joule heating in practice.\nMagnetic insulators have recently been utilized to\nachieve low-dissipation magnetization control by combin-\ning the insulator with heavy metals hosting prominent\nspin Hall e\u000bect that injects a spin current into the insu-\nlator [9, 10]. An electric \feld can also directly manipulate\nmagnetization in an insulator without Joule heating by\nmeans of spin-orbit torques mediated by occupied elec-\ntronic states [11{13]. In particular, mesoscopic transport\ntheories proposed the exchange gapped edge states of a\ntwo dimensional topological insulator in hybrid with a\nmagnet as a unique platform for studying the magnetic\nThouless motor [14, 15], which works as the inverse mode\nof the adiabatic charge pumping by a cyclic magnetic\nmotion [16{18] under an applied voltage. By using the\nscattering matrix approach [19], previous works showed\nquantized electrical energy transfer into the magnet if the\nmagnetization accomplishes a cyclic motion [14, 15]. On\nthe other hand, as the Berry curvature in the mixed space\nof crystal momentum and magnetization con\fguration\nunderlies the magnetic Thouless pumping, it would be in-\nteresting to reveal the relation between nonzero electrical\nenergy input into magnetic dynamics and the topological\ncharacteristics in the mixed parameter space [20]. More-\n\u0003These authors contributed equally to this work.over, it has not been shown whether the electric driving\nof sustained magnetic motion, which enables a motor,\ncan be realized in the presence of magnetic damping due\nto coupling of magnetization to other degrees of freedom\nthan electrons.\nIn this study we show the possibility of nondissipative\ndriving of magnetization dynamics with steady energy\npumping by a static electric \feld in insulators. This is\nmotivated by the fact that electric polarization is not al-\nways a single-valued quantity [21, 22], and the adiabatic\ncurrent pumped by cyclic motion of the magnetization\ncan acquire a quantized net amount of energy from a\nstatic electric \feld under certain topological conditions\nof the valence electronic states. We exploit this idea\nin a model system of edge states of a two-dimensional\ntopological insulator gapped by hybrid with a magnetic\nwire, and show explicitly sustained magnetization mo-\ntion when a constant electric \feld is applied to overcome\nGilbert damping.\nOur results can also be generalized to the case of slowly\nvarying magnetic textures. There is a topological current\nbilinear in the gradient and time derivative of the mag-\nnetization density [23], a sort of anomalous Hall current\ninduced by the arti\fcial electric \feld from the time de-\npendent magnetic texture [24{27]. This current provides\na channel of nondissipative drive of the magnetic tex-\nture by an external static electric \feld. In topologically\nnontrivial cases this drive is nonconservative and capa-\nble of delivering a nonzero and quantized amount of en-\nergy when the magnetic texture wraps around the Bloch\nsphere in time. In topologically trivial cases, where the\nelectric polarization induced by magnetization gradient\nis well de\fned, the drive is conservative because it can\nbe identi\fed as originating from a polarization energy\ndensity whose susceptibility to the magnetization gradi-\nent gives the electric \feld induced Dzyaloshinskii-Moriya\ninteraction (DMI) [28{31].\nThe rest of the paper is organized as follows. In Sec. II\nwe focus on the electric-\feld induced generalized force on\na homogeneous magnetization in insulators, and study its\nnonconservative nature which is related to certain topo-\nlogical conditions of the occupied electronic states. These\ngeneral rationales are illustrated in a hybrid insulator in\nSec. III, in which the electric driving of sustained mag-arXiv:2101.07164v3  [cond-mat.mes-hall]  23 Jun 20212\nnetic motion is also demonstrated. Section IV is devoted\nto the electrical generalized force on the magnetization in\ninhomogeneous insulators and its relation to an electric-\n\feld induced DMI. Finally, we concludes the paper in\nSec. V.\nII. ELECTRICAL GENERALIZED FORCE ON\nMAGNETIZATION\nIn the language of analytic mechanics, an generalized\nforce is an amount of work done on the system per\nunit displacement in the dynamical variable (the mag-\nnetization here). Considering a system with a homoge-\nneous magnetization mcoupled to an electronic insula-\ntor, change in the magnetization can in general pump an\nadiabatic electric current\nj=eZ\n[dk] \nkm\u0001_m; (1)\nwhere \n kmis the electronic Bloch-state Berry curvature\nin the parameter space of the magnetization and crystal\nmomentumk(set~= 1 unless otherwise noted), with its\nCartesian components given by \u00002Imh@kiuj@mjui. Here\njuiis the periodic part of the Bloch wave, and the band\nindexnis omitted for simple notation. [ dk]\u0011ddk=(2\u0019)d\nwithdas the spatial dimension, and the summation over\nvalence bands is implied. Through this adiabatic cur-\nrent, an external electric \feld can deliver work on the\nsystem, with the work density \u000ew=E\u0001jdt, which is\nproportional to \u000em=_m\u0001dt. Therefore, we obtain the\nelectrical generalized force density on magnetization as\nEe\nm\u0011\u000ew\n\u000em=eE\u0001Z\n[dk] \nkm: (2)\nThis electrical generalized force is nondissipative be-\ncause of the lack of conduction electrons for joule heat-\ning, and is in fact topological in the sense that it delivers\na quantized amount of energy over a cycle of the mag-\nnetization motion. For simplicity, we \frst consider an\ninsulator with zero Chern numbers in the Brillouin zone\nat each point over the path of m, such that one can take\nak-space periodic gauge to locally de\fne an electronic\npolarizationP=\u0000eR\n[dk]Ak[21], with Ak=huji@kui.\nThen the electrical work density delivered over the cycle\ncan be written in terms of the change of this polarization\nw=I\ndm\u0001Ee\nm=E\u0001\u0001P: (3)\nThis change is quantized in units of \u0001 P=\u0000ea=V0with\nabeing a discrete lattice vector (including the null vec-\ntor) andV0the volume of a unit cell. When this change is\nzero, so that the polarization is globally de\fned, the elec-\ntrical generalized force is conservative in the sense that\nits work can be regarded as a change in the globally well\nde\fned polarization energy density \u0000E\u0001P. When this\nFIG. 1. A ferromagnetic wire (blue bar) hybridizes and gaps\nthe edge states of a two-dimensional topological insulator\n(green region). When the magnetization mmoves around\n(blue circle on the right) on the Bloch sphere, the pumped\nadiabatic current jalong the edge couples to an applied elec-\ntric \feldEto provide energy to overcome Gilbert damping.\nA static magnetic \feld His applied to help preparing the\nsystem into a sustained motion of limit cycle.\nchange is nonzero, the electrical generalized force is non-\nconservative and capable of supporting sustained magne-\ntization motion even in the presence of Gilbert damping\ndue to other dissipative channels.\nSome comments are in order. First, if the electronic in-\nsulator is one dimensional, then the electrical work (per\nunit length) over a cycle of the magnetization reduces to\neEtimes the Chern number over the torus of the com-\nbined space of crystal momentum and the magnetization\n(along its path), corresponding to the quantized number\nof electrons pumped over the cycle. Second, quantization\nof electrical work over the cycle of magnetization also ap-\nplies to insulators with nonzero k-space Chern numbers\nby a simple argument. Although one cannot take a peri-\nodic gauge in k-space, one can always choose a periodic\ngauge over a \fxed one dimensional path of the magneti-\nzation. It is then clear that the electrical work over the\ncycle equals the Brillouin-zone integral of the k-gradient\nof the Berry's phase over the cycle. Topological quantiza-\ntion of this work then follows from the multi-valuedness\nof the Berry's phase. Third, using the Bianchi identity\non Berry curvatures, one can easily show that the electri-\ncal generalized force is curl-free @m\u0002Ee\nm= 0 everywhere\ninm-space, except the singular points where the energy\ngap above the \flled states of the electron system closes.\nWhen can the electrical work on magnetization be\nnonzero? Quantization of its value implies that the elec-\ntrical work is invariant if the path in m-space is deformed\nwithout closing the energy gap. In particular, within a\nsingly connected region where the gap is open, the elec-\ntrical work is zero on all closed paths. This applies for\nexample to the north or south hemispheres of magneti-\nzation in the two dimensional ferromagnetic Dirac model\nstudied in [32], where one can de\fne polarization ener-\ngies separately for each region, although cannot globally\nbecause of gap closing on the equator. Consequently, this\nmodel system cannot provide nonzero electrical work for\nsustained magnetization motion. It is therefore clear that\nthe singular points of gap closing have to be arranged to\nde\fne multiply connected regular regions, where electri-3\ncal work can possibly be nonzero on topologically non-\ntrivial paths.\nIII. A MODEL FOR SUSTAINED\nMAGNETIZATION MOTION\nHere we propose a one dimensional model system,\nwhere the gap closes on the two poles of the magne-\ntization Bloch sphere, and the electrical work per unit\nlength iseEtimes the winding number of the path\naround the poles. The system is constructed by inter-\nfacing a magnetic wire with the topological edge states\nof a two-dimensional topological insulator (Fig. 1). The\nexchange coupling renders the electronic system insulat-\ning by opening a gap in the Dirac spectrum. The relevant\nlow-energy Hamiltonian is\n^h=~vk^\u001by+J^\u001b\u0001m; (4)\nwherevis the Fermi velocity, ^\u001bis the Pauli matrix, and\nJis the coupling constant. The magnetization mis as-\nsumed to have a \fxed magnitude and is parameterized by\nthe polar angle \u0012relative to the yaxis and the azimuthal\nangle\u001eas shown in Fig. 1. The energy gap is open\neverywhere except at the north and south poles of the\nBloch sphere with my=\u0006m(red dots). Assuming that\nthe lower band is \flled and the electric \feld is applied\nalong the magnetic wire (positive xdirection), we can\nevaluate the formula for the electrical generalized force\nto \fnd\nEe=\u0000eE\n2\u0019m^e\u001e\nsin\u0012=\u0000eE@m\u001e\n2\u0019: (5)\nIt is singular at the poles and is a gradient of the multiple-\nvalued azimuthal angle, so that the electrical work den-\nsity over a closed path on the Bloch sphere is quantized\nin terms of the winding number of the path\nI\ndm\u0001Ee=\u0000NteE; (6)\nin line with the aforementioned general topological ar-\nguments. The winding number Ntcounts how many\ntimes the closed path wraps around the yaxis counter-\nclockwise.\nIn such a one dimensional insulator it is also interest-\ning to understand the electrical generalized force from\nthe polarization as Ee=E@mP, where the polarization\nis not single-valued and can only be determined to be\nP=\u0000e\u001e=2\u0019up to an uncertainty quantum \u0000e. Con-\nsistently, the two gap closing poles are singular points of\nthe polarization, and the change of polarization upon a\nclosed path on the Bloch sphere is \u0000eNt.\nWe now proceed to study the dynamics of the mag-\nnetization to see the e\u000bect of this generalized force. In\nthe absence of coupling to the electronic system, we can\nrewrite the Landau-Lifshitz-Gilbert equation of the fer-\nromagnet in the form of \u0000@mG0+_m\u0002\n0\nm\u0000\u00110_m= 0, as\nFIG. 2. Free energy contours in the angular space and typical\nevolution trajectories on the Bloch sphere in the absence (top\npanels) and presence (middle panels) of an electric \feld, and\nin the presence of both electric and magnetic \felds (bottom\npanels). In the last case, a limit cycle emerges.\nbalancing out a conserved force from the free energy G0,\na Lorentz type force from the m-space Berry curvature\n\n0\nm[33], and a frictional force with a scalar damping\ncoe\u000ecient\u00110. We will model the free energy density as\nG0=\u0000K0^m2\nx\u0000Hmywith an easy axis anisotropy and\nan applied static magnetic \feld H. Them-space Berry\ncurvature is given in terms of the gyromagnetic ratio \r0\nas\n0\nm=m=(m2~\r0). The damping coe\u000ecient is related\nto the Gilbert number \u0015as\u0015= (\r0)2\u00110.\nIn the presence of coupling to the electronic system,\nthe equation of motion becomes\nEe\nm\u0000@mG+_m\u0002\nm\u0000\u0011_m= 0; (7)\nwhere the electrical generalized force enters as an ex-\ntra term along with electronic modi\fcations to the other\nterms. The gap opening in the electronic system con-\ntributes a lowering of the free energy Ge=Ke( ^m2\ny\u00001)\nthat we model as a hard-axis anisotropy. The electronic\ncontribution to the m-space Berry curvature is given by\n\ne\nm=R\n[dk] \nm=m=(m2~\re), where \n m=@m\u0002Am\nis derived from Am=huji@mui, and\re= 2\u0019v=J . Fi-4\nnally, we assume that the gap of the electronic system\nremains open during the course of dynamics, so there\nis no electronic contribution to the damping coe\u000ecient\n\u0011=\u00110.\nRepresentative results of the magnetization motion\nare presented in Fig. 2, where we take \re=\r0=\u0019,\nKe=(m=\r0) =K0=(m=\r0) = 1 GHz and \u0011= 0:2=(m\r0).\nShown in the top and middle panels ( H= 0), there are\ntwo types of energy conserved motion in the absence of\ndamping and external \felds, divided by the contour of\nzero energy (the white dashed curve). In the area enclos-\ning the two points of lowest energy, mrotates around the\nxaxis, whereas in the upper and lower areas outside of\nthe zero-energy contour mgoes around the yaxis. This\nsituation is changed in the presence of damping, as shown\nin the top panel, where two points ( \u001e= 0;\u0012= 0:3\u0019)\nand (\u001e= 0;\u0012= 0:7\u0019) outside of the zero-energy contour\nevolve to di\u000berent points of lowest energy. In the middle\npanels, an electric \feld eE=2\u0019= 0:1K0is applied, which\ngives a force in the clockwise (negative \u001e) direction. The\nblue trajectory starting from ( \u001e= 0;\u0012= 0:7\u0019) falls faster\nto the +mxaxis, while the red trajectory starting from\n(\u001e= 0;\u0012= 0:3\u0019) extends for 3 =4 circle before the \f-\nnal decay into the same energy minimum as the other\ntrajectory.\nThe lower panels show the situation where limit cycle\nmotion is found. We found it important to prepare the\nsystem with predominantly around- my-axis energy con-\ntours, so that the non-conservative electrical force can\nbe best utilized. We therefore apply a static magnetic\n\feld inydirection with the magnitude H=K0=mto\nchange the energy landscape. We also switched the di-\nrection of the electric \feld so that the electrical force\ngoes along the directions of the energy contours. We\nfound that all initial points in a wide region, between the\ntwo dashed circles shown on the right of the lower panels\nof Fig. 2, fall into the same limit cycle. For instance,\nthe blue curve starts from ( \u001e= 0;\u0012= 0:4\u0019) and evolves\ninto the right handed limit cycle under an electric \feld\neE=2\u0019=\u00000:1K0. Figure 3 shows how the limit cycle\nmotion is reached in time for two trajectories (blue and\nred) from di\u000berent initial angles, along with one (black)\nthat falls into an energy minimum. On the limit cycle, we\nfound that the energy input from the electrical force bal-\nances out the energy dissipation from the Gilbert damp-\ning,H\ndm\u0001(Ee\nm\u0000\u0011_m) = 0, as can be easily derived from\nthe equation of motion.\nIV. ELECTRICAL DMI FORCE\nSo far we have been concentrating on nondissipative\nelectrical driving on a uniform magnetization. When the\nmagnetization is nonuniform, the electrical generalized\nforce Eq. (2) still applies as a local force density, but\nthere will be additional contributions due to the magne-\ntization gradients. In metals, the electric-current induced\nDMI have been discussed recently [34{36], which is simi-\nFIG. 3. Time dependence of the polar angle for di\u000berent\ninitial conditions, \u001e= 0,\u0012= 0:001\u0019(black),\u0012= 0:4\u0019(red),\n\u0012= 0:8\u0019(blue). Correspondingly on the Bloch sphere shown\nin the inset, the red and blue trajectories evolve into a right-\nhanded limit cycle, whereas the black trajectory evolves into\nthe point of lowest energy.\nlar to the current induced orbital magnetization [37, 38].\nThe intrinsic analog, the electric-\feld induced nondissi-\npative DMI [1, 39], remains elusive in the band picture,\nbut should be well de\fned in insulators as we show now.\nTo \frst order in the gradient, there is an adiabatic\ncurrent pumped by the magnetization dynamics [23] j=\neR\n[dk] \nk[kr]m\u0001_minvolving the second Chern form of\nBerry curvatures \n ks[kr]mj\u0011\nkski\nrimj+\nksri\nmjki+\n\nksmj\nkiri. Through this current, an external electric\n\feld can produce a work density \u000ew=E\u0001jdtpropor-\ntional of\u000em, implying an electrical generalized force lin-\near in the magnetization gradient\nEe\nm=eE\u0001Z\n[dk] \nk[kr]m: (8)\nFor reasons to be discussed later, we will call this an\nelectrical DMI force, although it is nonconservative in\ngeneral and capable of sustained driving of magnetization\ntextures.\nBecause the second Chern form is antisymmetric in the\ncrystal momentum, a nonzero result demands that the\nelectronic system is more than one dimensional. Consider\nfor simplicity a two-dimensional system with the mag-\nnetization gradient in the ydirection (one-dimensional\ndomain wall or a spiral) and an electric \feld applied in\nthe transverse xdirection. The electrical work per unit\ntransverse width over one pumping period may be writ-\nten as\nW=eExNytZ\nT2d2k\n2\u0019Z\nS2d\u0012d\u001e\n2\u0019\nkxky\u0012\u001e=eExNytC2\n(9)\nwhich is topological and quantized in terms of the sec-\nond Chern number C2in the space [40] spanned by the5\nBrillouin zone and the Bloch sphere, and the winding\nnumberNyt=1\n4\u0019R\ndydt^m\u0001(@y^m\u0002@t^m) for the map-\nping ^m(y;t) of theytspace-time onto the Bloch sphere\n[41] ( ^m=m=m). This winding number has previously\nappeared in discussion of quantized electromotive force\ninduced by a moving domain wall [42], the so called ferro-\nJosephson e\u000bect, and the second Chern number may be\nregarded as the quantum measure of the anomalous Hall\nresponse to this emf [43]. The quantized electrical work\nis therefore a result of this quantized Hall current in the\ndirection of the applied electric \feld.\nThe same second Chern number has also been intro-\nduced in study of electric charges carried by magnetic\ntextures such as a skyrmion [30], where it may be un-\nderstood as the quantum measure of charge response to\nthe quantized \rux of arti\fcial magnetic \feld [43]. This is\na sort of Streda dual e\u000bect of the quantum Hall current\nresponse to the arti\fcial electric \feld of the magnetic tex-\nture. This relationship becomes especially clear in the ab-\nsence of spin-orbit coupling, where \n kxky\u0012\u001e= \nkxky\n\u0012\u001e\nandC2reduces to the \frst Chern number in k-space [44]\nwhich characterizes the usual quantum anomalous Hall\ninsulators.\nIn non-Chern insulators where one may choose a pe-\nriodic gauge in k-space, the electrical generalized force\nmay be written as a \feld derivative of the polarization\nenergy, Ee\nm=\u0000\u000emU, with [30, 31]\nU=\u0000Z\ndrE\u0001P=Z\ndrDil@iml; (10)\nwherePis the electric polarization induced by magne-\ntization gradient, including a topological Chern-Simons\npart [23] for which\nDil=e\n2EjZ\n[dk] (Akj\nkiml+Aki\nmlkj+Aml\nkjki):\n(11)\nHowever, this expression for the DMI coe\u000ecient is only\nlocally de\fned because of the gauge dependence of the\nChern-Simons form [45].\nOn the other hand, the electrical DMI force Ee\nm[Eq.\n(8)] is not only gauge invariant and single valued but also\nwell de\fned for Chern insulators. In practice, such a force\nenters directly in determining the static and dynamic be-\nhavior of the magnetic textures. For example, we show\nin the following that the width of a chiral Neel wall may\nbe tuned by such a force, as would normally be antici-\npated from DMI e\u000bects [34, 35]. Speci\fcally, we consider\na chiral Neel wall with easy axis in the zdirection in a\nmodel of the insulating transition metal dichalcogenide\nmonolayer materials with magnetic proximity e\u000bect, and\nshow that its width would be enhanced (decreased) when\nan electric \feld is applied in the x(\u0000x) direction. We\nemploy the model Hamiltonian ^h=^h0+J^\u001b\u0001m, where\n^h0is a six-band tight-binding Hamiltonian suitable forthe low-energy physics in monolayers of AB 2(A = Mo,\nW; B = S, Se, Te), as was detailed in Ref. [46]. Consider\na right-handed up-down Neel-type wall with easy axis in\nFIG. 4. Spin generation due to the electrical generalized force\nin a model of chiral Neel wall of ferromagnetic transition metal\ndichalcogenide monolayer.\nthezdirection, the induced spin is plotted in Fig. 4 un-\nder an electric \feld in the positive xdirection. With the\nlowest two bands \flled, the \frst Chern form contribution\nvanishes. The dominant component, \u000esx, is antisymmet-\nric on the two sides of the domain wall center. Thus the\ntorque exerted on magnetization \u000e\u001c=\u000es\u0002mis in the\npositiveydirection on both sides, increasing the width\nof the domain wall. Apparently, when the electric \feld\nis reversed, the width of the domain wall is decreased.\nV. CONCLUSION\nIn conclusion, we have studied nondissipative electric\ndriving of magnetization motion in uniform and nonuni-\nform magnetic insulators due to nontrivial topologies of\noccupied Bloch states in the combined space of crystal\nmomentum and magnetization con\fguration. The resul-\ntant nonconservative electrical generalized force is capa-\nble of supporting sustained magnetization motion even in\nthe presence of Gilbert damping. A minimal model has\nbeen exploited to show explicitly the limit-cycle behav-\nior of magnetic evolution. For magnetic textures, there is\nan additional nonconservative and nondissipative electri-\ncal generalized force, related to a Chern-Simons DMI for\nnon-Chern insulators in the presence of an electric \feld.\nACKNOWLEDGMENTS\nWe thank Hua Chen, Peng Yan, Yunshan Cao, Liang\nDong and Tianlei Chai for useful discussions. This work\nis supported by NSF (EFMA-1641101) and Welch Foun-\ndation (F-1255).6\n[1] H. Katsura, N. Nagaosa, and A. V. Balatsky, Phys. Rev.\nLett.95, 057205 (2005).\n[2] M. Mostovoy, Phys. Rev. Lett. 96, 067601 (2006).\n[3] Y. Yamasaki, H. Sagayama, T. Goto, M. Matsuura, K.\nHirota, T. Arima, and Y. Tokura, Phys. Rev. Lett. 98,\n147204 (2007).\n[4] I. \u0014Zuti\u0013 c, J. Fabian, and S. Das Sarma, Rev. 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B 84, 245123\n(2011).\n[45] Under a gauge transformation of the wave functions, the\nBerry connections acquire additional terms given by the\ngradients of a phase \feld in the combined kandmspace.\nThis phase is modulo 2 \u0019in thekspace because of the pe-\nriodic gauge, and the kintegral of these additional terms\nreduces to a partial derivative of some scalar \feld in the\nmspace. Ordinarily, such a derivative term for the DMI\ncoe\u000ecient is permissible, because that would result in\nan immaterial total gradient in the DMI energy density.\nWhen singularities occur as discussed in the example of\nthe previous section, this scalar \feld can be multi-valued\nover multiply connected regular regions, rendering the\nDMI energy also multi-valued.\n[46] G. B. Liu, W. Y. Shan, Y. Yao, W. Yao, and D. Xiao,\nPhys. Rev. B 88, 085433 (2013)."    },    {        "title": "1912.00310v3.Coarse_graining_in_micromagnetic_simulations_of_dynamic_hysteresis_loops.pdf",        "content": "Coarse-graining in micromagnetic simulations of\ndynamic hysteresis loops\nR Behbahani1;2, M L Plumer1and I Saika-Voivod1;2\n1 Department of Physics and Physical Oceanography, Memorial University of\nNewfoundland, Canada\n2 Department of Applied Mathematics, University of Western Ontario, London,\nOntario, Canada, N6A 3K7\nE-mail: saika@mun.ca\nAbstract. We use micromagnetic simulations based on the stochastic Landau-\nLifshitz-Gilbert equation to calculate dynamic magnetic hysteresis loops at \fnite\ntemperature that are invariant with simulation cell size. As a test case, we simulate\na magnetite nanorod, the building block of magnetic nanoparticles that have been\nemployed in preclinical studies of hyperthermia. With the goal to e\u000bectively simulate\nloops for large iron-oxide-based systems at relatively slow sweep rates on the order\nof 1 Oe/ns or less, we modify and employ a previously derived renormalization group\napproach for coarse-graining (Grinstein and Koch, Phys. Rev. Lett. 20, 207201, 2003).\nThe scaling algorithm is shown to produce nearly identical loops over several decades\nin the model cell volume. We also demonstrate sweep-rate scaling involving the Gilbert\ndamping parameter that allows orders of magnitude speed-up of the loop calculations.\nKeywords : Landau-Lifshitz-Gilbert equation, micromagnetics, coarse-graining, mag-\nnetic hyperthermia, nanorods\nThe fundamental premise of micromagnetics is that the physics of interest can\nbe modeled by a macrospin representing a collection of atomic spins within a small\n\fnite volume, or cell. The approximation that all spins within a cell point in the same\ndirection is valid at temperature T= 0, so long as cells remain smaller than the exchange\nlength [1]. A limiting factor for micromagnetic computer simulations is the number of\ncells used to model the system; using larger cells is computationally advantageous.\nAt \fniteT, a few schemes have been proposed to account for how parameters\nused for modelling the magnetic properties of the material must vary with cell size\nin order to keep system properties invariant with cell size. For example, Kirschner\net al. [2, 3] suggested an approximate scaling of saturation magnetization Msbased\non the average magnetization of blocks of spins in atomistic Monte Carlo simulations,\nand subsequently scaling the exchange and uniaxial anisotropy constants AandKto\npreserve the exchange length and anisotropy \feld. Feng and Visscher [4] proposed that\nthe damping parameter \u000b, which models the dynamics of magnetic energy loss [5], should\nscale with cell size, arguing that using larger cells is analogous to having more degrees of\nfreedom for energy absorption; see also [6] for e\u000borts related to \u000b. The renormalizationarXiv:1912.00310v3  [cond-mat.mtrl-sci]  8 Nov 2021Coarse-graining in micromagnetic simulations of dynamic hysteresis loops 2\ngroup (RG) approach of Grinstein and Koch [7], based on mapping a Fourier space\nanalysis of the non-linear sigma model to ferromagnets in order to scale A,K, \feldH\nand magnetization M, has garnered signi\fcant attention. However, to the best of our\nknowledge, no scaling theory has been applied to the calculation of magnetization-\feld\n(MH) hysteresis loops [8], which are the foundation of experimental characterization of\nmagnetic systems.\nIn this Letter, we modify and employ the approach proposed by Grinstein and\nKoch [7] to the test case of calculating MH loops for magnetite nanorods at sweep rates\nrelevant to magnetic hyperthermia, allowing us to make estimates of speci\fc loss power\nthat would otherwise be computationally impractical.\nThe magnetite nanorods we simulate are the building-blocks of the nanoparticles\nthat were shown by Dennis et al to successfully treat cancerous tumours in mice via\nhyperthermia [9]. It is reasonable to choose the smallest micromagnetic cell to be\nthe cubic unit cell, which is of length a0= 0:839 nm and contains 24 magnetic\nFe ions. We set the exchange sti\u000bness constant to A0= 0:98\u000210\u000011J/m, which\nfor cell length a0yields an e\u000bective exchange constant between neighbouring cells of\nJe\u000b=a0A0= 8:222\u000210\u000021J, which in turn yields a bulk critical temperature of\nTc= 1:44Je\u000b=kB= 858 K for the bulk 3D-Heisenberg-model version of our system.\nThis value of A0is close to what can be theoretically determined by considering the\natomic-level exchange interactions across the faces of neighbouring unit cells [10], and\nis in reasonable agreement with experimental values [11, 12, 13, 14, 15, 16, 17]. The\nnanorod dimensions are approximately 6.7 nm \u000220 nm\u000247 nm (8a0\u000224a0\u000256a0),\nwith its length along the z-axis. We set Ms= 480 kA/m [11, 18, 19], the bulk value\nfor magnetite. We do not consider magnetostatic interactions explicitly, but rather\nimplicitly through an e\u000bective uniaxial anisotropy. For the purposes of this study, we\nchoose a strength of K0= 10 kJ/m3, which is consistent with other studies of iron\noxide nanoparticles [20, 21], and for which a more precise estimate can be obtained by\nconsidering the nanorod's demagnetization tensor [19, 22, 23, 24, 25, 26], maghemite\ncontent [9], and the e\u000bect of neighbouring nanorods within a nanoparticle. We\nomit cubic crystalline anisotropy as it has negligible e\u000bects on the hysteresis loops of\nmagnetite nanoparticles with even modest aspect ratios, as discussed in Refs. [19, 26]\n(we have also veri\fed that adding cubic anisotropy of strength 10 kJ/m3has no impact\non the loops presented here). Anisotropy is set along the z-axis with a 5\u000edispersion to\nmimic lattice disorder [21]. For convenience we set \u000b= 0:1, a choice consistent with\nprevious studies [21, 27] and with magnetite thin \flms [28].\nWhile hysteretic heating is at the heart of magnetic nanoparticle hyperthermia,\npreventing eddy current heating of healthy tissue limits the frequency fand amplitude\nHmaxof the external \feld such that the sweep rate SR = 4 Hmaxfis less than a target\nvalue of 0:25 Oe/ns [29, 18]. For our simulation, we set Hmax= 500 Oe, which for\nthe target SR implies a target value of f= 125 kHz, a value large enough to restrict\nunwanted Brownian relaxation [18].\nTo model the dynamics of the magnetization of a cell Mof \fxed magnitude Ms,Coarse-graining in micromagnetic simulations of dynamic hysteresis loops 3\n𝑎\"𝝁𝑎\"\nNanoparticle\nSingleBlock𝑎$=2𝑎\"(b=2)1344 Cells 𝑎=𝑎\"(b=1) 10752 cells𝑎'=4𝑎\"(b=4)168 Cells 𝑎)=8𝑎\"(b=8)21 Cells \nFigure 1. Coarse-grained modelling of a magnetite nanorod. The smallest\nmicromagnetic cell models the atomic spins within a cubic unit cell of length a0=\n0:839 nm with a single magnetic moment. Our goal is to model the system using\na smaller number of larger cells (of length ab=ba0forb > 1) with appropriately\nscaled parameters. The number of cells drawn and their sizes are only approximate.\nIllustrative spins for half of the tetrahedral Fe3+sites (FCC sites) are drawn over a\nspinel unit cell taken from Ref. [30].\nwe solve the Landau-Lifshitz-Gilbert (LLG) equation [22, 5, 31],\ndM\ndt=\u0000\r1M\u0002He\u000b\u0000\u000b\r1\nMsM\u0002(M\u0002He\u000b) (1)\nwheretis time,\r1=\u00160\re=(1 +\u000b2),\re= 1:76\u00021011rad/(s.T) is the gyromagnetic\nratio for an electron, \u00160is the vacuum permeability, and He\u000bis due to the combination\nof an external \feld, uniaxial anisotropy, exchange interactions and a thermal \feld. We\nperform our simulations using OOMMF (Object Oriented Micromagnetic Framework)\nsoftware [32]. In particular, we include the Theta Evolve module [33] used for simulations\nat \fniteTvia a stochastic thermal \feld [31].\nWe simulate the rod using cubic cells of length ba0, withbtaking on values 1,\n2, 4 and 8. See Fig. 1. For b= 1, 10752 cells make up the rod. For b= 2,\nthere are 10752 =23= 1344 cells. The volume of the rod is \fxed for all simulations\nat 10752a3\n0\u0019(22a0)3. Additionally, we simulate the rod as a single cell { a single\nrectangular prism, or block. While there is some ambiguity in assigning a single length\nscale to represent a rectangular prism, we choose b= 22 from the geometrical mean,\ni.e., the side length of the cube of the same volume as the rod.Coarse-graining in micromagnetic simulations of dynamic hysteresis loops 4\nThe goal of coarse-graining is to determine A(b) andK(b), i.e., how the exchange\nand anisotropy parameters should change with bto keep system properties invariant with\nb. Theb= 22 case is a practical limit where all the atomic spins are represented by a\nsingle macrospin, where exchange interactions are no longer required in the simulations,\nand which provides for an interesting test of a coarse-graining procedure in predicting\nK(b). In calculating hysteresis loops for a system with cell length ba0, we apply an\nexternal \feld along the zaxis ofH(b) =Hmaxsin (2\u0019ft), and report the z-component\nof the average (over cells) magnetization unit vector mH=\u0016Mz=Ms, averaged over 88 to\n100 independent simulations for b>1. Forb= 1 we use 250 simulations.\nIn Fig. 2a we plot hysteresis loops at T= 310 K using di\u000berent cell sizes (varying\nb) while keeping the exchange and anisotropy parameters \fxed at A0andK0. A value\nof SR = 2:5 Oe/ns is chosen to make the simulations computationally feasible at b= 1.\nBoth the coercivity Hcand the remanence increase with increasing b. The increasing\nloop area is consistent with the stronger exchange coupling ( Je\u000b=ba0A0) between\nmagnetization vectors of adjacent cells. For b\u00154, it appears that the exchange is\nstrong enough for the system to be nearly uniformly magnetized, and so Hcremains\nlargely unchanged for b\u00154 sinceKis constant. This means that for b= 1, at this T\nand for our rod size, exchange is not strong enough to be able to treat the nanorod as\na single macrospin in a trivial way. Clearly, varying cell size changes the loops and a\ncoarse-graining procedure is required.\nIn their coarse-graining procedure, Grinstein and Koch introduced a reduced\ntemperature T\u0003, which for a three dimensional system is given by,\nT\u0003=kBT\u0003\nA: (2)\nwhere \u0003 = 2 \u0019=ba 0is a high wave-number cut-o\u000b that re\rects the level of coarse-graining.\nSimilarly, the reduced parameters for \feld and anisotropy constants are de\fned as,\nh=\u00160MsH\nA\u000321000\n4\u0019; g =K\nA\u00032; (3)\nwithHgiven in Oe. Introducing the parameter l= ln(b), they gave the following set of\nequations for calculating the reduced parameters as functions of cell size,\ndT\u0003(l)\ndl= [\u00001 +F(T\u0003(l);h(l);g(l))]T\u0003(l)\ndh(l)\ndl= 2h(l)\ndg(l)\ndl= [2\u00002F(T\u0003(l);h(l);g(l))]g(l)(4)\nwhere\nF(T\u0003;h;g) =T\u0003\n2\u0019(1 +h+g): (5)\nAdditionally, the magnetization of the coarse-grained system is scaled via,\nM(T\u0003;h) =\u0010(l)\u0002M(T\u0003(l);h(l)) (6)Coarse-graining in micromagnetic simulations of dynamic hysteresis loops 5\nwhere\n\u0010(l) =e\u0000Rl\n0F(T\u0003(l0);h(l0);g(l0))dl0: (7)\nFor our system parameters and range of H, bothg\u001c1 andh\u001c1, and so\nF'T=2\u0019, which makes the numerical solution of Eq. 4 practically indistinguishable\nfrom the approximate analytic solution, which we \fnd to be,\nA(b) =\u0010(b)\u0002A0 (8)\nK(b) =\u0010(b)3\u0002K0 (9)\nH(b) =\u0010(b)\u0002H0 (10)\nM0=\u0010(b)\u0002M(b) (11)\nwheret=T=Tcand\u0010(b) =t=b+ 1\u0000t. AtT= 310 K,t= 0:3613,\u0010(2) = 0:8193,\n\u0010(4) = 0:7290,\u0010(8) = 0:6839, and\u0010(22) = 0:6551.\nEqs. 8 and 9 provide a prescription for changing material parameters with b, while\nEqs. 10 and 11 provide the prescription for scaling HandMafter a loop calculation.\nHowever, we \fnd that the prescription does not yield loops that are invariant with b,\non account of Eq. 11; the correction of the coarse-grained values of Mback to those\ncorresponding to the unscaled system is too large (the corrected remanance is too small),\nas we show in Fig. 2b. In Fig. 2c, we apply a correction to Eq. 11 and obtain good\nagreement between the reference ( b= 1) and coarse-grained ( b>1) loops.\nTo motivate our correction to the rescaling of the magnetization, we begin by\nnoting that the same value of T\u0003in Eq. 2 can be achieved by either having a rescaled\ntemperature T(b) or having a rescaled A(b). Combining this idea with Eq. 8 yields,\nT(b) =T0\nb\u0010(b;T0); (12)\nwhich together with Eq. 11 [after solving for M(b)] predicts an overly simple dependence\nofMonT, parametrically through b: a line passing through M0andT0atb= 1 and\nthroughM= 0 andT=Tcasb!0.\nTo obtain a model that better matches the data, we introduce a phenomenolgical\ncorrection to Eq. 11, one in which M0is a weighted average of M(b) and the RG\nexpression for M0,\nM0=\u000e\u0010(b;T0)M(b) + (1\u0000\u000e)M(b): (13)\nWe use\u000eas a free parameter to \ft the M(T) data for the nanorod. This yields a value\nof\u000e= 0:511, which we use in rescaling mHin Fig. 2c. The \ft reasonably recovers M(T)\nin theTrange corresponding to values of bbetween 1 and 22, as shown in Fig. 3.\nThe collapse of the data in Fig. 2c is remarkable, with the biggest discrepancy\narising between b= 1, corresponding to the most \fne-grained simulation, and b= 2, the\n\frst step in coarse-graining. The di\u000berence lies most noticeably in the shoulder region\nwhere magnetization begins to change, where the microscopic details likely matter most.\nLoss of some detail is expected with coarse-graining and consistent with previous studies\ninvolving atomic-level magnetization switching in a grain [34]. The magnetization inCoarse-graining in micromagnetic simulations of dynamic hysteresis loops 6\n400\n 200\n 0 200 400\nH(b) (Oe)1.00\n0.75\n0.50\n0.25\n0.000.250.500.751.00\nmH(a)\nb=1\nb=2\nb=4\nb=8\nblock\n400\n 200\n 0 200 400\nH(b)/ (Oe)\n1.00\n0.75\n0.50\n0.25\n0.000.250.500.751.00×mH\n(b)\nb=1\nb=2\nb=4\nb=8\nblock\n400\n 200\n 0 200 400\nH(b)/ (Oe)\n1.00\n0.75\n0.50\n0.25\n0.000.250.500.751.00((1 )+)×mH\n(c)\nb=1\nb=2\nb=4\nb=8\nblock\nFigure 2. Application of RG coarse graining to nanorod MH loops at T= 310 K\nand SR= 2 :5 Oe/ns. (a) Changing cell length ( a=ba0) without changing magnetic\nparameters. (b) AandKare scaled according to Eqs. 8 and 9, respectively, and\nmHandHare scaled according to Eqs. 11 and 10, respectively. (c) As in panel (b),\nexceptmHis scaled according to Eq. 13 with \u000e=0.511. \u0001t= 1 fs for all simulations.\nHorizontal error bars shown for Hcrepresent one standard error and are vertically\ndisplaced to avoid overlap. Uncertainty in Hcis approximately 7 to 13%.\nthe shoulder areas appears to diminish with increasing b. The behavior of b= 22 runs\ncounter to this trend, but at this level of coarse-graining, there is only a single cell. It is\nsigni\fcant, however, that scaling seems to hold even in this limit. (We note that in this\nlimit, even though there are no exchange interactions in the simulations, the value of\nthe e\u000bective anisotropy still depends on exchange through the dependence of TconA0.)\nThe loop areas for b= 1, 2 , 4, 8 and 22 are 495, 488, 443, 432 and 472 Oe, respectively.\nThe smallest loop area (for b= 8) is 13% smaller than the area for b= 1.\nWe note that the unrenormalized exchange length for our simulated material is\nlex;0=q\n2A0\n\u00160M2s= 8:23 nm, which is longer than a8= 6:712 nm, and so only our b= 22\nsingle block simulations scale the cell size beyond lex;0. Under renormalization, however,\nthe exchange length becomes lex;b=q\n2\u0010(b)A0\n\u00160M2s, which decreases with increasing b, and\ntakes on values 7.45, 7.02, 6.80 and 6.66 nm for b= 2;4;8;and 22, respectively. Thus\nforb= 8, the cell length and the exchange length are approximately the same.\nWe now turn our attention to speeding up simulations by considering the\nrelationship between SR and \u000b. A larger value of \u000bsigni\fes a faster loss of energy\nand a shorter relaxation time for alignment of the magnetic moments to the \feld, and\nresults in a smaller hysteresis loop. Likewise, a slower SR is equivalent to a longer\nmeasurement time and consequently a smaller hysteresis loop. To build on these ideas,\nwe recall Sharrock's equation for Hcas a function of T[35],\nHc=HK\"\n1\u0000s\nkBT\nKVln\u0012f0\u001c\nln 2\u0013#\n: (14)\nSharrock derived this equation by calculating the time required for half of the\nmagnetization vectors in the system, which are initially anti-aligned with the \feld,\nto overcome an energy barrier that grows with KV and align with a \feld of strength\nHc. In this context, \u001cis the relaxation time. In the context of hysteresis loops, Hc\nis the \feld required to \rip half of the magentization vectors in an observation time \u001c,Coarse-graining in micromagnetic simulations of dynamic hysteresis loops 7\n0.0 0.2 0.4 0.6 0.8 1.0\nT/Tc0.00.20.40.60.81.0m\nb=1b=2b=4b=8b=22Simulations\nm(b), =0.511\nFigure 3. Determining a scaling function for M(b) from the Tdependence of the\nnanorod magnetization. \u000eis used as a \ftting parameter to match nanorod data,\nyielding a value of 0.511. Vertical dot-dash lines indicate reduced temperatures\ncorresponding to di\u000berent values of b.\nwhich is related to SR via \u001c/1=SR.f0is the so-called attempt frequency, for which\nBrown [31, 36, 37, 38, 39] derived an expression in the high-barrier limit. At small \u000b,\nf0/\u000b, and so the product f0\u001c/\u000b=SR, implying that so long as SR =\u000b= constant, Hc\nshould remain the same.\nIn Fig. 4 we show loops calculated for SR =\u000b= 2:5 (Hmax = 500 Oe, and\nf= 125 kHz), the ratio obtained using a clinically relevant SR = 0 :25 Oe/ns and\nthe estimate of \u000b= 0:1. Data for b= 4 and 8 and for various SR- \u000bpairs show good\nagreement. At 0 :25 Oe/ns, simulations using b= 1 are prohibitively long, taking several\nmonths on available computing resources. The results shown here combine the RG\napproach to reduce the number of cells, the ability to use a larger time step \u0001 tfor\nlarger cells in solving the LLG equation [6], and the SR =\u000bscaling to employ a faster\nSR, all to dramatically reduce simulation time { by a factor of 43to 83for reducing\nthe number of cells, a factor of at least 5 for the time step, and a factor of up to 1000\nwhen using the fastest SR. The average area of the \fve loops for b= 4 in Fig. 4 is\nS= 171:3\u00062:8 Oe, translating to a speci\fc loss power of f\u001601000\n4\u0019MsS=\u001a= 207 W/g\n\u000610% (using \u001a= 5:17 g/cm3), which is consistent with clinical expectations [40]. The\nloop area for b= 8 is 13% lower at 149 :4 Oe.\nIn summary, we show that our modi\fcation to the RG approach of Grinstein and\nKoch [7] yields a scaling of exchange and anisotropy parameters and \fnite temperature\nnanorod hysteresis loops that are, to approximately 10-15%, invariant with cell size. WeCoarse-graining in micromagnetic simulations of dynamic hysteresis loops 8\n400\n 200\n 0 200 400\nH(b)/ (Oe)\n1.00\n0.75\n0.50\n0.25\n0.000.250.500.751.00((1 )+)×mH\nSR=0.25,=0.1,b=8\nSR=0.25,=0.1\nSR=2.5  ,=1\nSR=25   ,=10\nSR=50   ,=20\nSR=250 ,=100\nFigure 4. Invariance of MH loops. We combine RG scaling of magnetic quantities,\nlarger time step with block size, and SR =\u000bscaling to predict the behaviour of\nprohibitively long \fne-grain ( b= 1) simulations. b= 4 unless otherwise noted.\nnote that the coarse-graining of magnetostatic interactions is beyond the framework of\nRef. [7]. We are currently investigating magnetostatic scaling, and intend to report on\nit in future work.\nScaling results hold even to the point where the nanorod is represented by a single\nmagnetization vector that experiences anisotropy only. Whether this limit holds for\nsystems with weaker exchange remains to be studied. This reduction to an e\u000bective\nStoner-Wohlfarth (SW) model [41] should facilitate comparison with experiments on\nnanorods, since an analytic solution to the SW model at \fnite Tand SR exists [27]. It\nshould also simplify computational studies of nanoparticles (nanorod composites) and\ncollections of nanoparticles used in a wide variety of applications and hence facilitate\ncomparison with experimental MH loops and quanti\fcation of system properties through\nsimulations.\nIn addition to the computational speedup resulting from the use of fewer\nmicromagnetic cells, the invariance of loops when SR =\u000bis \fxed provides another avenue\nfor computational speedup by allowing one to use a larger SR than the target value.\nWe caution, however, that the theoretical motivation for this invariance stems from\nconsidering the Sharrock equation for only small \u000b. While both SR and \u000bset time\nscales, we have not provided any reasoning for why the invariance should hold as well\nas it does for larger \u000b.\nThe data that support the \fndings of this study are available from theCoarse-graining in micromagnetic simulations of dynamic hysteresis loops 9\ncorresponding author upon reasonable request.\nAcknowledgments\nWe thank Johan van Lierop, Rachel Nickel and Mikko Karttunen for enlightening\ndiscussions, and Martin D. Leblanc for guidance in using OOMMF. R.B. and I.S.-V.\nthank Mikko Karttunen and Styliani Consta for hosting our stay at Western University.\nWe acknowledge the \fnancial support from the Natural Sciences and Engineering\nResearch Council (Canada). Computational resources were provided by ACENET and\nCompute Canada.\nReferences\n[1] Abo G S, Hong Y K, Park J, Lee J, Lee W and Choi B C 2013 IEEE Trans. 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Rev. 1301677\n[32] Donahue M J and Porter D G 1999 OOMMF User's Guide, Version 1.0, Interagency Report\nNISTIR 6376 National Institute of Standards and Technology Gaithersburg, MD URL https:\n//math.nist.gov/oommf/\n[33] Lemcke O 2004 ThetaEvolve for OOMMF releases: 1.2a3, see\nhttps://math.nist.gov/oommf/contrib/oxsext/oxsext.html URL http://www.nanoscience.\nde/group_r/stm-spstm/projects/temperature/download.shtml\n[34] Mercer J, Plumer M, Whitehead J and Van Ek J 2011 Appl. Phys. Lett. 98192508\n[35] Sharrock M and McKinney J 1981 IEEE Trans. Magn. 173020{3022 ISSN 0018-9464\n[36] Garc\u0013 \u0010a-Palacios J L and L\u0013 azaro F J 1998 Phys. Rev. B 5814937\n[37] Breth L, Suess D, Vogler C, Bergmair B, Fuger M, Heer R and Brueckl H 2012 J. Appl. Phys. 112\n023903\n[38] Taniguchi T and Imamura H 2012 Phys. Rev. B 85184403\n[39] Leliaert J, Vansteenkiste A, Coene A, Dupr\u0013 e L and Van Waeyenberge B 2015 Med. Biol. Eng. 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A 240599{642"    },    {        "title": "2103.05871v3.Anisotropic_superconducting_spin_transport_at_magnetic_interfaces.pdf",        "content": "Anisotropic superconducting spin transport at magnetic interfaces\nYuya Ominato1, Ai Yamakage2, and Mamoru Matsuo1;3;4;5\n1Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China\n2Department of Physics, Nagoya University, Nagoya 464-8602, Japan\n3CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n4Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan and\n5RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n(Dated: October 18, 2022)\nWe present a theoretical investigation of anisotropic superconducting spin transport at a magnetic\ninterface between a p-wave superconductor and a ferromagnetic insulator. Our formulation describes\nthe ferromagnetic resonance modulations due to spin current generation depending on spin-triplet\nCooper pair, including the frequency shift and enhanced Gilbert damping, in a uni\fed manner. We\n\fnd that the Cooper pair symmetry is detectable from the qualitative behavior of the ferromagnetic\nresonance modulation. Our theory paves the way toward anisotropic superconducting spintronics.\nIntroduction.| Use of spin-triplet Cooper pairs as car-\nriers for spin currents in the emergent \feld of super-\nconducting spintronics is challenging1,2. Previous stud-\nies have demonstrated spin transport mediated by spin-\ntriplet Cooper pairs that formed at the s-wave supercon-\nductor (SC)/ferromagnet interfaces of Josephson junc-\ntions. The spin-singlet pairs in SCs are converted into\nspin-triplet pairs in half-metallic CrO 23. However, pre-\nvious studies on spin-triplet pairs at magnetic interfaces\nhave been limited to cases induced by the proximity ef-\nfect.\nOne promising candidate material system for investi-\ngation of spin-triplet currents to enable more active use\nof spin-triplet pairs is the p-wave SC/ferromagnetic in-\nsulator (FI) bilayer thin \flm system4,5. Tunneling of the\nspins is driven by the magnetization dynamics excited\nby ferromagnetic resonance (FMR) in the ferromagnetic\nmaterial via interfacial exchange coupling between the\nmagnetization in the FI and the electron spins in the\np-wave SC, and a spin-triplet current is expected to be\ngenerated. Furthermore, as a backaction of spin injec-\ntion, both the FMR frequency and the Gilbert damping\nof the FI should be modulated6{8. Although similar sce-\nnarios have already been studied vigorously in s-wave\nSC/ferromagnet systems, most previous studies have fo-\ncused on the Gilbert damping modulation due to spin\ninjection9{22. To gain an in-depth understanding of the\nspin-triplet transport mechanisms, the FMR modulation\nprocesses, including both the frequency shift and the en-\nhanced Gilbert damping, should be formulated micro-\nscopically in a systematic manner.\nDetermination of the pairing symmetry of the spin-\ntripletp-wave SCs within the same framework is also\ndesirable. Despite many years of research based on sev-\neral experimental techniques that detect the pairing sym-\nmetry, including nuclear magnetic resonance23, polar-\nized neutron scattering24{26, and muon-spin resonance\ntechniques27, there are few established candidate systems\nfor spin-triplet SCs28{32. The FMR modulation has been\nobserved in various nanoscale magnetic multilayers. Ac-\ncordingly, the technique is widely used to investigate a\n(c) FMR modulation due to the coupling between\n     spin-triplet Cooper pair and magnetization\nH\nH0H0+ δH\n(b) Spin-triplet Cooper pair\n(i) Chiral p-wave (ii) Helical p-wave\nα+ δα\nH\nH0αH0H0+ δHFISC\nFIxz\nY, yθ(a) System\nθZ\nXS-HFISC\nFIG. 1. Mechanism of FMR modulation due to anisotropic\nsuperconducting spin transport at magnetic interfaces. (a)\nPrecession axis located on the x-zplane, where the angle\nbetween the precession axis and the zaxis is\u0012(where 0\u0014\u0012\u0014\n\u0019=2). (b) Two types of spin-triplet Cooper pairs considered\nin this work. (c) FMR signal modulation in the SC/FI bilayer\nsystem compared with the signal in the FI monolayer.\nspin transport property in a variety of nanoscale thin\n\flm systems because it is highly sensitive. Thus one can\nexpect that the FMR measurements in p-wave SC/FI bi-\nlayer systems provide useful information about pairing\nsymmetry.\nIn this Letter, we investigate anisotropic superconduct-\ning spin transport at the magnetic interfaces of hybrid\nsystems composed of p-wave SC/FI thin \flms theoret-\nically, as illustrated in Fig. 1(a). The two-dimensional\nbulk SC is placed on the FI, where the FMR occurs. The\nprecession axis is rotated by an angle \u0012from the direc-\ntion perpendicular to the interface. Here, we use two\ncoordinate systems: ( x;y;z ) and (X;Y;Z ). Thezaxis is\nperpendicular to the interface and the xandyaxes arearXiv:2103.05871v3  [cond-mat.supr-con]  15 Oct 20222\nalong the interface. The ( X;Y;Z ) coordinate is obtained\nby rotating the angle \u0012around the yaxis, so that the\nprecession axis and the Zaxis are parallel. Figure 1(b)\nshows a schematic image of the spin-triplet Cooper pairs\nfor the chiral and helical p-wave SCs considered in this\nwork. Figure 1(c) shows a schematic image of the FMR\nsignal in the FI monolayer and the SC/FI bilayer. The\nFMR frequency and linewidth in the SC/FI bilayer are\nboth modulated because of the spin transfer occurring at\nthe interface.\nUsing the nonequilibrium Green's function method,\nwe formulate the FMR modulations due to the back\naction of the spin-triplet transport process systemati-\ncally. The main advantage of using the nonequilibrium\nGreen's function is dealing with both a spectral function\nand a nonequilibrium distribution function. Indeed, the\ninterface spin current is given by the expression using\nthe nonequilibrium distribution function, which shows\nthat the interface spin current by the spin pumping and\nthe enhanced Gilbert damping are proportional to each\nother. Furthermore, as an advantage of \feld theoretical\ntreatment, the frequency shift and the enhanced Gilbert\ndamping are both described in a uni\fed manner. Addi-\ntionally, it is shown that the symmetry of the spin-triplet\npairs can be extracted from the FMR modulations. The\nresults presented here o\u000ber a pathway toward develop-\nment of anisotropic superconducting spintronics.\nModel Hamiltonian.| The FMR modulation due to the\nSC adjacent to the FI is calculated microscopically using\nthe spin tunneling Hamiltonian method9{11,33{38. The\ne\u000bect of the SC on the FI is treated as a perturbation\nand suppression of ferromagnetism with the onset of su-\nperconductivity is assumed to be negligible, which is con-\nsistent with the results of spin pumping experiments in\nmagnetic multilayer thin \flms. The details of the model\nHamiltonians and the formulations are described in the\nSupplemental Material39. In the main text, we focus on\ngiving an overview of the model Hamiltonians and the\nformulations.\nThe total Hamiltonian H(t) comprises three terms\nH(t) =HFI(t) +HSC+Hex: (1)\nThe \frst term HFI(t) describes the bulk FI,\nHFI(t) =X\nk~!kby\nkbk\u0000h+\nac(t)by\nk=0\u0000h\u0000\nac(t)bk=0;(2)\nwhereby\nkandbkdenote the creation and annihilation op-\nerators of magnons with the wave vector k= (kx;ky;kz),\nrespectively. We assume the parabolic dispersion ~!k=\nDk2\u0000~\rH, where\r(<0) is the electron gyromagnetic ra-\ntio. The coupling between the microwave radiation and\nthe magnons is given by h\u0006\nac(t) = ~\rhacp\nSN=2e\u0007i!t,\nwherehacand!are the amplitude and the frequency of\nthe microwave radiation, respectively. Sis the magni-\ntude of the localized spin and Nis the number of sites\nin the FI. Note that the precession axis for the localized\nspin is \fxed along the Zaxis [see Fig. 1(a)].The second term HSCdescribes the two-dimensional\nbulk SCs,\nHSC=1\n2X\nkcy\nkHBdGck; (3)\nwhere we use the four-component notations\ncy\nk= (cy\nk\";cy\nk#;c\u0000k\";c\u0000k#); (4)\nck= (ck\";ck#;cy\n\u0000k\";cy\n\u0000k#)T: (5)\nHere,cy\nksandcksdenote creation and annihilation op-\nerators, respectively, of electrons with the wave vector\nk= (kx;ky) and thezcomponent of the spin s=\";#.\nThe Bogoliubov-de Gennes Hamiltonian HBdGis a 4\u00024\nmatrix given by\nHBdG=\u0012\n\u0018k\u001b0\u0001k\n\u0000\u0001\u0003\n\u0000k\u0000\u0018k\u001b0\u0013\n; (6)\nwhere\u0018krepresents the energy of the electrons as mea-\nsured from their chemical potential, \u001b0is a 2\u00022 unit\nmatrix, and the pairing potential \u0001 kis also a 2\u00022 ma-\ntrix. We consider three pairing potential types, including\nthe spin-singlet s-wave pairing \u0001 k= \u0001i\u001byand two spin-\ntripletp-wave pairings \u0001 k= (dk\u0001\u001b)i\u001by, where their d\nvectors are given by\ndk=(\n\u0001(0;0;ei\u001ek) : Chiral p\u0000wave\n\u0001(\u0000sin\u001ek;cos\u001ek;0) : Helical p\u0000wave(7)\nwhere\u001ek= arctan(ky=kx) is an azimuth angle. The\nphenomenological form of the gap function is assumed\n\u0001 = 1:76kBTctanh\u0010\n1:74p\nTc=T\u00001\u0011\n; (8)\nwithTcthe superconducting transition temperature. By\ndiagonalizing HBdG, the quasiparticle energy is given by\nEk=p\n\u00182\nk+ \u00012for all SCs considered here. There-\nfore, one cannot distinguish them by the energy spectrum\nalone, and they are simple models suitable for studying\nthe di\u000berence of the magnetic responses due to the pair-\ning symmetry40.\nThe third term Hexrepresents the proximity exchange\ncoupling that occurs at the interface, which describes the\nspin transfer between the SC and the FI10,33,\nHex=X\nq;k\u0000\nJq;k\u001b+\nqS\u0000\nk+ h:c:\u0001\n; (9)\nwhereJq;kis the matrix element for the spin transfer pro-\ncesses,\u001b\u0006\nq= (\u001bX\nq\u0006i\u001bY\nq)=2 represent the spin-\rip opera-\ntors for the electron spins in the SCs, and S\u0000\n\u0000k=p\n2Sby\nk\nandS+\nk=p\n2Sbkrepresent the Fourier component of the\nlocalized spin in the FI. Note that the precession axis is\nalong theZaxis, so that the Zcomponent of the spin\nis injected into the SC when the FMR occurs. Using3\nthe creation and annihilation operators of electrons and\nmagnons,Hexis written as\nHex=X\nq;k;k0;s;s0\u0010p\n2SJq;k\u001b+\nss0cy\nk0sck0+qs0by\n\u0000k+ h:c:\u0011\n:\n(10)\nFrom the above expression, one can see that Hexde-\nscribes electron scattering processes with magnon emis-\nsion and absorption.\nModulation of FMR.| The FMR modulation can be\nread from the retarded component of the magnon Green's\nfunction33, which is given by\nGR\nk(!) =2S=~\n!\u0000!k+i\u000b!\u0000(2S=~)\u0006R\nk(!); (11)\nwhere the Gilbert damping constant \u000bis introduced\nphenomenologically41{43. In the second-order perturba-\ntion calculation with respect to the matrix element Jq;k,\nthe self-energy caused by proximity exchange coupling is\ngiven by\n\u0006R\nk(!) =\u0000X\nqjJq;kj2\u001fR\nq(!); (12)\nwhere the dynamic spin susceptibility of the SCs is de-\n\fned as\n\u001fR\nq(!) :=Z\ndtei(!+i0)ti\n~\u0012(t)h[\u001b+\nq(t);\u001b\u0000\n\u0000q(0)]i:(13)\nThe pole of GR\nk(!) indicates the FMR modulation, i.e.,\nthe shift of resonance frequency and the enhancement of\nthe Gilbert damping. By solving the equation\n!\u0000!k=0\u0000(2S=~)Re\u0006R\nk=0(!) = 0; (14)\nat a \fxed microwave frequency !, one obtains the mag-\nnetic \feld at which the FMR occurs. The imaginary part\nof the self-energy gives the enhancement of the Gilbert\ndamping. Consequently, the frequency shift and the en-\nhanced Gilbert damping are given by\n\u000eH=2S\n\r~Re\u0006R\nk=0(!); \u000e\u000b =\u00002S\n~!Im\u0006R\nk=0(!):(15)\nFrom the above equations and Eq. (12), one can see that\nthe FMR modulation provides information about both\nthe interface coupling properties and the dynamic spin\nsusceptibility of the SCs.\nThe form of matrix element Jq;k=0depends on the\ndetails of the interface. In this work, we assume the\ninterface with uncorrelated roughness. jJq;k=0j2is given\nby\njJq;k=0j2=J2\n1\nN\u000eq;0+J2\n2l2\nNA; (16)\nwhere the \frst and second terms describe averaged uni-\nform contribution and uncorrelated roughness contribu-\ntion, respectively39.J1andJ2correspond to the meanvalue and variance, respectively. Ais the area of the in-\nterface, which is equal to the system size of the SC. lis\nan atomic scale length. Using Eq. (16), the self-energy\nfor the uniform magnon mode is given by\n\u0006R\nk=0(!) =\u0000J2\n1\nN\u001fR\nuni(!)\u0000J2\n2l2\nNA\u001fR\nloc(!); (17)\nwhere the uniform and local spin susceptibilities are de-\n\fned as\n\u001fR\nuni(!) := lim\njqj!0\u001fR\nq(!); \u001fR\nloc(!) :=X\nq\u001fR\nq(!):(18)\nThe self-energy \u0006R\nk=0(!) consists of two terms originating\nfrom the uniform and roughness contributions, so that\nboth\u001fR\nuni(!) and\u001fR\nloc(!) contribute to \u000eHand\u000e\u000b.\nHere, we discuss the FI thickness dependence on the\nFMR modulation44. From Eqs. (15), and (17), one can\nsee that the FMR modulation is inversely proportional\nto the FI thickness ( /A=N ) because\u001fR\nuni(!)/Aand\n\u001fR\nloc(!)/A2. This is consistent with the experiments on\nthe spin pumping in Y 3Fe5O12=Pt heterostructures45. In\norder to observe the FMR modulation experimentally, it\nis necessary to prepare a sample that is su\u000eciently thin,\ne.g., typically, the thickness of several tens of nanometers.\nNumerical results.| In the following, we consider a \rat\ninterface where J2= 0, so that the behavior of the FMR\nmodulation is determined by \u001fR\nuni(!). The roughness\ncontribution proportional to \u001fR\nloc(!) is discussed later.\nFigure 2 shows the frequency shift \u000eHand the enhanced\nGilbert damping \u000e\u000bas a function of temperature and fre-\nquency. Here, we set \u0012= 0 and \u0000=kBTc= 0:05, where \u0000\nis a constant level broadening of the quasiparticle intro-\nduced phenomenologically39.\nFirst, we explain the qualitative properties of \u000eHand\n\u000e\u000bfor the chiral p-wave SC. In the low frequency re-\ngion, where ~!=kBTc\u00141,\u000eHis \fnite and remains al-\nmost independent of !near the zero temperature and\n\u000e\u000bdecreases and becomes exponentially small with the\ndecrease of the temperature. In the high frequency re-\ngion, where ~!=kBTc\u00151, a resonance peak occurs at\n~!= 2\u0001 for both \u000eHand\u000e\u000b. The qualitative proper-\nties of\u000eHand\u000e\u000bfor the helical p-wave SC are the same\nas those of the chiral p-wave SC.\nNext, we explain the qualitative properties of \u000eHand\n\u000e\u000bfor thes-wave SC. In the low frequency region, where\n~!=kBTc\u00141, both\u000eHand\u000e\u000bdecrease and become\nexponentially small with the decrease of the temperature.\nIn the high frequency region, where ~!=kBTc\u00151, both\n\u000eHand\u000e\u000bvanish.\nThep-wave SCs show two characteristic properties\nthat thes-wave SC does not show: a \fnite \u000eHatT= 0\nand a resonance peak of \u000eHand\u000e\u000b. These properties\ncan be understood by the analogy between SCs and band\ninsulators as follows. The uniform dynamic spin suscepti-\nbility consists of contributions from intraband transitions\nwithin particle (hole) bands and interband transitions\nbetween particles and holes. In the low temperature or4\n(a) (b)Γ/kBTc=0.05 Chiral p-wave\n(c) (d)Γ/kBTc=0.05 Helical p-wave\n(e) (f)Γ/kBTc=0.05 s-waveT/T\nc\nhω/kBTcδα/δα1\n0.00.51.0 0\n2\n410\n5\n0\nT/T\nc\nhω/kBTcδH/δH1\n0.00.51.0 0\n2\n410\n5\n0\nT/T\nc\nhω/kBTcδα/δα1\n0.00.51.0 0\n2\n410\n5\n0\nT/T\nc\nhω/kBTcδH/δH1\n0.00.51.0 0\n2\n410\n5\n0\nT/T\nc\nhω/kBTcδH/δH1\n0.00.51.0 0\n2\n42\n1\n0\nT/T\nc\nhω/kBTcδα/δα1\n0.00.51.0 0\n2\n410\n5\n0\nFIG. 2. The frequency shift \u000eHand the enhanced Gilbert\ndamping\u000e\u000bas a function of temperature and frequency nor-\nmalized by the characteristic values \u000eH1=\u0000SJ2\n1DF=(N\r~)\nand\u000e\u000b1=SJ2\n1DF=(NkBTc) in the normal state. DF(/A)\nis the density of states at the Fermi level in the normal state.\nWe set\u0012= 0 and \u0000=kBTc= 0:05. The sign of \u000eHcorresponds\nto the sign of Re \u001fR\nuni(!), which can be positive and negative\nat low and high frequencies, respectively. In contrast, \u000e\u000bis\npositive at any frequency.\nhigh frequency region, the intraband contribution is neg-\nligible and the interband contribution is dominant. In\nthe case of the s-wave SC, the interband transitions are\nforbidden because the Hamiltonian and the spin operator\ncommute. As a result, there is no spin response in the\nlow-temperature or high-frequency regions. In contrast,\nthe Hamiltonian for the p-wave SCs and the spin operator\ndo not commute. Therefore, \u000eHhas a \fnite value near-\nzero temperature due to the interband contribution. In\naddition, a resonance peak occurs when ~!= 2\u0001 because\nthe density of states diverges at the band edge E=\u0006\u0001.\nA detailed proof of the above statement is given in the\nSupplemental Material39.\nThe angle dependences of \u000eHand\u000e\u000bare distinct for\nchiral and helical p-wave SCs, as shown in Fig. 3. In\nboth cases, we set ~!=kBTc= 3:0 as the typical values\nat high frequencies, where the main contribution of the\nuniform spin susceptibility is the interband transitions.\nIn the chiral p-wave SC,\u000eHand\u000e\u000btend to decrease and\nare halved at a \fxed temperature when \u0012increases from\n0 to\u0019=2. Conversely, in the helical p-wave SC, the qual-\n0.4\nT/Tc1.0 0.2 0.08\n4\n0\n0.8 0.6δH/δH1\n2\n-210\n60.4\nT/Tc1.0 0.2 0.08\n4\n0\n0.8 0.6δH/δH1\n2\n-210\n6\nHelical p-waveChiral p-wave\nθ=0\nπ/4\nπ/2\nθ=0π/4π/2\n0.4\nT/Tc1.0 0.2 0.02\n0\n0.8 0.6δα/δα1\n13θ=0\nπ/4\nπ/2\nθ=0π/4π/20.4\nT/Tc1.0 0.2 0.02\n0\n0.8 0.6δα/δα1\n13Γ/kBTc=0.05, hω/kBTc=3.0\n(a) (b)\n(c) (d)Γ/kBTc=0.05, hω/kBTc=3.0FIG. 3. Frequency shift and the enhanced Gilbert damping\nas a function of temperature at angles of \u0012= 0;\u0019=4;\u0019=2. The\nupper and lower panels show the characteristics for the chiral\nand helical p-wave SCs, respectively.\nitative behavior shows the opposite trend. \u000eHand\u000e\u000b\nboth tend to increase and become 1 :5 times larger at a\n\fxed temperature when \u0012increases from 0 to \u0019=2. In\nfact, the angle dependences are approximately obtained\nto be/1 +cos2\u0012and 1 +(sin2\u0012)=2 for chiral and helical\np-wave SCs, respectively39. Therefore, the spin con\fg-\nuration of the Cooper pair can be detected from the \u0012\ndependence data for the FMR modulation.\nThe FMR modulation properties of the three SCs are\nsummarized in Table I. All SCs considered here can be\ndistinguished based on three properties: the frequency\nshift in the low temperature limit, the presence of their\nresonance peak, and their \u0012dependence. For the s-wave\nSC,\u000eHbecomes exponentially small in T!0, while for\nthep-wave SCs, \u000eHis \fnite in T!0. For the s-wave\nSC,\u000eHand\u000e\u000bshow no resonance and no \u0012dependence,\nwhile for the chiral and helical p-wave SCs, both \u000eHand\n\u000e\u000bexhibit a resonance at ~!= 2\u0001 and a \u0012dependence.\nIn addition, these two p-wave SCs can be distinguished\nfrom their\u0012dependences of \u000eHand\u000e\u000b, which are char-\nacterized by @\u0012(\u000eH) and@\u0012(\u000e\u000b), respectively. Here, it\nshould be emphasized that the pairing symmetry can be\ncharacterized by the sign of @\u0012(\u000eH) and@\u0012(\u000e\u000b). These\nproperties are summarized in the Table I.\nSpin-triplet current generation.| The relationship be-\ntween the enhanced Gilbert damping discussed above5\nand the spin-triplet current generation must also be dis-\ncussed. The enhancement of the Gilbert damping is\nknown to originate from the spin current generation at\nthe magnetic interface6,33. The interface spin current in-\nduced by FMRhISiSPis given by39\nhISiSP=N(~\rhac)2\n2\u000b\u0002\n\u0000ImGR\nk=0(!)\u0003\n\u000e\u000b: (19)\nOne can see that hISiSPand\u000e\u000bare proportional to each\nother. In our setup, the enhanced Gilbert damping \u000e\u000b\nwill lead to the generation of both the Cooper pair spin-\ntriplet current and the quasiparticle spin current. Since\nthe angular dependence of \u000e\u000bre\rects the direction of\nthe Cooper pair spins, it is expected that the spin-triplet\ncurrent can be controlled by varying the magnetization\ndirection of the FI.\nDiscussion.| We have considered a \rat SC/FI inter-\nface. In the presence of roughness, the correction term\nproportional to \u001fR\nloc(!) contributes to the FMR mod-\nulation, as shown in Eq. (17). In the rough limit,\nJ2\n1\u001cJ2\n2,\u001fR\nloc(!) dominates to make the FMR modu-\nlation isotropic, due to the angle average by summation\noverq. Namely, the anisotropy peculiar to p-wave SC\nis smeared by the roughness. The detailed behavior of\n\u001fR\nloc(!) is shown in the Supplemental Material39. This\nresult implies that it is crucial to control the interface\nroughness. In principle, the roughness of the interface\ncan be observed using transmission electron microscopy\nof interfaces46{48and it is possible to detect whether the\ninterface of the sample is \rat or rough. More detailed\nspectroscopy can be obtained from the FMR modulation\nby using a \rat interface.\nOur results show that the pairing symmetry can be\ndetected by the sign of @\u0012(\u000eH) and@\u0012(\u000e\u000b) around the\nin-plane magnetic \feld ( \u0012\u0018\u0019=2), where the vortices are\nnegligible. When the external magnetic \feld has a large\nout-of-plane component, the vortex formation may cause\nproblems in observing the angular dependence. The qual-\nitative behavior is expected to change when the out-of-\nplane magnetic \feld approaches the upper critical \feld\n(H\u0018Hc2\u00181T). This is because the coherence length\nof the Cooper pair and the distance between the vor-\ntices can become comparable. Indeed, it has been exper-\nimentally reported that the vortex formation suppresses\nthe characteristic properties in the spin pumping into\nSCs20. Therefore, the out-of-plane magnetic \feld should\nbe as small as possible when FMR measurements are per-\nformed for H\u0018Hc2.\nTABLE I. FMR modulation properties for the \rat SC/FI in-\nterface where J16= 0 andJ2= 0.\nPairing symmetry s Chiral Helical\n\u000eHin the limit of T!0 0 \fnite \fnite\nResonance peak of \u000eH,\u000e\u000b { X X\n@\u0012(\u000eH),@\u0012(\u000e\u000b) 0 negative positiveRecent experiments have reported that UTe 2is a can-\ndidate material for spin-triplet p-wave SCs31, which has\nattracted a great deal of attention. Various experi-\nments, including spectroscopic measurements, are now\nin progress to investigate the pairing symmetry of UTe 2,\nand indicated that the superconducting transition tem-\nperature is about 1K \u001830 GHz. Therefore, the resonance\ncondition ~!= 2\u0001 shown above is accessible to recent\nbroadband FMR measurements.\nIn addition, experiments on spin pumping into d-wave\nSCs have recently been reported49and a theoretical in-\nvestigation of the enhancement of the Gilbert damp-\ning in ad-wave SC/FI bilayer system has recently been\npresented50. Thus anisotropic superconducting spintron-\nics can be expected to develop as a new research direc-\ntion.\nWe should emphasize two important aspects of the\nFMR method presented here: the spectroscopic probe\nmethod for the p-wave SC thin \flms and the versa-\ntile spin injection method. First, the FMR measure-\nment procedure can provide a new spin-sensitive mea-\nsurement method that will complement other measure-\nment methods to enable a breakthrough in the discovery\nof spin-triplet SCs. Second, the FMR method represents\na promising way to generate spin-triplet currents in p-\nwave SC thin \flms.\nConclusions.| We have investigated the anisotropic\nsuperconducting spin transport at magnetic interfaces\ncomposed of a p-wave SC and an FI based on a micro-\nscopic model Hamiltonian. The FMR signal in these p-\nwave SC/FI bilayer systems is modulated via spin trans-\nfer at the interface, which generates spin-triplet currents.\nWe have shown that the pairing symmetry of the SCs\ncan be extracted from the FMR modulation character-\nistics. Our approach provides a unique way to explore\nanisotropic superconducting spintronics, which will be\nuseful for application to emerging device technologies.\nNote added.| After the submission of this manuscript,\nwe became aware of a closely related work, where a way\nto convert spin-triplet currents to magnon spin currents\nin SC/FI bilayer systems is discussed51.\nWe thank R. Ohshima, M. Shiraishi, H. Chudo, G.\nOkano, K. Yamanoi, and Y. Nozaki for helpful discus-\nsions. This work was supported by the Priority Pro-\ngram of the Chinese Academy of Sciences under Grant\nNo. XDB28000000, and by JSPS KAKENHI under\nGrants Nos. JP20K03835, JP20H04635, JP20H01863,\nJP21H04565, and JP21H01800.6\nSUPPLEMENTAL MATERIAL\nI. MODEL HAMILTONIAN\nIn this section, we describe the derivation and details of the model Hamiltonian used in the main text.\nA. Ferromagnetic Heisenberg model\nThe ferromagnetic Heisenberg model with the transverse AC magnetic \feld due to the microwave radiation is given\nby\nHFI(t) =\u0000JX\nhi;jiSi\u0001Sj+~\rHX\njSZ\nj\u0000~\rhacX\nj\u0000\nSX\njcos!t\u0000SY\njsin!t\u0001\n; (S.1)\nwhereJ >0 is the exchange coupling constant, hi;jirepresents summation over all nearest-neighbor sites, Sjis the\nlocalized spin at site jin the ferromagnetic insulator (FI), \r(<0) is the gyromagnetic ratio, His a static magnetic\n\feld,hacis an amplitude of an transverse oscillating magnetic \feld due to the microwave radiation with a frequency\n!. The rotated coordinates ( X;Y;Z ) are shown in Fig. 1(a).\nIt is convenient to introduce the boson creation and annihilation operators in order to formulate the problem in\nterms of the quantum \feld theory. In the current problem, we perturbatively treat the excitation of the FI. In this\ncase, the Holstein-Primako\u000b transformation is useful, where the localized spin can be described using boson creation\nand annihilation operators bj;by\njin Hilbert space constrained to 2 S+ 1 dimensions. The spin operators are written as\nS+\nj=SX\nj+iSY\nj=\u0010\n2S\u0000by\njbj\u00111=2\nbj; (S.2)\nS\u0000\nj=SX\nj\u0000iSY\nj=by\nj\u0010\n2S\u0000by\njbj\u00111=2\n; (S.3)\nSZ\nj=S\u0000by\njbj; (S.4)\nwhere we require [ bi;by\nj] =\u000ei;j;in order that the S+\nj,S\u0000\nj, andSZ\njsatisfy the commutation relation of angular\nmomentum. The deviation of SZ\njfrom its ground-state value Sis quanti\fed by the boson particle number.\nWe consider low-energy excitation in the FI, where the deviation of SZ\njfrom the ground state is small hby\njbji=S\u001c1.\nThe ladder operators S\u0006\njare approximated as\nS+\nj\u0019(2S)1=2bj; (S.5)\nS\u0000\nj\u0019(2S)1=2by\nj; (S.6)\nwhich is called spin-wave approximation. Here, we de\fne the magnon operators\nbk=1p\nNX\nje\u0000ik\u0001rjbj; (S.7)\nby\nk=1p\nNX\njeik\u0001rjby\nj; (S.8)\nwhereNis the number of sites and k= (kx;ky;kz). The inverse transformation is then given by\nbj=1p\nNX\nkeik\u0001rjbk; (S.9)\nby\nj=1p\nNX\nke\u0000ik\u0001rjby\nk: (S.10)\nThe magnon operators satisfy [ bk;by\nk0] =\u000ek;k0and describe the quantized collective excitations. Using the spin-wave\napproximation and the magnon operators, the Hamiltonian HFI(t) is written as\nHFI(t)\u0019X\nk~!kby\nkbk\u0000h+\nac(t)by\nk=0\u0000h\u0000\nac(t)bk=0; (S.11)7\nwhere ~!k=Dk2\u0000~\rHwithD= 2JSa2and the lattice constant a,h\u0006\nac(t) =~\rhacp\nSN=2e\u0007i!t, and constant\nterms are omitted.\nB. BCS Hamiltonian\nWe derive a mean-\feld Hamiltonian, which describes a bulk superconductor (SC), and we diagonalize the mean-\feld\nHamiltonian with the Bogoliubov transformation. At the end of this section, the spin density operators of the SC are\nwritten in terms of the Bogoliubov quasiparticle creation and annihilation operators.\nWe start with the e\u000bective Hamiltonian in momentum space\nHSC=X\nk;s\u0018kcy\nkscks+1\n2X\nk;k0;s1;s2;s3;s4Vs1;s2;s3;s4(k;k0)cy\n\u0000ks1cy\nks2ck0s3c\u0000k0s4; (S.12)\nwhere\u0018kis the band energy measured relative to the chemical potential, and cy\nksandcksare the creation and\nannihilation operators of electrons with the wave vector k= (kx;ky) and thezcomponent of the spin s=\";#. The\nmatrix elements satisfy\nVs1;s2;s3;s4(k;k0) =\u0000Vs2;s1;s3;s4(\u0000k;k0); (S.13)\nVs1;s2;s3;s4(k;k0) =\u0000Vs1;s2;s4;s3(k;\u0000k0); (S.14)\nbecause of the anticommutation relation of fermions, and\nVs1;s2;s3;s4(k;k0) =V\u0003\ns4;s3;s2;s1(k0;k); (S.15)\nbecause of the Hermitianity of the Hamiltonian. We consider a mean-\feld, which is called a pair potential\n\u0001k;ss0=\u0000X\nk0;s3;s4Vs0;s;s 3;s4(k;k0)hck0s3c\u0000k0s4i; (S.16)\nand its conjugate\n\u0001\u0003\n\u0000k;ss0=X\nk0;s1;s2Vs1;s2;s0;s(k0;k)hcy\n\u0000k0s1cy\nk0s2i: (S.17)\nHere, we consider a mean-\feld approximation where the interaction term is replaced as follows\ncy\n\u0000ks1cy\nks2ck0s3c\u0000k0s4!cy\n\u0000ks1cy\nks2hck0s3c\u0000k0s4i+hcy\n\u0000ks1cy\nks2ick0s3c\u0000k0s4\u0000hcy\n\u0000ks1cy\nks2ihck0s3c\u0000k0s4i; (S.18)\nso that the interaction term is rewritten as\nX\nk;k0;s1;s2;s3;s4Vs1;s2;s3;s4(k;k0)cy\n\u0000ks1cy\nks2ck0s3c\u0000k0s4!X\nk;s1;s2h\n\u0001k;s1s2cy\nks1cy\n\u0000ks2\u0000\u0001\u0003\n\u0000k;s1s2c\u0000ks1cks2i\n; (S.19)\nwhere an constant term is omitted. Consequently, we derive a mean-\feld Hamiltonian\nHSC=X\nk;s\u0018kcy\nkscks+1\n2X\nk;s1;s2\u0002\n\u0001k;s1s2cy\nks1cy\n\u0000ks2\u0000\u0001\u0003\n\u0000k;s1s2c\u0000ks1cks2\u0003\n: (S.20)\nUsing a four-component notation\ncy\nk= (cy\nk\";cy\nk#;c\u0000k\";c\u0000k#); (S.21)\nck= (ck\";ck#;cy\n\u0000k\";cy\n\u0000k#)T; (S.22)\nthe mean-\feld Hamiltonian is written as\nHSC=1\n2X\nkcy\nkHBdGck: (S.23)8\nHBdGis the 4\u00024 matrix\nHBdG= \n\u0018k\u001b0\u0001k\n\u0000\u0001\u0003\n\u0000k\u0000\u0018k\u001b0!\n; (S.24)\nwhere\u001b0is the 2\u00022 unit matrix and \u0001 kis the 2\u00022 matrix given as\n\u0001k= \n\u0001k;\"\"\u0001k;\"#\n\u0001k;#\"\u0001k;##!\n: (S.25)\nIn principle, the pair potential is obtained by solving the gap equation self-consistently for an explicit form of the\nmatrix elements Vs1;s2;s3;s4(k;k0). In this work, we do not solve the gap equation, but instead assume an explicit\nform of the pair potential and perform calculations using a phenomenological gap function. For the singlet pairing,\nthe pair potential is given by\n\u0001k= ki\u001by; (S.26)\nwith an even function  k= \u0000k. For ans-wave SC, the pair potential is given by\n\u0001k= \u0001 \n0 1\n\u00001 0!\n: (S.27)\nFor the triplet pairing, the pair potential is given by\n\u0001k= [dk\u0001\u001b]i\u001by; (S.28)\nwith an odd vectorial function dk=\u0000d\u0000k. For a chiral p-wave SC and a helical p-wave SC,dkis given by\ndk=(\n\u0001(0;0;ei\u001ek) : chiral p\u0000wave\n\u0001(\u0000sin\u001ek;cos\u001ek;0) : helical p\u0000wave(S.29)\nwith\u001ek= arctan(ky=kx), so that the pair potential is given by\n\u0001k=8\n>>>><\n>>>>:\u0001 \n0ei\u001ek\nei\u001ek0!\n: chiralp\u0000wave\n\u0001 \nie\u0000i\u001ek0\n0iei\u001ek!\n: helicalp\u0000wave(S.30)\nThe phenomenological gap function is given by\n\u0001 = 1:76kBTctanh\u0010\n1:74p\nTc=T\u00001\u0011\n: (S.31)\nThe Bogoliubov transformation to diagonalize HBdGis given by\nUk= \nukvk\nv\u0003\n\u0000ku\u0003\n\u0000k!\n; (S.32)\nUy\nk= \nuk\u0000vk\n\u0000v\u0003\n\u0000ku\u0003\n\u0000k!\n; (S.33)\nwith the 2\u00022 matricesukandvkgiven by\nuk=s\n1\n2\u0012\n1 +\u0018k\nEk\u0013\n\u001b0; (S.34)\nvk=\u0000s\n1\n2\u0012\n1\u0000\u0018k\nEk\u0013\u0001k\n\u0001; (S.35)9\nwhereEkis the eigenenergy\nEk=q\n\u00182\nk+ \u00012: (S.36)\nUsing the Bogoliubov transformation Uk, the 4\u00024 matrixHBdGis diagonalized as\nUy\nkHBdGUk=0\nBBB@Ek0 0 0\n0Ek0 0\n0 0\u0000Ek0\n0 0 0\u0000Ek1\nCCCA: (S.37)\nThe excitation of HSCis described by the creation and annihilation operators of the Bogoliubov quasiparticles \r(y)\nk\n\ry\nk= (\ry\nk\";\ry\nk#;\r\u0000k\";\r\u0000k#); (S.38)\n\rk= (\rk\";\rk#;\ry\n\u0000k\";\ry\n\u0000k#)T; (S.39)\nwhere they are obtained by the Bogoliubov transformation\n\rk=Uy\nkck; (S.40)\n\ry\nk=cy\nkUk: (S.41)\nThe spin density operators \u001ba(r) (a=x;y;z ) is de\fned as\n\u001ba(r) :=1\nAX\nk;k0;s;s0e\u0000i(k\u0000k0)\u0001r\u001ba\nss0cy\nksck0s0; (S.42)\nwhereAis the area of the system. \u001ba(r) (a=x;y;z ) is expanded in Fourier series\n\u001ba(r) =1\nAX\nqeiq\u0001r\u001ba\nq; (S.43)\nand the Fourier coe\u000ecient is given by\n\u001ba\nq=Z\ndre\u0000iq\u0001r\u001ba(r) =X\nk;s;s0\u001ba\nss0cy\nksck+qs0: (S.44)\nUsing the Bogoliubov transformation Uk, the above expression is rewritten as\n\u001ba\nq=X\nk;s;s0\"\u0010\nsa(1)\nk;k+q\u0011\ns;s0\ry\nks\rk+qs0+\u0010\nsa(2)\nk;k+q\u0011\ns;s0\r\u0000ks\ry\n\u0000k\u0000qs0+\u0010\nsa(3)\nk;k+q\u0011\ns;s0\ry\nks\ry\n\u0000k\u0000qs0+\u0010\nsa(4)\nk;k+q\u0011\ns;s0\r\u0000ks\rk+qs0#\n;\n(S.45)\nwith the 2\u00022 matricessa(i)\nk;k+qgiven by\nsa(1)\nk;k+q=uy\nk\u001bauk+q; (S.46)\nsa(2)\nk;k+q=vy\nk\u001bavk+q; (S.47)\nsa(3)\nk;k+q=uy\nk\u001bavk+q; (S.48)\nsa(4)\nk;k+q=vy\nk\u001bauk+q: (S.49)\nThe \frst and second terms describe the intraband transition from particle-to-particle and from hole-to-hole, respec-\ntively. The third and fourth terms describe the interband transition from hole-to-particle and from particle-to-hole,\nrespectively.10\nC. Proximity exchange coupling at interface\nWe start with a model for the proximity exchange coupling given by\nHex=Z\ndrX\njJ(r;rj)\u001b(r)\u0001Sj: (S.50)\nWe rewrite the above expression in the real space into the expression in the wave space. The proximity exchange\ncoupling is rewritten as\nHex=Z\ndrX\njJ(r;rj)1\nAp\nNX\nq;kei(q\u0001r+k\u0001rj)\u0000\n\u001b+\nqS\u0000\nk+\u001b\u0000\nqS+\nk\u0001\n+Z\ndrX\njJ(r;rj)\u001bZ(r)SZ\nj; (S.51)\nwhere the Fourier series are given by\n\u001b(r) =1\nAX\nqeiq\u0001r\u001bq; (S.52)\nSj=1p\nNX\nkeik\u0001rjSk; (S.53)\nwith the area of the SC, A, and the number of sites in the FI, N, and the ladder operators are given by\n\u001b\u0006=1\n2(\u001bX\u0006i\u001bY); (S.54)\nS\u0006=SX\u0006iSY: (S.55)\nThe matrix element is given by\nJq;k=1\nAp\nNZ\ndrX\njJ(r;rj)ei(q\u0001r+k\u0001rj): (S.56)\nConsequently, the exchange coupling which we use in the main text is derived as\nHex=X\nq;k\u0000\nJq;k\u001b+\nqS\u0000\nk+J\u0003\nq;k\u001b\u0000\n\u0000qS+\n\u0000k\u0001\n; (S.57)\nwhere we use a relation J\u0000q;\u0000k=J\u0003\nq;k, and we omit the last term\nZ\ndrX\njJ(r;rj)\u001bZ(r)SZ\nj; (S.58)\nin order to focus on the spin transfer at the interface. For the uniform magnon mode jkj= 0, the matrix element is\ngiven by\nJq;k=0=1\nAp\nNZ\ndrX\njJ(r;rj)eiq\u0001r: (S.59)\nII. TIME DEPENDENT QUANTUM AVERAGE\nIn this section, we show that the ferromagnetic resonance (FMR) frequency and linewidth are read from the\nmagnon Green's function. We consider the Hamiltonian H(t) composed of the unperturbed Hamiltonian H0and the\nperturbation V(t)\nH(t) =H0+V(t): (S.60)\nThe time-dependent quantum average of a physical quantity Ois calculated as\nhO(t)i=hSy(t;\u00001)~O(t)S(t;\u00001)i; (S.61)11\nwhere ~O(t) is the interaction picture and the S matrix S(t;t0) is given by\nS(t;t0) =Texp Zt\nt0dt0~V(t0)\ni~!\n: (S.62)\nThe time-dependent quantum average hO(t)iis written as\nhO(t)i=hOieq+\u000ehO(t)i; (S.63)\nwherehOieq= Tr (\u001aeqO) is the equilibrium value and \u000ehO(t)iis deviation from the equilibrium. When the perturbation\nis written as V(t) =\u0000AF(t), the \frst order perturbation calculation gives\n\u000ehO(t)i=\u0000Zt\n\u00001dt01\ni~h[~O(t);~A(t0)]iF(t0)\n=\u0000Z1\n\u00001dt0GR(t0)F(t\u0000t0); (S.64)\nwhere we de\fne the retarded Green's function\nGR(t) =1\ni~\u0012(t)h[~O(t);~A(0)]i: (S.65)\nWhen the external force is written as F(t) =Fe\u0000i(!+i0)t,\u000ehO(t)iis written as\n\u000ehO(t)i=\u0000Fe\u0000i(!+i0)tZ1\n\u00001dt0ei(!+i0)t0GR(t0)\n=\u0000Fe\u0000i!tGR(!): (S.66)\nUsing the above formula, the dynamics of \u000ehS+\nk=0(t)iis written as\n\u000ehS+\nk=0(t)i=\u0000~\rhacp\nN\n2e\u0000i!tGR\nk=0(!); (S.67)\nwhereGR\nk(!) is the Fourier transform of the retarded component of the magnon Green's function GR\nk(t). They are\nde\fned as\nGR\nk(t) :=1\ni~\u0012(t)h[S+\nk(t);S\u0000\n\u0000k(0)]i; (S.68)\nGR\nk(!) :=Z1\n\u00001dt0ei(!+i0)t0GR\nk(t0): (S.69)\nFrom Eq. (S.67), one can see that the FMR frequency and linewidth are read from GR\nk(!).\nIII. MAGNON GREEN'S FUNCTION\nIn this section, we perform perturbative calculation for the magnon Green's function. We treat the proximity\nexchange coupling as a perturbation. The Hamiltonian is written as\nH=H0+V; (S.70)\nwhereH0is the unperturbed Hamiltonian\nH0=X\nk~!kby\nkbk+X\nk;sEk\ry\nks\rks; (S.71)\nandVis the perturbation\nV=X\nq;k\u0000\nJq;k\u001b+\nqS\u0000\nk+ h:c:\u0001\n: (S.72)12\n(e) Vertex(b) Keldysh contour\n(d) Self-energytime(a) Magnon Green’s function (c) Dyson equation\nk/uni2032 +qs/uni2032 −k\n−k/uni2032 −qs/uni2032 /uni03C3+\nqS−\nk= + + +−k −k −k−k/uni2032 s k/uni2032 s k/uni2032 s\n−k/uni2032 −qs/uni2032 k/uni2032 +qs/uni2032 −k/uni2032 sGk(/uni03C4,/uni03C4/uni2032 )= = + /uni03A3\n/uni03A3 = + + +\nk/uni2032 +qs/uni2032 −k/uni2032 −qs/uni2032 −k/uni2032 s k/uni2032 s k/uni2032 s\n−k/uni2032 −qs/uni2032 k/uni2032 +qs/uni2032 −k/uni2032 s\nBogoliubov quasiparticle: Magnon:\nFIG. 4. (a) The Feynman diagram for the magnon Green's function. (b) Keldysh contour to perform perturbative calculations.\n(c) The Feynman diagram for the Dyson equation. (d) The self-energy within the second-order perturbation is given by the\ndynamic spin susceptibility of the SCs. (e) The Feynman diagrams for the vertex \u001b+\nqS\u0000\nk, which represent scattering of a\nBogoliubov quasiparticle with magnon emission. The solid and wavy lines represent a Bogoliubov quasiparticle and a magnon,\nrespectively.\nWe de\fne the magnon Green's function\nGk(\u001c;\u001c0) :=1\ni~hTCS+\nk(\u001c)S\u0000\n\u0000k(\u001c0)i; (S.73)\nwhereTCis the time-ordering operator on the Keldysh contour (see Figs. 4(a) and (b)). To perform the perturbative\ncalculation, we introduce interaction picture. The perturbation is written as\n~V(t) =X\nq;k\u0010\nJq;k~\u001b+\nq(t)~S\u0000\nk(t) + h:c:\u0011\n: (S.74)\nThe magnon Green's function is given by\nGk(\u001c;\u001c0) =1\ni~hTCSC~S+\nk(\u001c)~S\u0000\n\u0000k(\u001c0)iconn; (S.75)\nwhereh\u0001\u0001\u0001i connmeans the connected diagrams and the S matrix is given by\nSC=TCexp Z\nCd\u001c~V(\u001c)\ni~!\n: (S.76)\nThe above expressions lead to the Dyson equation (see Fig. 4(c))\nGk(\u001c;\u001c0) =G(0)\nk(\u001c;\u001c0) +Z\nCd\u001c1Z\nCd\u001c2G(0)\nk(\u001c;\u001c1)\u0006k(\u001c1;\u001c2)Gk(\u001c2;\u001c0); (S.77)\nwhereG(0)\nk(\u001c;\u001c0) is the unperturbed magnon Green's function\nG(0)\nk(\u001c;\u001c0) =1\ni~hTC~S+\nk(\u001c)~S\u0000\n\u0000k(\u001c0)i; (S.78)\nand \u0006 k(\u001c1;\u001c2) is the self-energy. Within the second-order perturbation, the self-energy is given by (see Fig. 4(d))\n\u0006k(\u001c;\u001c0) =1\ni~X\nqjJq;kj2hTC~\u001b+\nq(\u001c)~\u001b\u0000\n\u0000q(\u001c0)i: (S.79)\nThe Feynman diagram for the vertex is shown in Fig. 4(e). Substituting the ladder operators expressed in terms of13\n\r(y)\nks, the self-energy is written as\n\u0006k(\u001c;\u001c0) =\u0000i~X\nqjJq;kj2X\nk0;s;s0\"\u0012\f\f\f(s+(1)\nk0;k0+q)s;s0\f\f\f2\n\u0000(s+(1)\nk0;k0+q)s;s0(s\u0000(2)\n\u0000k0\u0000q;\u0000k0)\u0003\ns0;s\u0013\ngk0;s(\u001c0;\u001c)gk0+q;s0(\u001c;\u001c0)\n+\u0012\f\f\f(s+(2)\nk0;k0+q)s;s0\f\f\f2\n\u0000(s+(2)\nk0;k0+q)s;s0(s\u0000(1)\n\u0000k0\u0000q;\u0000k0)\u0003\ns0;s\u0013\ng\u0000k0;s(\u001c;\u001c0)g\u0000k0\u0000q;s0(\u001c0;\u001c)\n\u0000\u0012\f\f\f(s+(3)\nk0;k0+q)s;s0\f\f\f2\n\u0000(s+(3)\nk0;k0+q)s;s0(s\u0000(3)\n\u0000k0\u0000q;\u0000k0)\u0003\ns0;s\u0013\ngk0;s(\u001c0;\u001c)g\u0000k0\u0000q;s0(\u001c0;\u001c)\n\u0000\u0012\f\f\f(s+(4)\nk0;k0+q)s;s0\f\f\f2\n\u0000(s+(4)\nk0;k0+q)s;s0(s\u0000(4)\n\u0000k0\u0000q;\u0000k0)\u0003\ns0;s\u0013\ng\u0000k0;s(\u001c;\u001c0)gk0+q;s0(\u001c;\u001c0)#\n;\n(S.80)\nwhere the quasiparticle Green's function is de\fned as\ngk;s(\u001c;\u001c0) :=1\ni~hTC~\rks(\u001c)~\ry\nks(\u001c0)i: (S.81)\nThe \frst and second terms give the intraband contribution, and the third and fourth terms give the interband\ncontribution. Evaluating the Dyson equation, the retarded component of the magnon Green's function is given by\nGR\nk(!) =1\nh\nG(0)R\nk(!)i\u00001\n\u0000\u0006R\nk(!); (S.82)\nwhere the unperturbed Green's function is written as\nG(0)R\nk(!) =2S=~\n!\u0000!k+i\u000b!: (S.83)\nHere, we introduce the phenomenological dimensionless damping parameter \u000b. Using Eq.(S.83), the retarded Green's\nfunction is written as\nGR\nk(!) =2S=~\n!\u0000!k+i\u000b!\u0000(2S=~)\u0006R\nk(!): (S.84)\nFrom the above expression, the frequency shift at a \fxed !is given by\n\u000eH=2S\n\r~Re\u0006R\nk(!); (S.85)\nand the enhanced Gilbert damping is given by\n\u000e\u000b=\u00002S\n~!Im\u0006R\nk(!): (S.86)\nThe Fourier transform of the self-energy is given as\n\u0006R\nk(!) =Z\ndtei(!+i0)t\u0006R\nk(t) =\u0000X\nqjJq;kj2\u001fR\nq(!); (S.87)\nwhere the dynamic spin susceptibility of the SC is de\fned as\n\u001fR\nq(!) :=Z\ndtei(!+i0)ti\n~\u0012(t)h[~\u001b+\nq(t);~\u001b\u0000\n\u0000q(0)]i: (S.88)\nEvaluating the self-energy Eq. (S.87), one can obtain the information of the FMR modulation, \u000eHand\u000e\u000b. Using the\nsystem's symmetry, the dynamic spin susceptibility \u001fR\nq(!) can be written as\n\u001fR\nq(!) = cos2\u0012\u001fxx\nq(!) +\u001fyy\nq(!) + sin2\u0012\u001fzz\nq(!); (S.89)\nwhich means that both \u000eHand\u000e\u000bshow a dependence on \u0012when the dynamic spin susceptibility is anisotropic.14\nIV. SPIN CURRENT AT THE INTERFACE\nIn this section, we derive the general expression of spin current at the interface. We treat the tunneling Hamiltonian\nas a perturbation and the other terms as the unperturbed Hamiltonian\nH(t) =H0(t) +Hex; (S.90)\nH0(t) =HFI(t) +HSC: (S.91)\nThe operator of spin current \rowing from the SC to the FI at the interface is de\fned by\nIS:=\u0000~\n2_\u001bZ\ntot=\u0000~\n21\ni~[\u001bZ\ntot;Hex] =i\n2[\u001bZ\ntot;Hex]; (S.92)\nwhere\u001bZ\ntotis given by\n\u001bZ\ntot=Z\ndr\u001bZ(r): (S.93)\nCalculating the commutation relation, we obtain the following expression\nIS=iX\nq;k\u0000\nJq;k\u001b+\nqS\u0000\nk\u0000h:c:\u0001\n: (S.94)\nThe time-dependent quantum average of ISis written as\nhIS(t)i= Re2\n42iX\nq;kJq;kh\u001b+\nq(t)S\u0000\nk(t)i3\n5; (S.95)\nwhereh\u0001\u0001\u0001i = Tr[\u001a0\u0001\u0001\u0001] denotes the statistical average with an initial density matrix \u001a0. In order to develop the\nperturbation expansion, we introduce the interaction picture\nhIS(\u001c1;\u001c2)i= Re2\n42iX\nq;kJq;khTCSC~\u001b+\nq(\u001c1)~S\u0000\nk(\u001c2)i3\n5: (S.96)\nSCand ~O(t) are given by\nSC=TCexp Z\nCd\u001c~Hex(\u001c)\ni~!\n; (S.97)\nand\n~O(t) =Uy\n0(t;t0)OU0(t;t0); (S.98)\nwhere\nU0(t;t0) =Texp\u0012Zt\nt0dt0H0(t0)\ni~\u0013\n: (S.99)\nExpandingSCas\nSC\u00191 +Z\nCd\u001cTC~Hex(\u001c)\ni~; (S.100)\nthe spin current is given by\nhIS(\u001c1;\u001c2)i=X\nq;kjJq;kj2Re\"\n2\n~Z\nCd\u001chTC~\u001b+\nq(\u001c1)~\u001b\u0000\n\u0000q(\u001c)ihTC~S+\n\u0000k(\u001c)~S\u0000\nk(\u001c2)i#\n: (S.101)15\nUsing the contour ordered Green's functions\n\u001fq(\u001c1;\u001c) =\u00001\ni~hTC~\u001b+\nq(\u001c1)~\u001b\u0000\n\u0000q(\u001c)i; (S.102)\nGk(\u001c;\u001c2) =1\ni~hTC~S+\nk(\u001c)~S\u0000\n\u0000k(\u001c2)i; (S.103)\nthe above equation is rewritten as\nhIS(\u001c1;\u001c2)i=X\nq;kjJq;kj2Re\"\n2~Z\nCd\u001c\u001fq(\u001c1;\u001c)G\u0000k(\u001c;\u001c2)#\n: (S.104)\nWe put\u001c2on the forward contour and \u001c1on the backward contour to describe spin transfer at the interface in\nappropriate time order. Assuming a steady state, the spin current is written as\nhISi= 2~X\nq;kjJq;kj2Re\"Z1\n\u00001d!0\n2\u0019\u0010\n\u001fR\nq(!0)G<\n\u0000k(!0) +\u001f<\nq(!0)GA\n\u0000k(!0)\u0011#\n: (S.105)\nWe introduce the distribution functions as\n\u001f<\nq(!) =fSC\nq(!)\u0002\n2iIm\u001fR\nq(!)\u0003\n; (S.106)\nG<\nk(!) =fFI\nk(!)\u0002\n2iImGR\nk(!)\u0003\n: (S.107)\nThe formula of the spin current at the interface is derived as\nhISi= 4~X\nq;kjJq;kj2Z1\n\u00001d!0\n2\u0019Im\u001fR\nq(!0)\u0002\n\u0000ImGR\n\u0000k(!0)\u0003\u0002\nfFI\n\u0000k(!0)\u0000fSC\nq(!0)\u0003\n: (S.108)\nWhen both the SC and the FI are in equilibrium, the di\u000berence of the distribution functions is zero (i.e. fFI\n\u0000k(!0)\u0000\nfSC\nq(!0) = 0), so that no spin current is generated. Under the microwave irradiation, the distribution function of\nthe FI deviates from equilibrium, which generates the interface spin current. Performing a second-order perturbation\ncalculation, the deviation of the distribution function of the FI, \u000efFI\n\u0000k(!0), is given by\n\u000efFI\n\u0000k(!0) =2\u0019NS (\rhac=2)2\n\u000b!0\u000ek;0\u000e(!0\u0000!): (S.109)\nConsequently, the interface spin current is written as\nhISiSP= 4~X\nq;kjJq;kj2Z1\n\u00001d!0\n2\u0019Im\u001fR\nq(!0)\u0002\n\u0000ImGR\n\u0000k(!0)\u0003\n\u000efFI\n\u0000k(!0): (S.110)\nFinally, one can show that the spin current is proportional to the enhanced Gilbert damping\nhISiSP= 4~NS(\rhac=2)2\n\u000b!\u0002\n\u0000ImGR\nk=0(!)\u0003X\nqjJq;k=0j2Im\u001fR\nq(!);\n=N(~\rhac)2\n2\u000b\u0002\n\u0000ImGR\nk=0(!)\u0003\n\u000e\u000b: (S.111)\nV. MODEL FOR INTERFACE CONFIGURATIONS\nIn order to calculate Eq. (S.87), one needs to set up an explicit expression for jJq;k=0j2. We consider an interface\nwith uncorrelated roughness. To model this interface, we assume that J(r;rj) satis\fes\nDX\njJ(r;rj)E\nave=J1; (S.112)\nDX\nj;j0J(r;rj)J(r0;rj0)E\nave=J2\n1+J2\n2l2\u000e(r\u0000r0); (S.113)16\nwhereh\u0001\u0001\u0001i avemeans interface con\fguration average. The spatially averaged J(r;rj) is given by a constant J1as\nshown in Eq. (S.112). Equation (S.113) means that the interface roughness is uncorrelated and J2\n2l2is a variance.\nJ1andJ2are coupling constants with dimension of energy, and are independent of the system size. lis introduced\nbecause the Hamiltonian of the SCs is treated as a continuum model. Performing the interface con\fguration average,\nand using Eq. (S.112) and (S.113), one can obtain the expression for jJq;k=0j2in the main text.\nVI. DYNAMIC SPIN SUSCEPTIBILITY OF SC\nEvaluating the retarded component of the self-energy Eq. (S.80), the dynamic spin susceptibility of the SC is given\nby\n\u001fR\nq(!) =\u0000Z1\n\u00001dEf(E)X\n\u0015;k(\nM\u0015;\u0015(a)\nk;k+q\u0014\n\u00001\n\u0019ImgR\n\u0015;k(E)gR\n\u0015;k+q(E+~!)\u00001\n\u0019ImgR\n\u0015;k+q(E)gA\n\u0015;k(E\u0000~!)\u0015\n+M\u0015;\u0000\u0015(a)\nk;k+q\u0014\n\u00001\n\u0019ImgR\n\u0015;k(E)gR\n\u0000\u0015;k+q(E+~!)\u00001\n\u0019ImgR\n\u0000\u0015;k+q(E)gA\n\u0015;k(E\u0000~!)\u0015)\n;\n(S.114)\nwhereM\u0015;\u00150(a)\nk;k+qwitha=s;c;andhare given by\nM\u0015;\u00150(s)\nk;k+q=(\u0018k+\u0015Ek)(\u0018k+q+\u00150Ek+q)\n4\u0015Ek\u00150Ek+q+\u00012\n4\u0015Ek\u00150Ek+q; (S.115)\nM\u0015;\u00150(c)\nk;k+q=(\u0018k+\u0015Ek)(\u0018k+q+\u00150Ek+q)\n4\u0015Ek\u00150Ek+q\u0000\u00012e\u0000i(\u001ek\u0000\u001ek+q)\n4\u0015Ek\u00150Ek+qcos2\u0012; (S.116)\nM\u0015;\u00150(h)\nk;k+q=(\u0018k+\u0015Ek)(\u0018k+q+\u00150Ek+q)\n4\u0015Ek\u00150Ek+q\u0000\u00012sin\u001eksin\u001ek+q\n4\u0015Ek\u00150Ek+qsin2\u0012: (S.117)\n\u0015;\u00150=\u0006give a sign, and a=s;c;andhcorrespond to matrix elements for s-wave, chiral p-wave, and helical p-wave\nSCs, respectively. In Eq. (S.114), the terms multiplied by M\u0015;\u0015(a)\nk;k+qdescribe the intraband transition processes, i.e.,\ntransition processes from particle to particle and from hole to hole, and the terms multiplied by M\u0015;\u0000\u0015(a)\nk;k+qdescribe\nthe interband transition processes, i.e., transition processes from particle to hole and vice versa. The retarded and\nadvanced Green's functions of the quasiparticles gR=A\n\u0015;k(E) are given by\ngR\n\u0015;k(E) =1\nE\u0000\u0015Ek+i\u0000; (S.118)\ngA\n\u0015;k(E) =1\nE\u0000\u0015Ek\u0000i\u0000; (S.119)\nwhere \u0000 is a constant level broadening introduced phenomenologically. \u0000 is introduced to incorporate the intraband\ncontribution in the calculation of the uniform spin susceptibility. The details are explained in the next section.\nThe sum over kis replaced by the integral near the Fermi energy\nX\nkF(k)!DFZ1\n0dEDs(E)X\n\u0011=\u0006F\u0011(E); (S.120)\nX\nkF(k) sin2\u001ek!DFZ1\n0dEDs(E)X\n\u0011=\u00061\n2F\u0011(E); (S.121)\nwhereDFis the density of states near the Fermi energy in the normal state and Ds(E) is the density of states of\nquasiparticles\nDs(E) =jEjp\nE2\u0000\u00012\u0012(jEj\u0000\u0001): (S.122)\nF\u0011(E) means to assign \u0011p\nE2\u0000\u00012to\u0018contained in F(k).17\nVII. UNIFORM SPIN SUSCEPTIBILITY\nIn this section, we explain three properties related to the calculation of the uniform spin susceptibility. First, the\nmatrix element's properties are explained, which is essential to understand the qualitative di\u000berence between spin-\nsinglets-wave and spin-triplet p-wave SCs. Second, the reason to introduce the constant level broadening \u0000. Third,\nthe analytical expression for the uniform spin susceptibility of the p-wave SCs is given.\nPerforming the angular integral and replacing the sum over kby theEintegral, the matrix elements are replaced\nby\nM\u0015;\u00150(s)\nk;k!1 +\u0015\u00150\n4\u0015\u00150; (S.123)\nM\u0015;\u00150(c)\nk;k!(1 +\u0015\u00150)E2\u0000(1 + cos2\u0012)\u00012\n4\u0015\u00150E2; (S.124)\nM\u0015;\u00150(h)\nk;k!(1 +\u0015\u00150)E2\u0000(1 +1\n2sin2\u0012)\u00012\n4\u0015\u00150E2: (S.125)\nHere, the \frst-order terms in \u0018kare omitted because they vanish in the Eintegral. From the above expressions, the\nintraband matrix elements become \fnite for all SCs considered here, while the interband matrix elements vanish in\nthes-wave SC and becomes \fnite in the p-wave SCs. The above properties of the intraband and interband matrix\nelements can be understood using the commutation relation between the Hamiltonian and the spin operators. We\nintroduce the BdG form of the spin operators \u001ba\nBdG(a=x;y;z ) as below\n\u001ba\nBdG= \n\u001ba0\n0\u0000(\u001ba)T!\n: (S.126)\nThe commutation relation of HBdGand\u001ba\nBdGis given by\n[HBdG;\u001ba\nBdG] = 0 :s\u0000wave; (S.127)\n[HBdG;\u001ba\nBdG]6= 0 :p\u0000wave: (S.128)\nEquation (S.127) means that both the Hamiltonian and the spin operator are diagonalized simultaneously, so that the\nmatrix elements of the spin operator between a particle and a hole with the same wave-number vanish. This is because\nthes-wave SC is spin singlet. Therefore, the interband matrix elements vanishes in the s-wave SC. In contrast, in\nthep-wave SCs, the commutation relation between the Hamiltonian and the spin operator is \fnite as shown in Eq.\n(S.128), so that the matrix elements of the spin operator between a particle and a hole with the same wave-number\nis \fnite. This is because the p-wave SCs are spin triplet. As a result, the interband matrix elements are \fnite.\nHere, we explain the reason to introduce the constant level broadening \u0000 for gR=A\n\u0015;k(E). The intraband and interband\ntransitions are schematically shown in Fig. 5. The quasiparticles are scattered due to the magnon emission or\nabsorption. The scattering process conserves the wave-number. Consequently, in the case of the intraband transition,\nthe transition process is forbidden when \u0000 = 0. In order to incorporate the intraband processes, one needs to introduce\n\u0000, otherwise the intraband contribution vanishes, which can be directly shown by calculating Eq. (S.114).\nWhen \u0000 = 0, the uniform spin susceptibility for the chiral p-wave SCs is given by\nRe\u001fR\nuni(!) =2DFZ1\n\u0001dEEp\nE2\u0000\u00012(1 + cos2\u0012)\u00012\n4E2(f(E)\u0000f(\u0000E))\u00121\n2E+~!+1\n2E\u0000~!\u0013\n; (S.129)\nand\nIm\u001fR\nuni(!) =2\u0019DFj~!=2jp\n(~!=2)2\u0000\u00012(1 + cos2\u0012)\u00012\n(~!)2(f(\u0000~!=2)\u0000f(~!=2)): (S.130)\nFrom the above expressions, one can show that both the real part and imaginary part of the uniform spin susceptibility\ndiverge at ~!= 2\u0001, leading a resonance peak. The expressions for the helical p-wave SC can be obtained by replacing\ncos2\u0012with1\n2sin2\u0012. Therefore, \u0012dependence of \u001fR\nuni(!) explained in the main text is obtained from the above\nexpressions.18\nΓintra\ninter\nk+Ek\n−EkE E\nSpectral\nfunction\n2/uni0394/uni210F/uni03C9\n/uni210F/uni03C9Γ\nFIG. 5. Schematic image of intraband transition and interband transitions. The intraband transition gives contribution to the\nuniform spin susceptibility when the excitation energy is comparable to or smaller than the level broadening, ~!.\u0000. The\ninterband contribution is dominant when the excitation energy is comparable to the superconucting gap, ~!\u00192\u0001.\nVIII. LOCAL SPIN SUSCEPTIBILITY\nPerforming the angular integral and replacing the sum over k;qby theE;E0integral, the matrix elements are\nreplaced by\nM\u0015\u00150(s)\nk;q!1\n4+\u00012\n4\u0015E\u00150E0; (S.131)\nM\u0015\u00150(c)\nk;q!1\n4; (S.132)\nM\u0015\u00150(h)\nk;q!1\n4: (S.133)\nThe matrix elements for the chiral and helical p-wave SCs are identical. From the above expressions, one can see\nthat the interband contribution in the s-wave SC is suppressed. Unlike the uniform spin susceptibility, the intraband\ncontribution for the local spin susceptibility is \fnite even when \u0000 = 0. This is because the transition processes\nconsidered here leads to momentum transfer and the intraband transition is not forbidden. Therefore, we calculate\nthe local spin susceptibility at \u0000 = 0. The local spin susceptibility for the s-wave SC is given by\n\u001fR\nloc(!) =\u0000D2\nFZ1\n\u00001dEZ1\n\u00001dE0Ds(E)Ds(E0)\u0012\n1 +\u00012\nEE0\u0013f(E)\u0000f(E0)\nE\u0000E0+~!+i0; (S.134)\nand the local spin susceptibility for the p-wave SCs is given by\n\u001fR\nloc(!) =\u0000D2\nFZ1\n\u00001dEZ1\n\u00001dE0Ds(E)Ds(E0)f(E)\u0000f(E0)\nE\u0000E0+~!+i0: (S.135)\nIX. FMR MODULATION: ROUGH INTERFACE\nIn this section, we show the numerical results and summarize the characteristic properties of the FMR modulation\nfor the rough interface limit. In the following calculations, we set J1= 0 and assume that only \u001fR\nloc(!) contributes to\n\u000eHand\u000e\u000b.\nFigures 6 show (a) \u000eHand (b)\u000e\u000bfor the chiral and helical p-wave SCs as a function of frequency and temperature.\n\u000eHis \fnite inT!0 and has a resonance peak at ~!= 2\u0001.\u000e\u000bexhibits a coherence peak just below the transition\ntemperature in the su\u000eciently low frequency region, where ~!=kBTc\u001c1.\u000e\u000bdrops abruptly at ~!= 2\u0001.\u000e\u000bis\nalmost independent of both frequency and temperature when ~!>2\u0001.\nFigures 6 show (c) \u000eHand (d)\u000e\u000bfor thes-wave SC as a function of frequency and temperature. In the low\nfrequency region, where ~!=kBTc\u00141,\u000eHat a \fxed frequency decreases by about thirty percent with the decrease of\nthe temperature, and \u000eHis \fnite inT!0. As the frequency increases, \u000eHis almost independent of the temperature.\n\u000e\u000bshows a coherence peak just below the transition temperature in the su\u000eciently low frequency, where ~!=kBTc\u001c1.19\nThe coherence peak in the s-wave SC is larger than the corresponding coherence peak in the p-wave SCs. \u000e\u000bhas a\nkink structure at ~!= 2\u0001.\nNote that the cuto\u000b energy Ecwas introduced here to cause the integral for Re \u001fR\nloc(!) to converge. Although\nRe\u001fR\nloc(!) is approximately proportional to Ec, the qualitative properties explained above are independent of Ec.\nThe FMR modulation properties of the three SCs are summarized in Table II. In the case of the rough interface\nlimit, the pairing symmetry can be detected from either the absence or the existence of the resonance peak of \u000eH. The\npairing symmetry may also be detected from the properties of \u000e\u000b, the height of the coherence peak, and the structure\nat~!= 2\u0001. When compared with the resonance peak for \u000eH, however, the properties of \u000e\u000bare too ambiguous to\nallow the pairing symmetry to be distinguished clearly.\n(c) (d)s-wave(a) (b)Chiral & Helical p-wave\nT/T\nchω/kBTcδH/δH2\n0.00.51.00\n2\n410\n8\n6\nT/T\nchω/kBTcδα/δα2\n0.00.51.00\n2\n42\n1\n0\nT/T\nchω/kBTcδH/δH2\n0.00.51.00\n2\n410\n8\n6\nT/T\nchω/kBTcδα/δα2\n0.00.51.00\n2\n42\n1\n0\nFIG. 6. (a) The frequency shift and (b) the enhanced Gilbert damping as a function of both frequency and temperature for the\np-wave SCs. (c) The frequency shift and (d) the enhanced Gilbert damping as a function of both frequency and temperature\nfor thes-wave SC. The terms \u000eH2and\u000e\u000b2are given by \u000eH2=\u00002\u0019SJ2\n2l2D2\nFkBTc=(NA\r ~) and\u000e\u000b2= 2\u0019SJ2\n2l2D2\nF=(NA),\nwhere they are characteristic values in the normal state. The cuto\u000b energy is set to be Ec=kBTc= 10.\nTABLE II. 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Linder,\nPhysical Review Letters 127, 207001 (2021)."    },    {        "title": "1805.01776v2.Superparamagnetic_Relaxation_Driven_by_Colored_Noise.pdf",        "content": "arXiv:1805.01776v2  [cond-mat.stat-mech]  7 May 2018Superparamagnetic Relaxation Driven by Colored Noise\nJ. G. McHugh,1R. W. Chantrell,1I. Klik,2and C. R. Chang2\n1Department of Physics, The University of York, York, YO10 5D D, UK\n2Department of Physics, National Taiwan University, Taipei , Taiwan\nA theoretical investigation of magnetic relaxation proces ses in single domain particles driven\nby colored noise is presented. Two approaches are considere d; the Landau-Lifshitz-Miyazaki-Seki\nequation, which is a Langevin dynamics model based on the int roduction of an Ornstein-Uhlenbeck\ncorrelated noise into the Landau-Lifshitz-Gilbert equati on and a Generalized Master Equation ap-\nproach whereby the ordinary Master Equation is modified thro ugh the introduction of an explicit\nmemory kernel. It is found that colored noise is likely to bec ome important for high anisotropy\nmaterials where the characteristic system time, in this cas e the inverse Larmor precession frequency,\nbecomes comparable to the correlation time. When the escape time is much longer than the corre-\nlation time, the relaxation profile of the spin has a similar e xponential form to the ordinary LLG\nequation, while for low barrier heights and intermediate da mping, for which the correlation time is\na sizable fraction of the escape time, an unusual bi-exponen tial decay is predicted as a characteristic\nof colored noise. At very high damping and correlation times , the time profile of the spins exhibits\na more complicated, noisy trajectory.\nI. INTRODUCTION\nThermally-activated magnetization reversal over an\nanisotropic energy barrier is the driving force for switch-\ning in magnetic materials. Theoretical understanding\nwasfirst developed by N´ eel1basedon the transition state\ntheory (TST) leading to an Arrhenius-like relaxation\ntime proportional to exp( EB/kBT) whereEBis the en-\nergy barrier, kBthe Boltzmann constant and Tthe tem-\nperature. Brown2provided further insight through the\nconstruction ofthe Langevinequation for the problem by\nthe introduction of white-noise fields into the Landau-\nLifshitz equation with Gilbert damping, leading to the\nstochastic Landau-Lifshitz-Gilbert (LLG) equation, An\nexpression for the relaxation time of thermally-driven\nescape over the energy barrier is then found through\nthe lowest eigenvalue of the correspondingFokker-Planck\nequation(FPE)governingthetime-evolutionoftheprob-\nability density function of the magnetization orientation.\nThe routeto the Arrhenius-likerelaxationtime expres-\nsion is one of two directions leading from the Langevin\nequation. The second, Langevin Dynamics (LD) ap-\nproach is the direct numerical solution of the Langevin\nequation3–6. There is a natural separation of timescales,\nwith LD used for high frequency applications such as\nmagnetic recording and the Arrhenius-like relaxation\ntimeusedforslowdynamicbehaviorarisingfromthermal\nactivationoverenergybarriers. The twoapproacheshave\nbeen compared by Kalmykov et. al.,7who calculated es-\ncape times for both cases giving excellent agreement for\nthe variation of escape time with damping constant and\ndemonstrating the importance of starting the LD calcu-\nlations from the correct thermal equilibrium distribution\nwithin the energy minimum.The LLG equation for a single spin takes the well-\nknown form\ndS\ndt=−γ\n1+α2/parenleftbig\nS×H+αS×(S×H)/parenrightbig\n,(1)\nwhereαis the phenomenological damping constant, γ=\n1.7611T−1s−1andSisaunitvectorinthedirectionofthe\nspin,S=µ/µs. The local magnetic field, H, is derived\nfrom the first derivative of the spin Hamiltonian Hwith\nrespect to the spin degree of freedom,\nH=−1\nµs∂H\n∂S. (2)\nThermal fluctuations are necessary to incorporate the\ndeviations of a particular spin from the average tra-\njectory. This is done via the formal inclusion of ran-\ndom fields in the LLG equation. In order to realize the\nFluctuation-Dissipation theorem for this system, these\nthermal fields must also be proportionalto the same phe-\nnomenological damping constant, αthat occurs in the\ndamping. The moments of the thermal field are then\ngiven by\n∝angbracketleftHth,i(t)∝angbracketright= 0 (3)\n∝angbracketleftHth,i(t)Hth,j(t′)∝angbracketright=2αkBT\nγµsδ(t−t′)δij(4)\nwherei,jlabel the spin components.\nIn all numerical simulations, we interpret the stochas-\ntic equation in the Stratonovich sense and employ the\nHeun method An implicit assumption of this approach is\nthe presence of white noise, which exists in the zero cor-\nrelation time limit for some physical noise process with a2\nwell-defined correlation time. Such a colored noise may\nbe implemented for a magnetic system through the use\nof the Landau-Lifshitz-Miyazaki-Seki pair of Langevin\nequations, which take the form\ndS\ndt=γS×/parenleftbig\nH+η/parenrightbig\n, (5)\ndη\ndt=−1\nτc(η−χS)+R, (6)\nwhereτcis the correlation time and χis a spin-bath cou-\npling which is related to the phenomenological damping\nparameter as α=γχτcin the limit of small correlation\ntimes. The autocorrelation of the white noise field, R, is\ngiven by\n∝angbracketleftRi(t)Rj(t′)∝angbracketright=2χkBT\nτcµsδijδ(t−t′).(7)\nThis pair of Langevin equations leads to a frequency-\ndependent damping of the spin together with an expo-\nnentially correlated noise term in the spin-only space,\n∝angbracketleftˆηi(t)ˆηj(t′)∝angbracketright=χkBT\nµse−(t−t′)\nτcδij=χkBT\nµsK(t−t′)δij\n(8)\nwhereK(t) = exp−(t−t′)\nτcis the exponential memory ker-\nnel. For completeness, additional background on the\nLLMS Langevin equation and colored noise is included\nin Appendix A.\nAn alternative approach to the Langevin equation is\nthe discrete orientation approximation, whereby, in the\nlimit oflarge barriers, the detailed dynamics are replaced\nby phenomenological rate equations describing transi-\ntions between the minima of the magnetic potential. We\nmay augment this description by the introduction of a\nmemory kernel into the rates, thus replacing the master\nequation description with a generalized master equation\nwhich explicitly incorporates the retardation effect into\nthe rate equations.\nHere we investigate the introduction of colored noise\ninto the calculation of escape rates. This leads to sig-\nnificant effects for materials with large magnetocrys-\ntalline anisotropy energies, including the prediction of\nbi-exponential behavior at intermediate damping, when\nthe characteristic time of the relaxation process becomes\ncomparable to the heat bath correlationtime. The paper\nis organized as follows. We first outline thermally acti-\nvated escape times for single nanoparticles, followed by\nanintroductionofcolorednoiseintotheLangevinformal-\nism via the LLMS equations. We then derive the relax-\nation profile from the non-Markovian generalized exten-\nsion of the rate equation, followed by a systematic inves-\ntigationoftheeffects ofthebarrierheightandcorrelation\ntimes on the relaxation profile from LLMS simulations.A. Thermally-Assisted Magnetization Reversal\nWe will investigate here the effect that colored noise\nhas on the dynamics of the thermal escape problem for a\nmagnetic nanoparticle. The spin Hamiltonian of the sys-\ntem contains both an applied field and anisotropy term,\ntaking the form\nH=−KVS2\nz−µs/vectorH·S, (9)\nwhereKis the anisotropy constant and Vis the particle\nvolume. For the escape problem we have a spin energy\npotential of the form\nV(θ,φ) =σβ−1/parenleftbig\nsin2θ−2h(cosψcosθ(10)\n+sinψsinθcosφ)/parenrightbig\n,\nwhereθ,φare respectively the polar and azimuthal\ncomponents of the spin in spherical coordinates, σ=\nKV/k BTis the reduced barrier height parameter, h=\nH/2σis the reduced field, β= (kBT)−1andψis\nthe angle between the easy-axis and the applied field.\nThis potential has a bistable character under the condi-\ntion than the critical applied field value, h < h c(ψ) =\n((cos2/3ψ+sin2/3ψ)−3/213, in which case there are local\nand global minima in the north and south polar regions,\nwith an equatorial saddle point between them. We are\nthen interested in the calculation of the characteristic es-\ncape time of a spin initialized in one such minimum.\nFor the special case of aligned field and easy axis, for\nwhichψ= 0 the potential is\nV(θ) =σβ−1/parenleftbig\nsin2θ−2hcosθ/parenrightbig\n. (11)\nIn this case the escape time takes the Arrhenius form,\nwhere the barrier energy, EB, is proportional to the\nanisotropy energy, leading to an escape time\nτ∝f−1\n0eKV/k BT(12)\nwheref0is the attempt frequency, the frequency of Lar-\nmor gyromagnetic precession at the bottom of the well.\nWeinvestigatetheescapetimeinthecoloredandwhite\nnoise cases through repeated numerical integration of the\nLangevin equations for a spin initialized in a potential\nminimum. An important consideration for such simula-\ntions is the choice of initial and switching condition for\nthe spin. We will initialize the spins with the Boltz-\nmann distribution at the bottom of the well in order to\navoidinconsistenciesat lowdamping, while the switching\ncondition is chosen such that Sz<−0.5, with the spin\ninitialized in the positive z-direction, so that the spin is\nsufficiently deep in the well such that it has escaped.\nII. LLMS ESCAPE TIMES & COLORED NOISE\nA. System time τsvs.τccharacteristic bath time.\nFor the uniaxial escape problem the external field in\nthe LLMS will consist of an external applied part and an3\nanisotropy contribution\nH=Ha+H0 (13)\nthe magnitude of the anisotropic contribution depends\non the orientation of the spin and is given by Ha=\n2ku\nµs/vectorSz·/vectorz=Hk/vectorSz·/vectorzwhere/vectorzis the direction of easy\nmagnetization and kuis the anisotropy energy. To gain\nintuition into the relevant timescales for the relaxation\nproblem, we will assume the uniaxial case in the follow-\ning, where the external field is applied along the same\ndirection as the easy axis, such that both fields only have\ncomponents in the z-direction.\nWe note that the anisotropic field contribution varies\nwith the projection of the spin on to the easy-axis as\nHa=Hk(S·z)/vectorz= (Hkcosθ)/vectorz, (14)\nThelargest field magnitude and consequently the fastest\ntimescale of the problem is set by the value for which\nthe anisotropic field contribution is at its largest, which\nis when the spin and the easy-axis precisely coalign. For\nany other orientation, the field will be smaller and the\ntimescale of oscillation hence slower. We may then take\nthe spin-only Langevin equation,\ndS\ndt=γS(t)×/parenleftbig\n(Hkcos(θ))/vectorz+¯η−χ/integraldisplayt\n−∞dt′K(t−t′)dS(t′)\ndt′/parenrightbig\n,\n(15)\nand proceed to scale this equation by the maximum\nanisotropy field value. Defining the system time for the\nspin asτs= (γHk)−1then\ndS\ndt=1\nτsS(t)×/parenleftbig\ncos(θ)/vectorz+H−1\nk¯η\n−H−1\nkχ/integraldisplayt\n−∞dt′K(t−t′)dS(t′)\ndt′/parenrightbig\n.(16)\nWe may scale the time variable in the Langevin equa-\ntionsothatthesystemtimeisremovedbytaking ζ=τst.\nThen we have\ndS\ndζ=S(ζ)×cos(θ)/vectorz+S(ζ)×/parenleftbigg\nH−1\nk¯η(ζ)\n+H−1\nkχ/integraldisplayζ′\n−∞dζ′e−(ζ−ζ′)τs\nτcdS(ζ′)\ndζ′/parenrightbigg\n(17)\nThe autocorrelation of the noise is similarly transformed\nto become\n∝angbracketleft¯η(ζ)¯η(ζ′)∝angbracketright=τs\nτc¯De−ζ−ζ′)τs\nτc=¯D\nτe−(ζ−ζ′)/τ(18)\nwhereτs/τc=τand¯D=D/τs=χτkBT/µs. We\ncan then write the coupling as ¯ χ=χ/Hk, and ab-\nsorb theHkfactor into the diffusion constant for the\nthermal field. Since the thermal fields are given by\n¯η(ζ) =√\n2D\nτ/integraltextζ\n−∞K(ζ−ζ′)Γ(ζ′), the diffusion constant\nbecomes\n¯D=χτkBT\nµsH2\nk=¯χτkBT\n2ku=¯χτ\n2σ(19)11.21.41.61.82\n0.01 0.1 1τ/τLLG\nγ Hk τcσ = 2\nσ = 5\nσ = 7.5\nFIG. 1. Escape time, normalized to the uncorrelated LLG\nescape time vs correlation time, from LLMS simulations for a\nCo nanoparticle with α= 0.05 and different reduced barrier\nheights, σ,\nwhereσ=ku/kBT. The final expression for the\nLangevin equation is then\ndS(ζ)\ndt=S(ζ)×/parenleftbig\ncos(θ)/vectorz+¯η−¯χ/integraldisplayζ\n−∞dζ′K(ζ−ζ′)dS(ζ′)\ndζ′/parenrightbig\n.\n(20)\nIn the case that τ≪1 andτc≪τs, the memory\nkernels appearing in the noise and damping terms are\nreduced to delta functions and the white noise behav-\nior is restored. Additionally the bath coupling and the\nstrength of the thermal fluctuations are reduced by the\nanisotropy field, so that in the event of a very large\nanisotropy the precessional dynamics of the spin dom-\ninate the thermal and damping parts. We then conclude\nthat the condition τc/greaterorsimilar(γHk)−1dictates whether the\neffect of correlations are relevant in the system dynamics\nin the high barrier limit.\nThis prediction is borne out in numerical simulations\nof the LLMS equation. Figure 1 depicts the escape time\ncalculated using the LLMS model for a Co nanoparti-\ncle of volume V= 8×10−27m3, with anisotropy en-\nergyKV= 1.12×1021J, and a magnetic moment µs=\n1.12×10−20J/T.wherethe correlationtime is normalized\nby the inverse of the Larmor precession frequency, and\nthe escape time in the LLMS is normalized by the escape\ntime calculated from the Markovian LLG equation. The\nescape rate departs from the LLG escape rate only once\nthe correlation time is some significant fraction of the\nLarmor time, and for increasing barrier height the corre-\nlation time must be a larger fraction of the gyromagnetic\nprecession before the escape rate departs from the LLG\nprediction.\nFigure 2 shows a comparison of the escape time for the\nCo nanoparticle and a SmCo 5nanoparticle of the same4\n110\n1.0E-15 1.0E-14 1.0E-13 1.0E-12 1.0E-11 1.0E-10τ/τLLG\nτcSmCo\nCo\nFIG. 2. Comparison of simulation results for systems with pa -\nrameters chosen tobe similar toSmCo 5and Co nanoparticles,\nrespectively, for large reduced barriers σ= 13.5, and a fixed\nα= 0.05. The higher anisotropy SmCo 5exhibits departure\nfrom LLG behavior at smaller correlation times.\nvolume. The SmCo 5material parameters are taken to be\nµs= 6.4×10−18J/T, and anisotropy KV= 2.16×10−16,\na much higher anisotropy energy density than Co. This\nhigher anisotropy gives the nanoparticle a faster system\ntime, which causes the LLMS to depart from the LLG\nfor smaller bath correlation times, τc, on the order of\n50−100fsfor the SmCo 5particle, while it is approxi-\nmately 1psfor the Co nanoparticle. The fact that the\nsystem time is inversely proportional to the magnitude\nof the anisotropy field is exhibited in the simulations by\nthe difference between LLMS and LLG escape rates at\nsmaller values of the bath correlation time for the mate-\nrial with higher magnetic anisotropy.\nB. Arrhenius Behavior\nCrucially, it is found that the Arrhenius behavior of\nthe escape rate is recovered from LLMS simulations in\nthe limit of large barrier height. In figure 3 we show the\ntemperature- dependence of the escape time vs reduced\nbarrier height.\nIn the high damping case, we see that the escape rates\nbegin to convergeas the temperature tends towardszero.\nAs the escape time between the wells becomes much\nlonger than the bath correlation time, the detailed dy-\nnamics of the spin within the well becomes less relevant.\nAt low damping, the LLMS and LLG appear not to\nconverge even at the larger barrier heights considered\nhere. We attribute this difference to the difference in\ndamping regimes and the physically distinct mechanisms\ninvolved in the escape process between the two regimes.\nEscape at high damping is mediated by thermal fluctu-0.010.1110100100010000100000\n0246810121416τkr γ Hk\nσLLMS, h=0.2, α = 0.01\nLLG, h=0.2, α = 0.01 Ψ = 1/4\n0.010.1110100100010000\n0246810121416τkr γ Hk\nσLLMS, h=0.3, α = 1, Ψ = 1/4\nLLG, h=0.3, α = 0.01, Ψ = 1/4\n0.010.1110100100010000\n0246810121416τkr γ Hk\nσLLMS, h=0.2, α = 1\nLLG, h=0.2, α = 1 Ψ = 1/4\n0.010.11101001000\n0246810121416τkr γ Hk\nσLLMS, h=0.3, α = 1, Ψ = 1/4\nLLG, h=0.3, α = 1, Ψ = 1/4\nFIG. 3. Escape time, τγHkvs reduced barrier height, σ,\nfrom LLMS and LLG simulations, for different values of the\napplied field h=µsH/σand damping, α, with a fixed angle\nof Ψ = π/4 between the applied field and the easy axis of\nmagnetization. 1:Low damping, h= 0.2,2:Low damping,\nh= 0.3.3:High damping, h= 0.2,4:High damping,\nh= 0.3.\nations, which liberate the bound spin. In the limit of\nvanishing temperature the infrequency of thermal oscil-\nlations of sufficient energy dominate the escape behavior\nand the escape rates converge.\nIn contrast, the energy-controlled diffusion regime is\ncharacterized by the almost-free precessional motion of\nthe spin in the well. In the highly correlated case, the5\nsimple damping is replaced with a frequency-dependent\ndamping, an effect which increases the overall effective\ndamping. In the limit T→0, this inhibits the escape\nratebetweenthewellsbydecreasingtherateatwhichthe\nspin is able to attain a trajectory with sufficient escape\nenergy.\nIII. RATE EQUATIONS FOR\nTHERMALLY-ACTIVATED MAGNETIZATION\nREVERSAL\nA. Master Equation\nThe master equation is a phenomenological set of first-\norder differential rate equations for a multi-level system,\nwhich takes the form\ndni\ndt= Γij(t)nj(t), (21)\nwhereniis a probability vector representing the proba-\nbility that the system is in one of a discrete set of states,\nandi,jlabel those discrete states, while the matrix of\ncoefficients Γ i,jdictates the transition rate from state i\nto the state jof the system.\nThe dynamics of the thermally-assisted escape prob-\nlem in a magnetic system may be approximated by such\na master equation under the condition that the energy\nbarrier is large compared to the thermal energy, σ >1,\nbut not too large such that it would inhibit inter-well\ntransitions. This approximation to the Langevin dynam-\nics is called the discrete orientation approximation . The\nspin orientations are assumed to be restricted only to the\n2 minima of the potential energy dictated by the spin\nHamiltonian. The time evolution of the occupation of\neach state follows from Eq. 21, where i,j= 1,2. The\ntransition matrix elements follow from the applied field,\nanisotropy and temperature. In particular, we will as-\nsume a fixed applied field, such that the transition rates\nare constant in time and the matrix takes the form\nΓij=/parenleftbigg\n−κ12κ21\nκ12−κ21/parenrightbigg\n, (22)\nIn the uniaxial case these rates are given by κ1→2=\nκ12=f0exp(−σ(1+h)2) andκ2→1=κ21=\nf0exp(−σ(1−h)2), whereσandhare the reduced bar-\nrier height and applied field, respectively. The time evo-\nlution of the population of the state n1is then explicitly\ngiven by\ndn1\ndt=−κ12n1+κ21n2= (κ12+κ21)n1+κ21.(23)\nThe time-evolution of the magnetization follows from\nthe individual rates for the two wells, where the magne-\ntization is given by m(t) =n1(t)−n2(t) and is subject\nto the normalization condition n1(t) +n2(t) = 1. The\ndifferential equation for the magnetization is then\ndm\ndt=−Γ1m(t)−Γ2, (24)where Γ 1=κ12+κ21and Γ 2=κ12−κ21. This is the\nsame form as the rate for the individual wells, Eq. 23.\nFor an initial magnetization m0=n1(t= 0)−n2(t=\n0), the magnetization as a function of time is a simple\nexponential,\nm(t) =e−Γ1t(Γ1m0+Γ2)\nΓ1−Γ2\nΓ1,(25)\nwhich tends to the value\n−Γ1\nΓ2=κ21−κ12\nκ12+κ21. (26)\nIn the long-time limit, the steady state magnetization\ncorresponding to the difference in the transition rates\nbetween the wells, if κ2→1> κ1→2, the transition rate\ninto well 1 is greater than the rate out, and we have a\npositive magnetization, as expected.\nB. Generalized Master Equation\nThe non-Markovian extension of the master equation\nformalismiswhat iscalledageneralizedmasterequation.\nUnder this model, the set of i×jrates represented in the\ntransition matrix in Eq. 21 are promoted to a set of i×j\nmemory kernels for the transitions between the wells i,j,\nreplacing the set of first-order differential equations with\na set of integro-differential equations for the population\nof each well,\ndni\ndt=/integraldisplay∞\n0Mij(t−τ)n(τ)dτ. (27)\nWe will consider the simplified case\nMij(t) =e−t/Θ\nΘAij=K(t)Γij, (28)\nwhere Γ ijare the same constant transition rates consid-\nered in the Markovian master equations, now modified\nby a simple exponential kernel over the recent popula-\ntion of the well. The integro-differential expression for\nthe magnetization then becomes\ndm\ndt=−Γ1/integraldisplay∞\n0K(t−τ)m(τ)dτ−Γ2/integraldisplay∞\n0K(t−τ)dτ.\n(29)\nWhere we note that for the exponential kernel, K(t) =\ne−t/Θ\nΘ, the uncorrelated form of the master equation\nis recovered in the limit of vanishing correlation time,\nlimΘ→0K(t) =δ(t).\nThe Laplace transform of this equation is\nωm(ω)−m0=−Γ1K(ω)m(ω)−Γ2\nωK(ω),(30)\nwhereK(ω) =L(K(t)) is the Laplace transform of the\nmemory kernel,\nK(ω) =Θ−1\nω+Θ−1=1\n1+Θω, (31)6\nwe then have\nm(ω) =−Γ2\nωK(ω)+m0\nω+Γ1K(ω). (32)\nAfter inserting the expression for the Laplace transform\nof the kernel we find\nm(ω) =−Γ2\nω+m0(1+Θω)\nΘω2+ω+Γ1. (33)\nFinally we solve for the time-dependence of the mag-\nnetization by taking the inverse Laplace transform,\nm(t) =L−1[(1+Θω)\nΘω2+ω+Γ1] =φ(t)(Γ1m0+Γ2)\nΓ1−Γ2\nΓ1,\n(34)\nwe note that this bears a strong resemblance to the\nMarkovian expression, Eq. 25, with the exponential be-\ning replaced by the function φ(t), which is\nφ(t) =1\n2β/parenleftBig\n(β−1)e−t(1+β)/2Θ+(β+1)e−t(1−β)/2Θ/parenrightBig\n,\n(35)\nwhereβ=√1−4Γ1Θ. In the limit t→ ∞, the value of\nthe magnetization again tends to−Γ2\nΓ1. To see that this\nagrees with the uncorrelated solution for small correla-\ntion times, we may expand βin Θ for small Θ, hence\nβ= 1−2Γ1Θ, inserting into the magnetization it be-\ncomes\nm(t) =β−1\n2βe−t/2ΘeΓ1t+(β+1)\n2βe−Γ1t.(36)\nAs Θ→0,β→1, and only the second term in the\nexpression for the magnetization remains, m(t) =e−Γ1t,\nso the small correlation time limit of the spin evolution\nagrees with the non Markovian master equation.\n00.10.20.30.40.50.60.70.80.91\n00.511.522.53m(t)\nΓ1tR < 0.01\nR=0.1\nR = 0.24\nFIG. 4. m(t) vst, forR= 0,0.1,0.2, under the initial condi-\ntionm= 1, with transition rates κ12= 1,κ21= 0Finally, we note that the solution for the magnetiza-\ntion breaks down into two regimes. First, we note that\nthe expression for βdepends only on the product of the\ncorrelation time, Θ, and the rate Γ 1, and not on their\nspecific individual values. We may then discuss the be-\nhavior of the model in terms of only the ratio parameter\nR= Γ1Θ = Θ/Γ−1\n1, which gives the ratio of the well\ncorrelation time to the escape time. Rewriting the Eq.35\nfor the spin vs time,\nm(t) =(Γ1m0+Γ2)\nΓ1/parenleftBig\n(e−t/2Θ([eβt/2Θ(37)\n−e−βt/2Θ]/2β+[e−βt/2Θ+eβt/2Θ]/2)/parenrightBig\n−Γ2\nΓ1,\nwhich may be simplified in terms of hyperbolic trigono-\nmetric functions,\nm(t) =e−t/2Θ/parenleftBigsinh(βt/2Θ)\nβ+cosh(βt/2Θ)/parenrightBig\n.(38)\nFor smaller R <1\n4, we have a real value of β=√1−4R, and the time-dependence of the spin corre-\nsponds to Eq. 38. In Figure 4, we plot the time-evolution\nfor values of R <1\n4. Once the correlation time is some\nsizable fraction of the escape time, the behavior begins\nto depart from the simple exponential behaviorpredicted\nin the Markovian system. At early times the magnetiza-\ntion decays more slowly than the exponential decay and\nat later times it decays more quickly, while the timescale\nover which the decay occurs (Γ 1) remains the same. The\neffect of the increasing correlation time between the pop-\nulations of the wells is then to shift the process to differ-\nent, lower frequencies.\nIn the case that R >1\n4, we have an imaginary argu-\nment to sinh and cosh, we then have an expression for\nm(t)\nm(t) =e−t/2Θ(sin(bt/2Θ)\nb+cos(bt/2Θ)) (39)\nwhereb=√\n4R−1. We note that the solutions take\nthe form of damped oscillations which tends toward the\nequilibrium value of the magnetization. However, these\nsolutions are unphysical as the occupation in individual\nwells maybecome lessthan 0 for these values. This is not\nsurprising, as for longer correlation times the generalized\nmaster equation will overestimate the population in each\nwell and generate a time evolution which will continue to\nreduce the population of a well, even when that well is\npresentlyempty. Itisalsounclearwhatitwouldmeanfor\nthe correlation time of the well population to exceed or\nbe on the order of the overall escape time, as this would\nimplythatthetimescaleoverwhichthespinpopulationis\ncorrelatedexceeds the overallescape time for the system,\nwhich is itself determined by changes in the individual\nwell populations.7\nIV. COMPARISON\nWe may now directly compare the magnetic relaxation\nprofiles calculated from explicit numerical integration of\nEqs. 5, 6 at various barrier heights, damping and cor-\nrelation times, to the biexponential decay predicted by\nthe generalized master equation. In all of the present\nsimulations we again use simulation parameters compa-\nrable to the Co nanoparticle of volume V= 8×10−27m3,\nanisotropy energy density K= 4.2×105J/m3giving an\nanisotropy energy KV= 1.12×1021J, and a magnetic\nmomentµs= 1.12×10−20J/T, while no external applied\nfield is assumed, Hext= 0.\n00.20.40.60.81\n0e+002e-094e-096e-098e-091e-08M(t)/Mr\nt (s)σ = 2, α = 0.01\ne-t/τ\n00.20.40.60.81\n0e+002e-094e-096e-098e-091e-08M(t)/Mr\nt (s)σ = 6, α = 0.5\ne-t/τ\nFIG. 5. Spin relaxation profiles from LLG simulations for\nTOP:σ= 2,α= 0.01, giving an exponential decay with\ncharacteristic escape time τ= 5×10−9sandBOTTOM :\nσ= 6,α= 0.5,τ= 4.5×10−9s\n.\nThe spins are initialized in the equilibrium Boltz-\nmann distribution in one of the minima of the po-\ntential energy, according to the distribution P(θ)∝\nsin(θ)exp(−ku/kBTsin2(θ)). To ensure that the noise\nis equilibrated with the spin at the correct temperature,\nthe noise is initially set to ηi,j,k= 0, and is then evolved\nin the presence of the equilibrium distribution in the well\nuntiltheycomeintothermalequilibrium. Theinitialcon-\ndition of the noise is important, as, for example, a choice\nofη(t= 0) = 0, will result in a field which quickly alignswith the spins in the potential minimum and give an un-\nphysical increase in the well population from equilibrium\nat short times.\nThe time-evolution of the magnetization, M(t) =\n∝angbracketleftSz(i)∝angbracketrightis then plotted, normalized by the initial rema-\nnent magnetization inside of the well, Mr=M(0).\n00.20.40.60.81\n0e+002e-094e-096e-098e-091e-08M(t)/Mr\nt (s)τc = 1, σ = 2, α = 0.01\ne-t/τ\n00.20.40.60.81\n0e+002e-094e-096e-098e-091e-08M(t)/Mr\nt (s)τc = 1, σ = 6, α = 0.5\ne-t/τ\nFIG. 6. Exponential behavior from LLMS simulations, for\nTOP:τc= 1,σ= 2.α= 0.01 we have an exponential decay\nwith escape time τ= 5.5×10−9, andBOTTOM :τc= 1,\nσ= 6.α= 0.5τ= 5.3×0−9s For low damping and large\nbarrier heights, the correlation time is much smaller than t he\nescape time.\nIn Figure 5, we depict the numerical calculation of the\nrelaxation profile from the LLG. This gives rise to an ex-\nponential behavior with a single relaxation time, which\nis directly comparable to the exponential decay of the\nmaster equation. In general, the relaxation profile from\nthe LLG may be non-exponential, with both the inte-\ngral relaxation time and the decay profile depending on\nthe higher-order eigenvalues of the Fokker-Planck opera-\ntor and the equilibrium correlation functions of the spin,\nτi\nint=/summationtext\nkτi\nkλk. However, the relaxation is dominated\nby the first eigenvalue in the high-barrier limit and for\nsmall applied fields , for σ>1, with good agreement be-\ntween the LLG and exponential decay for σas low as 2,\nas is shown in Figure 5.\nFigure 6 shows the relaxation from LLMS simulations\nof the Co nanoparticle, where the correlation time is cho-8\n0.10.20.30.40.50.60.70.80.91\n0e+005e-111e-101e-102e-103e-10M(t)/Mr\nt (s)τc = 1, σ = 2, α = 0.5\ne-t/τ\nFIG. 7. Biexponential behavior from LLMS simulations for\nτ= 1,σ= 2,α= 0.5 andτ= 1.48x10−10\nsen to be of the order of the inverse Larmor precession\ntime such that τc≈(γHk)−1. In both the cases of low\ndamping and higher barriers, we see that the ordinary\nexponential behavior of the LLG is retained. In this case\nthe escape time is much larger than the correlation time\nofthenoise,andtherelaxationaldynamicsareunaffected\nby the intra-well dynamics of the spin which occur on a\nmuch faster timescale than the relaxation, τc/τ≈0.01\nfor both simulations.\nIn the intermediate-to-high damping and high damp-\ning regimes, the behavior of the magnetization becomes\nmuchmoreinterestinganddepartsfromthe LLG. In par-\nticular,forarelativelysmallbarrierof σ= 2,α= 0.5and\nacorrelationtimeagainoftheorderoftheinverseLarmor\nfrequency. In this case the ratio of the escape to the cor-\nrelationtime is τc/τ= 9.4×10−12s/1.48×10−10s≈0.06.\nThe influence of the spin correlation is now visible in the\nrelaxation profile of the escape, as shown in Figure 7,\nwhich is similar to the biexponential deviation predicted\nby the generalized master equation, with the relaxation\nproceeding more slowly at earlier times and speeding up\nat later times.\nFinally, for very long correlationtimes and high damp-\ning, the correlation time remains a sizable fraction of\nthe escape time. However the biexponential behavior\nis no longer evident as shown in figure 8. The decay\nremains approximately exponential with a highly noisy\npath, a possible indication that the precise decay profile\nis extremely dependent on the initial conditions for such\nstrong coupling between the spin and bath.\nV. CONCLUSIONS\nWe have investigated thermal relaxation in magnetic\nnanoparticles introducing colored noise. Two models00.20.40.60.81\n0e+002e-104e-106e-108e-101e-09M(t)/Mr\nt (s)τc = 5, σ = 2, α = 0.5\ne-t/τ\n00.20.40.60.81\n0e+001e-102e-103e-104e-105e-10M(t)/Mr\nt (s)τc = 5, σ = 2, α = 5\ne-t/τ\nFIG. 8. LLMS simulations at high damping and long corre-\nlation times. The behavior continues to depart from a purely\nexponential decay, but now exhibits a noisy, more compli-\ncated time-dependence. TOP:τc= 5,σ= 2 ,α= 0.5 and\nτ= 4.5×10−10,BOTTOM :τc= 5,σ= 2 ,α= 5 and\nτ= 1.8×10−10\n.\nare considered. The first is an approach based on the\nnumerical solution of the Landau-Lifshitz-Miyazaki-Seki\n(LLMS) model, which replaces the white noise approx-\nimation associated with the use of LLB-equation based\nmodels. Due to computational requirements the LLMS\napproach is useful for relatively short timescales, conse-\nquently a second approach is derived based on a general-\nizedmasterequationapproachinvolvingtheintroduction\nof a memory kernel. We find that the importance of col-\nored noise is determined by the ratio of the correlation\ntimeτcto the characteristic system time τs= (γHk)−1,\nwhich is essentially the Larmor precession time. Con-\nsequently correlated noise should become important for\nmaterials with large magnetic anisotropy such as SmCo 5\nwhere the characteristic time approaches femtoseconds.\nBoth models, the LLMS-based approach and the mas-\nter equation, although derived for different timescales,\nexhibit an unusual bi-exponential decay of the magneti-\nzation, which represents an interesting signature of the\npresence of colored noise.9\nAppendix A: Colored Noise\nIn this appendix we present some relevant background\nmaterial on the LLMS equation and colored noise.\n1. Landau-Lifshitz-Miyazaki-Seki\nThe LLMS equations constitute an implementation of\na colored noise in a system with a thermalization con-\ndition represented through the Fluctuation-Dissipation\ntheorem. We reproduce here the original derivation by\nMiyazaki and Seki10, of the spin-only expression of the\nLLMS, which allows us to compare the LLMS thermal\nfluctuations directly to the Ornstein-Uhlenbeck. The\ntime evolution of the LLMS noise term is similar to the\nOU, with an additional term which couples explicitly to\nthe spin,\ndη\ndt=−1\nτc/parenleftBig\nη(t)−χS(t)/parenrightBig\n+R. (A1)\nTakingD=χkBT\nµs, then the autocorrelation of the field\nRis∝angbracketleftR(t)R(t′)∝angbracketright= 2D\nτcδ(t−t′), and proceeding to solve\nas a first-order linear differential equation in the same\nmanner as the OU noise, we have\nη(t) =χ\nτc/integraldisplayt\n−∞dt′K(t−t′)S(t′) (A2)\n+/radicalbigg\n2D\nτc/integraldisplayt\n−∞dt′K(t−t′)Γ(t).\nAfter integrating the first term by parts, we have\nη(t) =/radicalbigg\n2D\nτc/integraldisplayt\n−∞dt′K(t−t′)Γ(t) (A3)\n−χ/integraldisplayt\n−∞dt′K(t−t′)dS(t′)\ndt′,\nand by inserting this into the precessional equation for\nthe spin, we get the spin-only form for the LLMS equa-\ntion,\ndS\ndt=γS(t)×/parenleftBig\nH+¯η−χ/integraldisplayt\n−∞dt′K(t−t′)dS(t′)\ndt′/parenrightBig\n,(A4)\nwhere we now label the thermal fluctuations by ¯η(t),\n¯η(t) =/radicalbigg\n2D\nτc/integraldisplayt\n−∞dt′K(t−t′)Γ(t′).(A5)\nThe autocorrelation of this thermal field is\n∝angbracketleft¯η(t)¯η(t′)∝angbracketright=DK(t−t′) (A6)\n=χkBT\nµsK(t−t′) =β−1\nµsχK(t−t′),\nRecognizing χK(t−t′) as the damping term, we see that\nthis is a representation of the Fluctuation-Dissipationtheorem for the colored noise, where the additional fac-\ntor ofµsarises from the spin normalization. Taking the\nzero correlation time limit,\nlim\nτc→0∝angbracketleft¯η(t)¯η(t′)∝angbracketright= 2Dτcδ(t−t′).(A7)\n00.20.40.60.81\n00.511.522.533.5P(θ)\nθLLMS, σ = 1\nAnalytical\n00.20.40.60.81\n00.511.522.533.5P(θ)\nθLLMS, σ = 10\nAnalytical\nFIG. 9. P(θ) vsθ, from numerical simulations of the LLMS\nequation for TOP:σ= 1 and BOTTOM :σ= 10, with\nτcγHk= 2.\n.\nWe note that the LLMS thus derived from the physi-\ncal consideration of the spin-field interaction is not im-\nmediately comparable with the typical expression for the\nOrnstein-Uhlenbeck colored noise, owing to the fact that\nthe 1/τcterm has been implicitly absorbed in the white\nnoise term. If we rescale the driving noise such that\nQ(t) =τcR(t), we then have a pair of Langevin equa-\ntions\ndS\ndt=γ(S×(H+η)), (A8)\nwhile the noise evolves as,\ndη\ndt=−1\nτc/parenleftBig\nη(t)−χS(t)+Q/parenrightBig\n. (A9)\nThe autocorrelation of the white noise is\n∝angbracketleftQ(t)Q(t′)∝angbracketright=2χτckBT\nµsδ(t−t′) = 2Dδ(t−t′),(A10)10\nwithD=χτckBT\nµs, while the limit of the autocorrelation\nof the thermal term in the spin-only expression is now,\nlim\nτc→0∝angbracketleft¯Q(t)¯Q(t′)∝angbracketright=D\nτcδ(t−t′),(A11)\nwhich is directly comparable to the Ornstein-Uhlenbeck\nform of the colored noise. The expression of the LLMS\nin terms of the bath variable Qhas the additional benefit\nthat/bracketleftbig\nQ/bracketrightbig\n=Tand so we can interpret Qas the thermal\nmagnetic field contribution to the evolution of the bath\nfield.\nFinally, we may see that the limit of the LLMS equa-\ntion for vanishing correlation time is the LLG equation.\nFor small correlation times we can then take the Taylor\nexpansion about the time tint′, so that the damping\nterm becomes,\n/integraldisplayt\n−∞K(t−t′)dS(t′)\ndt′dt′=/bracketleftBig/integraldisplayt\n−∞K(t′)dt′/bracketrightBigdS(t)\ndt+...\n(A12)\nHence the spin and memory kernel decouple in the small\ncorrelation time limit, and the Langevin equation be-\ncomes\ndS\ndt=γS(t)×/parenleftBig\nH+¯η−/bracketleftBig\nχ/integraldisplayt\n−∞dt′K(t−t′)/bracketrightBigdS(t)\ndt/parenrightBig\n,\n(A13)\nAfter performing the integration over t′, the damping is\nχ/integraldisplayt\n−∞dt′e−(t−t′)/τc=χτc. (A14)\nand by direct comparison of the damping terms in this\nexpression and in Gilbert’s equation we have the rela-\ntionship of the phenomenological damping to the LLMS\nparameters α=χγτc. We note also that this expression\ncan be seen if we identify the driving white noise in the\nbath field of the LLMS with the thermal magnetic fieldsof the LLG.\n∝angbracketleftQ(t)Q(t′)∝angbracketright=2χτckBT\nµsδ(t−t′)\n=2αkBT\nγµsδ(t−t′)\n=∝angbracketleftHth(t)Hth(t′)∝angbracketright(A15)\nunder the assumption that α=γχτc.\n2. Thermalization\nAs a quantitative evaluation of the LLMS model and\nour implementation thereof, we compare the equilibrium\nbehavior to the appropriate analytical Boltzmann distri-\nbution, which the Markovian LLG equation also satis-\nfies. We simulate a single spin under the influence of\nanisotropy only. The Boltzmann distribution for such a\nsystem is\nP(θ)∝sinθexp(−kusin2θ\nkBT) (A16)\nwhereθis the angle between the spin and the easy-\naxis and the factor of sin θarises from normalizing the\nprobability distribution on the sphere. WE initialize the\nspin along the easy-axis direction, then allow the spin to\nevolve for 108steps after equilibration and evaluate the\nprobability distribution by recording the number of steps\nthe spin spends at each angle to the easy-axis.\nIn Figure 9, we compare the numerical results to the\nanalytical expression for both the LLMS model and the\nstandard LLG augmented by Ornstein-Uhlenbeck fields\nof the type generated by the Langevin equation in Eq. 4.\nThe simulations using the LLMS model agree with the\nanticipated Boltzmann distribution at equilibrium, while\nthe LLG with Ornstein-Uhlenbeck fails to reproduce the\ncorrect distribution. This is because, as we have argued,\nthis does not comprise a correct implementation of the\nFluctuation-Dissipation theorem, with deviations corre-\nsponding to the missing high-frequency components of\nthe damping.\n1L. N´ eel, Ann. G´ eophys. C.N.R.S. 5, 99 (1949).\n2W. F. Brown, Jr., Phys. Rev. 130, 1677 (1963).\n3A. Lyberatos, D.V. Berkov and R.W. Chantrell, J Phys:\nCondens. Matter 5, 8911 (1993)\n4A. Lyberatos and R. W. Chantrell, J. Appl. Phys. 73, 6501\n(1993).\n5J. L. Garc´ ıa-Palacios and F. J. L´ azaro, Phys. Rev. B 58,\n14937 (1998).\n6D. V. Berkov, IEEE Trans. Magn. 38, 2489 (2002).7Y. P. Kalmykov, W. T. Coffey, U. Atxitia, O. Chubykalo-\nFesenko, P. M. D´ ejardin, and R. W. Chantrell, Phys. Rev.\nB82, 024412 (2010).\n8R. Street and J. C. Woolley, Proc. Phys. Soc. A 62, 562\n(1949).\n9U Atxitia, O. Chubykalo-Fesenko, R. W. Chantrell, U\nNowak and A. Rebei, Phys. Rev. Lett. 102, 057203 (2009).\n10K. Miyazaki and K. Seki, J. Appl. Phys. 112, 121301\n(2012).11\n11U. Atxitia and O. Chubykalo-Fesenkoo, Phys. Rev. B 84,\n144414 .(2011)\n12P. H¨ anggi, P. Jung, Adv. Chem. Phys. 89, 239 (1995).\n13U. Nowak, Annual Reviews of Computational Physics IX,\npg. 105-151 , World Scientific (2001).\n14U. Nowak, R. W. Chantrell, and E. C. Kennedy, Phys.\nRev. Lett. 84, 163 (2000).15W. T. Coffey and Y. P. Kalmykov, J. Appl. Phys. 112,\n121301 (2012).\n16R. F. L. Evans, W. J. Fan, P. Chureemart, T. A. Ostler,\nM. O. A. Ellis and R. W. Chantrell, J. Phys.: Condens.\nMatter26, 103202 (2014).\n17I. M. Sokolov, Phys. Rev. E 66, 041101 (2002).\n18I. M. Sokolov, Phys. Rev. E 63, 056111 (2001)."    },    {        "title": "1806.04782v3.Dynamical_and_current_induced_Dzyaloshinskii_Moriya_interaction__Role_for_damping__gyromagnetism__and_current_induced_torques_in_noncollinear_magnets.pdf",        "content": "arXiv:1806.04782v3  [cond-mat.other]  9 Dec 2020Dynamical and current-induced Dzyaloshinskii-Moriya int eraction: Role for damping,\ngyromagnetism, and current-induced torques in noncolline ar magnets\nFrank Freimuth1,2,∗Stefan Bl¨ ugel1, and Yuriy Mokrousov1,2\n1Peter Gr¨ unberg Institut and Institute for Advanced Simula tion,\nForschungszentrum J¨ ulich and JARA, 52425 J¨ ulich, German y and\n2Institute of Physics, Johannes Gutenberg University Mainz , 55099 Mainz, Germany\nBoth applied electric currents and magnetization dynamics modify the Dzyaloshinskii-Moriya in-\nteraction (DMI), which we call current-induced DMI (CIDMI) and dynamical DMI (DDMI), respec-\ntively. We report a theory of CIDMI and DDMI. The inverse of CI DMI consists in charge pumping\nbyatime-dependentgradient ofmagnetization ∂2M(r,t)/∂r∂t, while theinverseofDDMIdescribes\nthe torque generated by ∂2M(r,t)/∂r∂t. In noncollinear magnets CIDMI and DDMI depend on\nthe local magnetization direction. The resulting spatial g radients correspond to torques that need\nto be included into the theories of Gilbert damping, gyromag netism, and current-induced torques\n(CITs) in order to satisfy the Onsager reciprocity relation s. CIDMI is related to the modification of\norbital magnetism induced by magnetization dynamics, whic h we call dynamical orbital magnetism\n(DOM), and spatial gradients of DOM contribute to charge pum ping. We present applications of\nthis formalism to the CITs and to the torque-torque correlat ion in textured Rashba ferromagnets.\nI. INTRODUCTION\nSince the Dzyaloshinskii-Moriya interaction (DMI)\ncontrols the magnetic texture of domain walls and\nskyrmions, methods to tune this chiral interaction by\nexternal means have exciting prospects. Application of\ngatevoltage[1–3]orlaserpulses[4]arepromisingwaysto\nmodify DMI. Additionally, theory predicts that in mag-\nnetic trilayer structures the DMI in the top magnetic\nlayer can be controlled by the magnetization direction\nin the bottom magnetic layer [5]. Moreover, methods to\ngeneratespin currentsmaybe usedto induceDMI, which\nis predicted by the relations between the two [6, 7]. Re-\ncent experiments show that also electric currents modify\nDMI in metallic magnets, which leads to large changes in\nthe domain-wallvelocity [8, 9]. However,a rigoroustheo-\nretical formalism for the investigation of current-induced\nDMI (CIDMI) in metallic magnets has been lacking so\nfar, and the development of such a formalism is one goal\nof this paper.\nRecently, a Berry phase theory of DMI [6, 10, 11]\nhas been developed, which formally resembles the mod-\nern theory of orbital magnetization [12–14]. Orbital\nmagnetism is modified by the application of an electric\nfield, which is known as the orbital magnetoelectric re-\nsponse [15]. In the case of insulators it is straightfor-\nward to derive the expressions for the magnetoelectric\nresponse directly. However, in metals it is much easier\nto derive expressions instead for the inverse of the mag-\nnetoelectric response, i.e., for the generation of electric\ncurrents by time-dependent magnetic fields [16]. The in-\nverse current-induced DMI (ICIDMI) consists in charge\npumping by time-dependent gradients of magnetization.\nDue to the analogies between orbital magnetism and the\nBerryphasetheoryofDMIonemayexpectthatinmetals\nit is convenient to obtain expressions for ICIDMI, which\ncanthen be used todescribe the CIDMI byexploitingthereciprocity between CIDMI and ICIDMI. We will show\nin this paper that this is indeed the case.\nIn noncentrosymmetric ferromagnets spin-orbit inter-\naction (SOI) generates torques on the magnetization –\nthe so-called spin-orbit torques (SOTs) – when an elec-\ntric current is applied [17]. The Berry phase theory of\nDMI [6, 10, 11] establishes a relation to SOTs. The for-\nmal analogies between orbital magnetism and DMI have\nbeen shown to be a very useful guiding principle in the\ndevelopment of the theory of SOTs driven by heat cur-\nrents [18]. In particular, it is fruitful to consider the DMI\ncoefficients as a spiralization, which is formally analo-\ngous to magnetization. In the theory of thermoelectric\neffects in magnetic systems the curl of magnetization de-\nscribes a bound current, which cannot be measured in\ntransport experiments and needs to be subtracted from\nthe Kubo linear response in order to obtain the mea-\nsurable current [19–21]. Similarly, in the theory of the\nthermal spin-orbit torque spatial gradients of the DMI\nspiralization, which result from the temperature gradi-\nent together with the temperature dependence of DMI,\nneed to be subtracted in order to obtain the measurable\ntorqueandto satisfyaMott-likerelation[10, 18]. In non-\ncollinear magnets the question arises whether gradients\nof the spiralization that are due to the magnetic texture\ncorrespond to torques like those from thermal gradients.\nWe will show that indeed the spatial gradients of CIDMI\nneed to be included into the theory of current-induced\ntorques (CITs) in noncollinear magnets in order to sat-\nisfy the Onsager reciprocity relations [22].\nWhen the system is driven out of equilibrium by mag-\nnetization dynamics rather than electric current one may\nexpect DMI to be modified as well. The inverse effect of\nthis dynamical DMI (DDMI) consists in the generation\nof torques by time-dependent magnetization gradients.\nIn noncollinear magnets the DDMI spiralization varies\nin space. We will show that the resulting gradient cor-2\nresponds to a torque that needs to be considered in the\ntheory of Gilbert damping and gyromagnetism in non-\ncollinear magnets.\nThis paper is structured as follows. In section IIA\nwe give an overview of CIT in noncollinear magnets and\nintroduce the notation. In section IIB we describe the\nformalism used to calculate the response of electric cur-\nrent to time-dependent magnetization gradients. In sec-\ntion IIC we show that current-induced DMI (CIDMI)\nand electric current driven by time-dependent magneti-\nzation gradients are reciprocal effects. This allows us\nto obtain an expression for CIDMI based on the formal-\nism of section IIB. In section IID we discuss that time-\ndependent magnetization gradients generate additionally\ntorques on the magnetization and show that the inverse\neffect consists in the modification of DMI by magnetiza-\ntion dynamics, which we calldynamical DMI (DDMI). In\nsection IIE we demonstrate that magnetization dynam-\nics induces orbital magnetism, which we call dynamical\norbitalmagnetism (DOM) and showthat DOM is related\nto CIDMI. In section IIF we explain how the spatial gra-\ndients of CIDMI and DOM contribute to the direct and\nto the inverse CIT, respectively. In section IIG we dis-\ncuss how the spatial gradients of DDMI contribute to the\ntorque-torque correlation. In section IIH we complete\nthe formalism used to calculate the CIT in noncollinear\nmagnets by adding the chiral contribution of the torque-\nvelocity correlation. In section III we finalize the theory\nof the inverse CIT by adding the chiral contribution of\nthe velocity-torque correlation. In section IIJ we fin-\nish the computational formalism of gyromagnetism and\ndamping by adding the chiral contribution of the torque-\ntorque correlation and the response of the torque to the\ntime-dependent magnetization gradients. In section III\nwe discuss the symmetry properties of the response to\ntime-dependent magnetizationgradients. In section IVA\nwe present the results for the chiral contributions to the\ndirectand the inverseCITin the Rashbamodel andshow\nthat both the perturbation by the time-dependent mag-\nnetization gradient and the spatial gradients of CIDMI\nand DOM need to be included to ensure that they are\nreciprocal. In section IVB we present the results for the\nchiralcontributiontothe torque-torquecorrelationinthe\nRashba model and show that both the perturbation by\nthe time-dependent magnetization gradient and the spa-\ntialgradientsofDDMI need tobe included toensurethat\nit satisfies the Onsager symmetry relations. This paper\nends with a summary in section V.II. FORMALISM\nA. Direct and inverse current-induced torques in\nnoncollinear magnets\nEven in collinearmagnets the application of an electric\nfieldEgenerates a torque TCIT1on the magnetization\nwhen inversion symmetry is broken [17, 23]:\nTCIT1\ni=/summationdisplay\njtij(ˆM)Ej, (1)\nwheretij(ˆM) is the torkance tensor, which depends on\nthe magnetization direction ˆM. This torque is called\nspin-orbit torque (SOT), but we denote it here CIT1,\nbecause it is one contribution to the current-induced\ntorques (CITs) in noncollinear magnets. Inversely, mag-\nnetization dynamics pumps a charge current JICIT1ac-\ncording to [24]\nJICIT1\ni=/summationdisplay\njtji(−ˆM)ˆej·/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\n,(2)\nwhereˆejis a unit vector that points into the j-th spa-\ntial direction. Generally, JICIT1can be explained by\nthe inverse spin-orbit torque [24] or the magnonic charge\npumping [25]. We denote it here by ICIT1, because it\nis one contribution to the inverse CIT in noncollinear\nmagnets. In the special case of magnetic bilayers one im-\nportantmechanism responsiblefor JICIT1arisesfrom the\ncombination of spin pumping and the inverse spin Hall\neffect [26, 27].\nIn noncollinear magnets there is a second contribution\nto the CIT, which is proportional to the spatial deriva-\ntives of magnetization [28]:\nTCIT2\ni=/summationdisplay\njklχCIT2\nijklEjˆek·/bracketleftBigg\nˆM×∂ˆM\n∂rl/bracketrightBigg\n.(3)\nThe description of noncollinearity by the derivatives\n∂ˆM/∂rlisonlyapplicablewhenthe magnetizationdirec-\ntion changes slowly in space like in magnetic skyrmions\nwith large radius and in wide magnetic domain walls. In\norder to treat noncollinear magnets such as Mn 3Sn [29],\nwhere the magnetization direction varies strongly on the\nscale of one unit cell, Eq. (3) needs to be modified, which\nis beyond the scope of the present paper. The adia-\nbatic and the non-adiabatic [30] spin transfer torques\nare two important contributions to χCIT2\nijkl, but the in-\nterplay between broken inversion symmetry, SOI, and\nnoncollinearity can lead to a large number of additional\nmechanisms [22, 31]. Similarly, the current pumped\nby magnetization dynamics contains a contribution that\nis proportional to the spatial derivatives of magnetiza-3\ntion [22, 32, 33]:\nJICIT2\ni=/summationdisplay\njklχICIT2\nijklˆej·/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\nˆek·/bracketleftBigg\nˆM×∂ˆM\n∂rl/bracketrightBigg\n.\n(4)\nTCIT2\niandJICIT2\nican be considered as chiral contribu-\ntionsto the CIT and to the ICIT, respectively, because\nthey distinguish between left- and right-handed spin spi-\nrals. Due to the reciprocity between direct and inverse\nCIT [22, 24] the coefficients χICIT2\nijklandχCIT2\njiklare related\naccording to\nχICIT2\nijkl(ˆM) =χCIT2\njikl(−ˆM). (5)\nB. Response of electric current to time-dependent\nmagnetization gradients\nIn order to compute JICIT2based on the Kubo lin-\near response formalism it is necessary to split it into\ntwo contributions, JICIT2aandJICIT2b. While JICIT2a\nis obtained as linear response to the perturbation by\natime-dependent magnetization gradient in a collinear\nferromagnet, JICIT2bis obtained as linear response to\nthe perturbation by magnetization dynamics in a non-\ncollinear ferromagnet. Therefore, as will become clear\nbelow,JICIT2acan be expressed by a correlation func-\ntion of two operators, because it describes the response\nof the current to a time-dependent magnetization gradi-\nent: A time-dependent magnetization gradient is a single\nperturbation, which is described by a single perturbing\noperator. In contrast, JICIT2binvolves the correlation\nof three operators, because it describes the response of\nthe current to magnetization dynamics in the presence of\nperturbation by noncollinearity. These are twoperturba-\ntions: One perturbation by the magnetization dynamics,\nandasecondperturbationtodescribethenoncollinearity.\nIn the Kubo formalism the expressions for the response\nonethe onehand toatime-dependent magnetizationgra-\ndient, which is described by a single perturbing operator,\nand the response on the other hand to a time-dependent\nmagnetization in the presence of a magnetization gradi-\nent, which is described by two perturbing operators, are\ndifferent. Therefore, we split JICIT2into these two con-\ntributions, which we call JICIT2aandJICIT2b. In the\nremainder of this section we discuss the calculation of\nthe contribution JICIT2a. The contribution JICIT2bis\ndiscussed in section III below.\nJICIT2ais determined by the second derivative of mag-\nnetization with respect to time and space variables and\ncan be written as\nJICIT2a\ni=/summationdisplay\njkχICIT2a\nijk∂2ˆMj\n∂rk∂t. (6)\nAnonzerosecondderivative∂2ˆMj\n∂rk∂tis what we referto asa\ntime-dependent magnetization gradient . Wewillshowbe-low that in special cases∂2ˆMj\n∂rk∂tcan be expressed in terms\nof the products∂ˆMl\n∂rk∂ˆMl\n∂t, which will allow us to rewrite\nJICIT2a\niin the form of Eq. (4) in the cases relevant for\nthe chiral ICIT. However, as will become clear below,\nEq. (6) is the most general expression for the response\nto time-dependent magnetization gradients, and it can-\nnot generally be rewritten in the form of Eq. (4): This\nis only possible when it describes a contribution to the\nchiral ICIT.\nJICIT2aoccurs in two different situations, which need\nto be distinguished. In one case the magnetization gra-\ndient varies in time like sin( ωt) everywhere in space. An\nexample is\nˆM(r,t) =\nηsin(q·r)sin(ωt)\n0\n1\n, (7)\nwhereηis the amplitude and the derivatives at t= 0 and\nr= 0 are\n∂ˆM(r,t)\n∂ri/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=t=0=∂ˆM(r,t)\n∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=t=0= 0 (8)\nand\n∂2ˆM(r,t)\n∂ri∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=t=0=\nηqiω\n0\n0\n. (9)\nIn the other case the magnetic texture varies like a\npropagating wave, i.e., proportional to sin( q·r−ωt). An\nexample is given by\nˆM(r,t) =\nηsin(q·r−ωt)\n0\n1−η2\n2sin2(q·r−ωt)\n,(10)\nwhere the derivatives at t= 0 and r= 0 are\n∂ˆM(r,t)\n∂ri/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=t=0=\nηqi\n0\n0\n, (11)\n∂ˆM(r,t)\n∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=t=0=\n−ηω\n0\n0\n (12)\nand\n∂2ˆM(r,t)\n∂ri∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nr=t=0=\n0\n0\nη2qiω\n. (13)\nIn the latter example, Eq. (10), the second derivative,\nEq. (13), is along the magnetization ˆM(r= 0,t= 0),\nwhile in the former example, Eq. (7), the second deriva-\ntive, Eq. (9), is perpendicular to the magnetization when\nr= 0 andt= 0.4\nWe assume that the Hamiltonian is given by\nH(r,t) =−/planckover2pi12\n2me∆+V(r)+µBˆM(r,t)·σΩxc(r)+\n+1\n2ec2µBσ·[∇V(r)×v],\n(14)\nwhere the first term describes the kinetic energy, the sec-\nond term is a scalar potential, Ωxc(r) in the third term is\nthe exchange field, and the last term describes the spin-\norbit interaction. Around t= 0 and r= 0 we can de-\ncompose the Hamiltonian as H(r,t) =H0+δH(r,t),\nwhereH0is obtained from H(r,t) by replacing ˆM(r,t)\nbyˆM(r= 0,t= 0) and\nδH(r,t) =∂H0\n∂ˆMxηsin(q·r)sin(ωt)\n=µBΩxc(r)σxηsin(q·r)sin(ωt)(15)\nin the case of the first example, Eq. (7). In the case of\nthe second example, Eq. (10),\nδH(r,t)≃∂H\n∂ˆMxηsin(q·r−ωt)\n+∂H\n∂ˆMzη2sin(q·r)sin(ωt),(16)\nwhereforsmall randtonlythesecondterm ontheright-\nhand side contributes to∂2H(r,t)\n∂rk∂t. We consider here only\nthe time-dependence of the exchange field direction and\nignore the time-dependence of the exchange field mag-\nnitude Ωxc(r) that is induced by the time-dependence\nof the exchange field direction. While the variation of\nthe exchange field magnitude drives currents and torques\nas well, as shown in Ref. [34], the variation of the ex-\nchange field magnitude is a small response and therefore\nthese secondary responses are suppressed in magnitude\nwhen compared to the direct primary responses of the\ncurrent and torque to the variation in the exchange field\ndirection. We will use the perturbations Eq. (15) and\nEq. (16) in order to compute the response of current and\ntorque within the Kubo response formalism. An alterna-\ntive approach for the calculation of the response to time-\ndependent fields is variational linear-response, which has\nbeen applied to the spin susceptibility by Savrasov [35].\nThe perturbation by the time-dependent gradient can\nbe written as\nδH=∂H\n∂ˆM·∂2ˆM\n∂ri∂tsin(qiri)\nqisin(ωt)\nω,(17)\nwhich turns into Eq. (15) when Eq. (9) is inserted. When\nEq. (13) is inserted it turns into the second term in\nEq. (16).\nIn Appendix A we derive the linear response to pertur-\nbations of the type of Eq. (17) and show that the corre-\nsponding coefficient χICIT2a\nijkin Eq. (6) can be expressedas\nχICIT2a\nijk=ie\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nviRvkRROjR+viRRvkROjR+\n−viRROjRvkR−viRvkROjAA\n+viROjAvkAA+viROjAAvkA\n−viRvkRROjA−viRRvkROjA\n+viRROjAvkA+viAvkAOjAA\n−viAOjAvkAA−viAOjAAvkA/bracketrightBig\n,(18)\nwhereR=GR\nk(E) andA=GA\nk(E) are shorthands for the\nretarded and advanced Green’s functions, respectively,\nandOj=∂H/∂ˆMj.e >0 is the positive elementary\ncharge.\nIn the case of the perturbation of the type Eq. (7) the\nsecond derivative∂2ˆM\n∂ri∂tis perpendicular to M. In this\ncase it is convenient to rewrite Eq. (6) as\nJICIT2a\ni=/summationdisplay\njkχICIDMI\nijkˆej·/bracketleftBigg\nˆM×∂2ˆM\n∂rk∂t/bracketrightBigg\n,(19)\nwhere the coefficients χICIDMI\nijkare given by\nχICIDMI\nijk=ie\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nviRvkRRTjR+viRRvkRTjR+\n−viRRTjRvkR−viRvkRTjAA\n+viRTjAvkAA+viRTjAAvkA\n−viRvkRRTjA−viRRvkRTjA\n+viRRTjAvkA+viAvkATjAA\n−viATjAvkAA−viATjAAvkA/bracketrightBig\n,(20)\nand\nT=ˆM×∂H\n∂ˆM(21)\nis the torque operator. In Sec. IIC we will explain that\nχICIDMI\nijkdescribes the inverse of current-induced DMI\n(ICIDMI).\nIn the case of the perturbation of the type of Eq. (10)\nthe second derivative∂2ˆMj\n∂rk∂tmay be rewritten as product\nof the first derivatives∂ˆMl\n∂tand∂ˆMl\n∂rk. This may be seen5\nas follows:\n∂H\n∂ˆM·∂2ˆM\n∂ri∂t=∂2H\n∂t∂ri=\n=∂\n∂t/bracketleftBigg/parenleftbigg\nˆM×∂H\n∂ˆM/parenrightbigg\n·/parenleftBigg\nˆM×∂ˆM\n∂ri/parenrightBigg/bracketrightBigg\n=\n=/bracketleftBigg/parenleftBigg\n∂ˆM\n∂t×∂H\n∂ˆM/parenrightBigg\n·/parenleftBigg\nˆM×∂ˆM\n∂ri/parenrightBigg/bracketrightBigg\n=\n=/bracketleftBigg/parenleftBigg/parenleftBigg\nˆM×∂ˆM\n∂t/parenrightBigg\n׈M/parenrightBigg\n×∂H\n∂ˆM/bracketrightBigg\n·/bracketleftBigg\nˆM×∂ˆM\n∂ri/bracketrightBigg\n=\n=−/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\n·/bracketleftBigg\nˆM×∂ˆM\n∂ri/bracketrightBigg/bracketleftbigg\nˆM·∂H\n∂ˆM/bracketrightbigg\n=\n=−∂ˆM\n∂t·∂ˆM\n∂ri/bracketleftbigg\nˆM·∂H\n∂ˆM/bracketrightbigg\n.\n(22)\nThis expression is indeed satisfied by Eq. (11), Eq. (12)\nand Eq. (13):\n∂ˆM\n∂ri·∂ˆM\n∂t=−∂2ˆM\n∂ri∂t·ˆM (23)\natr= 0,t= 0. Consequently, Eq. (6) can be rewritten\nas\nJICIT2a\ni=/summationdisplay\njkχICIT2a\nijk∂2ˆMj\n∂rk∂t=\n=−/summationdisplay\njklχICIT2a\nijk∂ˆMl\n∂rk∂ˆMl\n∂t[1−δjl]\n=/summationdisplay\njklχICIT2a\nijklˆej·/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\nˆek·/bracketleftBigg\nˆM×∂ˆM\n∂rl/bracketrightBigg\n,\n(24)\nwhere\nχICIT2a\nijkl=−/summationdisplay\nmχICIT2a\niml[1−δjm]δjk.(25)\nThus, Eq. (24) and Eq. (25) can be used to express\nJICIT2a\niin the form of Eq. (4).\nC. Direct and inverse CIDMI\nEq. (20) describes the response of the electric current\nto time-dependent magnetization gradients of the type\nEq. (15). The reciprocal process consists in the current-\ninduced modification of DMI. This can be shown by ex-\npressing the DMI coefficients as [10]\nDij=1\nV/summationdisplay\nnf(Ekn)/integraldisplay\nd3r(ψkn(r))∗Dijψkn(r)\n=1\nV/summationdisplay\nnf(Ekn)/integraldisplay\nd3r(ψkn(r))∗Ti(r)rjψkn(r),\n(26)where we defined the DMI-operator Dij=Tirj. Using\nthe Kubo formalism the current-induced modification of\nDMI may be written as\nDCIDMI\nij=/summationdisplay\nkχCIDMI\nkijEk (27)\nwith\nχCIDMI\nkij=1\nVlim\nω→0/bracketleftBige\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tDij;vk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightBig\n,(28)\nwhere\n∝an}bracketle{t∝an}bracketle{tDij;vk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) =−i∞/integraldisplay\n0dteiωt∝an}bracketle{t[Dij(t),vk(0)]−∝an}bracketri}ht(29)\nis the Fourier transform of a retarded function and Vis\nthe volume of the unit cell.\nSince the position operator rin the DMI operator\nDij=Tirjis not compatible with Bloch periodic bound-\nary conditions, we do not use Eq. (28) for numerical\ncalculations of CIDMI. However, it is convenient to use\nEq. (28) in order to demonstrate the reciprocity between\ndirect and inverse CIDMI.\nInverseCIDMI (ICIDMI) describes the electric current\nthat responds to the perturbation by a time-dependent\nmagnetization gradient according to\nJICIDMI\nk=/summationdisplay\nijχICIDMI\nkijˆei·/bracketleftBigg\nˆM×∂2ˆM\n∂t∂rj/bracketrightBigg\n.(30)\nThe perturbation by a time-dependent magnetization\ngradient may be written as\nδH=−/summationdisplay\njm·∂2ˆM\n∂t∂rjrjΩxc(r)sin(ωt)\nω=\n=/summationdisplay\njT·/bracketleftBigg\nˆM×∂2ˆM\n∂t∂rj/bracketrightBigg\nrjsin(ωt)\nω\n=/summationdisplay\nijDijˆei·/bracketleftBigg\nˆM×∂2ˆM\n∂t∂rj/bracketrightBigg\nsin(ωt)\nω.(31)\nConsequently, the coefficient χICIDMI\nkijis given by\nχICIDMI\nkij=1\nVlim\nω→0/bracketleftBige\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tvk;Dij∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightBig\n.(32)\nUsing\n∝an}bracketle{t∝an}bracketle{tDij;vk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω,ˆM) =−∝an}bracketle{t∝an}bracketle{tvk;Dij∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω,−ˆM) (33)\nwe find that CIDMI and ICIDMI are related through the\nequations\nχCIDMI\nkij(ˆM) =−χICIDMI\nkij(−ˆM). (34)\nIn order to calculate CIDMI we use Eq. (20) for ICIDMI\nand then use Eq. (34) to obtain CIDMI.6\nThe perturbation Eq. (16) describes a different kind\nof time-dependent magnetization gradient, for which the\nreciprocaleffect consists in the modification of the expec-\ntation value ∝an}bracketle{tσ·ˆMrj∝an}bracketri}ht. However, while the modification\nof∝an}bracketle{tTirj∝an}bracketri}htby an applied current can be measured [8, 9]\nfrom the change of the DMI constant Dij, the quantity\n∝an}bracketle{tσ·ˆMrj∝an}bracketri}hthas not been considered so far in ferromagnets.\nIn noncollinear magnets the quantity ∝an}bracketle{tσrj∝an}bracketri}htcan be used\ntodefinespintoroidization[36]. Therefore,whiletheper-\nturbation of the type of Eq. (15) is related to CIDMI and\nICIDMI, which are both accessible experimentally [8, 9],\nin the case of the perturbation of the type of Eq. (16)\nwe expect that only the effect of driving current by the\ntime-dependent magnetization gradient is easily accessi-\nble experimentally, while its inverse effect is difficult to\nmeasure.\nD. Direct and inverse dynamical DMI\nNot only applied electric currents modify DMI, but\nalso magnetization dynamics, which we call dynamical\nDMI (DDMI). DDMI can be expressed as\nDDDMI\nij=/summationdisplay\nkχDDMI\nkijˆek·/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\n.(35)\nIn Sec. IIG we will show that the spatial gradient of\nDDMI contributes to damping and gyromagnetism in\nnoncollinear magnets. The perturbation used to describe\nmagnetization dynamics is given by [24]\nδH=sin(ωt)\nω/parenleftBigg\nˆM×∂ˆM\n∂t/parenrightBigg\n·T.(36)\nConsequently, the coefficients χDDMI\nkijmay be written as\nχDDMI\nkij=−1\nVlim\nω→0/bracketleftbigg1\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tDij;Tk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightbigg\n.(37)\nSince the position operator in Dijis not compatible\nwith Bloch periodic boundary conditions, we do not use\nEq. (37) for numerical calculations of DDMI, but instead\nwe obtain it from its inverse effect, which consists in the\ngeneration of torques on the magnetization due to time-\ndependent magnetization gradients. These torques can\nbe written as\nTIDDMI\nk=/summationdisplay\nijχIDDMI\nkijˆei·/bracketleftBigg\nˆM×∂2ˆM\n∂t∂rj/bracketrightBigg\n,(38)\nwhere the coefficients χIDDMI\nkijare\nχIDDMI\nkij=1\nVlim\nω→0/bracketleftbigg1\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tTk;Dij∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightbigg\n,(39)becausethe perturbationby the time-dependent gradient\ncan be expressed in terms of Dijaccording to Eq. (31)\nand because the torque on the magnetizationis described\nby−T[23]. Consequently,DDMIandIDDMIarerelated\nby\nχDDMI\nkij(ˆM) =−χIDDMI\nkij(−ˆM). (40)\nFor numerical calculations of IDDMI we use\nχIDDMI\nijk=i\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nTiRvkRRTjR+TiRRvkRTjR+\n−TiRRTjRvkR−TiRvkRTjAA\n+TiRTjAvkAA+TiRTjAAvkA\n−TiRvkRRTjA−TiRRvkRTjA\n+TiRRTjAvkA+TiAvkATjAA\n−TiATjAvkAA−TiATjAAvkA/bracketrightBig\n,(41)\nwhichisderivedinAppendix A. InordertoobtainDDMI\nwecalculateIDDMIfromEq.(41)andusethereciprocity\nrelation Eq. (40).\nEq.(38)is validfortime-dependent magnetizationgra-\ndients that lead to perturbations of the type of Eq. (15).\nPerturbations of the second type, Eq. (16), will induce\ntorques on the magnetization as well. However, the in-\nverse effect is difficult to measure in that case, because it\ncorresponds to the modification of the expectation value\n∝an}bracketle{tσ·ˆMrj∝an}bracketri}htby magnetization dynamics. Therefore, while\nin the case of Eq. (15) both direct and inverse response\nare expected to be measurable and correspond to ID-\nDMI and DDMI, respectively, we expect that in the case\nof Eq. (16) only the direct effect, i.e., the response of the\ntorque to the perturbation, is easy to observe.\nE. Dynamical orbital magnetism (DOM)\nMagnetization dynamics does not only induce DMI,\nbut also orbital magnetism, which we call dynamical or-\nbital magnetism (DOM). It can be written as\nMDOM\nij=/summationdisplay\nkχDOM\nkijˆek·/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\n,(42)\nwhere we introduced the notation\nMDOM\nij=e\nV∝an}bracketle{tvirj∝an}bracketri}htDOM, (43)\nwhich defines a generalized orbital magnetization, such\nthat\nMDOM\ni=1\n2/summationdisplay\njkǫijkMDOM\njk (44)7\ncorresponds to the usual definition of orbital magnetiza-\ntion. The coefficients χDOM\nkijare given by\nχDOM\nkij=−1\nVlim\nω→0/bracketleftBige\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tvirj;Tk∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightBig\n,(45)\nbecause the perturbation by magnetization dynamics is\ndescribed by Eq. (36). We will discuss in Sec. IIF that\nthe spatial gradient of DOM contributes to the inverse\nCIT. Additionally, we will show below that DOM and\nCIDMI are related to each other.\nIn order to obtain an expression for DOM it is conve-\nnient to consider the inverse effect, i.e., the generation of\natorquebythe applicationofa time-dependent magnetic\nfieldB(t) that actsonly onthe orbitaldegreesoffreedom\nof the electrons and not on their spins. This torque can\nbe written as\nTIDOM\nk=1\n2/summationdisplay\nijlχIDOM\nkijǫijl∂Bl\n∂t, (46)\nwhere\nχIDOM\nkij=−1\nVlim\nω→0/bracketleftBige\n/planckover2pi1ωIm∝an}bracketle{t∝an}bracketle{tTk;virj∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)/bracketrightBig\n,(47)\nbecause the perturbation by the time-dependent mag-\nnetic field is given by\nδH=−e\n2/summationdisplay\nijkǫijkvirj∂Bk\n∂tsin(ωt)\nω.(48)\nTherefore, thecoefficientsofDOMandIDOM arerelated\nby\nχDOM\nkij(ˆM) =−χIDOM\nkij(−ˆM). (49)\nIn Appendix A we show that the coefficient χIDOM\nijkcan\nbe expressed as\nχIDOM\nijk=−ie\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nTiRvkRRvjR+TiRRvkRvjR+\n−TiRRvjRvkR−TiRvkRvjAA\n+TiRvjAvkAA+TiRvjAAvkA\n−TiRvkRRvjA−TiRRvkRvjA\n+TiRRvjAvkA+TiAvkAvjAA\n−TiAvjAvkAA−TiAvjAAvkA/bracketrightBig\n.(50)\nEq. (50) and Eq. (20) differ only in the positions of\nthe two velocity operators and the torque operator be-\ntween the Green functions. As a consequence, IDOM\nare ICIDMI are related. In Table I and Table II we list\nthe relations between IDOM and ICIDMI for the Rashba\nmodel Eq. (83). We will explain in Sec. III that IDOM\nandICIDMI arezeroin the Rashbamodel when themag-\nnetization is along the zdirection. Therefore, we discussin Table I the case where the magnetization lies in the xz\nplane, and in Table II we discuss the case where the mag-\nnetization lies in the yzplane. According to Table I and\nTable II the relation between IDOM and ICIDMI is of\nthe formχIDOM\nijk=±χICIDMI\njik. This is expected, because\nthe indexiinχIDOM\nijkis connected to the torque operator,\nwhile the index jinχICIDMI\nijkis connected to the torque\noperator.\nTABLEI:Relations betweentheinverseofthemagnetization -\ndynamics induced orbital magnetism (IDOM) and inverse\ncurrent-inducedDMI (ICIDMI)in the 2d Rashbamodel when\nˆMlies in the zxplane. The components of χIDOM\nijk(Eq. (50))\nandχICIDMI\nijk(Eq. (20)) are denoted by the three indices ( ijk).\nICIDMI IDOM\n(211) (121)\n(121) (211)\n-(221) (221)\n(112) (112)\n-(212) (122)\n-(122) (212)\n(222) (222)\n(231) (321)\n(132) (312)\n-(232) (322)\nTABLE II: Relations between IDOM and ICIDMI in the 2d\nRashba model when ˆMlies in the yzplane.\nICIDMI IDOM\n(111) (111)\n-(211) (121)\n-(121) (211)\n(221) (221)\n-(112) (112)\n(212) (122)\n(122) (212)\n-(131) (311)\n(231) (321)\n(132) (312)\nF. Contributions from CIDMI and DOM to direct\nand inverse CIT\nIn electronic transport theory the continuity equation\ndetermines the current only up to a curl field [37]. The\ncurl of magnetization corresponds to a bound current\nthat cannot be measured in electron transport experi-\nments such that\nJ=JKubo−∇×M (51)\nhastobeusedtoextractthetransportcurrent Jfromthe\ncurrentJKuboobtained from the Kubo linear response.8\nThe subtraction of ∇×Mhas been shown to be impor-\ntant when calculating the thermoelectric response [37]\nand the anomalous Nernst effect [20]. Similarly, in the\ntheory of the thermal spin-orbit torque [10, 18] the gra-\ndients of the DMI spiralization have to be subtracted in\norder to obtain the measurable torque:\nTi=TKubo\ni−/summationdisplay\nj∂\n∂rjDij, (52)\nwhere the spatial derivative of the spiralization arises\nfrom its temperature dependence and the temperature\ngradient.\nSince CIDMI and DOM depend on the magnetization\ndirection, they vary spatially in noncollinear magnets.\nSimilar to Eq. (52) the spatial derivatives of the current-\ninduced spiralization need to be included into the theory\nof CIT. Additionally, the gradients of DOM correspond\ntocurrentsthatneedtobeconsideredinthetheoryofthe\ninverse CIT, similar to Eq. (51). In section IV we explic-\nitly show that Onsager reciprocity is violated if spatial\ngradients of DOM and CIDMI are not subtracted from\nthe Kubo response expressions. By trial-and-error we\nfind that the following subtractions are necessary to ob-\ntain response currents and torques that satisfy this fun-\ndamental symmetry:\nJICIT\ni=JKubo\ni−1\n2/summationdisplay\nj∂ˆM\n∂rj·∂MDOM\nij\n∂ˆM(53)\nand\nTCIT\ni=TKubo\ni−1\n2/summationdisplay\nj∂ˆM\n∂rj·∂DCIDMI\nij\n∂ˆM,(54)\nwhereJICIT\niis the current driven by magnetization dy-\nnamics, and TCIT\niis the current-induced torque.\nInterestingly, we find that also the diagonal elements\nMDOM\niiare nonzero. This shows that the generalized def-\ninition Eq. (43) is necessary, because the diagonal ele-\nmentsMDOM\niido not contribute in the usual definition\nofMiaccording to Eq. (44). These differences in the\nsymmetry properties between equilibrium and nonequi-\nlibrium orbital magnetism can be traced back to sym-\nmetry breaking by the perturbations. Also in the case\nof the spiralization tensor Dijthe nonequilibrium cor-\nrectionδDijhas different symmetry properties than the\nequilibrium part (see Sec. III).\nThe contribution of DOM to χICIT2\nijklcan be written as\nχICIT2c\nijkl=−1\n2ˆek·/bracketleftBigg\nˆM×∂χDOM\njil\n∂ˆM/bracketrightBigg\n(55)\nand the contribution of CIDMI to χCIT2\nijklis given by\nχCIT2b\nijkl=−1\n2ˆek·/bracketleftBigg\nˆM×∂χCIDMI\njil\n∂ˆM/bracketrightBigg\n.(56)G. Contributions from DDMI to gyromagnetism\nand damping\nThe response to magnetization dynamics that is de-\nscribed by the torque-torque correlation function con-\nsists of torques that are related to damping and gyro-\nmagnetism [24]. The chiral contribution to these torques\ncan be written as\nTTT2\ni=/summationdisplay\njklχTT2\nijklˆej·/bracketleftBigg\nˆM×∂ˆM\n∂t/bracketrightBigg\nˆek·/bracketleftBigg\nˆM×∂ˆM\n∂rl/bracketrightBigg\n,\n(57)\nwhere the coefficients χTT2\nijklsatisfy the Onsager relations\nχTT2\nijkl(ˆM) =χTT2\njikl(−ˆM). (58)\nSinceDDMIdependsonthemagnetizationdirection,it\nvaries spatially in noncollinear magnets and the resulting\ngradients of DDMI contribute to the damping and to the\ngyromagnetic ratio:\nTTT\ni=TKubo\ni−1\n2/summationdisplay\nj∂ˆM\n∂rj·∂DDDMI\nij\n∂ˆM.(59)\nThe resulting contribution of the spatial derivatives of\nDDMI to the coefficient χTT2\nijklis\nχTT2c\nijkl=−1\n2ˆek·/bracketleftBigg\nˆM×∂χDDMI\njil(ˆM)\n∂ˆM/bracketrightBigg\n.(60)\nH. Current-induced torque (CIT) in noncollinear\nmagnets\nThe chiral contribution to CIT consists of the spatial\ngradient of CIDMI, χCIT2b\nijklin Eq. (56), and the Kubo\nlinear response of the torque to the applied electric field\nin a noncollinear magnet, χCIT2a\nijkl:\nχCIT2\nijkl=χCIT2a\nijkl+χCIT2b\nijkl. (61)\nIn orderto determine χCIT2a\nijkl, we assume that the magne-\ntization direction ˆM(r) oscillates spatially as described\nby\nˆM(r) =\nηsin(q·r)\n0\n1\n1/radicalBig\n1+η2sin2(q·r),(62)\nwherewewilltakethelimit q→0attheendofthecalcu-\nlation. Since the spatial derivative of the magnetization\ndirection is\n∂ˆM(r)\n∂ri=\nηqicos(q·r)\n0\n0\n+O(η3),(63)9\nthe chiralcontributiontothe CIToscillatesspatiallypro-\nportional to cos( q·r). In order to extract this spatially\noscillating contribution we multiply with cos( q·r) and\nintegrate over the unit cell. The resulting expression for\nχCIT2a\nijklis\nχCIT2a\nijkl=−2e\nVηlim\nq→0lim\nω→0/bracketleftBigg\n1\nql/integraldisplay\ncos(qlrl)Im∝an}bracketle{t∝an}bracketle{tTi(r);vj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n/planckover2pi1ωd3rd3r′/bracketrightBigg\n,\n(64)\nwhereVis the volume of the unit cell, and\nthe retarded torque-velocity correlation function\n∝an}bracketle{t∝an}bracketle{tTi(r);vj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) needs to be evaluated in the\npresence of the perturbation\nδH=Tkηsin(q·r) (65)\ndue to the noncollinearity (the index kin Eq. (65) needs\nto match the index kinχCIT2a\nijkl).\nIn Appendix B we show that χCIT2a\nijklcan be written as\nχCIT2a\nijkl=−2e\n/planckover2pi1Im/bracketleftBig\nW(surf)\nijkl+W(sea)\nijkl/bracketrightBig\n,(66)\nwhere\nW(surf)\nijkl=1\n4π/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)Tr/bracketleftBigg\nTiGR\nk(E)vlGR\nk(E)vjGA\nk(E)TkGA\nk(E)\n+TiGR\nk(E)vjGA\nk(E)vlGA\nk(E)TkGA\nk(E)\n−TiGR\nk(E)vjGA\nk(E)TkGA\nk(E)vlGA\nk(E)\n+/planckover2pi1\nmeδjlTiGR\nk(E)GA\nk(E)TkGA\nk(E)/bracketrightBigg(67)\nis a Fermi surface term ( f′(E) =df(E)/dE) and\nW(sea)\nijkl=1\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)/bracketleftBigg\n−Tr[TiRvlRRvjRTkR]−Tr[TiRvlRTkRRvjR]\n−Tr[TiRRvlRvjRTkR]−Tr[TiRRvjRvlRTkR]\n+Tr[TiRRvjRTkRvlR]+Tr[TiRRTkRvjRvlR]\n+Tr[TiRRTkRvlRvjR]−Tr[TiRRvlRTkRvjR]\n−Tr[TiRvlRRTkRvjR]+Tr[TiRTkRRvjRvlR]\n+Tr[TiRTkRRvlRvjR]+Tr[TiRTkRvlRRvjR]\n−/planckover2pi1\nmeδjlTr[TiRRRTkR]−/planckover2pi1\nmeδjlTr[TiAAATkA]\n−/planckover2pi1\nmeδjlTr[TiAATkAA]/bracketrightBigg(68)\nis a Fermi sea term.I. Inverse CIT in noncollinear magnets\nThe chiral contribution JICIT2(see Eq. (4)) to the\ncharge pumping is described by the coefficients\nχICIT2\nijkl=χICIT2a\nijkl+χICIT2b\nijkl+χICIT2c\nijkl,(69)\nwhereχICIT2a\nijkldescribes the response to the time-\ndependentmagnetizationgradient(seeEq.(18),Eq.(25),\nand Eq. (24)) and χICIT2c\nijklresults from the spatial gra-\ndient of DOM (see Eq. (55)). χICIT2b\nijkldescribes the re-\nsponseto the perturbation bymagnetizationdynamics in\na noncollinear magnet. In order to derive an expression\nforχICIT2b\nijklwe assume that the magnetization oscillates\nspatially as described by Eq. (62). Since the correspond-\ning response oscillates spatially proportional to cos( q·r),\nwe multiply by cos( q·r) and integrate over the unit cell\nin order to extract χICIT2b\nijklfrom the retarded velocity-\ntorque correlation function ∝an}bracketle{t∝an}bracketle{tvi(r);Tj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω), which\nis evaluated in the presence of the perturbation Eq. (65).\nWe obtain\nχICIT2b\nijkl=2e\nVηlim\nq→0lim\nω→0/bracketleftBigg\n1\nql/integraldisplay\ncos(qlrl)Im∝an}bracketle{t∝an}bracketle{tvi(r);Tj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n/planckover2pi1ωd3rd3r′/bracketrightBigg\n,\n(70)\nwhich can be written as (see Appendix B)\nχICIT2b\nijkl=2e\n/planckover2pi1Im/bracketleftBig\nV(surf)\nijkl+V(sea)\nijkl/bracketrightBig\n,(71)\nwhere\nV(surf)\nijkl=1\n4π/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)Tr/bracketleftBig\nviGR\nk(E)vlGR\nk(E)TjGA\nk(E)TkGA\nk(E)\n+viGR\nk(E)TjGA\nk(E)vlGA\nk(E)TkGA\nk(E)\n−viGR\nk(E)TjGA\nk(E)TkGA\nk(E)vlGA\nk(E)/bracketrightBig(72)\nis the Fermi surface term and\nV(sea)\nijkl=1\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\n−Tr[viRvlRRTjRTkR]−Tr[viRvlRTkRRTjR]\n−Tr[viRRvlRTjRTkR]−Tr[viRRTjRvlRTkR]\n+Tr[viRRTjRTkRvlR]+Tr[viRRTkRTjRvlR]\n+Tr[viRRTkRvlRTjR]−Tr[viRRvlRTkRTjR]\n−Tr[viRvlRRTkRTjR]+Tr[viRTkRRTjRvlR]\n+Tr[viRTkRRvlRTjR]+Tr[viRTkRvlRRTjR]/bracketrightBig(73)\nis the Fermi sea term.\nIn Eq. (70) we use the Kubo formula to describe the\nresponse to magnetization dynamics combined with per-\nturbation theory to include the effect of noncollinearity.10\nThereby, the time-dependent perturbation and the per-\nturbation by the magnetization gradient are separated\nand perturbations of the form of Eq. (15) or Eq. (16)\nare not automatically included. For example the flat cy-\ncloidal spin spiral\nˆM(x,t) =\nsin(qx−ωt)\n0\ncos(qx−ωt)\n (74)\nmoving inxdirection with speed ω/qand the helical spin\nspiral\nˆM(y,t) =\nsin(qy−ωt)\n0\ncos(qy−ωt)\n (75)\nmovinginydirectionwith speed ω/qbehavelikeEq.(10)\nwhentandraresmall. Thus, these movingdomainwalls\ncorrespond to the perturbation of the type of Eq. (10)\nand the resulting contribution JICIT2afrom the time-\ndependent magnetization gradient is not described by\nEq. (70) and needs to be added, which we do by adding\nχICIT2a\nijklin Eq. (69).\nJ. Damping and gyromagnetism in noncollinear\nmagnets\nThe chiral contribution Eq. (57) to the torque-torque\ncorrelation function is expressed in terms of the coeffi-\ncient\nχTT\nijkl=χTT2a\nijkl+χTT2b\nijkl+χTT2c\nijkl, (76)\nwhereχTT2c\nijklresults from the spatial gradient of DDMI\n(see Eq. (60)), χTT2a\nijkldescribes the response to a time-\ndependent magnetization gradient in a collinear magnet,\nandχTT2b\nijkldescribes the response to magnetization dy-\nnamics in a noncollinear magnet.\nIn order to derive an expression for χTT2b\nijklwe as-\nsume that the magnetization oscillates spatially accord-\ning to Eq. (62). We multiply the retarded torque-torque\ncorrelation function ∝an}bracketle{t∝an}bracketle{tTi(r);Tj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) with cos(qlrl)\nand integrate over the unit cell in order to extract the\npart of the response that varies spatially proportional to\ncos(qlrl). We obtain:\nχTT2b\nijkl=2\nVηlim\nql→0lim\nω→0/bracketleftBigg\n1\nql/integraldisplay\ncos(qlrl)Im∝an}bracketle{t∝an}bracketle{tTi(r);Tj(r′)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n/planckover2pi1ωd3rd3r′/bracketrightBigg\n.\n(77)\nIn Appendix B we discuss how to evaluate Eq. (77) in\nfirst order perturbation theory with respect to the per-\nturbation Eq.(65) and showthat χTT2b\nijklcan be expressedas\nχTT2b\nijkl=2\n/planckover2pi1Im/bracketleftBig\nX(surf)\nijkl+X(sea)\nijkl/bracketrightBig\n,(78)\nwhere\nX(surf)\nijkl=1\n4π/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)Tr/bracketleftBigg\nTiGR\nk(E)vlGR\nk(E)TjGA\nk(E)TkGA\nk(E)\n+TiGR\nk(E)TjGA\nk(E)vlGA\nk(E)TkGA\nk(E)\n−TiGR\nk(E)TjGA\nk(E)TkGA\nk(E)vlGA\nk(E)/bracketrightBigg(79)\nis a Fermi surface term and\nX(sea)\nijkl=1\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBigg\n−(TiRvlRRTjRTkR)−(TiRvlRTkRRTjR)\n−(TiRRvlRTjRTkR)−(TiRRTjRvlRTkR)\n+(TiRRTjRTkRvlR)+(TiRRTkRTjRvlR)\n+(TiRRTkRvlRTjR)−(TiRRvlRTkRTjR)\n−(TiRvlRRTkRTjR)+(TiRTkRRTjRvlR)\n+(TiRTkRRvlRTjR)+(TiRTkRvlRRTjR)/bracketrightBigg\n(80)\nis a Fermi sea term.\nThe contribution χTT2a\nijklfrom the time-dependent gra-\ndients is given by\nχTT2a\nijkl=−/summationdisplay\nmχTT2a\niml[1−δjm]δjk,(81)\nwhere\nχTT2a\niml=i\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nTiRvlRROmR+TiRRvlROmR+\n−TiRROmRvlR−TiRvlROmAA\n+TiROmAvlAA+TiROmAAvlA\n−TiRvlRROmA−TiRRvlROmA\n+TiRROmAvlA+TiAvlAOmAA\n−TiAOmAvlAA−TiAOmAAvlA/bracketrightBig\n,(82)\nwithOm=∂H/∂ˆMm(see Appendix A).\nIII. SYMMETRY PROPERTIES\nIn this section we discuss the symmetry properties of\nCIDMI, DDMI and DOM in the case of the magnetic\nRashba model\nHk(r) =/planckover2pi12\n2mek2+α(k׈ez)·σ+∆V\n2σ·ˆM(r).(83)11\nAdditionally, we discuss the symmetry properties of the\ncurrents and torques induced by time-dependent magne-\ntization gradients of the form of Eq. (10).\nWe consider mirror reflection Mxzat thexzplane,\nmirror reflection Myzat theyzplane, and c2 rotation\naround the zaxis. When ∆ V= 0 these operations leave\nEq. (83) invariant, but when ∆ V∝ne}ationslash= 0 they modify the\nmagnetization direction ˆMin Eq. (83), as shown in Ta-\nble III. At the same time, these operations affect the\ntorqueTandthecurrent Jdrivenbythe time-dependent\nmagnetization gradients (see Table III). In Table IV and\nTable V we show how ˆM×∂ˆM/∂rkis affected by the\nsymmetry operations.\nAflat cycloidalspin spiralwith spinsrotatingin the xz\nplane is mapped by a c2 rotation around the zaxis onto\nthe same spin spiral. Similarly, a flat helical spin spiral\nwith spins rotating in the yzplane is mapped by a c2 ro-\ntationaroundthe zaxisontothesamespinspiral. There-\nfore, when ˆMpoints inzdirection, a c2 rotation around\nthezaxis does not change ˆM×∂ˆM/∂ri, but it flips the\nin-plane current Jand the in-plane components of the\ntorque,TxandTy. Consequently, ˆM×∂2ˆM/∂ri∂tdoes\nnot induce currents or torques, i.e., ICIDMI, CIDMI, ID-\nDMI and DDMI are zero, when ˆMpoints inzdirection.\nHowever, they become nonzero when the magnetization\nhas an in-plane component (see Fig. 1).\nSimilarly, IDOM vanishes when the magnetization\npoints inzdirection: In that case Eq. (83) is invariant\nunder the c2 rotation. A time-dependent magnetic field\nalongzdirection is invariant under the c2 rotation as\nwell. However, TxandTychange sign under the c2 rota-\ntion. Consequently, symmetryforbidsIDOM inthiscase.\nHowever, when the magnetization has an in-plane com-\nponent, IDOM and DOM become nonzero (see Fig. 2).\nThat time-dependent magnetization gradients of the\ntype of Eq. (7) do not induce in-plane currents and\ntorqueswhen ˆMpoints inzdirectioncan alsobe seendi-\nrectly from Eq. (7): The c2 rotation transforms q→ −q\nandMx→ −Mx. Since sin( q·r) is odd in r, Eq. (7) is in-\nvariantunder c2rotation, whilethe in-planecurrentsand\ntorques induced by time-dependent magnetization gradi-\nents change sign under c2 rotation. In contrast, Eq. (10)\nis not invariant under c2 rotation, because sin( q·r−ωt)\nis not odd in rfort>0. Consequently, time-dependent\nmagnetization gradients of the type of Eq. (10) induce\ncurrents and torques also when ˆMpoints locally into\nthezdirection. These currents and torques, which are\ndescribed by Eq. (24) and Eq. (82), respectively, need to\nbe added to the chiral ICIT and the chiral torque-torque\ncorrelation. While CIDMI, DDMI, and DOM are zero\nwhen the magnetization points in zdirection, their gra-\ndients are not (see Fig. 1 and Fig. 2). Therefore, the gra-\ndients of CIDMI, DOM, and DDMI contribute to CIT, to\nICIT and to the torque-torque correlation, respectively,\neven when ˆMpoints locally into the zdirection.TABLE III: Effect of mirror reflection Mxzat thexzplane,\nmirror reflection Myzat theyzplane, and c2 rotation around\nthezaxis. The magnetization Mand the torque Ttransform\nlike axial vectors, while the current Jtransforms like a polar\nvector.\nMxMyMzJxJyTxTyTz\nMxz−MxMy−MzJx−Jy−TxTy−Tz\nMyzMx−My−Mz−JxJyTx−Ty−Tz\nc2-Mx-MyMz-Jx−Jy−Tx−TyTz\nTABLE IV: Effect of symmetry operations on the magneti-\nzation gradients. Magnetization gradients are described b y\nthree indices ( ijk). The first index denotes the magnetiza-\ntion direction at r= 0. The third index denotes the di-\nrection along which the magnetization changes. The second\nindex denotes the direction of ∂ˆM/∂rkδrk. The direction of\nˆM×∂ˆM/∂rkis specified by the number below the indices\n(ijk).\n(1,2,1) (1,3,1) (2,1,1) (2,3,1) (3,1,1) (3,2,1)\n3-2 -3 1 2-1\nMxz(-1,2,1)(-1,-3,1) (2,-1,1) (2,-3,1) (-3,-1,1) (-3,2,1)\n-3 -2 3 -1 2 1\nMyz(1,2,1) (1,3,1)(-2,-1,1) (-2,3,1) (-3,-1,1) (-3,2,1)\n3-2 -3 -1 2 1\nc2(-1,2,1)(-1,-3,1) (-2,1,1) (-2,-3,1) (3,1,1) (3,2,1)\n-3 -2 3 1 2-1\n.\nTABLE V: Continuation of Table IV\n(1,2,2) (1,3,2) (2,1,2) (2,3,2) (3,1,2) (3,2,2)\n3 -2 -3 1 2-1\nMxz(-1,-2,2) (-1,3,2) (2,1,2) (2,3,2)(-3,1,2)(-3,-2,2)\n3 2-3 1-2 -1\nMyz(1,-2,2) (1,-3,2) (-2,1,2)(-2,-3,2) (-3,1,2)(-3,-2,2)\n-3 2 3 1-2 -1\nc2(-1,2,2) (-1,-3,2) (-2,1,2)(-2,-3,2) (3,1,2) (3,2,2)\n-3 -2 3 1 2-1\nA. Symmetry properties of ICIDMI and IDDMI\nInthefollowingwediscusshowTableIII,TableIV,and\nTable V can be used to analyze the symmetry of ICIDMI\nandIDDMI.AccordingtoEq.(19)thecoefficient χICIDMI\nijk\ndescribes the response of the current JICIT2a\nito the time-\ndependent magnetization gradient ˆej·[ˆM×∂2ˆM\n∂rk∂t]. Since\nˆM×∂2ˆM\n∂rk∂t=∂\n∂t[ˆM×∂ˆM\n∂rk] fortime-dependent magnetiza-\ntion gradients of the type Eq. (7) the symmetry proper-\nties ofχICIDMI\nijkfollow from the transformation behaviour\nofˆM×∂ˆM\n∂rkandJunder symmetry operations.\nWe consider the case with magnetization in xdirec-\ntion. The component χICIDMI\n132describes the current in x\ndirection induced by the time-dependence of a cycloidal\nmagnetizationgradientin ydirection(withspinsrotating12\nFIG. 1: ICIDMI in a noncollinear magnet. (a) Arrows illus-\ntrate the magnetization direction. (b) Arrows illustrate t he\ncurrentJyinduced by a time-dependent magnetization gra-\ndient, which is described by χICIDMI\n221. When ˆMpoints in z\ndirection, χICIDMI\n221andJyare zero. The sign of χICIDMI\n221and\nofJychanges with the sign of Mx.\nFIG. 2: DOM in a noncollinear magnet. (a) Arrows illustrate\nthe magnetization direction. (b) Arrows illustrate the orb ital\nmagnetization induced by magnetization dynamics (DOM).\nWhenˆMpoints in zdirection, DOM is zero. The sign of\nDOM changes with the sign of Mx.\nin thexyplane).Myzflips both ˆM×∂ˆM\n∂yandJx, but\nit preserves ˆM.Mzxpreserves ˆM×∂ˆM\n∂yandJx, but it\nflipsˆM. A c2 rotation around the zaxis flips ˆM×∂ˆM\n∂y,\nˆMandJx. Consequently, χICIDMI\n132(ˆM) is allowed by\nsymmetry and it is even in ˆM. The component χICIDMI\n122\ndescribes the current in xdirection induced by the time-\ndependence of a helical magnetization gradient in ydi-\nrection (with spins rotating in the xzplane).Myzflips\nˆM×∂ˆM\n∂yandJx, but it preserves ˆM.MzxflipsˆM×∂ˆM\n∂y\nandˆM, but it preserves Jx. A c2 rotation around the z\naxis flipsJxandˆM, but it preserves ˆM×∂ˆM\n∂y. Conse-\nquently,χICIDMI\n122is allowed by symmetry and it is odd in\nˆM. The component χICIDMI\n221describes the current in y\ndirection induced by the time-dependence of a cycloidal\nmagnetization gradient in xdirection (with spins rotat-\ning in thexzplane).Mzxpreserves ˆM×∂ˆM\n∂x, but it flipsJyandˆM.Myzpreserves ˆM,Jy, andˆM×∂ˆM\n∂x. The\nc2 rotation around the zaxis preserves ˆM×∂ˆM\n∂x, but\nit flipsˆMandJy. Consequently, χICIDMI\n221is allowed by\nsymmetry and it is odd in ˆM. The component χICIDMI\n231\ndescribes the current in ydirection induced by the time-\ndependence of a cycloidal magnetization gradient in xdi-\nrection (with spins rotating in the xyplane).Mzxflips\nˆM×∂ˆM\n∂x,ˆM, andJy.Myzpreserves ˆM×∂ˆM\n∂x,ˆMand\nJy. The c2 rotation around the zaxis flips ˆM×∂ˆM\n∂x,Jy,\nandˆM. Consequently, χICIDMI\n231is allowed by symmetry\nand it is even in ˆM.\nThese properties are summarized in Table VI. Due to\nthe relations between CIDMI and DOM (see Table I and\nTable II), they can be used for DOM as well. When the\nmagnetization lies at a general angle in the xzplane or in\ntheyzplaneseveraladditionalcomponentsofCIDMIand\nDOMarenonzero(seeTableIandTableII,respectively).\nTABLE VI: Allowed components of χICIDMI\nijkwhenˆMpoints\ninxdirection. + components are even in ˆM, while - compo-\nnents are odd in ˆM.\n132 122 221 231\n+ - - +\nSimilarly, one can analyze the symmetry of DDMI. Ta-\nble VII lists the components of DDMI, χDDMI\nijk, which are\nallowed by symmetry when ˆMpoints inxdirection.\nTABLEVII:Allowedcomponentsof χDDMI\nijkwhenˆMpointsin\nxdirection. +componentsareevenin ˆM, while -components\nare odd in ˆM.\n222 232 322 332\n- + + -\nB. Response to time-dependent magnetization\ngradients of the second type (Eq. (10))\nAccording to Eq. (13) the time-dependent magneti-\nzation gradient is along the magnetization. Therefore,\nin contrast to the discussion in section IIIA we can-\nnot use ˆM×∂2ˆM\n∂rk∂tin the symmetry analysis. Eq. (24)\nand Eq. (25) show that χICIT2a\nijjldescribes the response of\nJICIT2a\nitoˆej·/bracketleftBig\nˆM×∂ˆM\n∂t/bracketrightBig\nˆej·/bracketleftBig\nˆM×∂ˆM\n∂rl/bracketrightBig\nwhileχICIT2a\nijkl=\n0 forj∝ne}ationslash=k. According to Eq. (23) the symmetry prop-\nerties of/bracketleftBig\nˆM×∂ˆM\n∂t/bracketrightBig\n·/bracketleftBig\nˆM×∂ˆM\n∂rl/bracketrightBig\nagree to the symmetry\nproperties of ˆM·∂2ˆM\n∂rl∂t. Therefore, in order to under-\nstand the symmetry properties of χICIT2a\nijjlwe consider\nthe transformation of JandˆM·∂2ˆM\n∂rl∂tunder symmetry\noperations.\nWe consider the case where ˆMpoints inzdirection.\nχICIT2a\n1jj1describes the current driven in xdirection, when13\nthe magnetization varies in xdirection. MxzflipsˆM,\nbut preserves JxandˆM·∂2ˆM/(∂x∂t).MyzflipsˆM,Jx,\nandˆM·∂2ˆM/(∂x∂t). c2 rotation flips ˆM·∂2ˆM/(∂x∂t)\nandJx, but preserves ˆM. Consequently, χICIT2a\n1jj1is al-\nlowed by symmetry and it is even in ˆM.\nχICIT2a\n2jj1describes the current flowing in ydirection,\nwhen magnetization varies in xdirection. MxzflipsˆM\nandJy, but preserves ˆM·∂2ˆM/(∂x∂t).MyzflipsˆM,\nandˆM·∂2ˆM/(∂x∂t), but preserves Jy. c2 rotation\nflipsˆM·∂2ˆM/(∂x∂t) andJy, but preserves ˆM. Conse-\nquently,χICIT2a\n2jj1is allowed by symmetry and it is odd in\nˆM.\nSimilarly, one can show that χICIT2a\n1jj2is odd in ˆMand\nthatχICIT2a\n2jj2is even in ˆM.\nAnalogously, one can investigate the symmetry prop-\nerties ofχTT2a\nijjl. We find that χTT2a\n1jj1andχTT2a\n2jj2are odd\ninˆM, whileχTT2a\n2jj1andχTT2a\n1jj2are even in ˆM.\nIV. RESULTS\nIn the following sections we discuss the results for the\ndirect and inverse chiral CIT and for the chiral torque-\ntorque correlation in the two-dimensional (2d) Rashba\nmodel Eq. (83), and in the one-dimensional (1d) Rashba\nmodel [38]\nHkx(x) =/planckover2pi12\n2mek2\nx−αkxσy+∆V\n2σ·ˆM(x).(84)\nAdditionally, we discuss the contributions of the time-\ndependent magnetization gradients, and of DDMI, DOM\nand CIDMI to these effects.\nWhile vertex corrections to the chiral CIT and to\nthe chiral torque-torque correlation are important in the\nRashba model [38], the purpose of this work is to show\nthe importance ofthe contributionsfrom time-dependent\nmagnetization gradients, DDMI, DOM and CIDMI. We\ntherefore consider only the intrinsic contributions here,\ni.e., we set\nGR\nk(E) =/planckover2pi1[E −Hk+iΓ]−1, (85)\nwhere Γ is a constant broadening, and we leave the study\nof vertex corrections for future work.\nThe results shown in the following sections are ob-\ntained for the model parameters ∆ V= 1eV,α=2eV˚A,\nand Γ = 0 .1Ry = 1.361eV, when the magnetization\npoints inzdirection, i.e., ˆM=ˆez. The unit of χCIT2\nijkl\nis charge times length in the 1d case and charge in the\n2d case. Therefore, in the 1d case we discuss the chiral\ntorkance in units of ea0, wherea0is Bohr’s radius. In the\n2d case we discuss the chiral torkance in units of e. The\nunit ofχTT2\nijklis angular momentum in the 1d case and\nangular momentum per length in the 2d case. Therefore,\nwe discussχTT2\nijklin units of /planckover2pi1in the 1d case, and in units\nof/planckover2pi1/a0in the 2d case.-2 -1 0 1 2\nFermi energy [eV]-0.02-0.0100.010.020.030.040.05χijklCIT2 [ea0]2121\n1121\n2121 (gauge-field)\n1121 (gauge-field)\nFIG. 3: Chiral CIT in the 1d Rashba model for cycloidal gra-\ndients vs. Fermi energy. General perturbation theory (soli d\nlines) agrees to the gauge-field approach (dashed lines).\nA. Direct and inverse chiral CIT\nIn Fig. 3 we show the chiral CIT as a function of the\nFermi energyfor cycloidalmagnetization gradients in the\n1d Rashba model. The components χCIT2\n2121andχCIT2\n1121are\nlabelled by 2121 and 1121, respectively. The component\n2121ofCITdescribesthe non-adiabatictorque, while the\ncomponent 1121 describes the adiabatic STT (modified\nby SOI). In the one-dimensional Rashba model, the con-\ntributionsχCIT2b\n2121andχCIT2b\n1121(Eq. (56)) from the CIDMI\nare zero when ˆM=ˆez(not shown in the figure). For cy-\ncloidal spin spirals, it is possible to solve the 1d Rashba\nmodel by a gauge-field approach [38], which allows us to\ntest the perturbation theory, Eq. (66). For comparison\nwe show in Fig. 3 the results obtained from the gauge-\nfield approach, which agree to the perturbation theory,\nEq. (66). This demonstrates the validity of Eq. (66).\nIn Fig. 4 we show the chiral ICIT in the 1d Rashba\nmodel. The components χICIT2\n1221andχICIT2\n1121are labelled\nby 1221and 1121, respectively. The contribution χICIT2a\n1221\nfrom the time-dependent gradient is of the same order of\nmagnitude as the total χICIT2\n1221. Comparison of Fig. 3 and\nFig. 4 shows that CIT and ICIT satisfy the reciprocity\nrelationsEq. (5), that χCIT2\n1121is odd in ˆM, and thatχCIT2\n2121\nis even in ˆM, i.e.,χCIT2\n2121=χICIT2\n1221andχCIT2\n1121=−χICIT2\n1121.\nThe contribution χICIT2a\n1221from the time-dependent gradi-\nents is crucial to satisfy the reciprocity relations between\nχCIT2\n2121andχICIT2\n1221.\nIn Fig. 5 and Fig. 6 we show the CIT and the ICIT, re-\nspectively, for helical gradients in the 1d Rashba model.\nThe components χCIT2\n2111andχCIT2\n1111are labelled 2111 and\n1111, respectively, in Fig. 5, while χICIT2\n1211andχICIT2\n1111\nare labelled 1211 and 1111, respectively, in Fig. 6. The\ncontributions χCIT2b\n2111andχCIT2b\n1111from CIDMI are of the14\n-2 -1 0 1 2\nFermi energy [eV]-0.0200.020.04χijklICIT2 [ea0]1221 \n1121\nχ1221ICIT2a\nFIG. 4: Chiral ICIT in the 1d Rashba model for cycloidal\ngradients vs. Fermi energy. Dashed line: Contribution from\nthe time-dependent gradient.\nsame order of magnitude as the total χCIT2\n2111andχCIT2\n1111.\nSimilarly, the contributions χICIT2c\n1211andχICIT2c\n1111from\nDOM are of the same order of magnitude as the to-\ntalχICIT2\n1211andχICIT2\n1111. Additionally, the contribution\nχICIT2a\n1111from the time-dependent gradient is substantial.\nComparisonofFig.5andFig.6showsthatCITandICIT\nsatisfy the reciprocity relation Eq. (5), that χCIT2\n2111is odd\ninˆM, and thatχCIT2\n1111is even in ˆM, i.e.,χCIT2\n1111=χICIT2\n1111\nandχCIT2\n2111=−χICIT2\n1211. These reciprocity relations be-\ntween CIT and ICIT are only satisfied when CIDMI,\nDOM, and the response to time-dependent magnetiza-\ntion gradients are included. Additionally, the compar-\nison between Fig. 5 and Fig. 6 shows that the contri-\nbutions of CIDMI to CIT ( χCIT2b\n1111andχCIT2b\n2111) are re-\nlated to the contributions of DOM to ICIT ( χICIT2c\n1111and\nχICIT2c\n1211). These relations between DOM and ICIT are\nexpected from Table I.\nIn Fig. 7 and Fig. 8 we show the CIT and the ICIT,\nrespectively, for cycloidal gradients in the 2d Rashba\nmodel. In this case there are contributions from CIDMI\nand DOM in contrast to the 1d case with cycloidal gra-\ndients (Fig. 3). Comparison between Fig. 7 and Fig. 8\nshows that χCIT2\n1121andχCIT2\n2221are odd in ˆM, thatχCIT2\n1221\nandχCIT2\n2121are even in ˆM, and that CIT and ICIT sat-\nisfy the reciprocity relation Eq. (5) when the gradients\nof CIDMI and DOM are included, i.e., χCIT2\n1121=−χICIT2\n1121,\nχCIT2\n2221=−χICIT2\n2221,χCIT2\n1221=χICIT2\n2121, andχCIT2\n2121=χICIT2\n1221.\nχCIT2\n1121describesthe adiabatic STT with SOI, while χCIT2\n2121\ndescribes the non-adiabatic STT. Experimentally, it has\nbeen found that CITs occur also when the electric field\nis applied parallel to domain-walls (i.e., perpendicular to\ntheq-vector of spin spirals) [39]. In our calculations, the\ncomponents χCIT2\n2221andχCIT2\n1221describe such a case, where\nthe applied electric field points in ydirection, while the-2 -1 0 1 2\nFermi energy [eV]-0.04-0.0200.020.040.06χijklCIT2 [ea0]1111\n2111\nχ1111CIT2b\nχ2111CIT2b\nFIG. 5: Chiral CIT for helical gradients in the 1d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nCIDMI.\n-2 -1 0 1 2\nFermi energy [eV]-0.0200.020.040.06χijklICIT2 [ea0]1111\n1211\nχ1111ICIT2a\nχ1111ICIT2c\nχ1211ICIT2c\nFIG. 6: Chiral ICIT for helical gradients in the 1d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nDOM. Dashed-dotted line: Contribution from the time-\ndependent magnetization gradient.\nmagnetization direction varies with the xcoordinate.\nIn Fig. 9 and Fig. 10 we show the chiral CIT and\nICIT, respectively, for helical gradients in the 2d Rashba\nmodel. The component χCIT2\n2111describes the adiabatic\nSTT with SOI and the component χCIT2\n1111describes the\nnon-adiabatic STT. The components χCIT2\n2211andχCIT2\n1211\ndescribe the case when the applied electric field points\ninydirection, i.e., perpendicular to the direction along\nwhich the magnetization direction varies. Comparison\nbetween Fig. 9 and Fig. 10 shows that χCIT2\n1111andχCIT2\n2211\nare even in ˆM, thatχCIT2\n1211andχCIT2\n2111are odd in ˆMand\nthat CIT andICIT satisfythe reciprocityrelationEq.(5)\nwhenthegradientsofCIDMIandDOMareincluded, i.e.,15\n-2 -1 0 1 2\nFermi energy [eV]-0.00200.0020.0040.006χijklCIT2 [e]1121\n2221\n1221\n2121\nχ2221CIT2b\nχ1221CIT2b\nχ2121CIT2b\nFIG. 7: Chiral CIT for cycloidal gradients in the 2d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nCIDMI.\n-2 -1 0 1 2\nFermi energy [eV]-0.00200.0020.0040.006χijklICIT2 [e]1121\n1221\n2121\n2221\nχ2221ICIT2a\nχ1221ICIT2a\nχ2121ICIT2c\nχ1221ICIT2c\nχ2221ICIT2c\nFIG. 8: Chiral ICIT for cycloidal gradients in the 2d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nDOM. Dashed-dotted lines: Contributions from the time-\ndependent gradients.\nχCIT2\n1111=χICIT2\n1111,χCIT2\n2211=χICIT2\n2211,χCIT2\n1211=−χICIT2\n2111, and\nχCIT2\n2111=−χICIT2\n1211.\nB. Chiral torque-torque correlation\nIn Fig. 11 we show the chiral contribution to the\ntorque-torque correlation in the 1d Rashba model for\ncycloidal gradients. We compare the perturbation the-\nory Eq. (78) plus Eq. (82) to the gauge-field approach\nfrom Ref. [38]. This comparison shows that perturba-\ntion theory provides the correct answer only when the\ncontribution χTT2a\nijkl(Eq. (82)) from the time-dependent-2 -1 0 1 2\nFermi energy [eV]-0.00200.0020.0040.006χijklCIT2 [e]2211\n1111\n1211\n2111\nχ2111CIT2b\nχ1211CIT2b\nχ2211CIT2b\nχ1111CIT2b\nFIG. 9: Chiral CIT for helical gradients in the 2d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nCIDMI.\n-2 -1 0 1 2\nFermi energy [eV]-0.004-0.00200.0020.0040.006χijklICIT2 [e]1111\n1211\n2111\n2211\nχ1111ICIT2a\nχ2221ICIT2a\nχ1111ICIT2c\nχ2111ICIT2c\nχ1211ICIT2c\nχ2211ICIT2c\nFIG. 10: Chiral ICIT for helical gradients in the 2d Rashba\nmodel vs. Fermi energy. Dashed lines: Contributions from\nDOM. Dashed-dotted lines: Contributions from the time-\ndependent gradient.\ngradients is taken into account. The contributions χTT2a\n1221\nandχTT2a\n2221fromthe time-dependent gradientsarecompa-\nrable in magnitude to the total values. In the 1d Rashba\nmodel the DDMI-contribution in Eq. (60) is zero for cy-\ncloidal gradients (not shown in the figure). The compo-\nnentsχTT2\n2121andχTT2\n1221describe the chiral gyromagnetism\nwhile the components χTT2\n1121andχTT2\n2221describe the chi-\nral damping [38, 40, 41]. The components χTT2\n2121and\nχTT2\n1221are odd in ˆMand they satisfy the Onsagerrelation\nEq. (58), i.e., χTT2\n2121=−χTT2\n1221.\nIn Fig. 12 we show the chiral contributions to the\ntorque-torque correlation in the 1d Rashba model for\nhelical gradients. In contrast to the cycloidal gradients16\n-2 -1 0 1 2\nFermi energy [eV]-0.00500.0050.01χijklTT2 [h_]2121\n1221\n2221\n1121\nχ1221TT2a\nχ2221TT2a\n2121 (gf)\n1221 (gf)\n1121 (gf)\n2221 (gf)\nFIG. 11: Chiral contribution to the torque-torque correla-\ntion for cycloidal gradients in the 1d Rashba model vs. Fermi\nenergy. Perturbation theory (solid lines) agrees to the gau ge-\nfield (gf) approach (dotted lines). Dashed lines: Contribut ion\nfrom the time-dependent gradient.\n(Fig. 11) there are contributions from the spatial gra-\ndients of DDMI (Eq. (60)) in this case. The Onsager\nrelation Eq. (58) for the components χTT2\n2111andχTT2\n1211is\nsatisfied only when these contributions from DDMI are\ntaken into account, which are of the same order of mag-\nnitude as the total values. The components χTT2\n2111and\nχTT2\n1211are even in ˆMand describe chiral damping, while\nthe components χTT2\n1111andχTT2\n2211are odd in ˆMand de-\nscribe chiral gyromagnetism. As a consequence of the\nOnsager relation Eq. (58) we obtain χTT2\n1111=χTT2\n2211= 0\nfor the total components: Eq. (58) shows that diagonal\ncomponents of the torque-torque correlation function are\nzero unless they are even in ˆM. However, χTT2a\n1111,χTT2c\n1111,\nandχTT2b\n1111=−χTT2a\n1111−χTT2c\n1111are individually nonzero.\nInterestingly, the off-diagonal components of the torque-\ntorquecorrelationdescribechiraldampingforhelicalgra-\ndients, while for cycloidal gradients the off-diagonal ele-\nments describe chiral gyromagnetism and the diagonal\nelements describe chiral damping.\nIn Fig. 13 we show the chiral contributions to the\ntorque-torque correlation in the 2d Rashba model for cy-\ncloidal gradients. In contrast to the 1d Rashba model\nwith cycloidal gradients (Fig. 11) the contributions from\nDDMIχTT2c\nijkl(Eq.(60))arenonzerointhiscase. Without\nthesecontributionsfromDDMI theOnsagerrelation(58)\nχTT2\n2121=−χTT2\n1221is violated. The DDMI contribution is\nof the same order of magnitude as the total values. The\ncomponents χTT2\n2121andχTT2\n1221are odd in ˆMand describe\nchiral gyromagnetism, while the components χTT2\n1121and\nχTT2\n2221are even in ˆMand describe chiral damping.\nIn Fig. 14 we show the chiral contributions to the\ntorque-torque correlation in the 2d Rashba model for he-\nlical gradients. The components χTT2\n1211andχTT2\n2111are even-2 -1 0 1 2\nFermi energy [eV]-0.00500.0050.01χijklTT2 [h_]1111\n2111\n1211\n2211\nχ1111TT2c\nχ2111TT2c\nχ1211TT2c\nχ2211TT2c\nχ1111TT2a\nχ2111TT2a\nFIG. 12: Chiral contribution to the torque-torque correla-\ntion for helical gradients in the 1d Rashba model vs. Fermi\nenergy. Dashed lines: Contributions from DDMI. Dashed-\ndotted lines: Contributions from the time-dependent gradi -\nents.\n-2 -1 0 1 2\nFermi energy [eV]-0.000500.00050.001χijklTT2 [h_/a0]1121\n2121\n1221\n2221\nχ1221TT2a\nχ2221TT2a\nχ2121TT2c\nχ1221TT2c\nFIG. 13: Chiral contribution to the torque-torque correla-\ntion for cycloidal gradients in the 2d Rashba model vs. Fermi\nenergy. Dashed lines: Contributions from DDMI. Dashed-\ndotted lines: Contributions from the time-dependent gradi -\nents.\ninˆMand describe chiral damping, while the compo-\nnentsχTT2\n1111andχTT2\n2211are odd in ˆMand describe chiral\ngyromagnetism. The Onsager relation Eq. (58) requires\nχTT2\n1111=χTT2\n2211= 0 andχTT2\n2111=χTT2\n1211. Without the\ncontributions from DDMI these Onsager relations are vi-\nolated.17\n-2 -1 0 1 2\nFermi energy [eV]-0.000500.00050.001χijklTT2 [h_ /a0]1111\n2111\n1211\n2211\nχ1111TT2a\nχ2111TT2a\nχ1111TT2c\nχ1211TT2c\nχ2211TT2c\nFIG. 14: Chiral contribution to the torque-torque correla-\ntion for helical gradients in the 2d Rashba model vs. Fermi\nenergy. Dashed lines: Contributions from DDMI. Dashed-\ndotted lines: Contributions from the time-dependent gradi -\nents.\nV. SUMMARY\nFinding ways to tune the Dzyaloshinskii-Moriya inter-\naction (DMI) by external means, such as an applied elec-\ntriccurrent,holdsmuchpromiseforapplicationsinwhich\nDMI determines the magnetic texture of domain walls or\nskyrmions. In order to derive an expression for current-\ninduced Dzyaloshinskii-Moriya interaction (CIDMI) we\nfirst identify its inverse effect: When magnetic textures\nvary as a function of time, electric currents are driven by\nvarious mechanisms, which can be distinguished accord-\ningtotheirdifferentdependenceonthetime-derivativeof\nmagnetization, ∂ˆM(r,t)/∂t, and on the spatial deriva-\ntive∂ˆM(r,t)/∂r: One group of effects is proportional\nto∂ˆM(r,t)/∂t, a second group of effects is propor-\ntional to the product ∂ˆM(r,t)/∂t ∂ˆM(r,t)/∂r, and\na third group is proportional to the second derivative\n∂2ˆM(r,t)/∂r∂t. We show that the response of the elec-\ntric current to the time-dependent magnetization gradi-\nent∂2ˆM(r,t)/∂r∂tcontais the inverse of CIDMI. We\nestablish the reciprocity relation between inverse and di-\nrectCIDMI and therebyobtainan expressionforCIDMI.\nWe find that CIDMI is related to the modification of\norbital magnetism induced by magnetization dynamics,\nwhich we call dynamical orbital magnetism (DOM). We\nshow that torques are generated by time-dependent gra-\ndients of magnetization as well. The inverse effect con-\nsists in the modification of DMI by magnetization dy-\nnamics, which we call dynamical DMI (DDMI).\nAdditionally, we develop a formalism to calculate the\nchiral contributions to the direct and inverse current-\ninduced torques (CITs) and to the torque-torque correla-tion in noncollinear magnets. We show that the response\nto time-dependent magnetization gradients contributes\nsubstantially to these effects and that the Onsager reci-\nprocityrelationsareviolated when it is not takeninto ac-\ncount. InnoncollinearmagnetsCIDMI,DDMIandDOM\ndepend on the local magnetization direction. We show\nthat the resulting spatial gradients of CIDMI, DDMI\nand DOM have to be subtracted from the CIT, from\nthe torque-torque correlation, and from the inverse CIT,\nrespectively.\nWe apply our formalism to study CITs and the torque-\ntorque correlation in textured Rashba ferromagnets. We\nfind that the contribution of CIDMI to the chiral CIT is\noftheorderofmagnitudeofthe totaleffect. Similarly, we\nfind that the contribution of DDMI to the chiral torque-\ntorque correlation is of the order of magnitude of the\ntotal effect.\nAcknowledgments\nWeacknowledgefinancialsupportfromLeibnizCollab-\norative Excellence project OptiSPIN −Optical Control\nofNanoscaleSpin Textures. Weacknowledgefundingun-\nder SPP 2137 “Skyrmionics” of the DFG. We gratefully\nacknowledge financial support from the European Re-\nsearch Council (ERC) under the European Union’s Hori-\nzon 2020 research and innovation program (Grant No.\n856538, project ”3D MAGiC”). The work was also sup-\nported by the Deutsche Forschungsgemeinschaft (DFG,\nGerman Research Foundation) −TRR 173 −268565370\n(project A11). We gratefully acknowledge the J¨ ulich\nSupercomputing Centre and RWTH Aachen University\nfor providing computational resources under project No.\njiff40.\nAppendix A: Response to time-dependent gradients\nIn this appendix we derive Eq. (18), Eq. (20), Eq. (41),\nand Eq. (82), which describe the response to time-\ndependent magnetization gradients, and Eq. (50), which\ndescribesthe responsetotime-dependentmagneticfields.\nWe consider perturbations of the form\nδH(r,t) =Bb1\nqωsin(q·r)sin(ωt).(A1)\nWhenweset B=∂H\n∂ˆMkandb=∂2ˆMk\n∂ri∂t, Eq.(A1)turnsinto\nEq. (17), while when we set B=−eviandb=1\n2ǫijk∂Bk\n∂t\nwe obtain Eq. (48). We need to derive an expression for\nthe response δA(r,t) of an observable Ato this pertur-\nbation, which varies in time like cos( ωt) and in space like\ncos(q·r), because∂2ˆM(r,t)\n∂ri∂t∝cos(q·r)cos(ωt). There-\nfore, weusethe Kubolinearresponseformalismtoobtain18\nthe coefficient χin\nδA(r,t) =χcos(q·r)cos(ωt), (A2)\nwhich is given by\nχ=i\n/planckover2pi1qωV/bracketleftBig\n∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n−∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htR(−/planckover2pi1ω)/bracketrightBig\n,(A3)\nwhere∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) is the retarded\nfunction at frequency ωandVis the volume of the unit\ncell.\nThe operator Bsin(q·r) can be written as\nBsin(q·r) =1\n2i/summationdisplay\nknm/bracketleftBig\nB(1)\nknmc†\nk+nck−m−B(2)\nknmc†\nk−nck+m/bracketrightBig\n,\n(A4)\nwherek+=k+q/2,k−=k−q/2,c†\nk+nis the cre-\nation operator of an electron in state |uk+n∝an}bracketri}ht,ck−mis the\nannihilation operator of an electron in state |uk−m∝an}bracketri}ht,\nB(1)\nknm=1\n2∝an}bracketle{tuk+n|[Bk++Bk−]|uk−m∝an}bracketri}ht(A5)\nand\nB(2)\nknm=1\n2∝an}bracketle{tuk−n|[Bk++Bk−]|uk+m∝an}bracketri}ht.(A6)\nSimilarly,\nAcos(q·r) =1\n2/summationdisplay\nknm/bracketleftBig\nA(1)\nknmc†\nk+nck−m+A(2)\nknmc†\nk−nck+m/bracketrightBig\n,\n(A7)\nwhere\nA(1)\nknm=1\n2∝an}bracketle{tuk+n|/bracketleftbig\nAk++Ak−/bracketrightbig\n|uk−m∝an}bracketri}ht(A8)\nand\nA(2)\nknm=1\n2∝an}bracketle{tuk−n|/bracketleftbig\nAk++Ak−/bracketrightbig\n|uk+m∝an}bracketri}ht.(A9)\nIt is convenient to obtain the retarded response func-\ntion in Eq. (A3) from the correspondingMatsubarafunc-\ntion in imaginary time τ\n1\nV∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htM(τ) =\n=1\n4i/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/bracketleftBig\nA(1)\nknmB(2)\nkn′m′Z(1)\nknmn′m′(τ)\n−A(2)\nknmB(1)\nkn′m′Z(2)\nknmn′m′(τ)/bracketrightBig\n,\n(A10)\nwhered= 1,2 or 3 is the dimension,\nZ(1)\nknmn′m′(τ) =∝an}bracketle{tTτc†\nk+n(τ)ck−m(τ)c†\nk−n′(0)ck+m′(0)∝an}bracketri}ht\n=−GM\nm′n(k+,−τ)GM\nmn′(k−,τ),\n(A11)Z(2)\nknmn′m′(τ) =∝an}bracketle{tTτc†\nk−n(τ)ck+m(τ)c†\nk+n′(0)ck−m′(0)∝an}bracketri}ht\n=−GM\nm′n(k−,−τ)GM\nmn′(k+,τ),\n(A12)\nand\nGM\nmn′(k+,τ) =−∝an}bracketle{tTτck+m(τ)c†\nk+n′(0)∝an}bracketri}ht(A13)\nis the single-particle Matsubara function. The Fourier\ntransform of Eq. (A10) is given by\n1\nV∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htM(iEN) =\n=i\n4/planckover2pi1β/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\np/bracketleftBig\nA(1)\nknmB(2)\nkn′m′GM\nm′n(k+,iEp)GM\nmn′(k−,iEp+iEN)\n−A(2)\nknmB(1)\nkn′m′GM\nm′n(k−,iEp)GM\nmn′(k+,iEp+iEN)/bracketrightBig\n,\n(A14)\nwhereEN= 2πN/βandEp= (2p+ 1)π/βare bosonic\nandfermionicMatsubaraenergypoints, respectively, and\nβ= 1/(kBT) is the inverse temperature.\nIn order to carry out the Matsubara summation over\nEpwe make use of\n1\nβ/summationdisplay\npGM\nmn′(iEp+iEN)GM\nm′n(iEp) =\n=i\n2π/integraldisplay\ndE′f(E′)GM\nmn′(E′+iEN)GM\nm′n(E′+iδ)\n+i\n2π/integraldisplay\ndE′f(E′)GM\nmn′(E′+iδ)GM\nm′n(E′−iEN)\n−i\n2π/integraldisplay\ndE′f(E′)GM\nmn′(E′+iEN)GM\nm′n(E′−iδ)\n−i\n2π/integraldisplay\ndE′f(E′)GM\nmn′(E′−iδ)GM\nm′n(E′−iEN),(A15)\nwhereδis a positive infinitesimal. The retarded function\n∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htR(ω) is obtained from the Mat-\nsubara function ∝an}bracketle{t∝an}bracketle{tAcos(q·r),Bsin(q·r)∝an}bracketri}ht∝an}bracketri}htM(iEN) by the\nanalytic continuation iEN→/planckover2pi1ωto real frequencies. The\nright-hand side of Eq. (A15) has the following analytic\ncontinuation to real frequencies:\ni\n2π/integraldisplay\ndE′f(E′)GR\nmn′(E′+/planckover2pi1ω)GR\nm′n(E′)\n+i\n2π/integraldisplay\ndE′f(E′)GR\nmn′(E′)GA\nm′n(E′−/planckover2pi1ω)\n−i\n2π/integraldisplay\ndE′f(E′)GR\nmn′(E′+/planckover2pi1ω)GA\nm′n(E′)\n−i\n2π/integraldisplay\ndE′f(E′)GA\nmn′(E′)GA\nm′n(E′−/planckover2pi1ω).(A16)\nTherefore, we obtain\nχ=−i\n8π/planckover2pi12qω/integraldisplayddk\n(2π)d[Zk(q,ω)−Zk(−q,ω)\n−Zk(q,−ω)+Zk(−q,−ω)],(A17)19\nwhere\nZk(q,ω) =\n=/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−(E′+/planckover2pi1ω)BkGR\nk+(E′)/bracketrightBig\n+/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−(E′)BkGA\nk+(E′−/planckover2pi1ω)/bracketrightBig\n−/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−(E′+/planckover2pi1ω)BkGA\nk+(E′)/bracketrightBig\n−/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGA\nk−(E′)BkGA\nk+(E′−/planckover2pi1ω)/bracketrightBig\n.(A18)\nWe consider the limit lim q→0limω→0χ. In this limit\nEq. (A17) may be rewritten as\nχ=−i\n2π/planckover2pi12/integraldisplayddk\n(2π)d∂2Zk(q,ω)\n∂q∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq=ω=0.(A19)\nThe frequency derivative of Zk(q,ω) is given by\n1\n/planckover2pi1∂Zk\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0=/integraldisplay\ndE′f(E′)Tr/bracketleftBigg\nAk∂GR\nk−(E′)\n∂E′BkGR\nk+(E′)/bracketrightBigg\n−/integraldisplay\ndE′f(E′)Tr/bracketleftBigg\nAkGR\nk−(E′)Bk∂GA\nk+(E′)\n∂E′/bracketrightBigg\n−/integraldisplay\ndE′f(E′)Tr/bracketleftBigg\nAk∂GR\nk−(E′)\n∂E′BkGA\nk+(E′)/bracketrightBigg\n+/integraldisplay\ndE′f(E′)Tr/bracketleftBigg\nAkGA\nk−(E′)Bk∂GA\nk+(E′)\n∂E′/bracketrightBigg\n.\n(A20)\nUsing∂GR(E)/∂E=−GR(E)GR(E)//planckover2pi1we obtain\n∂Zk\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0=−/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−GR\nk−BkGR\nk+/bracketrightBig\n+/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−BkGA\nk+GA\nk+/bracketrightBig\n+/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGR\nk−GR\nk−BkGA\nk+/bracketrightBig\n−/integraldisplay\ndE′f(E′)Tr/bracketleftBig\nAkGA\nk−BkGA\nk+GA\nk+/bracketrightBig\n.\n(A21)\nMaking use of\nlim\nq→0∂GR\nk+\n∂q=1\n2GR\nkv·q\nqGR\nk (A22)we finally obtain\nχ=−i\n2π/planckover2pi12/integraldisplayddk\n(2π)dlim\nq→0lim\nω→0∂2Z(q,ω)\n∂q∂ω=\n=−i\n4π/planckover2pi12q\nq·/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBig\nAkRvRRBkR+AkRRvRBkR\n−AkRRBkRvR−AkRvRBkAA\n+AkRBkAvAA+AkRBkAAvA\n−AkRvRRBkA−AkRRvRBkA\n+AkRRBkAvA\n+AkAvABkAA−AkABkAvAA\n−AkABkAAvA/bracketrightBig\n,(A23)\nwhere we use the abbreviations R=GR\nk(E) andA=\nGA\nk(E). When we substitute B=∂H\n∂ˆMj,A=−evi, and\nq=qkˆek, we obtain Eq. (18). When we substitute B=\nTj,A=−evi, andq=qkˆek, we obtain Eq. (20). When\nwe substitute A=−Ti,B=Tj, andq=qkˆek, we obtain\nEq. (41). When we substitute B=−evj,A=−Ti,\nandq=qkˆek, we obtain Eq. (50). When we substitute\nB=∂H\n∂ˆMj,A=−Ti, andq=qkˆek, we obtain Eq. (82).\nAppendix B: Perturbation theory for the chiral\ncontributions to CIT and to the torque-torque\ncorrelation\nIn this appendix we derive expressionsfor the retarded\nfunction\n∝an}bracketle{t∝an}bracketle{tAcos(q·r);C∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω) (B1)\nwithin first-orderperturbation theory with respect to the\nperturbation\nδH=Bηsin(q·r), (B2)\nwhich may arise e.g. from the spatial oscillation of the\nmagnetization direction. As usual, it is convenient to ob-\ntain the retarded response function from the correspond-\ning Matsubara function\n∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(τ) =−∝an}bracketle{tTτcos(q·r)A(τ)C(0)∝an}bracketri}ht.\n(B3)\nThe starting point for the perturbative expansion is\nthe equation\n−∝an}bracketle{tTτcos(q·r)A(τ1)C(0)∝an}bracketri}ht=\n=−Tr/bracketleftbig\ne−βHTτcos(q·r)A(τ1)C(0)/bracketrightbig\nTr[e−βH]=\n=−Tr/braceleftbig\ne−βH0Tτ[Ucos(q·r)A(τ1)C(0)]/bracerightbig\nTr[e−βH0U],(B4)20\nwhereH0is the unperturbed Hamiltonian and we con-\nsider the first order in the perturbation δH:\nU(1)=−1\n/planckover2pi1/integraldisplay/planckover2pi1β\n0dτ1Tτ{eτ1H0//planckover2pi1δHe−τ1H0//planckover2pi1}.(B5)\nThe essentialdifference between Eq. (A3) and Eq. (B4) is\nthat in Eq. (A3) the operator Benters together with the\nfactor sin( q·r)sin(ωt) (see Eq. (A1)), while in Eq. (B4)\nonly the factor sin( q·r) is connected to Bin Eq. (B2),\nwhile the factor sin( ωt) is coupled to the additional op-\neratorC.\nWe use Eq. (A4) and Eq. (A7) in order to express\nAcos(q·r) andBsin(q·r) in terms of annihilation and\ncreation operators. In terms of the correlators\nZ(3)\nknmn′m′n′′m′′(τ,τ1) =\n∝an}bracketle{tTτc†\nk−n(τ)ck+m(τ)c†\nk+n′(τ1)ck−m′(τ1)c†\nk−n′′ck−m′′∝an}bracketri}ht\n(B6)\nand\nZ(4)\nknmn′m′n′′m′′(τ,τ1) =\n∝an}bracketle{tTτc†\nk−n(τ)ck+m(τ)c†\nk+n′(τ1)ck−m′(τ1)c†\nk+n′′ck+m′′∝an}bracketri}ht\n(B7)\nand\nZ(5)\nknmn′m′n′′m′′(τ,τ1) =\n∝an}bracketle{tTτc†\nk+n(τ)ck−m(τ)c†\nk−n′(τ1)ck+m′(τ1)c†\nk+n′′ck+m′′∝an}bracketri}ht\n(B8)\nand\nZ(6)\nknmn′m′n′′m′′(τ,τ1) =\n∝an}bracketle{tTτc†\nk+n(τ)ck−m(τ)c†\nk−n′(τ1)ck+m′(τ1)c†\nk−n′′ck−m′′∝an}bracketri}ht\n(B9)\nEq. (B4) can be written as\n∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(τ1) =\n=ηV\n4i/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay/planckover2pi1β\n0dτ/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′/bracketleftBigg\n−B(2)\nknmA(1)\nkn′m′Ck−n′′m′′Z(3)\nknmn′m′n′′m′′(τ,τ1)\n−B(2)\nknmA(1)\nkn′m′Ck+n′′m′′Z(4)\nknmn′m′n′′m′′(τ,τ1)\n+B(1)\nknmA(2)\nkn′m′Ck+n′′m′′Z(5)\nknmn′m′n′′m′′(τ,τ1)\n+B(1)\nknmA(2)\nkn′m′Ck−n′′m′′Z(6)\nknmn′m′n′′m′′(τ,τ1)/bracketrightBigg(B10)\nwithin first-order perturbation theory, where we de-\nfinedCk−n′′m′′=∝an}bracketle{tuk−n′′|C|uk−m′′∝an}bracketri}htandCk+n′′m′′=\n∝an}bracketle{tuk+n′′|C|uk+m′′∝an}bracketri}ht.\nNote that Z(5)can be obtained from Z(3)by replac-\ningk−byk+andk+byk−. Similarly, Z(6)can be\nobtained from Z(4)by replacing k−byk+andk+by\nk−. Therefore, we write down only the equations forZ(3)andZ(4)in the following. Using Wick’s theorem\nwe find\nZ(3)\nknmn′m′n′′m′′(τ,τ1) =\n=−GM\nm′n(k−,τ1−τ)GM\nmn′(k+,τ−τ1)GM\nm′′n′′(k−,0)\n+GM\nmn′(k+,τ−τ1)GM\nm′′n(k−,−τ)GM\nm′n′′(k−,τ1)\n(B11)\nand\nZ(4)\nknmn′m′n′′m′′(τ,τ1) =\n=−GM\nmn′(k+,τ−τ1)GM\nm′n(k−,τ1−τ)GM\nm′′n′′(k+,0)\n+GM\nmn′′(k+,τ)GM\nm′n(k−,τ1−τ)GM\nm′′n′(k+,−τ1).\n(B12)\nThe Fourier transform\n∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(iEN) =\n=/integraldisplay/planckover2pi1β\n0dτ1ei\n/planckover2pi1ENτ1∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(τ1)(B13)\nof Eq. (B10) can be written as\n∝an}bracketle{t∝an}bracketle{tcos(q·r)A;C∝an}bracketri}ht∝an}bracketri}htM(iEN) =\n=ηV\n4i/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′/bracketleftBigg\n−B(2)\nknmA(1)\nkn′m′Ck−n′′m′′Z(3a)\nknmn′m′n′′m′′(iEN)\n−B(2)\nknmA(1)\nkn′m′Ck+n′′m′′Z(4a)\nknmn′m′n′′m′′(iEN)\n+B(1)\nknmA(2)\nkn′m′Ck+n′′m′′Z(5a)\nknmn′m′n′′m′′(iEN)\n+B(1)\nknmA(2)\nkn′m′Ck−n′′m′′Z(6a)\nknmn′m′n′′m′′(iEN)/bracketrightBigg(B14)\nin terms of the integrals\nZ(3a)\nknmn′m′n′′m′′(iEN) =/integraldisplay/planckover2pi1β\n0dτ/integraldisplay/planckover2pi1β\n0dτ1ei\n/planckover2pi1ENτ1×\n×GM\nmn′(k+,τ−τ1)GM\nm′′n(k−,−τ)GM\nm′n′′(k−,τ1) =\n=1\n/planckover2pi1β/summationdisplay\npGM\nk+mn′(iEp)GM\nk−m′′n(iEp)GM\nk−m′n′′(iEp+iEN)\n(B15)\nand\nZ(4a)\nknmn′m′n′′m′′(iEN) =/integraldisplay/planckover2pi1β\n0dτ/integraldisplay/planckover2pi1β\n0dτ1ei\n/planckover2pi1ENτ1×\n×GM\nmn′′(k+,τ)GM\nm′n(k−,τ1−τ)GM\nm′′n′(k+,−τ1) =\n=1\n/planckover2pi1β/summationdisplay\npGM\nk+mn′′(iEp)GM\nk−m′n(iEp)GM\nk+m′′n′(iEp−iEN),\n(B16)\nwhereEN= 2πN/βis a bosonic Matsubara energy point\nand we used\nGM(τ) =1\n/planckover2pi1β∞/summationdisplay\np=−∞e−iEpτ//planckover2pi1GM(iEp),(B17)21\nwhereEp= (2p+1)π/βis a fermionic Matsubara point.\nAgain Z(5a)is obtained from Z(3a)by replacing k−by\nk+andk+byk−andZ(6a)is obtained from Z(4a)in\nthe same way.\nSummation overMatsubarapoints Epin Eq.(B15) and\nin Eq. (B16) and analytic continuation iEN→/planckover2pi1ωyields\n2πi/planckover2pi1Z(3a)\nknmn′m′n′′m′′(/planckover2pi1ω) =\n−/integraldisplay\ndEf(E)GR\nk+mn′(E)GR\nk−m′′n(E)GR\nk−m′n′′(E+/planckover2pi1ω)\n+/integraldisplay\ndEf(E)GA\nk+mn′(E)GA\nk−m′′n(E)GR\nk−m′n′′(E+/planckover2pi1ω)\n−/integraldisplay\ndEf(E)GA\nk+mn′(E−/planckover2pi1ω)GA\nk−m′′n(E−/planckover2pi1ω)GR\nk−m′n′′(E)\n+/integraldisplay\ndEf(E)GA\nk+mn′(E−/planckover2pi1ω)GA\nk−m′′n(E−/planckover2pi1ω)GA\nk−m′n′′(E)\n(B18)\nand\n2πi/planckover2pi1Z(4a)\nknmn′m′n′′m′′(/planckover2pi1ω) =\n−/integraldisplay\ndEf(E)GR\nk+mn′′(E)GR\nk−m′n(E)GA\nk+m′′n′(E−/planckover2pi1ω)\n+/integraldisplay\ndEf(E)GA\nk+mn′′(E)GA\nk−m′n(E)GA\nk+m′′n′(E−/planckover2pi1ω)\n−/integraldisplay\ndEf(E)GR\nk+mn′′(E+/planckover2pi1ω)GR\nk−m′n(E+/planckover2pi1ω)GR\nk+m′′n′(E)\n+/integraldisplay\ndEf(E)GR\nk+mn′′(E+/planckover2pi1ω)GR\nk−m′n(E+/planckover2pi1ω)GA\nk+m′′n′(E).\n(B19)\nIn the next step we take the limit ω→0 (see Eq. (64),\nEq. (70), and Eq. (77)):\n−1\nVlim\nω→0Im∝an}bracketle{t∝an}bracketle{tAcos(q·r);C∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n/planckover2pi1ω=\n=η\n4/planckover2pi1Im/bracketleftBig\nY(3)+Y(4)−Y(5)−Y(6)/bracketrightBig\n,(B20)where we defined\nY(3)=1\ni/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′B(2)\nknmA(1)\nkn′m′Ck−n′′m′′×\n×∂Z(3a)\nknmn′m′n′′m′′(/planckover2pi1ω)\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0,\nY(4)=1\ni/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′B(2)\nknmA(1)\nkn′m′Ck+n′′m′′×\n×∂Z(4a)\nknmn′m′n′′m′′(/planckover2pi1ω)\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0,\nY(5)=1\ni/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′B(1)\nknmA(2)\nkn′m′Ck+n′′m′′×\n×∂Z(5a)\nknmn′m′n′′m′′(/planckover2pi1ω)\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0,\nY(6)=1\ni/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nnm/summationdisplay\nn′m′/summationdisplay\nn′′m′′B(1)\nknmA(2)\nkn′m′Ck−n′′m′′×\n×∂Z(6a)\nknmn′m′n′′m′′(/planckover2pi1ω)\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nω=0,\n(B21)\nwhich can be expressed as Y(3)=Y(3a)+Y(3b)and\nY(4)=Y(4a)+Y(4b), where\n2π/planckover2pi1Y(3a)=1\n/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)×\n×Tr/bracketleftBigg\nAkGR\nk−(E)Ck−GA\nk−(E)BkGA\nk+(E)GA\nk+(E)\n+AkGR\nk−(E)GR\nk−(E)Ck−GA\nk−(E)BkGA\nk+(E)\n+AkGR\nk−(E)Ck−GA\nk−(E)GA\nk−(E)BkGA\nk+(E)/bracketrightBigg\n=/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)×\n×Tr/bracketleftBig\nAkGR\nk−(E)Ck−GA\nk−(E)BkGA\nk+(E)/bracketrightBig\n(B22)\nand\n2π/planckover2pi1Y(3b)=−1\n/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)×\n×Tr/bracketleftBigg\nAkGA\nk−(E)Ck−GA\nk−(E)BkGA\nk+(E)GA\nk+(E)\n+AkGR\nk−(E)GR\nk−(E)Ck−GR\nk−(E)BkGR\nk+(E)\n+AkGA\nk−(E)Ck−GA\nk−(E)GA\nk−(E)BkGA\nk+(E)/bracketrightBigg\n.(B23)22\nSimilarly,\n2π/planckover2pi1Y(4a)=1\n/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)×\n×Tr/bracketleftBigg\nAkGR\nk−(E)BkGR\nk+(E)Ck+GA\nk+(E)GA\nk+(E)\n−AkGR\nk−(E)GR\nk−(E)BkGR\nk+(E)Ck+GA\nk+(E)\n−AkGR\nk−(E)BkGR\nk+(E)GR\nk+(E)Ck+GA\nk+(E)/bracketrightBigg\n=/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)×\n×Tr/bracketleftBig\nAkGR\nk−(E)BkGR\nk+(E)Ck+GA\nk+(E)/bracketrightBig\n(B24)\nand\n2π/planckover2pi1Y(4b)=−1\n/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)×\n×Tr/bracketleftBigg\nAkGA\nk−(E)BkGA\nk+(E)Ck+GA\nk+(E)GA\nk+(E)\n+AkGR\nk−(E)GR\nk−(E)BkGR\nk+(E)Ck+GR\nk+(E)\n+AkGR\nk−(E)BkGR\nk+(E)GR\nk+(E)Ck+GR\nk+(E)/bracketrightBigg\n.(B25)\nWe call Y(3a)andY(4a)Fermi surface terms and Y(3b)\nandY(4b)Fermi sea terms. Again Y(5)is obtained from\nY(3)by replacing k−byk+andk+byk−andY(6)is\nobtained from Y(4)in the same way.\nFinally, we take the limit q→0:\nΛ =−2\n/planckover2pi1VηIm lim\nq→0lim\nω→0∂\n∂ω∂\n∂qi∝an}bracketle{t∝an}bracketle{tAcos(q·r);C∝an}bracketri}ht∝an}bracketri}htR(/planckover2pi1ω)\n=1\n2/planckover2pi1lim\nq→0∂\n∂qiIm/bracketleftBig\nY(3)+Y(4)−Y(5)−Y(6)/bracketrightBig\n=1\n2/planckover2pi1Im/bracketleftBig\nX(3)+X(4)−X(5)−X(6)/bracketrightBig\n,\n(B26)\nwhere we defined\nX(j)=∂\n∂qi/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq=0Y(j)(B27)\nforj= 3,4,5,6. Since Y(4)andY(6)are related by\nthe interchange of k−andk+it follows that X(6)=\n−X(4). Similarly, since Y(3)andY(5)arerelated by the\ninterchange of k−andk+it follows that X(5)=−X(3).\nConsequently, we need\nΛ =1\n/planckover2pi1Im/bracketleftBig\nX(3a)+X(3b)+X(4a)+X(4b)/bracketrightBig\n,(B28)\nwhere X(3a)andX(4a)are the Fermi surface terms and\nX(3b)andX(4b)are the Fermi sea terms. The Fermisurface terms are given by\nX(3a)=−1\n4π/planckover2pi1/integraldisplayddk\n(2π)d/integraldisplay\ndEf′(E)Tr/bracketleftBigg\nAkGR\nk(E)vkGR\nk(E)CkGA\nk(E)BkGA\nk(E)\n+AkGR\nk(E)CkGA\nk(E)vkGA\nk(E)BkGA\nk(E)\n−AkGR\nk(E)CkGA\nk(E)BkGA\nk(E)vkGA\nk(E)\n+AkGR\nk(E)∂Ck\n∂kGA\nk(E)BkGA\nk(E)/bracketrightBigg(B29)\nand\nX(4a)=−/bracketleftBig\nX(3a)/bracketrightBig∗\n. (B30)\nThe Fermi sea terms are given by\nX(3b)=−1\n4π/planckover2pi12/integraldisplayddk\n(2π)d/integraldisplay\ndEf(E)Tr/bracketleftBigg\n−(ARvRRCRBR)+(AACAABAvA)\n−(ARRvRCRBR)−(ARRCRvRBR)\n+(ARRCRBRvR)−(AAvACABAA)\n−(AACAvABAA)+(AACABAvAA)\n+(AACABAAvA)−(AAvACAABA)\n−(AACAvAABA)−(AACAAvABA)\n−(ARR∂C\n∂kRBR)−(AA∂C\n∂kAABA)\n−(AA∂C\n∂kABAA)/bracketrightBigg(B31)\nand\nX(4b)=−/bracketleftBig\nX(3b)/bracketrightBig∗\n. (B32)\nIn Eq. (B31) we use the abbreviations R=GR\nk(E),A=\nGA\nk(E),A=Ak,B=Bk,C=Ck. It is important\nto note that Ck−andCk+depend on qthrough k−=\nk−q/2 andk+=k+q/2 . Theqderivative therefore\ngenerates the additional terms with ∂Ck/∂kin Eq. (B29)\nand Eq. (B31). In contrast, AkandBkdo not depend\nlinearly on q.\nEq. (B28) simplifies due to the relations Eq. (B30) and\nEq. (B32) as follows:\nΛ =2\n/planckover2pi1Im/bracketleftBig\nX(3a)+X(3b)/bracketrightBig\n. (B33)\nIn order to obtain the expression for the chiral con-\ntribution to the torque-torque correlation we choose the\noperators as follows:\nB→ Tk\nA→ −Ti\nC→ Tj\n∂C\n∂k= 0\nv→vl.(B34)23\nThis leads to Eq. (78), Eq. (79) and Eq. (80) of the main\ntext.\nIn order to obtain the expression for the chiral contri-\nbution to the CIT, we set\nB→ Tk\nA→ −Ti\nC→ −evj\n∂C\n∂k→ −e/planckover2pi1\nmδjl\nv→vl.(B35)\nThis leads to Eq. (66), Eq. (67) and Eq. (68).\nIn order to obtain the expression for the chiral contri-\nbution to the ICIT, we set\nB→ Tk\nA→ −evi\nC→ Tj\n∂C\n∂k→0\nv→vl.(B36)\nThis leads to Eq. (71), Eq. (72) and Eq. (73).\n∗Corresp. author: f.freimuth@fz-juelich.de\n[1] K. Nawaoka, S. 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Krivorotov \nDepartment of Physics and Astronomy, Univers ity of California–Irvine, Irvine, California \n92697, USA \ndmitri.e.nikonov@intel.com \nAbstract \nSchemes of switching nanomagnetic memories via the effect of spin torque with various \npolarizations of injected electrons are studied. Simulations based on macrospin and \nmicromagnetic theories are performed and compared. We demonstrate that switching \nwith perpendicularly polarized current by short pulses and free precession requires \nsmaller time and energy than spin torque switching with collinear in plane spin \npolarization; it is also found to be superior to other ki nds of memories. We study the \ntolerances of switching to the magnitude of  current and pulse duration. An increased \nGilbert damping is found to improve tolera nces of perpendicular switching without \nincreasing the threshold current , unlike in plane switching.  Page2\n \n1. Introduction \nResearch in spintronics resulted in huge  technological impact  via development of \nextremely high capacity hard  drives and magnetic RAM1. The basic paradigm of such \ndevices is a stack of two ferromagnetic (FM)  layers separated by a non-magnetic layer, \nsee Fig. 1a. Giant magnetoresistance2,3 (GMR) – dependence of resi stance of such a stack \non the relative direction of magne tization in the FM layers – provides the crucial path to \ninterfacing magnetic and electroni c states in the device. If the non-magnetic layer is a \ndielectric, this effect is calle d tunneling magnetoresistance (TMR)4,5,6,7. \nA better way of switching ma gnetization in such devices  was first theoretically \npredicted8,9 followed by the demonstration10,11,12,13,14 of spin transfer torque effect. This \neffect consists of precession of magnetization of one of the FM layers as current flows \nacross the stack. As this happens, the angul ar momentum of spin-polarized current \noriginating in one FM layer is transferred to  the magnetization of another FM layer, see \nFig 1a. Since this original work, a tremendous  amount of research has been conducted in \nthe field (see review15). Good values of the performance metrics have been achieved: the \nmemory cell size has been reduced to a few square microns, the switching time to a few \nnanoseconds, and the switching current to a few milliamps. Spin transfer torque random \naccess memory (STTRAM) has been prototyped16 and is close to commercialization. \nStill to be competitive with the incumbent memory technologies, such as SRAM, DRAM, \nand flash, STTRAM has to surpass them in the majority of a set of metrics (size, speed,  Page3\nenergy). The fact that STTRAM is non-volatile is an important advantage, but would not \nalone ensure commercial success. For this r eason it is necessary to devise ways to \ndecrease STTRAM’s threshold switching curren t and time, and thus the overall switching \nenergy. One of the possible pathways to th is end is conducting sw itching by short pulses \nrather than quasi-steady currents. \nA previous theoretical treatment17 of pulsed currents envisioned two pulses of opposite \npolarity without a gap between them. Experi mental demonstration of switching with \npulses was performed with single pulses 18,19 or double pulses 20 of constant polarity.   \nIn the present work we attempt to give a comprehensive view of the pulsed switching of \nnanomagnetic memories. We consider a variet y of cases of spin polarization of the \ncurrent injected from the fixed FM layer to  the free FM layer. In considering various \ndynamical switching strategies, we first define  the energy landscape of the system and \nrepresent a specific switching strategy as a specific path across th e topological surface \nwhich defines the energy landscape   The anal ysis contained here identifies the optimal \nswitching strategy to be along the path of steep est accent (unlike the tr aditional, in-plane \ncollinear polarization switching which pro ceeds along the path of minimum accent). We \nperform simulations both in the macrospi n and micromagnetic approximations and \ncompare them side by side in order to make conclusions on applicabil ity of these methods \nin each of the cases. In these simulations we se parate the contribution of spin transfer and \nfield-like torques 21,22,23 to draw a conclusion on how they  affect switching in each of the \ncases. Finally we perform multiple simulations over a range of switching parameters in \norder to predict the tolerances of the memori es relative to the va riations in both the  Page4\nmagnitude of current and the pulse duration. Su ch variations of curre nt and pulse duration \nare bound to exist in elec tronic circuits due to fabricati on variability and the temperature \ndrift. Counter-intuitively, we find that Gilbert damping is be neficial for the tolerance of \nswitching and increases neither the thres hold current nor the switching duration.  \nThe paper is organized as follows. Section 2 co ntains the description of the mathematical \nmodels of the macrospin approximation and micromagnetic simulator OOMMF used in \nthis paper. Section 3 analyzes the energy dependence on the direction of magnetization \nand various strategies for switching based on it. Time dependence of magnetization in \nvarious strategies of switching is also ex emplified. Results of multiple simulation runs, \npresented in Section 4, establish tolerances of  switching relative to current magnitude and \npulse duration. In Section 5 we compare th e results of macrospin and micromagnetic \nsimulations side by side. The conclusions of this work ar e summarized in Section 6.  \n2. Mathematical models \nThe mathematical model of macrospin dynami cs is based on the Landau-Lifshitz-Gilbert \n(LLG) equation (see the review24). In addition it assumes that spatial variation of \nmagnetization can be neglected, and the w hole magnetic moment of the nanomagnet \n(“macrospin”) can be represented by a single average vector of magnetization M, or \ndimensionless magnetization /sM=mM , where the saturation magnetization of the \nmaterial is sM. Thus the LLG equation, containing the spin torque terms Γ, is   Page5\n 0 effdd\ndt dtγµ α⎡⎤⎡⎤=− + +⎣⎦ ⎢⎥⎣⎦mmm×H m× Γ, (1) \nwhere the gyromagnetic constant is Bgµγ==, the Bohr magneton is Bµ, and the Lande \nfactor is g, the permeability of vacuum is 0µ, and the Gilbert damping factor is α. The \neffective magnetic field originates from all contributions to the energy E per unit volume \nof the nanomagnet: \n 01\neffEδ\nµδ=−HM (2) \nThe energy of the nanomagnet includes the demagnetization term – coming from the \ninteraction of magnetic dipoles between them selves, the material anisotropy, and the \nZeeman energy due to an external magnetic fiel d. In this article we disregard the latter \ntwo contributions for the free FM layer, and focus on the former. The demagnetization \nterm (synonymous with the shape an isotropy of the nanomagnet) is \n 0\n2Eµ= MNM , (3) \nThe demagnetization tens or is diagonal and =1xx yy zzNNN++ , if the coordinate axes \ncoincide with the principal axes of the nanomagnet:  Page6\n 00\n0000xx\nyy\nzzN\nN\nN⎛⎞\n⎜⎟=⎜⎟⎜⎟⎝⎠N . (4) \nFor the shape of typical nanomagnets, ellipti cal cylinder, the demagnetization tensor is \ncalculated according to Ref. 25. A heuristic rule for demagneti zation energy is that it is \nlowest when magnetization points along the l ongest axis of a nanomagnet and highest \nwhen it points along the shortest axis. For exam ple, in the case considered here of the \nnanomagnet with dimensions of 120nm*60nm*3nm, the demagnetization tensor elements \nare =0.0279,  0.0731,  =0.8990xx yy zzNN N = . \nThe spin torque contribution is described by th e spin transfer (Slonczewski) and field like \nterms (in the brackets of the following equation) \n [][]() '\nsJ\nMe tγεε=+ ⎡⎤⎣⎦Γ m× p×m p×m=, (5) \nwhere t is the thickness of the nanomagnet,  e is the absolute value of the electron charge, \nJ is the density of current perpendicu lar to the plane of the nanomagnet, p is the unit \nvector of polarizati on of electrons , and P is the degree of polari zation of these electrons. \nIn this paper we disregard the angular depende nce in the prefactor fo r simplicity, so that \n 2Pε= (6) \nThe mathematical model of micromagnetics is realized in a widely used simulator \nOOMMF 26. It is based on the same LLG equation (1). The main difference from the  Page7\nmacrospin model is that now magnetization M is considered a function of spatial \ncoordinates (discretized on a gr id) and its spatial variation plays an important role. The \nexchange interaction of between spin s is included as the energy density \n ( )2 22\nxyz EAm m m= ∇ +∇ +∇  (7) \nand the demagnetization energy is calculat ed by explicit summation of dipole-dipole \ninteractions between the parts of the nanomagnet, see Ref. 26. In this paper we will use \nthe following set of typical parameters a nd hope that the reader  agrees that the \nconclusions of the paper do not depend on this  particular choice of numerical values: \nsaturation magnetization 1 /sM MA m= , Lande factor 2g=, spin polarization 0.8P= , \nand the exchange constant 1121 0 /A Jm−=⋅ . In the cases when we include the field-like \ntorque, we set its constant '0 . 3εε=  to be in approximate agr eement with the results of \nRefs. 21 and 22. \n3. Energy profile and strategies for switching \nWe are applying the above mathematical model to treat various cases of spin transfer \ntorque switching, Fig. 1. In these schemes of  nanopillars, the upper blue layer designates \na free nanomagnet. We introduce th e coordinate axes as follow s: x along the long axis of \nthe nanomagnet, in plane of the chip, y perp endicular to x in plane of the chip, and z \nperpendicular to the plane of the chip. We  also introduce the angles to specify the \nmagnetization direction: θ, the angle from the x-axis, and φ, the angle of the projection \non the yz-plane from the y axis. The free nanomagnet has two stable (lowest energy)  Page8\nstates when magnetization points al ong +x or –x directions, i.e. 0, θπ= . The goal of \nmemory engineering is to switch magnetiza tion between these two states. The bottom \nblue layer designates the fixed nanomagnet. Even though the spin torque acts on this \nlayer as well, one keeps its magnetization fr om switching by coupling it to an adjacent \n(“pinning”) antiferromagnetic la yer (not shown in the pictur e). The magnetization of the \nfixed nanomagnet can be set in various direc tions by fabricating it with the right shape \nand magnetocrystalline anisotropy. One cas e is when the fixed magnetization is \napproximately along the x-axis, Fig. 1a. If both magnetizations were exactly along the x-\naxis, the spin torque acting on the free laye r would be zero, and sw itching would not take \nplace. In fact, thermal fluctuations cause the angles of both free and fixed nanomagnets to \nhave random values around 0θ=. Therefore in the macrospi n simulations, we formally \nset the initial angle of the free nanomagnet to 0.1θ= radians, and set the angle of the \nfixed layer to 0θ=. Another case, Fig 1b, is that of th e fixed magnetization in plane of \nthe chip, with various θ and 0φ=. Finally, the case in Fig. 1c is that of perpendicular \nmagnetization of the fixed nanomagnet /2φπ=± . We will see further on that the \ndirections of magnetization of the fixed layer,  which we consider identical to directions \nof spin polarization of the el ectrons injected into the free layer, correspond to different \nstrategies of switching. \nIn order to gain an intuitive understandi ng of the process of switching, one needs to \nvisualize the “energy landscape” – the patter n of demagnetization energy in the phase \nspace of magnetization angles, according to Eq. (3). The map of this sphere of angles on \nthe plane is shown in Fig. 2. For a different sh ape of the nanomagnet, or in the presence \nof material anisotropy and exte rnal field, this dependence wi ll be quantitatively different,  Page9\nbut the same qualitative approach applies. The salient features of the energy landscape \nare: a) the stretched ellipse-shaped “basins” close to the poles – th e stable equilibrium \nstates; b) the two “valleys” stretching from one pole to the other – the states with in-plane \nmagnetization; c) they have “mount ain passes”, or saddle points at /2θπ=  and 0,φπ=   \nd) two peaks corresponding to magnetiza tion perpendicular to the plane, at /2θπ=  and \n/2φπ=± . The iso-energy lines are shown in the contour plot, Fig. 2. In the absence of \ndamping and spin torque, they would coincide with closed orbits of magnetization. There \nare orbits of low energy precessing around one of the poles, and orbits of high energy \noscillating between the two poles. In the presence of damping, the nanomagnet will \nevolve to one of the basins and eventually to  the pole inside it. Spin torque can cause the \nnanomagnet to gain or lose energy, under some  conditions moving between the basins, i.e. \nswitching. \nAt this point, let us agree on the definitions for switching time. We will consider \nswitching under the action of rectan gular pulse currents of magnitude I and duration puτ \n(marked in the following plots). These valu es are relevant for the switching charge \npuIτand energy swp uE UIτ=  dissipated in switching under the voltage bias U. Therefore \nthe pulse duration puτ is the time important in the technological sense for optimizing the \nenergy per writing a bit. From simulations  we obtain the switching time, which \ncharacterizes how fast the nanomagnet responds to the current pulse. We customarily \ndefine the switching time swτ as the time over which the magne tization is switched from   \n10% to 90% of its limiting values.  In our part icular case it is the interval between the \nfirst time the projection of magnetization xm goes below 0.8 till the last time it is over -  Page\n10\n0.8. The 10-90% time gives the lower bound of switching time. Its importance is to \ncharacterize the switchin g time pertinent to the strategy, ra ther than influence of initial \nconditions. The reason that we us e it instead of 0-100% time is  that, in the collinear in-\nplane case, the latter strongly depends on the choice of the initial angle of magnetization \n(see discussion below). The total write time needs to be longer than the largest of the two \ntime measures.  \nIn the following, we will plot the switching times swτ resulting from the simulation of \nmagnetization dynamics over certain time interv als, typically 1, 2.5, 3, or 4ns. When the \nswitching time reaches this constant value, it really means that switching does not occur \nover the simulation time, and, with high probabi lity, even after any duration of evolution. \nThus these limiting constant values on th e plot are just tokens for “no switching \noccurring”. \nCollinear polarization spin torque switching is done in a configuration of Fig 1a. Since \nthe injected spin polarization is in plane close to 0θ=, the resulting spin transfer torque \npushes the magnetization to rotate in plane, along the slope of sl owest accent in energy, \nwhich might seem at first glance like the optimal strategy of switching. An example of \nswitching dynamics for this case is shown in  Fig. 3. From the time evolution plot we see \nthat magnetization performs an oscillatory mo tion close to in plan e position with slowly \nincreasing amplitude. Torque increases with the angle from the x-axis, and this reinforces \nthe growth of this angle. At some point the projections on the x-axis abruptly switches to \na negative value and then the amplit ude of the oscillations is damped27,28. From the \ntrajectory in phase space of magnetization direction, we see that the switching happens by   Page\n11\nmoving along one of the valleys and crossing over a saddle point. The duration of the \npulse is sufficient if it is l onger than the time necessary to go to the other side of this \nsaddle point. \nAnother strategy is the in th e configuration of Fig. 1b. It is similar to the previous \nstrategy, with a few modifications. In case of th e spin polarization 90 degree in plane, i.e. \n/2θπ= , see Fig. 4, the torque is maximal in the initial instant when the magnetization is \nat 0θ=. At very large current values, when th e magnetization reaches the saddle point, \nthe torque turns to zero and the nanomagnet dwells in an unstable equilibrium until the \nend of the current pulse. At this point it falls towards one of the equilibriums; the choice \nof which is governed by the randomness of its  position at that moment. Overall, this \nlooks like an unreliable method of switching. Also it requires a much higher current than \ncollinear polarization switching a nd the switching time is longer29. The situation is \ndrastically improved for the case of injected spin polarization at a different angle, e.g. \n135 degree in plane, 3/ 4θπ= , shown in Fig. 5. The path of switching still goes along \nthe energy valley and over the saddle point. But in that case, torque is not zero at the \ninitial instant, and it does not turn to zero  at the position of th e energy saddle point. \nSwitching time proves to be shorter. But one im portant similarity is that the pulse time \nneeds to be relatively long in order to cross over the saddle point. The strategy of \nswitching with perpendicular spin polarization, as shown in Fig 1c, turns out to be very \ndifferent from collinear polari zation switching. It stems from  the fact that the spin \ntransfer torque acts in the direction perpe ndicular to the sample plane as well. In a \ncounter-intuitive manner, it pushes the nanomagne t along the path of steepest accent. The \nway to take advantage of this situation is to use a very short pulse, which will supply   Page\n12\nsufficient energy to the nanomagnet. The adva ntage of a short pulse is that it requires \nsmaller switching energy supplied by the curren t. After the initial short current pulse, the \nnanomagnet precesses due to the torque from the shape anisotropy at zero spin torque, see Fig 6 for an example of such evolu tion. Under the condition of a small Gilbert \ndamping, its trajectory will be close to the iso-energy line. A necessary condition for switching is that the nanomagnet has sufficient energy to be on the trajectory that crosses \nto the other basin, even with the account of  loss to damping. The sufficient condition of \nsuccessful switching is that by the time magne tization reaches the other basin, it loses \nenough energy so that it cannot cross back to the original basin. If this condition is \nsatisfied, the nanomagnet slowly loses energy to damping and approaches the equilibrium. \nDue to the small value of damping the switching time turns out to be long.   \nA variation on this strategy is to apply a nother pulse of curren t after the nanomagnet \ncrosses to the other basin, Fig. 7. Th is pulse can have the same duration \npuτ and start \nafter a delay time, gτ, after the trailing edge of the fi rst pulse but must have the opposite \npolarity of the current. Such a tw o-pulse sequence with the total time 2tp u gτττ=+ , will \nefficiently decrease the energy and avoid the process of slow energy damping. As a result \nthe switching time swτ becomes very short, ~0.2ns, comparable topuτ. The downside of \nthis strategy is that two pulses of course require twice the energy of one pulse with the \nsame current magnitude and duration. Also it re quires a more complicated circuit to time \nthe pulses of opposite polarity. We note that th e difference of the strategy considered here \nfrom the one of Kent et al.17 is that in their approach the two pulses of the opposite \npolarity did not have a gap between them, so  the stage of free precession was absent.   Page\n13\nWe make the following approximate estimate of the pulse parameters for the pulsed \nswitching. The energy that the nanomagnet gains in  the first pulse must be larger than the \nenergy necessary to cros s over the saddle point.  \n 2\nzz z y y x xmN N N=−  (8) \nOn the other hand, the out of plane projection at the end of the pulse is obtained from Eq.  \n 2Bp u\nz\nsgI PmMeVµτ= . (9) \nFor the parameters used in this paper, it amounts to the switching charge of 81puIf Cτ=  \nper bit. This is much smalle r than the best achieved values  for the collinear polarization \nswitching, ~5pC per bit. Moreover, if we  compare the switching time and energy \nprojected here with incumbent type s of memory, see Table 3 in Ref. 30, we see that \nSTTRAM with pulsed switching is superior  to all other types of memory. From \nprototypes of STTRAM 16 we also know that its density can be comparable to that of \nDRAM. Therefore the proposed improvement gives it the crucial performance boost to \npotentially be the universal memory and to replace all other kinds. \n4. Tolerances of switching and in fluence of field-like torque \nSpin transfer torque memories operate in el ectronic circuits. The ci rcuits naturally have \nvariability originating from fabrication imperfections. Also the state of the circuit is subject to electronic noise and temperature drif t. Obviously it is not  possible to guarantee \nprecise values of switching current and time fo r elements of memories. Therefore it is   Page\n14\nespecially important to study the tolerances  of memory operation relative to external \nparameters. We believe such studies have not been conducted up to now. Here we run \nmultiple simulations over a wide set of parameters to draw some conclusions about these \ntolerances.  \nAt the same time we study the effect of the field-like torque (FLT) contribution. \nExperiments on separate measurements of the spin transfer (Slonczewski) and field like \ntorques have been conducted 21,22. But the implications of th ese contributions to current \ninduced magnetization switching have not been  sufficiently clarified. The experiments \nshow that FLT increases with increasing appl ied voltage. To account for this we consider \ntwo cases – no field-like torque '0ε= and large field-like torque '0 . 3εε= . \nThe contour plots of switching time vs. current and pulse duration for collinear \npolarization switching are shown in Fig. 8. It is common to think of spin torque switching \nas having a threshold (or critical) current cI. However from this plot we see that, for \nsufficiently short pulse du ration, the switching current 1/cp uIIτ−∼  is much larger than \nthe critical current. Therefore it is the threshold charge, 1.2puI pCτ≈ , that determines \nwhether the memory state is sw itched.  Above this threshold,  switching can be quite fast, \n~0.2ns, but the total write time is limited by the pulse duration instead. One can notice \nthat the shapes of the swit ching time dependence with and without FLT are remarkably \nsimilar, but they appear to be shifted. For th at reason, one needs to be cautious of the fact \nthat for a specific values of current and pul se duration, the dynami cs may be different \nwith and without FLT. The reason for similarity  is that for this cas e FLT plays a role of \nan effective magnetic field in the z-direction,  in addition to a large effective field from   Page\n15\ndemagnetization. There are curious geometri cal features of the switching threshold \nborder. Even though they are persistent in simula tion, we believe that they are artifacts of \nthe choice of initial magnetization and of os cillations (“ringing”) of magnetization after \nswitching. In reality, thermal fluctuation wi ll vary the initial magnetization, and the \nfeatures on the plot will be washed away for the thermally averaged switching time. \nOverall, this strategy of switching gives an excellent tolerance when the switching \ncurrent and pulse duration are se t high enough above the threshold. \nThe switching time diagrams for the 90 degree in  plane polarization ar e shown in Fig. 9. \nIn this case, the threshold current is actua lly a good criterion of  switching; and this \nthreshold turns out to be very high, ~10mA. Without FLT, even above threshold there are \ntightly interlaced regions of successful and uns uccessful switching. This attests to the \nunstable nature of such switchi ng. It cannot be used for a pr actical device. The situation \nis different with FLT. There are large regions of small sw itching time, and therefore good \ntolerance to parameters, above th e threshold. The reason for this  stabilization is that even \nthough the Slonczewski torque vanishes at /2θπ= , FLT is still finite and it succeeds in \npushing the nanomagnet over the energy saddl e point. However at  excessively high \ncurrent we encounter the regions of unsucce ssful switching that memory designers need \nto avoid. \nThe switching diagrams for 135degree in plan e polarization are shown in Fig 10. The \nthreshold behavior is in be tween the collinear polarizati on and 90degree in plane cases. \nThe threshold current is not c onstant, and it is ~2-4mA, which is lower than that for 90 \ndegrees. But it is not inversely proportional to the pulse duration either. The threshold   Page\n16\ncharges are in the range 0.6 1.6puI pC τ≈÷ . Overall it has the same excellent tolerance to \ncurrent and pulse variation as the collinear polarization switc hing, but in the absence of \nFLT, regions of unsuccessful switchi ng are observed at higher current. \nThe switching diagrams for out of plane sp in polarization and one  switching pulse are \nshown in Fig. 11. The areas of successful sw itching are seen as na rrow strips across the \nplot. They are interlaced with stripes of unsuccessful switching. The reason for such \nbehavior is the precession character of  magnetization dynamics for this switching \nstrategy. If too much energy is transferre d from the current to the nanomagnet, it \novershoots and returns to  the basin around the initial ma gnetization state. The set of \nsuccessful switching correspond to 0.5, 1.5, 2.5 etc. full turns of magnetization. The \nlowest of these stripes corresponds  to the threshol d of switching, 100puIf C τ≈ . Though \nthe threshold is only approximately given by the product of the current and the pulse \nduration. This is in a very good agreemen t with the analytical estimate (9).  \nThis is the first case when we encounter the problem of tolerances in earnest. From the \nleft plot in Fig. 11 for Gilbert damping of 0. 01, we estimate the tolerances to be 1ps and \n0.1mA. This is likely too tight for a realistic memory circuit. The right plot is calculated \nfor Gilbert damping 0.03. We see that, c ontrary to what is known about collinear \npolarization switching, the thre shold is almost unchanged at a higher Gilbert damping. At \nthe same time, the tolerances are much rela xed, to 5ps and 0.7mA. The switching stripes \nbecame wider, and the next order of switc hing with 1.5 turns is moved to a higher \nswitching charge. This seems to be the fi rst occasion when increasing damping is \nbeneficial for the device performance. We note that these simulations are done with   Page\n17\ninclusion of FLT. The results with zero F LT (not included in this  paper) are almost \nindistinguishable from those. The reason for this is that the current pulses act when \nmagnetization is close to the poles, and FLT has projections mostly in-plane of the \nnanomagnet, which contribute only negligibly  to the precessional type of switching. \nThe switching diagrams for out of plane polariz ation and two pulses are shown in Fig. 12. \nThe total pulse time is fixed at 0.2t nsτ=    , while the pulse time puτ  is given on the \nhorizontal axis of the plot.  Their overall character is sim ilar to those for a single pulse. \nThe threshold condition is approximately the sa me. The first stripe is very narrow, with \nthe tolerances 0.3ps and 30uA. Surprisingly the second stripe  corresponding to 1.5 turns, \nhas a much larger and acceptable value of to lerance of 4ps and 0.4mA. We do not have \nan intuitive explanation for this difference between the two switching cases. For smaller \nGilbert damping 0.01 the switching time is quite short, ~0.2ps. This is due to the fact that the second pulse eliminates ringing of magnetizat ion, as it was discussed in the previous \nsection. With increase of Gilbert damping to 0.03, the tolerances in the first stripe improve to 2ps and 0.2mA. This is manifested  as appearances of several satellite strips \naround the first one, meaning that at highe r damping the condition of timing the gap \nbetween the pulses becomes less crucial fo r successful switchi ng. Conversely, the \nswitching time gets slower, ~0.5ps. This is because the magnetization oscillations are not \neliminated as efficiently is the pulses are not  finely timed.   Like for the single pulse, the \ninclusion of FLT changes the results very insignificantly.   Page\n18\n5. Comparison of macrospin an d micromagnetic simulations \nThe simulation of micromagnetic dynamics is a more rigorous model and presumably \ngives a better approximation to reality that the macrospin approximation. However it is \nalso much more computationally demanding. Fo r this reason it is us eful to compare the \nresults of macrospin and micromagnetic m odels. We compare the switching time at \nvarious pulse durations but fixe d values of current. Microma gnetic simulations start with \nan initial state obtained by rela xing its energy to a minimum. It has the general direction \nof 0θ=, and a “leaf state” pattern, i.e. magnetization is closer to being parallel to the \nlonger sides of the ellipse. \nThe comparison for the collinear polarization switching is shown in Fig. 13. Here we set \nthe direction of the sp in polarization of the fixed layer to be 0.1θ= . This value is an \nestimate of the r.m.s deviation of the angle of magnetization due to thermal distribution \nof energy. We make this choice of the initia l angle only in the case of collinear in-plane \nswitching, because spin torque would vanish for 0θ=. For other cases the spin torque \ndoes not vanish at the zero initial angle.  The agreement is surprisingly good. Both \nmodels give approximately the same value of  threshold charge. This may be because we \nare focusing on the evolu tion starting from angle 0.1θ= . The evolution around 0θ= \noccurs under a much smaller torque and carries more un certainty. Macrospin model \nexhibits more oscillations close to the thre shold. Micromagnetic model predicts shorter \nswitching time above threshold, probably due to more efficient damping of higher modes \nof magnetization.   Page\n19\nThe comparison in the case of 90degree in plan e polarization, Fig. 14, shows essentially \nthe same lower envelope of switching time. The micromagnetic model shows fewer areas \nof failed switching. We speculate that this  is due to stronger effect of FLT on non-\nuniform magnetization. \nThe disagreement is more pronounced in the cas e of single pulse out of plane polarization, \nFig. 15. The similarities are the same positi on on the time scale of  successful switching \nstripes for 0.5 and 3.5 turns. The regions of  unsuccessful switching for 1 and 3 turns are \nabsent in the micromagnetic model, probabl y because the barrier for crossing into the \nother basin is increased for a non-uniform ma gnetization pattern.   The switching times \nare generally predicted to be shor ter in the micromagnetic model.  \nA similar situation is observed in out of plane pol arization switching with two pulses, see \nFig. 16. The positions of successful switching on the time scale are shifted. This is \nprobably due to the fact that  the nanomagnet energy is di fferent with the account of \nexchange and dipole-dipole interaction, and thus the time of free precession of \nmagnetization is different too.  As before, micromagnetics pr edict shorter switching times, \nas well as giving better tolerance for 0.5 turn switching. In fact, micromagnetic \nsimulations for one and two pulses look quite si milar. This points to higher significance \nof damping than of the second pulse in bri nging the magnetization to  its final state.  \nThis paper includes just a few examples of co mparison. A more complete set of data is \navailable 31. All of this supports the conclusion th at macrospin simulati ons give generally \nthe same qualitative dependence of switchi ng time on the current a nd pulse duration as   Page\n20\nOOMMF simulations. The former can be used to  infer general trends . The latter should \nbe saved for obtaining quanti tatively precise results. \n6. Conclusions \nWe have compared various strategies of spin torque switching. The tolerances of \nswitching, performance limits of out-of plane pol arization devices, and the effect of field-\nlike torque have been comprehensively studied  for the first time. In summary, switching \nwith short pulses of out of plane polarization is the prefer red strategy. It has a much \nlower threshold charge than other strategies  of switching, and suffers less from low \ntolerance to current magnitude and pulse dura tion. This problem of low tolerance can be \nresolved by increasing Gilbert damping in the nanomagnet. Field-like torque significantly \nchanges the results for non-co llinear in-plane switching a nd happens to produce a minor \neffect for other switching strategies. Implem entation of this strategy would put STTRAM \nin a position of a technological leadership  among memories. We find that macrospin \ngives good qualitative prediction of the dyna mics, though micromagnetic models should \nbe used to get better quantitative precision. \n8. Acknowledgements \nG. R. and I. N. K. gratefully acknowle dge the support of DARPA, NSF (grants DMR-\n0748810 and ECCS-0701458) and the Nanoelectr onic Research Ini tiative through the \nWestern Institute of Nanoelectronics. Co mputations supporting this paper were \nperformed on the BDUC Compute Cluster donated by Broadcom, co-run by the UCI OIT \nand the Bren School of Inform ation and Computer Sciences.   Page\n21\nReferences \n                                                 \n1  A. Fert, “Nobel Lecture: Origin, developmen t, and future of spintronics”, Rev. Mod. \nPhys. 80, 1517-30 (2008).  \n2 M. N. Baibich, J. M. Broto, A. Fert, F. N.  Vandau, F. Petroff, P.  Eitenne, G. Creuzet, A. \nFriederich, J. Chazelas, “Giant Magneotres istance of (100) Fe / (100) Cr magnetic \nsuperlattices”, Phys. Rev. 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Buhrman, \n“Time-Resolved Measurements of Nanoma gnet Dynamics Driven by Spin-Transfer \nTorques”, Science 307, 228 (2005).  \n28 T. Devolder, J. Hayakawa, K. Ito, H. Taka hashi, S. Ikeda, P. Crozat, N. Zerounian, \nJoo-Von Kim, C. Chappert, and H. Ohno, “S ingle-Shot Time-Resolved Measurements of \nNanosecond-Scale Spin-Transfer Induced Switc hing: Stochastic Versus Deterministic \nAspects” , Phys. Rev. Lett. 100, 057206 (2008).  \n29 F. B. Mancoff, R. W. Dave, N. D. Rizzo, T. C. Eschrich, B. N. Engel, S. Tehrani, \n“Angular dependence of spin-tra nsfer switching in a magnetic nanostructure”, Appl. Phys. \nLett. 83, 1596 (2003).  \n30 J. Gallagher and S. S. P. Parkin, “Devel opment of the magnetic tunnel junction MRAM \nat IBM: From first junctions to a 16-Mb MRAM demonstrator chip”, IBM J. Res. & Dev. \n50, 5-23 (2006).  \n31 D. E. Nikonov, G. I. Bour ianoff, G. Rowlands, I. N. Krivorotov, ”Comparisons of \nmacrospin and OOMMF simulations”, online www.nanohub.org  (2010).    Page\n25\n \n \n \nFigure 1. The geometry of the spin tor que memory nanopillars. Red layer – tunneling \nbarrier oxide, orange – non-ma gnetic electrodes, blue with  double-sided arrow – free \nlayer with in-plane magnetization (easy x-axis),  blue with one sided arrow – fixed layer. \nPolarizations: a) in plane collin ear with x-axis (left), b) in plane at an angle to x-axis \n(middle), c) perpendicula r to the plane (right). \n a b c  Page\n26\n \n \nFigure 2. Map of the demagnetization ener gy of the nanomagnet (normalized) in the \nmacrospin model. θ – angle of magnetization from the x-axis (easy axis), φ – angle of \nprojection of magnetization within  the yz-plane (hard plane). \n   Page\n27\n \n \n  \n \nFigure 3. Magnetization projections vs. time and the trajectory of magnetization for \ncollinear polarization switching (0  degree in plane polarization). \n   Page\n28\n \n \n  \n \nFigure 4. Magnetization projections vs. time a nd the trajectory of magnetization for 90 \ndegree in plane polarization.    Page\n29\n \n \n  \n \nFigure 5. Magnetization projection vs. time a nd the trajectory of magnetization for 135 \ndegree in plane polarization.   Page\n30\n \n \n  \n \nFigure 6. Magnetization projections vs. time and the trajectory of magnetization for \nperpendicular out of plane polarization, one pulse.   Page\n31\n \n \n  \n \nFigure 7. Magnetization projections vs. time and the trajectory of magnetization for \nperpendicular out of plane polarization, two pulses with a total time of 0.2ns. \n   Page\n32\n \n \n  \n \nFigure 8. Contour maps of switching time, left  without the field like torque and right with \n0.3 factor of field like tor que, for collinear polarization switching (0 degree in plane \npolarization). Simulation time 2ns.    Page\n33\n \n \n  \n \nFigure 9. Contour maps of switching time, left  without the field like torque and right with \n0.3 factor of field like torque, for 90 degree in plane polarization. Simulation time 4ns. \n   Page\n34\n \n \n  \n \nFigure 10. Contour maps of switching time, le ft without the field like torque and right \nwith 0.3 factor of field like  torque, for 135 degree in plan e polarization. Simulation time \n4ns.    Page\n35\n \n \n  \n \nFigure 11. Contour maps of switching time, left α=0.01 and right α =0.03, with 0.3 \nfactor of field like torque, for perpendicular out of pl ane polarization, one pulse. \nSimulation time 1ns.    Page\n36\n \n \n  \n \nFigure 12. Contour maps of switching time, left α=0.01 and right α =0.03, with 0.3 \nfactor of field like torque, fo r perpendicular out of plane pol arization, two pulses with a \ntotal time of 0.2ns. Simulation time 1 ns.    Page\n37\n \n \n  \n \nFigure 13. Comparison of the switching time vs. pulse length simulated by a macrospin \nmodel (left) and OOMMF (right ) for collinear polarization switching at I=2.5mA, field-\nlike torque 0.3, and polarization angle in  plane of 10 degrees. Simulation time 3ns. \n   Page\n38\n \n \n  \n \nFigure 14. Same as Fig. 13 at I=20mA, field- like torque 0.3, and polarization angle in \nplane of 90 degrees. Simulation time 2.4ns.    Page\n39\n \n \n  \n \nFigure 15. Same as Fig. 13, single puls e at I=4mA, field-like torque 0.3, α=0.01, and \npolarization angle perpendi cular to plane of 90 degrees. Simulation time 1.8ns, \n   Page\n40\n \n \n  \n \nFigure 16. Same as Fig. 13 for two pulses with  total duration of 0. 2ns, at I=4mA, field-\nlike torque 0.3, α=0.01, and polarization angle perp endicular to plane of 90 degrees. \nSimulation time 1.8ns. \n "    },    {        "title": "1011.5868v1.Dependence_of_nonlocal_Gilbert_damping_on_the_ferromagnetic_layer_type_in_FM_Cu_Pt_heterostructures.pdf",        "content": "arXiv:1011.5868v1  [cond-mat.mes-hall]  26 Nov 2010Draft\nDependence of nonlocal Gilbert damping on the ferromagneti c layer type in\nFM/Cu/Pt heterostructures\nA. Ghosh, J.F. Sierra, S. Auffret, U. Ebels1and W.E. Bailey2\n1)SPINTEC, UMR(8191) CEA / CNRS / UJF / Grenoble INP ; INAC,\n17 rue des Martyrs, 38054 Grenoble Cedex, France\n2)Dept. of Applied Physics & Applied Mathematics, Columbia Un iversity,\nNew York NY 10027, USA\n(Dated: 7 September 2021)\nWe have measured the size effect in nonlocal Gilbert relaxation rate in FM(tFM) /\nCu (5nm) [/ Pt (2nm)] / Al(2nm) heterostructures, FM = {Ni81Fe19, Co60Fe20B20,\npure Co}. Common behavior is observed for three FM layers, where the addit ional\nrelaxation obeys botha strict inverse power law dependence ∆ G=Ktn,n=−1.04±\n0.06 and a similar magnitude K= 224±40 Mhz·nm. As the tested FM layers span\nan order of magnitude in spin diffusion length λSDL, the results are in support of\nspin diffusion, rather than nonlocal resistivity, as the origin of the e ffect.\n1Theprimarymaterialsparameter which describes thetemporal res ponseofmagnetization\nMto applied fields His the Gilbert damping parameter α, or relaxation rate G=|γ|Msα.\nUnderstanding of the Gilbert relaxation, particularly in structures of reduced dimension, is\nan essential question for optimizing the high speed / Ghz response o f nanoscale magnetic\ndevices.\nExperiments over the last decade have established that the Gilbert relaxation of ferro-\nmagnetic ultrathin films exhibits a size effect, some component of whic h is nonlocal. Both\nα(tFM) =α0+α′(tFM) andG(tFM) =G0+G′(tFM) increase severalfold with decreasing FM\nfilm thickness tFM, from near-bulk values α0,G0fortFM>∼20 nm. Moreover, the damp-\ning size effect can have a nonlocal contribution responsive to layers or scattering centers\nremoved, through a nonmagnetic (NM) layer, from the precessing FM. Contributed Gilbert\nrelaxation has been seen from other FM layers1as well as from heavy-element scattering\nlayers such as Pt.2\nThe nonlocal damping size effect is strongly reminiscent of the electr ical resistivity in\nferromagnetic ultrathin films. Electrical resistivity ρis size-dependent by a similar factor\noverasimilarrangeof tFM; theresistivity ρ(tFM)issimilarlynonlocal,dependentuponlayers\nnot in direct contact.3–5. It isprima facie plausible that the nonlocal damping and nonlocal\nelectrical resistivity share a common origin in momentum scattering ( with relaxation time\nτM) by overlayers. If the nonlocal damping arises from nonlocal scat teringτ−1\nM, however,\nthere should be a marked dependence upon FM layer type. Damping in materials with\nshort spin diffusion length λSDLis thought to be proportional to τ−1\nM(ref.6); the claim for\n”resistivity-like” damping hasbeenmadeexplicitly forNi 81Fe19byIngvarsson7et al. ForFM\nwith along λSDL, onthe other hand, relaxation Giseither nearly constant withtemperature\nor ”conductivity-like,” scaling as τM.\nInterpretation of the nonlocal damping size effect has centered in stead on a spin current\nmodel8advanced by Tserkovnyak et al9. An explicit prediction of this model is that the\nmagnitude of the nonlocal Gilbert relaxation rate ∆ Gis only weakly dependent upon the\nFM layer type. The effect has been calculated10as\n∆G=|γ|2¯h/4π/parenleftBig\ng↑↓\neff/S/parenrightBig\nt−1\nFM (1)\n, where the effective spin mixing conductance g↑↓\neff/Sis given in units of channels per area.\nAb-initio calculationspredictaveryweakmaterialsdependencefortheinter facialparameters\n2g↑↓/S, with±10% difference in systems as different as Fe/Au and Co/Cu, and neglig ible\ndependence on interfacial mixing.11\nIndividual measurements exist of the spin mixing conductance, thr ough the damping,\nin FM systems Ni 81Fe1912, Co13, and CoFeB14. However, these experiments do not share\na common methodology, which makes a numerical comparison of the r esults problematic,\nespecially given that Gilbert damping estimates are to some extent mo del-dependent15. In\nour experiments, we have taken care to isolate the nonlocal dampin g contribution due to Pt\noverlayers only, controlling for growth effects, interfacial interm ixing, and inhomogeneous\nlosses. The only variable in our comparison of nonlocal damping ∆ G(tFM), to the extent\npossible, has been the identity of the FM layer.\nGilbert damping αhas been measured through ferromagnetic resonance (FMR) fro m\nω/2π= 2-24 Ghz using a broadband coplanar waveguide (CPW) with broad c enter conduc-\ntor width w= 400µm, using field modulation and lock-in detection of the transmitted signa l\nto enhance sensitivity. The Gilbert damping has been separated fro minhomogeneous broad-\nening inthe filmsmeasured using the well-known relation∆ Hpp(ω) = ∆H0+/parenleftBig\n2/√\n3/parenrightBig\nαω/|γ|.\nWe have fit spectra to Lorenzian derivatives with Dysonian compone nts at each frequency,\nfor each film, to extract the linewidth ∆ Hppand resonance field Hres;αhas been extracted\nusing linear fits to ∆ H(ω).\nFor the films, six series of heterostructures were deposited of th e form Si/ SiO 2/\nX/ FM(tFM)/ Cu(3nm)[ /Pt(3nm)]/ Al(3nm), FM = {Ni81Fe19(”Py”), Co 60Fe20B20\n(”CoFeB”), pure Co }, andtFM= 2.5, 3.5, 6.0, 10.0, 17.5, 30.0 nm, for 36 heterostruc-\ntures included in the study. For each ferromagnetic layer type FM, one thickness series tFM\nwas deposited with the Pt overlayer and one thickness series tFMwas deposited without the\nPt overlayer. This makes it possible to record the additional damping ∆α(tFM) introduced\nby the Pt overlayer alone, independent of size effects present in th e FM/Cu layers deposited\nbelow. In the case of pure Co, a X=Ta(5nm)/Cu(5nm) underlayer w as necessary to sta-\nbilize low-linewidth films, otherwise, depositions were carried out direc tly upon the in-situ\nion-cleaned substrate.\nField-for-resonance data are presented in Figure 1. The main pane l showsω(H/bardbl\nB) data\nfor Ni 81Fe19(tFM). Note that there is a size effect in ω(H/bardbl\nB): the thinner films have a\nsubstantially lower resonance frequency. For tFM= 2.5 nm, the resonance frequency is\ndepressed by ∼5 Ghz from ∼20 Ghz resonance HB≃4 kOe. The behavior is fitted through\n3the Kittel relation (lines) ω(H/bardbl\nB) =|γ|/radicalbigg/parenleftBig\nH/bardbl\nB+HK/parenrightBig/parenleftBig\n4πMeff\ns+H/bardbl\nB+HK/parenrightBig\n, and the inset\nshows a summary of extracted 4 πMeff\ns(tFM) data for the three different FM layers. Samples\nwith (open symbols) and without (closed symbols) Pt overlayers sho w negligible differences.\nLinear fits according to 4 πMeff\ns(tFM) = 4πMs−(2Ks/Ms)t−1\nFMallow the extraction of bulk\nmagnetization 4 πMsand surface anisotropy Ks; we find 4 πMPy\ns= 10.7 kG, 4 πMCoFeB\ns=\n11.8 kG, 4 πMCo\ns= 18.3 kG, and KPy\ns= 0.69 erg/cm2,KCoFeB\ns= 0.69 erg/cm2,KCo\ns=\n1.04 erg/cm2. The value of gL/2 =|γ|/(e/mc),|γ|= 2π·(2.799 Mhz/Oe) ·(gL/2) is found\nfrom the Kittel fits subject to this choice, yielding gPy\nL= 2.09,gCoFeB\nL= 2.07,gCo\nL= 2.15.\nThe 4πMsandgLvalues, taken to be size-independent, are in good agreement with b ulk\nvalues.\nFMR linewidth as a function of frequency ∆ Hpp(ω) is plotted in Figure 2. The data\nfor Py show a near-proportionality, with negligble inhomogeneous co mponent ∆ H0≤4 Oe\neven for the the thinnest layers, facilitating the extraction of intr insic damping parameter\nα. The size effect in in α(tFM) accounts for an increase by a factor of ∼3, fromαPy\n0=\n0.0067 (GPy\n0= 105 Mhz) for the thickest films ( tFM= 30.0 nm) to α= 0.021 for the\nthinnest films ( tFM= 2.5 nm). The inset shows the line shapes for films with and without\nPt, illustrating the broadening without significant frequency shift o r significant change in\npeak asymmetry.\nA similar analysis has been carried through for CoFeB and Co (not pict ured). Larger\ninhomogeneous linewidths are observed for pure Co, but homogene ous linewidth still ex-\nceeds inhomogeneous linewidth by a factor of three over the frequ ency range studied, and\ninhomogeneous linewidths agree within experimental error for the t hinnest films with and\nwithout Pt overlayers. We extract for these films αCoFeB\n0= 0.0065 ( GCoFeB\n0= 111 Mhz)\nandαCo\n0= 0.0085( GCo\n0= 234 Mhz). The latter value is in very good agreement with the\naverage of easy- and hard-axis values for epitaxial FCC Co films mea sured up to 90 Ghz,\nGCo\n0= 225 Mhz.16\nWe isolate the effect of Pt overlayers on the damping size effect in Figu re 3. Values\nofαhave been fitted for each deposited heterostructure: each FM t ype, at each tFM,\nfor films with and without Pt overlayers. We take the difference ∆ α(tFM) for identical\nFM(tFM)/Cu(5nm)/Al(3nm) depositions with and without the insertion of Pt (3nm) after\nthe Cu deposition. Data, as shown on the logarithmic plot in the main pa nel, are found\n4to obey a power law ∆ α(tFM) =Ktn, withn= -1.04±0.06. This is excellent agreement\nwith an inverse thickness dependence ∆ α(tFM) =KFM/tFM, where the prefactor clearly\ndepends on the FM layer, highest for Py and lowest for Co. Note tha t efforts to extract\n∆α(tFM) =Ktnwithout the FM( tFM)/Cu baselines would meet with significant errors;\nnumerical fits to α(tFM) =KtFMnfor the FM( tFM)/Cu/Pt structures yield exponents\nn≃1.4.\nExpressing now the additional Gilbert relaxation as ∆ G(tFM) =|γ|Ms∆α(tFM) =\n|γFM|MFM\nsKFM/tFM, we plot ∆ G·tFMin Figure 4. We find ∆ G·tPy= 192±40 Mhz,\n∆G·tCoFeB= 265±40 Mhz, and ∆ G·tCo= 216±40 Mhz. The similarity of values for\n∆G·tFMis in good agreement with predictions of the spin pumping model in Equa tion 1,\ngiven that interfacial spin mixing parameters are nearly equal in diffe rent systems.\nThe similarity of the ∆ G·tFMvalues for the different FM layers is, however, at odds\nwith expectations from the ”resistivity-like” mechanism. In Figure 4 ,inset, we show the\ndependence of ∆ G·tFMupon the tabulated λSDLof these layers from Ref17. It can be seen\nthatλCo\nSDLis roughly an order of magnitude longer than it is for the other two FM layers,\nPy and CoFeB, but the contribution of Pt overlayers to damping is ve ry close to their\naverage. Since under the resistivity mechanism, only Py and CoFeB s hould be susceptible\nto a resistivity contribution in ∆ α(tFM), the results imply that the contribution of Pt to\nthe nonlocal damping size effect has a separate origin.\nFinally, we compare the magnitude of the nonlocal damping size effect with that pre-\ndicted by the spin pumping model in Ref.10. According to ∆ G·tFM=|γ|2¯h/4π=\n25.69 Mhz ·nm3(gL/2)2/parenleftBig\ng↑↓\neff/S/parenrightBig\n, our experimental ∆ G·tFMandgLdata yield effective\nspin mixing conductances g↑↓\neff/S[Py/Cu/Pt ] = 6.8 nm−2,g↑↓\neff/S[Co/Cu/Pt ] = 7.3 nm−2,\nandg↑↓\neff/S[CoFeB/Cu/Pt ] = 9.6 nm−2. The Sharvin-corrected form, in the realistic limit\nofλN\nSDL≫tN11is (g↑↓\neff/S)−1= (g↑↓\nF/N/S)−1−1\n2(g↑↓\nN,S/S)−1+ 2e2h−1ρ tN+ (˜g↑↓\nN1/N2/S)−1.\nUsing conductances 14.1nm−2(Co/Cu), 15.0nm−2(Cu), 211nm−2(bulkρCu,tN= 3nm), 35\nnm−2(Cu/Pt) would predict a theoretical g↑↓\neff,th./S[Co/Cu/Pt ] = 14.1 nm−2. Reconciling\ntheory and experiment would require an order of magnitude larger ρCu≃20µΩ·cm, likely\nnot physical.\nTo summarize, a common methodology, controlling for damping size eff ects and intermix-\ning in single films, has allowed us to compare the nonlocal damping size eff ect in different\nFM layers. We observe, for Cu/Pt overlayers, the same power law in thickness t−1.04±0.06,\n5the same materials independence, but roughly half the magnitude th at predicted by the spin\npumping theory of Tserkovnyak10. The rough independence on FM spin diffusion length,\nshown here for the first time, argues against a resistivity-based in terpretation for the effect.\nWe would like to acknowledge the US NSF-ECCS-0925829, the Bourse Accueil Pro n◦\n2715 of the Rhˆ one-Alpes Region, the French National Research A gency (ANR) Grant ANR-\n09-NANO-037, and the FP7-People-2009-IEF program no 252067 .\nREFERENCES\n1R. Urban, G. Woltersdorf, and B. Heinrich, “Gilbert damping in single a nd multilayer\nultrathin films: role of interfaces in nonlocal spin dynamics,” Physical Review Letters 87,\n217204–7 (2001).\n2S. Mizukami, Y. Ando, and T. Miyazaki, “Effect of spin diffusion onGilber t damping for a\nverythinpermalloylayer inCu/permalloy/Cu/Pt films,”Phys. Rev. B 66, 104413 (2002).\n3B. Dieny, J. Nozieres, V. Speriosu, B. Gurney, and D. Wilhoit, “Chan ge in conductance\nis the fundamental measure of spin-valve magnetoresistance,” Ap plied Physics Letters 61,\n2111–3 (1992).\n4W. H. Butler, X. G. Zhang, D. M. C. Nicholson, T. C. Schulthess, and J. M. MacLaren,\n“Giant magnetoresistance from an electron waveguide effect in cob alt-copper multilayers,”\nPhysical Review Letters 76, 3216–19 (1996).\n5W. E. Bailey, S. X. Wang, and E. Y. Tsymbal, “Electronic scattering\nfrom Co/Cu interfaces: In situ measurement and comparison with t heory,”\nPhys. Rev. B 61, 1330–1335 (2000).\n6V. Kambersk´ y, “On the landau-lifshitz relaxation in ferromagnetic metals,” Canadian\nJournal of Physics 48, 2906 (1970).\n7S. Ingvarsson, L. Ritchie, X. Liu, G. Xiao, J. Slonczewski, P. Trouillo ud, and R. Koch,\n“Role of electron scattering in the magnetization relaxation of thin N i81Fe19films,”\nPhys. Rev, B, Condens, Matter Mater. Phys. (USA) 66, 214416 – 1 (2002/12/01).\n8R. Silsbee, A. Janossy, and P. Monod, “Coupling between ferromag netic and conduction-\nspin-resonancemodesataferromagneticnormal-metalinterfac e,”PhysicalReviewB(Con-\ndensed Matter) 19, 4382 – 99 (1979).\n9Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, “Enhanced Gilbe rt damping in thin\n6ferromagnetic films,” Phys. Rev. Lett. 88, 117601 (2002).\n10Y. Tserkovnyak, A. Brataas, G. Bauer, and B. Halperin, “Nonloca l magnetization dy-\nnamics in ferromagnetic heterostructures,” Reviews in Modern Phy sics77, 1375 – 421\n(2005).\n11M. Zwierzycki, Y. Tserkovnyak, P. J. Kelly, A. Brataas, and G. E. W . Bauer, “First-\nprinciples study of magnetization relaxation enhancement and spin t ransfer in thin mag-\nnetic films,” Phys. Rev. B 71, 064420 (2005).\n12S. Mizukami, Y. Ando, and T. Miyazaki, “Magnetic relaxation of norma l-metal\nNM/NiFe/NM films,” Journal of Magnetism and Magnetic Materials 239, 42 – 4 (2002);\n“The study on ferromagnetic resonance linewidth for NM/ 80NiFe/ N M, (NM=Cu, Ta,\nPd and Pt) films,” Japanese Journal of Applied Physics, Part 1 (Regu lar Papers, Short\nNotes and Review Papers) 40, 580 – 5 (2001).\n13J. Beaujour, W. Chen, A. Kent, and J. Sun, “Ferromagnetic reso nance study of poly-\ncrystalline cobalt ultrathin films,” Journal of Applied Physics 99, 08N503 (2006); J.-M.\nBeaujour, J. Lee, A. Kent, K. Krycka, and C.-C. Kao, “Magnetiza tion damping in ultra-\nthin polycrystalline co films: evidence for nonlocal effects,” Physical Review B (Condensed\nMatter and Materials Physics) 74, 214405 – 1 (2006).\n14H.Lee,L.Wen, M.Pathak, P.Janssen, P.LeClair,C.Alexander, C .Mewes, andT.Mewes,\n“Spin pumping in Co 56Fe24B20multilayer systems,” Journal of Physics D: Applied Physics\n41, 215001 (5 pp.) – (2008).\n15R. McMichael and P. Krivosik, “Classical model of extrinsic ferroma gnetic resonance\nlinewidth in ultrathin films,” IEEE Transactions on Magnetics 40, 2 – 11 (2004).\n16“Gilbert damping and g-factor in Fe xCo1−xalloy films,” Solid State Communications 93,\n965 – 968 (1995).\n17J. Bass and J. Pratt, W.P., “Spin-diffusion lengths in metals and alloys, and spin-flipping\nat metal/metal interfaces: an experimentalist’s critical review,” Jo urnal of Physics: Con-\ndensed Matter 19, 41 pp. –(2007); C. Ahn, K.-H. Shin, andW. Pratt, “Magnetotran sport\nproperties of CoFeB and Co/Ru interfaces in the current-perpen dicular-to-plane geome-\ntry,” Applied Physics Letters 92, 102509 – 1 (2008).\n7FIGURES\nω/ 2π (Ghz)\nH  (Oe)B     1 / t      (nm  ) FM -1 \nFIG. 1. Fields for resonance ω(HB) for in-plane FMR, FM=Ni 81Fe19, 2.5 nm ≤tFM≤30.0 nm;\nsolid lines are Kittel fits. Inset:4πMeff\nsfor all three FM/Cu, with and without Pt overlayers.\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48\n/s49/s52/s48/s48 /s49/s53/s48/s48 /s49/s54/s48/s48 /s49/s55/s48/s48/s45/s48/s46/s48/s48/s48/s49/s48/s46/s48/s48/s48/s48/s48/s46/s48/s48/s48/s49/s48/s46/s48/s48/s48/s50/s48/s46/s48/s48/s48/s51\n/s49/s50/s32/s71/s72/s122/s72\n/s114/s101/s115\n/s72\n/s112/s112/s32 /s32/s78/s105\n/s56/s49/s70/s101\n/s49/s57/s40/s54/s110/s109/s41\n/s32 /s32/s78/s105\n/s56/s49/s70/s101\n/s49/s57/s40/s54/s110/s109/s41/s45/s80/s116\n/s32 /s32/s67/s111\n/s54/s48/s70/s101\n/s50/s48/s66\n/s50/s48/s40/s54/s110/s109/s41\n/s32 /s32/s67/s111\n/s54/s48/s70/s101\n/s50/s48/s66\n/s50/s48/s40/s54/s110/s109/s41/s45/s80/s116/s39/s39/s47 /s72/s40/s97/s46/s117/s46/s41\n/s70/s105/s101/s108/s100/s32/s40/s79/s101/s41\n/s32/s50/s46/s53/s110/s109\n/s32/s51/s46/s53/s110/s109\n/s32/s54/s110/s109\n/s32/s49/s48/s110/s109\n/s32/s49/s55/s46/s53/s110/s109\n/s32/s51/s48/s110/s109/s112/s112/s40/s79/s101/s41\n/s84/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41\nFIG. 2. Frequency-dependent peak-to-peak FMR linewidth ∆ Hpp(ω) for FM=Ni 81Fe19,tFM\nas noted, films with Pt overlayers. Inset:lineshapes and fits for films with and without Pt,\nFM=Ni 81Fe19, CoFeB.\n8      t     (nm)FM       t     (nm) FM α\nFIG. 3. Inset:αno Pt(tFM) andαPtfor Py, after linear fits to data in Figure 2. Main panel:\n∆α(tFM) =αPt(tFM)−αno Pt(tFM) for Py, CoFeB, and Co. The slopes express the power law\nexponent n= -1.04±0.06.\nλPy \nCoFeB\nCo \n      t     (nm)FM ∆G·t      (Mhz·nm) FM \n∆G·t     FM \n(nm) SDL \nFIG. 4. The additional nonlocal relaxation due to Pt overlay ers, expressed as a Gilbert relaxation\nrate - thickness product ∆ G·tFMfor Py, CoFeB, and Co. Inset:dependence of ∆ G·tFMon spin\ndiffusion length λSDLas tabulated in17.\n9"    },    {        "title": "1811.00020v2.Anisotropic_and_controllable_Gilbert_Bloch_dissipation_in_spin_valves.pdf",        "content": "Anisotropic and controllable Gilbert-Bloch dissipation in spin valves\nAkashdeep Kamra,1,\u0003Dmytro M. Polishchuk,2Vladislav Korenivski,2and Arne Brataas1\n1Center for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, Trondheim, Norway\n2Nanostructure Physics, Royal Institute of Technology, Stockholm, Sweden\nSpin valves form a key building block in a wide range of spintronic concepts and devices from\nmagnetoresistive read heads to spin-transfer-torque oscillators. We elucidate the dependence of the\nmagnetic damping in the free layer on the angle its equilibrium magnetization makes with that in\nthe \fxed layer. The spin pumping-mediated damping is anisotropic and tensorial, with Gilbert- and\nBloch-like terms. Our investigation reveals a mechanism for tuning the free layer damping in-situ\nfrom negligible to a large value via the orientation of \fxed layer magnetization, especially when the\nmagnets are electrically insulating. Furthermore, we expect the Bloch contribution that emerges\nfrom the longitudinal spin accumulation in the non-magnetic spacer to play an important role in a\nwide range of other phenomena in spin valves.\nIntroduction. { The phenomenon of magnetoresistance\nis at the heart of contemporary data storage technolo-\ngies [1, 2]. The dependence of the resistance of a multi-\nlayered heterostructure comprising two or more magnets\non the angles between their respective magnetizations has\nbeen exploited to read magnetic bits with a high spatial\nresolution [3]. Furthermore, spin valves comprised of two\nmagnetic layers separated by a non-magnetic conductor\nhave been exploited in magnetoresistive random access\nmemories [2, 4, 5]. Typically, in such structures, one\n`free layer' is much thinner than the other `\fxed layer'\nallowing for magnetization dynamics and switching in\nthe former. The latter serves to spin-polarize the charge\ncurrents \rowing across the device and thus exert spin-\ntorques on the former [6{9]. Such structures exhibit a\nwide range of phenomena from magnetic switching [5] to\noscillations [10, 11] driven by applied electrical currents.\nWith the rapid progress in taming pure spin cur-\nrents [12{20], magnetoresistive phenomena have found\na new platform in hybrids involving magnetic insulators\n(MIs). The electrical resistance of a non-magnetic metal\n(N) was found to depend upon the magnetic con\fgura-\ntion of an adjacent insulating magnet [21{24]. This phe-\nnomenon, dubbed spin Hall magnetoresistance (SMR),\nrelies on the pure spin current generated via spin Hall\ne\u000bect (SHE) in N [25, 26]. The SHE spin current accu-\nmulates spin at the MI/N interface, which is absorbed\nby the MI depending on the angle between its magne-\ntization and the accumulated spin polarization. The\nnet spin current absorbed by the MI manifests as ad-\nditional magnetization-dependent contribution to resis-\ntance in N via the inverse SHE. The same principle of\nmagnetization-dependent spin absorption by MI has also\nbeen exploited in demonstrating spin Nernst e\u000bect [27],\ni.e. thermally generated pure spin current, in platinum.\nWhile the ideas presented above have largely been ex-\nploited in sensing magnetic \felds and magnetizations,\ntunability of the system dissipation is a valuable, un-\nderexploited consequence of magnetoresistance. Such\nan electrically controllable resistance of a magnetic wire\nFIG. 1. Schematic depiction of the device under investigation.\nThe blue arrows denote the magnetizations. The \fxed layer\nF2magnetization remains static. The free layer F 1magneti-\nzation precesses about the z-axis with an average cone angle\n\u0002\u001c1. The two layers interact dynamically via spin pumping\nand back\row currents.\nhosting a domain wall [28] has been suggested as a ba-\nsic circuit element [29] in a neuromorphic computing [30]\narchitecture. In addition to the electrical resistance or\ndissipation, the spin valves should allow for controlling\nthe magnetic damping in the constituent magnets [31].\nSuch an in-situ control can be valuable in, for example,\narchitectures where a magnet is desired to have a large\ndamping to attain low switching times and a low dissipa-\ntion for spin dynamics and transport [13, 16]. Further-\nmore, a detailed understanding of magnetic damping in\nspin valves is crucial for their operation as spin-transfer-\ntorque oscillators [10] and memory cells [5].\nInspired by these new discoveries [21, 27] and previous\nrelated ideas [31{34], we suggest new ways of tuning the\nmagnetic damping of the free layer F 1in a spin valve\n(Fig. 1) via controllable absorption by the \fxed layer\nF2of the spin accumulated in the spacer N due to spin\npumping [31, 35]. The principle for this control over spin\nabsorption is akin to the SMR e\u000bect discussed above and\ncapitalizes on altering the F 2magnetization direction.\nWhen spin relaxation in N is negligible, the spin lost by\nF1is equal to the spin absorbed by F 2. This lost spin\nappears as tensorial Gilbert [36] and Bloch [37] damp-arXiv:1811.00020v2  [cond-mat.mes-hall]  10 Apr 20192\ning in F 1magnetization dynamics. In its isotropic form,\nthe Gilbert contribution arises due to spin pumping and\nis well established [31{33, 35, 38{40]. We reveal that\nthe Bloch term results from back\row due to a \fnite dc\nlongitudinal spin accumulation in N. Our results for the\nangular and tensorial dependence of the Gilbert damping\nare also, to best of our knowledge, new.\nWe show that the dissipation in F 1, expressed in terms\nof ferromagnetic resonance (FMR) linewidth, varies with\nthe angle\u0012between the two magnetizations (Fig. 3).\nThe maximum dissipation is achieved in collinear or or-\nthogonal con\fgurations depending on the relative size\nof the spin-mixing g0\nrand longitudinal spin glconduc-\ntances of the NjF2subsystem. For very low gl, which\ncan be achieved employing insulating magnets, the spin\npumping mediated contribution to the linewidth vanishes\nfor collinear con\fgurations and attains a \u0012-independent\nvalue for a small non-collinearity. This can be used to\nstrongly modulate the magnetic dissipation in F 1electri-\ncally via, for example, an F 2comprised by a magneto-\nelectric material [41].\nFMR linewidth. { Disregarding intrinsic damping for\nconvenience, the magnetization dynamics of F 1including\na dissipative spin transfer torque arising from the spin\ncurrent lost IIIs1may be expressed as:\n_^mmm=\u0000j\rj(^mmm\u0002\u00160HHHe\u000b) +j\rj\nMsVIIIs1: (1)\nHere, ^mmmis the unit vector along the F 1magnetization\nMMMtreated within the macrospin approximation, \r(<0)\nis the gyromagnetic ratio, Msis the saturation magneti-\nzation,Vis the volume of F 1, andHHHe\u000bis the e\u000bective\nmagnetic \feld. Under certain assumptions of linearity\nas will be detailed later, Eq, (1) reduces to the Landau-\nLifshitz equation with Gilbert-Bloch damping [36, 37]:\n_^mmm=\u0000j\rj(^mmm\u0002\u00160HHHe\u000b) + ( ^mmm\u0002GGG)\u0000BBB: (2)\nConsidering the equilibrium orientation ^mmmeq=^zzz, Eq. (2)\nis restricted to the small transverse dynamics described\nbymx;y\u001c1, while the z-component is fully determined\nby the constraint ^mmm\u0001^mmm= 1. Parameterized by a diagonal\ndimensionless tensor \u0014 \u000b, the Gilbert damping has been in-\ncorporated via GGG=\u000bxx_mx^xxx+\u000byy_my^yyyin Eq. (2). The\nBloch damping is parametrized via a diagonal frequency\ntensor \u0014\n asBBB= \n xxmx^xxx+ \nyymy^yyy. A more familiar,\nalthough insu\u000ecient for the present considerations, form\nof Bloch damping can be obtained by assuming isotropy\nin the transverse plane: BBB= \n 0(^mmm\u0000^mmmeq). This form,\nrestricted to transverse dynamics, makes its e\u000bect as a\nrelaxation mechanism with characteristic time 1 =\n0ev-\nident. The Bloch damping, in general, captures the so-\ncalled inhomogeneous broadening and other, frequency\nindependent contributions to the magnetic damping.\nConsidering uniaxial easy-axis and easy-plane\nanisotropies, parametrized respectively by Kzand\n0 30 60 9000.10.20.30.40.5FIG. 2. Normalized damping parameters for F 1magneti-\nzation dynamics vs. spin valve con\fguration angle \u0012(Fig.\n1). ~\u000bxx6= ~\u000byysigni\fes the tensorial nature of the Gilbert\ndamping. The Bloch parameters ~\nxx\u0019~\nyyare largest for\nthe collinear con\fguration. The curves are mirror symmetric\nabout\u0012= 90\u000e. ~g0\nr= 1, ~gl= 0:01, \u0002 = 0:1,!0= 10\u00022\u0019\nGHz, and!ax= 1\u00022\u0019GHz.\nKx[42], the magnetic free energy density Fmis ex-\npressed as: Fm=\u0000\u00160MMM\u0001HHHext\u0000KzM2\nz+KxM2\nx;with\nHHHext=H0^zzz+hhhrfas the applied static plus microwave\n\feld. Employing the e\u000bective \feld \u00160HHHe\u000b=\u0000@Fm=@MMM\nin Eq. (2) and switching to Fourier space [ \u0018exp(i!t)],\nwe obtain the resonance frequency !r=p\n!0(!0+!ax).\nHere,!0\u0011j\rj(\u00160H0+ 2KzMs) and!ax\u0011j\rj2KxMs.\nThe FMR linewidth is evaluated as:\nj\rj\u00160\u0001H=(\u000bxx+\u000byy)\n2!+t(\nxx+ \nyy)\n2\n+t!ax\n4(\u000byy\u0000\u000bxx); (3)\nwhere!is the frequency of the applied microwave \feld\nhhhrfand is approximately !rclose to resonance, and t\u0011\n!=p\n!2+!2ax=4\u00191 for a weak easy-plane anisotropy.\nThus, in addition to the anisotropic Gilbert contribu-\ntions, the Bloch damping provides a nearly frequency-\nindependent o\u000bset in the linewidth.\nSpin \row. { We now examine spin transport in the\ndevice with the aim of obtaining the damping parame-\nters that determine the linewidth [Eq. (3)]. The N layer\nis considered thick enough to eliminate static exchange\ninteraction between the two magnetic layers [31, 40]. Fur-\nthermore, we neglect the imaginary part of the spin-\nmixing conductance, which is small in metallic systems\nand does not a\u000bect dissipation in any case. Disregarding\nlongitudinal spin transport and relaxation in the thin free\nlayer, the net spin current IIIs1lost by F 1is the di\u000berence\nbetween the spin pumping and back\row currents [31]:\nIIIs1=gr\n4\u0019\u0010\n~^mmm\u0002_^mmm\u0000^mmm\u0002\u0016\u0016\u0016s\u0002^mmm\u0011\n; (4)\nwheregris the real part of the F 1jN interfacial spin-\nmixing conductance, and \u0016\u0016\u0016sis the spatially homogeneous3\nspin accumulation in the thin N layer. The spin current\nabsorbed by F 2may be expressed as [31]:\nIIIs2=g0\nr\n4\u0019^mmm2\u0002\u0016\u0016\u0016s\u0002^mmm2+gl\n4\u0019(^mmm2\u0001\u0016\u0016\u0016s)^mmm2;\n\u0011X\ni;j=fx;y;zggij\n4\u0019\u0016sj^iii; (5)\nwhereglandg0\nrare respectively the longitudinal spin\nconductance and the real part of the interfacial spin-\nmixing conductance of the N jF2subsystem, ^mmm2denotes\nthe unit vector along F 2magnetization, and gij=gji\nare the components of the resulting total spin conduc-\ntance tensor. glquanti\fes the absorption of the spin\ncurrent along the direction of ^mmm2, the so-called longi-\ntudinal spin current. For metallic magnets, it is domi-\nnated by the di\u000busive spin current carried by the itin-\nerant electrons, which is dissipated over the spin re-\nlaxation length [31]. On the other hand, for insulat-\ning magnets, the longitudinal spin absorption is domi-\nnated by magnons [43, 44] and is typically much smaller\nthan for the metallic case, especially at low tempera-\ntures. Considering ^mmm2= sin\u0012^yyy+ cos\u0012^zzz(Fig. 1),\nEq. (5) yields gxx=g0\nr,gyy=g0\nrcos2\u0012+glsin2\u0012,\ngzz=g0\nrsin2\u0012+glcos2\u0012,gxy=gyx=gxz=gzx= 0,\nandgyz=gzy= (gl\u0000g0\nr) sin\u0012cos\u0012.\nRelegating the consideration of a small but \fnite spin\nrelaxation in the thin N layer to the supplemental ma-\nterial [45], we assume here that the spin current lost by\nF1is absorbed by F 2, i.e.,IIIs1=IIIs2. Imposing this spin\ncurrent conservation condition, the spin accumulation in\nN along with the currents themselves can be determined.\nWe are primarily interested in the transverse (x and y)\ncomponents of the spin current since these fully deter-\nmine the magnetization dynamics ( ^mmm\u0001^mmm= 1):\nIs1x=1\n4\u0019grgxx\ngr+gxx(\u0000~_my+mx\u0016sz);\nIs1y=1\n4\u0019\u0014grgyy\ngr+gyy(~_mx+my\u0016sz) +gyz\u0016sz(1\u0000ly)\u0015\n;\n\u0016sz=~gr(lxmx_my\u0000lymy_mx\u0000p_mx)\ngzz\u0000pgyz+gr\u0000\nlxm2x+lym2y+ 2pmy\u0001;\n(6)\nwherelx;y\u0011gxx;yy=(gr+gxx;yy) andp\u0011gyz=(gr+gyy).\nThe spin lost by F 1appears as damping in the magneti-\nzation dynamics [Eqs. (1) and (2)] [31, 35].\nWe pause to comment on the behavior of \u0016szthus ob-\ntained [Eq. (6)]. Typically, \u0016szis considered to be \frst\nor second order in the cone angle, and thus negligibly\nsmall. However, as discussed below, an essential new\n\fnding is that it becomes independent of the cone an-\ngle and large under certain conditions. For a collinear\ncon\fguration and vanishing gl,gzz=gyz= 0 results\nin ~\u0016sz\u0011\u0016sz=~!!1 [38]. Its \fnite dc value con-\ntributes to the Bloch damping [Eq. (6)] [38]. For a\nnon-collinear con\fguration, \u0016sz\u0019\u0000~grp_mx=(gzz\u0000pgyz)\n0 45 90 135 18000.10.20.30.40.50.6FIG. 3. Normalized ferromagnetic resonance (FMR)\nlinewidth of F 1for di\u000berent values of the longitudinal spin\nconductance ~ gl\u0011gl=grof NjF2bilayer. The various parame-\nters employed are ~ g0\nr\u0011g0\nr=gr= 1, \u0002 = 0:1 rad,!0= 10\u00022\u0019\nGHz, and!ax= 1\u00022\u0019GHz.grandg0\nrare the spin-mixing\nconductances of F 1jN and NjF2interfaces respectively. Only\nthe spin pumping-mediated contribution to the linewidth has\nbeen considered and is normalized to its value for the case of\nspin pumping into a perfect spin sink [31].\nand contributes to Gilbert damping via Is1y[Eq. (6)].\nThus, in general, we may express the spin accumulation\nas\u0016sz=\u0016sz0+\u0016sz1[46], where \u0016sz0is the dc value\nand\u0016sz1/_mxis the linear oscillating component. \u0016sz0\nand\u0016sz1contribute, respectively, to Bloch and Gilbert\ndamping.\nGilbert-Bloch dissipation. { Equations (1) and (6) com-\npletely determine the magnetic damping in F 1. However,\nthese equations are non-linear and cannot be captured\nwithin our linearized framework [Eqs. (2) and (3)]. The\nleading order e\u000bects, however, are linear in all but a nar-\nrow range of parameters. Evaluating these leading or-\nder terms within reasonable approximations detailed in\nthe supplemental material [45], we are able to obtain the\nGilbert and Bloch damping tensors \u0014 \u000band\u0014\n. Obtaining\nthe general result numerically [45], we present the ana-\nlytic expressions for two cases covering a large range of\nthe parameter space below.\nFirst, we consider the collinear con\fgurations in the\nlimit of ~gl\u0011gl=gr!0. As discussed above, we obtain\n~\u0016sz0\u0011\u0016sz0=~!!1 and ~\u0016sz1\u0011\u0016sz1=~!!0 [Eq. (6)].\nThus the components of the damping tensors can be di-\nrectly read from Eq. (6) as ~ \u000bxx;yy\u0011\u000bxx;yy=\u000bss=ly;x=\ng0\nr=(gr+g0\nr) = ~g0\nr=(1+~g0\nr);and~\nxx;yy\u0011\nxx;yy=(\u000bss!) =\n\u0000lx;y\u0016sz0=(~!) =\u0000g0\nr=(gr+g0\nr) =\u0000~g0\nr=(1 + ~g0\nr). Here,\nwe de\fned ~ g0\nr\u0011g0\nr=grand\u000bss\u0011~grj\rj=(4\u0019MsV) is the\nGilbert constant for the case of spin-pumping into an\nideal spin sink [31, 35]. Substituting these values in Eq.\n(3), we \fnd that the linewidth, or equivalently damping,\nvanishes. This is understandable since the system we\nhave considered is not able to relax the z component of\nthe spin at all. There can, thus, be no net contribution to4\nFIG. 4. Normalized FMR linewidth of F 1for very small ~ gl.\nThe squares and circles denote the evaluated points while the\nlines are guides to the eye. The linewidth increases from being\nnegligible to its saturation value as \u0012becomes comparable to\nthe average cone angle \u0002. ~ g0\nr= 1,!0= 10\u00022\u0019GHz, and\n!ax= 1\u00022\u0019GHz.\nmagnetic damping. \u0016sz0accumulated in N opposes the\nGilbert relaxation via a negative Bloch contribution [38].\nThe latter may also be understood as an anti-damping\nspin transfer torque due to the accumulated spin [6].\nNext, we assume the system to be in a non-collinear\ncon\fguration such that ~ \u0016sz0!0 and may be disre-\ngarded, while ~ \u0016sz1simpli\fes to:\n~\u0016sz1=\u0000_mx\n!(~gl\u0000~g0\nr) sin\u0012cos\u0012\n~g0r~gl+ ~glcos2\u0012+ ~g0rsin2\u0012; (7)\nwhere ~gl\u0011gl=grand ~g0\nr\u0011g0\nr=gras above. This in turn\nyields the following Gilbert parameters via Eq. (6), with\nthe Bloch tensor vanishing on account of ~ \u0016sz0!0:\n~\u000bxx=~g0\nr~gl\n~g0r~gl+ ~glcos2\u0012+ ~g0rsin2\u0012;~\u000byy=~g0\nr\n1 + ~g0r;(8)\nwhere ~\u000bxx;yy\u0011\u000bxx;yy=\u000bssas above. Thus, ~ \u000byyis\u0012-\nindependent since ^mmm2lies in the y-z plane and the x-\ncomponent of spin, the absorption of which is captured\nby ~\u000byy, is always orthogonal to ^mmm2. ~\u000bxx, on the other\nhand, strongly varies with \u0012and is generally not equal\nto ~\u000byyhighlighting the tensorial nature of the Gilbert\ndamping.\nFigure 2 depicts the con\fgurational dependence of nor-\nmalized damping parameters. The Bloch parameters are\nappreciable only close to the collinear con\fgurations on\naccount of their proportionality to \u0016sz0. The\u0012range over\nwhich they decrease to zero is proportional to the cone\nangle \u0002 [Eq. (6)]. The Gilbert parameters are described\nsu\u000eciently accurately by Eq. (8). The linewidth [Eq.\n(3)] normalized to its value for the case of spin pump-\ning into a perfect spin sink has been plotted in Fig. 3.\nFor low ~gl, the Bloch contribution partially cancels the\nGilbert dissipation, which results in a smaller linewidthclose to the collinear con\fgurations [38]. As ~ glincreases,\nthe relevance of Bloch contribution and \u0016sz0diminishes,\nand the results approach the limiting condition described\nanalytically by Eq. (8). In this regime, the linewidth\ndependence exhibits a maximum for either collinear or\northogonal con\fguration depending on whether ~ gl=~g0\nris\nsmaller or larger than unity. Physically, this change in\nthe angle with maximum linewidth is understood to re-\n\rect whether transverse or longitudinal spin absorption\nis stronger.\nWe focus now on the case of very low ~ glwhich can\nbe realized in structures with electrically-insulating mag-\nnets. Figure 4 depicts the linewidth dependence close to\nthe collinear con\fgurations. The evaluated points are\nmarked with stars and squares while the lines smoothly\nconnect the calculated points. The gap in data for very\nsmall angles re\rects the limited validity of our linear\ntheory, as discussed in the supplemental material [45].\nAs per the limiting case ~ gl!0 discussed above, the\nlinewidth should vanish in perfectly collinear states. A\nmore precise statement for the validity of this limit is\nre\rected in Fig. 4 and Eq. (6) as ~ gl=\u00022!0. For su\u000e-\nciently low ~ gl, the linewidth changes sharply from a neg-\nligible value to a large value over a \u0012range approximately\nequal to the cone angle \u0002. This shows that systems com-\nprised of magnetic insulators bearing a very low ~ glare\nhighly tunable as regards magnetic/spin damping by rel-\natively small deviation from the collinear con\fguration.\nThe latter may be accomplished electrically by employ-\ning magnetoelectric material [41] for F 2or via current\ndriven spin transfer torques [6, 9, 47].\nDiscussion. { Our identi\fcation of damping contribu-\ntions as Gilbert-like and Bloch-like [Eq. (6)] treats \u0016sz\nas an independent variable that may result from SHE,\nfor example. When it is caused by spin pumping cur-\nrent and\u0016sz/!, this Gilbert-Bloch distinction is less\nclear and becomes a matter of preference. Our results\ndemonstrate the possibility of tuning the magnetic damp-\ning in an active magnet via the magnetization of a passive\nmagnetic layer, especially for insulating magnets. In ad-\ndition to controlling the dynamics of the uniform mode,\nthis magnetic `gate' concept [48] can further be employed\nfor modulating the magnon-mediated spin transport in a\nmagnetic insulator [43, 44]. The anisotropy in the result-\ning Gilbert damping may also o\u000ber a pathway towards\ndissipative squeezing [49] of magnetic modes, comple-\nmentary to the internal anisotropy-mediated `reactive'\nsqueezing [50, 51]. We also found the longitudinal accu-\nmulated spin, which is often disregarded, to signi\fcantly\na\u000bect the dynamics. This contribution is expected to\nplay an important role in a wide range of other phenom-\nena such as spin valve oscillators.\nSummary. { We have investigated the angular modu-\nlation of the magnetic damping in a free layer via control\nof the static magnetization in the \fxed layer of a spin\nvalve device. 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Polishchuk, Vladislav Korenivski and Arne Brataas\nCOLLINEAR CONFIGURATION WITHOUT LONGITUDINAL SPIN RELAXATION\nIn order to appreciate some of the subtleties, we \frst examine the collinear con\fguration in the limit of vanishing\nlongitudinal spin conductance. \u0012= 0;\u0019andgl= 0 imply the following values for the various parameters:\ngxx=gyy=g0\nr; g zz=gyz=p= 0; lx;y=g0\nr\ngr+g0r\u0011l; (S1)\nwhence we obtain:\n\u0016sz\n~=(mx_my\u0000my_mx)\nm2x+m2y; (S2)\n=!0+!ax\n1 +!ax\n2!0[1\u0000cos(2!t)]; (S3)\nwhere we have assumed magnetization dynamics as given by the Landau-Lifshitz equation without damping, and\nthe phase of mxis treated as the reference and set to zero. In order to obtain analytic expressions, we make the\nassumption !ax=!0\u001c1 such that we have:\n\u0016sz=\u0016sz0+\u0016sz2; with (S4)\n\u0016sz0=~\u0010\n!0+!ax\n2\u0011\n; (S5)\n\u0016sz2=~!ax\n4\u0000\ne\u0000i2!t+ei2!t\u0001\n: (S6)\nIn contrast with our assumptions in the main text, a term oscillating with 2 !appears. Furthermore, it yields\ncontributions to the Bloch damping via products such as my\u0016sz, which now have contributions oscillating at !due\nto the\u0016sz0as well as\u0016sz2. We obtain:\n~\u000bxx= ~\u000byy=l; (S7)\n~\nxx=\u0000l!0+3!ax\n4\n!0+!ax\n2and ~\nyy=\u0000l!0+!ax\n4\n!0+!ax\n2; (S8)\nsubstituting which into Eq. (3) from the main text yields a vanishing linewidth and damping. This is expected from\nthe general spin conservation argument that there can be no damping in the system if it is not able to dissipate the\nz-component of the spin. In fact, in the above considerations, \u0016sz2contributed with the opposite sign to ~\nxxand\n~\nyy, and thus dropped out of the linewidth altogether. This also justi\fes our ignoring this contribution in the main\ntext.\nFigure 1 depicts the dependence of the accumulated z-polarized spin and the normalized linewidth for small but\n\fniteglin the collinear con\fguration. The accumulated longitudinal (z-polarized) spin increases with the cone angle\nand the linewidth accordingly decreases to zero [38].\nNUMERICAL EVALUATION\nDespite the additional complexity in the previous section, we could treat the dynamics within our linearized frame-\nwork. However, in the general case, \u0016szhas contributions at all multiples of !and cannot be evaluated in a simple\nmanner. A general non-linear analysis must be employed which entails treating the magnetization dynamics numer-\nically altogether. Such an approach prevents us from any analytic description of the system, buries the underlying\nphysics, and is thus undesirable.\nFortunately, the e\u000bects of non-linear terms are small for all, but a narrow, range of parameters. Hence, we make\nsome simplifying assumptions here and continue treating our system within the linearized theory. We only show2\n10-310-210-100.10.20.30.40.5\nFIG. 1. Ferromagnetic resonance linewidth and the dc spin accumulation created in the spacer as a function of the average\ncone angle in the collinear con\fguration. Depending on ~ gl, there is a complementary transition of the two quantities between\nsmall and large values as the cone angle increases. ~ g0\nr= 1,!0= 10\u00022\u0019GHz, and!ax= 1\u00022\u0019GHz.\nresults in the parameter range where our linear analysis is adequate. Below, we describe the numerical routine for\nevaluating the various quantities. To be begin with the average cone angle \u0002 is de\fned as:\n\u00022=\nm2\nx+m2\ny\u000b\n; (S9)\nwhereh\u0001idenotes averaging over time. The spin accumulation is expressed as \u0016sz=\u0016sz0+\u0016sz1with:\n\u0016sz0=*\n~gr(lxmx_my\u0000lymy_mx\u0000p_mx)\ngzz\u0000pgyz+gr\u0000\nlxm2x+lym2y+ 2pmy\u0001+\n; (S10)\n\u0016sz1=\u0000*\ngrp\ngzz\u0000pgyz+gr\u0000\nlxm2x+lym2y+ 2pmy\u0001+\n~_mx: (S11)\nThe above expressions combined with the equations for the spin current \row (Eqs. (6) in the main text) directly yield\nthe Gilbert and Bloch damping tensors.\nVARIATION WITH ADDITIONAL PARAMETERS\nHere, we discuss the dependence of the FMR linewidth on the easy-plane anisotropy and the spin-mixing conduc-\ntanceg0\nrof the NjF2interface. The results are plotted in Fig. 2. A high easy-plane anisotropy is seen to diminish\nthe con\fguration dependence of the linewidth and is thus detrimental to the dissipation tunability. The easy-axis\nanisotropy, on the other hand, is absorbed in !0and does not need to be examined separately. We also see an increase\nin the con\fguration dependence of the damping with an increasing g0\nr. This is understood simply as an increased\ndamping when the spin is absorbed more e\u000eciently due to a larger g0\nr. The damping is expected to reach the case of\nspin pumping into a perfect spin sink in the limit of ~ g0\nr!1 and\u0012= 0;\u0019.\nEFFECT OF SPIN RELAXATION IN THE SPACER LAYER\nWe now address the role of the small but \fnite spin relaxation in the non-magnetic spacer layer. To this end, we\nconsider that a part of the spin current injected into N by F 1is lost as the \\spin-leakage current\" IIIsl, as depicted in\nFig. 3, such that IIIs1=IIIs2+IIIsl. In order to evaluate the leakage, we consider the spin di\u000busion equation in N which\nreads [31]:\nD@2\nx\u0016\u0016\u0016s=\u0016\u0016\u0016s\n\u001csf; (S12)3\n0 45 90 135 18000.10.20.30.40.50.6\n(a)\n0 45 90 135 18000.20.40.60.81 (b)\nFIG. 2. Normalized ferromagnetic resonance (FMR) linewidth of F 1. (a) Same as Fig. 3 in the main text with additional plots\nfor a large easy-plane anisotropy. (b) Linewidth dependence for di\u000berent spin-mixing conductances of N jF2interface. The\nparameters employed are the same as Fig. 2 in the main text.\nFIG. 3. Schematic depiction of the spin currents \rowing through the device, including the spin-leakage current IIIslthat is lost\non account of a \fnite spin relaxation in the spacer layer N.\nwhereDand\u001csfare di\u000busion constant and spin-\rip time, respectively. We now integrate the equation over the\nthickness of N:\nZ\nd(D@x\u0016\u0016\u0016s) =Zd\n0\u0016\u0016\u0016s\n\u001csfdx: (S13)\nSince the N-layer thickness dis typically much smaller than the spin di\u000busion length in N (e.g., a few nm versus a\nfew hundred nm for Cu), we treat \u0016\u0016\u0016son the right hand side as a constant. Furthermore, in simplifying the left hand\nside, we invoke the expression for the spin current [31]: IIIs= (\u0000~NSD=2)@x\u0016\u0016\u0016s, withNthe one-spin density of states\nper unit volume and Sthe interfacial area. Thus, we obtain\n2\n~NS(IIIs1\u0000IIIs2) =d\n\u001csf\u0016\u0016\u0016s; (S14)\nwhich simpli\fes to the desired relation IIIs1=IIIs2+IIIslwith\nIIIsl=~NVN\n2\u001csf\u0016\u0016\u0016s\u0011gsl\n4\u0019\u0016\u0016\u0016s; (S15)\nwhereVNis the volume of the spacer layer N.\nIt is easy to see that accounting for spin leakage, as derived in Eq. (S15), results in the following replacements to\nEqs. (6) of the main text:\ngxx!gxx+gsl; g yy!gyy+gsl; g zz!gzz+gsl: (S16)4\nSince all our speci\fc results are based on Eqs. (6) of the main text, this completes our assessment of the role played\nby spin relaxation in N. Physically, this new result means that the condition for no spin relaxation in the system,\nwhich was previously treated as gl!0, is now amended to gl+gsl!0. This, however, does not a\u000bect the generality\nand signi\fcance of the key results presented in the main text."    },    {        "title": "2204.10596v2.A_short_circuited_coplanar_waveguide_for_low_temperature_single_port_ferromagnetic_resonance_spectroscopy_set_up_to_probe_the_magnetic_properties_of_ferromagnetic_thin_films.pdf",        "content": "arXiv:2204.10596v2  [cond-mat.mtrl-sci]  19 Jul 2022A short-circuited coplanar waveguide for low-temperature single-port ferromagnetic\nresonance spectroscopy set-up to probe the magnetic proper ties of ferromagnetic thin\nfilms\nSayani Pal, Soumik Aon, Subhadip Manna and Chiranjib Mitra∗\nIndian Institute of Science Education and Research Kolkata ,\nWest Bengal, India\nA coplanar waveguide shorted in one end is proposed, designe d, and implemented successfully to\nmeasure the properties of magnetic thin films as a part of the v ector network analyser ferromag-\nnetic resonance (VNA-FMR) spectroscopy set-up. Its simple structure, potential applications and\neasy installation inside the cryostat chamber made it advan tageous especially for low-temperature\nmeasurements. It provides a wide band of frequencies in the g igahertz range essential for FMR\nmeasurements. Our spectroscopy set-up with short-circuit ed coplanar waveguide has been used to\nextract Gilbert damping coefficient and effective magnetizat ion values for standard ferromagnetic\nthin films like Py and Co. The thickness and temperature depen dent studies of those magnetic\nparameters have also been done here for the afore mentioned m agnetic samples.\nINTRODUCTION\nIn recent years, extensive research on microwave mag-\nnetization dynamics in magnetic thin films[1–3], planar\nnanostructures[4–6] and multi-layers[7–9] havebeen per-\nformedduetotheirpotentialapplicationsinvariousfields\nof science and technology. Spintronics is one such emerg-\ning discipline that encompasses the interplay between\nmagnetization dynamics and spin transport. It also in-\ncludes fields like spin-transfer torque [10–13], direct and\ninversespin hall effect [14–18], spin pumping [19, 20] etc.,\nwhich are crucial in industrial applications for develop-\ning devices like magnetic recording head[21], magnetic\ntunnel junction(MTJ) sensors [22, 23], magnetic memory\ndevices[24, 25] andspin-torquedevices[26, 27]. Thus ex-\nploring more about the static and dynamic properties of\nmagnetic materials in itself is an interesting subject. Fer-\nromagnetic resonance spectroscopy(FMR) is a very ba-\nsic and well-understood technique that is used to study\nthe magnetization dynamics of ferromagnets[28, 29, 31].\nNowadays, most advanced FMR spectroscopy methods\nuse a vector network analyzer (VNA)[30, 31] as the mi-\ncrowave source and detector. We have used VNA in our\nset-up too.\nTo determine the magnetic parameters of the ferromag-\nnetic materials using the VNA-FMR spectroscopy, one\nneeds to carry out the measurements at a wide range of\nfrequencies. Since the microwave magnetic field in the\ncoplanar waveguide (CPW) is parallel to the plane, it\nservesthepurposeofexploringthemagneticpropertiesof\nthe concernedsystem overabroadfrequencyrangein the\nGHz region. The advantage of using CPW in the spec-\ntroscopy system lies in the fact that we no longer need\nto remount samples at different waveguides or cavities\nforeveryotherfrequency measurements, which consumes\n∗Corresponding author:chiranjib@iiserkol.ac.ina lot of time and effort in an experiment[32, 33]. Re-\nsearchers design and use different types of CPW for vari-\nous other purposes like micron-sized CPW in microwave-\nassisted magnetic recording; two-port CPW in antenna;\nshorted CPW in ultra-wideband bandpass-filter and per-\nmeability measurements [34–36]. However, in broadband\nFMR spectroscopy two-port CPW jigs have most com-\nmonly been used till date. Using two-port CPW in FMR\nspectroscopy, one gets absorption spectra in terms of\nthe transmissioncoefficient of scatteringparameters, and\nfrom there magnetic parameters of the samples can be\ndetermined. The use of two-port CPW in VNA-FMR\ncan be replaced by one-port CPW where the reflection\ncoefficient of scattering parameters of the FMR spectra\ncan be used to determine the magnetic parameters of\nthe sample. One port reflection geometry is a lot more\nconvenient in terms of easy design, calibration, installa-\ntion, and sample loading. This is especially true when\nthe whole CPW arrangement is kept inside the cryostat\nchamber for low-temperature measurements and the sys-\ntem becomes very sensitive to vibration and any kind\nof magnetic contacts, one port CPW seems very con-\nvenient to operate rather than the two-port one. Previ-\nously, manyhavedesignedandusedshort-circuitedCPW\njigs for other purposes but to the best our knowledge it\nhas not been used for low-temperature VNA-FMR spec-\ntroscopy measurements before.\nIn this work, we report the development of short-\ncircuited CPW based low-temperature broadband VNA-\nFMR spectroscopy set-up to study the magnetic param-\neters of standard ferromagnetic samples. For measure-\nments, we chose the permalloy(Py) thin films as ferro-\nmagnetic (FM) material which has greatly been used in\nresearchfields like spintronics and industrial applications\ndue to its interesting magnetic properties like high per-\nmeability, large anisotropy magnetoresistance, low coer-\ncivity, and low magnetic anisotropy. We have also con-\nsidered another standard magnetic thin film, Co of thick-\nness 30nm as a standard for ascertaining the measure-2\nment accuracy. In our system, we swept the magnetic\nfield keeping the frequencies constant, and got the FMR\nspectra for several frequencies. From there we found the\nvariation of resonance fields and field linewidths with\nthe resonance frequencies. We have used the linear fit\nfor resonance frequencies vs field line-widths data to\ncalculate the Gilbert damping coefficient( α). We fit-\nted the set of resonance frequencies vs resonance fields\ndata to the Kittel formula [59] to obtain the effec-\ntive magnetization(4 πMeff). Subsequently, we investi-\ngated the thickness and temperature-dependent studies\nof 4πMeffandαfor FM thin films of different thickness\ninthetemperaturerangeof7.5Kto300K.Tocharacterise\nthe measurement set-up using short-circuited CPW, we\ncompared the previous measurements in the literature\nwith ourresults and there wasa good agreementbetween\nthe two[36, 41].\nEXPERIMENTAL DETAILS\nA short-circuited CPW has been designed and fab-\nricated as a part of our low-temperature VNA-FMR\nspectroscopy set-up. To make the CPW we have used\nRogers AD1000, a laminated PCB substrate with copper\ncladding on both sides of the dielectric. The thickness of\nthe dielectric and the copper layer are 1.5 mm and 17.5\nmicrons respectively and the dielectric constant of the\nsubstrate is 10.7. The main concern about the design of\nthe CPW is to match its characteristic impedance with\nthe impedance of the microwave transmission line con-\nnected to it. We haveused the line calculatorto calculate\nthe dimensions of CPW. For a CPW with a characteris-\ntic impedance of 50 ohms, the line calculator calculated\nthe width of the signal line and the gap to be 900 mi-\ncrons and 500 microns respectively. The fabrication is\ndone using optical lithography which is described in de-\ntail in the literature[49]. Other components of our mea-\nCryostatVNA\nElectromagnetSample\nCPWCoaxial Transmission Line\nFIG. 1. The schematic diagram of measurement system and\nthe arrangement inside the cryostat with the sample on top\nof the CPW\nsurement system are a)Vector Network Analyser(VNA),\nwhich is a microwave source as well as a detector, b)theelectromagnet that generates the external magnetic field,\ni.e., Zeemanfieldand, c)optistatdrycryogen-freecooling\nsystem from Oxford instruments which is used for low-\ntemperature measurements. One end of the CPW signal\nline is shorted to the ground, and the other end is con-\nnected to the VNA through a SMA connector and coax-\nial cable (fig 3b). On top of the CPW, thin-film samples\nhave been placed face down after wrapping them with\nan insulating tape to electrically isolate them. For low-\ntemperature measurements, the sample has been glued\nto the CPW using a low-temperature adhesive to ensure\ncontact of sample and resonator at all times, in spite of\nthe vibration caused by the cryostat unit. This whole ar-\nrangementis then placed inside the twopole pieces of the\nelectromagnet as we can see from the diagram in fig 1.\nTherearetwostandardmethods ofgettingFMR spectra:\nsweeping the frequency keeping the field constant and\nsweeping the magnetic field while keeping the frequency\nconstant. We have adopted the second method. We have\nworked in the frequency range from 2.5GHz to 5.5GHz\nand in the magnetic field range from 0 Oe to roughly\naround 500 Oe. We have used 1mW of microwave power\nthroughout the experiment. From the FMR spectra, we\nhavedeterminedeffectivemagnetizationanddampingco-\nefficient of FM thin films and studied their variation with\ntemperature and sample thickness.\nSAMPLE PREPARATION AND\nCHARACTERIZATION\nPy (Ni80Fe20) and Co thin films were fabricated by\nthermal evaporation technique on Si/SiO 2substrates,\nfrom commercially available pellets (99 .995%pure) at\nroom temperature. The substrates were cleaned with\nacetone, IPA and DI water respectively in ultrasonica-\ntor and dried with a nitrogen gun. The chamber was\npumped down to 1 ×10−7torr using a combination of\na scroll pump and turbo pump. During the deposition,\npressure reached upto 1 ×10−6torr. Thin films were fab-\nricated at a rate of 1 .2˚A/swhere thickness can be con-\ntrolled by Inficon SQM 160 crystal monitor. For our\nexperiments a series of Py thin films of different thick-\nnesses were fabricated by keeping the other parameters\nlike base pressure, deposition pressure and growth rate\nconstant. Film thickness and morphology was measured\nby using atomic force microscopy technique as shown in\nfig 2(a). We have used Py films with thicknesses 10nm,\n15nm, 34nm, 50nm, and 90nm with a surface roughness\nof around 1nm and one Co film of thickness 30nm. X-ray\ndiffraction experiment confirms the polycrystalline struc-\nture of the samples as shown in fig 2b and fig 2c for Py\nand Co respectively.3\n2µm\n2µm\n(a)\n/s51/s53 /s52/s48 /s52/s53 /s53/s48 /s53/s53 /s54/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48\n/s52/s52/s46/s51/s54/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s105/s116/s97/s114/s121/s32/s117/s110/s105/s116/s41\n/s50 /s113 /s32/s40/s100/s101/s103/s114/s101/s101/s41/s80/s121/s32/s40/s49/s53/s110/s109/s41\n(b)\n/s51/s53 /s52/s48 /s52/s53 /s53/s48 /s53/s53/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48/s56/s48/s48/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s65/s114/s98/s105/s116/s97/s114/s121/s32/s117/s110/s105/s116/s41\n/s50 /s113 /s32/s40/s100/s101/s103/s114/s101/s101/s41/s67/s111/s32/s40/s51/s48/s110/s109/s41\n(c)\nFIG. 2. (a)Atomic force microscope (AFM) image of 30 nm\nthick Py thin film with a surface roughness of 1 nm . X-ray\ndiffraction peak of (b)15nm thick Py film and (c)30nm Co\nprepared by thermal evaporation.\nRESULTS AND DISCUSSION\nWe have calculated the dimensions of the short-\ncircuited CPW using the line calculator of the CST Stu-\ndio Suite software as mentioned in the experimental de-\ntails section. Using those dimensions we have also done\nthe full-waveelectromagneticsimulation in CST software\nto get the electric and magnetic field distribution of the\nCPW. One can see from the simulation result displayed\nin figure 3a that the farther it is from the gap, the weaker\nthe intensity of the magnetic field, and the magnitude of\nthe field in the gap area is one order of greater than that\non the signal line. When placing the thin film sample\non top of the CPW, the dimension of the sample shouldDielectricSampleSignal  Line\nGap\nMagnetic field lines\nElectric field lines\na) b)\nc)\nFIG. 3. (a) Schematic diagram of the cross-sectional view of\nCPW. (b) Top view of the short-circuited CPW after fabri-\ncation. (c)Intensity distribution of microwave magnetic fi eld\nin the one end shorted CPW at 5GHz (top view)\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s45/s48/s46/s53/s45/s48/s46/s52/s45/s48/s46/s51/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s83\n/s49/s49/s40/s100/s66/s41\n/s72/s32/s40/s79/s101/s41/s32/s32/s32 /s102/s114/s101/s113/s117/s101/s110/s99/s121\n/s32/s50/s46/s53/s71/s72/s122\n/s32/s51/s46/s53/s71/s72/s122\n/s32/s52/s46/s53/s71/s72/s122\n/s32/s53/s46/s53/s71/s72/s122/s49/s53/s110/s109/s32/s80/s121\n/s84/s61/s51/s48/s48/s75\nFIG. 4. Ferromagnetic Resonance spectra of absorption at\nfrequencies 2.5 GHz, 3.5 GHz, 4.5 GHz, 5.5 GHz for 15nm Py\nthin films at room temperature after background subtraction\nbe such that it can cover the gap area on both sides of\nthe signal line of the CPW because the magnetic field is\nmost intense in that area. This microwave magnetic field\ncirculatingthe signal line ofthe CPW is perpendicular to4\n/s50 /s51 /s52 /s53 /s54 /s55/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s84/s61/s51/s48/s48/s75\n/s32/s67/s111/s32/s40/s51/s48/s110/s109/s41\n/s32/s80/s121/s32/s40/s51/s52/s110/s109/s41/s68 /s72/s32/s40/s79/s101/s41\n/s102/s32/s40/s71/s72/s122/s41\n(a)/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/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/s97\n/s116/s32\n/s80/s121 /s32/s40/s110/s109/s41/s32/s84/s61/s51/s48/s48/s75\n(b)\n/s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s50/s51/s52/s53/s54/s55\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s84/s61/s32/s51/s48/s48/s75\n/s32/s67/s111/s32/s40/s51/s48/s110/s109/s41\n/s32/s80/s121/s32/s40/s51/s52/s110/s109/s41/s102/s32/s40/s71/s72/s122/s41\n/s72/s32/s40/s79/s101/s41\n(c)/s48/s46/s48/s48 /s48/s46/s48/s50 /s48/s46/s48/s52 /s48/s46/s48/s54 /s48/s46/s48/s56 /s48/s46/s49/s48/s56/s57/s49/s48/s49/s49/s52 /s112 /s77\n/s101/s102/s102/s32/s40/s107/s71/s41\n/s116/s32/s45/s49\n/s32/s80/s121/s32/s40/s110/s109/s45/s49\n/s41/s32/s84/s61/s51/s48/s48/s75\n(d)\nFIG. 5. a)Field linewidth variation with resonance frequen cies at 300K for 34nm Py and 30nm Co thin films. Equation 1 has\nbeen used for fitting the curve and to determine the Gilbert da mping coefficient; b)thickness dependence of Gilbert dampin g\ncoefficient at room temperature for Py thin films; c)resonance field variation with resonance frequencies at 300K for 34 nm P y\nand 30 nm Co thin films. Kittel formula (eqn-3)has been used fo r fitting the curve and to determine the effective magnetizati on;\nd)thickness dependence of effective magnetization for Py th in films at room temperature.\nthe external magnetic field and both the magnetic fields\nare parallel to the film surface as can be seen from fig\n3a and 3b. On account of the static magnetic field, the\nmagnetic moment will undergo a precession around the\nstatic magnetic field at a frequency called the Larmor\nprecession frequency. Absorption of electromagnetic en-\nergy happens when the frequency of the transverse mag-\nnetic field (microwave) is equal to the Larmor frequency.\nFig4exhibitsthe absorptionspectrafor15nmbarePy\nfilm after subtraction of a constant background for four\ndifferent frequencies, 2.5 GHz, 3.5 GHz, 4.5 GHz and 5.5\nGHz at room temperature in terms of S-parameter re-\nflection coefficient ( S11) vs. external magnetic field. We\nfitted these experimental results to the Lorentz equation\n[56]. We extracted the field linewidth at half maxima\nfrom the FMR spectra at different frequencies and fitted\nthem using equation 1 to obtain αas one can see from\nfig 5a and fig 6a. The experimental values of the absorp-tion linewidth (∆ H) contains both the effect of intrinsic\nGilbert damping and the extrinsic contribution to the\ndamping. Linewidth due to Gilbert damping is directly\nproportional to the resonance frequency and follows the\nequation:\n∆H= (2π\nγ)αf+∆H0 (1)\nwhereγis the gyromagneticratio, αis the Gilbert damp-\ning coefficient and ∆ H0is the inhomogeneous linewidth.\nA number of extrinsic contributions to the damping coef-\nficient like magnetic inhomogeneities, surface roughness,\ndefects of the thin films bring about the inhomogeneous\nlinewidth broadening [55]. αhas been determined using\nthe above equation only. Damping coefficient values ob-\ntainedhereareintherangeofabout0 .005to0.009forPy\nsamplesofthicknessescoveringthe whole thin film region\ni.e., 10nm to 90nm at room temperature. These values5\n/s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53 /s53/s46/s48 /s53/s46/s53 /s54/s46/s48/s51/s48/s51/s53/s52/s48/s52/s53/s53/s48/s53/s53/s54/s48\n/s32/s49/s48/s110/s109/s32/s84/s61/s51/s48/s48/s75\n/s32/s49/s48/s110/s109/s32/s84/s61/s52/s53/s75/s80/s121/s68 /s72/s32/s40/s79/s101/s41\n/s102/s32/s40/s71/s72/s122/s41/s32/s49/s53/s110/s109/s32/s84/s61/s51/s48/s48/s75\n/s32/s49/s53/s110/s109/s32/s84/s61/s52/s53/s75/s80/s121\n(a)/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s46/s48/s48/s54/s48/s46/s48/s48/s56/s48/s46/s48/s49/s48/s48/s46/s48/s49/s50\n/s84/s32/s40/s75/s41/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s116 /s32\n/s80/s121 \n/s32/s49/s53/s110/s109\n/s32/s49/s48/s110/s109\n(b)\n/s54/s48 /s49/s50/s48 /s49/s56/s48 /s50/s52/s48 /s51/s48/s48 /s51/s54/s48 /s52/s50/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48/s52/s46/s53/s53/s46/s48/s53/s46/s53/s54/s46/s48\n/s32/s49/s53/s110/s109/s32/s84/s61/s51/s48/s48/s75\n/s32/s49/s53/s110/s109/s32/s84/s61/s32/s52/s53/s75/s80/s121/s102/s32/s40/s71/s72/s122/s41\n/s72/s32/s40/s79/s101/s41/s32/s49/s48/s110/s109/s32/s84/s61/s51/s48/s48/s75\n/s32/s49/s48/s110/s109/s32/s84/s61/s52/s53/s75/s80/s121\n(c)/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s56/s46/s48/s56/s46/s50/s56/s46/s52/s56/s46/s54/s56/s46/s56/s57/s46/s48/s57/s46/s50/s57/s46/s52/s57/s46/s54/s52 /s77\n/s101/s102/s102/s32/s40/s107/s71/s41\n/s84/s32/s40/s75/s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s116/s32\n/s80/s121 \n/s32/s49/s53/s110/s109\n/s32/s49/s48/s110/s109\n(d)\nFIG. 6. a)Field linewidth variation with resonance frequen cies at 300K and 45K for 10 nm and 15nm Py films. Equation 1\nhas been used for fitting the curve and to determine the Gilber t damping coefficient; b)temperature dependence of damping\ncoefficient for 10nm and 15nm Py thin films; c)resonance field va riation with resonance frequencies at 300K and 45K for 10nm\nand 15nm Py thin films. Kittel formula (eqn-3) has been used fo r fitting the curve and to determine the 4 πMeff; d)temperature\ndependence of 4 πMefffor 10nm and 15nm Py thin films.\nare pretty close to the values previously reported in the\nliterature [39–41, 43, 44]. For the Co film of thickness 30\nnm we have obtained the value of αto be 0.008 ±0.0004.\nBaratiet al.measured the damping value of 30nm Co\nfilm to be 0.004 [37, 38]. There are other literature also\nwhere Co multilayers have been studied where damping\ncoefficient value increasesbecause ofspin pumping effect.\nαis a veryinterestingparameterto investigatebecause it\nis used in the phenomenological LLG equation [57], [58]\nto describe magnetization relaxation:\nd/vectorM\ndt=−γ/vectorM×/vectorHeff+α\nMS/vectorM×d/vectorM\ndt(2)\nwhere,µBisBohrmagneton, /vectorMisthemagnetizationvec-\ntor,MSis the saturation magnetization and Heffis the\neffectve magnetic field which includes the external field,\ndemagnetization and crystalline anisotropy field. The in-troduction of the Damping coefficient in LLG equation is\nphenomenological in nature and the question of whether\nit has a physical origin or not has not been fully under-\nstood till date. We have measured 4 πMeffalso from\nthe absorption spectra. We have fitted the Kittel for-\nmula (equation 3) into resonance field vs. the resonance\nfrequency ( fres) data as shown in fig 5c and fig 6c.\nfres= (γ\n2π)[(H+4πMeff)H]1\n2 (3)\nwhere,His the applied magnetic field, and Meffis the\neffective magnetization which contains saturation mag-\nnetization and other anisotropic contributions. We ob-\ntained the 4 πMeffvalue for 30nm thick Co and 34nm Py\nto be 17.4 ±0.2kG and 9.6 ±0.09kG respectively at room\ntemperature. These values also agree quite well with the\nliterature. For a 10nm Co film, Beaujour et al.measured\nthe value to be around 16 kG[45] and for a 30nm Py the6\nvalue is 10 .4kGas measured by Zhao et al[41].\nWe tried to address here the thickness and tempera-\nture dependence of αand 4πMeffusing our measure-\nment set-up. The variation of the αwith thickness is\nshown here in figure 5b. It increases smoothly as film\nthickness decreases and then shows a sudden jump below\n15nm. Increased surface scattering could be the reason\nbehind this enhanced damping for thinner films. It has\nbeen previously observed [60] that damping coefficient\nand electrical resistivity follows a linear relation at room\ntemperature for Py thin film. It suggests a strong corre-\nlation between magnetization relaxation( α) and electron\nscattering. Magnetization relaxation could be explained\nby electron scattering by phonons and magnons. In the\nformer case, αis proportional to the electron scatter-\ning rate, τ−1and in the later case, α∼τ. Theoretical\npredictions by Kambersky [61] suggests that at higher\ntemperature α∼τ−1as electron scattering by phonons\nare predominant there. So, here in our case we can elim-\ninate the possibility of electron scattering by magnons as\nthickness dependent study has only been done at room\ntemperature where phonon scattering is prevalent. Ing-\nvasson et.al in their paper[60] also suggests that the re-\nlaxation of magnetization is similar to bulk relaxation\nwhere phonon scattering in bulk is replaced by surface\nand defect scattering in thin films.\nThicknessdependent studyof4 πMeffalsohasbeen done\nfor Py thin films at room temperature. As we can see\nfrom fig 5d, Meffis linear for thinner films and becomes\nalmost independent of thickness for thicker films. The\nchange in Meffwith thickness mainly comes from the\nsurface anisotropy,\nµ0Meff=µ0Ms−2Ks\nMsd(4)\nwhereMsis the saturation magnetization and2Ks\nMsdis\nthe surface anisotropy field. Surface anisotropy is higher\nfor thinner films and the anisotropy reduces as one in-\ncreases the film thickness. We have obtained saturation\nmagnetization(4 πMs) value of Py to be 10 .86kGusing\nthe linear fit (equation 4). Previously Chen et al.has re-\nported the 4 πMeffvalue for a 30nm Py film to be 12 kG\n[54] which includes both 4 πMsand anisotropy field.\nTemperature dependence of αfor 15nm and 10nm Py\nfilm is represented in figure 6b. The αvalue decreases\nmonotonically from room temperature value and reaches\na minimum value at around 100K and then starts to in-\ncrease with further decrease of temperature and reaches\na maximum value at 45K. Zhao et al.have seen this\nkind of damping enhancement at around50Kin their low\ntemperature experiment with Py thin films with differ-\nent types of capping layers and Rio et al.observed the\ndamping anomaly at temperature 25K when they have\nusedPtas a capping layer on Py thin film.[39, 41]. We\ndid not use any capping layer on Py film in our mea-\nsurement. So there is no question of interface effect for\nthe enhanced damping at 45K. A possible reason for the\nstrong enhancement of damping at 45K could be the/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48/s49/s50/s48/s49/s52/s48/s49/s54/s48/s49/s56/s48\n/s32/s57/s71/s72/s122\n/s32/s52/s71/s72/s122/s51/s48/s110/s109/s32/s67/s111/s68 /s72 /s32/s40/s79/s101/s41\n/s84/s32/s40/s75/s41\nFIG. 7. Temperature induced linewidth variation of 30nm Co\nthin film at two different frequencies 4GHz and 9GHz\nspin reorientation transition(SRT) on the Py surface at\nthat particular temperature [41, 42]. Previously it has\nbeen established that the competition between different\nanisotropy energies: magnetocrystalline anisotropy, sur-\nface anisotropy, shape anisotropy decides the magnetiza-\ntion direction in magnetic films. For thin films, the vari-\nation of temperature, film thickness, strain can alter the\ncompetition between shape and surface anisotropy. In\nour case, temperature variation could be the reason for\nthe spin reorientation transition on Py surface at around\n45K.Foradeeperunderstandingofthespinreorientation\nwe investigated the temperature dependence of 4 πMeff\nfor 15nm and 10nm Py film as shown in fig 6d. There\nMeffis showing an anomaly at around 45K, otherwise it\nis increasing smoothly with the decrease of temperature.\nSince there is no reason of sudden change in saturation\nmagnetization at this temperature, the possible reason\nfor the anomaly in Meffshould come from any change\ninmagneticanisotropy. Thatchangeofanisotropycanbe\nrelated to a spin reorientation at that particular temper-\nature value. Sierra et.al., [42], have also argued that in\nthe temperature dependent spin re-orientation (T-SRT),\nthe central effect of temperature on the magnetic prop-\nerties of Py films was to increase the in-plane uniax-\nial anisotropy and to induce a surface anisotropy which\norients the magnetization out of plane in the Py sur-\nface. They have verified this using X-Ray diffraction\nexperiments and high resolution transmission electron\nmicroscopy images. This establishes reasonably enough\nthat it is a spin re-orientation transition around 45K.\nLastly, for a 30nm Co thin film we have studied the\ntemperature variation of FMR linewidth(∆ H) at mi-\ncrowave frequencies 9GHz and 4GHz. One can see from\nfig7 that the linewidth does not change much in the tem-\nperature range 100 <T<300 but below 100K, ∆ Hstarts\nto increase significantly. This behaviour of ∆ Hhas been7\nobserved previously by Bhagat et al.[62]. They sug-\ngested that the increase of relaxation frequency at low\ntemp might be related to the rapid variation in the elec-\ntronicmeanfreepath. Thiscomparisonsuggeststhatour\nset-up with a single port CPW has the same sensitivity\nof the previously reported set-up.\nCONCLUSIONS\nShort-circuited coplanar waveguide can successfully be\nused for low-temperature single-port broadband FMR\nspectroscopy measurements. The magnetic parameters\nobtained here for the standard ferromagnetic materials,\nPy and Co are in good agreement with other experimen-\ntal works and theoretical predictions. Temperature de-\npendentstudiesof αand4πMeffforPyfilmshereexhibit\nspin reorientation phenomenon at low temperatures. We\nbelieve that the findings with Py sample will help in bet-\nter understanding of magnetic phenomena for other fer-\nromagnetic materials at low temperature. Though this\nsetup has limitations for angle-dependent FMR measure-\nments, itcanreadilybe usedforstudiesonmagnetization\ndynamics for multi-layered films and planar nanostruc-\ntures and is currently under way. In future we aim to in-\ntegrate electrical measurement facilities with the existing\nset-up thereby extending the possibilities of inverse spinhall(ISHE) measurements and spin-torque ferromagnetic\nresonance(ST-FMR)spectroscopyexperimentswhich are\nvery relevant for current scientific interests. We believe\nthe short-circuited CPW will serve its purpose conve-\nniently there too.\nACKNOWLEDGEMENTS\nTheauthorssincerelyacknowledgesMinistryofEduca-\ntion, Government of India and Science and Engineering\nResearch Board (SERB) (grant no:EMR/2016/007950)\nand Department of Science and Technology (grant\nno. DST/ICPS/Quest/2019/22) for financial sup-\nport. S.P. acknowledges Department of Science and\nTechnology(DST)-INSPIRE fellowship India, S. A. ac-\nknowledges Ministry of Education of Government of In-\ndia and S.M. acknowledges Council Of Scientific and\nIndustrial Research(CSIR),India for research fellowship.\nThe authors would like to thank Dr. Partha Mitra of the\nDepartment of Physical Sciences, Indian Institute of Sci-\nence Education and Research Kolkata for providing the\nlab facilities for sample deposition. The authors would\nalso like to thank Mr. Subhadip Roy, IISER Kolkata for\nhis help in fabricating the CPW structure using home-\nbuilt optical lithography set-up.\n[1]Barman, A. and Haldar, A., 2014. Time-domain study\nof magnetization dynamics in magnetic thin films and\nmicro-and nanostructures. Solid State Physics, 65, pp.1-\n108.\n[2]Silva, T.J., Lee, C.S., Crawford, T.M. and Rogers, C.T.,\n1999. Inductive measurement of ultrafast magnetization\ndynamics in thin-film Py. 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Ferro-\nmagnetic resonance studyof thickness-dependentmagne-tization precession in Ni80Fe20films. Journal ofapplied\nphysics. 2007 May 1;101(9):09C104.\n[55]Jiang, S., Sun, L., Yin, Y., Fu, Y., Luo, C., Zhai, Y.\nand Zhai, H., 2017. Ferromagnetic resonance linewidth\nand two-magnon scattering in Fe1-x Gd x thin films. AIP\nAdvances, 7(5), p.056029.\n[56]Celinski, Z., Urquhart, K.B. and Heinrich, B., 1997. Us-\ningferromagnetic resonance tomeasurethemagneticmo-\nments of ultrathin films. Journal of Magnetism and Mag-\nnetic Materials, 166(1-2), pp.6-26.\n[57]Landau LA, Lifshitz E. On the theory of the dispersion\nof magnetic permeability in ferromagnetic bodies. InPer-\nspectives in Theoretical Physics 1992 Jan 1 (pp. 51-65).\nPergamon.\n[58]Gilbert, T.L., 2004. A phenomenological theory of damp-\ning in ferromagnetic materials. IEEE transactions on\nmagnetics, 40(6), pp.3443-3449.\n[59]Kittel, C., 1948. On the theory of ferromagnetic reso-\nnance absorption. Physical review, 73(2), p.155.\n[60]Ingvarsson, S., Ritchie, L., Liu, X.Y., Xiao, G., Slon-\nczewski, J.C., Trouilloud, P.L. and Koch, R.H., 2002.\nRole of electron scattering in the magnetization relax-\nation ofthinNi81Fe19films.Physical ReviewB,66(21),\np.214416.\n[61]Kambersk´ y, V., 1976. Onferromagnetic resonance damp-\ning in metals. Czechoslovak Journal of Physics B, 26(12),\npp.1366-1383.\n[62]Bhagat SM, Lubitz P. Temperature variation of ferro-\nmagnetic relaxation in the 3 d transition metals. Physical\nReview B. 1974 Jul 1;10(1):179."    },    {        "title": "1603.07977v1.Large_spin_pumping_effect_in_antisymmetric_precession_of_Ni___79__Fe___21___Ru_Ni___79__Fe___21__.pdf",        "content": "Large spin pumping e\u000bect in antisymmetric precession of\nNi79Fe21/Ru/Ni 79Fe21\nH. Yang,1Y. Li,1and W.E. Bailey1,a)\nMaterials Science and Engineering, Dept. of Applied Physics and Applied Mathematics, Columbia University,\nNew York NY 10027\n(Dated: 16 September 2021)\nIn magnetic trilayer structures, a contribution to the Gilbert damping of ferromagnetic resonance arises from\nspin currents pumped from one layer to another. This contribution has been demonstrated for layers with\nweakly coupled, separated resonances, where magnetization dynamics are excited predominantly in one layer\nand the other layer acts as a spin sink. Here we show that trilayer structures in which magnetizations are\nexcited simultaneously, antisymmetrically, show a spin-pumping e\u000bect roughly twice as large. The antisym-\nmetric (optical) mode of antiferromagnetically coupled Ni 79Fe21(8nm)/Ru/Ni 79Fe21(8nm) trilayers shows a\nGilbert damping constant greater than that of the symmetric (acoustic) mode by an amount as large as\nthe intrinsic damping of Py (\u0001 \u000b'0.006). The e\u000bect is shown equally in \feld-normal and \feld-parallel to\n\flm plane geometries over 3-25 GHz. The results con\frm a prediction of the spin pumping model and have\nimplications for the use of synthetic antiferromagnets (SAF)-structures in GHz devices.\nPumped spin currents1,2are widely understood to in-\n\ruence the magnetization dynamics of ultrathin \flms\nand heterostructures. These spin currents increase the\nGilbert damping or decrease the relaxation time for thin\nferromagnets at GHz frequencies. The size of the e\u000bect\nhas been parametrized through the e\u000bective spin mixing\nconductance g\"#\nr, which relates the spin current pumped\nout of the ferromagnet, transverse to its static (time-\naveraged) magnetization, to its precessional amplitude\nand frequency. The spin mixing conductance is inter-\nesting also because it determines the transport of pure\nspin current across interfaces in quasistatic spin trans-\nport, manifested in e.g. the spin Hall e\u000bect.\nIn the spin pumping e\u000bect, spin current is pumped\naway from a ferromagnet / normal metal (F 1/N) in-\nterface, through precession of F1, and is absorbed else-\nwhere in the structure, causing angular momentum loss\nand damping of F1. The spin current can be absorbed\nthrough di\u000berent processes in di\u000berent materials. When\ninjected into paramagnetic metals (Pt, Pd, Ru, and oth-\ners), the spin current relaxes exponentially with para-\nmagnetic layer thickness3{5. The relaxation process has\nbeen likened to spin-\rip scattering as measured in CPP-\nGMR, where spin-\rip events are localized to heavy-metal\nimpurities6and the measurement reveals the spin di\u000bu-\nsion length \u0015SD. When injected into other ferromagnets\n(F2in F 1/N/F 2), the spin current is absorbed through\nits torque on magnetization5,7. A similar process appears\nto be relevant for antiferromagnets as well8.\nIn F 1/N/F 2structures, only half of the total possible\nspin pumping e\u000bect has been detected up until now. For\nwell-separated resonances of F1andF2, only one layer\nwill precess with large amplitude at a given frequency\n!, and spin current is pumped from a precessing F1into\na staticF2. If both layers precess symmetrically, with\na)Electronic mail: Contact author. web54@columbia.eduthe same amplitude and phase, equal and opposite spin\ncurrents are pumped into and out of each layer, causing\nno net e\u000bect on damping. The di\u000berence between the\nsymmetric mode and the uncoupled mode, increased by\na spin pumping damping \u000bspwas detected \frst in epi-\ntaxial Fe/Au/Fe structures9. However, if the magnetiza-\ntions can be excited with antisymmetric precession, the\ncoupled mode should be damped by twice that amount,\n2\u000bsp. Takahashi10has published an explicit prediction of\nthis \"giant spin pumping e\u000bect\" very recently, including\nan estimate of a fourfold enhanced spin accumulation in\nthe central layer.\nIn this paper, we show that a very large spin pump-\ning e\u000bect can be realized in antisymmetric precession of\nPy(8 nm)/Ru(0.70-1.25 nm)/Py(8 nm) synthetic antifer-\nromagnets (SAF, Py=Ni 79Fe21). The e\u000bect is roughly\ntwice that measured in Py trilayers with uncoupled\nprecession. Variable-frequency ferromagnetic resonance\n(FMR) measurements show, for structures with magne-\ntization saturated in the \flm plane or normal to the \flm\nplane, that symmetric (acoustic mode) precession of the\ntrilayer has almost no additional damping, but the an-\ntisymmetric (optical mode) precession has an additional\nGilbert damping of \u00180.006, compared with an uncou-\npled Py(8nm) layer in a F 1/N/F 2structure of\u00180.003.\nThe interaction stabilizes the antiparallel magnetization\nstate of SAF structures, used widely in di\u000berent elements\nof high-speed magnetic information storage, at GHz fre-\nquencies.\nMethod: Ta(5 nm)/Cu(3 nm)/Ni 79Fe21(8\nnm)/Ru(tRu)/Ni 79Fe21(8 nm)/Cu(3 nm)/SiO 2(5 nm),\ntRu= 0.7 - 1.2 nm heterostructures were deposited by\nultrahigh vacuum (UHV) sputtering at a base pressure\nof 5\u000210\u00009Torr on thermally oxidized Si substrates.\nThe Ru thckness range was centered about the second\nantiferromagnetic maximum of interlayer exchange\ncoupling (IEC) for Py/Ru/Py superlattices, 8-12 \u0017A,\nestablished \frst by Brillouin light scattering (BLS)\nmeasurement11. Oscillatory IEC in this system, as aarXiv:1603.07977v1  [cond-mat.mtrl-sci]  25 Mar 20162\nfunction of tRu, is identical to that in the more widely\nstudied Co/Ru( tRu)/Co superlattices12, 11.5 \u0017A, but is\nroughly antiphase to it. An in-plane magnetic \feld bias\nof 200 G, rotating in phase with the sample, was applied\nduring deposition as described in13.\nThe \flms were characterized using variable fre-\nquency, swept-\feld, magnetic-\feld modulated ferromag-\nnetic resonance (FMR). Transmission measurements\nwere recorded through a coplanar waveguide (CPW) with\ncenter conductor width of 300 \u0016m, with the \flms placed\ndirectly over the center conductor, using a microwave\ndiode signal locked in to magnetic \feld bias modulation.\nFMR measurements were recorded for magnetic \feld bias\nHBapplied both in the \flm plane (parallel condition, pc)\nand perpendicular to the plane (normal condition, nc.)\nAn azimuthal alignment step was important for the nc\nmeasurements. For these, the sample was rotated on twoaxes to maximize the \feld for resonance at 3 GHz.\nFor all FMR measurements, the sample magnetization\nwas saturated along the applied \feld direction, simplify-\ning extraction of Gilbert damping \u000b. The measurements\ndi\u000ber in this sense from low-frequency measurements of\nsimilar Py/Ru/Py trilayer structures by Belmenguenai et\nal14, or broadband measurements of (sti\u000ber) [Co/Cu] \u000210\nmultilayers by Tanaka et al15. In these studies, e\u000bects\non\u000bcould not be distinguished from those on inhomo-\ngeneous broadening.\nModel: In the measurements, we compare the mag-\nnitude of the damping, estimated by variable-frequency\nFMR linewidth through \u0001 H1=2= \u0001H0+ 2\u000b!=j\rj, and\nthe interlayer exchange coupling (IEC) measured through\nthe splitting of the resonances. Coupling terms between\nlayersiandjare introduced into the Landau-Lifshitz-\nGilbert equations for magnetization dynamics through\n_mi=\u0000mi\u0002(\riHe\u000b+!ex;imj) +\u000b0mi\u0002_mi+\u000bsp;i(mi\u0002_ mi\u0000mj\u0002_ mj) (1)\nincgsunits, where we de\fned magnetization unit vec-\ntors as m1=M1=Ms;1,m2=M2=Ms;2withMs;ithe\nsaturation moments of layer i. The coupling constants\nare, for the IEC term, !ex;i\u0011\riAex=(Ms;iti), where\nthe energy per unit area of the system can be written\nuA=\u0000Aexmi\u0001mj, andtiis the thickness of layer i. Pos-\nitive values of Aexcorrespond to ferromagnetic coupling,\nnegative values to antiferromagnetic coupling. The spin\npumping damping term is \u000bsp;i\u0011\r\u0016h~gFNF\n\"#=(4\u0019Ms;iti),\nwhere ~gFNF\n\"# is the spin mixing conductance of the tri-\nlayer in units of nm\u00002.\u000b0is the bulk damping for the\nlayer.\nThe collective modes of 1 ;2 are found from small-\namplitude solutions of Equations 1 for i= 1;2. General\nsolutions for resonance frequencies with arbitrary mag-\nnetization alignment, not cognizant of any spin pump-\ning damping or dynamic coupling, were developed by\nZhang et al12. In our experiment, to the extent possi-\nble, layers 1 ;2 are identical in deposited thickness, mag-\nnetization, and interface anisotropy (each with Cu the\nopposite side from Ru). Therefore if !irepresents the\nFMR frequency (dependent on bias \feld HB) of each\nlayeri, the two layers have !1=!2=!0. In this\nlimit, there are two collective modes: a perfectly sym-\nmetric mode Sand a perfectly antisymmetric mode A\nwith complex frequencies f!S= (1\u0000i\u000b0)!0andf!A=\n(1\u0000i\u000b0\u00002i\u000bsp) (!0+ 2!ex). The Gilbert damping for\nthe modes, \u000bk=\u0000Im(f!k)=Re(!k\nf), wherek= (S;A),\nand the resonance \felds Hk\nBsatisfy\nHA\nB=HS\nB+ 2HexHex=\u0000Aex=(MstF) (2)\n\u000bA=\u000bS+ 2\u000bsp\u000bsp=\r\u0016h~gFNF\n\"#=(4\u0019Ms;iti) (3)\nand!ex=\rHex. Note that there is no relationshipin this limit between the strength of the exchange cou-\nplingAexand the spin-pumping damping 2 \u000bspexpressed\nin the antisymmetric mode. The spin pumping damping\nand the interlayer exchange coupling can be read sim-\nply from the di\u000berences in the the Gilbert damping \u000b\nand resonance \felds between the antisymmetric and sym-\nmetric modes. The asymmetric mode will have a higher\ndamping by 2 \u000bspfor anyAexand a higher resonance\n\feld forAex<0, i.e. for antiferromagnetic IEC: because\nthe ground state of the magnetization is antiparallel at\nzero applied \feld, antisymmetric excitation rotates mag-\nnetizations towards the ground state and is lower in fre-\nquency than symmetric excitation.\nResults: Samplepc\u0000andnc\u0000FMR data are shown in\nFigure 1. Raw data traces (lock-in voltage) as a func-\ntion of applied bias \feld HBat 10 GHz are shown in the\ninset. We observe an intense resonance at low \feld and\nresonance weaker by a factor of 20-100 at higher \feld. On\nthe basis of the intensities, as well as supporting MOKE\nmeasurements, we assign the lower-\feld resonance to the\nsymmetric, or \"acoustic\" mode and the higher-\feld res-\nonance to the antisymmetric, or \"optical\" mode. Similar\nbehavior is seen in the nc- andpc\u0000FMR measurements.\nIn Figure 1a) and c), which summarizes the \felds-\nfor-resonance !(HB), there is a rigid shift of the\nantisymmetric-mode resonances to higher bias \felds HB,\nas predicted by the theory. The lines show \fts to\nthe Kittel resonance, !pc=\rr\nHeff\u0010\nHeff+ 4\u0019Meff\ns\u0011\n,\n!nc=\r\u0000\nHeff\u00004\u0019Meff\ns\u0001\nwith an additional e\u000bective\n\feld along the magnetization direction for the antisym-\nmetric mode: Heff;S =HB, andHeff;A =HB\u0000\n8\u0019Aex=(4\u0019MstF).\nIn Figure 2, we show coupling parameters, as a func-3\nHBHeffm(t)\nHBHeff m(t)(a)\n(c)(b)\n(d)10 GHz10 GHz\n1/21/2\nFIG. 1. FMR measurement of Ni 79Fe21(8\nnm)/Ru(tRu)/Ni 79Fe21(8 nm) trilayers; example shown\nfortRu= 1.2 nm. Inset : lock-in signal, transmitted power\nat 10 GHz, as a function of bias \feld HB, for a) pc-FMR\nand c) nc-FMR. A strong resonance is observed at lower HB\nand a weaker one at higher HB, attributed to the symmetric\n(S) and antisymmetric (A) modes, respectively. a), c): Field\nfor resonance !(HB) for the two resonances. Lines are \fts\nto the Kittel resonance expression, assuming an additional,\nconstant, positive \feld shift for !A,Hex=\u00002Aex=(MstF)\ndue to antiferromagnetic interlayer coupling Aex<0. b)\npc-FMR and d) nc-FMR linewidth as a function of frequency\n\u0001Hpp(!) for \fts to Gilbert damping \u000b.\ntion of Ru thickness, extracted from the FMR measure-\nments illustrated in Figure 1. Coupling \felds are mea-\nsured directly from the di\u000berence between the symmet-\nric and antisymmetric mode positions and plotted in\nFigure 2a. We convert the \feld shift to antiferromag-\nnetic IEC constant Aex<0 through Equation 2, us-\ning the thickness tF= 8 nm and bulk magnetization\n4\u0019Ms= 10.7 kG4. The extracted exchange coupling\nstrength in pc-FMR has a maximum antiferromagnetic\nvalue ofAex=\u00000.2 erg/cm2, which agrees to 5% with\nthat measured by Fassbender et al11for [Py/Ru] Nsu-\nperlattices.\nThe central result of the paper is shown in Figure 2 b).\nWe compare the damping \u000bof the symmetric ( S) and an-\ntisymmetric ( A) modes, measured both through pc-FMR\nandnc-FMR. The values measured in the two FMR ge-\nometries agree closely for the symmetric modes, for which\nsignals are larger and resolution is higher. They agree\nroughly within experimental error for the antisymmetric\nmodes, with no systematic di\u000berence. The antisymmetric\nmodes clearly have a higher damping than the symmetric\nmodes. Averaged over all thickness points, the enhanced\ndamping is roughly \u000bA\u0000\u000bS= 0.006.\nDiscussion: The damping enhancement of the anti-\nsymmetric ( A\u0000) mode over the symmetric ( S\u0000) mode,\nshown in Figure 2b), is a large e\u000bect. The value is close\nto the intrinsic bulk damping \u000b0\u00180.007 for Ni 79Fe21.\n0.7 0.8 0.9 1.0 1.1 1.2 1.301002003004005006002Hex (Oe)a)\nnc,Hexpc,HexMOKE,Hex\n0.7 0.8 0.9 1.0 1.1 1.2 1.3\ntRu (nm)0.0060.0080.0100.0120.0140.0160.018α\nα0α0+αspα0+2αspb)pc, S\nnc, Spc, A\nnc, A0.000.050.100.150.20\n−Aex (erg/cm2)FIG. 2. Coupling parameters for Py/Ru/Py trilayers. a):\nInterlayer (static) coupling from resonance \feld shift of an-\ntisymmetric mode; see Fig 1 a),c). The antiferromagnetic\nexchange parameter Aexis extracted through Eq 2, in agree-\nment with values found in Ref11. The line is a guide to the\neye. b) Spin pumping (dynamic) coupling from damping of\nthe symmetric (S) and antisymmetric (A) modes; see Fig 1\nb), d). The spin pumping damping for uncoupled layer pre-\ncession in Py/Ru/Py, \u000bspis shown for comparsion. Dotted\nlines show the possible e\u000bect of \u0018100 Oe detuning for the two\nPy layers. See text for details.\nWe compare the value with the value 2 \u000bspexpected from\ntheory for the antisymmetric mode and written in Eq 3.\nThe interfacial spin mixing conductance for Ni 79Fe21/Ru,\nwas found in Ref.16to be ~gFN\n\"#= 24 nm\u00002. For a F/N/F\nstructure, in the limit of ballistic transport with no spin\nrelaxation through N, the e\u000bective spin mixing conduc-\ntance is ~gFN\n\"#=2: spin current must cross two interfaces\nto relax in the opposite Flayer, and the conductance re-\n\rects two series resistances17. This yields \u000bsp= 0:0027.\nThe observed enhancement matches well with, and per-\nhaps exceeds slightly, the \"giant\" spin pumping e\u000bect of\n2\u000bsp, as shown.\nLittle dependence of the Gilbert damping enhancement\n\u000bA\u0000\u000bSon the resonance \feld shift HA\u0000HScan be ob-\nserved in Figure 2 a,b. We believe that this independence\nre\rects close tuning of the resonance frequencies for Py\nlayers 1 and 2, as designed in the depositions. For \f-\nnite detuning \u0001 !de\fned through !2=!0+ \u0001!and\n!1=!0\u0000\u0001!, the modes change. Symmetric and anti-\nsymmetric modes become hybridized as S0andA0, and4\nthe di\u000berence in damping is reduced. De\fning g\u0001!2=\n(1\u0000i\u000b0\n) (1\u0000i\u000b0\n\u00002i\u000bsp\n) \u0001!2, it is straightfor-\nward to show that for the nc-case, the mode frequen-\ncies are!S0;A0= (f!S+f!A)=2\u0006q\n(f!S\u0000f!A)2=4 +g\u0001!2.\nThe relevant parameter is the frequency detuning nor-\nmalized to the exchange (coupling) frequency, z\u0011\n\u0001!=(2!ex); ifz\u001d1, the layers have well-separated\nmodes, and each recovers the uncoupled damping en-\nhancement of \u000bsp,\u000bS0;A0=\u000b0+\u000bspidenti\fed in Refs5,9.\nThe possibility of \fnite detuning, assuming ( !2\u0000\n!1)=\r= 100 Oe, is shown in Figure 2b), with the dot-\nted lines. The small \u0000zlimit for detuning \fnds sym-\nmetric e\u000bects on damping of the S0andA0modes, with\n\u000bS0=\u000b0+ 2\u000bspz2and\u000bA0=\u000b0+ 2\u000bsp(1\u0000z2), respec-\ntively, recovering perfect symmetric and antisymmetric\nmode values for z= 0. We assume that the \feld split-\nting shown in Figure 2 a) gives an accurate measure of\n2!ex=\r, as supported by the MOKE results. This value\ngoes into the denominator of z. We \fnd a reasonable \ft\nto the dependence of SandAdamping on Ru thickness,\nimplicit in the coupling. For the highest coupling pionts,\nthe damping values closely reach the low- zlimit, and we\nbelieve that the \"giant\" spin pumping result of 2 \u000bspis\nevident here.\nWe would like to point out next that it was not a-\npriori obvious that the Py/Ru/Py SAF would exhibit\nthe observed damping. Ru could behave in two limits in\nthe context of spin pumping: either as a passive spin-\nsink layer, or as a ballistic transmission layer supporting\ntransverse spin-current transmission from one Py layer to\nthe other. Our results show that Ru behaves as the latter\nin this thickness range. The symmetric-mode damping of\nthe SAF structure, extrapolated back to zero Ru thick-\nness, is identical within experimental resolution ( \u001810\u00004)\nto that of a single Py \flm 16 nm thick measured in nc-\nFMR (\"\u000b0\" line in Fig 2b.) If Ru, or the Py/Ru interface,\ndepolarized pumped spin current very strongly over this\nthickness range as has been proposed for Pt18, we would\nexpect an immediate increase in damping of the acous-\ntic mode by the amount of \u0018\u000bsp. Instead, the volume-\ndependent Ru depolarization in spin pumping has an (ex-\nponential) characteristic length of \u0015SD\u001810 nm5, and\nattenuation over the range explored of \u00181 nm is negli-\ngible.\nPerspectives: Finally, we would like to highlight some\nimplications of the study. First, as the study con\frms\nthe prediction of a \"giant\" spin pumping e\u000bect as pro-\nposed by Takahashi10, it is plausible that the greatly en-\nhanced values of spin accumulation predicted there may\nbe supported by Ru in Py/Ru/Py synthetic antiferro-\nmagnets (SAFs). These spin accumulations would di\u000ber\nstrongly in the excitation of symmetric and antisymmet-\nric modes, and may then provide a clear signature intime-resolved x-ray magnetooptical techniques19, similar\nto the observation of static spin accumulation in Cu re-\nported recently20.\nSecond, in most device applications of synthetic an-\ntiferromagnets, it is not desirable to excite the antisym-\nmetric (optical) mode. SAFs are used in the pinned layer\nof MTJ/spin valve structures to increase exchange bias\nand in the free layer to decrease (magnetostatic) stray\n\felds. Both of these functions are degraded if the opti-\ncal, or asymmetric mode of the SAF is excited. Accord-\ning to our results, at GHz frequencies near FMR, the\nsusceptibility of the antisymmetric mode is reduced sub-\nstantially, here by a factor of two (from 1/ \u000b) for nc-FMR ,\ndue to spin pumping. This reduction of \u001fon resonance\nwill scale inversely with layer thickness. The damping,\nand susceptibility, of the desired symmetric (acoustic)\nmode is unchanged, on the other hand, implying that\nspin pumping favors the excitation the symmetric mode\nfor thin Ru, the typical operating point.\nWe acknowledge NSF-DMR-1411160 for support.\n1Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev.\nLett. 88, 117601 (2002).\n2Y. Tserkovnyak, A. Brataas, G. Bauer, and B. 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Kukreja, S. Bonetti, Z. Chen, D. Backes, Y. Acremann, J. A.\nKatine, A. D. Kent, H. A. D urr, H. Ohldag, and J. St ohr, Phys.\nRev. Lett. 115, 096601 (2015)."    },    {        "title": "1907.02734v1.Theory_for_shift_current_of_bosons__Photogalvanic_spin_current_in_ferrimagnetic_and_antiferromagnetic_insulators.pdf",        "content": "Theory for shift current of bosons: Photogalvanic spin current\nin ferrimagnetic and antiferromagnetic insulators\nHiroaki Ishizuka1and Masahiro Sato2\n1Department of Applied Physics, The University of Tokyo, Bunkyo, Tokyo, 113-8656, JAPAN\n2Department of Physics, Ibaraki University, Mito, Ibaraki, 310-8512, JAPAN\n(Dated: July 8, 2019)\nWe theoretically study the optical generation of dc spin current (i.e., a spin-current solar cell) in\nordered antiferromagnetic and ferrimagnetic insulators, motivated by a recent study on the laser-\ndriven spinon spin current in noncentrosymmetric quantum spin chains [H. Ishizuka and M. Sato,\nPhys. Rev. Lett. 122, 197702 (2019)]. Using a non-linear response theory for magnons, we analyze\nthe dc spin current generated by a linearly-polarized electromagnetic wave (typically, terahertz or\ngigahertz waves). Considering noncentrosymmetric two-sublattice magnets as an example, we \fnd\na \fnite dc spin current conductivity at T= 0, where no thermally-excited magnons exist; this is in\ncontrast to the case of the spinon spin current, in which the optical transition of the Fermi degenerate\nspinons plays an essential role. We \fnd that the dc spin-current conductivity is insensitive to the\nGilbert damping, i.e., it may be viewed as a shift current carried by bosonic particles (magnons).\nOur estimate shows that an electric-\feld intensity of E\u0018104\u0000106V/cm is su\u000ecient for an\nobservable spin current. Our theory indicates that the linearly-polarized electromagnetic wave\ngenerally produces a dc spin current in noncentrosymmetric magnetic insulators.\nI. INTRODUCTION\nMaterials subject to an intense incident light shows\nrich behaviors which are studied in the context of non-\nlinear response and non-equilibrium phenomena. An ex-\nample of such is electric shift current in noncentrosym-\nmetric semiconductors and ferroelectrics [1{7], where a\nnon-trivial shift of electron position during its optical\ntransition produces a macroscopic electric current. Re-\ncent studies revealed that the shift current exhibits strik-\ningly di\u000berent behaviors from the ordinary photocurrent;\nthe shift current shows unique light-position dependence\nwhen it is excited locally [8{10], and propagates faster\nthan the Fermi velocity of electrons [10{12]. On the other\nhand, in correlated materials, lower-energy excitations of-\nten emerge due to the interaction e\u000bect; a typical exam-\nple is magnetic excitations in Mott insulators. The op-\ntical transition of these emergent particles may produce\nnon-trivial phenomena, especially, transport phenomena,\nrelated to the nonlinear response of the emergent excita-\ntions.\nSeveral recent studies in opto-spintronics and magneto-\noptics [13{15] implies that the intensity and coherence of\ncurrently-available electromagnetic waves are su\u000ecient\nfor the control of magnetic excitations or magnetism.\nTypical results are the following: Magnetization switch-\ning by a circularly-polarized laser in ferrimagnets [16{19],\nlaser-driven demagnetization [20{22], the spin pumping\nby gigahertz (GHz) or terahertz (THz) waves [23, 24],\nfocused-laser driven magnon propagation [25, 26], intense\nTHz-laser driven magnetic resonance [27, 28], spin con-\ntrol by THz-laser driven electron transitions [29], dichro-\nisms driven by THz vortex beams [30], angular mo-\nmentum transfer between photons and magnons in cav-\nities [31{35], a ultrafast detection of spin Seebeck ef-\nfect [36], a phonon-mediated spin dynamics with THzlaser [37], etc. Moreover, recent theoretical works have\nproposed several ways of optical control of magnetism:\nTHz-wave driven inverse Faraday e\u000bect [38, 39], Floquet\nengineering of magnetic states such as chirality ordered\nstates [40, 41] and a spin liquid state [42], generation\nof magnetic defects with laser-driven heat [43, 44], ap-\nplications of topological light waves to magnetism [44{\n47], control of exchange couplings in Mott insulators\nwith high- [48] and low-frequency [49] waves, optical\ncontrol of spin chirality in multiferroic materials [50],\nrecti\fcation of dc spin currents in magnetic insulators\nwith electromagnetic waves [51{53]. These studies are\npartly supported by recent developments in THz laser\nscience [54, 55] which realized high-intensity light beams\nwith the photon energy comparable to those of magnetic\nexcitations. Despite these developments, the optical con-\ntrol of the current carried by magnetic excitations is lim-\nited to some theoretical proposals.\nAmong the proposals, a recent theory proposes a mech-\nanism for producing a dc spin current in quantum spin\nchains without the angular-momentum transfer [52]; it\nis distinct from the known mechanisms in which the an-\ngular momentum of photons are transferred to the mag-\nnet [23, 24, 51, 53, 56]. The mechanism in Ref. 52 is anal-\nogous to that of the shift-current photovoltaic e\u000bect [2].\nThe close relation between two phenomena are clear from\nthe Jordan-Wigner fermion representation of spin chain;\nthe ground state of the spin chain is a band insulator of\nJordan-Wigner fermions, and the photovoltaic response\nis related to the optical transition of the fermions by the\nlinearly-polarized THz light. However, the relation of\nthis mechanism to the fermion excitations casts doubt\non the generality because the low-energy excitations of\nthe ordered magnets are usually magnons, i.e., bosonic\nexcitations.\nIn this work, we theoretically show that a dc spin cur-\nrent similar to that of the spin chain [52] also appearsarXiv:1907.02734v1  [cond-mat.mes-hall]  5 Jul 20192\nin ordered antiferromagnetic (AFM) and ferrimagnetic\n(FRM) insulators by applying a linearly-polarized elec-\ntromagnetic wave. The symmetry argument in Sec. III\nshows that the creation of dc spin current with linearly-\npolarized waves is possible only if both site- and bond-\ncenter inversion symmetries are broken. AFM and FRM\ninsulators violate the bond-center inversion symmetry\nand thereby they naturally satisfy half of the required\nsymmetry condition. The staggered moment is an ad-\nvantage of considering AFM/FRM insulators for gen-\nerating a dc spin current. As an example, we con-\nsider two-sublattice models with N\u0013 eel type ground state.\nBosonic particles describe the low-energy excitations of\nthese models, i.e., magnons; the ground state is the zero-\nmagnon state. This ground state is very di\u000berent from\nthat of noncentrosymmetric S= 1=2 spin chains [52]\nwhich are described by a Fermi degenerated state of\nspinons. Despite the di\u000berence, our calculation using\na nonlinear response theory \fnds a \fnite photovoltaic\nspin current similar to that of the spinons. We dis-\ncuss that it is related to the zero-point \ructuation of\nthe quantum magnets. Our theory also indicates that\nthe magnon spin current is shift-current like, i.e., it is\ninsensitive to the magnon lifetime as in the spinon case.\nThis mechanism allows generation of spin current using\na linearly-polarized electromagnetic wave and ordinary\nAFM or FRM insulators.\nThe remaining part of the paper is organized as fol-\nlows. In Sec. II, we introduce the nonlinear-response\ntheory for two-species magnons, which we will use in the\nfollowing sections. The main results of this paper are in\nSecs. III and IV. Section III focuses on the photo-induced\nspin current in AFM and FRM insulators with a strong\none dimensionality, while we study the three-dimensional\n(3D) magnets in Sec. IV. E\u000bective experimental setups\nand signatures for investigating the proposed mechanism\nare discussed in Sec. V. Section VI is devoted to the sum-\nmary and discussions.II. NONLINEAR RESPONSE THEORY\nWe calculate the nonlinear response coe\u000ecients for\nthe photo-induced spin current by extending the linear-\nresponse theory to the quadratic order in the perturba-\ntion. A similar method for fermions is used to calculate\nthe photovoltaic current in semiconductors [57, 58] and\nthe spin current of spinons [52]. The derivation of the\nformula is summarized in Appendix A. We here summa-\nrize the outline of the derivation. We also discuss the\nphysical implications.\nWe consider a two-sublattice AFM/FRM insulator\nwith two species of magnons. The e\u000bective Hamiltonian\nfor the magnons is\nH=X\nk\"\u000b(k)\u000by\nk\u000bk+\"\f(k)\fk\fy\nk; (1)\nwhere\u000bk(\u000by\nk) and\fk(\fy\nk) are the boson annihila-\ntors (creators) for the magnons with the momentum\nk= (kx;ky;kz) and\"a(k) (a=\u000b;\f) is the energy\nof the magnons in the a=\u000b;\f branch with momen-\ntumk. We here consider a general perturbation (spin-\nelectromagnetic-wave coupling)\nH0=\u0000X\n\u0016;kZd!\n2\u0019h\u0016\n!ei!t y\nk\u0012\n(B\u0016\nk)\u000b\u000b(B\u0016\nk)\u000b\f\n(B\u0016\nk)\f\u000b(B\u0016\nk)\f\f\u0013\n k\n+ h.c.; (2)\nand spin-current operator\nJ=X\nk y\nk\u0012\n(Jk)\u000b\u000b(Jk)\u000b\f\n(Jk)\f\u000b(Jk)\f\f\u0013\n k: (3)\nHere,!is the frequency of ac light, h\u0016\n!is the spin-\nlight coupling constant for the \u0016direction, and  k=\n(\u000bk;\fy\n\u0000k)T.\nThe nonlinear conductivity is de\fned by\nhJi(\n) =X\n\u0016;\u0017Z\nd!\u001b\u0016\u0017(\n;!;\n\u0000!)h\u0016\n!h\u0017\n\n\u0000!;(4)\nwherehJi(\n)\u0011R\ndthJi(t)e\u0000i\ntis the Fourier transform\nof the expectation value of the spin current hJi(t). For\nthe two-sublattice model, the formula for nonlinear spin\ncurrent conductivity reads\n\u001b\u0016\u0017(\n;!;\n\u0000!) =1\n2\u0019X\nk;ai=\u000b;\fsgn(a3)(~\u001ak;a1sgn(a2)\u0000sgn(a1)~\u001ak;a2)(B\u0016\nk)a1a2\n!\u0000~\"a2(k) + ~\"a1(k)\u0000i=(2\u001ck)\n\u0002\u0014(B\u0017\nk)a2a3(Jk)a3a1\n\n + ~\"a1(k)\u0000~\"a3(k)\u0000i=(2\u001ck)\u0000(Jk)a2a3(B\u0017\nk)a3a1\n\n + ~\"a3(k)\u0000~\"a2(k)\u0000i=(2\u001ck)\u0015\n; (5)\nwhere\n~\"a(k) =sgn(a)\"a(k); (6)\nsgn(a) =\u001a\n1 (a=\u000b)\n\u00001 (a=\f); (7)\n~\u001ak;a=\u001ah\u000by\nk\u000bki0(a=\u000b)\nh\f\u0000k\fy\n\u0000ki0(a=\f): (8)The relaxation time of magnons, \u001ck, was introduced in3\n(a) (b)\nω ω\n001234\n0\nE\nπ −π −π π(c) (d)\nk kαk, βkαk+ β −k\nαkαk+ β −k\nβk\nω\nFIG. 1. (Color online) Schematic pictures of the noncen-\ntrosymmetric magnets. A quasi-one-dimensional magnet\nconsisting of weakly-coupled spin chains (a) and a three-\ndimensional magnet with two-sublattice order (b). Each sub-\nlattice (blue and orange) has a di\u000berent environment, e.g., dif-\nferentgfactors, uniaxial anisotropy, etc., and with the bond\ndimerization (shown by the thick bond). The two-sublattice\norder and bond dimerization respectively breaks the inversion\nsymmetry on the bond center and sites. Magnetic excitation\nand nonlinear spin current conductivity of the spin chain. The\nmagnon band dispersions of the model in Eq. (11a) for (c)\nh+= 0 and (d) h+= 1=100. Parameters h\u0006are de\fned in\nEq. (17). When h+= 0, two magnon dispersions are degen-\nerate. The THz light produces two magnons, one on each\nbranch as schematically shown in panel (d).\nEq. (5), andh\u0001\u0001\u0001i 0is the expectation value of \u0001\u0001\u0001in the\nequilibrium state of the Hamiltonian in Eq. (1). The\nconductivity for dc spin current corresponds to the \n = 0\ncase,\u001b\u0016\u0017(0;!;\u0000!). In the rest of this work, we focus\non the case B\u0016\nk=B\u0017\nk=Bkbecause we are interested\nin the response to a linearly polarized light. Hence, we\nabbreviate the subscripts in the nonlinear conductivity,\n\u001b\u0016\u0017(0;!;\u0000!) =\u001b(0;!;\u0000!).\nWe note that the conductivity in Eq. (5) remains non-\nzero atT= 0. The substitutions of ~ \u001ak;\u000b= 0 and ~\u001ak;\f= 1\nin Eq. (5) reduce the formula to\n\u001b(0;!;\u0000!) =X\nk\n\u00001\n\u0019\u0014(1 +i2\u001ck!)j(Bk)\f\u000bj2((Ak)\u000b\u000b+ (Ak)\f\f)\n(!\u0000i=2\u001ck)2\u0000(\"\u000b(k) +\"\f(k))2\u0015\n+1\n2\u0019(Bk)\f\u000b(Ak)\u000b\f((Bk)\f\f+ (Bk)\u000b\u000b)\n(!\u0000i=2\u001ck\u0000\"\u000b(k)\u0000\"\f(k))(\"\u000b(k) +\"\f(k) +i=2\u001ck)\n+1\n2\u0019(Bk)\u000b\f(Ak)\f\u000b((Bk)\f\f+ (Bk)\u000b\u000b)\n(!\u0000i=2\u001ck+\"\u000b(k) +\"\f(k))(\"\u000b(k) +\"\f(k)\u0000i=2\u001ck):\n(9)Because of ~ \u001ak;\f= 1, the terms involving the o\u000b-diagonal\ncomponent of Bkremains at T= 0. In other words, the\ntwo-magnon creation/annihilation process plays a crucial\nrole as shown in Fig. 1(d). We focus on the T= 0 case in\nthe rest of this paper as this process is dominant in the\nlow temperature limit.\nFrom a di\u000berent viewpoint, Eq. (9) implies the zero-\npoint \ructuation plays a key role in the photovoltaic\nresponse of magnons. In our formalism, the zero-point\n\ructuation is manifested in the Bogoliubov transforma-\ntion of Holstein-Primakov bosons. This transformation\ncreates\fk\fy\nkand\u000by\nk\fy\n\u0000kterms which contribute to the\nphotovoltaic response in the ground state. ~ \u001ak;\f= 1 is\nanother consequence of the Bogoliubov transformation.\nThe importance of the zero-point \ructuation resembles\nthe spinon spin current [52], in which the Fermi degen-\neracy of spinons represents the quantum \ructuation of\nspins. A crucial di\u000berence in the current case is the\nabsence of Fermi degeneracy. However, in the case of\nthe AFMs/FRMs, the condensate of Holstein-Primakov\nbosons plays a similar role to the Fermi degeneracy. The\npair-creation process represented by \u000by\nk\fy\nkgenerates pho-\ntovoltaic response of the magnons which is manifested\nin the denominator of Eq. (9); the sum of eigenener-\ngies,\"\u000b(k) +\"\f(k), represents creation/annihilation of a\nmagnon pair. These features implies that the zero-point\n\ructuation is necessary for the shift current response at\nT= 0.\nThe \frst term in Eq. (9) vanishes when the ground\nstate has a certain symmetry. For example, collinear\nmagnetic orders with the moments parallel to Szaxis\nare often symmetric with respect to G=TMs\nx, which is\nthe product of time-reversal operation ( T) and the mir-\nror operation for the spin degrees of freedom about xaxis\n(Ms\nx). In this case, the real part of \u001b(0;!;\u0000!) reads\nRe [\u001b(0;!;\u0000!)] =\n\u00001\n\u0019X\nkRe\u001a(Bk)\f\u000b(Ak)\u000b\f((Bk)\f\f+ (Bk)\u000b\u000b)\n!2\u0000(\"\u000b(k) +\"\f(k) +i=2\u001ck)2\u001b\n:\n(10)\nThe conductivities for the models considered in the fol-\nlowing sections are calculated using this formula.\nIII. SPATIALLY-ANISOTROPIC MAGNET\nIn this section, we apply the above formula to a spin\nchain with AFM or FRM order, which corresponds to\na quasi-one-dimensional (quasi-1D) magnetic compound\nwith a negligible inter-chain interaction. The spins are\ncoupled to the electromagnetic wave through the Zee-\nman coupling. To make the problem theoretically well-\nde\fned, we consider a model which conserves the spin\nangular momentum Sz; the model has an easy axis and\nthe applied ac magnetic \feld is parallel to the ordered\nmoments. The conservation of Szallows us to unam-\nbiguously de\fne the spin current operator from the con-4\ntinuity equation. This setup is in contrast to those of\nusual magnetic resonances and spin pumping [23, 24], in\nwhich the ac \feld is perpendicular to the magnetic mo-\nment. We use the standard spin-wave approximation to\ndescribe magnetic excitations (magnons).\nA. Model\nWe consider an ordered noncentrosymmetric spin chain\nwith a two-sublattice unit cell [Fig. 1(a)], whose Hamil-\ntonian is given by\nHtot=H0+H(!)\nZ; (11a)\nH0\u0011X\nry;rzH1D(ry;rz); (11b)\nH1D(ry;rz)\u0011X\nrxJ(1 +\u000e)SA(r)\u0001SB(r)\n+J(1\u0000\u000e)SA(r+ ^x)\u0001SB(r)\n\u0000(D+Ds) [Sz\nA(r)]2\u0000(D\u0000Ds) [Sz\nB(r)]2\n\u0000h[gASz\nA(r) +gBSz\nB(r)]; (11c)\nH(!)\nZ=\u0000(h!ei!t+ h.c.)X\nrgASz\nA(r) +gBSz\nB(r);\n(11d)\nwhereH1Dis the spin-chain Hamiltonian with the stag-\ngered nearest-neighbor exchange interaction (i.e., dimer-\nization) along the xdirection,H0is the bundle of all\nthe chains, and H(!)\nZis the Zeeman coupling between\nthe spins and the external electromagnetic wave. Here,\nSa(r)\u0011(Sx\na(r);Sy\na(r);Sz\na(r)) (a=A;B) is the spin- Sa\noperator on the asublattice of the unit cell at position\nr= (rx;ry;rz). Symbols ^ x, ^y, and ^zstand for the unit\nvectors along the x,y, andzdirections, respectively. The\nparameters in the Hamiltonian H1Dare as follows: J >0\nis the antiferromagnetic exchange interaction along the\nspin-chain ( x) direction, \u000eis the dimerization, D > 0\n(Ds) is the uniform easy-axis (staggered) anisotropy, gA\n(gB) is thegfactor for the spins on A(B) sublattice,\nandhis the external static magnetic \feld along the Sz\naxis. In the spin-light coupling H(!)\nZ,jh!jand arg(h!)\nare respectively the magnitude and the phase of the ac\nmagnetic \feld of the linearly-polarized electromagnetic\nwave. We assume jDsj<D, andj\u000ej<1.\nWhenSA6=SB, the ground state of the model in\nEq. (11c) is a FRM-ordered state with magnetization\njSA\u0000SBjper a unit cell [59, 60]. The ground state is a\nN\u0013 eel ordered when SA=SB. The classical ground state\nofH0has a collinear order with spins pointing along the\nSzaxis because of the easy axis anisotropy D[Fig. 1(a)].\nThe anisotropy also produces the spin gap in the excita-\ntion spectrum [Fig. 2(a) and 1(b)]. We discuss the e\u000bect\nof the gap and its relation to the frequency dependence\nof the nonlinear spin conductivity in the next section.\nHere we de\fne the spin current for Sz. Since\nthe model H0conserves the zcomponent of to-tal spin angular momentum, the spin current for\nSzcan be de\fned from the continuity equations\n@tSz\nA=Jz\nx(rx\u00001;B;rx;A)\u0000Jz\nz(rx;A;rx;B) and\n@tSz\nB=Jz\nx(rx;A;rx;B)\u0000Jz\nz(rx;B;rx+ 1;A), in which\nJ\u000b\n\f(r;a;r0;b) is the local spin- S\u000bcurrent operator be-\ntween two neighboring sites ( r;a) and (r0;b) and it \rows\nalong the\fdirection. The above continuity equation is\nobtained from Heisenberg equation of motion for local\nspins. With these procedures, we \fnd the uniform cur-\nrent operator for H1Dreads\nJz\nx=J\n2NX\nr(1 +\u000e)fSx\nB(r)Sy\nA(r)\u0000Sy\nB(r)Sx\nA(r)g\n+ (1\u0000\u000e)fSx\nA(r+ ^x)Sy\nB(r)\u0000Sy\nA(r+ ^x)Sx\nB(r)g;\n(12)\nwhereNis the total number of unit cells.\nB. Linear spin-wave approximation\nHereafter, we assume that in the ground state of H1D,\nthe spins on the Asublattice points up while those on\nBsublattice are down [see Fig. 1 (a)]. The low-energy\nexcitations of H0is calculated by linear spin-wave ap-\nproximation. Using the Holstein-Primakov bosons, the\nspin operators are given by\nSz\nA=SA\u0000^nA(r); (13a)\nS+\nA(r) =p\n2SA\u0012\n1\u0000^nA(r)\n2SA\u00131\n2\na(r); (13b)\nS\u0000\nA=p\n2SAay(r)\u0012\n1\u0000^nA(r)\n2SA\u00131\n2\n; (13c)\nfor theAsublattice and\nSz\nB=^nB(r)\u0000SB; (14a)\nS+\nB(r) =p\n2SBby(r)\u0012\n1\u0000^nB(r)\n2SB\u00131\n2\n; (14b)\nS\u0000\nB=p\n2SB\u0012\n1\u0000^nB(r)\n2SB\u00131\n2\nb(r); (14c)\nfor theBsublattice. Up to the linear order in SAand\nSB,H0reads\nH0\u0018X\nk\u0012ak\nby\n\u0000k\u0013y\u0012\nh0\nk+hz\nkhx\nk\u0000ihy\nk\nhx\nk+ ihy\nkh0\nk\u0000hz\nk\u0013\u0012ak\nby\n\u0000k\u0013\n+ const. (15)\nwhere the wave number along the chain ( x) direction\nis simply represented by k,ak\u0011(1=p\nN)P\nra(r)eik\u0001r,\nbk\u0011(1=p\nN)P\nrb(r)eik\u0001(r+^x=2)are the Fourier trans-\nformation of Holstein-Primakov bosons. The matrix ele-\nments of the magnon Hamiltonian (15) are calculated as5\nh0\nk=h++J(SA+SB); (16a)\nhx\nk=2Jp\nSASBcos(k=2); (16b)\nhy\nk=\u00002J\u000ep\nSASBsin(k=2); (16c)\nhz\nk=h\u0000\u0000J(SA\u0000SB); (16d)\nwhere\nh+=D(SA+SB\u00001) +Ds(SA\u0000SB) +h\n2(gA\u0000gB);\n(17a)\nh\u0000=D(SA\u0000SB) +Ds(SA+SB\u00001) +h\n2(gA+gB):\n(17b)\nThe quadratic Hamiltonian (15) is diagonalized by the\nBogoliubov transformation:\nak= cosh \u0002 k\u000bk+ sinh \u0002 k\fy\n\u0000k; (18)\nby\n\u0000k= sinh \u0002 kei\bk\u000bk+ cosh \u0002 kei\bk\fy\n\u0000k; (19)\nwhere\u000bk(\u000by\nk) and\fk(\fy\nk) are bosonic annihilation (cre-\nation) operators. By choosing\nei\bk=hx\nk+ ihy\nkp\n(hx\nk)2+ (hy\nk)2(20a)\nand\ncosh(2\u0002 k) =h0\nkp\n(h0\nk)2\u0000(hx\nk)2\u0000(hy\nk)2; (20b)\nsinh(2\u0002 k) =\u0000p\n(hx\nk)2+ (hy\nk)2\np\n(h0\nk)2\u0000(hx\nk)2\u0000(hy\nk)2; (20c)the Hamiltonian becomes\nH0=X\nk\"\u000b(k)\u000by\nk\u000bk+\"\f(k)\fy\n\u0000k\f\u0000k; (21)\nwhere\n\"\u000b(k) =hz\nk+q\n(h0\nk)2\u0000(hx\nk)2\u0000(hy\nk)2; (22a)\n\"\f(k) =\u0000hz\nk+q\n(h0\nk)2\u0000(hx\nk)2\u0000(hy\nk)2: (22b)\nHere, we ignored the constant term in H0. We note that\nthe dispersions \"\u000b;\f(k) and the phases (\u0002 k;\bk) are all\nindependent of kyandkzbecause we now consider the\n1D modelH0. Using the same transformation, we \fnd\nH(!)\nZ=hX\nk(gAcosh2\u0002k\u0000gBsinh2\u0002k)\u000by\nk\u000bk\n+ (gAsinh2\u0002k\u0000gBcosh2\u0002k)\f\u0000k\fy\n\u0000k\n+gA\u0000gB\n2sinh(2\u0002 k)(\u000by\nk\fy\n\u0000k+\f\u0000k\u000bk)\n+gB+hN(gBSB\u0000gASA): (23)\nand\nJz\nx=Jp\nSASBX\nksinh(2\u0002 k)\u0012\nsink\n2cos \b k+\u000ecosk\n2sin \b k\u0013\u0010\n\u000by\nk\u000bk+\f\u0000k\fy\n\u0000k\u0011\n+\u0014\u001a\ncosh(2\u0002 k)\u0012\ncos \b ksink\n2\u0000\u000esin \b kcosk\n2\u0013\n+ i\u0012\nsin \b ksink\n2+\u000ecos \b kcosk\n2\u0013\u001b\n\u000by\nk\fy\n\u0000k+ h.c.\u0015\n:(24)\nC. Spin current conductivity\nCombining the magnon representation of ( \u000bk;\fk) with the formula (10), we compute the nonlinear dc spin-current\nconductivity for the model Htotunder the application of THz laser. We \frst study the nonlinear conductivity in the\nclean limit with in\fnite relaxation time \u001ck!1 . The analytic solution for the conductivity Re [ \u001b(0;!;\u0000!)] obtained\nfrom Eq. (10) reads\nRe [\u001b(0;!;\u0000!)] =(gA\u0000gB)2\u000e(h++J(SA+SB))\u0000\n!2\u00004(h++J(SA+SB))2\u00002J2SASB(1 +\u000e2)\u0001\n8\u0019(1\u0000\u000e2)!2p\n4J4S2\nAS2\nB(1\u0000\u000e2)2\u0000f(!=4)2+ 2J2SASB(1 +\u000e2)\u0000(h++J(SASB))2g2;(25)\nwhen!2[!c1;!c2] and zero otherwise. Here,\n!c1\u0011\"\u000b(0) +\"\f(0)\n=2p\n(h++J(SA+SB))2\u00004J2SASB; (26)corresponds to the energy for the band bottom of the6\n(a)\nσ(0;ω,−ω) (a.u.) \nω/J(b)\n-0.15-0.10-0.050.000.05\n10-410-310-210-1100101\n 0  1  2  3  4σ(ω)\nα=5x10-3\nα=1x10-2\nα=1x10-1\n0.01 0.10 1\n0.0010.0100.1001J┴=0\nJ┴=1/4\nJ┴=1δω|σ(0;ω,−ω)| (a.u.)\nFIG. 2. (Color online) Frequency dependence of the non-\nlinear spin current conductivity \u001b(0;!;\u0000!). (a) Analytic re-\nsult for the small Gilbert damping limit \u000b!0 and (b) numer-\nical results for a \fnite \u000b. The inset in (a) is the \u000e!\u0011!\u0000!c1\nfor di\u000berent J?. The calculations are done using a chain with\nN= 2048\u000032768 unit cells. All results are for J= 1,\u000e= 1=4,\nSA=SB= 1,gA= 1,gB= 1=2,h\u0000= 0, andh+= 1=100\nunless noted explicitly.\npair excitation and\n!c2\u0011\"\u000b(\u0019) +\"\f(\u0019)\n=2p\n(h++J(SA+SB))2\u00004\u000e2J2SASB;(27)\nis that for the top of the pair excitation [See Fig. 1(c)\nand (d)]. The frequency dependence of the conductivity\nis shown in Fig. 2(a).\nEquation (25) is an odd function of \u000e. This re\rects\nthe fact that the inversion-symmetry breaking is neces-\nsary for the spin current. H0has two inversion centers\nwhen\u000e= 0,Ds= 0,gA=gB, andSA=SB: one at\nthe center of the bond and the another on the site. The\ninversion center on the site is broken by the dimerization\n\u000e. To see the dependence of \u001b(0;!;\u0000!) on the model pa-\nrameters, we explicitly write down the nonlinear conduc-\ntivity as a function of the parameters, i.e., \u001b(0;!;\u0000!) =\n\u001b(!;\u000e;gA\u0000gB;Ds;m), wherem=hSz\nr2Ai\u0000hSz\nr2Biis\nthe order parameter of the AFM or FRM insulators. A\nsymmetry argument on the transport coe\u000ecient \fnds\n\u001b(!;\u000e;gA\u0000gB;Ds;m) =\u0000\u001b(!;\u0000\u000e;gA\u0000gB;Ds;m) for\nthe site-center inversion operation. This result is identi-\ncal to the spinon case in Ref. [52].\nOn the other hand, the magnetic order changes the\nparameter dependence of \u001b(0;!;\u0000!), which is related\nto the bond-center inversion operation. The inversionoperation about the center of the bonds is broken by\nthe N\u0013 eel ordering, Ds6= 0 orgA6=gB. Therefore, the\nsymmetry operation indicates \u001b(!;\u000e;gA\u0000gB;Ds;m) =\n\u0000\u001b(!;\u000e;\u0000gA+gB;\u0000Ds;\u0000m). In addition, the trans-\nlation operation about half a unit cell switches Aand\nBsublattices and m!\u0000m;\u001b(!;\u000e;gA\u0000gB;Ds;m) =\n\u0000\u001b(!;\u000e;gA\u0000gB;Ds;m). Hence, the conductivity in the\nordered phase is an even function of gA\u0000gBandDs. This\nis a di\u000berent behavior from the spinon case, in which the\nconductivity is an odd function of the staggered magnetic\n\feld (corresponds to gA\u0000gBin our case).\nThe conductivity diverges when !approaches !c1. The\nasymptotic form reads\nRe [\u001b(0;!;\u0000!)]\u0019\n\u0000(gA\u0000gB)2J2\u000eSASB(h++J(SA+SB))\n8\u0019Jf(h++J(SA+SB))2\u00004J2SASBg5\n4\n\u00021p\n(1\u0000\u000e2)SASB\u000e!; (28)\nwhere\u000e!\u0011!\u0000!c1. A similar feature is also found in\nthe spinon case, in which the divergence is related to the\nsingularity of the density of states [52]. On the other\nhand, the asymptotic form around !=!c2reads\nRe [\u001b(0;!;\u0000!)]\u0019\n(gA\u0000gB)2J2\u000eSASB(h++J(SA+SB))\n8\u0019Jf(h++J(SA+SB))2\u00004J2\u000e2SASBg5\n4\n\u00021p\n(1\u0000\u000e2)SASBj\u000e!j: (29)\nThe sign of the conductivity is the opposite of that in the\nlower frequency regime. This is in contrast to the spinon\ncase [52], in which the sign of the nonlinear conductivity\nremains the same for all frequencies !2[!c1;!c2].\nD. Relaxation-time dependence\nWe next study the damping (relaxation time) depen-\ndence of the spin current. In the study of photovoltaic\ne\u000bect, the relaxation-time dependence re\rects the micro-\nscopic mechanism behind the photovoltaic e\u000bect [2]; it\nis called shift current when \u001b(0;!;\u0000!)/\u001c0while is\nan injection current when \u001b(0;!;\u0000!)/\u001c. In bosonic\nsystems, a slight di\u000berence appears in the momentum\ndependence of the single-particle relaxation time [61]; it\nis inversely proportional to the momentum for the Gold-\nstone modes. Therefore, we assume the momentum de-\npendence of damping term as \u001ck= 1=(\u000b0\"\f(k)) so that\nthe momentum dependence is consistent with the \feld\ntheoretic requirement ( \u000b0is the damping factor). Phys-\nically, the assumed form of \u001ckcorresponds to the phe-\nnomenological Gilbert damping.\nWe substitute \u001ck= 1=(\u000b0\"\f(k)) in Eq. (10) in or-\nder to estimate the relaxation-time dependence of the7\nkzkxEk\nFIG. 3. (Color online) Schematic \fgure of the magnon disper-\nsion for the 3D model in Eq. (30). We set ky= 0. The blue\nand orange planes are the dispersions of two magnon branches\nand the green transparent plane is that of two-magnon exci-\ntation. The plot is for J= 1,J?= 1,\u000e= 1=4,SA= 1,\nSB= 1,h+= 1=100, andh\u0000= 1=10.\nspin-current conductivity. Figure 2(b) shows the \u000b0de-\npendence of \u001b(0;!;\u0000!). Our numerical result shows\n\u001b(0;!;\u0000!) is insensitive to the damping. A slight dif-\nference, however, appears in the high-frequency region,\nwhere the smearing due to the damping is more distinct\nthan that in the low-frequency region. This behavior is\nrelated to the momentum dependence of \u001ck, which is in-\nversely proportional to the energy of the magnon. The\ninsensitivity is a signature of a shift-current type photo-\ninduced current [2]; this is a similar feature to the spinon\ncase [52].\nIV. THREE-DIMENSIONAL MAGNETIC\nINSULATORS\nIn this section, we consider a three-dimensional (3D)\nmagnet which consists of coupled spin chains H1Dwith a\nnon-negligible inter-chain interaction [See Fig. 1(b)]. We\nparticularly focus on the limit in which !is close to the\nband gap of two-magnon excitations. The procedure of\nthe calculation is the same as the 1D case in the previous\nsection. The static part of the Hamiltonian reads\nH(3D)\n0\u0011X\nrJ(1 +\u000e)SA(r)\u0001SB(r)\n+J(1\u0000\u000e)SA(r+ ^x)\u0001SB(r)\n\u0000(D+Ds) [Sz\nA(r)]2\u0000(D\u0000Ds) [Sz\nB(r)]2\n\u0000J?[SA(r)\u0001SA(r+ ^y) +SA(r)\u0001SA(r+ ^z)\n+SB(r)\u0001SB(r+ ^y) +SB(r)\u0001SB(r+ ^z)]\n\u0000h[gASz\nA(r) +gBSz\nB(r)]: (30)\nThe spin chains is parallel to the xdirection, while the y\nandzdirections are perpendicular to the chains. The fer-\nromagnetic coupling J?>0 denotes the strength of theinter-chain exchange interaction. We study this model\nwithin the linear spin-wave approximation using Holsten-\nPrimakov transformation in Sec. III B. Focusing on the\nlower edge of the magnon dispersion, we \frst expand\nthe matrix elements ha\nkof the magnon Hamiltonian [See\nEq. (16)] up to second order in k:\nh0\nk'h++J(SA+SB) +J?(SA+SB)\n2(k2\ny+k2\nz);\n(31a)\nhx\nk'Jp\nSASB(2\u00001\n4k2\nx); (31b)\nhy\nk'\u0000Jp\nSASB\u000ekx; (31c)\nhz\nk'h\u0000+J(SA\u0000SB) +J?(SA\u0000SB)\n2(k2\ny+k2\nz):\n(31d)\nWe note that the magnon dispersions depend on both\nintra- and inter-chain wave numbers di\u000berently from the\n1D case. The dispersion around \u0000 point k=0is shown in\nFig. 3. Using the momentum gradient of the low-energy\nHamiltonian with h0;x;y;z\nk, we can de\fne the spin current\noperator; this approximation is essentially equivalent to\nexpanding the lattice spin-current operator in Eq. (24)\nup to the linear order in k:\nJz\nz=Jp\nSASBX\nksinh(2\u0002 k)\u0012kx\n2cos \b k+\u000esin \b k\u0013\n\u0002\u0010\n\u000by\nk\u000bk+\f\u0000k\fy\n\u0000k\u0011\n+\u001a\ncosh(2\u0002 k)\u0012\ncos \b kkx\n2\u0000\u000esin \b k\u0013\n+i\u0012kx\n2sin \b k+\u000ecos \b k\u0013\u001b\n\u000by\nk\fy\n\u0000k+ h.c.:\n(32)\nThese equations corresponds to the k\u0001pexpansion of the\nlattice model. Therefore, it should be a good approxima-\ntion for the lattice model when !is close to the gap for\ntwo-magnon excitations.\nThe spin current conductivity is calculated using the\nformula of Eq. (10). A calculation similar to the 1D\nmodel considered in Sec. III gives\nRe [\u001b(0;!;\u0000!)] =\n\u0000J2\u000eSASB(gA\u0000gB)2\n(4\u0019)22J?!2(SA+SB)\u0000\n8kx\u0000k3\nx\u0001\nkx=KX;(33)\nwhere\nKX=\"\n8(1\u0000\u000e2)\n\u00004s\n(h++J(SA+SB))2\u0000(!=2)2\nJ2SASB+\u000e2(\u000e2\u00002)3\n51\n2\n:\n(34)8\nWhen!is close to the lower edge, i.e.,\n!\u0018!c1\u00112p\n(h++J(SA+SB))2\u00004J2SASB;(35)\nKXbecomes\nKX\u0019s\n2p\n(h++J(SA+SB))2\u00004J2SASB\u000e!\n(1\u0000\u000e2)J2SASB;(36)\nwhere\u000e!=!\u0000!c1. Therefore, the asymptotic form of\nRe [\u001bABB(0;!;\u0000!)] is\nRe [\u001b(0;!;\u0000!)]\u0019\u0000(gA\u0000gB)2\u000epSASB\n16\u00192p\n1\u0000\u000e2(SA+SB)\n\u0002Jp\n\u000e!\nJ?f(h++J(SA+SB))2\u00004J2SASBg3\n4:(37)\nUnlike the 1D case, in which the conductivity diverges\nat the band edge !c1, the 3D result in Eq. (37) decreases\nproportionally top\n\u000e!when approaching !c1. The result\nis plotted in the inset of Fig. 2(a) with the results for\nthe 1D limit. This di\u000berence is a consequence of the\ndi\u000berence in the density of states: it diverges in the 1D\nmodel while it is proportional top\n\u000e!in the present 3D\ncase.\nThe approximation we used in this section is accurate\nwhen!is close to the magnon gap at the \u0000 point in the\nBrillouin zone. In our model, the band bottom for the\ntwo-magnon excitations are at \u0000 point, and the band-\nwidth of two-magnon excitation along the xandydirec-\ntions are in the order of J?and that for zdirection is in\nthe order of J. Therefore, our approximation is accurate\nwhen\u000e!\u001cJ;J?. This condition is manifested in J?\nin the denominator of Eq. (37), which implies the diver-\ngence of Re [ \u001b(0;!;\u0000!)] atJ?!0. WhenJ?is very\nsmall, we expect Re [ \u001b(0;!;\u0000!)] to behave like that of\nthe 1D case. On the other hand, Re [ \u001b(0;!;\u0000!)] looks\nlike Eq. (37) when J?is su\u000eciently large, e.g., when\nJ?\u0018J. Therefore, the 1d result and the result in this\nsection corresponds to the two limits of the 3D magnet.\nV. EXPERIMENTAL OBSERVATION\nIn this section, we discuss experimental methods for\ndetecting signatures of a directional spin current in our\nmechanism.\nA. Setup\nWe here discuss experimental setups for the observa-\ntion of the spin current generated by linearly-polarized\nlight. The mechanism studied here produces a directional\n\row of the spin current, which is a distinct feature from\nthe spin pumping [23, 24]. Therefore, the observation of\nthe directional \row should provide an evidence for our\n(a) (b)\n(c)\n(d)\nFIG. 4. (Color online) Schematic \fgure of the experimental\nsetups for measuring photo-induced spin current: All-optical\nsetup [(a) and (b)] and two-terminal setup (c). (a) The all-\noptical setup irradiates the isolated magnet using THz light.\nThe optically-induced spin current accumulates the angular\nmomentum at the end of the magnet which is depicted by the\nclouds; it produces the asymmetric distribution of the angular\nmomentum in the magnet. (b) A similar observation by at-\ntaching a thin layer of a soft ferromagnet at the two ends. The\nphotovoltaic spin current is injected to or absorbed from the\nsoft ferromagnets. (c) The two-terminal setup observes the\ndirectional \row of spin current using the inverse spin Hall ef-\nfect. The optically-induced spin current \rows along a certain\ndirection of the system. Therefore, inverse spin Hall voltage\nof the two leads has the same sign. These setups are di\u000berent\nfrom that of spin pumping of panel (d), in which a trans-\nverse AC \feld is applied to the magnet and the spin current\nis di\u000busively expanded.\nmechanism. We discuss two di\u000berent mechanisms: First\none is an all-optical setup using Kerr rotation or Fara-\nday e\u000bect, and the second is a two-terminal setup using\ninverse spin-Hall e\u000bect.\nObservation of the spatial distribution of angular mo-\nmentum in the open circuit setup provides a direct\nevidence for the optically-generated spin current [See\nFig. 4(a)]. In an isolated magnet, the spin current pro-\nduced by a THz light \rows along a direction de\fned by\nthe magnetic order and the crystal symmetry. There-\nfore, if the system becomes close enough to a laser-driven\nnon-equilibrium steady state, the angular momentum ac-\ncumulates at the two ends in an open circuit setup in\nFig. 4(a); positive angular momentum on one end and\nnegative on the other end. The angular momentum dis-\ntribution is anti-symmetric along the direction of the spin\ncurrent. This distribution is strikingly di\u000berent from the\nspin pumping case in which the distribution is symmetric\nand its di\u000berence from the equilibrium state is larger at\nthe focal area of the laser than at the ends.\nAn all-optical setup using Kerr rotation or Faraday\ne\u000bect would be a useful setup for the observation of such a9\nspatial distribution. Measurement of magnetic moments\nand its spatial distribution using the optical probe is a\ncommonly used technique for observing the spin current.\nFor instance, this method is used to observe the spin\nHall e\u000bect [62]. Similarly, observing the magnetization\nof soft magnet layers attached to the two ends is another\npossible setup for the experiment [Fig. 4(b)].\nThe observation of spin current in a two-terminal setup\nin Fig. 4(c) also enables us to see the directional \row of\nspin current and to distinguish it from the spin pumping\ne\u000bect. This setup consists of a noncentrosymmetric mag-\nnetic insulator which is sandwiched between two metallic\nleads; the two leads detect spin current via inverse spin\nHall e\u000bect [63{65]. In the photovoltaic mechanism, the\nspin current in the two leads \rows toward the same di-\nrection. Therefore, the inverse spin Hall voltage of the\ntwo leads has the same sign. In contrast, in the spin\npumping, the spin current di\u000busively \rows outward from\nthe magnet; the inverse spin Hall voltage is positive on\none side and negative on the other. Therefore, the rela-\ntive sign of the inverse spin Hall voltage of the two leads\ncan make a distinction between the spin pump and our\nmechanism.\nFinally, we shortly comments on heating e\u000bect of ap-\nplied electromagnetic waves. When we try to detect the\nphotovoltaic spin current with the above setups, spin\npumping might also occur due to the heating e\u000bect of the\napplied laser. For such a case, extracting the asymmet-\nric part of the angular-momentum distribution or inverse\nspin Hall voltage is important to detect an evidence for\nour mechanism.\nB. Required intensity of AC \feld\nWe next estimate the required ac electromagnetic \feld\nfor generating an observable spin current. We here as-\nsume a spin current of Js= 10\u000016J/cm2is observable.\nThis estimate is based on a Boltzmann theory calcula-\ntion for spin Seebeck e\u000bect in a ferromagnet [52, 69].\nThe details of the estimate is brie\ry explained in Ap-\npendix B. We use the following parameters as a typical\nvalue for 1D insulating magnets: J= 100kBJ,\u000e= 0:1,\nSA=SB= 1,gA\u0000gB= 0:1\u0016BJ/T,h+= 10kBJ, and a\nthe light with a frequency which is ~\u000e!= 6\u0019~\u00021011Hz\nabove the band gap. Here, ~is the Planck constant.\nWith these parameters, the conductivity for the 1D\nAFM/FRM chain is Re [ \u001b(0;!;\u0000!)]\u001810\u000014J/(cm2T2).\nTherefore, the required magnitude of oscillating magnetic\n\feld to produce a spin current of Js= 10\u000016J/cm2is\nB\u0018q\nJs\njRe[\u001b(0;!;\u0000!)]j\u00180:1 T. This corresponds to the\nelectric \feld E=cB\u0018104\u0000105V/cm under the as-\nsumption of c= 108m/s which is a typical value of\nspeed of light in insulators. Similar estimate for the\n3D magnet with J= 100kBJ,J?= 10kBJ,\u000e= 0:1,\nSA=SB= 1,gA\u0000gB= 0:1\u0016BJ/T,h+= 10kBJ,\nand!= 2\u0019\u00021012Hz gives Re [ \u001b(0;!;\u0000!)]\u001810\u000011J/(cm2T2) andE=cB\u0018105\u0000106V/cm. Our estimate\npredicts that the photovoltaic spin current is experimen-\ntally observable by using a moderate-intensity THz light.\nC. Candidate material\nWe believe the photovoltaic spin current should be seen\ngenerically in noncentrosymmetric magnets. In a recent\nwork [52], the authors \fnd three kinds of spin-light cou-\nplings induce the spin current in a spin chain, and this\nwork presents photovoltaic spin current in ordered mag-\nnets. These results imply the generation of photovoltaic\nspin current is a universal phenomenon in noncentrosym-\nmetric magnetic insulators. One such material is ferri-\nmagnetic diamond chains [66{68]. These materials often\nhave a distortion associated with trimerization, which\nbreaks the inversion symmetry [66]. Also, a large den-\nsity of states for the magnon excitations is expected in\nthis material because it is a quasi-1D magnet. Thus the\nferrimagnetic phase of the diamond chain is a promising\ncandidate for studying the spin current.\nVI. SUMMARY AND DISCUSSION\nTo summarize, we studied the spin current genera-\ntion through the shift current mechanism in ferrimag-\nnetic/antiferromagnetic insulators. Our theory uses a\nnonlinear response theory, which is a natural generaliza-\ntion of the linear response theory. Based on this method,\nwe \fnd that the illumination of a linearly-polarized\nlight produces the magnon current in noncentrosymmet-\nric magnets with antiferromagnetic/ferrimagnetic order.\nThe photovoltaic spin current appears even at the zero\ntemperature where no magnon excitation exists in the\nequilibrium; the current is related to the two-magnon\nexcitation process and not to the optical transition of\nexisting (thermally-excited) magnons. We stress that\nthe photo-induced spin current in our mechanism is car-\nried by electrically-neutral particles. The relaxation-time\ndependence of the spin current indicates that our pho-\ntovoltaic e\u000bect is a \\shift current\", i.e., the nonlinear\nconductivity is insensitive to the damping. Our theory\nclearly shows that the shift current mechanism, which is\nwell known in electron (fermion) systems, is also relevant\nto systems with bosonic excitations whose the ground\nstate is the vacuum of bosons (zero boson state).\nOur result implies the zero-point quantum \ructua-\ntion is a key for the shift-current type photocurrent. In\nthe spinon spin current [52], the optical transition of a\nfermionic excitation plays a crucial role for the photocur-\nrent. In contrast to these cases, the ground state of the\nordered magnets is the zero-magnon state. Therefore,\nno optical transition of the existing magnons. Despite\nthe crucial di\u000berence, we \fnd a \fnite photovoltaic spin\ncurrent at the zero temperature. The magnon photocur-\nrent we found is ascribed to the optical transition of the10\n\\condensed\" Holstein-Primakov bosons. In the antiferro-\nmagnets/ferrimagnets, the ground state is a condensate\nof Holstein-Primakov bosons, which is technically rep-\nresented by the Bogoliubov transformation. The optical\ntransition of the condensed Holstein-Primakov bosons al-\nlows generation of the shift-current type photocurrent\neven at the zero temperature. On the other hand, we\n\fnd that the nonlinear conductivity is zero at T= 0 for\nthe ferromagnetic version of the model considered here.\nFrom this viewpoint, the two-magnon creation is similar\nto the particle-hole pair creation in semiconductors; the\noptical transition of fermions from the valence band to\nthe conduction band is equivalent to the pair creation.\nAs the condensation of the Holstein-Primakov bosons is\na manifestation of zero-point \ructuation, the zero-point\n\ructuation is the essence for the shift-current type pho-\ntovoltaic e\u000bects in the magnetic insulators.\nOur results implies that the dc spin current generation\nusing linearly polarized light is generally possible in the\nmagnets without inversion symmetry.\nAppendix A: Derivation of Kraut-von Baltz formula\nfor Bosons\nHere, we shortly explain the derivation of the nonlin-\near conductivity in two-band boson systems. We used\nthe formula in Eq. (A10) for the analytic calculations\nand Eq. (A5) for numerical results with a \fnite Gilbert\ndamping.\nWe calculate the nonlinear response coe\u000ecients us-\ning a formalism similar to the linear response theory.\nWe assume a system with a time-dependent perturba-\ntionH0=\u0000P\n\u0016^B\u0016F\u0016(t), where ^B\u0016is an operator and\nF\u0016(t) is a time-dependent \feld; the Hamiltonian reads\nH=H0+H0. The expectation value of an observable\n^Areadsh^Ai(t) = Trh\n^\u001a(t)^Ai\n=Z;where\u001a(t) is the density\nmatrix at time tandZ\u0011Tr\u001a(t). By expanding \u001a(t) up\nto the second order in F\u0016(t), the Fourier transform of\nhAi(t),hAi(\n), reads\nhAi(\n) =X\n\u0016;\u0017Z\nd!\u001b\u0016\u0017(\n;!;\n\u0000!)F\u0016(!)F\u0017(\n\u0000!);\n(A1)\nwith the nonlinear conductivity\n\u001b\u0016\u0017(\n;!;\n\u0000!) =1\n2\u0019X\nn;m;l(\u001an\u0000\u001am)(B\u0016)nm\n!\u0000Em+En\u0000i=(2\u001cmn)\n\u0002\u0014(B\u0017)mlAln\n\n +En\u0000El\u0000i=(2\u001cmn)\u0000Aml(B\u0017)ln\n\n +El\u0000Em\u0000i=(2\u001cmn)\u0015\n:\n(A2)\nHere,Enis the eigenenergy of the many-body eigenstaten,\u001cmnis the relaxation time, and Onm(O=A;B\u0016;B\u0017)\nis the matrix element of ^Oin the eigenstate basis of H0.\nWe here consider a periodic free-boson system in which\nall matrices A,B\u0016, andB\u0017have the following form:\n^O=X\nk\u0000\n\u000by\nk\f\u0000k\u0001\nOk\u0012\u000bk\n\fy\n\u0000k\u0013\n; (A3)\n=X\nk\u0000\n\u000by\nk\f\u0000k\u0001\u0012\n(Ok)\u000b\u000b(Ok)\u000b\f\n(Ok)\f\u000b(Ok)\f\f\u0013\u0012\u000bk\n\fy\n\u0000k\u0013\n;\n(A4)\nwhere\u000bk(\u000by\nk) and\fk(\fy\nk) are the annihilation (creation)\noperators of the boson eigenstates with momentum k,\nandOk=Ak;B\u0016\nk;B\u0017\nk. The theory for spinwave exci-\ntations of many antiferromagnetic models with a N\u0013 eel-\ntype order reduces to the above form by using Holstein-\nPrimakov and Bogoliubov transformations.\nFor the two-band system, we can express Eq. (A2) us-\ning single-particle eigenstates. We note that A,B\u0016, and\nB\u0017for the two-band system above do not conserve the\nparticle number. However, all operators are quadratic in\nthe annihilation/creation operators and consists of only\nfor terms: \u000by\nk\u000bk,\f\u0000k\fy\n\u0000k,\f\u0000k\u000bk, and\u000by\nk\fy\n\u0000k. There-\nfore, only few terms out of the possible Wick decomposi-\ntion remain nonzero, similar to that of the systems with\nconserved particle number. Using these features, we \fnd\n\u001b(\n;!;\n\u0000!) =\n1\n2\u0019X\nk;ai=\u000b;\fsgn(a3)(~\u001ak;a1sgn(a2)\u0000sgn(a1)~\u001ak;a2)(B\u0016\nk)a1a2\n!\u0000~\"a2(k) + ~\"a1(k)\u0000i=(2\u001ck)\n\u0002\u0014(B\u0017\nk)a2a3(Ak)a3a1\n\n + ~\"a1(k)\u0000~\"a3(k)\u0000i=(2\u001ck)\n\u0000(Ak)a2a3(B\u0017\nk)a3a1\n\n + ~\"a3(k)\u0000~\"a2(k)\u0000i=(2\u001ck)\u0015\n:(A5)\nHere,\nsgn(a) =\u001a\n1 (a=\u000b)\n\u00001 (a=\f); (A6)\n~\"a(k) =sgn(a)\"a(k); (A7)\n~\u001ak;a=\u001ah\u000by\nk\u000bki0(a=\u000b)\nh\f\u0000k\fy\n\u0000ki0(a=\f); (A8)\nand we assumed the relaxation time only depends on k.\nIt is worth noting that the conductivity remains \fnite at\nT= 0 despite there are no excitations. Technically, this\nis a consequence of ~ \u001ak;\f, which is 1 at T= 0. Physically,\nthis is because the pair creation/annihilation processes\ncontribute to the spin current even at T= 0.\nWe here focus on the T= 0 limit. In this limit, ~ \u001ak;\u000b=\n0 and ~\u001ak;\f= 1. Using these results, we obtain11\n\u001b(0;!;\u0000!) =\u00001\n\u0019X\nk;ai=\u000b;\f\u0014(1 +i2\u001c!)jB\f\u000bj2(A\u000b\u000b+A\f\f)\n(!\u0000i=2\u001ck)2\u0000(\"\u000b(k) +\"\f(k))2\u0015\n+1\n2\u0019X\nk;ai=\u000b;\f\u001a(Bk)\f\u000b(Ak)\u000b\f((Bk)\f\f+ (Bk)\u000b\u000b)\n(!\u0000i=2\u001ck\u0000\"\u000b(k)\u0000\"\f(k))(\"\u000b(k) +\"\f(k) +i=2\u001ck)\n+(Bk)\u000b\f(Ak)\f\u000b((Bk)\f\f+ (Bk)\u000b\u000b)\n(!\u0000i=2\u001ck+\"\u000b(k) +\"\f(k))(\"\u000b(k) +\"\f(k)\u0000i=2\u001ck)\u001b\n: (A9)\nAs we discussed in the main text, certain symmetries\nrestricts the \frst term to be zero; this is the case for the\nmodels we consider in the main text. Assuming the \frst\nterm vanishes, we \fnd\nRe [\u001b(0;!;\u0000!)] =\n\u00001\n\u0019Re\u001a(Bk)\f\u000b(Ak)\u000b\f((Bk)\f\f+ (Bk)\u000b\u000b)\n!2\u0000(\"\u000b(k) +\"\f(k) +i=2\u001c)2\u001b\n:\n(A10)\nWe used this formula for the calculation of nonlinear con-\nductivity in the main text.\nAppendix B: Boltzmann theory for spin Seebeck\ne\u000bect\nThe magnitude of spin current Js= 10\u000016J/cm2is\nthe estimate for the spinon spin current produced by the\nspin Seebeck e\u000bect in a recent experiment [69]. We here\nsummarize the method and result discussed in a supple-\nmental material of a recent work [52].\nThe spin current is estimated from the Seebeck e\u000bect\nof magnons whose dispersion is given by\n\"(k) =JSk2+ 2DS+h: (B1)\nnear the \u0000 point of k=0. This magnon dispersion corre-\nsponds to that of a ferromagnetic heisenberg model with\nexchange interaction J, uniaxial anisotropy D, and the\nmagnetic \feld hparallel to the anisotropy. The current\nis calculated using the semiclassical Boltzmann theory,\nin which the current reads\nJs(r) =~Zdk\n(2\u0019)3vzfk(r): (B2)\nHere,fk(r) is the density of magnons with momentum\nkat position randvz\u0011@kz\"(k) is the group velocity of\nmagnons.fk(r) is calculated from the Boltzmann equa-\ntion with temperature gradient\nvk\u0001rrfk(r) =\u0000fk(r)\u0000f(0)\nk(r)\n\u001ck; (B3)wheref(0)\nk(r) is the density at the equilibrium. Here, the\nrelaxation-time approximation is used to simplify the cal-\nculation of collision integral on the right hand side. The\nspin current induced by the spin Seebeck e\u000bect is esti-\nmated by substituting the solution of fk(r) in Eq. (B3)\ninto the current formula in Eq. (B2)\nIn the Boltzmann theory, the spin current by the spin\nSeebeck e\u000bect reads\nJs(r)\u00183(6\u00192)2\n3J2\nHS2\n2\u000bkBaT(r)\u0001T\nT(r)F\u0012JHSa2\u00032\n2kBT(r);2DS+h\n2kBT(r)\u0013\n;\n(B4)\nwhere \u0003 = (6 \u00192)1=3=ais the cuto\u000b for magnon dispersion\nand\nF(a;b) =Z1\n0x4csch2(ax2+b)dx: (B5)\nUsing a set of typical parameters S= 1,JH= 100kBJ,\nD= 0 J,h=\u0016BJ,a= 4\u000210\u000010m,\u000b= 10\u00002,T= 100\nK, \u0001T= 3\u0002104K/m, we \fnd Js\u001810\u000012J/cm2for the\nferromagnet. We assume this value as the typical spin\ncurrent density in the insulating ferromagnets.\nA recent experiment on quasi-one-dimensional mag-\nnets observed a spin current which is 10\u00004of what is\ntypically observed in a ferromagnetic phase [69]. There-\nfore, we assume Js\u001810\u000016J/cm2as the experimental\nresolution for the spin current.\nACKNOWLEDGMENTS\nWe thank Ryosuke Matsunaga and Youtarou Taka-\nhashi for fruitful discussions. 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Khmelevskyi\nCenter for Computational Materials Science, Institute for Applied Physics,\nVienna University of Technology, Wiedner Hauptstrasse 8, A -1040 Vienna, Austria\n(Dated: July 24, 2019)\nThe high-entropy alloys Al xCrFeCoNi exist over a broad range of Al concentrations (0 < x <2).\nWith increasing Al content their structure is changed from t he fcc to bcc phase. We investigate\nthe effect of such structural changes on transport propertie s including the residual resistivity and\nthe anomalous Hall resistivity. We have performed a detaile d comparison of the first-principles\nsimulations with available experimental data. We show that the calculated residual resistivities for\nall studied alloy compositions are in a fair agreement with a vailable experimental data as concerns\nboth the resistivity values and concentration trends. We em phasize that a good agreement with\nexperiment was obtained also for the anomalous Hall resisti vity. We have completed study by\nestimation of the anisotropic magnetoresistance, spin-di sorder resistivity, and Gilbert damping. The\nobtained results prove that the main scattering mechanism i s due to the intrinsic chemical disorder\nwhereas the effect of spin polarization on the residual resis tivity is appreciably weaker.\nI. INTRODUCTION\nThe high-entropy alloys (HEA), the multicomponent\ncrystalline alloys, often also called multi-principal ele-\nmentalloyshaveattractedaquitesignificantandgrowing\ninterest in the last decade. Of the vast existing litera-\nture we just mention a recent book,1a critical review,2\nand an overview of possible theoretical approaches.3The\nhigh entropy of mixing of these multicomponent alloys\nsuppresses the formation of ordered intermetallic com-\npounds leading to well disordered phases with simple\nlattice structures such as the face-centered cubic (fcc)\nor the body-centered cubic (bcc) ones. Magnetic HEA’s\nthat consist of magnetic 3 d-elements are particularly in-\nteresting. A typical example is the so-called quinary\nCantor alloy (CrMnFeCoNi) consisting of equiconcentra-\ntion disordered mixture of magnetic Cr, Mn, Fe, Co, and\nNi elements with an fcc structure. Such alloy offers a\nrichness of magnetic properties depending on the sam-\nple preparation.4The structural change from the fcc to\nbcc phase was predicted by ab initio molecular dynamics\nsimulations5for Cantor-like alloys with Cr substituted\nby another metallic element with formula CoFeMnNiX\n(X = Al, Ga, and Sn).\nBy doping with sp-elements, which influences carrier\nconcentration in the conduction band and thus both the\nmagnetic and transport properties, and even the alloy\nstructure one can search for new functional properties.\nA typical example of such alloy is Al xCrFeCoNi alloy6,7\nwithxranging from x=0 tox=2. In particular, alloy-\ning with increasing Al content stabilizes the bcc phase\nfrom the original fcc phase of quaternary CrFeCoNi al-\nloy. Another interesting property, also present in theCantor alloy, is a large residual resistivity of the order\nof 100µΩcm which is in a striking contrast to a much\nsmallerresistivityofthebinaryfccNiFeorfccNiCocoun-\nterparts. The large values of the residual resistivity in\nHEA’s are caused by strong scattering on the intrinsic\nchemical disorder. In the present work we apply the\nalloy-specific first-principles methodology based on the\nKubo-Greenwood formula8which was successfully used\nfor binary alloys9,10also to HEA’s in order to estimate\nthe intrinsic contribution to the resistivity and compare\nthem to available experimental data.\nContrary to the experimental and theoretical studies\nof structural and thermodynamical properties of HEA’s\nthe studies of electronic transport are very rare.2The\ntransport properties, together with the electronic struc-\nture are among the most important material proper-\nties. The theoretical tools for the resistivity study are\nmore complicated and not so broadly available as elec-\ntronic structure codes focused on total energies, electron\ndensities and magnetic moments. Recently, theoretical\ntransport studies of the Cantor fcc CrMnFeCoNi alloy11\nand of a related medium-entropy fcc NiCoMn alloy12ap-\npeared which studied various possible scattering mecha-\nnisms contributing to the residual resistivity. However,\ntheoretical investigation of the effect of Al-doping on the\nresistivity, as well as of the role of different structures\n(bcc, fcc) for electron transport in AlCrFeCoNi, is still\nmissing. Moreover, in addition to residual resistivities\nalsothe anomalousHall resistivity(AHR) for both struc-\nturephaseswasdetermined experimentally.6,7Therefore,\nthese alloys are an obvious choice for ab initio based\nstudies of transport properties in HEA’s containing sp-\nelements. In the present study, also the anisotropic mag-2\nnetoresistance (AMR), the spin-disorder resistivity and\nthe Gilbert damping for both structures and typical Al\nconcentrations are calculated and discussed.\nII. FORMALISM\nThe disordered fcc and bcc phases of Al xCrFeCoNi al-\nloy with xranging from x=0 tox=2 were studied for\nexperimentally observed phases and lattice constants.6\nThe fcc phase exists roughly for x <0.5 while the bcc\nphase is stable for x >1.0, but boundaries are not well\ndefined. Duplex phase (a mixture of fcc and bcc phases)\nexists for the intermediate Al concentrations. We note\nthat sometimes the Al xCrFeCoNi alloy is presented as\nAl1−4yCryFeyCoyNiy, wherey= 1/(4+x) and the sum\nof all component concentrations is one.2\nThe spin-polarized electronic structure calculations\nwere done using the Green function formulation of\nthe tight-binding linear muffin-tin orbital (TB-LMTO)\nmethodintheatomicsphereapproximation(ASA).13We\nemploy the scalar-relativistic version of the TB-LMTO\nmethod and, in order to assess the importance of the rel-\nativistic effects, we also made calculations using the fully\nrelativistic version of the TB-LMTO method. In both\ncases the exchange-correlation potential of Vosko, Wilk\nand Nusair (VWN)14and the spd-basis set were used.\nThe alloy disorder in studied multicomponent alloys is\ndescribed in the framework of the coherent potential ap-\nproximation (CPA).15The use of the CPA allows us to\nwork very efficiently in small fcc or bcc unit cells. On\nthe contrary, large special-quasirandom structure (SQS)\nsupercells are needed in conventional density-functional-\ntheory (DFT) studies.4,5It should be noted that contin-\nuously varying Al content imposes additional non-trivial\nconstraintsonthechoiceofasuitableSQS-supercell. The\ncostwepayforusingtheCPAistheneglectofpossiblelo-\ncal environment and clustering effects in the alloy which,\non the other hand, are captured by the SQS-supercell\napproach. The CPA is a reliable approach in well disor-\ndered alloys, particularly when the concentration trends\nare concerned. Even more important advantage of the\nCPA is the fact that it provides naturally transport re-\nlaxation times which need not be taken from outside like,\ne.g., in the Boltzmann equation approach.\nThe transport properties are described by the conduc-\ntivity tensor σwith components σµν(µ,ν=x,y,z). The\nresistivity tensor ρwith components ρµνis obtained sim-\nply by inversion of the conductivity tensor, ρ=σ−1.\nThe conductivity tensor is determined in the framework\nof the Kubo-Greenwood (K-G) approach (only diagonal\nelements of σµνare non-zero in present cubic systems\nin the scalar-relativistic model).8The off-diagonal com-\nponents of σµνare needed for the AMR/AHR studies\nand they are calculated in the framework of the Kubo-\nBastin (K-B) formulation of the fully-relativistic trans-\nport in disordered magnetic alloys which includes both\nthe Fermi-surface and Fermi-sea terms on equal footing.9The Fermi-surface term contains contribution only from\nthe states at the Fermi energy and includes the most im-\nportant elastic scattering effects due to impurities. The\nFermi-sea term, on the contrary, depends on all occupied\nstates below the Fermi energy; this term contributes only\nto the antisymmetric part of the tensor σµν. Once the\ntransport tensor is determined, the AHR= ρxywhile the\nAMR=(ρzz−ρxx)/ρtot, whereρtotis the average value\nof diagonal components of the resistivity tensor. In rel-\nativistic calculations we assume that the magnetic mo-\nment points in the z-direction. The disorder-induced ver-\ntexcorrections,10whichdescribe thecorrelatedmotionof\ntwo electrons in a random alloy potential, are included.\nThey correspondto the backwardscatteringcontribution\nin the conventional Boltzmann equation approach.\nThe Gilbert damping (GD) constant is an important\nphenomenological parameter describing the magnetiza-\ntion dynamics. It is evaluated here with the help of a\nrecently developed approach using nonlocal torques16as\nanalternativetotheusuallocaltorqueoperatorsentering\nthe torque-correlation formula.17–19This leads to effec-\ntivetorquesthatarerepresentedasnon-site-diagonaland\nspin-independent matrices, which simplifies evaluation of\ndisorder-induced vertex corrections which play essential\nrole in the present formulation since their neglect would\nlead to quantitatively and physically incorrect results.16\nOur formulation gives results that compare well to\nother first-principles studies.17–19In this study we will\nconcentrate on the GD due to chemical disorder, espe-\ncially the effect of Al-doping. It should be noted that\nthere are other sources of damping, e.g., the tempera-\nture effects due to phonons and spin fluctuations which\nare neglected here.\nIII. RESULTS\nA. Electronic structure and magnetic moments\nThe results of electronic structure calculations serve\nas an input for transport calculations of Al xCrFeCoNi\nalloys.\nTo illustrate the underlying electronic structure, we\nshow in Figs. 1 and 2 the total and component-resolved\ndensities of states (DOS) for two typical alloys, namely,\nfcc Al 0.25CrFeCoNi and bcc Al 1.25CrFeCoNi. The fol-\nlowing conclusions can be done: (i) We note a typical\ntwo-peak-like total DOS characteristic of the bcc phase\nas compared to an essentially one-peak-like total DOS\nfor the fcc phase. In both cases the Fermi level is located\ndeep inside the valence band as it is typical for metal-\nlic systems in contrast, e.g., to doped semiconductors in\nwhich the clustering has a non-negligible effect on DOS\nclose to band edges. The CPA is thus a good approxima-\ntionforelectrontransportstudies; (ii)Wealsonotelarger\ntotal DOS at the Fermi energy for fcc phase as compared\nto the bcc phase. This indicates a larger amount of car-\nriers and thus a smaller resistivity/larger conductivity3\n-30-25-20-15-10-5 0 5 10 15 20 25 30\n-0.8-0.6-0.4-0.2  0 0.2Local DOS (states/Ry)\nEnergy (Ry)maj\nminCr\nFe\nCo\nNi\nAl-15-10-5 0 5 10 15 20Total DOS (states/Ry)fcc Al0.25CrFeCoNi maj\nmin\nFIG. 1: Calculated spin-resolved DOS’s for fcc\nAl0.25CrFeCoNi alloy: the total DOS (upper frame) and\natom-resolved DOS’s (lower frame) are shown. The vertical\nlines denote the position of the Fermi level.\nof fcc phases because the amount of disorder is similar\nin both phases (see atom-resolved DOS’s below and dis-\ncussion there); (iii) The Al-resolved DOS is free-electron\nlike with only small modifications in the energy region\nwhere it hybridizes with transition metal states; (iv) The\nmajority Ni-, Co-, and Fe-resolved DOS’s indicate only\nnegligible influence of the disorder: they all have similar\nshapesand centersofgravityand resemblecorresponding\ntotalDOS.Onthecontrary,themajorityCr-DOShasthe\ncenter of gravity shifted to higher energies and its shape\nis different from those of Ni-, Co-, and Fe-states. This is\ndue to a lower atomic number of Cr and thus a weaker\nCoulomb attraction in comparison with Fe, Co, and Ni.\nMajorityCrstatesthusintroduceasignificantdisorderin\nthe majority band. Below we show that in present alloys\nthe resistivity in both majority and minority channels\nis comparable (see also Ref.20); (v) The minority Ni-,-30-25-20-15-10-5 0 5 10 15 20 25 30\n-0.8-0.6-0.4-0.2  0 0.2Local DOS (states/Ry)\nEnergy (Ry)maj\nminCr\nFe\nCo\nNi\nAl-15-10-5 0 5 10 15 20Total DOS (states/Ry)bcc Al1.25CrFeCoNi maj\nmin\nFIG. 2: The same as in Fig. 1 but for bcc Al 1.25CrFeCoNi\nalloy.\nCo-, Fe-, and Cr-resolved DOS’s have centers of gravity\nshifted to different positions thus indicating the presence\nof disorder among all components as contrasted to the\nmajority states. It should be noted that the character of\ndisorder in both fcc and bcc phases is quite similar. The\npresence of non-negligible disorder in both majority and\nminority states is the reason of much larger resistivity of\nAlxCrFeCoNi alloys as compared to the Ni-rich NiFe and\nNiCo alloys in which the disorder effect in majority spin\nchannel is negligible.20The disorder effect is essential for\nboth, minority and majority spin channels, but operates\nin them differently. This observation provides us with a\nmotivation to study the magnetotransport phenomena.\nWe present magnetic properties in Table 1, where we\nshow the total and local magnetic moments for alloys\nwithx= 0.25 andx= 1.25. We can make the follow-\ning conclusions: (i) The induced local Al moments are\nvery small and negative. Also moments on Cr atoms\nare negative and their absolute value is reduced with in-4\nTABLE I: Calculated total magnetic moment ( Mtot) and\nlocal magnetic moments ( mX, X=Al, Cr, Fe, Co, Ni) for\nAlxCrFeCoNi alloys in the fcc ( xAl= 0.25) and bcc ( xAl=\n1.25) phases. Magnetic moments are in µB.\nxAlMtotmAlmCrmFemComNi\n0.25 (fcc) 0.606−0.054−0.6201.9331.0270.253\n1.25 (bcc) 0.646−0.045−0.1032.1171.2430.191\ncreasing Al content. The present alloys are thus ferri-\nmagnets. The values of Ni-local moments are strongly\nreduced as compared to the fcc Ni crystal; (ii) Dominat-\ning moments are those on Co and, first of all, on Fe sites\nwhich have values close to the values in bcc Fe crystal\nwhile moments on Co-sites are smaller as compared to\nfcc-Co crystal. Both moments depend weakly on the Al\ndoping; and (iii) Due to the character of local moments,\nboth alloys have non-zero total magnetization with total\nmoments slightly larger for the bcc phase and relatively\nsmall in their sizes, being of order 0.5 µB. We note a\ngood quantitative agreement of present moments with a\nrecent theoretical study.21\nThe present CPA calculations ignored any spin fluc-\ntuations in the ground state of the alloys. However, ex-\nisting studies of the Cantor CrMnFeCoNi alloy11and of\nthe ternary fcc NiCoMn system12revealed an instability\nof Mn atoms to form more complicated moment distribu-\ntions; we haveverifiedthis featurefor the quinaryCantor\nalloy by using the well-known CPA approach.22In order\ntoexamineasimilarinstabilityofCratomsinthepresent\nAlxCrFeCoNi systems, we have performed the CPA cal-\nculations which started with multiple Cr magnetic mo-\nments. For both structures (fcc and bcc), the iterations\nconverged always to the same single value of Cr moment.\nThis indicates that the present AlCrFeCoNi systems can\nbe reasonably described by assuming the same (average)\nlocal moment attached to each alloy species.\nB. Residual resistivities\nThe theoretical estimate of residual resistivity ρ0and\nits comparison with available experiments6,7,23is the\nmainresultofthepresentpaper. Wetreatthefccandbcc\nphases as disordered alloys described by the CPA. Corre-\nsponding electronic structure provides also naturally the\ntransportrelaxationtimes as used in the K-G formulafor\nestimate of resistivities. We note that the SQS-supercell\ncalculations (see, e.g., Refs.4,5) indicate the presence of\nlocal environment effects, both in atomic structure and\nspins, which are neglected here. On the other hand, local\nenvironment effects appear as fluctuations around aver-\nageatomiclevels. Although such fluctuations arelargein\nsome cases, they usually correspond to states with small\nweights. One can thus say that the intrinsic chemical\ndisorder due to many atomic components will dominate.\nConsidering further the fact that the CPA gives reliablyconcentration trends, we can conclude that present ap-\nproach represents reasonable first approximation to esti-\nmate resistivities in the present HEA’s.\nResults are shown in Fig. 3 for both fcc and bcc\nAlxCrFeCoNi alloys for which experiments6,7are avail-\nable.\nLet us start with experiment Ref.6 (see also Ref.23). It\nshould be noted that experiment was done at the room\ntemperature while calculations relate to T= 0 K. We\nnote an enhancement of the resistivity due to the lattice\nvibrations and spin fluctuations induced by a finite tem-\nperature. It is possibleto include, forsimple systems, the\neffect of temperature in the framework of the alloy anal-\nogymodelasformulatedintheCPA.24Itshouldbenoted\nthat although a success was recently reached by a modi-\nfied approach for fcc alloys with few alloy constituents25\nsuch a detailed study of temperature effects is beyond\nthe scope of the present paper. Under such situation we\ndecided just to scale the zero temperature results by an\nempirical constant to account for the finite temperature\neffects. We have chosen this factor to be 1.2 motivated\nby the experiment7in which resistivity ratio for 300 K\nto 0 K was measured for many samples of different com-\npositions. The purpose is just to see the effect of the\nfinite temperature. Results are shown in Figs. 3a,3b.\nAlready large values of ρ0exist for xAl= 0.0, i.e., for\nthe equiconcentration quaternary CrFeCoNi alloy, which\nagreesvery well with a recent theoretical study.11We ob-\nserve an increase of ρ0with increasing Al content in both\nthe fcc and bcc phases. This result as well as large values\nofρ0are due to the fact that d-states of Al are miss-\ning and thus Al has a low density of states around the\nFermi energy in comparison with other alloy components\nintroducing thus a strong scattering. For example, in\ndisordered bcc Fe 1−xAlxalloys26the experimental ρ0is\nabout 150 µΩcm atT=4 K for xAlaround 0.3. Similarly,\nthe random bcc V 0.75Al0.25alloy exhibits practically the\nsame resistivity.27Comparable values are obtained for\nAlxCrFeCoNi for large xAlin the bcc phase.\nThere is a good quantitative agreement of calculated\nandmeasured ρ0inbothfccandbccregions,inparticular\nfor the scaled model. In agreement with the experiment\nthe slope of the concentration dependence of ρ0is larger\nfor the fcc phase as compared to the bcc one, although\nthe effect is more pronounced in the experiment.\nIt is a well-known fact20that a significant increase of\nthe resistivity occurs in Ni-rich NiFe and NiCo fcc alloys\ndue to the mixing of spin-channels by the spin-orbit cou-\npling. We have therefore performed also fully-relativistic\ncalculations for xAl= 0.25 andxAl= 1.25 alloys with\nfcc and bcc phases, respectively. We have obtained\nonly a small changes of ρ0, namely, ρ0was 86.16 µΩcm\nvs 84.70 µΩcm for xAl= 0.25, and 137.42 µΩcm vs\n135.03µΩcm for xAl= 1.25. In both cases, higher val-\nues correspond to the fully-relativistic model. The ori-\ngin of large enhancement of ρ0by spin-orbit coupling\nin the above-mentioned binary alloys is the existence\nof disorder-free majority bands, which are missing here.5\n 0 25 50 75 100 125 150 175 200 225 250\n 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2ρο (µΩ cm)\nxAlAlxCrFeCoNi alloy (c)\nfcc fcc+bcc bcc 0 25 50 75 100 125 150 175 200 225 250\n         ρο (µΩ cm)(b) AlxCrFeCoNi alloy\nfcc fcc+bcc bcc 0 25 50 75 100 125 150 175 200 225 250\n         ρο (µΩ cm)AlxCrFeCoNi alloy (a)\nfcc fcc+bcc bcc\nFIG. 3: Calculated and experimental residual resistivitie sρ0\nas a function of Al concentration xAl. Dashed vertical lines\ndenote approximate boundaries between the fcc, duplex, and\nbcc phases. Note that the fcc phase extendsinto duplexphase\n(fcc + bcc) and similarly the bcc phase starts in duplexphase .\nFilled circles denote theoretical results while empty symb ols\ncorrespond to experimental data. (a) Experiment Ref.6 at\nroomtemperaturecomparedwiththeoretical resultsfor T= 0\nK. (b) The same as (a) but theoretical results are scaled by\na factor 1.2 to fit experiment done at room temperature (see\ntext). (c) Comparison of experiment, Ref.7, with theoretic al\nresults. Empty circles and empty boxes denote alloys ’as-ca st’\nand ’homogenized’, respectively (see experimental paper f or\ndetails). Both the theory and experiment now correspond to\nT= 0 K.Thecontributionsofthe spin-up( σ↑) andspin-down( σ↓)\nconductivity channels to the total conductivity are com-\nparable. For example, σ↑(σ↓) = 4.77 (7.04) kS/cm\nforxAl= 0.25, and σ↑(σ↓) = 3.85 (3.56) kS/cm for\nxAl= 1.25, respectively. We can thus conclude that rel-\nativistic effects are small in the studied alloy.\nTheoretical description of the duplex region in which\nboth phases co-exist is difficult because of the lack of\nstructural details. We therefore present separately re-\nsults for the fcc phase with Al concentrations extending\ninto the duplex (fcc+bcc) region, and similarly, we start\nthebcc phaseinthe duplexregion. Correspondinglattice\nconstants were taken from Ref.6. We have thus avoided\nany processing of results like, e.g., the serial or parallel\nresistivities, the arithmetic weighting, etc. Inhomogene-\nity of samples in the duplex region is obvious.\nRecently Singh et al.28have shown on the basis of cal-\nculations of chemical interatomic interactions that con-\nsideredAl xCrFeCoNialloyscanexhibitatendency tothe\nclustering in the fcc phase while in the bcc phase can ex-\nist ordering tendency. One can thus speculate that this\ncan be one of possible reasons for the smaller/larger cal-\nculatedresistivitiesforfcc/bccphasesascomparedtothe\nexperimental ones .\nNext we will discuss the experiment of Ref.7. Results\nare presented in Fig. 3c with the following comments: (i)\nContrary to the experiment6there is no clear concentra-\ntion trend. Fluctuating values of ρ0may indicate sample\ninhomogeneity due to its preparation. Largest fluctu-\nations are, as expected, in the duplex region. Larger\nfluctuations are also for ’as-cast’ samples as compared to\n’homogenized’ones; and (ii) Nevertheless, calculated and\nmeasured resistivity values are still in acceptable agree-\nment, as well as larger resistivities for higher xAl(bcc\nphase). Experiment gives no detailed structural data\nconcerning studied samples, just its Al-content so that\nmore detailed discussion of measured resistivity fluctu-\nations and their comparison with the experiment is not\npossible.\nWe note that the performed calculations of residual\nresistivity ignore the effect of local atomic relaxations,\nwhich provide an additional mechanism of electron scat-\ntering. Since we cannot determine the magnitude of the\nlocal relaxations quantitatively, we have performed only\na preliminary study in order to get rough estimation of\ntheir effect on the resistivity by employing the alloy anal-\nogy model in the CPA.24,29As a typical mean value of\nthe atomic displacement, we took ∆ u= 0.05˚A for fcc\nalloys (obtained for the fcc Cantor alloy)11and a slightly\nhigher value ∆ u= 0.075˚A for bcc alloys (because of\nthe more open bcc geometry). The resulting increase of\nthe residual resistivity ofAl xCrFeCoNi wassmall in both\nstructures, being about 2.8 % for x= 0.25 in the fcc case\nand about 1.4 % for x= 1.25 in the bcc case. These\nresults agree qualitatively with those of Ref.11 proving\nthe dominating effect of strong intrinsic chemical disor-\nder on the resistivity. A more systematic study of the\nrole of local atomic relaxations goes beyond the scope of6\nthe present work.\nOne can summarize that the CPA, despite of its sim-\nplicity, isabletoreproducethemainfeaturesofmeasured\nresistivitiesalsoinsuchcomplexalloyslikeAl xCrFeCoNi.\nClearly, the main reasonfor this successis the dominance\nofintrinsic chemical disorderin alloy. On the otherhand,\none should keep in mind that calculated values are influ-\nenced by the neglect of lattice relaxations. In general,\none could say that lattice relaxations roughly represent\nsite-off diagonal disorder which have much smaller effect\nas compared to the dominating chemical disorder related\nto different positions of atomic alloy levels.\nC. Spin-disorder resistivity (SDR)\nThe SDR is the resistivity caused by spin fluctuations\nthat exist at finite temperature in the paramagneticstate\nabove the Curie temperature. The local moments still\nexist but they are oriented randomly in such a way that\nthe total magnetic moment is zero. From the theoret-\nical point of view the SDR can be simulated success-\nfully in the framework of the CPA as the resistivity of\nan equiconcentration alloy of spin moments pointing in\nopposite directions (the disordered local moment (DLM)\nstate).30The fluctuating local moments are then deter-\nmined selfconsistently in the framework of the DFT. In\nthe fcc regiononlylocalmomentsonFe-sitesarenonzero,\nall other collapse to zero. In the bcc region, in addition,\nlocal moments on Co atoms survive. Such result, in gen-\neral, is not correct as, e.g., the local DLM moment in\nfcc Ni collapses to zero but the experiment indicates its\nnonzero value at the Curie temperature. These values\ncan be found theoretically not only for fcc Ni,31but also\nfor binary alloys.32The situation is much more compli-\ncated forthe presentmulticomponent alloy. We therefore\ndetermine just the lower and upper limits of the SDR.\nThe lower limit is the above DLM result, the upper limit\ncorresponds to the DLM state which is constructed on\nthe basis of an FM solution assuming the frozen Fermi\nenergy and frozen potential parameters.31It is denoted\nasρSDR\nmax.\nTABLE II: The spin disorder resistivity (SDR, the resistiv-\nity due to spin fluctuations in the paramagnetic state) of\nAlxCrFeCoNi for two values of Al concentrations, namely,\nxAl= 0.25 (fcc) and xAl= 1.25 (bcc) are shown. We present\nthe SDR results for two models, one in which the SDR is iden-\ntified with the resistivity of the DLM state ( ρSDR\nDLM) and the\nother (ρSDR\nmax) in which the DLM state is constructed from the\ncorresponding FM solutions with frozen Fermi energies and\nfrozen potential parameters. For a comparison we also show\nconventional resistivities ( ρFM, see Fig. 3) and resistivities of\nnon-magnetic phases ( ρNM). All values are in µΩcm.\nxAlρFMρNMρSDR\nDLMρSDR\nmax\n0.25 (fcc) 84.7072.1683.9889.97\n1.25 (bcc) 135.03117.17132.17137.49Results for two Al concentrations are summarized in\nTable 2 in which we have added for a comparison also\nresistivities of the reference FM state and resistivities of\ncorresponding non-magnetic phases. We have following\ncomments: (i) The non-magnetic phases have slightly\nsmaller resistivities as compared not only to the DLM\nphases but also as compared to the reference FM phase.\nThe effect of magnetic scatterings is thus less relevant\nthan the effect of different atom types and their differ-\nent potentials; (ii) Slightly larger values of the reference\nρFMas compared to ρSDR\nDLMare due to the fact that in the\nDLM state in the fcc/bcc phase are non-zero only Fe and\nperhaps also Co moments. Missing magnetic scattering\nthus leads to smaller resistivities (see also discussion in\n(i)); and (iii) On the contrary, the ρSDR\nmaxis slightly larger\ndue to the presence of fluctuating moments on Cr, Fe,\nCo, and Ni atoms. One can thus conclude that due to\nalready large resistivity of the reference FM state, the\nspin disorder influences resistivity only weakly.\nD. Anisotropic magnetoresistance, anomalous Hall\nresistivity, and Gilbert damping\nWe calculate further quantities which are due to the\nspin-orbit coupling, namely, the AMR and the AHR for\ntwo typical Al-concentrations, namely, xAl= 0.25 (fcc\nphase) and xAl= 1.25(bcc phase). The relativistic input\nis needed to solve the K-B transport equation.9While we\nhave found no experimental data for the AMR, the AHR\ndata are available for the above two alloys.6\nTABLE III: Calculated AMR and AHR for Al xCrFeCoNi al-\nloys in the fcc ( xAl= 0.25) and bcc ( xAl= 1.25) phases. The\nAMR values are in % while the AHR values are in µΩcm.\nxAlAMRAHRthAHRexp\n0.25 (fcc) 0.0310.879 0.5\n1.25 (bcc) 0.0441.699 1.5\nCalculated results are summarized in Table 3 with the\nfollowing conclusions: (i) The AMR is positive, but its\nvalues are very small, considering the fact that, e.g., for\nNi-rich NiFe the AMR can be as large as 15%.20,33It was\nshown that large values of the AMR in fcc Ni-rich alloys\nare due to essentially disorder-free majority bands. On\nthe contrary, the Ni-rich NiMn alloy has disorder in both\nthe majority and minority bands and significantly lower\nAMR than Ni-rich NiFe, but still few times larger than\nthe present alloys. In addition to very similar disorder\nin both channels in present alloys, the other reason of\nsuch small AMR can be the ferrimagnetic rather than\nthe FM character of present alloys with the antiparallel\nCr moments. Its role plays also small total moment of\nstudied HEA alloys; and (ii) There is a good agreement\nbetween calculated and measured AHR for the bcc phase\n(xAl= 1.25)while the agreementfor the fcc phase ( xAl=\n0.25) is worse but still reasonable. We note that a good7\nagreement of both calculated resistivities and AHR with\nexperiment is a non-trivial result.\nWehaveestimatedGDparametersforthe sametypical\nconcentrations as above for the AHR. Calculated values\nof the GD parameter for fcc ( xAl= 0.25) and bcc ( xAl=\n1.25) phases are, respectively, 0.00655and 0.00585. Both\nvalues are similar which is compatible with similar values\noftheDOSattheFermilevelandthetotalmagnetization\nwhose ratio is a rough estimate of the GD parameter,\nwhich explains also rather large values due to small total\nspin moments in both alloys. Calculated values of GD\nparameter are comparable to those in Ni-rich fcc NiFe\nalloy but are larger as compared to bcc-FeCo alloy.16\nIV. CONCLUSIONS\nTransport properties of the fcc and bcc phases of the\nhigh-entropy Al xCrFeCoNi alloys were calculated over a\nbroad range of Al concentrations using the DFT-based\nsimulations. The main conclusions from numerical stud-\nies can be summarized as follows: (i) The agreement\nof calculated residual resistivities with available exper-\nimental data is good for both fcc and bcc phase. In\nparticular, the resistivity values as well as larger resis-\ntivity of the bcc-phase as compared to the fcc one agree\nwith both experiments. Calculation even reproduce de-\ntails of concentration trends in one of the experiment.6\n(ii) The major contribution to the residual resistivity is\ndue to the intrinsic chemical disorder while the magneticdisorder has smaller effect. The increase of ρ0with in-\ncreasing Al concentration in both fcc and bcc phases and\nits large values in particular in the latter one are due to\nstrong scatterings on Al atoms; (iii) The calculated val-\nuesofanisotropicmagnetoresistancearepositivebutvery\nsmall being less than 0.05% for both fcc and bcc phases;\n(iv) The spin disorder influences resistivity only weakly\nbecause of already large resistivity of the reference FM\nstate; (v) Estimated values of the Gilbert damping are\ncomparable for chosen typical fcc and bcc phases and are\nrather large (of order 0.006) due to small total spin mo-\nments; and (vi) The estimated anomalous Hall resistivity\nagain agrees well for the bcc phase while agreement with\nthe experiment for the fcc phase is worse though still\nacceptable.\nThe present results thus suggest that the CPA cap-\ntures the main scattering mechanism due to intrinsic\nalloy disorder and gives acceptable description even for\nsuch complex alloys like the studied one.\nAcknowledgments\nThe work of J.K., V.D, F.M., and I.T. was supported\nby a Grant from the Czech Science Foundation (No. 18-\n07172S) and S.K. thanks for support from the Center for\nComputational Materials Science, Vienna University of\nTechnology. 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B 68, 012402\n(2003)."    },    {        "title": "1308.3578v1.Quantum_Gilbert_Varshamov_Bound_Through_Symplectic_Self_Orthogonal_Codes.pdf",        "content": "arXiv:1308.3578v1  [cs.IT]  16 Aug 2013Quantum Gilbert-Varshamov Bound Through\nSymplectic Self-Orthogonal Codes\nLingfei Jin and Chaoping Xing\nAbstract—It is well known that quantum codes can be con-\nstructed through classical symplectic self-orthogonal co des. In\nthis paper, we give a kind of Gilbert-Varshamov bound for\nsymplectic self-orthogonal codes firstand thenobtain theG ilbert-\nVarshamov bound for quantum codes. The idea of obtaining the\nGilbert-Varshamov bound for symplectic self-orthogonal c odes\nfollows from counting arguments.\nIndex Terms —Symplectic self-orthogonal, Quantum Gilbert-\nVarshamov bound, Symplectic distance.\nI. INTRODUCTION\nInthe pastfew years,thetheoryof quantumerror-correctin g\ncodes have been developed rapidly. Various constructions a re\ngiven through classical coding. However, it is still a great\nchallenge to construct good quantum codes. Shor and Steane\ngave the first construction of quantum codes through classic al\nself-orthogonalcodes. Subsequently,constructionsof qu antum\ncodes through classical codes with certain self-orthogona lity\nhave been extensively studied and investigated. For instan ce,\nquantum codes can be obtained from Euclidean, Hermitian\nself-orthogonal or Symplectic self-orthogonal codes (see [1],\n[2], [6], [11], [12], [13]).\nAs in the classical coding theory, the quantum Gilbert-\nVarshomov(GV,forshort)boundisalsoabenchmarkforgood\nquantum codes. The first quantum GV bound was obtained in\n[3]forthebinarycase.Oneyearlater,AshikhminandKnill[ 2]\ngeneralized the binary quantum GV bound to the q-ary case.\nIn 2004, Feng and Ma [11] derived a finite version of Gilbert-\nVaranmovboundfor classical Hermitian self-orthogonalco des\nand then applied to quantum codes to obtain a finite version\nof quantum GV bound.\nIn 1998, Macwilliams and Sloane [17] showed through\ncounting arguments that binary self-dual codes can achieve\nthe GV bound for classical case. In this paper, we use the\ncounting arguments as well to show that classical symplecti c\nself-orthogonal codes can achieve the GV bound and then\napply to quantum codes to obtain the asymptotic quantum GV\nbound.\nOurpaperis organizedasfollows.We first recallsome basic\nnotations and results of symplectic slef-orthogonal codes in\nSection II. In section III, we present a kind of GV bound for\nsymplectic self-orthogonal codes through counting argume nts.\nIn section IV, we apply our result obtained in Section III to\nquantum codes and derive the asymptotic quantum GV bound.\nII. PRELIMINARIES\nWith the development of classical error-correcting codes,\npeople have extensively studied the Euclidean inner produc tand investigated the Euclidean self-orthogonal codes. How -\never, due to applications to quantum codes in recent years,\nother inner products such as Hermitian and symplectic inner\nproductshaveattractedresearchersinthisarea andmanyin ter-\nsecting results have been obtained already. In this section , we\nintroduce some basic results and notations about symplecti c\ninner products and symplectic slef-orthogonal codes.\nLetFqbe a finite field with qelements, where qis\na prime power. For four vectors a= (a1,...,a n),b=\n(b1,...,b n),a′= (a′\n1,...,a′\nn),b′= (b′\n1,...,b′\nn)∈Fn\nq, the\nsymplectic inner product /a\\}⌊ra⌋ketle{t,/a\\}⌊ra⌋ketri}htSis defined by\n/a\\}⌊ra⌋ketle{t(a|b),(a′|b′)/a\\}⌊ra⌋ketri}htS=/a\\}⌊ra⌋ketle{ta,b′/a\\}⌊ra⌋ketri}htE−/a\\}⌊ra⌋ketle{tb,a′/a\\}⌊ra⌋ketri}htE,\nwhere/a\\}⌊ra⌋ketle{t,/a\\}⌊ra⌋ketri}htEis defined as the ordinary dot inner product (or\nEuclidean inner product). For an Fq-linear code CinF2n\nq,\ndefine the symplectic dual of Cby\nC⊥S={(x|y) :/a\\}⌊ra⌋ketle{t(x|y),(a|b)/a\\}⌊ra⌋ketri}htS= 0for all(a|b)∈C}.\nIt is easy to show that dimFq(C) + dim Fq(C⊥S) = 2n. A\ncodeCis said symplectic self-orthogonal if C⊆C⊥S, and\nself-dual if C=C⊥S.\nFor an vector (a|b) = (a1,...,a n|b1,...,b n)∈F2n\nq, the\nsymplectic weight is defined by :\nwtS(a|b) =|{i: (ai,bi)/\\e}atio\\slash= (0,0),i= 1...n}|\nFor two vectors (a|b),(a′|b′)∈Fq2n, the symplectic distance\nis defined by :\ndS((a|b),(a′|b′)) = wt S(a−a′|b−b′).\nThe symplectic minimumdistance of a linear code C∈F2n\nq\nis defined by\ndS(C) =: min{wtS(a|b) : (a|b)∈C−{(0|0)}}.\nThen it is straightforwardto verify that a [2n,k]-linear code\nCalso satisfies the symplectic Singleton bound:\nk+2dS(C)≤2n+2.\nIII. GILBERT-VARSHAMOV BOUND FOR SYMPLECTIC\nSELF-ORTHOGONAL CODES\nIn the previous section, we saw the symplectic Singleton\nbound already. Similarly, we can derive the symplectic GV\nbound. However, for application to quantum codes, we are\ninterested in a GV type bound for symplectic self-orthogona l\ncodes. The main goal of this section is to derive such a bound\nthrough counting argument.\nFirst, we have the following simple but useful observation.Lemma 3.1: Everyvectorin F2n\nqisorthogonalto itself with\nsymplectic inner product.\nLemma 3.2: The number of symplectic self-orthogonal\ncodes of length 2nand dimension kin the vector space F2n\nq\nis given by\n(q−1)k−1(q2n−2k+2−1)(q2n−2k+4−1)...(q2n−1)\n(qk−1)(qk−1−1)...(q−1).\n(III.1)\nProof:For convenience, denote by Akthe number of\nsymplectic self-orthogonal codes of length 2nand dimension\nkand denote by C2n,ka symplectic self-orthogonal code\nof length 2nand dimension koverFq. Then for k≤n,\nC2n,k−1can be extended to C2n,kby adding a vector u∈\nC⊥S\n2n,k−1\\C2n,k−1. Thus, we can obtain |C⊥S\n2n,k−1/C2n,k−1|−\n1 =q2n−2(k−1)−1distinct symplectic self-orthogonal codes\nof length 2nand dimension kfromC2n,k−1through this way.\nOn the other hand, by the above argument, we know\nthat every symplectic self-orthogonal code of length 2nand\ndimension k−1is contained in a symplectic self-orthogonal\ncode of length 2nand dimension k. Since every subspace of\ndimension k−1ofC2n,kis symplectic self-orthogonal and\nthere are (qk−1)/(q−1)subspaces of dimension k−1in\nC2n,k, we get a recursive formula\nAk=(q−1)(q2n−2(k−1)−1)\nqk−1Ak−1\nfork= 2,...n. The desired result follows from the above\nrecursive formula and the fact that A1=q2n−1\nq−1.\nLemma 3.3: Given a nonzero vector u∈F2n\nq, the number\nof symplectic self-orthogonalcodes containing uof length 2n\nand dimension kis given by\n(q−1)k−1(q2n−2k+2−1)(q2n−2k+4−1)...(q2n−2−1)\n(qk−1−1)(qk−2−1)...(q−1).\n(III.2)\nProof:Similarly, we denote by C2n,k(u)a symplectic\nself-orthogonalcode containing uof length 2nand dimension\nkoverFqand denote by Bk(u)the number of such C2n,k(u).\nUsing the similar argumentsas in lemma 3.2, we can establish\nthe following recursive formula for Bk(u),\nBk(u) =(q−1)(q2n−2(k−1)−1)\nqk−1−1Bk−1(u).\nThe desired result follows from the above recursive formula\nand the fact that B1(u) = 1.\nTheorem 3.4 (GV bound): For1≤k≤nandd≥1, there\nexists a[2n,k,d]symplectic self-orthogonal code over Fqif\nd−1/summationdisplay\ni=1/parenleftbiggn\ni/parenrightbigg\n(q2−1)i<q2n−1\nqk−1.\nProof:AsBk(u)is independent of u, we denote by Bk\nthe number given in (III.2).\nFirst, the numberof nonzerovectors uof symplectic weight\nless than dis given by\nV(2n,d) =:i/summationdisplay\ni=1/parenleftbiggn\ni/parenrightbigg\n(q2−1)i.On the other hand, there are at most/summationtext\nwtS(u)<dBk(u) =\nV(2n,d)Bksymplectic self-orthogonal codes with symplectic\ndistance less than d. Therefore,there is at least one symplectic\nself-orthogonal code with symplectic distance at least dif\nAk> V(2n,d)Bk. The desired result follows from this\ninequality and Lemmas 3.2 and 3.3.\nTheorem 3.4 shows existence of symplectic self-orthogonal\ncodes with good symplectic distance. However, to con-\nstruct good quantumcodes throughsymplectic self-orthogo nal\ncodes, we have to control symplectic dual distance for a give n\nsymplectic self-orthogonal code.\nLemma 3.5: For a given nonzero vector u∈F2n\nq, the\nnumberEkof symplectic self-orthogonal codes Cof length\n2nand dimension ksuch that C⊥Scontainusatisfies the\nfollowing recursive formula\nEk=(q−1)(q2n−2(k−1)−1−1)\nqk−1Ek−1+(q−1)2q2n−2(k−1)−1\nqk−1Bk−1\n(III.3)\nfor anyk≥2, whereBkis the quantity defined in (III.2).\nProof:LetWbe the symplectic dual space of /a\\}⌊ra⌋ketle{tu/a\\}⌊ra⌋ketri}ht. Then\nC⊥Scontainsuif and only if Cis a subspace of W. Thus,\nEkstands for the number of symplectic self-orthogonal codes\nCof length 2nand dimension ksuch that C⊆W. We denote\nbyDkthe number of symplectic self-orthogonal codes Cof\nlength2nand dimension ksuch that u∈C⊆W. We also\ndenote by Fkthe number of symplectic self-orthogonal codes\nCof length 2nand dimension ksuch that u/\\e}atio\\slash∈C⊆W. Then\nit is easy to see that Dk=BkandDk+Fk=Ek.\nWithout confusion, we denote by C2n,ka symplectic self-\northogonal code of length 2nand dimension koverFqsuch\nthatC⊆W. Then for k≤n,C2n,k−1can be extended\ntoC2n,kby adding a vector v∈W∩(C⊥S\n2n,k−1\\C2n,k−1).\nThus, we can obtain |C⊥S\n2n,k−1/C2n,k−1|−1 =q2n−2(k−1)−1\n(orq2n−2(k−1)−1−1, respectively) distinct symplectic self-\northogonal codes of length 2nand dimension kfromC2n,k−1\nthrough this way if u∈C2n,k−1(or ifu/\\e}atio\\slash∈C2n,k−1,\nrespectively).\nOn the other hand, by the above argument, we know\nthat every symplectic self-orthogonal codes of length 2nand\ndimension k−1is contained in a symplectic self-orthogonal\ncodes of length 2nand dimension k. Since every subspace of\ndimension k−1ofC2n,kis symplectic self-orthogonal and\nthere are (qk−1)/(q−1)subspaces of dimension k−1in\nC2n,k, we get a recursive formula\nqk−1\nq−1Ek= (q2n−2(k−1)−1)Dk−1+(q2n−2(k−1)−1−1)Fk−1.\nThe desired reclusive formula follows.\nCorollary 3.6: LetEkstands for the same number defined\nin Lemma 3.5. Then one has\nEk≤k(q−1)k−1(q2n−2k+1−1)···(q2n−1−1)\n(qk−1)···(q−1)(III.4)\nfor allk≥1.\nProof:We know that E1=q2n−1−1\nq−1. So it is true for\nk= 1.Now assume that the result is also true for k−1. Then by\nLemma 3.5, we have\nEk=(q−1)(q2n−2(k−1)−1−1)\nqk−1Ek−1+\n(q−1)2q2n−2(k−1)−1\nqk−1Bk−1\n≤(k−1)(q−1)k−2(q2n−2k+1−1)···(q2n−1−1)\n(qk−1)···(q−1)+\n(q−1)2q2n−2(k−1)−1\nqk−1Bk−1\n= (k−1)(q−1)k−2(q2n−2k+1−1)···(q2n−1−1)\n(qk−1)···(q−1)+\n(q−1)kq2n−2k+1(q2n−2k+4−1)···(q2n−2−1)\n(qk−1)···(q−1)\n≤k(q−1)k−1(q2n−2k+1−1)···(q2n−1−1)\n(qk−1)···(q−1).\nThis completes the proof.\nBy using the same arguments in the proof of Theorem 3.4. we\nobtain the following result.\nCorollary 3.7: For1≤k≤nandd≥1, there exists\na[2n,k]symplectic self-orthogonal code CoverFqwith\ndS(C⊥S)≥dif\nd−1/summationdisplay\ni=1/parenleftbiggn\ni/parenrightbigg\n(q2−1)i</producttextk−1\ni=0(q2n−2i−1)\nk/producttextk−1\ni=0(q2n−2i−1−1).\nIV. APPLICATION TO QUANTUM GILBERT-VARSHAMOV\nBOUND\nFor a quantum code Q, we denote by n(Q),K(Q),d(Q)\nlength,dimensionand minimumdistance of Q, respectively.A\nfundamental domain UQ\nq⊆[0,1]×[0,1]is defined as follows.\nDefinition 4.1: UQ\nqis a set which is consisted of ordered\npairs(δ,R)∈[0,1]×[0,1]such that there exists a family of\nq-ary quantum codes {Qi}∞\ni=1satisfying\nn(Qi)→ ∞,R= lim\ni→∞logqK(Qi)/n(Qi),δ= lim\ni→∞d(Qi)/n(Qi).\nAs in the classical coding theory, determining the domain UQ\nq\nis one of the central asymptotic problem for quantum coding\ntheory. One can imagine that it is very hard to completely\ndetermine UQ\nq. Nevertheless, some bounds on UQ\nqhave been\ngiven by many researchers [2], [9], [10]. A useful descripti on\nofUQ\nqthrough a function αQ\nqwas given by Feng-Ling-Xing\n[10]:\nthere exists a function αQ\nq(δ),δ∈[0,1], such that\nUQ\nqis the union of the domain\n{(δ,R)∈R2: 0≤R < αQ\nq(δ),0≤δ≤1}\nwith some points on the boundary αQ\nq(δ).\nThus, determining UQ\nqis almost equivalent to determining the\nfunctionαQ\nq(δ).\nTo apply our results in the previous section, let us establis h\na connection between symplectic self-orthogonal codes and\nquantum codes.Lemma 4.2: [2] IfCis aq-ary symplectic self-orthogonal\n[2n,k]code, then there exists a q-ary[[n,n−k,d]]quantum\ncode with d=dS(C⊥S).\nThus, by combining Corollary 3.7 with Lemma 4.2, we\nimmediately get the following result.\nCorollary 4.3: For1≤k≤nandd≥1, there exists a\nq-ary[[n,n−k,d]]quantum code Cif\nd−1/summationdisplay\ni=1/parenleftbiggn\ni/parenrightbigg\n(q2−1)i</producttextk−1\ni=0(q2n−2i−1)\nk/producttextk−1\ni=0(q2n−2i−1−1).\nFinally we are ready to derive the quantum GV bound.\nTheorem 4.4:\nαQ\nq(δ)≥1−δlogq(q+1)−Hq(δ).\nwhereHq(x)is theq-ary entropy function xlogq(q−1)−\nxlogqx−(1−x)logq(1−x).\nProof:Fixδ∈(0,1). For every n≥1/δ, denote by d\nthe quantity ⌊δn⌋. Thend/ntends toδwhenntends to∞.\nPut\nk=/floorleftBigg\nlogq/parenleftBiggd−1/summationdisplay\ni=1/parenleftbiggn\ni/parenrightbigg\n(q2−1)i/parenrightBigg\n+logqk/floorrightBigg\n+1.\nThen we have\nd−1/summationdisplay\ni=1/parenleftbiggn\ni/parenrightbigg\n(q2−1)i<qk\nk</producttextk−1\ni=0(q2n−2i−1)\nk/producttextk−1\ni=0(q2n−2i−1−1).\nHence, we have a q-ary quantum [[n,n−k,d]]-code by\nCorollary 4.3. Therefore, one has\nn−k\nn= 1−k\nn→1−δlogq(q+1)−Hq(δ).\nThe proof is completed.\nREFERENCES\n[1] S. A. Aly, A. Klappenecker and P. K. Sarvepalli, “On quant um and\nclassical BCH codes,” IEEE Trans. Inf. Theory, vol. 53, no. 3, pp. 1183–\n1188, Mar. 2007.\n[2] A. Ashikhmin and E. Knill, “Nonbinary quantum stablizer codes,”IEEE\nTrans. Inf. Theory, vol. 47, no. 7, pp. 3065–3072, Nov. 2001.\n[3] A. Ashikhmin, A. M. Barg, E. Knill and S. N. Litsyn, “Quant um error\ndetection II: Bounds,” IEEE Trans. Inf. Theory, vol. 46, no. 3, pp. 789–\n800, May. 2000.\n[4] A. Ashikhmin and S. Litsyn, “Upper bounds on the size of qu antum\ncodes,”IEEE Trans. Inf. Theory, vol. 45, no. 4, pp. 1206–1215, May\n1999.\n[5] A. Ashikhmin, S. Litsyn and M. A. Tsfasman, ”Aymptotical ly good\nquantum codes,” Phys. Rev. A, vol. 63, no. 3, p. 032311, Mar. 2 001\n[6] J. Bierbrauer and Y. Edel, “Quantum twisted codes,” J. Comb. Designs,\nvol. 8, pp. 174–188, 2000.\n[7] A. R. Calderbank, E. M. Rains, P. W. Shor and N. J. A. Sloane ,\n“Quantum error correction via codes over GF(4),” IEEE Trans. Inf.\nTheory,vol. 44, no. 4, pp. 1369–1387, July 1998.\n[8] H. Chen, S. Ling and C. Xing, “Quantum codes from concaten ated\nalgebraic-geometric codes,” IEEE Trans. Inf. Theory, vol. 51, no. 8, pp.\n2915–2920, Aug. 2005.\n[9] H. Chen, S. Ling, and C. P. Xing, ”Aymptoticially good qua ntum\ncodes exceeding the Ashikhmin-Litsyn-Tsfasman bound,” IEEE. Trans.\nInform. Theory, vol. 47, no. 5, pp. 2055-2058, Jul. 2001.\n[10] K. Q. Feng, S. Ling and C. P. Xing, ”Aymptotic Bounds on Qu antum\nCodes From Algebraic Geometry Codes,” IEEE. Trans. Inform. Theory,\nvol. 52, no. 3, pp. 986-991, Mar. 2006.[11] K. Q. Feng and Z. Ma, “A finite Gilbert-Varshmov Bound for pure\nstabilizer quantum codes,” IEEE. Trans. Inform. Theory, vol. 50, no. 12,\npp. 3323-3325, Dec. 2004.\n[12] M.Grassland T.Beth,“Quantum BCHcodes,” Internation al Symposium\non Theoretical Electrical Engineering, Magdeburg, 1999, p p. 207-212.\n[13] A. Ketkar, A. Klappenecker, S. Kumar and P. Sarvepalli, “Nonbinary\nstablizer codes over finite fields,” IEEE. Trans. Inform. Theory, vol. 52,\nno. 11, pp. 4892–4914, Nov. 2006.\n[14] E.Knilland R. Laflamme,“Atheory ofquantum error-corr ecting codes,”\nPhys. Rev. A, vol. 55, no. 2, pp. 900–911, 1997.\n[15] Z. Li, L. J. Xing and X. M. Wang, “A family of asymptotical ly good\nquantum codes based on code concatenation,” IEEE. Trans. Inform.\nTheory,vol. 55, no. 8, pp.3821–3824, Aug. 2009.\n[16] S. Ling and C. P. Xing, Coding Theory – A First Course, Cambridge\nUniversity Press, 2004.\n[17] F. J. Macwilliams and N. J. Sloane, ”Good Self-Dual code s Exists,”\nDiscrete Mathematics, vol. 3, no. 1,2,3, pp. 153-162, Sep, 1 972.\n[18] E. M. Rains, “Nonbinary quantum codes,” IEEE. Trans. Inform. Theory,\nvol. 45, no. 6, pp.1827–1832, Sept. 1999.\n[19] J. H. Silverman, The Arithmetic of Elliptic Curves , Springer, New York,\n1986.\n[20] A. M. Steane, “Enlargement of Calderbank-Shor-Steane quantum\ncodes,”IEEE. Trans. Inform. Theory, vol. 45, no. 7, pp. 2492–2495,\nNov. 1999.\n[21] M. A. Tsfasman and S. G. Vl ˘adut, ”Algebraic-Geometric Codes. Ams-\nterdam,” The Netherlands:Kluwer, 1991."    },    {        "title": "2111.15142v1.First_and_second_order_magnetic_anisotropy_and_damping_of_europium_iron_garnet_under_high_strain.pdf",        "content": "1 \n First and second order magnetic anisotropy and damping of \neuropium iron garnet under high strain  \n \nVíctor H. Ortiz1, Bassim Arkook1, Junxue Li1, Mohammed Aldosary1, Mason Biggerstaff1, Wei \nYuan1, Chad Warren2, Yasuhiro Kodera2, Javier E. Garay2, Igor Barsukov1*, and Jing Shi1* \n \n1Department of Physics and Astronomy, University of California, Riverside, CA 92521, \nUSA  \n2 Department of Mechanical and Aerospace Engineering, University of California, San \nDiego, CA 92093, USA  \n \n \nUnderstanding and tailoring static and dynamic properties of magnetic insulator thin films \nis important for spintronic device applications. Here, we grow atomically flat epitaxial europium \niron garnet (EuIG) thin films by pulsed laser deposition on (111) -oriented garnet sub strates with a \nrange of  lattice parameters. By controlling the lattice mismatch between EuIG and the substrates, \nwe tune the strain in EuIG films from compressive to tensile regime, which is characterized by X -\nray diffraction. Using ferromagnetic resonance , we find that in addition to the first -order \nperpendicular magnetic anisotropy which depends linearly on the strain, there is a significant \nsecond -order one that has a quadratic strain dependence. Inhomogeneous linewidth of the \nferromagnetic resonance inc reases notably with increasing strain, while the Gilbert damping \nparameter remains nearly constant (≈ 2× 10-2). The se results provide valuable insight into the  spin \ndynamics in ferrimagnetic insulators and useful guidance for material synthesis and engineer ing \nof next -generation spintronics applications.  \n \n \n*: Corresponding authors: Igor Barsukov ( igorb@ucr.edu ) and Jing Shi ( jing.shi@ucr.edu )   2 \n Ferrimagnetic insulators (FMIs) have played an important role in uncovering a series of \nnovel spintronic effects such as spin Seebeck effect (SSE) and spin Hall magnetoresistance (SMR). \nIn addition, FMI thin films have proved to be an excellent source of proximity -induced \nferromagnetism in adjacent layers (e.g., heavy metals [1], graphene  [1] and topological \ninsulators  [2]) and of pure spin currents  [3–6]. FMIs have also been shown to be a superb medium \nfor magnon spin currents with a long decay length  [7,8]. Among FMIs, rare earth iron  garnets \n(REIGs) in particular have a plethora of desirable properties for practical applications: high Curie \ntemperature (T c > 550  K), strong chemical stability, and relatively large band gaps (~ 2.8 eV).  \nCompared to other magnetic materials, REIGs are distinct owing to their magnetoelastic \neffect with the magnetostriction coefficient ranging from -8.5×106  to +21 ×106 at room \ntemperature  [9] and up to two orders of magnitude increases at low temperatures  [10]. This  unique  \nfeature allows for tailoring ma gnetic anisotropy in REIG thin films via growth, for example, by \nmeans of controlling lattice mismatch with substrates, film thickness, oxygen pressure, and \nchemical substitution. In thin films, the magnetization usually prefers to be in the film plane due  \nto magnetic shape anisotropy; however, the competing perpendicular magnetic anisotropy (PMA) \ncan be introduced by utilizing magneto -crystalline anisotropy or interfacial strain, both of which \nhave been demonstrated through epitaxial growth  [11–14]. In the  study of Tb 3Fe5O12 (TbIG) and \nEu3Fe5O12 (EuIG) thin films, the PMA field H2ꓕ was found to be as high as 7 T under interfacial \nstrain  [11], much stronger than the demagnetizing field. While using strain is proven to be an \neffective way of manipulating magn etic anisotropy, it often comes at a cost of increasing magnetic \ninhomogeneity and damping of thin films  [15,16].  \nIn this work, we investigate the effect of strain on magnetic properties of (111) -oriented \nEuIG thin films  for the following reasons: (1) The  spin dynamics in EuIG bulk crystals is \nparticularly interesting but has not been studied thoroughly in the thin film form. Compared to \nother REIGs, the Eu3+ ions occupying the dodecahedral sites (c -site) should have the J = 0 ground \nstate according to the Hund’s rules, which do not contribute to the total magnetic moment; \ntherefore, EuIG thin films can potentially have a ferromagnetic resonance (FMR) linewidt h as \nnarrow as that of Y 3Fe5O12 (YIG)  [17,18] or Lu 3Fe5O12 (LuIG)  [19]. In EuIG crystals, a very \nnarrow linewidth (< 1 Oe)  [20] was indeed observed at low temperatures, but it showed a nearly \ntwo orders of magnitude increase at high temperatures, which ra ises fundamental questions \nregarding the damping mechanism responsible for this precipitous change. (2) Although it has 3 \n been shown that the uniaxial anisotropy can be controlled by moderate strain for different substrate \norientations and even in polycrysta lline form  [21], the emergence of the higher -order anisotropy \nat larger strain, despite its technological significance, has remained elusive.  \nWe grow EuIG films by pulsed laser deposition (PLD) from a target densified by powders \nsynthesized using the meth od described previously  [22]. The films are deposited on (111) -oriented \nGd3Sc2Ga3O12 (GSGG), Nd 3Ga5O12 (NGG), Gd 2.6Ca0.4Ga4.1Mg 0.25Zr0.65O12 (SGGG), \nY3Sc2Ga3O12 (YSGG), Gd3Ga5O12 (GGG), Tb 3Ga5O12 (TGG) and Y 3Al5O12 (YAG) single crystal \nsubstrates, with the lattice mismatch 𝜂=𝑎𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒 −𝑎𝐸𝑢𝐼𝐺\n𝑎𝐸𝑢𝐼𝐺 (where 𝑎 represents the lattice parameter \nof the referred material) ranging from +0.45% (GSGG) to -3.95% (YAG) in the decreasing order \n(see Table I). After the standard solvent cleaning process, the substrates are annealed at 220 °C \ninside the PLD chamber with the base pressure lower than 10-6 Torr for 5 hours prior to deposition . \nThen the temperature is increased to ~ 600 °C in the atmosphere of 1.5 mT orr oxygen mixed with \n12% (wt.) ozone for 30 minutes. A 248 nm KrF excimer pulsed laser is used to ablate the target \nwith a power of 156 mJ and a repetition rate of 1 Hz. We crystalize the films by ex situ   annealing \nat 800 °C for 200 s in a steady flow of  oxygen using rapid thermal annealing  (RTA) .  \nReflection high energy electron diffraction (RHEED) is used to evaluate the crystalline \nstructural properties of the EuIG films grown on various substrates (Fig. 1a). Immediately after \nthe deposition, RHEED dis plays the absence of any crystalline order. After ex situ  rapid thermal \nannealing, all EuIG films turn into single crystals. We carry out atomic force microscopy (AFM) \non all samples and find that they show atomic flatness and good uniformity with root -mean-square \n(RMS) roughness < 2 Å (Fig. 1b). In addition, we perform X -ray diffraction (XRD) on all samples \nusing a Rigaku SmartLab with Cu K α radiation with a Ni filter and Ge(220) mirror as \nmonochromators, at room temperature in 0.002° steps over the 2  range from 10° to 90°  [23]. In \na representative XRD spectrum (Fig. 1c), two (444) Bragg peaks are present, one from the 50 nm \nthick EuIG film and the other from the YSGG substrate, which confirms the epitaxial growth and \nsingle crystal structure of the fi lm without evidence of any secondary phases. Other REIG films \ngrown under similar conditions , i.e., by PLD in oxygen mixed with ozone at ~600 °C during \nfollowed by RTA, have shown no observable interdiffusion across the interface from high \nresolution trans mission electron microscopy  and energy dispersive X -ray spectroscopy  (Fig. S1 , \n[24]). The EuIG Bragg peak ( a0 = 12.497 Å) is shifted with respect to the expected peak position \nof unstrained bulk crystal, indicating a change in the EuIG lattice parameter pe rpendicular to the 4 \n surface ( aꓕ). For the example shown in Fig. 1c, the EuIG (444) peak shifts to left with respect to \nits bulk value, indicating an out -of-plane tensile strain and therefore an in -plane compressive strain \nin the EuIG lattice.  \n \nA common app roach for inferring the in -plane strain ε|| of thin films from the standard −2 \nXRD measurements involves the following equation  [23],  \n \n𝜀∥= −𝑐11+2 𝑐12+4 𝑐44\n2𝑐11+4 𝑐12−4 𝑐44 𝜀⊥,  with 𝜀⊥=𝑎⊥−𝑎𝑜\n𝑎𝑜,    (1) \n \nwhere a0 is the lattice parameter of the bulk material, and aꓕ can be calculated using 𝑎⊥=\n𝑑ℎ𝑘𝑙√ℎ2+𝑘2+𝑙2 from the interplanar distance 𝑑ℎ𝑘𝑙  obtained from the XRD data (Fig. S 2, [25]), \nand cij are the elastic stiffness constants of the crystal which in most  cases can be found in the \nliterature  [9]. However, due to the wide range of strain values studied in this work and the \npossibility that the films may contain different amounts of crystalline defects, we perform \nreciprocal space mapping (RSM) measurements on a subset  of our EuIG samples (Fig. S 3, [26]) \nand compared the measured in -plane lattice parameters with the calculated ones using Eq. 1. We \nobserve that the average in -plane strain s measured by RSM has a  systematic difference of 40% \nfrom  the calculated values based on the  elastic properties (Fig. S 4, [26]). Given this nearly constant \nfactor  for all measured films, we find that the elastic stiffness constants of our EuIG films may \ndeviate  from the literature  reported bulk values , possibly due to stochiometric deviations  or slight \nunit cell distortion  in thin films . Here we adopt the reported lattice parameter value ( a0 = 12.497 \nÅ) as the reference due to the difficulty of  grow ing sufficiently thick, unstrained EuIG films usin g \nPLD .  \nIn the thickness -tuned magnetic anisotropy study [11], the anisotropy field in REIG films is \nfound to be proportional to η/(t+t o), which was attributed to the relaxation of strain as the film \nthickness  t increases. Here in EuIG samples with small lattice mismatch η (e.g., NGG/EuIG), the \nstrain is mostly preserved in 50 nm thick films (pseudomorphic regime), whereas for larger η (e.g., \nYAG/EuIG ), the lattice parameter of EuIG films shows nearly complete structural relaxation to \nthe bulk value. For this reason, in the samples with larger η (YAG = -3.95 %, GSGG = 0.45%), \nwe grow thinner EuIG films (20 nm) in order to retain a larger in -plane strain (compressive for 5 \n YAG, tensile for GSGG).  For EuIG films gr own on TGG and GGG substrates, the paramagnetic \nbackground of the substrates is too large to obtain a reliable magnetic moment measurement of the \nEuIG films; therefore, the results of thinner films on these two substrates are not included in this \nstudy.  \nRoom-temperature magnetic hysteresis curves for YSGG/EuIG sample are shown in Fig. 1d \nwith the magnetic field applied parallel and perpendicular to the film  [26]. The saturation field for \nthe out -of-plane loop (~1100 Oe) is clearly larger than that for the i n-plane loop, indicating that \nthe magnetization prefers to lie in the film plane. Moreover, since the demagnetizing field 4π Ms \n(≈ 920 Oe) is less than the saturation field in the out -of-plane loop (Fig. S 5, [27]), it suggests the \npresence of additional easy -plane anisotropy result ing from the magnetoelastic effect due to \ninterfacial strain. As shown in this example, we can qualitatively track the evolution of the \nmagnetic anisotropy in samples with different strains. However, this approach cannot provide a \nquantitative description when high -order anisotropy contributions are involved.  \nTo quantitatively determine magnetic anisotropy in all EuIG films, we perform polar angle  \n(H)-dependent FMR measurements using an X -band microwave cavity with f requency f = 9.32 \nGHz and field modulation. The samples are rotated from H = 0° to H = 180° in 10° steps, where \nH = 90° corresponds to the field parallel to the sample plane (Fig. 2a). The spectra at Η = 0° for \nall samples are displayed in Fig. 2b and show a single resonance peak which can be  well fitted by \na Lorentzian derivative. Despite different strains in all s amples, the resonance field Hres is lower \nfor the in -plane direction ( H = 90°) than for the out -of-plane direction ( H = 0°). A quick \ninspection reveals that the out -of-plane Hres shifts to larger values as η increases in the positive \ndirection (e.g., fro m YAG/EuIG to GSGG/EuIG), corresponding to stronger easy -plane \nanisotropy. Furthermore, the Hres values at θΗ = 0° show a large spread among the samples. Fig. 2c \nshows a comparison of FMR spectra at different polar angles between two representative samples: \nNGG/EuIG (small η) and YAG/EuIG (large η). \nFigs. 3a -c show  Hres vs. θH for three representative EuIG films . To evaluate  magnetic \nanisotropy, we fit the data using the Smit -Beljers  formalism by considering the first -order  \n−𝐾1cos2𝜃  and the second -order −1\n2𝐾2cos4𝜃  uniaxial anisotropy energy terms  [28]. From this \nfitting, we extract the parameters 4𝜋𝑀𝑒𝑓𝑓=4𝜋𝑀𝑠− 2𝐾1\n𝑀𝑠= 4𝜋𝑀𝑠- 𝐻2⊥ and 𝐻4⊥= 2𝐾2\n𝑀𝑠 (see Table \nI), her e 𝐻2⊥ and 𝐻4⊥ being the  first- and second - order anisotropy fields, respectively , and favoring  6 \n out-of-plane (in -plane) orientation of magnetization when they are positive (negative). The \nspectroscopic  g-factor is treated as a fitted parameter  which is found as a nearly constant , g = \n1.40 (Fig. S 6, [28]), in accordance to the previous results obtained by Miyadai  [31]. In Figs. 3d \nand 3e, we present 𝐻2⊥ and 𝐻4⊥ as functions of the measured out -of-plane strain 𝜀⊥ and in -plane \nstrain 𝜀∥. Clear ly, the magnitude of  4𝜋𝑀𝑒𝑓𝑓 is greater than the demagnetizing field for EuIG \n4𝜋𝑀𝑠=920  𝑂𝑒; therefore,  𝐻2⊥ is negative for all samples, i.e., favoring the in -plane orientation. \nAs shown in  Fig. 3d, |𝐻2⊥| increases linearly with increasing in -plane strain η. This is consistent \nwith the magnetoelastic effect in (111) -oriented EuIG films  [9]. As briefly discussed earlier,  due \nto the constant scaling factor between the calculated and measured 𝜀∥, we rewrite t he \nmagnetoelastic contribution to the first -order perpendicular anisotropy as −9𝛯\n3𝑀𝑠𝜀⊥, with the \nparameter 𝛯 containing the information related to the magnetoelastic constant λ111 and elastic \nstiffness cii. We fit the magnetoelastic equation in Ref.  [11] using the parameter 𝛯 and obtain 𝛯=\n−(7.06±0.95)×104 𝑑𝑦𝑛𝑒\n𝑐𝑚2 from the slope. On the other hand, based on  the reported literature \nvalues ( 𝜆111=+1.8×10−6, c11 = 25.10 ×1011 dyne/cm2, c12 = 10.70 ×1011 𝑑𝑦𝑛𝑒\n𝑐𝑚2, c44 = 7.62 ×1011 \n𝑑𝑦𝑛𝑒\n𝑐𝑚2) [10], we obtain  𝛯𝑙𝑖𝑡=−6.12×104 𝑑𝑦𝑛𝑒\n𝑐𝑚2. This result suggests  that even though  the actual \nelastic properties of our EuIG films may be different from the ones reported in for EuIG crystals  \ndue to  the thin film unit cell distortion (Table S1 , [32]), the pertaining parameter 𝛯 appears to be \nrelatively insensitive to variations of stoichiometry . The intercept of the  straight -line fit  should \ngive the magneto -crystalline anisotropy coefficient  of EuIG  Kc.  We find Kc = (+62.76 ± 0.18 ) × \n103 erg/cm3, which is differ ent from the previously reported values for EuIG bulk crystals in both \nthe magnitude and sign ( Kc = -38 × 103 erg/cm3) [31]. Similar growth -modified magneto -\ncrystalline anisotropy was observed in EuIG  films grown with relatively lo w temperatures \n(requiring post -deposition annealing to crystalize)  [10]. In the absence of  interfacial interdiffusion, \nthe anomalous anisotropy may be related to partial deviation from the chemical ordering of the \ngarnet structure  [31].  \n By comparing the first - and second -order anisotropy fields 𝐻2⊥ and 𝐻4⊥ vs. 𝜀∥ plotted in \nFigs. 3d and 3e, we find that the former dominates over the entire range of 𝜀∥ (except for \nYAG/EuIG). In contrast to the linear dependence for 𝐻2⊥, 𝐻4⊥ can be fitted well with a quadratic \n𝜀∥ dependence , which is not surprising for materials with large magnetostriction constants (such 7 \n as EuIG) under large strains. For relatively small 𝜀∥, the linear strain term in the magnetic \nanisotropy energy  dictates . For large 𝜀∥, higher -order strain terms may not be neglected. By \nincluding the ( 𝜀∥cos2θ)2 term, we obtain excellent fitting to the FMR data, indicating that the \nsecond -order expansion in 𝜀∥ is adequate. In contrast to  𝐻2⊥, 𝐻4⊥ is always positive, thus favoring \nout-of-plane magnetization orientation. It is worth pointing out that for YAG and TGG, the \nmagnitude of the 𝐻2⊥ becomes comparable with that of the 𝐻4⊥, but the sign differ s. Comparison \nof 𝐻4⊥ with 4𝜋𝑀𝑒𝑓𝑓 reveals that a coexistence (bi -stable) magnetic state can be realized when \n𝐻4⊥>4𝜋𝑀𝑒𝑓𝑓 [31, 33 -35]. The results are summarized in Table I.  \nThe above magnetic anisotropy energy analysis only deals with  the polar angle  dependence , \nbut in principle, it can also vary in the film plane and  therefore depend on the azimuthal angle.  To \nunderstand the latter, w e perform azimuthal angle dependent FMR measurements on all samples. \nWe indeed observe a six -fold in -plane anisotropy in Hres due to the crystalline symmetry  of EuIG  \n(111). However, the amplitude of  the six -fold Hres variation is less than 15 Oe, about two orders \nof magnitude smaller than the average value of Hres for most samples, thus we omit the in-plane \nanisotropy in our analysis.  \nBesides the Hres information, t he FMR spectra  in Fig. 2 c reveal s significant variations in \nFMR linewidth, which contains information  of magnetic inhomogeneity and Gilbert damping. To \ninvestigate these properties systematically, we perform broad -band (up to 15  GHz) FMR \nmeasurements with m agnetic field applied in the film plane, using a coplanar waveguide setup. \nFrom the frequency dependence of Hres, we obtain 4𝜋𝑀𝑒𝑓𝑓 and g independently via fitting the data \nwith the Kittel equation. These values agree very well with those previously found from the polar \nangle dependence. We plot the half width at half maximum, ∆𝐻, as a function of frequency  f in \nFig. 4a. While ∆𝐻 varies significantly across the samples, the data for  each sample fall \napproximately on a straight line and the slope of ∆𝐻 vs. 𝑓 appears to be visibly close  to each other. \nFor a quantitative evaluation of ∆𝐻, we consider the following contributions: the Gilbert damping \n∆𝐻𝐺𝑖𝑙𝑏𝑒𝑟𝑡 , two -magnon scatt ering ∆𝐻𝑇𝑀𝑆, and the inhomogeneous linewidth ∆𝐻0 [36], \n \n∆𝐻=∆𝐻𝐺𝑖𝑙𝑏𝑒𝑟𝑡 +∆𝐻𝑇𝑀𝑆 +∆𝐻0 .     (3) \n 8 \n The Gilbert term, ∆𝐻𝐺𝑖𝑙𝑏𝑒𝑟𝑡 =2𝜋𝛼𝑓\n|𝛾|, depends linearly on f, where α is the Gilbert damping \nparameter; the two -magnon term is described through ∆𝐻𝑇𝑀𝑆 =𝛤0𝑎𝑟𝑐𝑠𝑖𝑛  √√𝑓2+(𝑓𝑜\n2)2\n−𝑓𝑜\n2\n√𝑓2+(𝑓𝑜\n2)2\n+𝑓𝑜\n2  [37], \nwhere  𝛤0 denotes the magnitude of the two -magnon scattering, f0 = 2γMeff; and ∆𝐻0, the \ninhomogeneous linewidth w hich is frequency independent.  \nBy fitting Eq. (3) to the linewidth data, we obtain quantitative information on magnetic \ndamping through the Gilbert parameter and two -magnon scattering magnitude as well as the \nmagnetic inhomogeneity  [39–40]. In Fig. 4a, the overall linear behavior for all samples is an \nindication of a relatively small two -magnon scattering contribution ∆𝐻𝑇𝑀𝑆 which therefore may \nbe disregarded in the fitting process. Figs. 4b and 4c  show both  ∆𝐻0 and α vs. 𝜀∥. It is cl ear that \nfour of the samples with the smallest ∆𝐻0 (~ 10 Oe) are those with relatively low in -plane strain \n(|𝜀∥|<0.30% ). In the meantime, the XRD spectra  of these samples show  fringes characteristic of \nwell conformed crystal planes (Fig. S 2), and moreover,  the RSM plots (Fig. S 3) reveal a uniform \nstrain distribution in the films  [41]. On the compressive strain side, ∆𝐻0 increases steeply to 400 \nOe at 𝜀∥ ~ -0.40 %, and their XRD spectra  show no fringes and the RSM graphs indicate  non-\nuniform strain relaxation in the samples (Figs. S2 and S3 ). In sharp contrast to the ∆𝐻0 trend, the \nGilbert damping α remains about 2 ×10-2 over the entire range of 𝜀∥, sugges ting that the intrinsic \nmagneti c damping of EuIG films is nearly unaffected by the inhomogeneity. In fact, the magnitude \nof α is significantly larger than that of YIG  [17,18] or LuIG films  [19], which is somewhat \nunexpected for Eu3+ in EuIG with J = 0. A possible reason for this enhance d damping is that other \nvalence states of Eu such as Eu2+ (J =7/2) may be present, which leads to non -zero magnetic \nmoments of Eu ions in the EuIG  lattice and thus results in a larger damping constant, common to \nother REIG with non -zero 4f -moments  [42]. The X -ray photoelectron spectroscopy data taken on \nYSGG(111)/EuIG(50 nm) (Fig. S7 , [43]) indicates such a possibility. While the FMR linewidth \npresents large variations across the sample set, we have identified that the non-uniform strain \nrelaxation process caused by large lattice mismatch with the substrate is a main source of the \ninhomogeneity linewidth ∆𝐻0, but it does not affect the Gilbert damp ing α. The results raise \ninteresting questions on the mechanisms of intrinsic damping and the origin of magnetic \ninhomogeneity in EuIG thin films , both of which  warrant further investigations.  9 \n In summary, we find that uniaxial magnetic anisotropy in PLD -grown EuIG(111) thin films \ncan be tuned over a wide range via magnetostriction and lattice -mismatch induced strain. The first -\norder anisotropy field depends linearly on the strain and the second order anisotropy field has a \nquadratic dependence. While non -uniform strain relaxation significantly increases the magnetic \ninhomogeneity, the Gilbert damping remains nearly constant over a wide range of in -plane strain. \nThe results demonstrate broad tunab ility of magnetic properties in REIG films and provide \nguidance for implementation of EuIG for spintronic applications. Further studies to elucidate the \nrole of Eu2+ sites in magnetic damping  are called upon.  \n \nWe thank Dong Yan and Daniel Borchardt for the ir technical assistance. This work was supported \nas part of the SHINES, an Energy Frontier Research Center funded by the US Department of \nEnergy, Office of Science, Basic Energy Sciences under Award No. SC0012670. J.S. \nacknowledges support by DOE BES Award  No. DE -FG02 -07ER46351 and I.B. acknowledges \nsupport by the National Science Foundation under grant number NSF -ECCS -1810541.  \n  10 \n References  \n \n[1] Z. Wang, C. Tang, R. Sachs, Y. Barlas, and J. Shi, Proximity -Induced Ferromagnetism in \nGraphene Revealed by the Anomalous Hall Effect , Phys. Rev. Lett. 114, 016603 (2015).  \n[2] Z. Jiang, C. -Z. Chang, C. Tang, P. Wei, J. S. Moodera, and J. Shi, Independent Tuning of \nElectronic Properties and Induced Ferromagnetism in Topological Insulators with \nHeterostructure Approach , Nano Lett. 15, 5835 (2015).  \n[3] K. 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Interfaces 13, 20288 (2021).  \n[40] I. Barsukov, H. K. Lee, A. A. Jara, Y. J. Chen, A. M. Gonçalves, C. Sha, J. A. Katine, R. \nE. Arias, B. A. Ivanov, and I. N. Krivorotov, Giant Nonlinear Damping in Nanoscale \nFerromagnets , Sci. Adv. 5, 1 (2019).  \n[41] X. Guo, A. H. Tavakoli, S. Sutton, R. K. Kukkadapu, L. Qi, A. Lanzirotti, M. Newville, \nM. Asta, and A. Navrotsky, Cerium Substitution in Yttrium Iron Garnet: Valence State, \nStructure, and Energetics , (2013).  \n[42] C. Tang, P. Sellappan, Y. Liu, Y. Xu, J. E. Garay, and J. Shi, Anomalous Hall Hysteresis \nin Tm 3Fe5O12/Pt with Strain -Induced Perpendicular Magnetic Anisotropy , Phys. Rev. B \n94, 140403 (2016).  \n[43]    See Supplemental Information figure S7 at http://placeholder.html for the XPS spectra and \nanalysis.  \n \n 14 \n Figures  \n \n \n \n \n \n \n \n \nFigure  SEQ Figure \\* ARABIC 1 : Structural and magnetic property characterization of \nEuIG 50 nm film grown on YSGG(111) substrate. (a) Reflection high energy electron \ndiffraction (RHEED) pattern along the direction, displaying single crystal structure after \nrapid thermal anneal ing process. (b) 2 mm  2 mm atomic force microscope (AFM) surface \nmorphology scan, demonstrating a root -mean -square (RMS) roughness of 1.7 Å. (c) \nIntensity semi -log plot of \n  - 2\n XRD scan. The dashed line corresponds to the XRD peak \nfor bulk EuIG. (d) Mag netization hysteresis loops for field out -of-plane and in -plane \ndirections.  Figure 1: Structural and magnetic property characterization of EuIG 50 nm film grown on \nTGG(111)  substrate . (a) Reflection high  energy electron diffraction  (RHEED ) pattern along the \n⟨112⟩ direction, displaying single crystal structure after rapid thermal annealing process. (b) 5 mm \n 5 mm atomic force microscope ( AFM ) surface morphology scan, demonstrating  a root-mean -\nsquare (RMS) roughness of 1.8 Å. (c) Intensity semi -log plot of  - 2 XRD scan. The dashed line \ncorresponds to the XRD peak for bulk EuIG. (d) Magnetization hysteresis loops for field out -of-\nplane and in -plane directions.  15 \n  \n \nFigure 2  Polar angle dependent ferromagnetic resonance (FMR). (a) Coordinate system used for \nthe FMR measurement. (b) Room temperature FMR derivative absorption spectra for θH = 0° (out -\nof-plane configuration) for EuIG on different (111) substrates. (c) FMR derivative absorption \nspectra for 50 nm EuIG grown on NGG(111) ( 𝜀∥ ≈ 0)  and 20 nm EuIG on YAG(111) (𝜀∥< 0) with \npolar angle θH ranging from 0° (out -of-plane) to 90° ( in-plane) at 300 K, where 𝜀∥ is in-plane strain \nbetween the EuIG film and substrate.  \n \n \n \n16 \n  \nFigure 3  Polar angle dependent ferromagnetic resonance field Hres for (a) tensile in -plane strain \n(𝜀∥ > 0), (b) in -plane strain close to zero ( 𝜀∥ ≈ 0), and (c) compressive in -plane strain ( 𝜀∥ < 0). Solid \ncurves represent the best fitting results. In -plane strain dependence of the anisotropy fields H 2ꓕ (d) \nand  H 4ꓕ (e). \n \n \n \n \n \n17 \n  \nFigure 4  FMR linewidth and magnetic damping of EuIG films as a function of in -plane strain. (a) \nHalf width at half maximum ∆𝐻 vs. frequency f for EuIG films grown on different substrates, with \nthe corresponding fitting according to Eq. (3).  In -plane strain depen dence of inhomogeneous \nlinewidth ΔH0 (b) and Gilbert parameter α (c).  \n \n \n \n \n \n \n \n \n \n \n \n \n \n18 \n  \n \nSubstrate  asubstrate  \n(Å) η \n(%) t \n(nm)  𝜀∥ (%) 𝜀⊥ (%) g H2ꓕ \n(Oe) H4ꓕ \n(Oe) α (×10-2) ΔHo \n(Oe) Γo (Oe) \nGSGG  12.554  0.45 50 0.34 -0.16 1.40 -1394.2 \n± 44.9  339.79 \n± 6.59  2.46 ± \n0.03 21.4 ± \n1.3 2.61 \n   25 0.46 -0.21 1.41 -1543.6 \n± 39.7  709.47 \n± 27.5  1.58 ± \n0.06 10.2 ± \n1.7 6.05 \nNGG  12.508  0.06 50 0.12 -0.06 1.38 -1224.4 \n± 5.7  18.34 ± \n0.05 2.41 \n±0.01  8.9 ± \n0.7 0.20 \nSGGG  12.480  -\n0.14 50 -0.13 0.06 1.40 -909.6 \n± 15.2  164.8 ± \n1.36 2.13 ± \n0.01 5.6 ± \n0.4 0.50 \nYSGG  12.426  -\n0.57 50 -0.27 0.12 1.37 -709.4 \n± 22.0  377.3 ± \n5.09 2.47 ± \n0.03 9.9 ± \n1.8 2.47 \nGGG  12.383  -\n0.92 50 -0.45 0.21 1.38 -1015.0 \n± 81.3  887.2 ± \n37.27  2.20 ± \n0.14 412.2 \n± 8.4  3.35 \nTGG  12.355  -\n1.14 50 -0.38 0.18 1.38 -393.4 \n± 53.6  245.0 ± \n10.00  2.29 ± \n0.20 253.4 \n± 11.8  0.20 \nYAG  12.004  -\n3.95 20 -0.42 0.20 1.37 -36.8 ± \n47.1 424.8 ± \n20.91  1.86 ± \n0.20 217.0 \n± 22.6  0.20 \n \nTable 1  Structural and magnetic parameters for the EuIG thin films grown on different substrates.  "    },    {        "title": "1409.2340v1.Self_similar_solutions_of_the_one_dimensional_Landau_Lifshitz_Gilbert_equation.pdf",        "content": "arXiv:1409.2340v1  [math.AP]  8 Sep 2014Self-similar solutions of the one-dimensional\nLandau–Lifshitz–Gilbert equation\nSusana Gutiérrez1and André de Laire2\nAbstract\nWe consider the one-dimensional Landau–Lifshitz–Gilbert (LLG) equation, a model des-\ncribing the dynamics for the spin in ferromagnetic material s. Our main aim is the analytical\nstudy of the bi-parametric family of self-similar solution s of this model. In the presence of\ndamping, our construction provides a family of global solut ions of the LLG equation which\nare associated to a discontinuous initial data of infinite (t otal) energy, and which are smooth\nand have finite energy for all positive times. Special emphas is will be given to the behaviour\nof this family of solutions with respect to the Gilbert dampi ng parameter.\nWe would like to emphasize that our analysis also includes th e study of self-similar so-\nlutions of the Schrödinger map and the heat flow for harmonic m aps into the 2-sphere as\nspecial cases. In particular, the results presented here re cover some of the previously known\nresults in the setting of the 1d-Schrödinger map equation.\nKeywords and phrases: Landau–Lifshitz–Gilbert equation, Landau–Lifshitz equa tion, ferro-\nmagnetic spin chain, Schrödinger maps, heat-flow for harmon ic maps, self-similar solutions,\nasymptotics.\nContents\n1 Introduction and statement of results 2\n2 Self-similar solutions of the LLG equation 10\n3 Integration of the Serret–Frenet system 12\n3.1 Reduction to the study of a second order ODE . . . . . . . . . . . . . . . . . . . 12\n3.2 The second-order equation. Asymptotics . . . . . . . . . . . . . . . . . . . . . . . 15\n3.3 The second-order equation. Dependence on the parameter s . . . . . . . . . . . . 28\n3.3.1 Dependence on α. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28\n3.3.2 Dependence on c0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33\n4 Proof of the main results 34\n5 Some numerical results 37\n6 Appendix 41\n1School of Mathematics, University of Birmingham, Edgbasto n, Birmingham, B15 2TT, United Kingdom.\nE-mail:s.gutierrez@bham.ac.uk\n2Laboratoire Paul Painlevé, Université Lille 1, 59655 Ville neuve d’Ascq Cedex, France. E-mail:\nandre.de-laire@math.univ-lille1.fr\n11 Introduction and statement of results\nIn this work we consider the one-dimensional Landau–Lifshi tz–Gilbert equation (LLG)\n∂t/vectorm =β/vectorm×/vector mss−α/vectorm×(/vectorm×/vectormss), s∈R, t>0, (LLG)\nwhere/vectorm = (m 1,m2,m3) :R×(0,∞)−→S2is the spin vector, β≥0,α≥0,×denotes the\nusual cross-product in R3, andS2is the unit sphere in R3.\nHere we have not included the effects of anisotropy or an exter nal magnetic field. The first term\non the right in (LLG) represents the exchange interaction, w hile the second one corresponds to the\nGilbert damping term and may be considered as a dissipative t erm in the equation of motion. The\nparameters β≥0andα≥0are the so-called exchange constant and Gilbert damping coe fficient,\nand take into account the exchange of energy in the system and the effect of damping on the\nspin chain respectively. Note that, by considering the time -scaling/vectorm(s,t)→/vectorm(s,(α2+β2)1/2t),\nin what follows we will assume w.l.o.g. that\nα, β∈[0,1] andα2+β2= 1. (1.1)\nThe Landau–Lifshitz–Gilbert equation was first derived on p henomenological grounds by L. Lan-\ndau and E. Lifshitz to describe the dynamics for the magnetiz ation or spin /vectorm(s,t)in ferromag-\nnetic materials [24, 11]. The nonlinear evolution equation (LLG) is related to several physical\nand mathematical problems and it has been seen to be a physica lly relevant model for several\nmagnetic materials [19, 20]. In the setting of the LLG equati on, of particular importance is to\nconsider the effect of dissipation on the spin [27, 7, 6].\nThe Landau–Lifshitz family of equations includes as specia l cases the well-known heat-flow\nfor harmonic maps and the Schrödinger map equation onto the 2-sphere. Precisely, when β= 0\n(and therefore α= 1) the LLG equation reduces to the one-dimensional heat-flow equation for\nharmonic maps\n∂t/vectorm =−/vectorm×(/vectorm×/vectormss) =/vectormss+|/vectorms|2/vectorm (HFHM)\n(notice that |/vectorm|2= 1, and in particular /vectorm·/vectormss=−|/vectorms|2). The opposite limiting case of the\nLLG equation (that is α= 0, i.e. no dissipation/damping and therefore β= 1) corresponds to\ntheSchrödinger map equation onto the sphere\n∂t/vectorm =/vectorm×/vectormss. (SM)\nBoth special cases have been objects of intense research and w e refer the interested reader to\n[21, 14, 25, 13] for surveys.\nOf special relevance is the connection of the LLG equation wi th certain non-linear Schrödinger\nequations. This connection is established as follows: Let u s suppose that /vectormis the tangent vector\nof a curve in R3, that is/vectorm =/vectorXs, for some curve /vectorX(s,t)∈R3parametrized by the arc-length. It\ncan be shown [7] that if /vectormevolves under (LLG) and we define the so-called filament funct ionu\nassociated to /vectorX(s,t)by\nu(s,t) = c(s,t)ei/integraltexts\n0τ(σ,t)dσ, (1.2)\nin terms of the curvature cand torsion τassociated to the curve, then usolves the following\nnon-local non-linear Schrödinger equation with damping\niut+(β−iα)uss+u\n2/parenleftbigg\nβ|u|2+2α/integraldisplays\n0Im(¯uus)−A(t)/parenrightbigg\n= 0, (1.3)\nwhereA(t)∈Ris a time-dependent function defined in terms of the curvatur e and torsion\nand their derivatives at the point s= 0. The transformation (1.2) was first considered in the\n2undamped case by Hasimoto in [18]. Notice that if α= 0, equation (1.3) can be transformed\ninto the well-known completely integrable cubic Schröding er equation.\nThe main purpose of this paper is the analytical study of self -similar solutions of the LLG\nequation of the form\n/vectorm(s,t) =/vector m/parenleftbiggs√\nt/parenrightbigg\n, (1.4)\nfor some profile /vector m:R→S2, with emphasis on the behaviour of these solutions with resp ect to\nthe Gilbert damping parameter α∈[0,1].\nForα= 0, self-similar solutions have generated considerable inte rest [22, 21, 4, 15, 9]. We are\nnot aware of any other study of such solutions for α >0in the one dimensional case (see [10]\nfor a study of self-similar solutions of the harmonic map flow in higher dimensions). However,\nLipniacki [26] has studied self-similar solutions for a rel ated model with nonconstant arc-length.\nOn the other hand, little is known analytically about the effe ct of damping on the evolution\nof a one-dimensional spin chain. In particular, Lakshmanan and Daniel obtained an explicit\nsolitary wave solution in [7, 6] and demonstrated the dampin g of the solution in the presence\nof dissipation in the system. In this setting, we would like t o understand how the dynamics of\nself-similar solutions to this model is affected by the intro duction of damping in the equations\ngoverning the motion of these curves.\nAs will be shown in Section 2 self-similar solutions of (LLG) of the type (1.4) constitute a\nbi-parametric family of solutions {/vector mc0,α}c0,αgiven by\n/vectormc0,α(s,t) =/vector mc0,α/parenleftbiggs√\nt/parenrightbigg\n, c 0>0, α∈[0,1], (1.5)\nwhere/vector mc0,αis the solution of the Serret–Frenet equations\n\n\n/vector m′=c/vector n,\n/vector n′=−c/vector m+τ/vectorb,\n/vectorb′=−τ/vector n,(1.6)\nwith curvature and torsion given respectively by\ncc0,α(s) =c0e−αs2\n4, τc0,α(s) =βs\n2, (1.7)\nand initial conditions\n/vector mc0,α(0) = (1,0,0), /vector nc0,α(0) = (0,1,0),/vectorbc0,α(0) = (0,0,1). (1.8)\nThe first result of this paper is the following:\nTheorem 1.1. Letα∈[0,1],c0>01and/vector mc0,αbe the solution of the Serret–Frenet system\n(1.6)with curvature and torsion given by (1.7)and initial conditions (1.8). Define/vectormc0,α(s,t)by\n/vectormc0,α(s,t) =/vector mc0,α/parenleftbiggs√\nt/parenrightbigg\n, t> 0.\nThen,\n1The case c0= 0corresponds to the constant solution for (LLG), that is\n/vectormc0,α(s,t) =/vector m/parenleftbiggs√\nt/parenrightbigg\n= (1,0,0),∀α∈[0,1].\n3(i) The function /vectormc0,α(·,t)is a regular C∞(R;S2)-solution of (LLG) fort>0.\n(ii) There exist unitary vectors /vectorA±\nc0,α= (A±\nj,c0,α)3\nj=1∈S2such that the following pointwise\nconvergence holds when tgoes to zero:\nlim\nt→0+/vectormc0,α(s,t) =\n\n/vectorA+\nc0,α,ifs>0,\n/vectorA−\nc0,α,ifs<0,(1.9)\nwhere/vectorA−\nc0,α= (A+\n1,c0,α,−A+\n2,c0,α,−A+\n3,c0,α).\n(iii) Moreover, there exists a constant C(c0,α,p)such that for all t>0\n/bardbl/vectormc0,α(·,t)−/vectorA+\nc0,αχ(0,∞)(·)−/vectorA−\nc0,αχ(−∞,0)(·)/bardblLp(R)≤C(c0,α,p)t1\n2p, (1.10)\nfor allp∈(1,∞). In addition, if α>0,(1.10) also holds for p= 1. Here,χEdenotes the\ncharacteristic function of a set E.\nThe graphics in Figure 1 depict the profile /vector mc0,α(s)for fixedc0= 0.8and the values of\nα= 0.01,α= 0.2, andα= 0.4. In particular it can be observed how the convergence of /vector mc0,α\nto/vectorA±\nc0,αis accelerated by the diffusion α.\nm1m2m3\n(a)α= 0.01\nm1m2m3\n(b)α= 0.2\nm1m2m3\n(c)α= 0.4\nFigure 1: The profile /vector mc0,αforc0= 0.8and different values of α.\nNotice that the initial condition\n/vectormc0,α(s,0) =/vectorA+\nc0,αχ(0,∞)(s)+/vectorA−\nc0,αχ(−∞,0)(s), (1.11)\nhas a jump singularity at the point s= 0whenever the vectors /vectorA+\nc0,αand/vectorA−\nc0,αsatisfy\n/vectorA+\nc0,α/ne}ationslash=/vectorA−\nc0,α.\nIn this situation (and we will be able to prove analytically t his is the case at least for certain ranges\nof the parameters αandc0, see Proposition 1.5 below), Theorem 1.1 provides a bi-para metric\nfamily of global smooth solutions of (LLG) associated to a di scontinuous singular initial data\n(jump-singularity).\n4As has been already mentioned, in the absence of damping ( α= 0), singular self-similar\nsolutions of the Schrödinger map equation were previously o btained in [15], [22] and [4]. In this\nframework, Theorem 1.1 establishes the persistence of a jum p singularity for self-similar solutions\nin the presence of dissipation.\nSome further remarks on the results stated in Theorem 1.1 are in order. Firstly, from the\nself-similar nature of the solutions /vectormc0,α(s,t)and the Serret–Frenet equations (1.6), it follows\nthat the curvature and torsion associated to these solution s are of the self-similar form and given\nby\ncc0,α(s,t) =c0√\nte−αs2\n4t andτc0,α(s,t) =βs\n2√\nt. (1.12)\nAs a consequence, the total energy E(t)of the spin /vectormc0,α(s,t)found in Theorem 1.1 is expressed\nas\nE(t) =1\n2/integraldisplay∞\n−∞|/vectorms(s,t)|2ds=1\n2/integraldisplay∞\n−∞c2\nc0,α(s,t)ds\n=1\n2/integraldisplay∞\n−∞/parenleftbiggc0√\nte−αs2\n4t/parenrightbigg2\nds=c2\n0/radicalbiggπ\nαt, α> 0, t>0. (1.13)\nIt is evident from (1.13) that the total energy of the spin cha in at the initial time t= 0is infinite,\nwhile the total energy of the spin becomes finite for all posit ive times, showing the dissipation\nof energy in the system in the presence of damping.\nSecondly, it is also important to remark that in the setting o f Schrödinger equations, for fixed\nα∈[0,1]andc0>0, the solution /vectormc0,α(s,t)of (LLG) established in Theorem 1.1 is associated\nthrough the Hasimoto transformation (1.2) to the filament fu nction\nuc0,α(s,t) =c0√\nte(−α+iβ)s2\n4t, (1.14)\nwhich solves\niut+(β−iα)uss+u\n2/parenleftbigg\nβ|u|2+2α/integraldisplays\n0Im(¯uus)−A(t)/parenrightbigg\n= 0,withA(t) =βc2\n0\nt(1.15)\nand is such that at initial time t= 0\nuc0,α(s,0) = 2c0/radicalbig\nπ(α+iβ)δ0.\nHereδ0denotes the delta distribution at the point s= 0and√zdenotes the square root of a\ncomplex number zsuch that Im(√z)>0.\nNotice that the solution uc0,α(s,t)is very rough at initial time, and in particular uc0,α(s,0)\ndoes not belong to the Sobolev class Hsfor anys≥0. Therefore, the standard arguments (that\nis a Picard iteration scheme based on Strichartz estimates a nd Sobolev-Bourgain spaces) cannot\nbe applied at least not in a straightforward way to study the l ocal well-posedness of the initial\nvalue problem for the Schrödinger equations (1.15). The exi stence of solutions of the Scrödinger\nequations (1.15) associated to an initial data proportiona l to a Dirac delta opens the question\nof developing a well-posedness theory for Schrödinger equa tions of the type considered here to\ninclude initial data of infinite energy. This question was ad dressed by A. Vargas and L. Vega\nin [29] and A. Grünrock in [12] in the case α= 0and whenA(t) = 0 (see also [2] for a related\nproblem), but we are not aware of any results in this setting w henα >0(see [14] for related\nwell-posedness results in the case α >0for initial data in Sobolev spaces of positive index).\nNotice that when α>0, the solution (1.14) has infinite energy at the initial time, however the\n5energy becomes finite for any t>0. Moreover, as a consequence of the exponential decay in the\nspace variable when α>0,uc0,α(t)∈Hm(R), for allt>0andm∈N. Hence these solutions do\nnot fit into the usual functional framework for solutions of t he Schrödinger equations (1.15).\nAs already mentioned, one of the main goals of this paper is to study both the qualitative and\nquantitative effect of the damping parameter αand the parameter c0on the dynamical behaviour\nof the family {/vectormc0,α}c0,αof self-similar solutions of (LLG) found in Theorem 1.1. Pre cisely, in an\nattempt to fully understand the regularization of the solut ion at positive times close to the initial\ntimet= 0, and to understand how the presence of damping affects the dyn amical behaviour of\nthese self-similar solutions, we aim to give answers to the f ollowing questions:\nQ1: Can we obtain a more precise behaviour of the solutions /vector mc0,α(s,t)at positive times tclose\nto zero?\nQ2: Can we understand the limiting vectors /vectorA±\nc0,αin terms of the parameters c0andα?\nIn order to address our first question, we observe that, due to the self-similar nature of these\nsolutions (see (1.5)), the behaviour of the family of soluti ons/vectormc0,α(s,t)at positive times close to\nthe initial time t= 0is directly related to the study of the asymptotics of the ass ociated profile\n/vector mc0,α(s)for large values of s. In addition, the symmetries of /vector mc0,α(s)(see Theorem 1.2 below)\nallow to reduce ourselves to obtain the behaviour of the profi le/vector mc0,α(s)for large positive values\nof the space variable. The precise asymptotics of the profile is given in the following theorem.\nTheorem 1.2 (Asymptotics) .Letα∈[0,1],c0>0and{/vector mc0,α,/vector nc0,α,/vectorbc0,α}be the solution of\nthe Serret–Frenet system (1.6)with curvature and torsion given by (1.7)and initial conditions\n(1.8). Then,\n(i) (Symmetries). The components of /vector mc0,α(s),/vector nc0,α(s)and/vectorbc0,α(s)satisfy respectively that\n•m1,c0,α(s)is an even function, and mj,c0,α(s)is an odd function for j∈ {2,3}.\n•n1,c0,α(s)andb1,c0,α(s)are odd functions, while nj,c0,α(s)andbj,c0,α(s)are even func-\ntions forj∈ {2,3}.\n(ii) (Asymptotics). There exist an unit vector /vectorA+\nc0,α∈S2and/vectorB+\nc0,α∈R3such that the following\nasymptotics hold for all s≥s0= 4/radicalbig\n8+c2\n0:\n/vector mc0,α(s) =/vectorA+\nc0,α−2c0\ns/vectorB+\nc0,αe−αs2/4(αsin(/vectorφ(s))+βcos(/vectorφ(s)))\n−2c2\n0\ns2/vectorA+\nc0,αe−αs2/2+O/parenleftBigg\ne−αs2/4\ns3/parenrightBigg\n, (1.16)\n/vector nc0,α(s) =/vectorB+\nc0,αsin(/vectorφ(s))+2c0\ns/vectorA+\nc0,ααe−αs2/4+O/parenleftBigg\ne−αs2/4\ns2/parenrightBigg\n, (1.17)\n/vectorbc0,α(s) =/vectorB+\nc0,αcos(/vectorφ(s))+2c0\ns/vectorA+\nc0,αβe−αs2/4+O/parenleftBigg\ne−αs2/4\ns2/parenrightBigg\n. (1.18)\nHere,sin(/vectorφ)andcos(/vectorφ)are understood acting on each of the components of /vectorφ= (φ1,φ2,φ3),\nwith\nφj(s) =aj+β/integraldisplays2/4\ns2\n0/4/radicalbigg\n1+c2\n0e−2ασ\nσdσ, j∈ {1,2,3}, (1.19)\n6for some constants a1,a2,a3∈[0,2π), and the vector /vectorB+\nc0,αis given in terms of /vectorA+\nc0,α=\n(A+\nj,c0,α)3\nj=1by\n/vectorB+\nc0,α= ((1−(A+\n1,c0,α)2)1/2,(1−(A+\n2,c0,α)2)1/2,(1−(A+\n3,c0,α)2)1/2).\nAs we will see in Section 2, the convergence and rate of conver gence of the solutions /vectormc0,α(s,t)\nof the LLG equation established in parts (ii)and(iii)of Theorem 1.1 will be obtained as a con-\nsequence of the more refined asymptotic analysis of the assoc iated profile given in Theorem 1.2.\nWith regard to the asymptotics of the profile established in p art(ii)of Theorem 1.2, it is\nimportant to mention the following:\n(a) The errors in the asymptotics in Theorem 1.2- (ii)depend only on c0. In other words,\nthe bounds for the errors terms are independent of α∈[0,1]. More precisely, we use the\nnotationO(f(s))to denote a function for which exists a constant C(c0)>0depending on\nc0, but independent on α, such that\n|O(f(s))| ≤C(c0)|f(s)|,for alls≥s0. (1.20)\n(b) The terms /vectorA+\nc0,α,/vectorB+\nc0,α,B+\njsin(aj)andB+\njcos(aj),j∈ {1,2,3}, and the error terms in\nTheorem 1.2- (ii)depend continuously on α∈[0,1](see Subsection 3.3 and Corollary 3.14).\nTherefore, the asymptotics (1.16)–(1.18) show how the profi le/vector mc0,αconverges to /vector mc0,0as\nα→0+and to/vector mc0,1asα→1−. In particular, we recover the asymptotics for /vector mc0,0given\nin [15].\n(c) We also remark that using the Serret–Frenet formulae and the asymptotics in Theorem 1.2-\n(ii), it is straightforward to obtain the asymptotics for the der ivatives of/vectormc0,α(s,t).\n(d) Whenα= 0and for fixed j∈ {1,2,3}, we can write φjin (1.19) as\nφj(s) =aj+s2\n4+c2\n0ln(s)+C(c0)+O/parenleftbigg1\ns2/parenrightbigg\n,\nand we recover the logarithmic contribution in the oscillat ion previously found in [15].\nMoreover, in this case the asymptotics in part (ii)represents an improvement of the one\nestablished in Theorem 1 in [15].\nWhenα>0,φjbehaves like\nφj(s) =aj+βs2\n4+C(α,c0)+O/parenleftBigg\ne−αs2/2\nαs2/parenrightBigg\n, (1.21)\nand there is no logarithmic correction in the oscillations i n the presence of damping.\nConsequently, the phase function /vectorφdefined in (1.19) captures the different nature of the\noscillatory character of the solutions in both the absence a nd the presence of damping in\nthe system of equations.\n(e) Whenα= 1, there exists an explicit formula for /vector mc0,1,/vector nc0,1and/vectorbc0,1, and in particular\nwe have explicit expressions for the vectors /vectorA±\nc0,1in terms of the parameter c0>0in the\nasymptotics given in part (ii). See Appendix.\n7(f) At first glance, one might think that the term −2c2\n0/vectorA+\nc0,αe−αs2/2/s2in (1.16) could be\nincluded in the error term O(e−αs2/4/s3). However, we cannot do this because\ne−αs2/2\ns2>e−αs2/4\ns3, for all2≤s≤/parenleftbigg2\n3α/parenrightbigg1/2\n, α∈(0,1/8], (1.22)\nand in our notation the big- Omust be independent of α. (The exact interval where the\ninequality in (1.22) holds can be determined using the so-ca lled Lambert Wfunction.)\n(g) Let/vectorB+\nc0,α,sin= (Bjsin(aj))3\nj=1,/vectorB+\nc0,α,cos= (Bjcos(aj))3\nj=1. Then the orthogonality of\n/vector mc0,α,/vector nc0,αand/vectorbc0,αtogether with the asymptotics (1.16)–(1.18) yield\n/vectorA+\nc0,α·/vectorB+\nc0,α,sin=/vectorA+\nc0,α·/vectorB+\nc0,α,cos=/vectorB+\nc0,α,sin·/vectorB+\nc0,α,cos= 0,\nwhich gives relations between the phases.\n(h) Finally, the amplitude of the leading order term control ling the wave-like behaviour of the\nsolution/vector mc0,α(s)around/vectorA±\nc0,αfor values of ssufficiently large is of the order c0e−αs2/4/s,\nfrom which one observes how the convergence of the solution t o its limiting values /vectorA±\nc0,αis\naccelerated in the presence of damping in the system. See Fig ure 1.\nWe conclude the introduction by stating the results answeri ng the second of our questions. Pre-\ncisely, Theorems 1.3 and 1.4 below establish the dependence of the vectors /vectorA±\nc0,αin Theorem 1.1\nwith respect to the parameters αandc0. Theorem 1.3 provides the behaviour of the limiting\nvector/vectorA+\nc0,αfor a fixed value of α∈(0,1)and “small” values of c0>0, while Theorem 1.4 states\nthe behaviour of /vectorA+\nc0,αfor fixedc0>0andαclose to the limiting values α= 0andα= 1. Recall\nthat/vectorA−\nc0,αis expressed in terms of the coordinates of /vectorA+\nc0,αas\n/vectorA−\nc0,α= (A+\n1,c0,α,−A+\n2,c0,α,−A+\n3,c0,α) (1.23)\n(see part (ii)of Theorem 1.1).\nTheorem 1.3. Letα∈[0,1],c0>0, and/vectorA+\nc0,α= (A+\nj,c0,α)3\nj=1be the unit vector given in\nTheorem 1.2. Then /vectorA+\nc0,αis a continuous function of c0>0. Moreover, if α∈(0,1]the\nfollowing inequalities hold true:\n|A+\n1,c0,α−1| ≤c2\n0π\nα/parenleftbigg\n1+c2\n0π\n8α/parenrightbigg\n, (1.24)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleA+\n2,c0,α−c0/radicalbig\nπ(1+α)√\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤c2\n0π\n4+c2\n0π\nα√\n2/parenleftBigg\n1+c2\n0π\n8+c0/radicalbig\nπ(1+α)\n2√\n2/parenrightBigg\n+/parenleftbiggc2\n0π\n2√\n2α/parenrightbigg2\n,(1.25)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleA+\n3,c0,α−c0/radicalbig\nπ(1−α)√\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤c2\n0π\n4+c2\n0π\nα√\n2/parenleftBigg\n1+c2\n0π\n8+c0/radicalbig\nπ(1−α)\n2√\n2/parenrightBigg\n+/parenleftbiggc2\n0π\n2√\n2α/parenrightbigg2\n.(1.26)\nThe following result provides an approximation of the behav iour of/vectorA+\nc0,αfor fixedc0>0and\nvalues of the Gilbert parameter close to 0and1.\nTheorem 1.4. Letc0>0,α∈[0,1]and/vectorA+\nc0,αbe the unit vector given in Theorem 1.2. Then\n/vectorA+\nc0,αis a continuous function of αin[0,1], and the following inequalities hold true:\n|/vectorA+\nc0,α−/vectorA+\nc0,0| ≤C(c0)√α|ln(α)|,for allα∈(0,1/2], (1.27)\n|/vectorA+\nc0,α−/vectorA+\nc0,1| ≤C(c0)√\n1−α,for allα∈[1/2,1]. (1.28)\nHere,C(c0)is a positive constant depending on c0but otherwise independent of α.\n8As a by-product of Theorems 1.3 and 1.4, we obtain the followi ng proposition which asserts\nthat the solutions /vectormc0,α(s,t)of the LLG equation found in Theorem 1.1 are indeed associate d\nto a discontinuous initial data at least for certain ranges o fαandc0.\nProposition 1.5. With the same notation as in Theorems 1.1 and 1.2, the followi ng statements\nhold:\n(i) For fixed α∈(0,1)there exists c∗\n0>0depending on αsuch that\n/vectorA+\nc0,α/ne}ationslash=/vectorA−\nc0,α for allc0∈(0,c∗\n0).\n(ii) For fixed c0>0, there exists α∗\n0>0small enough such that\n/vectorA+\nc0,α/ne}ationslash=/vectorA−\nc0,α for allα∈(0,α∗\n0).\n(iii) For fixed 0<c0/ne}ationslash=k√πwithk∈N, there exists α∗\n1>0with1−α∗\n1>0small enough such\nthat\n/vectorA+\nc0,α/ne}ationslash=/vectorA−\nc0,α for allα∈(α∗\n1,1).\nRemark 1.6. Based on the numerical results in Section 5, we conjecture th at/vectorA+\nc0,α/ne}ationslash=/vectorA−\nc0,αfor\nallα∈[0,1)andc0>0.\nWe would like to point out that some of our results and their pr oofs combine and extend\nseveral ideas previously introduced in [15] and [16]. The ap proach we use in the proof of the\nmain results in this paper is based on the integration of the S erret–Frenet system of equations\nvia a Riccati equation, which in turn can be reduced to the stu dy of a second order ordinary\ndifferential equation given by\nf′′(s)+s\n2(α+iβ)f′(s)+c2\n0\n4e−αs2\n2f(s) = 0 (1.29)\nwhen the curvature and torsion are given by (1.7).\nUnlike in the undamped case, in the presence of damping no exp licit solutions are known\nfor equation (1.29) and the term containing the exponential in the equation (1.29) makes it\ndifficult to use Fourier analysis methods to study analytical ly the behaviour of the solutions to\nthis equation. The fundamental step in the analysis of the be haviour of the solutions of (1.29)\nconsists in introducing new auxiliary variables z,handydefined by\nz=|f|2, y= Re(¯ff′)andh= Im(¯ff′)\nin terms of solutions fof (1.29), and studying the system of equations satisfied by t hese key\nquantities. As we will see later on, these variables are the “ natural” ones in our problem, in the\nsense that the components of the tangent, normal and binorma l vectors can be written in terms\nof these quantities. It is important to emphasize that, in or der to obtain error bounds in the\nasymptotic analysis independent of the damping parameter α(and hence recover the asymptotics\nwhenα= 0andα= 1as particular cases), it will be fundamental to exploit the c ancellations\ndue to the oscillatory character of z,yandh.\nThe outline of this paper is the following. Section 2 is devot ed to the construction of the family\nof self-similar solutions {/vectormc0,α}c0,αof the LLG equation. In Section 3 we reduce the study of the\nproperties of this family of self-similar solutions to that of the properties of the solutions of the\ncomplex second order complex ODE (1.29). This analysis is of independent interest. Section 4\ncontains the proofs of the main results of this paper as a cons equence of those established in\n9Section 3. In Section 5 we give provide some numerical result s for/vectorA+\nc0,α, as a function of α∈[0,1]\nandc0>0, which give some inside for the scattering problem and justi fy Remark 1.6. Finally,\nwe have included the study of the self-similar solutions of t he LLG equation in the case α= 1\nin Appendix.\nAcknowledgements. S. Gutiérrez and A. de Laire were supported by the British proj ect\n“Singular vortex dynamics and nonlinear Schrödinger equat ions” (EP/J01155X/1) funded by\nEPSRC. S. Gutiérrez was also supported by the Spanish projec ts MTM2011-24054 and IT641-\n13.\nBoth authors would like to thank L. Vega for many enlightening conversations and for his\ncontinuous support.\n2 Self-similar solutions of the LLG equation\nFirst we derive what we will refer to as the geometric represe ntation of the LLG equation. To\nthis end, let us assume that /vectorm(s,t) =/vectorXs(s,t)for some curve /vectorX(s,t)inR3parametrized with\nrespect to the arc-length with curvature c(s,t)and torsion τ(s,t). Then, using the Serret–Frenet\nsystem of equations (1.6), we have\n/vectormss= cs/vectorn+c(−c/vectorn+τ/vectorb),\nand thus we can rewrite (LLG) as\n∂t/vectorm =β(cs/vectorb−cτ/vectorn)+α(cτ/vectorb+cs/vectorn), (2.1)\nin terms of intrinsic quantities c,τand the Serret–Frenet trihedron {/vectorm,/vectorn,/vectorb}.\nWe are interested in self-similar solutions of (LLG) of the f orm\n/vectorm(s,t) =/vector m/parenleftbiggs√\nt/parenrightbigg\n(2.2)\nfor some profile /vector m:R−→S2. First, notice that due to the self-similar nature of /vectorm(s,t)in (2.2),\nfrom the Serret–Frenet equations (1.6) it follows that the u nitary normal and binormal vectors\nand the associated curvature and torsion are self-similar a nd given by\n/vectorn(s,t) =/vector n/parenleftbiggs√\nt/parenrightbigg\n,/vectorb(s,t) =/vectorb/parenleftbiggs√\nt/parenrightbigg\n, (2.3)\nc(s,t) =1√\ntc/parenleftbiggs√\nt/parenrightbigg\nandτ(s,t) =1√\ntτ/parenleftbiggs√\nt/parenrightbigg\n. (2.4)\nAssume that /vectorm(s,t)is a solution of the LLG equation, or equivalently of its geom etric version\n(2.1). Then, from (2.2)–(2.4) it follows that the Serret–Fr enet trihedron {/vector m(·),/vector n(·),/vectorb(·)}solves\n−s\n2c/vector n=β(c′/vectorb−cτ/vector n)+α(cτ/vectorb+c′/vector n), (2.5)\nAs a consequence,\n−s\n2c=αc′−βcτ andβc′+αcτ= 0.\nThus, we obtain\nc(s) =c0e−αs2\n4andτ(s) =βs\n2, (2.6)\n10for some positive constant c0(recall that we are assuming w.l.o.g. that α2+β2= 1). Therefore,\nin view of (2.4), the curvature and torsion associated to a se lf-similar solution of (LLG) of the\nform (2.2) are given respectively by\nc(s,t) =c0√\nte−αs2\n4tandτ(s,t) =βs\n2t, c 0>0. (2.7)\nNotice that given (c,τ)as above, for fixed time t>0one can solve the Serret–Frenet system of\nequations to obtain the solution up to a rigid motion in the sp ace which in general may depend\nont. As a consequence, and in order to determine the dynamics of t he spin chain, we need\nto find the time evolution of the trihedron {/vectorm(s,t),/vectorn(s,t),/vectorb(s,t)}at some fixed point s∗∈R.\nTo this end, from the above expressions of the curvature and t orsion associated to /vectorm(s,t)and\nevaluating the equation (2.1) at the point s∗= 0, we obtain that /vectormt(0,t) =/vector0. On the other\nhand, differentiating the geometric equation (2.1) with res pect tos, and using the Serret–Frenet\nequations (1.6) together with the compatibility condition /vectormst=/vectormts, we get the following relation\nfor the time evolution of the normal vector\nc/vectornt=β(css/vectorb+c2τ/vectorm−cτ2/vectorb)+α((cτ)s/vectorb−ccs/vectorm+csτ/vectorb).\nThe evaluation of the above identity at s∗= 0together with the expressions for the curvature\nand torsion in (2.7) yield /vectornt(0,t) =/vector0. The above argument shows that\n/vectormt(0,t) =/vector0, /vectornt(0,t) =/vector0and/vectorbt(0,t) = (/vectorm×/vectorn)t(0,t) =/vector0.\nTherefore we can assume w.l.o.g. that\n/vectorm(0,t) = (1,0,0), /vectorn(0,t) = (0,1,0)and/vectorb(0,t) = (0,0,1),\nand in particular\n/vector m(0) =/vectorm(0,1) = (1,0,0), /vector n(0) =/vectorn(0,1) = (0,1,0),and/vectorb(0) =/vectorb(0,1) = (0,0,1).(2.8)\nGivenα∈[0,1]andc0>0, from the theory of ODE’s, it follows that there exists a uniq ue\n{/vector mc0,α(·),/vector nc0,α(·),/vectorbc0,α(·)} ∈/parenleftbig\nC∞(R;S2)/parenrightbig3solution of the Serret–Frenet equations (1.6) with\ncurvature and torsion (2.6) and initial conditions (2.8) su ch that\n/vector mc0,α⊥/vector nc0,α, /vector mc0,α⊥/vectorbc0,α, /vector nc0,α⊥/vectorbc0,α\nand\n|/vector mc0,α|2=|/vector nc0,α|2=|/vectorbc0,α|2= 1.\nDefine/vectormc0,α(s,t)as\n/vectormc0,α(s,t) =/vector mc0,α/parenleftbiggs√\nt/parenrightbigg\n. (2.9)\nThen,/vector mc0,α(·,t)∈ C∞/parenleftbig\nR;S2/parenrightbig\nfor allt>0, and bearing in mind both the relations in (2.3)–(2.4)\nand the fact that the vectors {/vector mc0,α(·),/vector nc0,α(·),/vectorbc0,α(·)}satisfy the identity (2.5), a straightfor-\nward calculation shows that /vector mc0,α(·,t)is a regular C∞(R;S2)-solution of the LLG equation for\nallt>0. Notice that the case c0= 0yields the constant solution /vector m0,α(s,t) = (1,0,0). Therefore\nin what follows we will assume that c0>0.\nThe rest of the paper is devoted to establish analytical prop erties of the solutions {/vectormc0,α(s,t)}c0,α\ndefined by (2.9) for fixed α∈[0,1]andc0>0. As already mentioned, due to the self-similar\nnature of these solutions, it suffices to study the properties of the associated profile /vector mc0,α(·)or,\nequivalently, of the solution {/vector mc0,α,/vector nc0,α,/vectorbc0,α}of the Serret–Frenet system (1.6) with curvature\nand torsion given by (2.6) and initial conditions (2.8). As w e will continue to see, the analysis\nof the profile solution {/vector mc0,α,/vector nc0,α,/vectorbc0,α}can be reduced to the study of the properties of the\nsolutions of a certain second order complex differential equ ation.\n113 Integration of the Serret–Frenet system\n3.1 Reduction to the study of a second order ODE\nClassical changes of variables from the differential geomet ry of curves allow us to reduce the nine\nequations in the Serret–Frenet system into three complex-v alued second order equations (see\n[8, 28, 23]). Theses changes of variables are related to ster eographic projection and this approach\nwas also used in [15]. However, their choice of stereographi c projection has a singularity at the\norigin, which leads to an indetermination of the initial con ditions of some of the new variables.\nFor this reason, we consider in the following lemma a stereog raphic projection that is compatible\nwith the initial conditions (2.8). Although the proof of the lemma below is a slight modification\nof that in [23, Subsections 2.12 and 7.3], we have included it s proof here both for the sake of\ncompleteness and to clarify to the unfamiliar reader how the integration of the Frenet equations\ncan be reduced to the study of a second order differential equa tion.\nLemma 3.1. Let/vector m= (mj(s))3\nj=1,/vector n= (nj(s))3\nj=1and/vectorb= (bj(s))3\nj=1be a solution of the Serret–\nFrenet equations (1.6)with positive curvature cand torsion τ. Then, for each j∈ {1,2,3}the\nfunction\nfj(s) =e1\n2/integraltexts\n0c(σ)ηj(σ)dσ,withηj(s) =(nj(s)+ibj(s))\n1+mj(s),\nsolves the equation\nf′′\nj(s)+/parenleftbigg\niτ(s)−c′(s)\nc(s)/parenrightbigg\nf′\nj(s)+c2(s)\n4fj(s) = 0, (3.1)\nwith initial conditions\nfj(0) = 1, f′\nj(0) =c(0)(nj(0)+ibj(0))\n2(1+mj(0)).\nMoreover, the coordinates of /vector m,/vector nand/vectorbare given in terms of fjandf′\njby\nmj(s) = 2/parenleftBigg\n1+4\nc(s)2/vextendsingle/vextendsingle/vextendsingle/vextendsinglef′\nj(s)\nfj(s)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenrightBigg−1\n−1, nj(s)+ibj(s) =4f′\nj(s)\nc(s)fj(s)/parenleftBigg\n1+4\nc(s)2/vextendsingle/vextendsingle/vextendsingle/vextendsinglef′\nj(s)\nfj(s)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenrightBigg−1\n.\n(3.2)\nThe above relations are valid at least as long as mj>−1and|fj|>0.\nProof. For simplicity, we omit the index j. The proof relies on several transformations that are\nrather standard in the study of curves. First we define the com plex function\nN= (n+ib)ei/integraltexts\n0τ(σ)dσ. (3.3)\nThenN′=iτN+ (n′+ib′)ei/integraltexts\n0τ(σ)dσ. On the other hand, the Serret–Frenet equations imply\nthat\nn′+ib′=−cm−iτNe−i/integraltexts\n0τ(σ)dσ.\nTherefore, setting\nψ=cei/integraltexts\n0τ(σ)dσ,\nwe get\nN′=−ψm. (3.4)\nUsing again the Serret–Frenet equations, we also obtain\nm′=1\n2(ψN+ψN). (3.5)\n12Let us consider now the auxiliary function\nϕ=N\n1+m. (3.6)\nDifferentiating and using (3.4), (3.5) and (3.6)\nϕ′=N′\n1+m−Nm′\n(1+m)2\n=N′\n1+m−ϕm′\n1+m\n=−ϕ2ψ\n2−ψ\n2(1+m)(2m+ϕN).\nNoticing that we can recast the relation m2+n2+b2= 1asNN= (1−m)(1+m)and recalling\nthe definition of ϕin (3.6), we have ϕN= 1−m, so that\nϕ′+ϕ2ψ\n2+ψ\n2= 0. (3.7)\nFinally, define the stereographic projection of (m,n,b)by\nη=n+ib\n1+m. (3.8)\nObserve that from the definitions of Nandϕ, respectively in (3.3) and (3.6), we can rewrite η\nas\nη=ϕe−i/integraltexts\n0τ(σ)dσ,\nand from (3.7) it follows that ηsolves the Riccati equation\nη′+iτη+c\n2(η2+1) = 0, (3.9)\n(recall that ψ=cei/integraltexts\n0τ(σ)dσ). Finally, setting\nf(s) =e1\n2/integraltexts\n0c(σ)η(σ)dσ, (3.10)\nwe get\nη=2f′\ncf(3.11)\nand equation (3.1) follows from (3.9). The initial conditio ns are an immediate consequence of\nthe definition of ηandfin (3.8) and (3.10).\nA straightforward calculation shows that the inverse trans formation of the stereographic pro-\njection is\nm=1−|η|2\n1+|η|2, n=2Reη\n1+|η|2, b=2Imη\n1+|η|2,\nso that we obtain (3.2) using (3.11) and the above identities .\nGoing back to our problem, Lemma 3.1 reduces the analysis of t he solution {/vector m,/vector n,/vectorb}of the\nSerret–Frenet system (1.6) with curvature and torsion give n by (2.6) and initial conditions (2.8)\nto the study of the second order differential equation\nf′′(s)+s\n2(α+iβ)f′(s)+c2\n0\n4e−αs2/2f(s) = 0, (3.12)\n13with three initial conditions: For (m1,n1,b1) = (1,0,0)the associated initial condition for f1is\nf1(0) = 1, f′\n1(0) = 0, (3.13)\nfor(m2,n2,b2) = (0,1,0)is\nf2(0) = 1, f′\n2(0) =c0\n2, (3.14)\nand for(m3,n3,b3) = (0,0,1)is\nf3(0) = 1, f′\n3(0) =ic0\n2. (3.15)\nIt is important to notice that, by multiplying (3.12) by ¯f′and taking the real part, it is easy to\nsee that\nd\nds/bracketleftbigg1\n2/parenleftbigg\neαs2\n2|f′|2+c2\n0\n4|f|2/parenrightbigg/bracketrightbigg\n= 0.\nThus,\nE(s) :=1\n2/parenleftbigg\neαs2\n2|f′|2+c2\n0\n4|f|2/parenrightbigg\n=E0,∀s∈R, (3.16)\nwithE0a constant defined by the value of E(s)at some point s0∈R. The conservation of the\nenergyE(s)allows us to simplify the expressions of mj,njandbjforj∈ {1,2,3}in the formulae\n(3.2) in terms of the solution fjto (3.12) associated to the initial conditions (3.13)–(3.1 5).\nIndeed, on the one hand notice that the energies associated t o the initial conditions (3.13)–\n(3.15) are respectively\nE0,1=c2\n0\n8, E 0,2=c2\n0\n4andE0,3=c2\n0\n4. (3.17)\nOn the other hand, from (3.16), it follows that\n/parenleftBigg\n1+4\nc2\n0e−αs2\n2|f′\nj|2(s)\n|fj|2(s)/parenrightBigg−1\n=c2\n0\n8E0,j|fj|2(s), j∈ {1,2,3}.\nTherefore, from (3.17), the above identity and formulae (3. 2) in Lemma 3.1, we conclude that\nm1(s) = 2|f1(s)|2−1, n 1(s)+ib1(s) =4\nc0eαs2/4¯f1(s)f′\n1(s), (3.18)\nmj(s) =|fj(s)|2−1, n j(s)+ibj(s) =2\nc0eαs2/4¯fj(s)f′\nj(s), j∈ {2,3}. (3.19)\nThe above identities give the expressions of the tangent, no rmal and binormal vectors in terms\nof the solutions {fj}3\nj=1of the second order differential equation (3.12) associated to the initial\nconditions (3.13)–(3.15).\nBy Lemma 3.1, the formulae (3.18) and (3.19) are valid as long a smj>−1, which is equivalent\nto the condition |fj| /ne}ationslash= 0. As shown in Appendix, for α= 1there is˜s>0such thatmj(˜s) =−1\nand then (3.18) and (3.19) are (a priori) valid just in a bound ed interval. However, the trihedron\n{/vector m,/vector n,/vectorb}is defined globally and fjcan also be extended globally as the solution of the linear\nequation (3.12). Then, it is simple to verify that the functi ons given by the l.h.s. of formulae\n(3.18) and (3.19) satisfy the Serret–Frenet system and henc e, by the uniqueness of the solution,\nthe formulae (3.18) and (3.19) are valid for all s∈R.\n143.2 The second-order equation. Asymptotics\nIn this section we study the properties of the complex-value d equation\nf′′(s)+s\n2(α+iβ)f′(s)+c2\n0\n4f(s)e−αs2/2= 0, (3.20)\nfor fixedc0>0,α∈[0,1),β >0such thatα2+β2= 1. We begin noticing that in the\ncaseα= 0, the solution can be written explicitly in terms of paraboli c cylinder functions or\nconfluent hypergeometric functions (see [1]). Another anal ytical approach using Fourier analysis\ntechniques has been taken in [15], leading to the asymptotic s\nf(s) =C1ei(c2\n0/2)ln(s)+C2e−is2/4\nse−i(c2\n0/2)ln(s)+O(1/s2), (3.21)\nass→ ∞, where the constants C1,C2andO(1/s2)depend on the initial conditions and c0.\nForα= 1, equation (3.20) can be also solved explicitly and the solut ion is given by\nf(s) =2f′(0)\nc0sin/parenleftbiggc0\n2/integraldisplays\n0e−σ2/4dσ/parenrightbigg\n+f(0)cos/parenleftbiggc0\n2/integraldisplays\n0e−σ2/4dσ/parenrightbigg\n.\nIn the case α∈(0,1), one cannot compute the solutions of (3.20) in terms of known functions\nand we will follow a more analytical analysis. In contrast wi th the situation when α= 0, it is\nfar from evident to use Fourier analysis to study (3.20) when α>0.\nFor the rest of this section we will assume that α∈[0,1). In addition, we will also assume that\ns>0and we will develop the asymptotic analysis necessary to est ablish part (ii)of Theorem 1.2.\nAt this point, it is important to recall the expressions give n in (3.18)–(3.19) for the coordinates\nof the tangent, normal and binormal vectors associated to ou r family of solutions of the LLG\nequation in terms f. Bearing this in mind, we observe that the study of the asympto tic behaviour\nof these vectors are dictated by the asymptotic behaviour of the variables\nz=|f|2, y= Re(¯ff′),andh= Im(¯ff′) (3.22)\nassociated to the solution fof (3.20).\nAs explained in the remark (a) after Theorem 1.2, we need to wo rk with remainder terms that\nare independent of α. To this aim, we proceed in two steps: first we found uniform es timates\nforα∈[0,1/2]in Propositions 3.2 and 3.3, then we treat the case α∈[1/2,1)in Lemma 3.6. In\nSubsection 3.3 we provide some continuity results that allo ws us to take α→1−and give the\nfull statement in Corollary 3.14. Finally, notice that thes e asymptotics lead to the asymptotics\nfor the original equation (3.20) (see Remark 3.9).\nWe begin our analysis by establishing the following:\nProposition 3.2. Letc0>0,α∈[0,1),β >0such thatα2+β2= 1, andfbe a solution of\n(3.20). Define z,yandhasz=|f|2andy+ih=¯ff′. Then\n(i) There exists E0≥0such that the identity\n1\n2/parenleftbigg\neαs2\n2|f′|2+c2\n0\n4|f|2/parenrightbigg\n=E0\nholds true for all s∈R. In particular, f,f′,z,yandhare bounded functions. Moreover,\nfor alls∈R\n|f(s)| ≤√8E0\nc0,|f′(s)| ≤/radicalbig\n2E0e−αs2/4, (3.23)\n|z(s)| ≤8E0\nc2\n0and|h(s)|+|y(s)| ≤8E0\nc0e−αs2/4. (3.24)\n15(ii) The limit\nz∞:= lim\ns→∞z(s)\nexists.\n(iii) Letγ:= 2E0−c2\n0z∞/2ands0= 4/radicalbig\n8+c2\n0. For alls≥s0, we have\nz(s)−z∞=−4\ns(αy+βh)−4γ\ns2e−αs2/2+R0(s), (3.25)\nwhere\n|R0(s)| ≤C(E0,c0)e−αs2/4\ns3. (3.26)\nProof. Part(i)is just the conservation of energy proved in (3.16). Next, us ing the conservation\nlaw in part (i), we obtain that the variables {z,y,h}solve the first-order real system\nz′= 2y, (3.27)\ny′=βs\n2h−αs\n2y+e−αs2/2/parenleftbigg\n2E0−c2\n0\n2z/parenrightbigg\n, (3.28)\nh′=−βs\n2y−αs\n2h. (3.29)\nTo show (ii), plugging (3.27) into (3.29) and integrating from 0to somes>0we obtain\nz(s)−1\ns/integraldisplays\n0z(σ)dσ=−4\nβs/parenleftbigg\nh(s)−h(0)+α\n2/integraldisplays\n0σh(σ)dσ/parenrightbigg\n. (3.30)\nAlso, using the above identity,\nd\nds/parenleftbigg1\ns/integraldisplays\n0z(σ)dσ/parenrightbigg\n=−4\nβs2/parenleftbigg\nh(s)−h(0)+α\n2/integraldisplays\n0σh(σ)dσ/parenrightbigg\n. (3.31)\nNow, since from part (i)|h(s)| ≤8E0\nc0e−αs2/4, bothhandα/integraltexts\n0σh(σ)dσare bounded functions,\nthus from (3.31) it follows that the limit of1\ns/integraltexts\n0zexists, ass→ ∞. Hence (3.30) and previous\nobservations conclude that the limit z∞:= lims→∞z(s)exists and furthermore\nz∞:= lim\ns→∞z(s) = lim\ns→∞1\ns/integraldisplays\n0z(σ). (3.32)\nWe continue to prove (iii). Integrating (3.31) between s>0and+∞and using integration\nby parts, we obtain\nz∞−1\ns/integraldisplays\n0z(σ)dσ=−4\nβ/integraldisplay∞\nsh(σ)\nσ2dσ+4\nβh(0)\ns−2α\nβ/bracketleftbigg1\ns/integraldisplays\n0σh(σ)dσ+/integraldisplay∞\nsh(σ)dσ/bracketrightbigg\n.(3.33)\nFrom (3.30) and (3.33), we get\nz(s)−z∞=−4\nβh(s)\ns+2α\nβ/integraldisplay∞\nsh(σ)dσ+4\nβ/integraldisplay∞\nsh(σ)\nσ2. (3.34)\nIn order to compute the integrals in (3.34), using (3.27) and (3.28), we write\nh=2\nβ/parenleftbiggy′\ns+α\n4z′−2E0\nse−αs2/2+c2\n0\n2sze−αs2/2/parenrightbigg\n.\n16Then, integrating by parts and using the bound for yin (3.24),\n/integraldisplay∞\nsh(σ) =2\nβ/parenleftBigg\n−y\ns+/integraldisplay∞\nsy\nσ2+α\n4(z∞−z)−2E0/integraldisplay∞\nse−ασ2/2\nσ+c2\n0\n2/integraldisplay∞\nsz\nσe−ασ2/2/parenrightBigg\n.(3.35)\nAlso, from (3.27) and (3.34), we obtain\n/integraldisplay∞\nsh(σ)\nσ2=2\nβ/parenleftBigg/integraldisplay∞\nsy′\nσ3+α\n2/integraldisplay∞\nsy\nσ2−2E0/integraldisplay∞\nse−ασ2/2\nσ3+c2\n0\n2/integraldisplay∞\nsz\nσ3e−ασ2/2/parenrightBigg\n.(3.36)\nMultiplying (3.34) by β2, using (3.35), (3.36) and the identity\nα/integraldisplay∞\nse−ασ2/2\nσn=e−αs2/2\nsn+1−(n+1)/integraldisplay∞\nse−ασ2/2\nσn+2,for allα≥0, n≥1,\nwe conclude that\n(α2+β2)(z−z∞) =−4\ns(αy+βh)−8E0\ns2e−αs2/2\n+8α/integraldisplay∞\nsy\nσ2+8/integraldisplay∞\nsy′\nσ3+2c2\n0/integraldisplay∞\nse−ασ2/2z/parenleftbiggα\nσ+2\nσ3/parenrightbigg\n. (3.37)\nFinally, using (3.27) and the boundedness of zandy, an integration by parts argument shows\nthat\n8α/integraldisplay∞\nsy\nσ2+8/integraldisplay∞\nsy′\nσ3=−4αz\ns2−8y\ns3−12z\ns4+8/integraldisplay∞\nsz/parenleftbiggα\nσ3−6\nσ5/parenrightbigg\n. (3.38)\nBearing in mind that α2+β2= 1, from (3.37) and (3.38), we obtain the following identity\nz−z∞=−4\ns(αy+βh)−8E0\ns2e−αs2/2−4αz\ns2−8y\ns3−12z\ns4+8/integraldisplay∞\nsz/parenleftbiggα\nσ3+6\nσ5/parenrightbigg\ndσ\n+2c2\n0/integraldisplay∞\nse−ασ2/2z/parenleftbiggα\nσ+2\nσ3/parenrightbigg\ndσ,(3.39)\nfor alls>0. In order to prove (iii), we first write z=z−z∞+z∞and observe that\n8α/integraldisplay∞\nsz\nσ3= 8α/integraldisplay∞\nsz−z∞\nσ3+4αz∞\ns2,\n/integraldisplay∞\nsz\nσ5=/integraldisplay∞\nsz−z∞\nσ5+z∞\n4s4and\n/integraldisplay∞\nse−ασ2/2z/parenleftbiggα\nσ+2\nσ3/parenrightbigg\n=/integraldisplay∞\nse−ασ2/2(z−z∞)/parenleftbiggα\nσ+2\nσ3/parenrightbigg\n+z∞\ns2e−αs2/2.\nTherefore, we can recast (3.39) as (3.25) with\nR0(s) =−4α(z−z∞)\ns2−8y\ns3−12(z−z∞)\ns4+8/integraldisplay∞\ns(z−z∞)/parenleftbiggα\nσ3+6\nσ5/parenrightbigg\ndσ\n+2c2\n0/integraldisplay∞\nse−ασ2/2(z−z∞)/parenleftbiggα\nσ+2\nσ3/parenrightbigg\ndσ.(3.40)\nLet us take s0≥1to be fixed in what follows. For t≥s0, we denote /bardbl · /bardbltthe norm of\nL∞([t,∞)). From the definition of R0in (3.40) and the elementary inequalities\nα/integraldisplay∞\nse−ασ2/2\nσn≤e−αs2/2\nsn+1,for allα≥0, n≥1, (3.41)\n17and/integraldisplay∞\nse−ασ2/2\nσn≤e−αs2/2\n(n−1)sn−1,for allα≥0, n>1, (3.42)\nwe obtain\n/bardblR0/bardblt≤8/bardbly/bardblt\nt3+4\nt2/parenleftBig\n8+c2\n0e−αt2/2/parenrightBig\n/bardblz−z∞/bardblt.\nHence, choosing s0= 4/radicalbig\n8+c2\n0, so that4\nt2/parenleftBig\n8+c2\n0e−αt2/2/parenrightBig\n≤1/2, from (3.24) and (3.25) we\nconclude that there exists a constant C(E0,c0)>0such that\n/bardblz−z∞/bardblt≤C(E0,c0)\nte−αt2/4,for allα∈[0,1)andt≥s0,\nwhich implies that\n|z(s)−z∞| ≤C(E0,c0)\nse−αs2/4,for allα∈[0,1), s≥s0. (3.43)\nFinally, plugging (3.24) and (3.43) into (3.40) and bearing in mind the inequalities (3.41) and\n(3.42), we deduce that\n|R0(s)| ≤C(E0,c0)e−αs2/4\ns3,∀s≥s0= 4/radicalBig\n8+c2\n0, (3.44)\nand the proof of (iii)is completed.\nFormula (3.25) in Proposition 3.2 gives zin terms of yandh. Therefore, we can reduce our\nanalysis to that of the variables yandhor, in other words, to that of the system (3.27)–(3.29).\nIn fact, a first attempt could be to define w=y+ih, so that from (3.28) and (3.29), we have\nthatwsolves/parenleftBig\nwe(α+iβ)s2/4/parenrightBig′\n=e(−α+iβ)s2/4/parenleftbigg\nγ−c2\n0\n2(z−z∞)/parenrightbigg\n. (3.45)\nFrom (3.43) in Proposition 3.2 and (3.45), we see that the lim itw∗= lims→∞w(s)e(α+iβ)s2/4\nexists (at least when α/ne}ationslash= 0), and integrating (3.45) from some s>0to∞we find that\nw(s) =e−(α+iβ)s2/4/parenleftbigg\nw∗−/integraldisplay∞\nse(−α+iβ)σ2/4/parenleftbigg\nγ−c2\n0\n2(z−z∞)/parenrightbigg\ndσ/parenrightbigg\n.\nIn order to obtain an asymptotic expansion, we need to estima te/integraltext∞\nse(−α+iβ)σ2/4(z−z∞), fors\nlarge. This can be achieved using (3.43),\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞\nse(−α+iβ)σ2/4(z−z∞)dσ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(E0,c0)/integraldisplay∞\nse−ασ2/2\nσdσ (3.46)\nand the asymptotic expansion\n/integraldisplay∞\nse−ασ2/2\nσdσ=e−αs2/2/parenleftbigg1\nαs2−2\nα2s4+8\nα3s6+···/parenrightbigg\n.\nHowever this estimate diverges as α→0. The problem is that the bound used in obtaining\n(3.46) does not take into account the cancellations due to th e oscillations. Therefore, and in\norder to obtain the asymptotic behaviour of z,yandhvalid for all α∈[0,1), we need a more\nrefined analysis. In the next proposition we study the system (3.27)–(3.29), where we consider\nthe cancellations due the oscillations (see Lemma 3.5 below ). The following result provides\nestimates that are valid for s≥s1, for somes1independent of α, ifαis small.\n18Proposition 3.3. With the same notation and terminology as in Proposition 3.2 , let\ns1= max/braceleftBigg\n4/radicalBig\n8+c2\n0,2c0/parenleftbigg1\nβ−1/parenrightbigg1/2/bracerightBigg\n.\nThen for all s≥s1,\ny(s) =be−αs2/4sin(φ(s1;s))−2αγ\nse−αs2/2+O/parenleftBigg\ne−αs2/2\nβ2s2/parenrightBigg\n, (3.47)\nh(s) =be−αs2/4cos(φ(s1;s))−2βγ\nse−αs2/2+O/parenleftBigg\ne−αs2/2\nβ2s2/parenrightBigg\n, (3.48)\nwhere\nφ(s1;s) =a+β/integraldisplays2/4\ns2\n1/4/radicalbigg\n1+c2\n0e−2αt\ntdt,\na∈[0,2π)is a real constant, and bis a positive constant given by\nb2=/parenleftbigg\n2E0−c2\n0\n4z∞/parenrightbigg\nz∞. (3.49)\nProof. First, notice that plugging the expression for z(s)−z∞in (3.25) into (3.28), the system\n(3.28)–(3.29) for the variables yandhrewrites equivalently as\ny′=s\n2(βh−αy)+2c2\n0\nse−αs2/2(βh+αy)+γe−αs2/2+R1(s), (3.50)\nh′=−s\n2(βy+αh), (3.51)\nwhere\nR1(s) =−c2\n0\n2e−αs2/2R0(s)+2c2\n0γe−αs2\ns2, (3.52)\nandR0is given by (3.40).\nIntroducing the new variables,\nu(t) =eαty(2√\nt), v(t) =eαth(2√\nt), (3.53)\nwe recast (3.50)–(3.51) as\n/parenleftbiggu\nv/parenrightbigg′\n=/parenleftbiggαK β(1+K)\n−β0/parenrightbigg/parenleftbiggu\nv/parenrightbigg\n+/parenleftbiggF\n0/parenrightbigg\n, (3.54)\nwith\nK=c2\n0e−2αt\nt, F=γe−αt\n√\nt+e−αt\n√\ntR1(2√\nt),\nwhereR1is the function defined in (3.52). In this way, we can regard (3 .54) as a non-autonomous\nsystem. It is straightforward to check that the matrix\nA=/parenleftbiggαK β(1+K)\n−β0/parenrightbigg\nis diagonalizable, i.e. A=PDP−1, with\nD=/parenleftbiggλ+0\n0λ−/parenrightbigg\n, P=/parenleftbigg−αK\n2β−i∆1/2−αK\n2β+i∆1/2\n1 1/parenrightbigg\n,\n19λ±=αK\n2±iβ∆1/2,and∆ = 1+K−α2K2\n4β2. (3.55)\nAt this point we remark that the condition t≥t1, witht1:=s2\n1/4ands1≥2c0(1\nβ−1)1/2, implies\nthat\n0<K/parenleftbigg1\nβ−1/parenrightbigg\n≤1,∀t≥t1, (3.56)\nso that\n∆ = 1+K−(1−β2)\n4β2K2=/parenleftbigg\n1+K\n2+K\n2β/parenrightbigg/parenleftbigg\n1+K\n2/parenleftbigg\n1−1\nβ/parenrightbigg/parenrightbigg\n≥1\n2,∀t≥t1.(3.57)\nThus, defining\nw= (w1,w2) =P−1(u,v), (3.58)\nwe get/parenleftBig\ne−/integraltextt\nt1Dw/parenrightBig′\n=e−/integraltextt\nt1D/parenleftBig\n(P−1)′Pw+P−1˜F/parenrightBig\n, (3.59)\nwith˜F= (F,0). From the definition of wand taking into account that uandvare real functions,\nwe have that w1= ¯w2and therefore the study of (3.59) reduces to the analysis of t he equation:\n/parenleftBig\ne−/integraltextt\nt1λ+w1/parenrightBig′\n=e−/integraltextt\nt1λ+G(t), (3.60)\nwith\nG(t) =iαK′\n4β∆1/2(w1+ ¯w1)−∆′\n4∆(w1−¯w1)+iF\n2∆1/2.\nFrom (3.60) we have\nw1(t) =e/integraltextt\nt1λ+/parenleftbigg\nw1(t1)+w∞−/integraldisplay∞\nte−/integraltextτ\nt1λ+G(τ)dτ/parenrightbigg\n, (3.61)\nwith\nw∞=/integraldisplay∞\nt1e−/integraltextτ\nt1λ+G(τ).\nSince\nw1=iu\n2∆1/2+v\n2+iαKv\n4β∆1/2, (3.62)\nwe recastGasG=i(G1+G2+G3)with\nG1=αK′v\n4β∆1/2−∆′\n4∆3/2/parenleftbigg\nu+αKv\n2β/parenrightbigg\n, G2=γe−αt\n2t1/2∆1/2andG3=e−αt\n2t1/2∆1/2R1(2t1/2).\nNow, from the definition of Kand∆, we have\nK′=−K/parenleftbigg\n2α+1\nt/parenrightbigg\n, K′′=K/parenleftBigg/parenleftbigg\n2α+1\nt/parenrightbigg2\n+1\nt2/parenrightBigg\n,\n∆′=K′/parenleftbigg\n1−α2K\n2β2/parenrightbigg\nand∆′′=K/parenleftBigg/parenleftbigg\n2α+1\nt/parenrightbigg2\n+1\nt2/parenrightBigg/parenleftbigg\n1−α2K\n2β2/parenrightbigg\n−α2\n2β2K2/parenleftbigg\n2α+1\nt/parenrightbigg2\n.\nAlso, since s1= max{4/radicalbig\n8+c2\n0,2c0(1/β−1)1/2}, for allt≥t1=s2\n1/4, we have in particular\nthatt≥8+c2\n0andt≥c2\n0(1/β−1), hence\nc2\n0\ntβ=c2\n0\nt/parenleftbigg1\nβ−1/parenrightbigg\n+c2\n0\nt≤2 (3.63)\n20and /vextendsingle/vextendsingle/vextendsingle/vextendsingle1−α2K\n4β2/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1+1\n4β/parenleftbiggc2\n0\ntβ/parenrightbigg\n≤2\nβ. (3.64)\nTherefore\n|K′| ≤c2\n0e−2αt/parenleftbigg2α\nt+1\nt2/parenrightbigg\n,|∆′| ≤2c2\n0e−2αt\nβ/parenleftbigg2α\nt+1\nt2/parenrightbigg\n(3.65)\nand\n|∆′′| ≤24c2\n0\nβe−2αt/parenleftbiggα\nt+1\nt2/parenrightbigg\n. (3.66)\nFrom Proposition 3.2, uandvare bounded in terms of the energy. Thus, from the definition o f\nG1and the estimates (3.56), (3.57) and (3.65), we obtain\n|G1(t)| ≤C(E0,c0)e−2αt\nβ2/parenleftbiggα\nt+1\nt2/parenrightbigg\n.\nSince /vextendsingle/vextendsingle/vextendsinglee±/integraltextτ\nt1λ+/vextendsingle/vextendsingle/vextendsingle≤2, (3.67)\nwe conclude that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞\nte−/integraltextτ\nt1λ+G1(τ)dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(E0,c0)\nβ2/integraldisplay∞\nte−2ατ/parenleftbiggα\nτ+1\nτ2/parenrightbigg\n≤C(E0,c0)e−2αt\nβ2t. (3.68)\nHere we have used the inequality\nα/integraldisplay∞\nte−2ασ\nσndσ≤e−2αt\n2tn, n≥1, (3.69)\nwhich follows by integrating by parts.\nIn order to handle the terms involving G2andG3, we need to take advantage of the oscillatory\ncharacter of the involved integrals, which is exploited in L emma 3.5. From (3.57), (3.65) and\n(3.66), straightforward calculations show that the functi on defined by f=γ/(2t1/2∆1/2)satisfies\nthe hypothesis in part (ii)of Lemma 3.5 with a= 1/2andL=C(E0,c0)/β. Thus invoking this\nlemma with f=γ/(2t1/2∆1/2)and noticing that\n1\n∆1/2= 1+/parenleftbigg1\n∆1/2−1/parenrightbigg\nand that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n∆1/2−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−∆\n∆1/2(∆1/2+1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤|K|/vextendsingle/vextendsingle/vextendsingle1−α2K\n4β2/vextendsingle/vextendsingle/vextendsingle\n|∆1/2(∆1/2+1)|≤2√\n2c2\n0\nβt,\nwhere we have used (3.57) and (3.64), we conclude that\n/integraldisplay∞\nte−/integraltextτ\nt1λ+G2(τ)dτ=γ\n2(α+iβ)t1/2e−/integraltextt\nt1λ+e−αt+R2(t), (3.70)\nwith\n|R2(t)| ≤C(E0,c0)e−αt\nβ2t3/2.\nForG3, we first write explicitly (recall the definition of R1in (3.52))\nG3(t) =−c2\n0R0(2√\nt)e−3αt\n4t1/2∆1/2+c2\n0γe−5αt\n4t3/2∆1/2:=G3,1(t)+G3,2(t). (3.71)\n21Using (3.44) and (3.57), we see that |G3,1(t)| ≤C(E0,c0)e−4αt/t2, so that we can treat this term\nas we did for G1to obtain\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞\nte−/integraltextτ\nt1λ+G3,1(τ)dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(E0,c0)e−4αt\nt. (3.72)\nFor the second term, using (3.57), (3.65) and (3.63), it is ea sy to see that the function fdefined\nbyf= (c2\n0γ)/(4t3/2∆1/2)satisfies\n|f(t)| ≤C(E0,c0)\nt3/2and|f′(t)| ≤C(E0,c0)/parenleftbiggα\nt3/2+1\nt5/2/parenrightbigg\n,\nas a consequence, invoking part (i)of Lemma 3.5, we obtain\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞\nte−/integraltextτ\nt1λ+G3,2(τ)dτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(E0,c0)e−5αt\nβt3/2. (3.73)\nFrom (3.61), (3.67), (3.68), (3.72) and (3.73), we deduce th at\nw1(t) =e/integraltextt\nt1λ+(w1(t1)+w∞)−γ(β+iα)\n2t1/2e−αt+R3(t)with|R3(t)| ≤C(E0,c0)e−αt\nβ2t.(3.74)\nNow we claim that\ne/integraltextt\nt1λ+=Cα,c0eiβI(t)+H(t),withI(t) =/integraldisplayt\nt1/radicalbig\n1+K(σ)dσ,|H(t)| ≤3c2\n0e−2αt\nt,(3.75)\nand\nCα,c0= exp/parenleftbiggα\n2/integraldisplay∞\nt1Kdσ/parenrightbigg\nexp/parenleftbigg\n−iα2\n4β/integraldisplay∞\nt1K2\n∆1/2+(1+K)1/2dσ/parenrightbigg\n.\nIndeed, recall that λ+=αK\n2+iβ∆1/2so that\ne/integraltextt\nt1λ+=eα/integraltextt\nt1K\n2eiβ/integraltextt\nt1∆1/2\n. (3.76)\nFirst, we notice that\nα/integraldisplayt\nt1K\n2=c2\n0α/integraldisplay∞\nt1e−2ασ\n2σ−c2\n0α/integraldisplay∞\nte−2ασ\n2σ,\nwhere both integrals are finite in view of (3.69). Moreover, b y combining with the fact that\n|1−e−x| ≤x, forx≥0, we can write\nexp/parenleftbigg\n−c2\n0α/integraldisplay∞\nte−2ασ\n2σ/parenrightbigg\n= 1+H1(t),\nwith\n|H1(t)| ≤c2\n0e−2αt\n4t,for allt≥c2\n0/4. (3.77)\nThe above argument shows that\neα/integraltextt\nt1K\n2=eα/integraltext∞\nt1K\n2(1+H1(t)), (3.78)\nwithH1(t)satisfying (3.77).\n22For the second term of the eigenvalue, using the definition of ∆in (3.55), we write\niβ/integraldisplayt\nt1∆1/2=iβ/integraldisplayt\nt1/parenleftBig\n∆1/2−√\n1+K/parenrightBig\n+iβ/integraldisplayt\nt1√\n1+K\n=−iα2\n4β/integraldisplayt\nt1K2\n∆1/2+(1+K)1/2+iβ/integraldisplayt\nt1√\n1+K.\nProceeding as before and using that |1−eix| ≤ |x|, forx∈R, and that\nα/integraldisplay∞\ntK2\n∆1/2+√\n1+K≤α/integraldisplay∞\ntK2(σ)dσ=αc4\n0/integraldisplay∞\nte−4ατ\nτ2≤c4\n0e−4αt\n4t2,\nwe conclude that\neiβ/integraltextt\nt1∆1/2\n=eiβI(t)e−iα2\n4β/integraltext∞\nt1K2\n∆1/2+(1+K)1/2(1+H2), (3.79)\nwith\n|H2(t)| ≤c4\n0e−4αt\n16βt2≤c2\n0e−4αt\n8t,\nbearing in mind (3.63). Therefore, from (3.76), (3.78) and ( 3.79),\ne/integraltextt\nt1λ+=Cα,c0eiβI(t)(1+H1(t))(1+H2(t)).\nThe claim follows from the above identity, the bounds for H1andH2, and the fact that Cα,c0\nsatisfies that |Cα,c0|=|e/integraltext∞\nt1λ+| ≤2(see (3.67)). From (3.74), the claim and writing\nCα,c0(w1(t1)+w∞) = (beia)/2 (3.80)\nfor some real constants aandbsuch thatb≥0anda∈[0,2π), it follows that\nw1(t) =b\n2ei(βI(t)+a)−γ(β+iα)\n2t1/2+Rw1(t)with|Rw1(t)| ≤C(E0,c0)e−αt\nβ2t. (3.81)\nThe above bound for Rw1(t)easily follows from the bounds for R3(t)andH(t)in (3.74) and\n(3.75) respectively, and the fact that\n|w1(t)| ≤C(E0,c0),∀t≥t1. (3.82)\nThis last inequality is a consequence of (3.53), (3.57), (3. 62), (3.63) and the bounds for yandh\nestablished in (3.24) in Proposition 3.2.\nGoing back to the definition of win (3.58), we have (u,v) =P(w1,w2), that is\nu=−αK\n2β(w1+ ¯w1)−i∆1/2(w1−¯w1) = 2Im(w1)+R4(t),\nv= (w1+ ¯w1) = 2Re(w1),(3.83)\nwith\n|R4(t)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle−αK\nβRe(w1)+2(∆1/2−1)Im(w1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤K\nβ|Re(w1)|+2|∆−1|\n∆1/2+1|Im(w1)|\n≤2c2\n0e−2αt\nβt(|Re(w1)|+|Im(w1)|)≤C(E0,c0)e−2αt\nβt,\n23where we have used (3.57), (3.64), and (3.82). From (3.81) an d (3.83), we obtain\nu(t) =bsin(βI(t)+a)−αγ\nt1/2e−αt+R5(t),\nv(t) =bcos(βI(t)+a)−βγ\nt1/2e−αt+R6(t),\nwith\n|R5(t)|+|R6(t)| ≤C(E0,c0)e−αt/(β2t).\nThe asymptotics for yandhgiven in (3.47) and (3.48) are a direct consequence of (3.53) and\nthe above identities and bounds.\nFinally, we compute the value of b. In fact, from (3.47) and (3.48)\nlim\ns→∞(y2(s)+h2(s))eαs2/2=b2.\nOn the other hand, since y+ih=¯ff′and using the conservation of energy (3.16)\n/parenleftbig\ny2(s)+h2(s)/parenrightbig\neαs2/2=|y+ih|2(s)eαs2/2=|f′|2|f|2eαs2/2= (2E0−c2\n0\n4|f|2)|f|2,\nso that, taking the limit as s→ ∞ and recalling that z=|f|2, (3.49) follows.\nRemark 3.4. From the definitions of bin(3.49), andbeiain(3.80) (in terms of Cα,c0,w1(t1)\nandw∞in(3.80)), it is simple to verify that bandbeiadepend continuously on α∈[0,1),\nprovided that z∞is a continuous function of α. In Subsection 3.3 we will prove that z∞depends\ncontinuously on α, forα∈[0,1], and establish the continuous dependence of the constants band\nbeiawith respect to the parameter αin Lemma 3.13 above.\nIn the proof of Proposition 3.3, we have used the following ke y lemma that establishes the\ncontrol of certain integrals by exploiting their oscillato ry character.\nLemma 3.5. With the same notation as in the proof of Proposition 3.2.\n(i) Letf∈C1((t1,∞))such that\n|f(t)| ≤L/taand|f′(t)| ≤L/parenleftbiggα\nta+1\nta+1/parenrightbigg\n,\nfor some constants L,a>0. Then, for all t≥t1andl≥1\n/integraldisplay∞\nte−/integraltextτ\nt1λ+e−lατf(τ)dτ=1\n(α+iβ)e−/integraltextt\nt1λ+e−lαtf(t)+F(t),\nwith\n|F(t)| ≤C(l,a,c0)Le−lαt\nβta. (3.84)\n(ii) If in addition f∈C2((t1,∞)),\n|f′(t)| ≤L/ta+1and|f′′(t)| ≤L/parenleftbiggα\nta+1+1\nta+2/parenrightbigg\n, (3.85)\nthen\n|F(t)| ≤C(l,a,c0)Le−lαt\nβta+1. (3.86)\n24HereC(l,a,c0)is a positive constant depending only on l,aandc0.\nProof. Defineλ=λ+. Recall (see proof of Proposition 3.2) that\nλ+=αK\n2+iβ∆1/2and∆ = 1+K−α2K2\n4β2,withK=c2\n0e−2αt\nt.\nSettingRλ= 1/λ−1/(iβ)and integrating by parts, we obtain\n/parenleftbigg\n1+lα\niβ/parenrightbigg/integraldisplay∞\nte−/integraltextτ\nt1λe−lατf(τ)dτ=e−/integraltextt\nt1λe−lαtf(t)/parenleftbigg1\niβ+Rλ/parenrightbigg\n+/integraldisplay∞\nte−/integraltextτ\nt1λe−lατ/parenleftbigg\n−lαfRλ+f′\nλ−fλ′\nλ2/parenrightbigg\ndτ,\nor, equivalently,\n/integraldisplay∞\nte−/integraltextτ\nt1λe−ατf(τ)dτ=1\nlα+iβe−/integraltextt\nt1λe−αtf(t)+F(t),\nwith\nF(t) =iβ\nlα+iβ/parenleftbigg\ne−/integraltextt\nt1λe−lαtRλf+/integraldisplay∞\nte−/integraltextτ\nt1λe−lατ/parenleftbigg\n−lαfRλ+f′\nλ−fλ′\nλ2/parenrightbigg\ndτ/parenrightbigg\n.\nUsing (3.57), (3.63) and (3.65), it is easy to check that for a llt≥t1\n|λ| ≥β√\n2and|λ′| ≤3c2\n0/parenleftbigg2α\nt+1\nt2/parenrightbigg\n. (3.87)\nOn the other hand,\n|Rλ|=/vextendsingle/vextendsingle/vextendsingle/vextendsingleiβ−λ\niβλ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤√\n2\nβ2/parenleftbigg\nβ|1−∆1/2|+αK\n2/parenrightbigg\n,\nwith, using the definition of ∆in (3.57) and (3.63),\nαK\n2≤c2\n0\n2tand|1−∆1/2|=|1−∆|\n1+∆1/2≤ |1−∆| ≤c2\n0\nt+c2\n0\n4βt/parenleftbiggc2\n0\nβt/parenrightbigg\n≤2c2\n0\nβt.\nPrevious lines show that\n|Rλ| ≤10c2\n0\nβ2t. (3.88)\nThe estimate (3.84) easily follows from the bounds (3.67), ( 3.69), (3.87), (3.88) and the hypothe-\nses onf. To obtain part (ii)we only need to improve the estimate for the term\n/integraldisplay∞\nte−/integraltextτ\nt1λe−lατf′\nλdτ\nin the above argument. In particular, it suffices to prove that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞\nte−/integraltextτ\nt1λe−lατf′\nλ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(l,c0,a)Le−lαt\nβ2ta+1.\nNow, consider the function g=f′/λ. Notice that from (3.63), (3.87) and the hypotheses on f\nin (3.85), we have\n|g(t)| ≤√\n2L\nβta+1\n25and\n|g′(t)| ≤√\n2\nβL/parenleftbiggα\nta+1+1\nta+2/parenrightbigg\n+6L\nβ/parenleftbiggc2\n0\nβt/parenrightbigg/parenleftbigg2α\nta+1+1\nta+2/parenrightbigg\n≤14L\nβ/parenleftbigg2α\nta+1+1\nta+2/parenrightbigg\n.\nTherefore, from part (i), we obtain\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay∞\nte−/integraltextτ\nt1λe−lατf′\nλ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(l,c0,a)Le−lαt/parenleftbigg1\nβta+1+1\nβ2ta+1/parenrightbigg\n≤C(l,c0,a)Le−lαt\nβ2ta+1,\nas desired.\nWe remark that if α∈[0,1/2], the asymptotics in Proposition 3.3 are uniform in α. Indeed,\nmax\nα∈[0,1/2]/braceleftBigg\n4/radicalBig\n8+c2\n0,2c0/parenleftbigg1\nβ−1/parenrightbigg1/2/bracerightBigg\n= 4/radicalBig\n8+c2\n0=s0.\nTherefore in this situation we can omit the dependence on s1in the function φ(s1;s), because\nthe asymptotics are valid with\nφ(s) :=φ(s0;s) =a+β/integraldisplays2/4\ns2\n0/4/radicalbigg\n1+c2\n0e−2αt\ntdt. (3.89)\nWe continue to show that the factor 1/β2in the big-Oin formulae (3.47) and (3.48) are due\nto the method used and this factor can be avoided if αis far from zero. More precisely, we have\nthe following:\nLemma 3.6. Letα∈[1/2,1). With the same notation as in Propositions 3.2 and 3.3, we hav e\nthe following asymptotics: for all s≥s0,\ny(s) =be−αs2/4sin(φ(s))−2αγ\nse−αs2/2+O/parenleftBigg\ne−αs2/2\ns2/parenrightBigg\n, (3.90)\nh(s) =be−αs2/4cos(φ(s))−2βγ\nse−αs2/2+O/parenleftBigg\ne−αs2/2\ns2/parenrightBigg\n. (3.91)\nHere, the function φis defined by (3.89) and the bounds controlling the error terms depend on\nc0, and the energy E0, and are independent of α∈[1/2,1)\nProof. Letα∈[1/2,1)and define w=y+ih. From Proposition 3.3 and (1.21), we have that\nfor allα∈[1/2,1)\nlim\ns→∞we(α+iβ)s2/4=bie−i˜a, (3.92)\nwhere˜a:=a+C(α,c0),aandbare the constants defined in Proposition 3.3 and C(α,c0)is the\nconstant in (1.21). Then, since wsatisfies\n/parenleftBig\nwe(α+iβ)s2/4/parenrightBig′\n=e(−α+iβ)s2/4/parenleftbigg\nγ−c2\n0\n2(z−z∞)/parenrightbigg\n, (3.93)\nintegrating the above identity between sand infinity,\nwe(α+iβ)s2/4=ibe−i˜a−/integraldisplay∞\nse(−α+iβ)σ2/4/parenleftbigg\nγ−c2\n0\n2(z−z∞)/parenrightbigg\ndσ.\n26Now, integrating by parts and using (3.41) (recall that 1≤2α), we see that\n/integraldisplay∞\nse(−α+iβ)σ2/4dσ= 2(α+iβ)e(−α+iβ)s2/4\ns+O/parenleftBigg\ne−αs2/4\ns3/parenrightBigg\n,∀s≥s0.\nNext, notice that from (3.43) in Proposition 3.2, we also obt ain\n/integraldisplay∞\nse(−α+iβ)σ2/4(z−z∞)dσ=O/parenleftBigg\ne−αs2/2\ns2/parenrightBigg\n,∀s≥s0.\nThe above argument shows that for all s≥s0\nw(s) =ibe−αs2/4e−i(˜a+βs2/4)−2(α+iβ)γ\nse−αs2/2+O/parenleftBigg\ne−αs2/2\ns2/parenrightBigg\n. (3.94)\nThe asymptotics for yandhin the statement of the lemma easily follow from (3.94) beari ng in\nmind thatw=y+ihand recalling that the function φbehaves like (1.21) when α>0.\nIn the following corollary we summarize the asymptotics for z,yandhobtained in this section.\nPrecisely, as a consequence of Proposition 3.2- (iii), Proposition 3.3 and Lemma 3.6, we have the\nfollowing:\nCorollary 3.7. Letα∈[0,1). With the same notation as before, for all s≥s0= 4/radicalbig\n8+c2\n0,\ny(s) =be−αs2/4sin(φ(s))−2αγ\nse−αs2/2+O/parenleftBigg\ne−αs2/2\ns2/parenrightBigg\n, (3.95)\nh(s) =be−αs2/4cos(φ(s))−2βγ\nse−αs2/2+O/parenleftBigg\ne−αs2/2\ns2/parenrightBigg\n, (3.96)\nz(s) =z∞−4b\nse−αs2/4(αsin(φ(s))+βcos(φ(s)))+4γe−αs2/2\ns2+O/parenleftBigg\ne−αs2/4\ns3/parenrightBigg\n, (3.97)\nwhere\nφ(s) =a+β/integraldisplays2/4\ns2\n0/4/radicalbigg\n1+c2\n0e−2αt\ntdt,\nfor some constant a∈[0,2π),\nb=z1/2\n∞/parenleftbigg\n2E0−c2\n0\n4z∞/parenrightbigg1/2\n, γ= 2E0−c2\n0\n2z∞ andz∞= lim\ns→∞z(s).\nHere, the bounds controlling the error terms depend on c0and the energy E0, and are independent\nofα∈[0,1).\nRemark 3.8. In the case when s<0, the same arguments to the ones leading to the asymptotics\nin the above corollary will lead to an analogous asymptotic b ehaviour for the variables z,hand\nyfors<0. As mentioned at the beginning of Subsection 3.2, here we hav e reduced ourselves to\nthe case of s >0when establishing the asymptotic behaviour of the latter qu antities due to the\nparity of the solution we will be applying these results to.\n27Remark 3.9. The asymptotics in Corollary 3.7 lead to the asymptotics for the solutions fof the\nequation (3.20), at least if |f|∞:=z1/2\n∞is strictly positive. Indeed, this implies that there exist s\ns∗≥s0such thatf(s)/ne}ationslash= 0for alls≥s∗. Then writing fin its polar form f=ρexp(iθ), we\nhaveρ2θ′= Im(¯ff′). Hence, using (3.22), we obtain ρ=z1/2andθ′=h/z. Therefore, for all\ns≥s∗,\nθ(s)−θ(s∗) =/integraldisplays\ns∗h(σ)\nz(σ)dσ. (3.98)\nHence, using the asymptotics for zandhin Corollary 3.7, we can obtain the asymptotics for f.\nIn the case that α∈(0,1], we can also show that the phase converges. Indeed, the asymp totics\nin Corollary 3.7 yield that the integral in (3.98) converges as s→ ∞forα>0, and we conclude\nthat there exists a constant θ∞∈Rsuch that\nf(s) =z(s)1/2exp/parenleftbigg\niθ∞−i/integraldisplay∞\nsh(σ)\nz(σ)dσ/parenrightbigg\n,for alls≥s∗.\nThe asymptotics for fis obtained by plugging the asymptotics in Corollary 3.7 int o the above\nexpression.\n3.3 The second-order equation. Dependence on the parameter s\nThe aim of this subsection is to study the dependence of the f,z,yandhon the parameters\nc0>0andα∈[0,1]. This will allow us to pass to the limit α→1−in the asymptotics in\nCorollary 3.7 and will give us the elements for the proofs of T heorems 1.3 and 1.4.\n3.3.1 Dependence on α\nWe will denote by f(s,α)the solution of (3.20) with some initial conditions f(0,α),f′(0,α)that\nare independent of α. Indeed, we are interested in initial conditions that depen d only onc0(see\n(3.13)–(3.15)). Moreover, in view of (3.17), we assume that the energyE0in (3.16) is a function\nofc0. In order to simplify the notation, we denote with a subindex αthe derivative with respect\ntoαand by′the derivative with respect to s. Analogously to Subsection 3.2, we define\nz(s,α) =|f(s,α)|2, y(s,α) = Re(¯f(s,α)f′(s,α)), h(s,α) = Im(¯f(s,α)f′(s,α)) (3.99)\nand\nz∞(α) = lim\ns→∞|f(s,α)|2.\nObserve that in Proposition 3.2- (ii), we proved the existence of z∞(α), forα∈[0,1). For\nα∈(0,1], the estimates in (3.24) hold true and hence z(s,α)is a bounded function whose\nderivative decays exponentially. Therefore, it admits a li mit at infinity for all α∈[0,1]and\nz∞(1)is well-defined.\nThe next lemma provides estimates for zα,hαandyα.\nLemma 3.10. Letα∈(0,1). There exists a constant C(c0), depending on c0but not onα, such\nthat for all s≥0,\n|zα(s,α)| ≤C(c0)min/braceleftBigg\ns2\n√1−α+s3,s2\n/radicalbig\nα(1−α),1\nα2√1−α/bracerightBigg\n, (3.100)\n|yα(s,α)|+|hα(s,α)| ≤C(c0)e−αs2/4min/braceleftBigg\ns2\n√1−α+s3,s2\n/radicalbig\nα(1−α)/bracerightBigg\n. (3.101)\n28Proof. Differentiating (3.12) with respect to α,\nf′′\nα+s\n2(α+iβ)f′\nα+c2\n0\n4fαe−αs2/2=g, (3.102)\nwhere\ng(s,α) =−/parenleftbigg\n1−iα\nβ/parenrightbiggs\n2f′+c2\n0s2\n8fe−αs2/2.\nAlso, since the initial conditions do not depend on α,\nfα(0,α) =f′\nα(0,α) = 0. (3.103)\nUsing the estimates in (3.23) and that α2+β2= 1, we obtain\n|g| ≤C(c0)/parenleftbiggs\nβe−αs2/4+s2e−αs2/2/parenrightbigg\n,for alls≥0. (3.104)\nMultiplying (3.102) by ¯f′\nαand taking real part, we have\n1\n2/parenleftbig\n|f′\nα|2/parenrightbig′+αs\n2|f′\nα|2+c2\n0\n8/parenleftbig\n|fα|2/parenrightbig′e−αs2/2= Re(g¯f′\nα). (3.105)\nMultiplying (3.105) by 2eαs2/2and integrating, taking into account (3.103),\n|f′\nα|2eαs2/2+c2\n0\n4|fα|2= 2/integraldisplays\n0eασ2/2Re(g¯f′\nα)dσ. (3.106)\nLet us define the real-valued function η=|f′\nα|eαs2/4. Then (3.106) yields\nη2(s)≤2/integraldisplays\n0eασ2/4|g|ηdσ, for alls≥0.\nThus, by the Gronwall inequality (see e.g. [3, Lemma A.5]),\nη(s)≤/integraldisplays\n0eασ2/4|g|,dσ, for alls≥0. (3.107)\nFrom (3.104), (3.106) and (3.107), we conclude that\n(|f′\nα|eαs2/4+c0\n2|fα|)2≤2(|fα|2eαs2/2+c2\n0\n4|fα|2)\n≤4/integraldisplays\n0eασ2/4|g|ηdσ≤4/parenleftBigg\nsup\nσ∈[0,s]η(σ)/parenrightBigg/parenleftbigg/integraldisplays\n0eασ2/4|g|dσ/parenrightbigg\n≤/parenleftbigg/integraldisplays\n0eασ2/4|g|dσ/parenrightbigg2\n.\nThus, using (3.104), from the above inequality it follows\n|f′\nα|eαs2/4+c0\n2|fα| ≤C(c0)/integraldisplays\n0/parenleftbiggσ\nβ+σ2e−ασ2/4/parenrightbigg\ndσ, for alls≥0. (3.108)\nIn particular, for all s≥0,\n|fα(s)| ≤C(c0)min/braceleftBigg\ns2\n√1−α+s3,s2\n/radicalbig\nα(1−α)/bracerightBigg\n,\n|f′\nα(s)| ≤C(c0)e−αs2/4min/braceleftBigg\ns2\n√1−α+s3,s2\n/radicalbig\nα(1−α)/bracerightBigg\n,(3.109)\n29where we have used that\n/integraldisplays\n0σ2e−ασ2/4dσ≤s2/integraldisplays\n0e−ασ2/4dσ≤s2/radicalbig\nπ/α.\nNotice that from (3.103) and (3.109),\n|fα(s)| ≤/integraldisplays\n0|f′\nα|dσ≤C(c0)/radicalbig\nα(1−α)/integraldisplays\n0σ2e−ασ2/4dσ,\nand /integraldisplay∞\n0σ2e−ασ2/4dσ=2√π\nα3/2, (3.110)\nso that\n|fα(s)| ≤C(c0)\nα2√1−α. (3.111)\nOn the other hand, differentiating the relations in (3.99) wi th respect to α,\n|zα| ≤2|fα||f|,|yα+ihα| ≤ |fα||f′|+|f||f′\nα|. (3.112)\nBy putting together (3.23), (3.109), (3.111) and (3.112), we obtain (3.100) and (3.101).\nLemma 3.11. The function z∞is continuous in (0,1]. More precisely, there exists a constant\nC(c0)depending on c0but not onα, such that\n|z∞(α2)−z∞(α1)| ≤C(c0)\nL(α2,α1)|α2−α1|,for allα1,α2∈(0,1], (3.113)\nwhere\nL(α2,α1) :=α2\n1α3/2\n2/parenleftBig\nα3/2\n1√\n1−α2+α3/2\n2√\n1−α1/parenrightBig\n.\nIn particular,\n|z∞(1)−z∞(α)| ≤C(c0)√\n1−α,for allα∈[1/2,1]. (3.114)\nProof. Letα1,α2∈(0,1],α1< α2. By classical results from the ODE theory, the functions\ny(s,α),h(s,α)andz(s,α)are smooth in R×[0,1)and continuous in R×[0,1](see e.g. [5, 17]).\nHence, integrating (3.27) with respect to s, we deduce that\nz∞(α2)−z∞(α1) = 2/integraldisplay∞\n0(y(s,α2)−y(s,α1))ds= 2/integraldisplay∞\n0/integraldisplayα2\nα1dy\ndµ(s,µ)dµds. (3.115)\nTo estimate the last integral, we use (3.101)\n/integraldisplayα2\nα1|dy\ndµ(s,µ)|dµ≤C(c0)s2\n√α1/integraldisplayα2\nα1e−µs2/4\n√1−µdµ. (3.116)\nNow, integrating by parts,\n/integraldisplayα2\nα1e−µs2/4\n√1−µdµ= 2/parenleftBig√\n1−α1e−α1s2/4−√\n1−α2e−α2s2/4/parenrightBig\n−s2\n2/integraldisplayα2\nα1/radicalbig\n1−µe−µs2/4dµ.\nTherefore, by combining with (3.115) and (3.116),\n|z∞(α2)−z∞(α1)| ≤C(c0)√α1/parenleftbigg√\n1−α1/integraldisplay∞\n0s2e−α1s2/4ds−√\n1−α2/integraldisplay∞\n0s2e−α2s2/4ds/parenrightbigg\n,\n30and bearing in mind (3.110), we conclude that\n|z∞(α2)−z∞(α1)| ≤C(c0)√α1/parenleftBigg√1−α1\nα3/2\n1−√1−α2\nα3/2\n2/parenrightBigg\n,\nwhich, after some algebraic manipulations and using that α1,α2∈(0,1], leads to (3.113).\nThe estimate for z∞near zero is more involved and it is based in an improvement of the\nestimate for the derivative of z∞.\nLemma 3.12. The function z∞is continuous in [0,1]. Moreover, there exists a constant\nC(c0)>0, depending on c0but not onαsuch that for all α∈(0,1/2],\n|z∞(α)−z∞(0)| ≤C(c0)√α|ln(α)|. (3.117)\nProof. As in the proof of Lemma 3.11, we recall that the functions y(s,α),h(s,α)andz(s,α)\nare smooth in any compact subset of R×[0,1). From now on we will use the identity (3.39)\nfixings= 1. We can verify that the two integral terms in (3.39) are conti nuous functions at\nα= 0, which proves that z∞is continuous in 0. In view of Lemma 3.11, we conclude that z∞is\ncontinuous in [0,1].\nNow we claim that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingledz∞\ndα(α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(c0)|ln(α)|√α,for allα∈(0,1/2]. (3.118)\nIn fact, once (3.118) is proved, we can compute\n|z∞(α)−z∞(0)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayα\n0dz∞\ndµ(µ)dµ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(c0)/integraldisplayα\n0|ln(µ)|√µdµ= 2C(c0)√α(|ln(α)|+2),\nwhich implies (3.117).\nIt remains to prove the claim. Differentiating (3.39) (recal l thats= 1) with respect to α,\nand using that y(1,·),h(1,·)andz(1,·)are continuous differentiable in [0,1/2], we deduce that\nthere exists a constant C(c0)>0such that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingledz∞\ndα(α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(c0)+8|I1(α)|+2c2\n0|I2(α)|, (3.119)\nwith\nI1(α) =/integraldisplay∞\n1z\nσ3+α/integraldisplay∞\n1zα\nσ3+6/integraldisplay∞\n1zα\nσ5(3.120)\nand\nI2(α) =−α\n2/integraldisplay∞\n1e−ασ2/2zσ+α/integraldisplay∞\n1e−ασ2/2zα\nσ+2/integraldisplay∞\n1e−ασ2/2zα\nσ3. (3.121)\nBy (3.24) and (3.100), zis uniformly bounded and zαgrows at most as a cubic polynomial,\nso that the first and the last integral in the r.h.s. of (3.120) are bounded independently of\nα∈[0,1/2]. In addition, (3.100) also implies that\n|zα|=|zα|1/2|zα|1/2≤C(c0)(s3)1/2/parenleftbigg1\nα2/parenrightbigg1/2\n=C(c0)s3/2\nα, (3.122)\nwhich shows that the remaining integral in (3.120) is bounde d.\n31Thus, the above argument shows that\n|I1(α)| ≤C(c0)for allα∈[0,1/2]. (3.123)\nThe same arguments also yield that the first two integrals in t he r.h.s. of (3.121) are bounded\nbyC(c0)α−1/2.Using once more that |zα| ≤C(c0)s2α−1/2, we obtain the following bounds for\nthe remaining two integrals in (3.121)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingleα/integraldisplays\n1e−ασ2/2zα\nσ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(c0)√α/integraldisplay∞\n1ασe−ασ2/2dσ=C(c0)√αe−α/2≤C(c0)√α\nand /vextendsingle/vextendsingle/vextendsingle/vextendsingle2/integraldisplay∞\n1e−ασ/2zα\nσ3dσ/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(c0)√α/integraldisplay∞\n1e−ασ2/2\nσdσ≤C(c0)|ln(α)|√α.\nIn conclusion, we have proved that\n|I2(α)| ≤C(c0)|ln(α)|√α,\nwhich combined with (3.119) and (3.123), completes the proo f of claim.\nWe end this section showing that the previous continuity res ults allow us to “pass to the limit”\nα→1−in Corollary 3.7. Using the notation b(α) =banda(α) =afor the constants defined for\nα∈[0,1)in Proposition 3.3 in Subsection 3.2, we have\nLemma 3.13. The valueb(α)is a continuous function of α∈[0,1]and the value b(α)eia(α)is\ncontinuous function of α∈[0,1)that can be continuously extended to [0,1]. The function a(α)\nhas a (possible discontinuous) extension for α∈[0,1]such thata(α)∈[0,2π).\nProof. By Lemma 3.12, we have the continuity of z∞in [0,1]. Therefore, in view of Remark 3.4,\nthe function beiais a continuous function of α∈[0,1)and by (3.49) bis actually well-defined\nand continuous in α∈[0,1].\nIt only remains to prove that the limit\nL:= lim\nα→1−b(α)eia(α)(3.124)\nexists. Ifb(1) = 0 , it is immediate that L= 0and we can give any arbitrary value in [0,2π)to\na(1). Let us suppose that b(1)>0. Integrating (3.93), we get\nw(s)e(α+iβ)s2/4=w(s0)e(α+iβ)s2\n0/4+/integraldisplays\ns0e(−α+iβ)σ2/4/parenleftbigg\nγ−c2\n0\n2(z−z∞)/parenrightbigg\ndσ,\nand this relation is valid for any α∈(0,1]. Letα∈(0,1). In view of (3.92), letting s→ ∞, we\nhave\nibei(a+C(α,c0))=w(s0)e(α+iβ)s2\n0/4+/integraldisplay∞\ns0e(−α+iβ)σ2/4/parenleftbigg\nγ−c2\n0\n2(z−z∞)/parenrightbigg\ndσ, (3.125)\nwhereC(α,c0)is the constant in (1.21). Notice that the r.h.s. of (3.125) i s well-defined for any\nα∈(0,1]and by the arguments given in the proof of Lemma 3.11 and the do minated convergence\ntheorem, the r.h.s. is also continuous for any α∈(0,1]. Therefore, the limit Lin (3.124) exists\nand is given by the r.h.s. of (3.125) evaluated in α= 1and divided by ieiC(1,c0). Moreover,\nlim\nα→1−eia(α)=L\nb(1),\nso that by the compactness of the the unit circle in C, there exists θ∈[0,2π)such thateiθ=\nL/b(1)and we can extend aby defining a(1) =θ.\n32The following result summarizes an improvement of Corollar y 3.7 to include the case α= 1\nand the continuous dependence of the constants appearing in the asymptotics on α. Precisely,\nwe have the following:\nCorollary 3.14. Letα∈[0,1],β≥0withα2+β2= 1andc0>0. Then,\n(i) The asymptotics in Corollary 3.7 holds true for all α∈[0,1].\n(ii) Moreover, the values bandbeiaare continuous functions of α∈[0,1]and each term in the\nasymptotics for z,yandhin Corollary 3.7 depends continuously on α∈[0,1].\n(iii) In addition, the bounds controlling the error terms de pend onc0and are independent of\nα∈[0,1].\nProof. Lets≥s0fixed. As noticed in the proof of Lemma 3.11, the functions y(s,α),h(s,α),\nz(s,α)are continuous in α= 1. In addition, by Lemma 3.13 beiais continuous in α= 1, using\nthe definition of φ, it is immediate that bsin(φ(s))andbcos(φ(s))are continuous in α= 1.\nTherefore the big- Oterms in (3.95), (3.96) and (3.97) are also are continuous in α= 1. The\nproof of the corollary follows by letting α→1−in (3.95), (3.96) and (3.97).\n3.3.2 Dependence on c0\nIn this subsection, we study the dependence of z∞as a function of c0, for a fixed value of α.\nTo this aim, we need to take into account the initial conditio ns given in (3.13)–(3.15). More\ngenerally, let us assume that fis a solution of (3.20) with initial conditions f(0)andf′(0)that\ndepend smoothly on c0, for anyc0>0, and that E0>0is the associated energy defined in\n(3.16). To keep our notation simple, we omit the parameter c0in the functions fandz∞. Under\nthese assumptions, we have\nProposition 3.15. Letα∈[0,1]andc0>0. Thenz∞is a continuous function of c0∈(0,∞).\nMoreover if α∈(0,1], the following estimate hold\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglez∞−/vextendsingle/vextendsingle/vextendsingle/vextendsinglef(0)+f′(0)√π√α+iβ/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤√2E0c0π\nα/vextendsingle/vextendsingle/vextendsingle/vextendsinglef(0)+f′(0)√π√α+iβ/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/parenleftbigg√2E0c0π\n2α/parenrightbigg2\n. (3.126)\nProof. Since we are assuming that the initial conditions f(0)andf′(0)depend smoothly on c0,\nby classical results from the ODE theory, the functions f,y,handzare smooth with respect to s\nandc0. From (3.39) with s= 1, we have that z∞can be written in terms of continuous functions\nofc0(the continuity of the integral terms follows from the domin ated convergence theorem), so\nthatz∞depends continuously on c0.\nTo prove (3.126), we multiply (3.20) by e(α+iβ)s2/4, so that\n(f′e(α+iβ)s2/4)′=−c2\n0\n4f(s)e(−α+iβ)s2/4.\nHence, integrating twice, we have\nf(s) =f(0)+G(s)+F(s), (3.127)\nwith\nG(s) =f′(0)/integraldisplays\n0e−(α+iβ)σ2/4dσandF(s) =−c2\n0\n4/integraldisplays\n0e−(α+iβ)σ2/4/integraldisplayσ\n0e(−α+iβ)τ2/4f(τ)dτdσ.\n33Since by Proposition 3.2 |f(s)| ≤2√2E0\nc0, we obtain\n|F(s)| ≤√2E0c0\n2/integraldisplays\n0e−ασ2/4/integraldisplayσ\n0e−ατ2/4dτdσ≤√2E0c0\n2·π\nα. (3.128)\nUsing (3.127) and the identity,\n|z1+z2|2=|z1|2+2Re(¯z1z2)+|z2|2, z1,z2∈C,\nwe conclude that z(s) =|f(s)|2satisfies\nz(s) =|f(0)+G(s)|2+2Re(¯F(s)(f(0)+G(s)))+|F(s)|2.\nTherefore, for all s≥0,\n|z(s)−|f(0)+G(s)|2| ≤2|F(s)||f(0)+G(s)|+|F(s)|2.\nHence we can use the bound (3.128) and then let s→ ∞. Noticing that\nlim\ns→∞G(s) =f′(0)/integraldisplay∞\n0e−(α+iβ)σ2/4dσ=f′(0)√π√α+iβ,\nthe estimate (3.126) follows.\n4 Proof of the main results\nIn Section 3 we have performed a careful analysis of the equat ion (3.12), taking also into con-\nsideration the initial conditions (3.13)–(3.15). Therefo re, the proofs of our main theorem consist\nmainly in coming back to the original variables using the ide ntities (3.18) and (3.19). For the\nsake of completeness, we provide the details in the followin g proofs.\nProof of Theorem 1.2 .Letα∈[0,1],c0>0and{/vector mc0,α(·),/vector nc0,α(·),/vectorbc0,α(·)}be the unique\nC∞(R;S2)-solution of the Serret–Frenet equations (1.6) with curvat ure and torsion (2.6) and\ninitial conditions (2.8). In order to simplify the notation , in the rest of the proof we drop the\nsubindexes c0andαand simply write {/vector m(·),/vector n(·),/vectorb(·)}for{/vector mc0,α(·),/vector nc0,α(·),/vectorbc0,α(·)}.\nFirst observe that if we define {/vectorM,/vectorN,/vectorB}in terms of {/vector m,/vector n,/vectorb}by\n/vectorM(s) = (m(−s),−m(−s),−m(−s)),\n/vectorN(s) = (−n(−s),n(−s),n(−s)),\n/vectorB(s) = (−b(−s),b(−s),b(−s)), s∈R,\nthen{/vectorM,/vectorN,/vectorB}is also a solution of the Serret system (1.6) with curvature a nd torsion (2.6).\nNotice also that\n{/vectorM(0),/vectorN(0),/vectorB(0)}={/vector m(0),/vector n(0),/vectorb(0)}.\nTherefore, from the uniqueness of the solution we conclude t hat\n/vectorM(s) =/vector m(s),/vectorN(s) =/vector n(s)and/vectorB(s) =/vectorb(s),∀s∈R.\nThis proves part (i)of Theorem 1.2.\n34Second, in Section 3 we have seen that one can write the compon ents of the Frenet trihedron\n{/vector m,/vector n,/vectorb}as\nm1(s) = 2|f1(s)|2−1, n 1(s)+ib1(s) =4\nc0eαs2/4¯f1(s)f′\n1(s), (4.1)\nmj(s) =|fj(s)|2−1, n j(s)+ibj(s) =2\nc0eαs2/4¯fj(s)f′\nj(s), j∈ {2,3}, (4.2)\nwithfjsolution of the second order ODE (3.12) with initial conditi ons (3.13)-(3.15) respectively,\nand associated initial energies (see (3.17))\nE0,1=c2\n0\n8andEj,1=c2\n0\n8,forj∈ {2,3}. (4.3)\nNotice that the identities (4.1)–(4.2) rewrite equivalent ly as\n\n\nm1,c0,α= 2z1−1, n1,c0,α=4\nc0eαs2/4y1, b1,c0,α=4\nc0eαs2/4h1,\nmj,c0,α=zj−1, nj,c0,α=2\nc0eαs2/4yj, bj,c0,α=2\nc0eαs2/4hj, j∈ {2,3},(4.4)\nin terms of the quantities {zj,yj,hj}defined by\nzj=|fj|2, yj= Re(¯fjf′\nj)andhj= Im(¯fjf′\nj).\nDenote by zj,∞,aj,bj,γjandφjthe constants and function appearing in the asymptotics of\n{yj,hj,zj}proved in Section 3 in Corollary 3.14.\nTaking the limit as s→+∞in (4.1)–(4.2), and since |/vector m(s)|= 1, we obtain that there exists\n/vectorA+= (A+\nj)3\nj=1∈S2with\nA+\n1= 2z1,∞−1, A+\nj=zj,∞−1,forj∈ {2,3}. (4.5)\nThe asymptotics stated in part (ii)of Theorem 1.2 easily follows from formulae (4.1)–(4.2) and the\nasymptotics for {zj,yj,hj}established in Corollary 3.14. Indeed, it suffices to observe that from\nthe formulae for bjandγjin terms of the initial energies E0,jandzj,∞given in Corollary 3.14,\n(4.3) and (4.5) we obtain\nb2\n1=c2\n0\n16(1−(A+\n1)2), b2\n2=c2\n0\n4(1−(A+\n2)2), b2\n3=c2\n0\n4(1−(A+\n3)2), (4.6)\nγ1=−c2\n0\n4A+\n1, γ2=−c2\n0\n2A+\n2, γ3=−c2\n0\n2A+\n3. (4.7)\nSubstituting these constants in (3.95), (3.96) and (3.97) i n Corollary 3.14, we obtain (1.16),\n(1.17) and (1.18). This completes the proof of Theorem 1.2- (ii).\nProof of Theorem 1.1 .Letα∈[0,1], andc0>0. As before, dropping the subindexes, we\nwill denote by {/vector m,/vector n,/vectorb}the unique solution of the Serret–Frenet equations (1.6) wi th curvature\nand torsion (2.6) and initial conditions (2.8). Define\n/vectorm(s,t) =/vector m/parenleftbiggs√\nt/parenrightbigg\n. (4.8)\n35As has been already mentioned (see Section 2), part (i)of Theorem 1.1 follows from the fact\nthat the triplet {/vector m,/vector n,/vectorb}is a regular- (C∞(R;S2))3solution of (1.6)-(2.6)-(2.8) and satisfies the\nequation\n−s\n2c/vector n=β(c′/vectorb−cτ/vector n)+α(cτ/vectorb+c′/vector n).\nNext, from the parity of the components of the profile /vector m(·)and the asymptotics established in\nparts(i)and(ii)in Theorem 1.2, it is immediate to prove the pointwise conver gence (1.9). In\naddition,/vectorA−= (A+\n1,−A+\n2,−A+\n3)in terms of the components of the vector /vectorA+= (A+\nj)3\nj=1.\nNow, using the symmetries of /vector m(·), the change of variables η=s/√\ntgives us\n/bardbl/vectorm(·,t)−/vectorA+χ(0,∞)(·)−/vectorA−χ(−∞,0)(·)/bardblLp(R)=3/summationdisplay\nj=1/parenleftbigg\n2t1/2/integraldisplay∞\n0|mj(η)−A+\nj|pdη/parenrightbigg1/p\n.(4.9)\nTherefore, it only remains to prove that the last integral is finite. To this end, let s0= 4/radicalbig\n8+c2\n0.\nOn the one hand, notice that since /vector mand/vectorA+are unitary vectors,\n/integraldisplays0\n0|mj(s)−Aj|pds≤2ps0. (4.10)\nOn the other hand, from the asymptotics for /vector m(·)in (1.16), (1.20), and the fact that the vectors\n/vectorA+and/vectorB+satisfy|/vectorA+|2= 1and|/vectorB+|2= 2, we obtain\n/parenleftbigg/integraldisplay∞\ns0|mj(s)−A+\nj|pds/parenrightbigg1/p\n≤2√\n2c0(α+β)/parenleftBigg/integraldisplay∞\ns0e−αs2p/4\nsp/parenrightBigg1/p\n+2c2\n0/parenleftBigg/integraldisplay∞\ns0e−αs2p/2\ns2p/parenrightBigg1/p\n+C(c0)/parenleftBigg/integraldisplay∞\ns0e−αs2p/4\ns3p/parenrightBigg1/p\n. (4.11)\nSince the r.h.s. of (4.11) is finite for all p∈(1,∞)ifα∈[0,1], and for all p∈[1,∞)if\nα∈(0,1], inequality (1.10) follows from (4.9), (4.10) and (4.11). T his completes the proof of\nTheorem 1.1.\nProof of Theorem 1.3 .The proof is a consequence of Proposition 3.15. In fact, reca ll the\nrelations (4.5) and (3.17), that is\nA+\n1= 2z1,∞−1,andA+\nj=zj,∞−1,forj∈ {2,3},\nand\nE0,1=c2\n0\n8, E 0,j=c2\n0\n4,forj∈ {2,3},\nThus the continuity of /vectorA+\nc0,αwith respect to c0, follows from the continuity of z∞in Proposi-\ntion 3.15.\nUsing the initial conditions (3.13)–(3.15), the values for the energies E0,jforj∈ {1,2,3}, and\nthe identity√π√α+iβ=√π√\n2/parenleftbig√\n1+α−i√\n1−α/parenrightbig\n,\nwe now compute\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglefj(0)+f′\nj(0)√π√α+iβ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n=\n\n1, ifj= 1,\n1+c2\n0π\n4+c0√π√\n2√1+α,ifj= 2,\n1+c2\n0π\n4+c0√π√\n2√1−α,ifj= 3.(4.12)\n36Then, substituting the values (4.12) in (3.126) and using th e above relations together with the\ninequality√1+x≤1+x/2forx≥0, we obtain the estimates (1.24)–(1.26).\nProof of Theorem 1.4 .Recall that the components of /vectorA+\nc0,αare given explicitly in (4.5) in\nterms of the functions zj,∞, forj∈ {1,2,3}. The continuity on [0,1]ofA+\nj,c0,αas a function\nofαforj∈ {1,2,3}follows from that of zj,∞established in Lemma 3.12. Notice also that the\nestimates (1.27) and (1.28) are an immediate consequence of (3.117) in Lemma 3.12 and (3.114)\nin Lemma 3.11, respectively.\nBefore giving the proof of Proposition 1.5, we recall that whe nα= 0orα= 1, the vector\n/vectorA+\nc0,α= (Aj,c0,α)3\nj=1is determined explicitly in terms of the parameter c0(see [15] for the case\nα= 0and Appendix for the case α= 1). Precisely,\nA1,c0,0=e−πc2\n0\n2, (4.13)\nA2,c0,0= 1−e−πc2\n0\n4\n8πsinh(πc2\n0/2)|c0Γ(ic2\n0/4)+2eiπ/4Γ(1/2+ic2\n0/4)|2, (4.14)\nA3,c0,0= 1−e−πc2\n0\n4\n8πsinh(πc2\n0/2)|c0Γ(ic2\n0/4)−2e−iπ/4Γ(1/2+ic2\n0/4)|2(4.15)\nand\n/vectorA+\nc0,1= (cos(c0√π),sin(c0√π),0). (4.16)\nProof of Proposition 1.5 .Recall that (see Theorem 1.1)\n/vectorA−\nc0,α= (A+\n1,c0,α,−A+\n2,c0,α,−A+\n3,c0,α), (4.17)\nwithA+\nj,c0,αthe components of /vectorA+\nc0,α. Therefore /vectorA+\nc0,α/ne}ationslash=/vectorA−\nc0,αiffA+\n1,c0,α/ne}ationslash= 1or−1.\nParts (ii)and(iii)follow from the continuity of A+\n1,c0,αin[0,1]established in Theorem 1.4\nbearing in mind that, from the expressions for A+\n1,c0,0in (4.13) and A+\n1,c0,1in (4.16), we have that\nA+\n1,c0,0/ne}ationslash=±1for allc0>0andA+\n1,c0,1/ne}ationslash=±1ifc0/ne}ationslash=k√πwithk∈N.\nIn order to proof part (i), we will argue by contradiction. Assume that for some α∈(0,1),\nthere exists a sequence {c0,n}n∈Nsuch thatc0,n>0,c0,n−→0asn→ ∞ and/vectorA+\nc0,n,α=/vectorA−\nc0,nα.\nHence from (4.17) the second and third component of /vectorA+\nc0,n,αare zero. Thus the estimate (1.25)\nin Theorem 1.3 yields\nc0,n/radicalbig\nπ(1+α)√\n2≤c2\n0,nπ\n4+c2\n0,nπ\nα√\n2/parenleftBigg\n1+c2\n0,nπ\n8+c0,n/radicalbig\nπ(1+α)\n2√\n2/parenrightBigg\n+/parenleftBigg\nc2\n0,nπ\n2√\n2α/parenrightBigg2\n.\nDividing by c0,n>0and letting c0,n→0asn→ ∞, the contradiction follows.\n5 Some numerical results\nAs has been already pointed out, only in the cases α= 0andα= 1we have an explicit formula\nfor/vectorA+\nc0,α(see (4.13)–(4.16)). Theorems 1.3 and 1.4 give information about the behaviour of /vectorA+\nc0,α\nfor small values of c0for a fixed valued of α, and for values of αnear to 0 or 1 for a fixed valued of\nc0. The aim of this section is to give some numerical results tha t allow us to understand the map\n37(α,c0)∈[0,1]×(0,∞)/ma√sto→/vectorA±\nc0,α∈S2. For a fixed value of α, we will discuss first the injectivity\nand surjectivity (in some appropriate sense) of the map c0/ma√sto→/vectorA±\nc0,αand second the behaviour of\n/vectorA+\nc0,αasc0→ ∞.\nFor fixedα, defineθc0,αto be the angle between the unit vectors /vectorA+\nc0,αand−/vectorA−\nc0,αassociated\nto the family of solutions /vectormc0,α(s,t)established in Theorem 1.1, that is θc0,αsuch that\ncos(θc0,α) = 1−2(A+\n1,c0,α)2. (5.1)\nIt is pertinent to ask whether θc0,αmay attain any value in the interval [0,π]by varying the\nparameterc0>0.\nIn Figure 2 we plot the function θc0,αassociated to the family of solutions /vectormc0,α(s,t)estab-\nlished in Theorem 1.1 for α= 0,α= 0.4andα= 1, as a function of c0>0. The curves θc0,0\nandθc0,1are exact since we have explicit formulae for A+\n1,c0,αwhenα= 0andα= 1(see (4.13)\nand (4.16)). We deduce that in the case α= 0, there is a bijective relation between c0>0and\nthe angles in (0,π). In the case α= 1, there are infinite values of c0>0that allow to reach\nany angle in [0,π]. Ifα∈(0,1), numerical simulations show that there exists θ∗\nα∈(0,π)such\nthat the angles in (θ∗\nα,π)are reached by a unique value of c0, but for angles in [0,θ∗\nα]there are\nat least two values of c0>0that produce them (See θc0,0.4in Figure 2).\nθc0,0\nπ\nc0\nθc0,0.4\nπ\nc0\nθc0,1\nπ\nc0\nFigure 2: The angles θc0,αas a function of c0forα= 0,α= 0.4andα= 1.\nThese numerical results suggest that, due to the invariance of (LLG) under rotations2, for a\nfixedα∈[0,1)one can solve the following inverse problem: Given any disti nct vectors /vectorA+,/vectorA−∈\nS2there exists c0>0such that the associated solution /vectormc0,α(s,t)given by Theorem 1.1 (possibly\nmultiplied by a rotation matrix) provides a solution of (LLG ) with initial condition\n/vectorm(·,0) =/vectorA+χ(0,∞)(·)+/vectorA−χ(−∞,0)(·). (5.2)\nNote that in the case α= 1the restriction /vectorA+/ne}ationslash=/vectorA−can be dropped.\nIn addition, Figure 2 suggests that /vectorA+\nc0,α/ne}ationslash=/vectorA−\nc0,αfor fixedα∈[0,1)andc0>0. Indeed,\nnotice that /vectorA+\nc0,α/ne}ationslash=/vectorA−\nc0,αif and only if A1/ne}ationslash=±1or equivalently cosθc0,α/ne}ationslash=−1, that isθc0,α/ne}ationslash=π,\nwhich is true if α∈[0,1)for anyc0>0(See Figure 2). Notice also that when α= 1, then the\nvalueπis attained by different values of c0.\nThe next natural question is the injectivity of the applicat ionc0−→θc0,α, for fixed α.\nPrecisely, can we generate the same angle using different val ues ofc0? In the case α= 0, the\n2In fact, using that\n(M/vector a)×(M/vectorb) = (det M)M−T(/vector a×/vectorb),for allM∈ M3,3(R), /vector a,/vectorb∈R3,\nit is easy to verify that if /vectorm(s,t)is a solution of (LLG) with initial condition /vectorm0, then/vectormR:=R/vectormis a solution\nof (LLG) with initial condition /vectorm0\nR:=R/vectorm0, for any R∈SO(3).\n38plot ofθc0,0in Figure 2 shows that the value of c0is unique, in fact one has following formula\nsin(θc0,0/2) =A1,c0,0=e−c2\n0\n2π(see [15]). In the case α= 1, we have sin(θc0,1/2) =A1,c0,1=\ncos(c0/radicalbig\nπ), moreover\n/vectorA+\nc0,1=/vectorA+\nc0+2k√π,1,for anyk∈Z. (5.3)\nAs before, if α∈(0,1)we do not have an analytic answer and we have to rely on numeric al\nsimulations. However, it is difficult to test the uniqueness o fc0numerically. Using the command\nFindRoot in Mathematica, we have found such values. For instance, for α= 0.4, we obtain that\nc0≈2.1749andc0≈6.6263give the same value of /vectorA+\nc0,0.4. The respective profiles /vector mc0,0.4(·)are\nshown in Figure 3. This multiplicity of solutions suggests t hat the Cauchy problem for (LLG)\nwith initial condition (5.2) is ill-posed, at least for cert ain values of c0. This interesting problem\nwill be studied in a forthcoming paper.\nm1m2m3\n(a)/vector mc0,0.4(·), withc0≈2.1749\nm1m2m3\n(b)/vector mc0,0.4(·), withc0≈6.6263\nFigure 3: Two profiles /vector mc0,0.4(·), with the same limit vector /vectorA+\nc0,0.4.\nThe rest of this section is devoted to give some numerical res ults on the behaviour of the\nlimiting vector /vectorA+\nc0,α. In particular, the results below aim to complement those es tablished in\nTheorem 1.3 on the behaviour of /vectorA+\nc0,αfor small values of c0, whenαis fixed.\nWe start recalling what it is known in the extremes cases α= 0andα= 1. Precisely, if\nα= 0, the explicit formulae (4.13)–(4.15) for /vectorA+\nc0,0allow us to prove that\nlim\nc0→0+A+\n3,c0,0= 0 andlim\nc0→∞A+\n3,c0,1= 1, (5.4)\nand also that {A+\n3,c0,0:c0∈(0,∞)}= (0,1). Whenα= 1the picture is completely different. In\nfactA+\n3,c0,1= 0for allc0>0, and the limit vectors remain in the equator plane S1×{0}. The\nnatural question is what happens with /vectorA+\nc0,αwhenα∈(0,1)as a function of c0.\nAlthough we do not provide a rigorous answer to this question , in Figure 4 we show some\nnumerical results. Precisely, Figure 4 depicts the curves /vectorA+\nc0,0.01,/vectorA+\nc0,0.4and/vectorA+\nc0,0.8as functions\nofc0, forc0∈[0,1000]. We see that the behaviour of /vectorA+\nc0,αchanges when αincreases in the sense\nthat the first and second coordinates start oscillating more and more as αgoes to 1. In all the\ncases the third component remains monotonically increasin g withc0, but the value of A+\n3,1000,α\nseems to be decreasing with α. At this point it is not clear what the limit value of A+\n3,c0,αas\n39c0→ ∞ is. For this reason, we perform a more detailed analysis of A+\n3,c0,αand we show the\ncurvesA+\n3,1,α,A+\n3,10,α,A+\n3,1000,α(for fixedα∈[0,1]) in Figure 5. From these results we conjecture\nthat{A+\n3,c0,·}c0>0is a pointwise nondecreasing sequence of functions that con verges to 1for any\nα<1asc0→ ∞. This would imply that, for α∈(0,1)fixed,A1,c0,α→0asc0→ ∞, and since\nA1,c0,α→1asc0→0(see (1.24)), we could conclude by continuity (see Theorem 1 .3) that for\nany angleθ∈(0,π)there exists c0>0such thatθis the angle between /vectorA+\nc0,αand−/vectorA+\nc0,α(see\n(5.1)). This provides an alternative way to justify the surj ectivity of the map c0/ma√sto→/vectorA+\nc0,α(in the\nsense explained above).\nA+\n1A+\n2A+\n3\n(a)/vectorA+\nc0,0.01\nA+\n1A+\n2A+\n3\n(b)/vectorA+\nc0,0.4\nA+\n1A+\n2A+\n3\n(c)/vectorA+\nc0,0.8\nFigure 4: The curves /vectorA+\nc0,0.01,/vectorA+\nc0,0.4and/vectorA+\nc0,0.8as functions of c0, forc0∈[0,1000].\n01\n1αA+\n3,1,αA+\n3,10,αA+\n3,1000,α\nFigure 5: The curves A+\n3,1,α,A+\n3,10,α,A+\n3,1000,αas functions of α, forα∈[0,1].\nThe curves in Figure 5 also allow us to discuss further the res ults in Theorem 1.4. In fact,\nwhenαis close to 1 the slope of the functions become unbounded and, roughly speaking, the\nbehaviour of A+\n3,c0,αis in agreement with the result in Theorem 1.4, that is\nA+\n3,c0,α∼C(c0)√\n1−α,asα→1−.\nNumerically, the analysis is more difficult when α∼0, because the number of computations\nneeded to have an accurate profile of A+\n3,c0,αincreases drastically as α→0+. In any case,\nFigure 5 suggests that A+\n3,c0,αconverges to A+\n3,c0,0faster than√α|ln(α)|. We think that this rate\nof convergence can be improved to α|ln(α)|. In fact, in the proof of Lemma 3.10 we only used\nenergy estimates. Probably, taking into account the oscill ations in equation (3.102) (as did in\nProposition 3.3), it would be possible to establish the nece ssary estimates to prove the following\nconjecture:\n|/vectorA+\nc0,α−/vectorA+\nc0,0| ≤C(c0)α|ln(α)|,forα∈(0,1/2].\n406 Appendix\nIn this appendix we show how to compute explicitly the soluti on/vectormc0,α(s,t)of the LLG equation\nin the case α= 1. As a consequence, we will obtain an explicit formula for the limiting vector\n/vectorA+\nc0,1and the other constants appearing in the asymptotics of the a ssociated profile established\nin Theorem 1.2 in terms of the parameter c0in the case when α= 1.\nWe start by recalling that if α= 1thenβ= 0. We need to find the solution {/vector m,/vector n,/vectorb}of the\nSerret–Frenet system (1.6) with c(s) =c0e−s2/4,τ≡0and the initial conditions (1.8). Hence,\nit is immediate that\nm3=n3≡0, b1=b2≡0andb3≡1.\nTo compute the other components, we use the Riccati equation (3.9) satisfied by the stereographic\nprojection of {mj,nj,bj}\nηj=nj+ibj\n1+mj,forj∈ {1,2}, (6.1)\nfound in the proof of Lemma 3.1. For the values of curvature an d torsionc(s) =c0e−s2/4and\nτ(s) = 0 the Riccati equation (3.9) reads\nη′\nj+iβs\n2ηj+c0\n2e−αs2/4(η2\nj+1) = 0. (6.2)\nWe see that when α= 1, and thusβ= 0, (6.2) is a separable equation that we write as:\ndηj\nη2\nj+1=−c0\n2e−αs2/4,\nso integrating, we get\nηj(s) = tan/parenleftBig\narctan(ηj(0))−c0\n2Erf(s)/parenrightBig\n, (6.3)\nwhereErf(s)is the non-normalized error function\nErf(s) =/integraldisplays\n0e−σ2/4dσ.\nAlso, using (1.8) and (6.1) we get the initial conditions η1(0) = 0 andη2(0) = 1 . In particular,\nifc0is small (6.3) is the global solution of the Riccati equation , but it blows-up in finite time if\nc0is large. As long as ηjis well-defined, by Lemma 3.1,\nfj(s) =ec0\n2/integraltexts\n0e−ασ2/4ηj(σ)dσ.\nThe change of variables\nµ= arctan(ηj(0))−c0\n2Erf(s)\nyields/integraldisplays\n0e−ασ2/4ηj(σ)dσ=2\nc0ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglecos/parenleftbig\narctan(ηj(0))−c0\n2Erf(s)/parenrightbig\ncos(arctan( ηj(0)))/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle,\nand after some simplifications, we obtain\nf1(s) =/vextendsingle/vextendsingle/vextendsinglecos/parenleftBigc0\n2Erf(s)/parenrightBig/vextendsingle/vextendsingle/vextendsingleandf2(s) =/vextendsingle/vextendsingle/vextendsinglecos/parenleftBigc0\n2Erf(s)/parenrightBig\n+sin/parenleftBigc0\n2Erf(s)/parenrightBig/vextendsingle/vextendsingle/vextendsingle.\nIn view of (3.18) and (3.19), we conclude that\nm1(s) = 2|f1(s)|2−1 = cos(c0Erf(s))andm2(s) =|f2(s)|2−1 = sin(c0Erf(s)).(6.4)\n41A priori, the formulae in (6.4) are valid only as long as ηis well-defined, but a simple verification\nshow that these are the global solutions of (1.6), with\nn1(s) =−sin(c0Erf(s))andn2(s) = cos(c0Erf(s)).\nIn conclusion, we have proved the following:\nProposition 6.1. Letα= 1, and thusβ= 0. Then, the trihedron {/vector mc0,1,/vector nc0,1,/vectorbc0,1}solution\nof(1.6)–(1.8)is given by\n/vector mc0,1(s) = (cos(c0Erf(s)),sin(c0Erf(s)),0),\n/vector nc0,1(s) =−(sin(c0Erf(s)),cos(c0Erf(s)),0),\n/vectorbc0,1(s) = (0,0,1),\nfor alls∈R. In particular, the limiting vectors /vectorA+\nc0,1and/vectorA−\nc0,1in Theorem 1.2 are given in\nterms ofc0as follows:\n/vectorA±\nc0,1= (cos(c0√π),±sin(c0√π),0).\nProposition 6.1 allows us to give an alternative explicit pr oof of Theorem 1.2 when α= 1.\nCorollary 6.2. [Explicit asymptotics when α= 1] With the same notation as in Proposition 6.1,\nthe following asymptotics for {/vector mc0,1,/vector nc0,1,/vectorbc0,1}holds true:\n/vector mc0,1(s) =/vectorA+\nc0,1−2c0\ns/vectorB+\nc0,1e−s2/4sin(/vector a)−2c2\n0\ns2/vectorA+\nc0,1e−s2/2+O/parenleftBigg\ne−s2/4\ns3/parenrightBigg\n,\n/vector nc0,1(s) =/vectorB+\nc0,1sin(/vector a)+2c0\ns/vectorA+\nc0,1e−s2/4−2c2\n0\ns2/vectorB+\nc0,1e−s2/2sin(/vector a)+O/parenleftBigg\ne−s2/4\ns3/parenrightBigg\n,\n/vectorbc0,1(s) =/vectorB+\nc0,1cos(/vector a),\nwhere the vectors /vectorA+\nc0,1,/vectorB+\nc0,1and/vector a= (aj)3\nj=1are given explicitly in terms of c0by\n/vectorA+\nc0,1= (cos(c0√π),sin(c0√π),0),/vectorB+\nc0,1= (|sin(c0√π)|,|cos(c0√π)|,1),\na1=/braceleftBigg\n3π\n2,ifsin(c0√π)≥0,\nπ\n2,ifsin(c0√π)<0,a2=/braceleftBigg\nπ\n2,ifcos(c0√π)≥0,\n3π\n2,ifcos(c0√π)<0,anda3= 0.\nHere, the bounds controlling the error terms depend on c0.\nProof. By Proposition 6.1,\n\n\n/vector mc0,1(s) = (cos(c0√π−c0Erfc(s)),sin(c0√π−c0Erfc(s)),0),\n/vector nc0,1(s) =−(sin(c0√π−c0Erfc(s)),cos(c0√π−c0Erfc(s)),0),\n/vectorbc0,1(s) = (0,0,1),(6.5)\nwhere the complementary error function is given by\nErfc(s) =/integraldisplay∞\nse−σ2/4dσ=√π−Erf(s).\nIt is simple to check that\nsin(c0Erfc(s)) =e−s2/4/parenleftbigg2c0\ns−4c0\ns3+24c0\ns5+O/parenleftBigc0\ns7/parenrightBig/parenrightbigg\n,\ncos(c0Erfc(s)) = 1+e−s2/2/parenleftbigg\n−2c2\n0\ns2+8c2\n0\ns4−56c2\n0\ns6+O/parenleftbiggc2\n0\ns8/parenrightbigg/parenrightbigg\n,\n42so that, using (6.5), we obtain that\nm1(s) =n2(s) = cos(c0√π)+2c0\nse−s2/4sin(c0√π)−2c2\n0\ns2e−s2/2cos(c0√π)+O/parenleftBigg\ne−s2/4\ns3/parenrightBigg\n,\nm2(s) =−n1(s) = sin(c0√π)−2c0\nse−s2/4cos(c0√π)−2c2\n0\ns2e−s2/2sin(c0√π)+O/parenleftBigg\ne−s2/4\ns3/parenrightBigg\n.\nThe conclusion follows from the definitions of /vectorA+\nc0,1,/vectorB+\nc0,1and/vector a.\nRemark 6.3. 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(9) , 80(10):1029–1044, 2001.\n45"    },    {        "title": "1906.10326v2.Conductivity_Like_Gilbert_Damping_due_to_Intraband_Scattering_in_Epitaxial_Iron.pdf",        "content": "1 \n Conductivity -Like Gilbert Damping due to Intraband Scattering in Epitaxial Iron  \n Behrouz Khodadadi1, Anish Rai2,3, Arjun Sapkota2,3, Abhishek Srivastava2,3, Bhuwan Nepal2,3, \nYoungmin  Lim1, David A. Smith1, Claudia Mewes2,3, Sujan Budhathoki2,3, Adam J. Hauser2,3, \nMin Gao4, Jie-Fang Li4, Dwight D. Viehland4, Zijian Jiang1, Jean J. Heremans1, Prasanna  V. \nBalachandran5,6, Tim Mewes2,3, Satoru Emori1* \n1 Department of Physics, Virginia Tech , VA 24061, U.S.A  \n2Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA  \n 3 Center for Materials for Information Technology (MINT), University of Alabama, Tuscaloosa, \nAL 35487, U .S.A. \n4  Department  of Material Science and Engineering, Virginia Tech ,  \n Blacksburg, VA 24061, U.S.A . \n5  Department of Material Science and Engineering, University of Virginia,  \n Charlottesville, VA 22904, U.S.A . \n6  Department of Mechanical and Aerospace Engineering , Univer sity of Virginia,  \n Charlottesville, VA 22904, U.S.A.  \n*email: semori@vt.edu  \n \nConfirming  the or igin of Gilbert damping by experiment has remained  a challenge  for \nmany decades , even for simple  ferromagnetic metals . In this Letter, we experimentally \nidentify  Gilbert damping that increases with decreasing electronic scattering  in epitaxial \nthin films of pure Fe . This observation of  conductivity -like damping, which cannot be  \naccounted for by  classical eddy current loss , is in excellent quantitative agreement with \ntheoretical  predictions of Gilbert damping due to intraband scatte ring. Our results  resolve 2 \n the longstanding question about a fundamental damping mechanism  and offer  hints for \nengineering low -loss magnetic metals for cryogenic spintronic s and quantum devices.    \n \nDamping determines how fast the magnetization relaxes towards the effective magnetic \nfield and plays a  central  role in many aspects  of magnetization dynamics  [1,2] . The magnitude of \nviscous Gilbert damping governs the  threshold current for spin -torque magnetic switching  and \nauto-oscillations  [3,4] , mobility of magnetic domain walls  [5,6] , and  decay leng ths of diffusive \nspin waves and superfluid -like spin current s [7,8] . To enable spintronic technologies with low \npower dissipation , there is currently  much interest in  minimizing Gilbert damping in thin films of \nmagnetic m aterials  [9–13], especially ferromagnetic metals  [14–23] that are compatible with \nconventional device fabrication schemes . Despite the fundamental and technological importance \nof Gilbert damping, its physical mechanisms  in various magnetic materials have yet to be \nconfirmed by experiment .   \nGilbert damping is generally attributed to spin-orbit coupling that ultimately dissipates  \nthe energy of the magnetic system  to the lattice  [1,2] . Kambersky’s torque correlation model  [24] \nqualitatively captures  the temperature dependence of damping in some experiments  [25–28] by \npartitioning Gilbert damping into two mechanisms due to spin -orbit coupling, namely interband \nand intraband scattering mechanisms, each with a distinct dependence  on the elect ronic \nmomentum scattering tim e e. For the interband scattering mechanism  where magnetization \ndynamics can excite electron -hole pairs across dif ferent bands, the resulting Gilbert damping is \n“resistivity -like” as its magnitude scales with e-1, i.e., increased  electronic scattering results in \nhigher damping  [29,30] . By contrast, the intraband scattering mechanism  is typically understood \nthrough the breathing Fermi surface mode l [31], where electron -hole pairs are excited in the 3 \n same band , yielding  “conductivity -like” Gilbert damping  that scales with  e, i.e., reduced \nelectronic scattering  results in  higher damping.  \nConductivity -like Gilbert damping was reported  experimentally more than  40 years ago \nin bulk crystals of pure Ni and Co at low temperatures , but surprisingly not in pure Fe [25]. The  \napparent absence  of co nductivity -like damping in Fe has been  at odds with many theoretical \npredictions that intraband scattering should dominate at low temperatures  [32–38], although \nsome theoretical studies have suggested that intraband scattering may be  absent alt ogether in \npure metals  [39,40] . To date, no experimental work has conclusively addressed  the role of \nintraband scattering in  pure Fe1. There thus remains  a significant gap in the fundamental \nunderstanding of damping in one of the simplest ferromagnetic metals.  Intrinsic conductivity -\nlike Gilbert damping in Fe is also technologically relevant,  since  minimizing damping in \nferromagnetic metals at low temperatures is crucial for cryogenic superconducting spintronic \nmemories  [41,42]  and quantum information  transduction schemes  [43,44] .  \nIn this Letter, we experimentally demonstrate the presence of conductivity -like Gilbert \ndamping due to intr aband scattering  in epitaxial thin films of body -centered -cubic (BCC) Fe. By \ncombining broadband ferromagnetic resonance (FMR) measurements with characterization of \nstructural and transport properties of these model -system thin films, we show that conductivity -\nlike Gilbert damping dominates at lo w temperatures in epitaxial Fe . These experimental  results \n                                                           \n1 Ref.  [36] includes experimental data that suggest the presence of conductivity -like Gilbert damping in an ultrathin \nFe film, although no detailed information is given about the sample and t he experimental results deviate \nconsiderably from the calculations. An earlier study by Rudd et al. also suggests an increase in Gilbert damping with \ndecreasing temperature  [27], but quantification of the Gilbert damping parameter in this experiment is difficult.  \n 4 \n agree remarkably well with the magnitude of Gilbert damping derived from first -principles \ncalculations  [32,33,36] , thereby  providing evidence for intraband scatterin g as a key mechanism \nfor Gilbert damping in pure BCC Fe. Our experiment  thus resolves the longstanding  question  \nregarding the origin of damping in the prototypical ferromagnetic metal . Our results also confirm  \nthat – somewhat counterintuitively – disorder can partially suppress intrinsic damping at low \ntemperatures in ferromagnetic metals, such that optimally disordered films  may be well suited \nfor cryogenic spintronic and quantum applications  [41–44].   \nEpitaxial BCC Fe thin films were sputter deposited on (001) -oriented MgAl 2O4 (MAO) \nand MgO single crystal substrates. The choices of substrates were inspired by the recent \nexperiment by Lee et al. [20], where epitaxial growth is enabled with t he [100] axis of a B CC \nFe-rich alloy oriented 45o with respect to  the [100] axis of MAO or MgO. MAO with a lattice \nparameter of a MAO /(2√2) = 0.2858  nm exhibits a lattice mismatch of less than 0.4% with Fe (a Fe \n≈ 0.287 nm) , whereas the lattice mismatch between MgO ( aMgO/√2 = 0.2978  nm) and Fe is of the \norder  4%. Here , we focus on 25 -nm-thick Fe films that were grown  simultaneously on MAO and \nMgO by confocal DC magnetron sputtering  [45].  In the Supplemental Material  [45], we report \non additional films depos ited by off -axis magnetron sputtering.  \nWe verified the crystalline quality of the epitaxial Fe films by X -ray diffraction, as s hown \nin Fig. 1( a-c). Only (00X )-type peaks of the substrate and film are found in each 2θ-ω scan, \nconsistent with the single -phase  epitaxial growth of the Fe films. The 2θ-ω scans reveal a larger \namplitude of film peak for MAO/Fe, suggesting higher crystalline quality  than that of MgO/Fe. \nPronounced Laue oscillations, indicative of atomically smooth film interfaces, are o bserved \naround  the film peak of MAO/Fe, whereas they are absent for MgO/Fe. The high crystalline \nquality of MAO/Fe is also evidenced  by its narrow film -peak rocking curve with a FWHM of 5 \n only 0.02o, comparable to the rocking curve F WHM of the substrate2. By contrast, the film -peak \nrocking curve of MgO/Fe has a FWHM of  1o, which indicates substantial mosaic  spread in the \nfilm due to the large  lattice mismatch with the MgO substrate.  \nResults of 2θ -ω scans for different film thicknesses  [45] suggest  that the 25 -nm-thick Fe \nfilm may be  coherently strained to the MAO substrate , consistent with  the smooth interfaces and \nminimal mosaic spread of MAO/Fe . By contrast, i t is likely that 25 -nm-thick Fe on MgO is \nrelaxed to accommodate the large film-substrate lattice mismatch. Static magnetometry provides \nfurther evidence that Fe is strained  on MAO a nd relaxed  on MgO  [45]. Since  strained MAO/Fe \nand relaxed MgO/Fe exhibit  distinct  crystalline quality,  as evidenced by an approximately 50 \ntimes narrower rocking FWHM for  MAO/Fe , we have two model systems that enable \nexperimental investigation of the impact of structural disorder on Gilbert damping.   \nThe residual electrical resistivity also reflects  the structural quality  of metal s. As shown \nin Fig. 1(d ), the residual resistivity is 20 % lower for MAO/Fe compared to MgO/Fe, which \ncorroborates the lower defect density in MAO/Fe.  The resistivity increases  by nearly an order of \nmagnitude with increasing  temperature, reaching  1.1×10-7  m for both samples  at room \ntemperature , consistent with behavior expected for pure metal  thin film s.  \nWe now examine how the difference in crystalline quality correlates with  magnetic \ndamping in MAO/Fe and MgO /Fe. Broadband FMR measurements were performed at room \ntemperature up to 65 GHz with a custom spectrometer that empl oys a coplanar waveguide \n(center conductor width  0.4 mm ) and an electromagnet (maximum field < 2 T) . For each \nmeasurement at a fixed excitat ion frequency, an external bias magnetic field was swept parallel \nto the film plane along the [110] axis of Fe , unless otherwise noted. I n the Supplemental \n                                                           \n2 The angular resolu tion of the diffractometer is 0.0068o.  6 \n Material  [45], we show similar results with the  field applied along the [110] and [100] axes of \nFe; Gilbert damping is essentially isotropic within the film plane for our epitaxial Fe films , in \ncontrast to a recent report of anisotropic damping in ultrathin epitaxial Fe  [22]. \nFigure 2  shows that the peak -to-peak FMR linewidth Hpp scales linearly with frequency \nf, enabling a precise  determination of  the measured  Gilbert damping parameter  𝛼𝑚𝑒𝑎𝑠  from the \nstandard equation,  \n𝜇0∆𝐻𝑝𝑝=𝜇0∆𝐻0+2\n√3𝛼𝑚𝑒𝑎𝑠\n𝛾′𝑓,    (1) \nwhere  Hpp,0 is the zero -frequency linewidth and 𝛾′=𝛾/2𝜋≈29.5 GHz/T is the reduced \ngyromagnetic ratio . Despite the difference  in crystalline  quality , we find essentially the same \nmeasured  Gilbert damp ing parameter of  𝛼𝑚𝑒𝑎𝑠  ≈ 2.3×10-3 for MAO/Fe and MgO/Fe. We note \nthat t his value of 𝛼𝑚𝑒𝑎𝑠  is comparable to the lowest damping parameters reported for epitaxial Fe \nat room temperature  [15–17]. Our results indicate  that Gilbert  damping at room temperature is \ninsensitive to the strain state or structural disorder  in epitaxial Fe.3  \n The measured damping parameter 𝛼𝑚𝑒𝑎𝑠  from  in-plane FMR can generally include a \ncontribution from non-Gilbert  relaxation , namely two -magnon scattering  driven by defects  [46–\n49]. However,  two-magnon scattering is suppressed when the film is magnetized out-of-\nplane  [19,48] . To isolate any two -magnon scattering contribution to d amping, we performed out-\nof-plane  FMR measurements under  a sufficiently large magnetic field (>4 T) for complete \nsaturation of  the Fe film, using a custom W-band shorted waveguide  combined with a \n                                                           \n3 However, the crystallographic texture of Fe has significant impact on damping; for example, non -epitaxial Fe films \ndeposited directly on amorphous SiO 2 substrates exhibit an order of magnitude wider linewidths, due to much more \npronounced non -Gilbert damping (e.g., two -magnon scattering), compared to (001) -oriented epitaxial Fe films.  \n 7 \n superconducting magnet. As shown in Fig. 2, the out -of-plane and in -plane FMR data yield  the \nsame slope  and hence 𝛼𝑚𝑒𝑎𝑠  (Eq. 1)  to within <  8%. This finding indicates  that two -magnon \nscattering is negligible and that frequency -dependent magnetic relaxation is dominated by \nGilbert damping in epitaxial Fe examined here.  \nThe insensitivity of Gilbert damping to disorder found in Fig. 2 can be explained by the \ndominance of the interband (resistivity -like) mechanis m at room temperature, with phonon \nscattering dominating over defect scattering. Indeed, since MAO/Fe and MgO/Fe have the same \nroom -temperature resistivity  (Fig. 1(d )), any contributions to Gilbert damping from  electronic \nscattering should be identical for  both samples at room temperature. Moreover, according to our \ndensity functional theory calculations  [45], the density of states of BCC Fe at the Fermi energy, \nD(EF), does not depend significantly on  the strain state of the crystal. Therefore, i n light of the \nrecent reports  that Gilbert damping is proportional to  D(EF) [18,50,51] , the different strain states \nof MAO/Fe and MgO/Fe are not expe cted to cause a significant  difference in Gilbert damping.   \n However, since MAO/Fe and MgO/Fe exhibit distinct resistivities (electronic scattering \ntimes e) at low temperatures, one might expect  to observe distinct temperature dependence in \nGilbert damping for these two samples. To this end, we performed variable -temperature FMR \nmeasurements using  a coplanar -waveguide -based spectrometer (maximum frequency 40 GHz, \nfield < 2 T)  equipped with  a clos ed-cycle cryostat4. Figure 3(a,b) shows that meas is enhanced \nfor both samples at lower temperatures. Notably, this damping enhancement with decreasing \ntemperature is significantl y greater for MAO/Fe . Thus, at low temperatures, we find a \n                                                           \n4 The W -band spectrometer for out -of-plane FMR (Fig. 2) could not be  cooled below room temperature  due to its \nlarge thermal mass , limiting us to in -plane FMR measurements at low temperatures.  8 \n conductivity -like damping increase that is evidently more pronounced in epitaxial Fe with less \nstructural disorder.  \nWhile this increased damping at low temperatures is reminiscent of intrinsic Gilbert \ndamping from intraband scattering  [31–38], we first consider other possible contributions. One \npossibility is two -magnon scattering  [46–49], which we have ruled out at room temperature (Fig. \n2) but could  be present in our low -temperature in-plane FMR measurements . From Fig. 3(a,b), \nthe zero -frequency linewidth   H0  (Eq. 1 ) – typically attributed to magnetic inhomogeneity – is \nshown to increase  along with meas at low temperatures  [45], which might  point to  the emergence \nof two -magnon scattering  [48,49] . However, our mean -field model calculations (see \nSupplemental Material  [45]) shows that H0  correlates with  meas due to  interactions among \ndifferent regions  of the inhomogeneous film  [52]. The increase of H0 at low temperatures is \ntherefore readily accounted for  by increased  Gilbert damping , rather than two -magnon scattering .  \nWe are also not aware of  any mechanism that enhance s two-magnon scattering with \ndecreasing temperature, particularly given that  the saturation magnetization (i.e., dipolar \ninteractions) is constant across  the measured temperature range  [45]. Moreover, the isotropic in -\nplane damping found in our study is inconsistent with typically anisotropic two-magnon \nscattering tied to the crystal symmetry of  epitaxial films  [46,47] , and the film thickness in our \nstudy (e.g., 25 nm) rules out t wo-magnon scattering of interfacial origin  [49]. As such, we \nconclude  that two -magnon scattering does not play any essential  role in our experimental \nobservations.  \n Another  possible contribution  is dissipation due to classical eddy current s, which \nincrease s proportionally with the increasing conductivity  𝜎 at lower temperatures . We estimate \nthe eddy current contribution to the measured Gilbert damping with [15,53]   9 \n 𝛼𝑒𝑑𝑑𝑦 =𝜎\n12𝛾𝜇02𝑀𝑠𝑡𝐹2,    (2) \nwhere  𝜇0𝑀𝑠≈2.0 T is the saturation magnetization  and tF is the film thickness . We find that  \neddy curr ent damping accounts  for only ≈20% (≈ 30%) of  the total measured  damping of  \nMAO/Fe (MgO/Fe)  even at the lowest measured temperature  (Fig. 3(c)) . Furthermore, a s shown \nin the Supplemental Material  [45], thinner MAO/Fe film s, e.g., tF = 11 nm , with negligible eddy \nstill exhibit a significant increase in damping with decreasing temperature. Our results thus \nindicate a substantial contribution to conductivity -like Gilbert damping that is not accounted for \nby classica l eddy current damping.  \n For further discussion , we subtract the eddy -current damping from the measured damping \nto denote the Gilbert damping parameter attributed to intrinsic spin-orbit coupling as \n𝛼𝑠𝑜= 𝛼𝑚𝑒𝑎𝑠 − 𝛼𝑒𝑑𝑑𝑦. To correlate electronic transport and magnetic damping across the entire \nmeas ured temperature range, we perform a phenomenological fit of the temperature dependence \nof Gilbert damping with  [26]  \n𝛼𝑠𝑜=𝑐𝜎(𝑇)\n𝜎(300  𝐾)+𝑑𝜌(𝑇)\n𝜌(300  𝐾),   (3) \nwhere  the conductivity -like (intraband) and resistivity -like (interband) terms are scaled by \nadjustable parameters c and d, respectively. As shown in Fig.  4(a),(b), t his simple \nphenomenological model  using the experimental transport results (Fig. 1(d))  agrees remarkably \nwell with the temperature dependence of Gilbert damping for both MAO/Fe and MgO/Fe.  \nOur fi nding s that Gilbert damping can  be phenomenologically partit ioned  into two \ndistinct contributions (Eq. 3 ) are in line with  Kambersky’s  torque correlation model . We \ncompare our experimental resul ts to first-principles calculations by Gilmore et al. [32,33]  that \nrelate  electronic momentum scattering rate e-1 and Gilbert damping through Kambersky’s torque \ncorrelation model. We use the  experimentally measured resistivity ρ (Fig. 1(d)) to convert the 10 \n temperature to e-1 by assuming the constant conversion factor ρ e = 1.30×10-21  m s [33]. To \naccount for the difference in electronic scattering time for the minority spin  and majority spin \n, we take the calculated curve from Gilmore et al. with / = 4 [33], which is close to the \nratio of D(EF) of the spin-split bands  for BCC Fe , e.g., derived from our density functional \ntheory calculations  [45]. For explicit comparison with Refs.  [32,33] , the Gilbert damping \nparameter in Fig. 4(c) is converted to the magnetic relaxation rate 𝜆= 𝛾𝛼𝑠𝑜𝜇0𝑀𝑠. The \ncalculated prediction is in excellent  quantitative agreement with our experimental results for both \nstrained MAO/Fe and relaxed MgO/Fe  (Fig. 4(c)) , providing additional experimental evidence \nthat intraband scattering predominately contribute s to Gilber t damping at low temperatures.   \n We also compare our experimental results to a more recent first -principles calculation \nstudy by Mankovsky et al., which  utilizes the linear response formalism  [36]. This approach  \ndoes not  rely on a phenomenological electronic scattering rate and instead  allows for explicitly \nincorporating thermal  effects  and structural disorder . Figure 4(d ) shows the calculated \ntemperature dependence of the Gilbert damping parameter for BCC Fe with a small density of \ndefects, i.e., 0.1% vacancies , adapted from Ref.  [36]. We again find good quantitative agreement \nbetween the ca lculations and our experimental results for MAO/Fe. On the other hand, the \nGilbert damping parameter s at low temperatures  for relaxed MgO/Fe are significantly below the \ncalculated values . This  is consistent with  the reduction  of intraband scattering  due to enhanced \nelectronic scattering (enhanced e-1) from defects in relaxed MgO/Fe .  \n Indeed, significant defect -mediated electronic scattering may explain  the absence of \nconductivity -like Gilbert damping for crystalline Fe in prior  experiments. For example, Ref.  [25] \nreports an upper limit of  only a two -fold increase of the estimated Gilbert damping parameter \nfrom T = 300 K to 4 K . This relatively small damping enhancement is similar to that for MgO/Fe 11 \n in our study  (Fig. 4(b)) , suggesting that intraband scattering may have been suppressed in Fe in \nRef. [25] due to a similar degree of structural disorder  to MgO/Fe. We therefore conclude that \nconductivity -like Gilbert damping from  intraband scattering is highly sensitive to disorder in \nferromagnetic metals.  \nMore generally , the presence of defects  in all real metals – evidenced by finite residual \nresistivity  – ensures that the Gilbert damping parameter is finite even in the zero -temperature \nlimit . This circumvents the theoretical deficiency of Kambersky’s  torque correlation model \nwhere Gilbert damping would diverge in a perfectly clean ferromagnetic metal at T  0 [39,40] . \nWe also remark that a fully quantum mechanical many -body theory of magnetization dynami cs \nyields finite Gilbert damping even in the clean, T = 0 limit  [54].  \n In summary, we have demonstrated the dominance of conductiv ity-like Gilbert damping \ndue to intraband scattering at low temperatures in high-quality epitaxial Fe . Our experimental \nresults  also validate  the longstanding theoretical prediction of  intraband scattering as an essential  \nmechanism for Gilbert damping in pure ferromagnetic metals  [32–38], thereby advancing the \nfundamental understanding of magnetic relaxation in real materials . Moreover, we have \nconfirmed that, at low temperatures, a ma gnetic metal with imperfect crystallinity can exhibit \nlower Gilbert damping (sp in decoherence) than its cleaner  counterpart. This somewhat \ncounterintuitive finding suggests that  magnetic thin film s with optimal structural or chemical \ndisorder may be useful for cryogenic spintronic memories  [41,42]  and spin-wave -driven  \nquan tum information systems  [43,44] .   \n \n \n 12 \n Acknowledgements  \nThis research was funded in part by 4 -VA, a collaborative partnership for advancing the  \nCommonwealth of Vir ginia , as well as by the ICTAS Junior Faculty Award . A. Sapkota and C. \nMewes would like to acknowledge support by NSF-CAREER  Award No. 1452670 , and A. \nSrivastava would like to acknowledge support by NASA Award No. CAN80NSSC18M0023 . \nWe thank M. D. Stiles , B. K. Nikolic , and F. Mahfouzi  for helpful discussions on theoretical \nmodels for computing Gilbert damping , as well as R. D. McMichael for his input on the mean -\nfield modeling of interactions in inhomogeneous ferromagnetic films.    \n \n \n1.  B. 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B \n96, 214421 (2017).  \n \n  19 \n  \nFigure 1. (a,b) 2θ -ω X-ray diffraction scans of MAO/Fe and MgO/Fe (a) over a wide angle range \nand (b) near the BCC Fe (002) film peak. (c) Rocking curve scans about the film peak. (d) \nTemperature dependence of resistivity plotted on a log -log scale.  \n \n \nFigure 2. Frequency dependence of FMR linewidth Hpp for MAO/Fe and MgO/Fe at room \ntemperature. Linewidths measured under in -plane field are shown as open symbols, whereas \nthose measured under out -of-plane (OP) field are shown as filled symbols .  \n62 64 66 68log(intensity) [a.u.]\n2q [deg.]\n30 40 50 60 70log(intensity) [a.u.]\n2q [deg.]\n-1.0 -0.5 0.0 0.5 1.0intensity [a.u.]\nw002 [deg.]\n10 10010-810-7r [ m]\nT [K] MgO/Fe\n MAO/Fe\nMAO (004)MgO (002)\nFe (002)MAO/Fe\nMgO /Fe\n( 4)(a) (b) (c) (d)MAO/Fe\nMgO /Fe\n(a)\n0 20 40 60 80 100 120024681012 MAO/Fe (OP)\nMgO/Fe\nMAO/Fem0Hpp [mT]\nf [GHz]20 \n  \nFigure 3. (a,b) Frequency dependence of FMR linewidth for MA O/Fe and MgO/Fe at (a) T = 100 \nK and (b) T = 10 K. (c) Temperature dependence of measured  Gilbert damping parameter meas \nand estimated eddy -current damping parameter eddy. \n \n \n0 50 100 150 200 250 3000246810\n meas MAO/Fe  \n eddy estimate\n meas MgO/Fe\n eddy estimatemeas, eddy [10-3]\nT [K]\n0 10 20 30 40051015\nMAO/Fe\nMgO/Fem0Hpp (mT)\nf (GHz)T = 100 K\n0 10 20 30 40051015m0Hpp (mT)\nf (GHz)T = 10 K(c)(a) (b)21 \n   \nFigure 4. (a,b) Temperature dependence of the spin-orbit -induced Gilbert damping parameter \nso, fit phenomenologically with the experimentally measured resistivity for (a) MAO/Fe and (b) \nMgO/Fe. The dashed and dotted curves indicate the conductivity -like and resistivity -like \ncontributions, respectively; the solid curve represents the fit curve for the total spin -orbit -induced \nGilbert damping parameter. (c,d) Comparison of our experimental results with calculated Gilbert \ndamping parameters by (c) Gilmore et al. [32,33]  and (d) Mankovsky et al. [36]. \n \n \n0 100 200 30002468\nr-liker-likeso [10-3]\nT [K]s-likeMAO/Fe\n0 100 200 30002468\nMgO/Fe\nr-likes-likeso [10-3]\nT [K](a) (b)\n0 100 200 30002468\n MAO/Fe\n MgO/Fe\n calculated [Mankovsky]so [10-3]\nT [K]\n0 50 1000123\nr-like MAO/Fe\n MgO/Fe\n calculated [Gilmore]l [109 s-1]\ne-1 [1012 s-1]s-like0.0 0.5 1.0\n02468\nso [10-3]r [10-7 m]\n(c)\n(d)"    },    {        "title": "1908.08629v2.Damping_enhancement_in_coherent_ferrite_insulating_paramagnet_bilayers.pdf",        "content": "Damping enhancement in coherent ferrite/insulating-paramagnet bilayers\nJacob J. Wisser,1Alexander J. Grutter,2Dustin A. Gilbert,3Alpha T. N'Diaye,4\nChristoph Klewe,4Padraic Shafer,4Elke Arenholz,4, 5Yuri Suzuki,1and Satoru Emori6,\u0003\n1Department of Applied Physics, Stanford University, Stanford, CA, USA\n2NIST Center for Neutron Research, Gaithersburg, MD, USA\n3Department of Materials Science and Engineering, University of Tennessee, Knoxville, TN, USA\n4Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, CA, USA\n5Cornell High Energy Synchrotron Source, Ithaca, NY, USA\n6Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA\n(Dated: October 29, 2019)\nHigh-quality epitaxial ferrites, such as low-damping MgAl-ferrite (MAFO), are promising\nnanoscale building blocks for all-oxide heterostructures driven by pure spin current. However, the\nimpact of oxide interfaces on spin dynamics in such heterostructures remains an open question. Here,\nwe investigate the spin dynamics and chemical and magnetic depth pro\fles of 15-nm-thick MAFO\ncoherently interfaced with an isostructural \u00191-8-nm-thick overlayer of paramagnetic CoCr 2O4\n(CCO) as an all-oxide model system. Compared to MAFO without an overlayer, e\u000bective Gilbert\ndamping in MAFO/CCO is enhanced by a factor of >3, irrespective of the CCO overlayer thickness.\nWe attribute this damping enhancement to spin scattering at the \u00181-nm-thick chemically disordered\nlayer at the MAFO/CCO interface, rather than spin pumping or proximity-induced magnetism. Our\nresults indicate that damping in ferrite-based heterostructures is strongly in\ruenced by interfacial\nchemical disorder, even if the thickness of the disordered layer is a small fraction of the ferrite\nthickness.\nI. INTRODUCTION\nEmerging spintronic device schemes leverage magnon\nspin currents in electrically insulating magnetic oxides\n(e.g., ferrites), unaccompanied by dissipative motion\nof electrons, for computing and communications\napplications1,2. Low-dissipation spintronic devices\nbecome particularly attractive if insulating ferrite thin\n\flms with low magnetic damping can serve as sources\nof magnon spin currents. Such low-damping ferrites\ninclude not only epitaxial garnet ferrites (e.g., YIG)3{11\nthat have been widely used in studies of insulating\nspintronics2{4,12{15, but also coherently strained epitaxial\nspinel ferrites16{18with crucial technical advantages over\ngarnets, such as lower thermal budget for crystallization,\nhigher magnon resonance frequencies, and potential to be\nintegrated coherently with other spinels and perovskites\nwith various functionalities19{22.\nIn general, low-damping ferrite thin \flms must be\ninterfaced with other materials to realize spintronic\ndevices. It is therefore essential to understand whether\nand how damping in the ferrite is impacted by the\nproximity to another material. For instance, to convert\nbetween electronic and magnonic signals through direct\nand inverse spin Hall or Rashba-Edelstein e\u000bects23,\nthe low-damping ferrite needs to be interfaced with\na nonmagnetic metal with strong spin-orbit coupling.\nSpin transport and enhanced damping through spin\npumping24in ferrite/spin-orbit-metal structures has\nalready been extensively studied3,4,12{15,25. Moreover,\nthe low-damping ferrite can be interfaced with an\ninsulating antiferromagnetic or paramagnetic oxide, in\nwhich signals can be transmitted as a pure magnon\nspin current26{40. While interfacing low-damping ferriteswith insulating anti/paramagnetic oxides has enabled\nprototypes of magnon spin valves37{39, the fundamental\nimpact of insulating oxide interfaces on spin dynamics\nhas remained mostly unexplored. In particular, it is an\nopen question whether or how damping of the ferrite is\nenhanced from spin dissipation within the bulk of the\nadjacent anti/paramagnetic oxide or from spin scattering\nat the oxide interface.\nHere, we investigate how room-temperature magnetic\ndamping in epitaxial ferrimagnetic spinel MgAl-ferrite\n(MgAl 1=2Fe3=2O4, MAFO) is impacted when interfaced\nwith an overlayer of insulating paramagnetic spinel\nCoCr 2O4(CCO)41,42. This epitaxial MAFO/CCO\nbilayer is an isostructural model system, possessing\na coherent interface with continuous crystal lattices\nbetween the spinel ferrite and paramagnet. We \fnd that\nthe presence of MAFO/CCO interface increases damping\nby more than a factor of >3 compared to MAFO without\nan overlayer. We attribute this damping enhancement {\nwhich is comparable to or greater than spin pumping\ne\u000bects reported for ferrite/spin-orbit-metal bilayers { to\nspin scattering by the ultrathin ( \u00181 nm) chemically\ndisordered layer at the MAFO/CCO interface. Our\n\fndings show that spin scattering at oxide interfaces\nhas a profound in\ruence on damping, even when the\nchemically disordered layer is a small fraction of the total\nmagnetic layer thickness.\nII. FILM GROWTH AND STRUCTURAL\nCHARACTERIZATION\nEpitaxial thin \flms of 15-nm-thick MAFO interfaced\nwith 1.3-8 nm of CCO overlayer were grown on as-arXiv:1908.08629v2  [cond-mat.mtrl-sci]  26 Oct 20192\n40 42 44 46MAO (004)\nMAFO/CCO\n      (004)\nMAFO (004)log10(Intensity) (arb. units)\n2q (deg)CCO (004)\n-0.180 -0.175 -0.1700.570.580.590.600.610.620.63-\n \n   (115)-(a) (c)\nqip(Å-1)qop(Å-1)\nCCO\n(25 nm)MAO(b)\n-0.2-0.1 0.00.10.2MAFOCCOIntensity (arb. units)\nDw004 (deg)MAFO/CCO\nFigure 1. (a) 2 \u0012-!scans of epitaxial MAFO(15 nm), CCO(25 nm), and MAFO(15 nm)/CCO(8 nm). The data are o\u000bset for\nclarity. (b) Rocking curve scans about the (004) \flm peak for the \flms shown in (a). (c) Reciprocal space map of epitaxial\nCCO(25 nm) coherently strained to the MAO substrate.\nreceived single-crystal MgAl 2O4(MAO) substrates via\npulsed laser deposition. A KrF 248 nm laser was\nincident on stoichiometric targets of MAFO and CCO\nwith \ruences of \u00191.5 J/cm2and\u00191.3 J/cm2,\nrespectively. Both \flms were grown in 10 mTorr (1.3\nPa) O 2and were cooled in 100 Torr (13 kPa) O 2.\nMAFO \flms were grown at 450\u000eC, whereas CCO \flms\nwere deposited at 300\u000eC in an attempt to minimize\nintermixing between the MAFO and CCO layers. These\ngrowth temperatures, much lower than >700\u000eC typically\nrequired for epitaxial garnets3{11, are su\u000ecient to fully\ncrystallize MAFO and CCO. The low crystallization\ntemperatures of the spinels o\u000ber an advantage over\nthe oft-studied garnets, with more opportunities for\nisostructural integration with coherent interfaces. The\nMAFO \flms exhibit a room-temperature saturation\nmagnetization of\u0019100 kA/m and a Curie temperature of\n\u0019400 K18. To obtain consistent ferromagnetic resonance\nresults, MAFO \flms were grown and subsequently\ncharacterized by ferromagnetic resonance (FMR) ex-situ;\nafter surface cleaning with ultrasonication in isopropanol,\nCCO overlayers were then deposited as described above.\nGrowth rates were calibrated via X-ray re\rectivity.\nOur structural characterization of MAFO and\nCCO shows high-quality, coherently strained \flms.\nIn symmetric 2 \u0012-!X-ray di\u000braction scans, only\npeaks corresponding to the (00 `) re\rections are\nobserved, indicating that the \flms are highly epitaxial.\nAdditionally, as seen in Fig. 1(a), Laue oscillations\naround the (004) Bragg re\rections in both single-layer\nMAFO and CCO layers as well as MAFO/CCO bilayers\ndenote smooth interfaces. Furthermore, MAFO, CCO,\nand MAFO/CCO samples all exhibit essentially the\nsame \flm-peak rocking curve widths (FWHM) of \u00190.06\u000e\n(Fig. 1(b)). Reciprocal space mapping of the ( \u00161\u001615)\nre\rection in 25-nm-thick single-layer CCO on MAO\n(Fig. 1(c)) reveals that the in-plane lattice parameter of\nthe \flm coincides with that of the substrate, indicating\nCCO is coherently strained to MAO. We note thatdespite the relatively large lattice mismatch between\nCCO and MAO of \u00193 %, coherently strained growth of\nCCO of up to 40 nm has been previously reported on\nMAO substrates41. For our CCO \flm, we calculate an\nout-of-plane lattice constant c\u00198:534\u0017A from the 2 \u0012-!\nscan; taking the in-plane lattice parameter a= 8:083\u0017A of\nthe MAO substrate, the resulting tetragonal distortion of\ncoherently strained CCO is c=a\u00191:055, similar to that\nfor coherently strained MAFO18.\nStructural characterization results underscore the\nquality of these epitaxial \flms grown as single layers and\nbilayers. Considering the comparable high crystalline\nquality for MAFO, CCO, and MAFO/CCO { as\nevidenced by the presence of Laue oscillations and narrow\n\flm-peak rocking curves { we conclude that MAFO/CCO\nbilayers (with the total thickness limited to \u001423 nm) are\ncoherently strained to the substrate. In these samples\nwhere the substrate and \flm layers are isostructural, we\nalso do not expect antiphase boundaries43{46. Indeed,\nwe \fnd no evidence for frustrated magnetism, i.e., high\nsaturation \feld and coercivity, that would arise from\nantiphase boundaries in spinel ferrites43{46; MAFO/CCO\nbilayers studied here instead exhibit soft magnetism, i.e.,\nsquare hysteresis loops with low coercivity <0.5 mT,\nsimilar to our previous report on epitaxial MAFO thin\n\flms18. Thus, MAFO/CCO is a high-quality all-oxide\nmodel system, which permits the evaluation of how spin\ndynamics are impacted by a structurally clean, coherent\ninterface.\nIII. FERROMAGNETIC RESONANCE\nCHARACTERIZATION OF DAMPING\nTo quantify e\u000bective damping in coherently strained\nMAFO(/CCO) thin \flms, we performed broadband\nFMR measurements at room temperature in a coplanar\nwaveguide setup using the same procedure as our prior\nwork16,18. We show FMR results with external bias3\nmagnetic \feld applied in the \flm plane along the [100]\ndirection of MAFO(/CCO); essentially identical damping\nresults were obtained with in-plane \feld applied along\n[110]47. Figure 2(a) shows the frequency fdependence of\nhalf-width-at-half-maximum (HWHM) linewidth \u0001 Hfor\na single-layer MAFO sample and a MAFO/CCO bilayer\nwith a CCO overlayer thickness of just 1.3 nm, i.e., less\nthan 2 unit cells. The linewidth is related to the e\u000bective\nGilbert damping parameter \u000beffvia the linear equation:\n\u0001H= \u0001H0+h\u000beff\ng\u00160\u0016Bf (1)\nwhere \u0001H0is the zero-frequency linewidth, his Planck's\nconstant,g\u00192:05 is the Land\u0013 e g-factor derived from the\nfrequency dependence of resonance \feld HFMR ,\u00160is the\npermeability of free space, and \u0016Bis the Bohr magneton.\nIt is easily seen from Fig. 2(a) that with the addition\nof ultrathin CCO, the damping parameter is drastically\nincreased, i.e., >3 times its value in bare MAFO.\nFigure 2(b) shows that the damping enhancement\nseen in MAFO/CCO is essentially independent of\nthe CCO thickness. This trend suggests that\nthe damping enhancement is purely due to the\nMAFO/CCO interface, rather than spin dissipation in\nthe bulk of CCO akin to the absorption of di\u000busive\nspin current reported in antiferromagnetic NiO26,35,48.\nWe note that other bulk magnetic properties of\nMAFO (e.g., e\u000bective magnetization, Land\u0013 e g-factor,\nmagnetocrystalline anisotropy) are not modi\fed by the\nCCO overlayer in a detectable way. We also rule\nout e\u000bects from solvent cleaning prior to CCO growth\nor thermal cycling in the deposition chamber up to\n300\u000eC, as subjecting bare MAFO to the same ex-\nsitu cleaning and in-situ heating/cooling processes as\ndescribed in Section II, but without CCO deposition,\nresults in no measurable change in damping. The\ndamping enhancement therefore evidently arises from the\nproximity of MAFO to the CCO overlayer.\nWe consider two possible mechanisms at the\nMAFO/CCO interface for the observed damping\nenhancement:\n(1) Spin current excited by FMR in MAFO\nmay be absorbed via spin transfer in an interfacial\nproximity-magnetized layer49of CCO, whose magnetic\nmoments may not be completely aligned with those of\nMAFO. While CCO by itself is paramagnetic at room\ntemperature, prior studies have shown that Co2+and\nCr3+cations in epitaxial CCO interfaced with a spinel\nferrite (e.g., Fe 3O4) can develop measurable magnetic\norder50. Such damping enhancement due to interfacial\nmagnetic layer is analogous to spin dephasing reported\nfor ferromagnets interfaced directly with proximity-\nmagnetized paramagnetic metal (e.g., Pt, Pd)49.\n(2) Even if CCO does not develop proximity-induced\nmagnetism, chemical disorder at the MAFO/CCO\ninterface may enhance spin scattering. For instance,\nchemical disorder may lead to an increase of Fe2+\n0 10 20 300246810HWHM Linewidth (mT)\nFrequency (GHz)(a)\n(b)MAFO/CCO\neff≈ 0.007\nMAFO\neff≈ 0.002\n0 2 4 6 80.0000.0020.0040.0060.0080.0100.012eff\nCCO thickness (nm)Figure 2. (a) HWHM FMR linewidth versus frequency\nfor MAFO(15 nm) and MAFO(15 nm)/CCO(1.3 nm). The\ne\u000bective Gilbert damping parameter \u000beffis derived from\nthe linear \ft. (b) \u000beffplotted against the CCO overlayer\nthickness. The dashed horizontal line indicates the average of\n\u000befffor MAFO without an overlayer.)\ncations at the MAFO surface, thereby increasing\nthe spin-orbit spin scattering contribution to Gilbert\ndamping in MAFO compared to its intrinsic composition\ndominated by Fe3+with weak spin-orbit coupling18,51.\nAnother possibility is that chemical disorder at the\nMAFO/CCO interface introduces magnetic roughness\nthat gives rise to additional spin scattering, perhaps\nsimilar to two-magnon scattering recently reported for\nferromagnet/spin-orbit-metal systems52.\nIn the following section, we directly examine interfacial\nproximity magnetism and chemical disorder to gain\ninsight into the physical origin of the observed damping\nenhancement in MAFO/CCO.\nIV. CHARACTERIZATION OF INTERFACE\nCHEMISTRY AND MAGNETISM\nTo evaluate the potential formation of a magnetized\nlayer in the interfacial CCO through the magnetic\nproximity e\u000bect, we performed depth-resolved\nand element-speci\fc magnetic characterization\nof MAFO/CCO bilayers using polarized neutron\nre\rectometry (PNR) and soft magnetic X-ray\nspectroscopy. PNR measurements were performed\nusing the PBR instrument at the NIST Center for\nNeutron Research on nominally 15-nm-thick MAFO\nlayers capped with either thick (5 nm) or thin (3 nm)4\nCCO overlayers. PNR measurements were performed in\nan in-plane applied \feld of 3 T at temperatures of 300\nK and 115 K, the latter case being slightly above the\nnominal 97 K Curie temperature of CCO41,42. Incident\nneutrons were spin-polarized parallel or anti-parallel to\nthe applied \feld both before and after scattering from\nthe sample, and the re\rected intensity was measured\nas a function of the perpendicular momentum transfer\nvector Q. The incident spin state of measured neutrons\nwere retained after scattering, corresponding to the\ntwo non-spin-\rip re\rectivity cross sections ( \"\"and##).\nSince all layers of the \flm are expected to saturate well\nbelow the applied \feld of 3 T, no spin-\rip re\rectivity is\nexpected and these cross sections were not measured.\nSince PNR is sensitive to the depth pro\fles of the\nnuclear and magnetic scattering length density (SLD),\nthe data can be \ftted to extract the chemical and\nmagnetic depth pro\fles of the heterostructure. In this\ncase, we used the Re\r1D software package for this\npurpose53. Figure 3(a,b) shows the 300 K re\rectivities\nand spin asymmetry curves of a nominal MAFO (15\nnm)/CCO (5 nm) sample alongside the depth pro\fle\n(Fig. 3(c)) used to generate the \fts shown. The\nbest \ft pro\fle (Fig. 3(c)) provides no evidence of a\nlayer with proximity-induced magnetization in the CCO.\nRather, we note that there appears to be a layer of\nmagnetization suppression near both the MAO/MAFO\nand MAFO/CCO interfaces. Further, the interfacial\nroughnesses of both the MAO/MAFO and MAFO/CCO,\n0.9(1) nm and 1.35(5) nm respectively, are signi\fcantly\nlarger than the CCO surface roughness of 0.27(3) nm\nand the bare MAFO surface roughness of <\u00180.5 nm54.\nThe interfacial roughnesses are signatures of chemical\nintermixing at the spinel-spinel interface leading to\ninterfacial suppression of the magnetization and/or Curie\ntemperature. Thus, we \fnd that the MAFO/CCO\ninterface, although structurally coherent, exhibits a\nchemically intermixed region on the order of one spinel\nunit cell thick on either side.\nTo obtain an upper limit of the proximity-induced\ninterfacial magnetization in CCO, we performed Markov-\nchain Monte-carlo simulations as implemented in the\nDREAM algorithm of the BUMPS python package.\nThese simulations suggest an upper limit (95% con\fdence\ninterval of) 7 emu/cc in the 1.5 nm of the CCO closest\nto the interface. In this case, the model evaluated the\nMAFO as a uniform structural slab but allowed for total\nor partial magnetization suppression at both interfaces,\nwhile the CCO layer was treated as a uniform slab with\nan allowed magnetization layer of variable thickness at\nthe interface.\nHowever, we note that equivalently good \fts are\nobtained using simpler models that \ft a single MAFO\nlayer with magnetically dead layers at the interfaces and\na completely nonmagnetic CCO layer. Equivalent results\nwere obtained for the thick CCO sample at 115 K and\nfor the thin CCO sample. We therefore conclude that the\nPNR results strongly favor a physical picture in which the\nFigure 3. (a) Spin-polarized neutron re\rectivity and (b)\nspin asymmetry of a MAFO (15 nm)/CCO (5 nm) bilayer\nalongside theoretical \fts. (c) Nuclear and magnetic scattering\n(scaled \u000210) length density pro\fle used to generate the \fts\nshown. Error bars represent \u00061 standard deviation.\nCCO is notmagnetized through the magnetic proximity\ne\u000bect.\nTo con\frm the PNR results and examine the e\u000bect\nof a CCO overlayer on the local environment of Fe\ncations in MAFO, we performed temperature-dependent\nX-ray absorption (XA) spectroscopy and X-ray magnetic\ncircular dichroism (XMCD) measurements at Beamline\n4.0.2 of the Advanced Light Source at Lawrence Berkeley\nNational Laboratory. We note that the detection\nmode (total electron yield) used here for XA/XMCD\nis sensitive to the top \u00195 nm of the sample, such that\nFe L edge signals from CCO-capped MAFO primarily\ncapture the cation chemistry near the MAFO/CCO\ninterface. Measurements were performed in an applied\n\feld of 400 mT along the circularly polarized X-ray beam,\nincident at 30\u000egrazing from the \flm plane. To minimize\ndrift e\u000bects during the measurement, multiple successive\nenergy scans were taken and averaged, switching both\napplied \feld direction and photon helicity so that all\nfour possible combinations of \feld direction and helicity\nwere captured at least once. XA and XMCD intensities\nwere normalized such that the pre-edge is zero and\nthe maximum value of the average of the (+) and\n(\u0000) intensities is unity. In the case of the Co L-\nedge, measurements were taken with energy sweeps\ncovering both Fe and Co edges, and for consistency\nboth edges were normalized to the highest XAS signal,\ncorresponding to the Fe L 3-edge.\nFigure 4(a) compares the XA of a bare MAFO \flm5\nFigure 4. (a) 300 K X-ray absorption spectra of MAFO and\nMAFO/CCO (3 nm) grown on MAO. (b) Photon helicity-\ndependent XA spectra and XMCD of the Fe L-edge for a\nMAFO/CCO (3 nm) bilayer at 300 K. (c) Co and (d) Cr\nL-edge XA and XMCD of the same bilayer.\nwith one capped by 3 nm of CCO. The two XA lineshapes\nare nearly identical, indicating the same average Fe\noxidation state and site-distribution in CCO-capped\nand uncapped MAFO \flms. It is therefore likely that\nthe reduced interfacial magnetization observed through\nPNR is a result of a defect-induced Curie temperature\nreduction, rather than preferential site-occupation of Co\nand Cr that might increase the Fe2+content in the\nintermixed interfacial region.\nWe further note that although a large XMCD signal\nis observed on the Fe-edge at 300 K (Fig. 4(b)), neither\nthe Co nor Cr L edges exhibit any signi\fcant magnetic\ndichroism, as shown in Figs. 4(c)-(d). Similar results\nare obtained on the Cr L edge at 120 K. Consistent\nwith the PNR results, we thus \fnd no evidence for\na net magnetization induced in the CCO through the\ninterfacial magnetic proximity e\u000bect.\nOur \fnding of suppressed interfacial magnetism\nin MAFO/CCO is reminiscent of earlier reports\nof magnetic dead layers in epitaxially-grown ferrite-\nbased heterostructures55{57. For example, prior\nPNR experiments have revealed magnetic dead layers\nat the interfaces of ferrimagnetic spinel Fe 3O4and\nantiferromagnetic rock-salt NiO or CoO, even when the\ninterfacial roughness is small (e.g., only 0.3 nm)55,56.\nA magnetic dead layer of 1 spinel unit cell has also\nbeen reported at the interface of Fe 3O4and diamagnetic\nrock-salt MgO grown by molecular beam epitaxy57.\nWe note that in these prior studies, the spinel ferrite\flms interfaced with the rock salts (NiO, CoO, MgO)\npossess antiphase boundaries. Suppressed magnetism\nis known to result from antiphase boundaries, as they\nfrustrate the long-range magnetic order and reduce\nthe net magnetization of the ferrite44. By contrast,\nthere is no evidence for antiphase boundaries in all-\nspinel MAFO/CCO grown on spinel MAO; therefore,\nthe suppressed magnetism at the MAFO/CCO interface\ncannot be attributed to antiphase-boundary-induced\nmagnetic frustration.\nAnother possible scenario is that magnetic dead layer\nformation is a fundamental consequence of the charge\nimbalance between di\u000berent lattice planes, as recently\nshown in a recent report of (polar) Fe 3O4undergoing\natomic reconstruction to avoid \\polar catastrophe\" when\ngrown on (nonpolar) MgO58. In our study on all-\nspinel heterostructures, there may also be some degree of\ncharge mismatch depending on the relative populations\nof cations on the tetrahedrally- and octahedrally-\ncoordinated sites at the MAFO/CCO interface, although\nthe charge mismatch is expected to be only \u0019\u00061, i.e.,\na factor of\u00195-6 smaller than that in MgO/Fe 3O458.\nThus, atomic reconstruction driven by charge imbalance\nappears unlikely as a dominant source of the magnetic\ndead layer in MAFO/CCO. We instead tentatively\nattribute the dead layer to atomic intermixing driven by\ndi\u000busion across the MAFO/CCO interface during CCO\noverlayer deposition.\nV. DISCUSSION\nOur PNR and XA/XMCD results (Section IV) indicate\nthat the damping enhancement observed in Section III\narises from chemical disorder, rather than proximity-\ninduced magnetism, at the MAFO/CCO interface.\nWe emphasize that this interfacial disordered layer\nis con\fned to within \u00192 spinel unit cells. We\nalso note that this interfacial disorder is due to\natomic intermixing, but not structural defects (e.g.,\ndislocations, antiphase boundaries), in this coherent\nbilayer system of MAFO/CCO. Nevertheless, this\nultrathin chemically disordered layer alone is evidently\nsu\u000ecient to signi\fcantly increase spin scattering.\nConsidering that the cation chemistry of Fe in MAFO\ndoes not change substantially (Fig. 4(a)), the interfacial\nspin scattering is likely driven by magnetic roughness,\nleading to a mechanism similar to two-magnon scattering\nthat accounts for a large fraction of e\u000bective damping in\nmetallic ferromagnet/Pt bilayers52.\nWe now put in context the magnitude of the damping\nenhancement \u0001 \u000beff, i.e., the di\u000berence in the e\u000bective\nGilbert damping parameter between CCO-capped and\nbare MAFO,\n\u0001\u000beff=\u000bbilayer\neff\u0000\u000bferrite\neff; (2)\nby comparing it with ferrite/spin-orbit-metal systems\nwhere spin pumping is often considered as the source6\n0.0000.0020.0040.0060.008\n MAFO/CCO\n   [this study] MAFO/W\n [Riddiford] MAFO/Pt\n [Riddiford]YIG/Pt\n[Wang]Daeff\nYIG/Pt\n [Sun]\nFigure 5. Comparison of the enhancement of the e\u000bective\nGilbert damping parameter \u0001 \u000befffor MAFO/CCO and\nferrite/spin-orbit-metal bilayers. YIG/Pt [Sun], YIG/Pt\n[Wang], and MAFO/Pt(W) [Riddiford] are adapted from\nRefs.59,60, and61respectively. The values of \u0001 \u000befffrom the\nliterature are normalized for the saturation magnetization\nof 100 kA/m and magnetic thickness of 15 nm for direct\ncomparison with our MAFO/CCO result.\nof damping enhancement. Since damping enhancement\nfrom spin pumping or interfacial scattering scales\ninversely with the product of the saturation of\nmagnetization Msand the magnetic layer thickness tm,\nthe values of \u0001 \u000befftaken from the literature59{61are\nnormalized for direct comparison with the MAFO \flms\nstudied here with Ms= 100 kA/m and tm= 15 nm.\nAs summarized in Fig. 5, \u0001 \u000befffor MAFO/CCO\nis comparable to { or even greater than { \u0001 \u000beff\nfor ferrite/metal bilayers. This \fnding highlights that\nthe strength of increased spin scattering in a ferrite\ndue to interfacial chemical disorder can be on par\nwith spin dissipation due to spin pumping in metallic\nspin sinks. More generally, this \fnding suggests that\nspecial care may be required in directly relating \u0001 \u000beff\nto spin pumping across bilayer interfaces (i.e., spin-\nmixing conductance52), particularly when the FMR-\ndriven magnetic layer is directly interfaced with a spin\nscatterer.\nFurthermore, the strong interfacial spin scattering {\neven when the oxide interface is structurally coherent\nand the chemically disordered layer is kept to just <\u00182\nunit cells { poses a signi\fcant challenge for maintaining\nlow damping in ferrite/insulator heterostructures. This\nchallenge is partially analogous to the problem of reduced\nspin polarization in tunnel junctions consisting of spinelFe3O4and oxide barriers (e.g., MgO)62{65, which is also\nlikely due to interfacial chemical disorder and magnetic\ndead layers. However, we emphasize that the problems of\nantiphase boundaries43{46and charge-imbalance-driven\natomic reconstruction58, which have posed intrinsic\nchallenges for devices with MgO/Fe 3O4interfaces, are\nlikely not applicable to all-spinel MAFO/CCO. It is\ntherefore possible that deposition schemes that yield\nsharper interfaces, e.g., molecular beam epitaxy, can be\nemployed to reduce interfacial imperfections and hence\nspin scattering at MAFO/CCO for low-loss all-oxide\ndevice structures.\nVI. CONCLUSIONS\nWe have shown that e\u000bective damping in epitaxial\nspinel MgAl-ferrite (MAFO) increases more than\nthreefold when interfaced coherently with an insulating\nparamagnetic spinel of CoCr 2O4(CCO). This damping\nenhancement is not due to spin pumping into the\nbulk of CCO. Our depth-resolved characterization of\nMAFO/CCO bilayers also reveals no proximity-induced\nmagnetization in CCO or signi\fcant change in the\ncation chemistry of MAFO. We attribute the giant\ndamping enhancement to spin scattering in an ultrathin\nchemically disordered layer, con\fned to within 2 spinel\nunit cells across the MAFO/CCO interface. Our results\ndemonstrate that spin dynamics in ferrite thin \flms are\nstrongly impacted by interfacial disorder.\nAcknowledgements - This work was supported in\npart by the Vannevar Bush Faculty Fellowship program\nsponsored by the Basic Research O\u000ece of the Assistant\nSecretary of Defense for Research and Engineering and\nfunded by the O\u000ece of Naval Research through grant\nno. N00014-15-1-0045. 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Phys.\nLett.105, 102410 (2014)."    },    {        "title": "1806.00658v1.Ultra_low_damping_insulating_magnetic_thin_films_get_perpendicular.pdf",        "content": "1 \n Ultra -low damping  insulating magnetic  thin films get perpendicular  \n \nLucile Soumah1, Nathan Beaulieu2, Lilia Qassym3, Cécile Carrétero1, Eric Jacquet1, Richard Lebourgeois3, \nJamal Ben Youssef2, Paolo Bortolotti1, Vincent Cros1, Abdelmadjid Anane1* \n \n1 Unité Mixte de Physique CNRS , Thales, Univ.  Paris -Sud, Université Paris Saclay, 91767 Palaiseau, France  \n2LABSTICC, UMR 6285 CNRS, Université de Bretagne Occidentale, 29238 Brest, France  \n3 Thales Research and Technology, Thales 91767 Palaiseau , France  \n* Email : madjid.anane@u -psud.fr  \n \nA magnetic material combining low losses and  large Perpendicular Magnetic Anisotropy  (PMA) is still a \nmissing brick in the magnonic  and spintronic field s. We report  here  on the growth of ultrathin Bismuth \ndoped Y 3Fe5O12 (BiYIG ) films on Gd 3Ga 5O12 (GGG) and substituted GGG (sGGG) (111) oriented \nsubstrates. A fine tuning of the PMA is obtained  using both epitaxial strain and growth induced \nanisotrop ies. Both spontaneously in -plane and out -of-plane magnetized thin films can be elaborated . \nFerromagnetic Resonance  (FMR)  measurement s demonstrate the  high dynamic quality of these BiYIG \nultrathin  films , PMA films with Gilbert damping values as low as 3 10-4 and FMR linewidth  of 0.3 mT at \n8 GHz are achieved  even for films that do not exceed 30 nm in thickness . Moreover, w e measure \nInverse Spin Hall Effect (ISHE) on Pt/BiYIG stack s showing  that the magnetic insulator ’s surface  is \ntransparent to  spin current  making it appealing  for spintronic applications . \n \n  2 \n Introduction.  \nSpintronic s exploit s the electron’s spin in ferromagnetic transition metal s for data storage and data \nprocessing. Interestingly, as spintronics  codes information in the angular momentum degre es of \nfreedom , charge transport and therefore the use of conducting materials is not a requirement, opening \nthus electronics to  insulators . In magnetic insulators  (MI),  pure spin currents  are described using \nexcitation  states of the ferromagnetic background named magnons (or spin waves). Excitation, \npropagation  and  detection  of magnons are   at  the  confluent of the emerging concepts of magnonics  \n1,2, caloritronics3 and  spin -orbitronics4. Magnons, and their classical counterpart , the  spin waves (SWs) \ncan carry information over distances as large as millimeters in high quality thick YIG films, with \nfrequencies extending from the GHz to the THz regime5–7. The main figure of merit for magnonic \nmaterials is the Gilbert damping  1,5,8 which  has to be as small as possible. This makes  the number of \nrelevant materials for SW propagation quite limited and none of them has yet been found to possess a \nlarge enough perpendicular magnetic anisotropy (PMA ) to induce  spontaneous out -of-plane \nmagnetization . We report  here  on the  Pulsed  Laser  Deposition  (PLD)  growth  of  ultra -low loss   MI  \nnanometers -thick films  with large PMA : Bi substituted Yttrium Iron Garnet ( BixY3-xFe5O12 or BiYIG ) where  \ntunability of the PMA  is achieved through epitaxial strain and Bi doping level.  The peak -to-peak FMR  \nlinewidth  (that characterize the losses)  can be as low as  𝜇0𝛥𝐻pp=0.3 mT at 8 GHz for 30 nm thick \nfilms. This material thus opens new perspectives for both spintronic s and magnonic s fields as the SW \ndispersion relation can now be easily tuned through magnetic anisotropy without the need of  a large \nbias magnetic field. Moreover, energy efficient data storage devices based on magn etic textures existing \nin PMA materials like magnetic bubble s, chiral domain walls  and magnetic skyrmions  would benefit from \nsuch a low loss material for efficient operation9. \nThe study of micron -thick YIG films grown by liquid phase epitaxy (LPE) was among the hottest topics in \nmagnetism few decades ago . At this time,  it has been already noticed that unlike rare earths (Thulium, \nTerbium, Dysprosium …) substitutions, Bi substitution does not overwhelmingly increase the magnetic \nlosses10,11 even though it induces high uniaxial magnetic anisotropy12–14 . Very recently, ultra -thin MI \nfilms showing PMA have been the subject of an increasing interes t 15,16: Tm 3Fe5O12 or BaFe 12O19 \n(respectively a garnet and an hexaferrite) have been used to demonstrate spin -orbit -torque \nmagnetization reversal using a Pt over -layer as a source of spin current 4,17,18. However, their large \nmagnetic losses prohibit their use as a spin -wave medium  (reported va lue o f 𝜇0𝛥𝐻pp of TIG is  16.7  mT at \n9.5 GHz)19. Hence, whether  it is possible to fabricate ultra -low loss thin films  with a large PMA  that can \nbe used for both magnonics and spintronics applications remains to be demonstrated . Not only l ow 3 \n losses are important for long range spin wave propagation but they are also necessary for spin transfer \ntorque oscillators (STNOs) as the threshold current scales with the Gilbert damping20.  \nIn the quest for the  optimal  material platform , we explore here the growth of Bi doped YIG ultra -thin \nfilms using PLD  with different substitution;  BixY3-xIG (x= 0.7, 1 and 1.5)  and having a thickness ranging \nbetween 8 and 50 nm. We demonstrate fine tuning of the magnetic anisotropy using epitaxial strain  and  \nmeasure  ultra low Gilbert damping values ( 𝛼=3∗10−4) on ultrathin films with PMA . \nResults  \nStructural and magnetic characterization s \nThe two substrates that are used are Gallium Gadolinium Garnet  (GGG)  which is best lattice matched to \npristine YIG and substituted GGG  (sGGG) which is  traditionally used to accommodate  substituted YIG \nfilms for photonics  applications . The  difference between  Bi and Y ionic radii ( rBi = 113 pm and rY = 102 \npm)21 leads to a linear increase of  the  BixY3-xIG bulk lattice  parameter  with Bi content  (Fig. 1 -(a) and Fig. \n1-(b)). In Fig. 1,  we present the  (2−) X-ray diffraction patterns (Fig.1 -(c) and 1 -(d)) and reciprocal \nspace maps (RSM) (Fig.1 -(e) and 1 -(f)) of BiYIG on sGGG(111) and GGG(111) substrates  respectively . The \npresence of ( 222) family peaks in the diffraction spectra shown in Fig. 1 -(b) and 1 -(c) is a signature of  the \nfilms’ epitaxial quality  and the presence of Laue fringes attest s the coherent crystal structure existing \nover the whole thickness. As expected, all films on GGG are under compressive strain, whereas films \ngrown on sGGG exhibit a transition from a tensile (for x= 0.7 and 1) towards a compressive ( x= 1.5) \nstrain . Reciprocal Space Mapping  of these BiYIG samples  shown in Fig.1 -(e) and 1 -(f) evidences the \npseudomorphic nature of the growth for all films , which confirms the good epitaxy.  \nThe static magnetic properties of the films have been characterized using SQUID ma gnetometry, Faraday \nrotation measurements and Kerr microscopy. As the Bi doping has the effect of enhancing the magneto -\noptical response 22–24, we measure on average a large  Faraday rotation coefficients reaching up to 𝜃F =\n−3 °.𝑚−1 @ 632 nm  for  x= 1 Bi doping level  and 15 nm film thickness . Chern et al .25 performed PLD \ngrowth of BixY3-xIG on GGG  and reported an increase of 𝜃F\n𝑥= −1.9 °.𝜇𝑚−1 per Bi substitution  x @ 632 \nnm. The Faraday rotation coefficients we find  are slightly larger  and m ay be due to the much lower \nthickness of our films as 𝜃F is also dependent on  the film  thickness26. The saturation magnetization ( Ms) \nremains constant for all Bi content  (see Table 1)  within the  10%  experimental errors . We observe a clear  \ncorrelation between the strain and the shape of the in -plane and out -of-plane hysteresis loop s reflecting \nchanges in the magnetic anisotropy. Wh ile films under compressive strain exh ibit in -plane anisotropy, \nthose under  tensile strain show a large out -of-plane anisotrop y that can eventually lead to an out -of-\nplane easy axis  for x= 0.7 and x= 1 grown on sGGG.  The transition can be either induced by ch anging the 4 \n substrate  (Fig.2 -(a)) or the Bi content ( Fig. 2-(b)) since both act on the misfit strain. We ascribe the \nanisotropy change  in our films to  a combination of magneto -elastic anisotropy and growth induced \nanisotropy, this later term being the domin ant one (see Supplementary Note 1).  \nIn Fig. 2 -(c), we  show the magnetic domains structures at remanance  observed using polar Kerr \nmicroscopy  for Bi1Y2IG films  after demagnetization : µm-wide maze -like magnetic domains demonstrate s \nunambiguously that the magnetic easy axis is perpendicular to the film surface . We observe  a decrease \nof the domain width (Dwidth) when the film thickness ( tfilm) increases as expected from magnetostatic  \nenergy considerations. In fact, a s Dwidth is severa l order s of magnitude larger than tfilm, a domain wall \nenergy of σDW  0.7 and 0.65  mJ.m-2 (for x= 0.7 and 1 Bi doping) can inferred using the Kaplan and \nGerhing model27  (the fitting procedure is detailed  in the Supplementary Note 2).  \n \nDynamical characterization and spin transparency . \nThe most striking feature of these large PMA films is their extremely low magne tic losses that we \ncharacterize  using Ferromagnetic Resonance (FMR) measurements. First of all, we quantify  by in-plane  \nFMR the anisotropy field HKU deduced from the effective magnetization ( Meff): HKU = M S – Meff (the \nprocedure to derive Meff from in plan e FMR is presented in Supplementary Note 3 ). HKU values for BiYIG \nfilms with different doping levels grown on various substrates  are summarized  in Table 1.  As expected \nfrom out-of-plane  hysteresis curves , we observe different signs for HKU. For spontaneously out -of-plane \nmagnetized samples , HKU is positive and large enough to fully compensate the demagnetizing field while \nit is negative for in -plane magnetized films. From these results , one can expect that fine tuning of the Bi \ncontent allow s fine tuning of the effective magnetization and consequently of the FMR resonan ce \nconditions. We measure magnetic losses on a 30nm thick Bi1Y2IG//sGGG film  under tensile strain with \nPMA (Fig. 3 -(a)). We use the FMR absorption line shape  by extracting  the peak -to-peak linewidt h (𝛥𝐻pp) \nat different out-of-plane  angle  for a 30nm thick perpendicularly magnetized Bi 1Y2IG//sGGG film at 8 GHz \n(Fig.  3-(b)). This  yields an optimal value of  𝜇0𝛥𝐻pp  as low as 0.3 mT  (Fig. 3-(c)) for 27° out-of-plane  polar \nangle. We stress here that state of the art PLD grown YIG//GGG films exhibit similar values for 𝛥𝐻pp at \nsuch resonant conditions28. This angular dependence of 𝛥𝐻pp that shows pronounced variations at \nspecific angle is characteristic of a two magnons scattering relaxation process with few \ninhomogenei ties29. The value of this angle is sample dependent  as it is related to the distribution of the \nmagnetic inhomogeneities .  The dominance in our films of those two i ntrinsic relaxation processes \n(Gilbert damping and two magnons  scattering) confirms  the high films quality . We also derived the \ndamping value of th is film (Fig.  3-(d)) by selecting the lowest linewidth (corresponding to a specific out of 5 \n plane angle) at each frequency, the spread of the out of plane angle is ±3.5 ° around 30.5 °. The obtained \nGilbert damping value is α = 3.10-4 and the peak -to-peak extrinsic linew idth 𝜇0𝛥𝐻0 =0.23 mT a re \ncomparable to the one obtained for  the best PLD grown YIG//GGG nanometer thick films28 (α =2.10-4). \nFor x= 0.7  Bi doping, the smallest observed FMR linewidth is 0.5 mT  at 8 G Hz. \nThe low magnetic losses of BiYIG films could open  new perspectives for magnetization dynamics control \nusing spin-orbit torques20,30,31. For such phenomenon  interface transparency to spin curr ent is then the \ncritical parameter which is defined using the effective spin -mixing conductance ( 𝐺↑↓). We use spin \npumping experiments to estimate the increase of the Gilbert damping due to Pt deposition on Bi1Y2IG \nfilms. The spin mixing conductance can thereafter be calculated using 𝐺↑↓=4𝜋𝑀s𝑡film\n𝑔eff𝜇B(𝛥𝛼) where 𝑀s and \n𝑡film are the BiYIG magnetization saturation and thickness, 𝑔eff  is the effective Landé factor ( 𝑔eff=2), \n𝜇B is the Bohr magneton and 𝛥𝛼 is the increase in the Gilbert damping constant induced by the Pt top \nlayer. We obtain  𝐺↑↓=3.9 1018m−2  which is comparable to what is obtained on PLD grown  YIG//GGG \nsystems  28,32,33. Consequently , the doping in Bi should  not alter the spin orbit -torque efficiency  and spin \ntorque devices made out of  BiYIG will be as energy efficient as their  YIG counterpart . To further confirm \nthat spin current cross es the Pt/BiYIG interface , we measure Inverse Spin Hall Effect (ISHE) in Pt for a Pt/  \nBi1.5Y1.5IG(20nm)/ /sGGG in -plane magnetized film (to fulfill the ISHE geometry requirements  the \nmagnetization needs to be in -plane and perp endicular to the measured voltage ). We measure a \ncharacteristic voltage peak due to ISHE  that reverses its sign when the static in-plane  magnetic field is \nreversed (Fig. 4). We emphasize here that the amplitude of the s ignal is similar to that of Pt/ YIG//GGG in \nthe same experimental conditions.  \nConclusion  \nIn summary, this new material platform will be highly beneficial for magnon -spintronics and  related \nresearch fields like  caloritronics. In many aspects , ultra -thin BiYIG films offer new leverages for  fine \ntuning of the magnetic properties with no  drawbacks compared to the reference materials of th ese \nfields: YIG.  BiYIG with its higher Faraday rotation coefficient (almost two orders of magnitude more than \nthat of  YIG) will increase the sensitivity of light based detection technics  that can be used  (Brillouin light \nspectroscopy (BLS) or time resolved Kerr microscopy34). Innovative scheme s for on -chip magnon -light \ncoupler could be now  developed bridging the field of magnonics to th e one  of photonics. From a \npractical point of view , the design of future active devices will be much more flexible as it is possible to \neasily engineer the spin waves dispersion relation through magnetic anis otropy tuning without the need \nof large bias magnetic fields. For instance, working in the forward volume waves configuration comes 6 \n now cost free, whereas in sta ndard in -plane magnetized media  one has to overcome the demagnetizing  \nfield. As the development  of PMA tunnel junctions  was key in developing today  scalable MRAM \ntechnology , likewise, we believe that P MA in nanometer -thick low loss  insulator s paves the path to new \napproaches where the magnonic medium material could also be used to store information locally \ncombining therefore the memory and computational functions, a most desirable feature for the brain -\ninspired neuromorphic paradigm . \n  7 \n Methods  \nPulsed Laser Deposition (PLD) growth  \nThe PLD growth of BiYIG films is realized using stoichiometric BiYIG  target. The laser used is a frequency \ntripled Nd:YAG laser ( λ =355nm), of a 2.5Hz repetition rate and a fluency E varying from 0.95 to 1.43 \nJ.cm-2 depending upon the Bi doping in the target. The distance between target and substrate is fixed at \n44mm. Pri or to the deposition the substrate is annealed at 700°C under 0.4 mbar of O 2. For the growth, \nthe pressure is set at 0.25 mbar O 2 pressure. The optimum growth temperature varies with the Bi \ncontent from 400 to 550°C. At the end of the growth, the sample is  cooled down under 300 mbar of O 2. \n \nStructural characterization  \nAn Empyrean diffractometer with Kα 1 monochromator  is used  for measurement in Bragg -Brentano  \nreflection mode  to derive the (111) interatomic plan distance. Reciprocal Space Mapping is performed on \nthe same diffractometer and we used the diffraction along the (642) plane direction which allow to gain \ninformation on the in-plane  epitaxy relation along [20 -2] direc tion.  \n \nMagnetic characterization  \nA quantum design SQUID magnetometer was used to measure the films’ magnetic moment ( Ms) by \nperforming hysteresis curves along the easy magnetic direction at room temperature. The linear \ncontribution of the paramagnetic (sGG G or GGG) substrate is linearly subtracted.  \nKerr microscope (Evico Magnetics) is used in the polar mode to measure out-of-plane  hysteresis curves \nat room temperature.  The same microscope is also used to image the magnetic domains structure after \na demagn etization procedure. The spatial resolution of the system is 300 nm.  \nA broadband FMR setup with a motorized rotation stage was used. Frequencies from 1 to 20GHz have \nbeen explored. The FMR is measured as the derivative of microwave power absorption via a low \nfrequency modulation of the DC magnetic field. Resonance spectra were recorded with the applied static \nmagnetic field oriented in different geometries (in plane or tilted of an angle 𝜃 out of the strip line \nplane).  For o ut of plane magnetized samples the Gilbert damping parameter has been obtained by \nstudying the  angular  linewidth dependence.  The procedure assumes that close to the minimum \nlinewidth (Fig 3a) most of the linewidth angular dependence is dominated by the inhomogeneous \nbroadening, thus opt imizing the angle for each frequency within few degrees allows to estimate  better  8 \n the intrinsic contribution. To do so we varied the out of plane angle of the static field from 2 7° to 34 ° for \neach frequency  and w e select the lowest value of   𝛥𝐻pp.  \nFor Inverse spin Hall effect measurements, the same FMR setup was used, however here the modulation \nis no longer applied to the magnetic field but to the RF power at a frequency of 5kHz. A Stanford  \nResearch SR860 lock -in was used a signal demodulator.  \nData  availability : \nThe data that support the findings of this study are available within the article or from the corresponding \nauthor upon reasonable request . \nAcknowledgements:  \n We acknowledge J. Sampaio for preliminary Faraday rotation measurements and N. Rey ren and A. \nBarthélémy for fruitful discussions. This research was supported by the ANR Grant ISOLYIG (ref 15 -CE08 -\n0030 -01). LS is partially supported by G.I.E III -V Lab. France.   \n \nAuthor Contributions : \nLS performed the growth, all the measurements, the da ta analysis and wrote the manuscript  with AA . NB \nand JBY conducted the quantitative Faraday Rotation measurements and participated in the FMR data \nanalysis. LQ and fabricated the PLD targets . RL supervised the target fabrication and participated in the \ndesign of the study . EJ participated in the optimization of the film growth conditions. CC supervised the \nstructural characterization experiments. AA conceived the study and w as in charge of overall direction . \nPB and VC contributed to the design and implem entation of the research . All authors discussed the \nresults and commented on the manuscript.  \nCompeting Financial Interests : \nThe authors declare no competing interest.  \n \n \n  9 \n References:  \n1. Karenowska, A. D., Chumak, A. V., Serga, A. A. & Hillebrands, B. Magnon spintronics. Handb. \nSpintron.  11, 1505 –1549 (2015).  \n2. Chumak, A. V., Serga, A. A. & Hillebrands, B. Magnonic crystals for data processing. J. Phys. D. \nAppl. Phys.  50, (2017).  \n3. Bauer , G. E. W., Saitoh, E. & Van Wees, B. J. Spin caloritronics. Nat. Mater.  11, 391–399 (2012).  \n4. Li, P. et al.  Spin -orbit torque -assisted switching in magnetic insulator thin films with perpendicular \nmagnetic anisotropy. Nat. Commun.  7, 12688 (2016).  \n5. Serga, A. A., Chumak, A. V. & Hillebrands, B. YIG magnonics. J. Phys. D. Appl. Phys.  43, (2010).  \n6. Seifert, T. et al.  Launching magnons at the terahertz speed of the spin Seebeck effect. Prepr. \nhttp//arxiv.org/abs/1709.00768  (2017).  \n7. Onbasli, M. C. et al. 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Magnetic anisotropies in ultrathin bismuth iron garnet films. J. Magn. Magn. \nMater.  335,  139–143 (2013).  \n14. Ben Youssef, J., , Legall, H. & . U. P. et M. C. Characterisation and physical study of bismuth \nsubstituted thin garnet films grown by liquid phase epitaxy (LPE). (1989).  \n15. Fu, J. et al.  Epitaxial growth of Y 3 Fe 5 O 12 thin films with perpe ndicular magnetic anisotropy. \nAppl. Phys. Lett.  110,  202403 (2017).  \n16. Wang, C. T. et al.  Controlling the magnetic anisotropy in epitaxial Y3Fe5O12 films by manganese \ndoping. Phys. Rev. B  96, 224403 (2017).  \n17. Avci, C. O. et al.  Fast switching and signature of efficient domain wall motion driven by spin -orbit \ntorques in a perpendicular anisotropy magnetic insulator/Pt bilayer. Appl. Phys. Lett.  111,  (2017).  \n18. Quindeau, A. et al.  Tm3Fe5O12/Pt Heterostructures with Perpendicular Magnetic Anisotropy for \nSpintronic Applications. Adv. Electron. Mater.  3, 1600376 (2017).  10 \n 19. Tang, C. et al.  Anomalous Hall hysteresis in Tm3Fe5O12/Pt with strain -induced perpendicular \nmagnetic anisotropy. Phys. Rev. B  94, 1–5 (2016).  \n20. Collet, M. et al. Generation of coherent spin -wave modes in yttrium iron garnet microdiscs by \nspin–orbit torque. Nat. Commun.  7, 10377 (2016).  \n21. Hansen, P., Klages, C. ‐P. & Witter, K. Magnetic and magneto ‐optic properties of praseodymium ‐ \nand bismuth ‐substituted yttr ium iron garnet films. J. Appl. Phys.  60, 721–727 (1986).  \n22. Robertson, J. M., Wittekoek, S., Popma, T. J. A. & Bongers, P. F. Preparation and optical properties \nof single crystal thin films of bismuth substituted iron garnets for magneto -optic applicatio ns. \nAppl. Phys.  (1973).  \n23. Hansen, P., Witter, K. & Tolksdorf, W. Magnetic and magneto -optic properties of lead - and \nbismuth -substituted yttrium iron garnet films. Phys. Rev. B  (1983).  \n24. Matsumoto, K. et al.  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Phys.  83, 4344 –4365 (1998).  \n30. Xiao, J. & Bauer, G. E. W. Spin -Wave Excitation in Mag netic Insulators by Spin -Transfer Torque. \nPhys. Rev. Lett.  108,  217204 (2012).  \n31. Safranski, C. et al.  Spin caloritronic nano -oscillator. Nat. Commun.  8, 117 (2017).  \n32. Takahashi, R. et al.  Electrical determination of spin mixing conductance at metal/ins ulator \ninterface using inverse spin Hall effect. J. Appl. Phys.  111,  07C307 (2012).  \n33. Heinrich, B. et al.  Spin Pumping at the Magnetic Insulator (YIG)/Normal Metal (Au) Interfaces. \nPhys. Rev. Lett.  107,  66604 (2011).  \n34. Stigloher, J. et al.  Snell’s Law for Spin Waves. Phys. Rev. Lett.  117,  1–5 (2016).  \n 11 \n Figures Captions :  \n \nFigure 1 -Structural properties  of ultra -thin BiYIG films . \n(a) and (b) : Evolution of the target  cubic lattice parameter of BixY3-xIG, the dashed line represents the \nsubstrate (sGGG  and GGG respectively) lattice parameter and allow s to infer the expected tensile or \ncompressive strain arising for each substrate/target combination.  \n(c) and ( d): 2𝜃−𝜔 X-Ray diffraction scan along the (111) out-of-plane  direction for BixY3-xIG films gr own \non sGGG (111) and GGG (111) respectively. From the film and substrate diffraction peak position , we can \nconclude about the nature of the strain. Compressive strain is observed for 1.5 doped films grown on \nsGGG substrate and for all films grown on GGG w hereas tensile strain occurs for films with x=  0.7 and x= \n1 Bi content grown on sGGG.  \n(e) and (f) : RSM along the evidence the (642) oblique plan showing pseudomorphic growth in films: both \nsubstrate and film the diffraction peak are aligned along the qx\\\\[20-2] direction. The relative position of \nthe diffraction peak of the film (up or down) along qx is related to the out-of-plane  misfit between the \nsubstrate and the film (tensile or compressive).  \nFigure 2 -Static magnetic  properties . \n(a) Out-of plane Kerr hysteresis loop performed in the polar mode for Bi0.7Y2.3IG films grown on the two \nsubstrates: GGG and sGGG  \n(b) Same measurement for BixY3-xIG grown on sGGG with  the three different Bi doping ( x= 0.7, 1 and 1.5) . \nBi0.7Y2.3IG//GGG is in -plane magnetized whereas perpendicular magnetic anisotropy (PMA) occurs for x= \n0.7 and x= 1 films grown on sGGG: square shaped loops with low saturation field ( µ0Hsat about 2.5 mT) are \nobserved. Those two films are experiencing tensile strain . Whereas the inset shows that the Bi1.5Y1.5IG \nfilm saturates at a much higher field wi th a curve characteristic of in -plane easy magnetization direction. \nNote that for Bi1.5Y1.5IG//sGGG µ0Hsat ≈290mT >µ0Ms≈162mT which points toward a negative uniaxial \nanisotropy term ( µ0HKU) of 128mT which is coherent with the values obtained from in plane FMR \nmeasurement .  \n(c) Magnetic domains structure imaged on Bi 1Y2IG//sGGG films of three different thicknesses at \nreman ant state after demagnetization . The scale bar, displayed in blue , equal s 20 µm . Periods of the \nmagnetic domains structure ( Dwidth) are derived using 2D Fast Fourier Transform . We obtained Dwidth =3.1,  \n1.6 and 0.4  µm for tBi1Y2IG= 32, 47 and 52  nm respectively. We note a decrease of Dwidth with increasing \ntBi1Y2IG  that is coherent with the Kaplan and Gehring model valide in the case  Dwidth>>tBiYIG.  12 \n Figure 3-Dynamical properties  of BiYIG films with PMA.  \n(a) Sketch of the epitaxial configuration for Bi1Y2IG films , films are grown under tensi le strain giving rise \nto tetragonal distortion of the unit cell.  \n(b) Out-of-plane  angular depend ence of the peak -to-peak FMR linewidth ( 𝛥𝐻pp) at 8 GHz on a 30 nm \nthick Bi1Y2IG//sGGG with PMA  (the continuous  line is a guide for the eye) . The geometry of the \nmeasurement is shown in top right of the graph. The wide disparity of the value for the peak to peak \nlinewidth 𝛥𝐻pp is attributed to the two magnons scattering process  and inhomogeneties in the sample . \n(c) FMR absorption linewidth o f 0.3 mT for the same film at measured at  𝜃=27°. (d) Frequency \ndependence of the FMR linewidth . The calculated  Gilbert damping parameter and the extrinsic linewidth \nare displayed  on the graph .  \nFigure 4- Inverse Spin Hall Effect  of BiYIG  films with in plane magnetic anisotropy.  \nInverse Spin Hall Effect (ISHE) voltage vs magnetic  field measured on the Pt / Bi1.5Y1.5IG//sGGG  sample in \nthe FMR resonant condition at 6 GHz proving  the interface transparency to spin current. The rf excitation \nfield is about 10-3 mT which corresponds to a linear regime of excitation. Bi1.5Y1.5IG//sGGG present s an in-\nplane  easy magnetization axis due to a growth under compressive strain.  \n  \n  13 \n Table 1 - Summary of the magnetic properties of BixY3-xIG films on GGG and sGGG \nsubstrates.  \nThe saturation magnetization is roughly unchanged. The effective magnetization Meff obtained through \nbroad -Band FMR measurements allow to deduce the out -of-plane anisotropy fields HKU (HKU =Ms-Meff) \nconfirming the dramatic changes of the out-of-plane magnetic anisotropy variations observed in the \nhysteresis curves . \n   \nBi doping  Substrate  µ0MS(mT)  µ0Meff(mT)  µ0HKU(mT)  \n0 GGG  157 200 -43 \n0.7 sGGG  180 -151 331 \n0.7 GGG  172 214 -42 \n1 sGGG  172 -29 201 \n1 GGG  160 189 -29 \n1.5 sGGG  162 278 -116 14 \n Figure 1  \n (f) \n0.5 1.0 1.51.2351.2401.2451.2501.255  Cubic Lattice parameter in nmBi content \n  sGGG\n45 50 55103106109\n Intensity (cps)\n2 angle(°)0.711.5\n0.711.5\n1.70 1.724.254.304.35qz //[444] (rlu)*10\nqx//[2-20] (rlu)*101E+01 1E+05\n1.74 1.754.234.274.30qz //[444] (rlu)*10\nqx//[2-20] (rlu)*101E+02 1E+05\n1.73 1.764.214.254.29qz //[444] (rlu)*10\nqx//[2-20] (rlu)*101E+02 1E+05\n0.7 1 1.5\nsGGGBi0.7Y2.3IG\nsGGGBi1Y2IG\nBi1.5Y1.5IGsGGG\n45 50 55102105108\n Intensity (cps)\n2 angle (°)\n0.5 1.0 1.51.2351.2401.2451.2501.255  Cubic Lattice parameter in nmBi content \n  \nGGG0.711.5\n0.7511.5\n1.75 1.774.204.304.40qz//[444] (rlu)*10\nqx//[2-20] (rlu)*101E+01 1E+05\n1.750 1.7754.284.324.35qz//[444] (rlu)*10\nqx//[2-20] (rlu)*101E+02 1E+05\n0.7\nGGG\nBi0.7Y2.3IGGGG\nBi1Y2IG1sGGG\nGGG(a)\n(b)(c)\n(d)(e)\n(f)15 \n Figure 2  \n \n \n \n16 \n Figure 3  \n \n \n  \n17 \n Figure 4  \n-130 -120 -110 110 120 130-4000400800V (nV)\nµ0H (mT)2 \n Supplementary Notes 1- Derivation of the magneto -elastic anisotropy  \n \nThe out -of-plan e anisotropy constant KU is ascribed to be a result  of, at least , two contributions : a \nmagneto -elastic anisotropy  term  induced by strain (𝐾MO) and a term that is due to preferential \noccupation of Bi atoms of non equivalent dodecahedral sites of the  cubic u nit cell. This last term is \nknown as the growth induced anisotropy term 𝐾GROWTH . From X -ray characterizations and f rom the \nknown properties of the thick BiYIG LPE grown films, it is possible to calculate the expected values of  \n𝐾MO in each doping/substra te combination. We thereafter deduce 𝐾GROWTH  from the relation  \n𝐾U=𝐾MO+ 𝐾GROWTH . KMO is directly proportional to the misfit between the film and the substrate:  \n𝐾MO=3\n2∙𝐸\n1−𝜇∙𝑎film −𝑎substrate\n𝑎film∙𝜆111 \nWhere E, µ and λ111 are respectively the Young modulus, the Poisson coefficient, and the \nmagnetostrictive constant along the (111) direction.  Those constant are well established for the bulk1: E= \n2.055.1011 J.m-3, µ= 0.29 . The magnetostriction  coefficient  λ111 for the thin film case is slightly higher \nthan that of the bulk and depends  upon the Bi rate x: 𝜆111(𝑥)=−2.819 ∙10−6(1+0.75𝑥) 2. The two \nlattice par ameter entering in the equation: afilm and asubstrate  correspond to the lattice parameter of the \nrelaxed film structure and of the subs trate. Under an elastic deformation afilm can be derived with the \nPoisson coefficient:  \n𝑎film =𝑎substrate −[1−𝜇111\n1+𝜇111]𝛥𝑎⊥  with  ∆𝑎⊥=4√3𝑑444film−𝑎substrate  \nAll values for the different target/substrate combinations are displayed in the Table S-1. We note here \nthat a negative (positive) misfit corresponding to a tensile (compressive) strain will favor an out-of-plane  \n(in-plane ) magnetic anisotropy which is coherent with what is observed in our samples. To estimate the \ncontribution to the magnetic  energy of the magneto elastic anisotropy term we compare it to the \ndemagnetizing field 𝜇0𝑀s that favors in-plane  magnetic anisotropy in thin films. Interestingly the \nmagneto elastic field ( 𝜇0𝐻MO) arising from 𝐾MO (𝜇0𝐻MO=2𝐾MO\n𝑀s) never exceed 30% o f the \ndemagnetizing fields and therefore cannot alon e be responsible of the observed PMA.  \n \nStudies on µm-thick BiYIG films  grown  by LPE showed that PMA in BiYIG arises due to the growth \ninduced anisotropy term 𝐾GROWTH  , this term is positive 3 for the case of Bi substitution . We have \ninferred 𝜇0𝐻GROWTH  values for all films using 𝐾U constants  measured by FMR. The results are \nsummarized in Supplementary Figure 1. One can clearly see that  𝐾GROWTH  is strongly substrate \ndependent and therefore does not depend sol ely on the Bi content. We conclude that strain play s a role \nin Bi3+ ion ordering within the unit cell . \n \n 3 \n  \n \nSupplementary Figure 1- Summary of the inferred values of the effective magnetic \nanisotropy out -of-plan fields.  \n \nHorizontal dash lines represent the magnitude of the demagnetization field µ0Ms. When µ0HKU is larger \nthan µ0Ms (dot line) films have a PMA, they are in -plane magnetized otherwise .   \n0200400µ0HKuvs Bi content \n HKu = Hstrain+ Hgrowth\nµ0Ms \n1.50.7µ0HKu (mT) µ0 Hstrain\n µ0 Hgrowth 1\nµ0Ms\n-1000100200\n0.7 µ0HKu (mT) µ0Hstrain\n µ0Hgrowth\n14 \n Supplementary Table 1 - Summary of the films’ calculated magneto -elastic \nanisotropy constant ( KMO) and the corresponding anisotropy field HMO \n \n \n  Bi content  substrate  afilm(Å) Δ a┴/afilm KMO (J.m-3) µ0 HMO(mT)  µ0 Hdemag (mT)  \n0.7 sGGG  12.45  0.6 5818  81 179 \n0.7 GGG  12.41  -0.4 -4 223  -61 172 \n1.0 sGGG  12.47  0.3 3 958  57 172 \n1.0 GGG  12.42  -0.6 -6 500  -102 160 \n1.5 sGGG  12.53  -0.6 -8 041  -124 157 5 \n Supplementary Notes 2- Derivation of the domain wall energy  \n \nTo derive the characteristic domain wall energy σDW for the maze shape like magnetic domains , we use \nthe Kaplan et al. model4. This model applies in our case as the ratio of the film thickness ( tfilm) to the  \nmagnetic domain width ( Dwidth) is small (𝑡film\n𝐷width~ 0.01). The domain wall width and the film thickness are \nthen expected to be linked by:  \n𝐷width =𝑡filme−π1.33𝑒𝜋𝐷0\n2𝑡film  where 𝐷0=2𝜎w\n𝜇0𝑀s2 is the dipolar length.  \nHence we expect a linear dependence of ln(𝐷width\n𝑡film) vs  1\n𝑡film : \nln(𝐷width\n𝑡film)=π𝐷0\n2∙ 1\n𝑡film+Cst.  (1) \nThe magnetic domain width of Bi xY3-xIG//sGGG ( x = 0.7 and 1) for several thicknesses are extracted from \n2D Fourier Transform of the Kerr microscopy images at remanence. In Supplementary Fig ure 2, we plot \nln(𝐷width\n𝑡film) vs  1\n𝑡film which follow s the expected linear dependence  of Equation (1) . We infer from the zero \nintercept an estimat ion of the dipolar length D0 of BiYIG films doped at 0.7 and 1 in Bi : D0 x=0.7= 16.5  µm \nand D0 x=1= 18.9µm. The corresponding domain wall energy are respectively 0.7 mJ .m-2 and 0.65 mJ .m-2. \nEven if the small  difference in domain wall energy between the two Bi content  may not be significant \nregarding the statistical fitting errors, it correlates to the decrease of the out-of-plane  anisotropy ( KU) \nwith increasing the Bi content.  6 \n Supplementary Figure 2- Evolution of the domain width vs film thickness  \n \nln(𝐷width\n𝑡film) vs  1\n𝑡film for Bi xY3-xIG//sGGG films doped at 0.7 (a) and 1 (b) in Bi. Dots correspond to the \nexperimental values. The dashed line is the linear fit that allows to extract the D0 parameter.  \n  \n25 50 752468\n  ln(Dwidth/tfilm)\ntfilm(µm-1)50 1002468\n  ln(Dwidth/tfilm)\n/tfilm(µm-1)Bi0.7Y2.3IG//sGGG\nD0=16.5µm\nBi1Y2IG//sGGG\nD0=18.9µm\n(a) \n(b) 7 \n  \nSupplementary Notes 3 - Damping and effective magnetic field derivation  \n \nFrom In Plane frequency dependent of FMR we can derive the effective magnetization  (𝑀eff) using the \nKittel law:  \n𝑓res=𝜇0𝛾√𝐻res(𝐻res+𝑀-eff) \nWhere  γ is the gyromagnetic ratio of the BiYIG  (assumed to be same as the one of the YIG) : γ=28 GHz.T-1. \n𝐻res  and 𝑓res are respectively the FMR resonant field and frequency . The uniaxial magnetic anisotropy \ncan thereafter  be derived using the saturation magnetizatio n form squid magnetometry using :   \nMeff=Ms-HKU. The Gilbert damping ( α) and the inhomogeneous linewidth ( ΔH 0) which are the two \nparamaters defining the magnetic relaxation are obtained from the evolution of the peak to peak \nlinewidth ( ΔHpp) vs the resonant frequency ( fres): \n𝛥𝐻pp=𝛥𝐻0+2\n√3𝛼𝑓res\n𝜇0𝛾  (2) \nThe first term is frequency independent and often attributed magnetic inhomogeneity’s (anisotropy, \nmagnetization).  \n  8 \n Supplementary Figure 3 - µ0ΔHpp vs fres on 18 nm thick Bi15.Y1.5IG//sGGG  \n \nThe l inewidth  frequency dependence from 5 to 19 GHz for in plane magnetized Bi 1.5Y1.5IG//sGGG  sample \nallow to ex tract the damping and the inhomogeneous linewidth parameter  using the Equation (2) . \n \n  \n0 5 10 150.00.51.01.5 \n Hpp(mT)\nfres (GHz)H0=0.5 mT\n=1.9*10-39 \n Supplementary References:  \n \n1. Hansen, P., Klages, C. ‐P. & Witter, K. Magnetic and magneto ‐optic properties of praseodymium ‐ \nand bismuth ‐substituted yttrium iron garnet films. J. Appl. Phys.  60, 721–727 (1986).  \n2. Ben Youssef, J., , Legall, H. & . U. P. et M. C. Characterisation and physical study of bismuth \nsubstituted thin garnet films grown by liquid phase epitaxy (LPE). (1989).  \n3. Fratello, V. J., Slusky, S. E. G., Brandle, C. D. & Norelli, M. P. Growth -induced anisotropy in \nbismuth: Rare -earth iron garnets. J. Appl. Phys.  60, 2488 –2497 (1986).  \n4. Kaplan, B. & Gehring, G. A. The domain structure in ultrathin magnetic films. J. Magn. Magn. \nMater.  128,  111–116 (1993).  \n "    },    {        "title": "1902.04608v1.Characterization_of_spin_wave_propagation_in__111__YIG_thin_films_with_large_anisotropy.pdf",        "content": "1 \n Characterization of spin wave propagation in (111) YIG thin films with \nlarge anisotropy  \nA. Krysztofik,1,b) H. Głowiński,1,a) P. Kuświk,1,2 S. Ziętek,3 L. E. Coy,4 J. N. Rychły ,5 \nS. Jurga,4 T. W. Stobiecki,3 J. Dubowik1 \n1Institute of Molecular Physics, Poli sh Academy of Sciences, M. Smoluchowskiego 17, PL -60-179 Poznań, Poland  \n2Centre of Advanced Technology, Adam  Mickiewicz University, Umultowska 89c, PL -61-614 Poznań, Poland  \n3Department of Electronics, AGH University of Science and Technology, Al. Mickiewic za 40, PL -30-059 Kraków, Poland  \n4NanoBioMedical Centre, Adam Mickiewicz University, Umultowska 85, PL -61-614 Poznań, Poland  \n5Faculty of Physics, Adam  Mickiewicz University, Umultowska 85, PL -61-614 Poznań, Poland  \n6Faculty of Physics and Applied Computer Sc ience, AGH University of Science and Technology, Al. Mickiewicza 30, PL -30-059 Kraków, \nPoland  \n \na)E-mail: hubert.glowinski@ifmpan.poznan.pl  \nb)E-mail: adam.krysztofik@ ifmpan.poznan.pl  \n \n \n \nAbstract  \nWe report on  long-range spin wave  (SW)  propagation in nanomete r-thick Yttrium \nIron Garnet (YIG) film with  an ultralow Gilbert damping. The knowledge of a wavenumber \nvalue  |𝑘⃗ | is essential  for design ing SW devices. Although determining  the wavenumber |𝑘⃗ | \nin experiments like Brillo uin light scattering spect roscopy is straightforward , quantifying \nthe wavenumber in all -electrical experiments has not been widely commented  so far. \nWe analyze magnetost atic spin wave (SW) propagation in  YIG films  in order to determine \nSW wavenumber |𝑘⃗ | excited by the coplanar waveguide . We show  that it is crucial  to \nconsider influence of magnetic anisotropy fields present in YI G thin films  for precise \ndetermination of SW wavenumber . With the proposed  methods  we find that experimentally \nderived v alues of |𝑘⃗ | are in perfect agreement with  that obtained from  electromagnetic \nsimulation only if anisotropy fields are included.  \n \n \n \n \n \n \n 2 \n  \nSpin wave (SW) propagation in magnetic thin film structures  has become intensively \ninvestigated topic in recent year s due to promising applications in modern electronics  [ 1, 2, \n3, 4 ]. The wavenumber (or equivalently – the wavelength 𝜆=2𝜋/|𝑘⃗ |) is an important \nparameter to  account for propagation characteristics.  For example, it is essential to choose \nSW wavenumber and correlate  it to certain device dimension in order to ensure observation \nof expected phenomena in SW devices e.g. i n magnonic crystals  [ 5, 6 ] or devices based on  \nwave interference such as SW transistor [  2 ], SW logic gates  [ 2 ], Mach -Zender  type \ninterferometer s [ 7 ]. The knowledge  of SW wavenumber is also very important  in the \nassessment of the  effective magnitude of Dzaloshinskii -Moriya interaction using collective \nspin-wave dynamics  [ 8 ]. \nIn propagating SW spectroscopy experiments two s horted coplanar waveguides  \n(CPW s) are commonly used as a transmitter and a receiver  [ 9 ]. Each CPW , integrated \nwithin the film , consists of a signal line and two ground lines conn ected at one end. When \na rf-current flows through the transmitter it induces  an oscillating magnetic field around the \nlines that exerts a torque and causes spin precession in the magnetic material beneath. The \ninverse  effect is then used for SW detection by the receiver . Since  the generated magnetic \nfield is not homogenous with reference to the film plane and solely depends on CPW \ngeometry , it determines the distribution of SW wavenumber that can be excited.  \nIt is assumed that the transmitter excites a broa d spectrum of SW wavevectors \nof wavenumber 𝑘 exten ding to 𝑘𝑚𝑎𝑥≈𝜋/𝑊 (𝑊 is a width of  CPW line) with a maximum \nof excitation amplitude approximately around 𝑘𝑚𝑎𝑥𝐴𝑚𝑝≈𝜋/2𝑊 [ 10 ]. The question now is : \nwhat is the actual wavenumber of the SW with the la rgest amplitude  detected by the receiver \nsituated at a certain distance f rom the transmitter. It appears that while in Brillo uin light \nscattering spect roscopy 𝑘 is easily accessible,  in all electrical spin wave spectroscopic \nexperiment s the determination of SW wavenumber  is rather challenging  [ 11 ]. \nWe aim to ans wer this question by analyzing our experimental results of  SW \npropagation  in yttrium iron garnet (Y 3Fe5O12, YIG)  thin film s. YIG films are known \nas possessing the lowe st Gilbert damping parameter enabl ing the SW transmission over  the \ndistances of several hundred micrometers  [ 2, 12 ]. However, YIG films synthesized by \npulsed laser deposition (PLD) exhibit substan tially disparate  values of  anisotropy fields and \nsaturation magnetization , depending on the growth process parameters and , consequently , \nstoichiometr y of the obtained film [ 13, 14, 15 ]. It has already been theoretically predicted 3 \n that anisotropy may significantly affect SW propagat ion and the transmission characteristics  \n[ 16, 17 ]. Therefore , for such  YIG films , SW spectra analysis requires careful consideration \nof anisotropic properties of a given film.  \nHere, we compare two methods of experimental determination of the SW \nwavenumber  which include anisotropy fields. The experimental results are then compared \nwith electromagnetic simulations.  \n \n \nFig. 1 . A θ-2θ XRD  scan of epitaxial YIG film on GGG (111) substrate near the GGG (444) reflection . \n \n \nYIG film was grown on a monocrystalline, 111-oriented Gadolinium Gallium \nGarnet substrate (Gd 3Ga5O12, GGG) by means of  PLD  technique . Substrate temperature was \nset to 650℃  and under  the 1.2×10−4 𝑚𝑏𝑎𝑟  oxygen pressure ( 8×10−8 𝑚𝑏𝑎𝑟  base \npressure)  thin film was deposited at the  0.8 𝑛𝑚/𝑚𝑖𝑛 growth rate using third harmonic \nof Nd:YAG Laser ( 𝜆=355 𝑛𝑚). After the growth, the sample was additionally ann ealed \nex situ at 800℃  for 5 𝑚𝑖𝑛. X-ray diffraction  and reflection  measurements showed that \nthe YIG film was single -phase, epitaxial with the GGG substrate  with the thickness of 82 𝑛𝑚 \nand RMS  roughness of 0.8 𝑛𝑚. XRD θ-2θ scan, presented in Fig. 1, c learly shows the high \ncrystallinity of the YIG film, displaying well defined Laue oscillations , typical for highly \nepitaxial films, which clearly point to the high quality and well textured YIG  (111) film \n[ 18 ]. Subsequently , a system of two CPW s made of 100 𝑛𝑚 thick alumin um was integrated  \nonto YIG film  (Fig. 2) using a maskless photolit hography techniqu e. The width 𝑊 of signal \nand ground lines was equal to  9.8 𝜇𝑚 and the gaps between them were 4 𝜇𝑚 wide. \nThe distance between the centers  of signal l ines was 150 𝜇𝑚. \n49.5 50.0 50.5 51.0 51.5 52.0 52.5101102103104105106Intensity [a.u.]\n2 [deg]YIG (444)\nGGG (444)4 \n  \nFig. 2. SEM  image of the integrated CPW s on the YIG film.  The distance 𝑑 between the transmitter and the \nreceiver  is equal to  150 𝜇𝑚. The depicted  Cartesian and crystallographic coordinate  system is used throughout \nthis paper. The width of signal and ground lines  is marked  with 𝑊. 𝐺 denotes the gap width between the lines.  \n \n \nTo investigate SW propagation we follow ed approach presented in Ref. [ 9 ] and \n[ 12 ]. Using a Vector  Network Analyzer  transmission signal S21 was measured for Damon -\nEshbach surface modes with  wavevector 𝑘⃗  perpendicular to the magnetization  for magnetic \nfields ranging from −310 𝑂𝑒 to +310 𝑂𝑒 (Fig.  3(a)). Exemplary S 21 signal s (imaginary \npart) , whic h are shown in Figs  3(b) and (c) , reveal  a series of oscillations as a function \nof frequency with a Gaussian -like envelope  corresponding to the excited SW wave number  \ndistribution.  Figure  3(c) shows  that frequency separation ∆𝑓 between two oscillation maxi ma \ndiffers noticeably in value  depending on the magnetic field . The decrease in signal amplitude \nis also observed  since SW decay length is inversely proportional to the frequency , so that the \nlow-frequency SWs propagate further away  [ 12, 19 ]. \n \n \n \n5 \n  \nFig. 3. (a) Color -coded SW propagation data S 21 measured at different magnetic fields.  \nWith a red line 𝑓(𝐻) dependence of the uniform excitation ( 𝑘=0) is depicted. The red line corresponds to \nthe maximum in S 11 signal  in (b). The blue dashed line represents a  dispersion relation with 𝐻𝑎=𝐻𝑢=0.     \n(b) Reflection (S 11, 𝑘=0) and transmission (S 21, 𝑘≠0) signals. The plot illustrates a magnified cross -section \nof (a) at 𝐻=−67.5 𝑂𝑒. (c) SW spectra measured  at different magnetic fields.  Color -coding in (b) and (c) \ncorresponds to the one defined in (a).  \n \n \nFor the frequencies of the highest signal am plitude, the wavenumber 𝑘𝑚𝑎𝑥𝐴𝑚𝑝  can \nbe determined according to the dispersion relation derived for (111) crystalline orientation of \nthe YIG film [ 16, 17 ]: \n6 \n  𝑓=𝜇𝐵\n2𝜋ℏ𝑔√(𝐻+2𝜋𝑀𝑠𝑡𝑘)(𝐻−1\n2𝐻𝑎−𝐻𝑢+4𝜋𝑀𝑠−2𝜋𝑀𝑠𝑡𝑘)−1\n2(𝐻𝑎sin (3𝜙))2, (1) \nwhere 𝑓 is the microwave frequency, 𝜇𝐵 – the Bohr magneton constant, ℏ – the reduced \nPlanck constant, 𝑔 – the spectroscopic splitting factor, 𝐻 – the ex ternal magnetic field, 𝑀𝑠 – \nthe saturation magnetization, 𝑡 – the film thickness, 𝑘 – the wavenumber, 𝐻𝑎 – the cubic \nanisotropy field and 𝐻𝑢 – the out -of-plane uniaxial anisotropy field. 𝐻𝑎=2𝐾𝑎\n𝑀𝑠 and 𝐻𝑢=2𝐾𝑢\n𝑀𝑠, \nwhere 𝐾𝑎 and 𝐾𝑢 are anisotropy constants. It should be highlighted that when 𝐻𝑎=𝐻𝑢=0, \nEq. 1 becomes equivalent to the one originally obtained by Damon and Eshbach  [ 20 ]. The \nazimuthal angle 𝜙 define s the in -plane orientation of magnetization direction with respect to \nthe (112̅) axis of YIG film. In our study the term −1\n2(𝐻𝑎sin (3𝜙))2 in Eq. 1 vanishes since \nmagnetic field 𝐻 is parallel to  (112̅) axis and 𝜙=0°. \nAs can be seen from Eq.  1, in order to determine wavenumber  𝑘 one need s \nto evaluate many material constants, namely 𝑔, 𝑀𝑠, 𝑡, 𝐻𝑎, 𝐻𝑢 in the first instance. \nThis problem can be partially  solved with a broadband ferromagnetic resonance measurement  \nof the film. Fo r 𝑘=0 Eq.1  simplifies to the form ula, which  allows for  the determination of \nthe spectroscopic factor 𝑔 and the effective magnetization 4𝜋𝑀𝑒𝑓𝑓∗=−1\n2𝐻𝑎−𝐻𝑢+4𝜋𝑀𝑠: \n 𝑓𝑘=0=𝜇𝐵\n2𝜋ℏ𝑔√𝐻(𝐻+4𝜋𝑀𝑒𝑓𝑓∗). (2) \nTherefore , within this approach , the film thickness and the saturation magnetization should be \ndetermined using  other experimental methods.  \nTo investigate ferromagnetic resonance of the YIG film , the reflection signal S 11 was \nmeasured. In order to avoid extrinsic  contribution to the resonance linewidth caused by non -\nmonochromatic excitation of the CPW (2𝜋∆𝑓𝑒𝑥𝑡𝑟=𝑣𝑔∆𝑘) [ 21 ] and, consequently , possible \nambiguities in  the interpret ation of  resonance peak position , it is recommended to perform \nthis measurement with the use of a wide CPW . Note that the full width a t half maximum of a \nCPW excitation spectra ∆𝑘≈𝑘𝑚𝑎𝑥𝐴𝑚𝑝  [ 21 ]. In our study  we used a CPW with signal and \nground lines  of the width equal to  450 𝜇𝑚 and with the  20 𝜇𝑚 wide gaps between  them. For \nsuch a CPW,  the simulated value of  𝑘𝑚𝑎𝑥𝐴𝑚𝑝  is equal to 49 𝑐𝑚−1 and, therefore, yields \nnegligible broadening that is of the order of a few MHz.  \nThe measured S 11 signal (imaginary part) is depicted in Fig.  3(a) with the red line. It \nappears to lie just below th e S 21 signal. Fitting to the experimental data with Eq.  2 gave  \nfollowing value  of the spectroscopic facto r  𝑔=2.010±0.001 and the effective 7 \n magnetization 𝑀𝑒𝑓𝑓∗=169±7 𝑒𝑚𝑢/𝑐𝑚3. A comparison of 4𝜋𝑀𝑒𝑓𝑓∗ with 4𝜋𝑀𝑠 (𝑀𝑠=\n120±19 𝑒𝑚𝑢/𝑐𝑚3 was measured  using  Vibrating Sample Magnetometry) gives  −1\n2𝐻𝑎−\n𝐻𝑢 of 616 𝑂𝑒, showing the substantial difference  between  obtained  values  of 𝑀𝑒𝑓𝑓∗ and 𝑀𝑠. \nThe determined value of −1\n2𝐻𝑎−𝐻𝑢 remain s in th e midst of the  range reported for  PLD -\ngrown YIG thin films , from 229 𝑂𝑒 up to 999 𝑂𝑒 [ 14, 22 ]. It is worth to mention that for \nfully stoichiometric , micrometer -thick  YIG films made by means of  liquid phase epitaxy  \n(LPE)  technique −1\n2𝐻𝑎−𝐻𝑢=101 𝑂𝑒 [ 14 ]. From  the analysis of resonance linewidth vs . \nfrequency  [ 23 ] we additionally  extracted Gilbert damping parameter of the YIG film , which \nequals to  𝛼=(5.5±0.6)×10−4 and impli es low damping of magnetization precession . \nSubstitution  of the 𝑔, 𝑀𝑒𝑓𝑓∗, 𝑀𝑠 and 𝑡 values  into Eq. 1 enabled the determination of \nwavenumber 𝑘𝑚𝑎𝑥𝐴𝑚𝑝=1980±102 𝑐𝑚−1. It sho uld be noted  that if anisotropy fields were \nneglected in the Eq.1 ( 𝐻𝑎=𝐻𝑢=0), yet only saturation magnetization was taken into \naccount , a fitting to the experimental data would  not converge . The calculated dispersion \nrelation with the derived valu e of 𝑘𝑚𝑎𝑥𝐴𝑚𝑝 , assuming 𝐻𝑎=𝐻𝑢=0 is depicted with blue \ndashed line in Fig. 3 (a). Omission of anisotropy fields in magnetization dynamic \nmeasurements  may therefore lead to the significant misinterpretation of  experimental results  \nfor YIG thin films.  \nTypical values of cubic magnetocrystalline anisotropy field 𝐻𝑎 range from  −18 𝑂𝑒 \nto −64 𝑂𝑒 for PLD grown YIG films  [ 14, 15, 22 ], what indicates that resonance \nmeasurements as well as spin wave propagation  are govern ed by the out -of-plane uniaxial \nanisotropy.  For the film employed in our study , the 𝐻𝑢 value is of about  −600 𝑂𝑒 in \nagreement with previous reports  [ 14, 15, 22 ]. For any more complex architecture  of \nmagnonic  waveguides and circuits  it is likewise  imperative t o investigate the  in-plane \nanisotropy properties  [ 24 ]. As  can be seen from Eq.  1 one would  expect  a six-fold \nanisotropy in the plane of (111) -oriented single crystals , that is common  among  rare-earth \nsubstituted YIG garnet s and LPE -YIG films  [ 18, 25, 26, 27 ]. To examine this issue , we \nperformed VSM and angular resolved ferromagnetic resonance measurements. Hysteresis \nloops for all measured in -plane directions exhibit no substantial differences regarding  \ncoercive field ( ≈1.2 𝑂𝑒), saturation field  and saturation magnetization (Fig.  4(a)). The \nangular resolved resonance measurement s confirm  this result  and show  that the (111) YIG \nfilm is isotropic in the film plane  (Fig. 4(b)). The main reason for this behavior is the low  \nvalue of cubic anisotropy field which cause s the resonance frequency modulation  by a value 8 \n of the fraction of  MHz. Such  small differences do not surpass  the experimental error, nor \nwould they  significantly affect the coherent SW propagation.  It is expecte d that t he SW \npropagation characteristics, measured for any other crystallographic orientation, would \ntherefore remain unaltered.  \n \n \n \n \nFig. 4. (a) VSM hysteresis loops measured in the film plane for three different  crystallographic directions. \nThe magneti zation is normalized to the saturation magnetization 𝑀𝑠=120±19 𝑒𝑚𝑢/𝑐𝑚3. A paramagnetic \ncontribution of the GGG substrate was subtracted for each loop.  (b) Resonance frequency as a function \nof azimuthal angle 𝜙 taken at 𝐻=150 𝑂𝑒. The red li ne depicts the calculated values of resonance frequency \naccording to Eq.1 for 𝑘=0, 𝐻𝑎=−30 𝑂𝑒 and 𝐻𝑢=−600 𝑂𝑒. \n \n \n \n-50 -40 -30 -20 -10 0 10 20 30 40 50-1.0-0.50.00.51.0M / MS\nMagnetic Field [Oe] \n \n (a)\n1.41.61.8\n0306090\n120\n150\n180\n210\n240\n2703003301.4\n1.6\n1.8\n(110)(101)(211)\no\n(112)o\n(011)o\n(121)o\noo (211)o (101)oo (110)\no (121)\no (011) Resonance frequency [GHz]\nH = 150 Oeo (112)(b)9 \n Another me thod of extracting SW wavenumber  involves the analysis of the SW \ngroup velocity  𝑣𝑔. Following Ref. [ 21 ], 𝑣𝑔 can be determined from frequency difference ∆𝑓 \nbetween two oscillation maxima in S 21 signal according to the relation:  \n 𝑣𝑔=𝑑∆𝑓, (3) \nwhere 𝑑 is the distance between two CPW s. To determine ∆𝑓 we chose two neighboring \noscillation maxima of the highes t S21 signal  amplitude  as it is shown in Fig. 3 (b) and (c) . \nIn Fig. 5 the derived values of group velocity are shown as a function of magnetic \nfield. It is found that 𝑣𝑔 reaches the  value of 7.6 𝑘𝑚/𝑠 for the field of 1.3 𝑂𝑒 (preferable in  \nmagnonic  information processing devices  of high efficiency ) and 1.4 𝑘𝑚/𝑠 for the field  of \n285 𝑂𝑒. It should be highlighted that such big difference s in 𝑣𝑔 values can be further utilized \nto design  tunable , impulse -response delay  lines  as 𝑣𝑔 changes up to five times  with the \nmagnetic field.  At a distance of 150 𝜇𝑚 between CPWs  it would allow to achieve 20 to \n110 𝑛𝑠 delay times of an impulse.  \n \n \nFig. 5. Spin wave group velocity  as a function of the external magnetic f ield. The red line represents a fit \naccording to Eq.  4. \n \n \nWith the red line in Fig. 5 a fitting is depicted according to:  \n 𝑣𝑔=2𝜋𝜕𝑓\n𝜕𝑘=𝜇𝐵\nℏ𝑔2𝜋𝑀𝑠𝑡(−1\n2𝐻𝑎−𝐻𝑢+4𝜋𝑀𝑠−4𝜋𝑀𝑠𝑡𝑘)\n2√(𝐻+2𝜋𝑀𝑠𝑡𝑘)(𝐻−1\n2𝐻𝑎−𝐻𝑢+4𝜋𝑀𝑠−2𝜋𝑀𝑠𝑡𝑘). (4) \nThe main advantage of extracting SW wavenumber  from 𝑣𝑔(𝐻) dependence is  that it does \nnot require additional measurement of 𝑀𝑠 which is often notably influenced by an error in the \nestimated  film volu me. Since  the saturation magnetization 𝑀𝑠 can be treated as a fitting \n-300 -200 -100 0 100 200 30012345678vg [km/s]\nH [Oe]10 \n parameter in Eq.  4, the derivation  of SW wavenumber  involves only S 11, S21 and thickness \nmeasurement s. The determined values of 𝑘𝑚𝑎𝑥𝐴𝑚𝑝=1690±53 𝑐𝑚−1 and 𝑀𝑠=116±\n2 𝑒𝑚𝑢/𝑐𝑚3 remain  in a good agreement with that obtained above  - directly derived from \ndispersion relation  (𝑘𝑚𝑎𝑥𝐴𝑚𝑝=1980±102 𝑐𝑚−1, 𝑀𝑠=120±19 𝑒𝑚𝑢/𝑐𝑚3). \nAs can be seen from Fig ure 5, SW group velocity attains the maximum va lue as the \nmagnetic field approaches 𝐻=0. The maximum value of 𝑣𝑔 is given by:   \n 𝑣𝑔(𝐻=0)≅𝜇𝐵\nℏ𝑔√𝜋𝑀𝑠𝑡\n2𝑘(−1\n2𝐻𝑎−𝐻𝑢+4𝜋𝑀𝑠[1−𝑡𝑘]). (5) \nThe zero -field region may therefore become the subject of interest  for magnonic applications. \nMoreover, Eq. 5 shows that the maximum value of 𝑣𝑔 depends on the anisotropy fields.  PLD -\ngrown YIG films possessing  a high anisotropy would allow faster information processing in \nSW circuits than LPE films for which  the value of  −1\n2𝐻𝑎−𝐻𝑢 is smaller (as it was pointed \nout above).  \nTo confront our experimental results with the expected, theoretical value of  \n𝑘𝑚𝑎𝑥𝐴𝑚𝑝 , we performed electromagnetic simulations in Comsol Multiphysics . Here,  CPW \nwas modeled accordin g to the geometry  of the performed CPW  (Fig.  2), assuming lossless \nconductor metallization, relative permittivity of the substrate 𝜀𝑟=12 and 50 𝛺 port \nimpedance. From the simulated in-plane  distribution of the dynamic magnetic field ℎ𝑥 (inset \nof Fig. 6) an excitation spectra of CPW was obtained  using d iscrete Fourier transformation  of \nℎ𝑥(𝑥). The highest excitation strength is observed for 𝑘𝑚𝑎𝑥𝐴𝑚𝑝=1838 𝑐𝑚−1, which \ncorrespond s well to the experimentally  obtained values  within 7% accuracy . The second \nobserved maxima is at 𝑘2=6770 𝑐𝑚−1. However,  as its amplitude is 2 0 times lower with \nrespect to  the amplitude of 𝑘𝑚𝑎𝑥𝐴𝑚𝑝  it is not observed in  the measured  S21 signal . \n \n 11 \n  \nFig. 6. Excitation spectrum  of the CPW with 9.8 𝜇𝑚 wide signal line s and 4 𝜇𝑚 gaps. The inset shows  in-plane \ncomponent of the dynamic  magnetic field excited by the CPW.  \n \n \nTo extend  our study , we performed a series of further simulations for the CPW \ndimensions , which are  achievable with electron - and photolithography.  We assumed equal \nwidths of signal and ground lines (𝑊) as well as equal widths of gaps between them  (𝐺). The \nresults are presented in Fig.  7. It is found  that for the width s 𝑊 ranging from 300 𝑛𝑚 to \n40 𝜇𝑚, the wavenumber 𝑘𝑚𝑎𝑥𝐴𝑚𝑝  vary between 70000 𝑐𝑚−1 and 250 𝑐𝑚−1, respectively , \nrevealing the CPW wavenumber probing limits. We also note that the gap width significantly \naffects  𝑘𝑚𝑎𝑥𝐴𝑚𝑝 . In order to accurately extrapolate its contribution to 𝑘𝑚𝑎𝑥𝐴𝑚𝑝 , we \ndeveloped empirical formula which incorporates width 𝐺: \n 𝑘𝑚𝑎𝑥𝐴𝑚𝑝=2.27\n𝑊+0.6 𝐺. (6) \nThe fittings , according to  the Eq. 6, are depicted in Fig.  7 with solid lines. We f ound that Eq. \n6 is valid  for gap  width 0.1 𝑊<𝐺<2 𝑊. For 𝐺=0.74 𝑊 this formula is equivalent to the \none previously proposed in Ref.  [ 10 ] (𝑘𝑚𝑎𝑥𝐴𝑚𝑝≈𝜋/2𝑊). \n \n0 4000 8000 120000.00.20.40.60.81.0\n-50 -25 0 25 50-2502550hx [Oe]\nx [m]\n-50 -25 0 25 50-2502550hx [Oe]\nx [m]\nk2= 6770 cm-1Amplitude [a.u.]\nk [cm-1]\n-50 -25 0 25 50-2502550hx [Oe]\nx [m]kmaxAmp= 1838 cm-112 \n  \nFig. 7. Wavenumber of the highest amplitude as a function of CPW signal line width. The solid lines represent \na fit according to Eq.  6. \n \nTo conclude, we  report ed on long-range spin wave propagation in the 82 𝑛𝑚 thick \nYIG film  over the distance as large as 150 𝜇𝑚. In order to precisely determine excited \nwavenumber by the coplanar antenna, it is essential to take in to account anisotropy fields \npresent in YIG films. We show ed that anisotropy significantly affect s SW propagation \ncharacteristics, namely it causes an increase in SW frequency as well as in SW group \nvelocity. The main contribution comes from  the out -of-plane uniaxial anisotropy field. T he \ncubic anisotropy field is neglig ibly small  in the YIG (111) film  and it does not affect \nmagnetization dynamics  in the film plane.  We explain ed that the wavenumber determination \nfrom group velocity vs . magnetic field depend ence requires only two types of measurement , \nthat is  broadband SW spectroscopy and the measurement of film thickness.  \n \n \nAcknowledgements  \nThis work was carried out within the Project NANOSPIN  PSPB -045/2010 supported by a \ngrant from Switzerland through the S wiss Contribution to the enlarged European Union . J. \nRychły and J. Dubowik would like to acknowledge support from the European Union’s \nHorizon 2020 MSCA -RISE -2014: Marie Skłodowska -Curie Research and Innovation Staff  \nExchange (RISE) Grant Agreement No. 644 348 (MagIC).  The authors would like to thank  \nProfessor Maciej Krawczyk  for thoughtful suggestions . We also acknowledge valuable \ncomments from Dr. Piotr Graczyk and Paweł Gruszecki .  \n1 10100010000100000\n G = W / 10\n G = W / 4\n G = W / 2\n G = W\n G = 2 WkmaxAmp [1/cm]\nW [m]13 \n References  \n \n[1] Jamali M , Kwon J H, Seo S -M, Lee K -J and Yang H 2016 Spin wave nonreciprocity \nfor logic device applications. Sci. Rep . 3, 3160  \n[2] Chumak A V, Serga A A and Hillebrands B 2014 Magnon transistor for all -magnon \ndata processing. Nat. Comun . 5, 4700  \n[3] Vogt K, Fradin F Y, Pearson J E, Sebastian T, Bader S D, Hille brands B, Hoffmann A \nSchultheiss H 2014 Realization of a spin -wave multiplexer. Nat. Comun . 5, 3727  \n[4] Gertz F, Kozhevnikov A V, Filimonov Y A, Nikonov D E and Khitun A 2015 \nMagnonic Holographic Memory: From Proposal to Device. IEEE J. Explor. Solid -State \nComputat. Devices Circuits  1, 67-75 \n[5] Serga A A, Chumak A V and Hillebrands B 2010 YIG magnonics J. Phys. 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Lett . 110 092408 \n[25] Onbasli M C , Beran  L, Zahradník  M, Kučera  M, Antoš  R, Mistrík  J, Dionne  G F, Veis \nM and Ross  C A 2016 Optical and magneto -optical behavior of Cerium Yttrium Iron \nGarnet thin films at wavelengths of 200 –1770 nm Sci. Rep . 6 23640  15 \n [26] Bonda A, Uba S  and Uba L 2012 Ultrafast Magneto -Optical and Magnetization -\nInduced Second Harmonic Generation Techniques for Studies of Magnetic \nNanostructures. Acta Phys. Polon. A  121, 1225  \n[27] Dubs C , Surzhenko  O, Linke  R, Danilewsky  A, Bruckner  U, Dellith  J 2016 Sub -\nmicrometer yttrium iron garnet LPE resonance losses. arXiv:1608.08043v1   \n[28] Perzlmaier K, Woltersdorf G and Back C H 2008 Observation of the propagation and \ninterference of spin waves in ferromagnetic thin films Phys. Rev. B  77, 054425  "    },    {        "title": "2103.03885v1.Universal_spin_wave_damping_in_magnetic_Weyl_semimetals.pdf",        "content": "arXiv:2103.03885v1  [cond-mat.str-el]  5 Mar 2021Universal spin wave damping in magnetic Weyl semimetals\nPredrag Nikoli´ c1,2\n1Department of Physics and Astronomy, George Mason Universi ty, Fairfax, VA 22030, USA and\n2Institute for Quantum Matter at Johns Hopkins University, B altimore, MD 21218, USA\n(Dated: March 9, 2021)\nWe analyze the decay of spin waves into Stoner excitations in magnetic Weyl semimetals. The\nlifetime of a mode is found to have a universal dependence on i ts frequency and momentum, and\non a few parameters that characterize the relativistic Weyl spectrum. At the same time, Gilbert\ndamping by Weyl electrons is absent. The decay rate of spin wa ves is calculated perturbatively\nusing the s-d model of itinerant Weyl or Dirac electrons coup led to local moments. We show that\nmany details of the Weyl spectrum, such as the momentum-spac e locations, dispersions and sizes\nof the Weyl Fermi pockets, can be deduced indirectly by probi ng the spin waves of local moments\nusing inelastic neutron scattering.\nI. INTRODUCTION\nWeyl semimetals are condensed matter realizations\nof massless fermions with a chiral relativistic three-\ndimensional spectrum1–3. Topologically protected gap-\nless Fermi “arc” states on the system boundaries, and\nunconventional transport properties such as the intrinsic\nanomalous Hall effect, set Weyl semimetals apart from\nother weakly interacting conductors. One way to ob-\ntain a Weyl spectrum involves breaking the time-reversal\nsymmetry in a material that has Dirac quasiparticles.\nThe presence of magnetization, for example, will remove\nthe spin degeneracy of a Dirac node by splitting it into\na dipole of opposite-chirality Weyl nodes in momentum\nspace. Magnetism then becomes intimately related to\nthe presence of Weyl electrons. Alternatively, Weyl spec-\ntrum of itinerant electrons can be created by a broken\ninversion symmetry, e.g. due to the crystal structure,\nand then coupled to magnetism if the material possesses\nadditional local moments or undergoes a spin density\nwave instability. Some of these theoretical scenarios are\nslowly finding their actualization in experimentally stud-\nied magnetic Weyl semimetals4–13.\nHere we analyze an important imprint of Weyl elec-\ntrons on the magnetic dynamics – the damping of spin\nwaves via particle-hole (Stoner) excitations. This basic\ninteractioneffect revealsthe definingfeatures ofthe Weyl\nspectrum, relativity and chirality. We will show that the\nlifetime of spin waves exhibits a universal dependence on\nthe modefrequency andmomentum whichcanbe used to\nextract detailed properties of the underlying Weyl elec-\ntrons. By measuring the mode lifetime throughout the\nfirst Brillouin zone, it is possible to discern the locations\nof the Weyl nodes in momentum space, their relative chi-\nralities, slope of the energy versusmomentum dispersion,\nand the size of the Fermi pockets on the Weyl nodes.\nThe spin wave lifetime is obtained from the width of the\nscattering intensity peaks in inelastic neutron scattering\nexperiments, provided that a sufficient energy resolution\nis available and other sources of decoherence (thermal\nbroadening, disorder, phonons) do not mask the elec-\ntronic source.\nEven though neutron scattering is a powerful Green’sfunction probe, its ability to detect fermionic quasipar-\nticles is normally ruined by the incoherent continuum\nof excitations that can absorb an angular momentum\nquantum. Interestingly, this problem is reduced in Weyl\nsemimetals14, and fortunately it is also possible to in-\ndirectly characterize the quasiparticles via collective ex-\ncitations. The latter has been achieved in the neutron\nstudies of samarium hexaboride (SmB 6)15,16, where the\nmeasured dispersion of a “spin exciton” has revealed a\nnon-trivial topology of the underlying electronic quasi-\nparticles. An energygap protects the exciton’s coherence\nin SmB 6, but the gaplessquasiparticlesin Weyl semimet-\nals will generally induce ubiquitous damping of collective\nmodes. Such a damping can in fact reveal the existence\nand properties of chiral fermionic quasiparticles. The\nWeyl electron characterization through damping could\npotentially overcome various issues that plague other ap-\nproaches, such as correlation effects in the case of band-\nstructure calculations, limited resolution in the case of\nARPES, sensitivity to conventional bands (that coexist\nwith Weyl nodes) in transport measurements, etc.\nClosely related to the physics we pursue here is the ex-\ntensively studied damping in metallic ferromagnets17–29.\nStoner excitations provide a mechanism for the decay\nof spin waves, and also typically give rise to Gilbert\ndamping30– the dissipated precession of uniform mag-\nnetization in an external magnetic field. Many works\nhave been devoted to the calculation of Gilbert damp-\ning since it is possible to measure it by ferromagnetic\nresonance31,32and time-resolved magneto-optical Kerr\neffect33,34. A careful consideration of the relativistic\nelectron dynamics has revealed that Gilbert damping\noriginates in the spin-orbit coupling and depends on\nthe electrons’ mass25. In the case of massless Weyl\nelectrons, we show here that Gilbert damping is ab-\nsent. However, spin waves unavoidably decay via Stoner\nexcitations35–39,41,42, and their damping features “non-\nreciprocity” – different polarization modes that carry\nthe same momentum have different damping rates. This\naccompanies the non-dissipative aspects of chiral spin-\nmomentumlocking44,45. Spinwave“non-reciprocity”has\nbeen anticipated in spiral magnets46, magnetic interfaces\nwith a Dzyaloshinskii-Moriya interaction derived from2\nthe Rashba spin-orbit coupling43,47–52, and observed in\nseveral experiments53–58. In the context of magnetic\nWeyl semimetals, initial theoretical studies have been fo-\ncused on the domain wall dynamics59,60.\nThe rest of this paper is organized as follows. Section\nII presentsthe approachand the main results ofthe anal-\nysis, focusing on the observable physical characteristics\nof the spin wave damping by Weyl electrons. Section III\nis devoted to the technical development of the damping\ntheory. It contains separate derivations of the dissipative\ntermsintheeffectivespinaction(IIIA),spinwavedamp-\ning (IIIB), and Gilbert damping from the semiclassical\nfield equation (IIIC). The last section IV summarizesthe\nconclusions and discusses the broader applicability and\nlimitations of the damping theory.\nII. SUMMARY OF THE RESULTS\nIn this paper, we work with the s-d model of Weyl\nelectrons coupled to local moments. We perturbatively\ncalculate the dissipative non-Hermitian parts of the mo-\nments’effectiveaction,whichdeterminetherate γofspin\nwavedamping. γalsodependsonthe magneticorderand\nthe wave’s propagation direction relative to the magneti-\nzation, but it is always controlled by the components of\nthe universal damping rate tensor given by\nγab\nmn(q) =a3J2\nKΩ2\n128πSv3fab\nmn/parenleftbigg|Ω|\nvq,|Ω|\n2|µ|;sign(µ,Ω)/parenrightbigg\n(1)\nfor ferromagnetic local moments of spin magnitude S.\nThe upper indices a,b∈ {x,y,z}refer to spin projec-\ntions. The universal scaling functions fab\nmnare dimen-\nsionless, the factor a3is the unit-cell volume of the local\nmoment’s lattice, JKis the Kondo or Hund coupling en-\nergy scale, vandµare the Fermi velocity and Fermi\nenergy of the Weyl electrons respectively, and Ω is the\nreal spin wave frequency (we use the units /planckover2pi1= 1). The\nspin wave momentum qin this expression is measured\nrelative to the difference ∆ Q=Qm−Qnbetween the\nwavevectors Qm,Qnof any two Weyl nodes in the first\nBrillouin zone. Coherent collective excitations that span\nthe entire first Brillouin zone can be used to separately\naddress many pairs of Weyl nodes – by tuning the total\nwavevector ∆ Q+qto the vicinity of ∆ Q. Representa-\ntive functions fab\nmnfor the Weyl nodes with finite Fermi\nsurfaces are plotted in Figures 1 and 2\nWemakeanalyticalprogressandgainvaluablephysical\ninsight through several idealizations: all Weyl nodes are\nassumed to be identical, sphericallysymmetric and living\nat the same node energy. Their chiralities χm=±1 and\nlocations Qmare arbitrary (as long as the total chirality\nin the first Brillouin zone vanishes). Under these condi-\ntions, only three tensor components of γabare finite and\nindependent, γ/bardbl/bardbl,γ⊥⊥andγ⊥⊥′. Here and throughout\nthe paper ∝bardblindicates the spin direction parallel to the\nmode’s wavevector q, and⊥,⊥′are the spin directions(a)\n(b)\nFIG. 1. The plots of functions (a) f⊥⊥and (b) f/bardbl/bardblfor the\ndamping rates of transverse and longitudinal spin waves re-\nspectively, contributed by the Fermi surfaces on a particul ar\npair of Weyl nodes. Solid red lines are for the same-chiralit y\nnodes, and the dashed blue lines are for the opposite-chiral ity\nnodes.|Ω|= 1.4|µ|was assumed in this example.\nFIG. 2. The plots of selected universal functions fabfeatured\nin the damping rate γ∼Ω2f(vq/|Ω|;xµ). The functions\nare parametrized by xµ= 2|µ/Ω|, with finer dashes corre-\nsponding to larger Weyl Fermi pockets (solid lines refer to\nthe Fermi level that crosses the Weyl nodes). Shown func-\ntionsincludetransverse( ⊥⊥)andchiral( ⊥⊥′)dampingchan-\nnels shaped by electron scattering between equal-chiralit y (+)\nandopposite-chirality ( −)Weyl nodes. Longitudinal channels\n(∝bardbl∝bardbl) are similar to the shown transverse channels, compare\nwith Fig.1.3\n(a)\n (b)\nFIG. 3. Examples of the damping rate map in momentum\nspace for (a) µ∝negationslash= 0 and (b) µ= 0 (with and without a Fermi\nsurface of Weylelectrons respectively). Brightness depic ts the\nrateγ(q) of spin wave damping, and the red crosshair shows\nthe reference ∆ Qfor the local wavevector q= 0. These are\nqz= 0 slices through the full 3D map. Observing patterns\nof this kind in the full Brillouin zone scan will indicate the\nWeyl-electron origin of damping and reveal the complete set\nof ∆Q=Qm−Qnwavevectors from which the individual\nnode wavevectors Qmcan be deduced (assuming, for exam-\nple,/summationtext\nmQm= 0). The bright outer ring, which shrinks and\ncloses when 2 |µ|<|Ω|, originates in the inter-band electron\nscattering and gains strength from the rapidly growing Weyl\nelectron density of states. Note that various details in the se\nmaps, such as the anisotropy and ring sizes, will generally\ndepend on the concrete spin-wave dispersion Ω( q+∆Q), po-\nlarization, type and orientation of magnetic order, as well as\nthe chiralities and symmetries of the Weyl nodes.\nwhich areperpendicular to qand eachother. The full ex-\npression for damping rates is presented in Section IIIB;\nin Weyl ferromagnets, it becomes\nγmn=γ⊥⊥\nmn±γ⊥⊥′\nmn (2)\nfor the two polarizations of spin waves propagating along\nthe magnetization direction.\nThe essential utility of the universal damping comes\nfrom its qualitative features that reflect the relativistic\nnature of Weyl electrons. If the Fermi energy µlies away\nfrom the energy of the Weyl nodes, Fermi surfaces will\nform. Then, the spin wave damping rate is expected to\nexhibitasetofminimumsandmaximumsasafunctionof\nthefrequencyΩandmomentum q. Thelocationsofthese\nextremums depend on the parameters that characterize\nthe Weyl nodes: Fermi velocity v, chemical potential µ\nand even their relative chiralities χmχn=±1. Fig.3\ndemonstrates how the locations Qmof Weyl nodes can\nbe extracted from the full Brillouin zone map of the spin\nwave’s damping rate γ(q). Once the wavevectors Qmare\nknown, Fig.4 illustrates how the observation of enough\nextremums enables indirect measurements of the Weyl\nelectron spectra on multiple Weyl nodes. The presence\nof Weyl Fermi pockets also introduces spin-momentum\nlocking into the damping rates ( γ⊥⊥′\nmn∝ne}ationslash= 0), but only on\nthe pairs of Weyl nodes with opposite chiralities. As a\nvqΩ\n02\u00012 \u0000\u0002\n\u0003Ω=vq\nΩ=vq-2\u0004\nFIG. 4. A density plot of the collective mode damping rate\nγ(q,Ω) induced by Weyl electrons. Thin solid green lines in-\ndicateγ= 0, and the thin dashed green line indicates the\nlocal maximum of γ. The thick dashed yellow line represents\nthe dispersion Ω( q+ ∆Q) of a hypothetical spin-wave exci-\ntation (note that the origin of the plot corresponds to the\nmomentum difference ∆ Qof two Weyl nodes in the first Bril-\nlouin zone). The spin-wave’s damping rate will exhibit loca l\nminimums and maximums at the shown red points, which are\ncharacteristic for the relativistic spectrum of Weyl elect rons.\nResolving two of these points is enough for the determinatio n\nof the Weyl Fermi velocity vand the chemical potential µof\nthe Weyl nodes addressed via ∆ Q. Resolving three points al-\nlows an independent verification that Weyl nodes are indeed\nresponsible for the damping. The two-parameter scaling of\nthe damping rate (1) across a range of energies is the most\ngeneral signature of Weyl electrons, and can be used to verif y\nthe Weyl-electron origin of damping even if the visible spin\nwave dispersion does not cross any of the shown characterist ic\npoints.\nconsequence, the two spin wave modes that carry oppo-\nsite spin currents at the same wavevector qhavedifferent\npeak widths in inelastic neutron scattering.\nThe above qualitative features of damping disappear if\nthe Fermi energy sits exactly at the Weyl nodes. How-\never, the damping rate then becomes a universalfunction\nofa single parameter |Ω|/vq. This kind of scalingis a sig-\nnature of the relativistic Weyl electrons – it is caused by\n“inter-band” transitions in which an electron below the\nWeyl nodeisexcited toastate abovethe Weyl node. The\nplots of universal functions fab\nmnthat appear in Eq. 1 at\nµ= 0 are shown in Fig.2.\nThe magnitude of the damping rate depends\non the Kondo/Hund scale JKwhich may not be\nknown. However, the spin wave damping caused\nby Weyl electrons is always related to the effec-\ntive strength Jof the Weyl-electron-induced Ruder-\nman–Kittel–Kasuya–Yosida(RKKY)interactionsamong4\nqqJKJK\nFIG. 5. The Feynman diagram for two-spin interactions.\nThick external lines represent local moment fields and thin\nlines represent Weyl electron propagators. The two-spin co u-\nplings include Heisenberg, Kitaev and Dzyaloshinskii-Mor iya\ninteractions, but the Weyl-electron origin of spin dynamic s\nalso creates a dissipation channel in which spin waves decay\ninto electron-hole pairs.\nthe local moments45:\nγ\nJ∼1\n(aΛ)3/parenleftBigq\nΛ/parenrightBig2\n×/parenleftbiggΩ\nvq/parenrightbigg2\n, J∼vΛ/parenleftbigga3Λ2JK\nv/parenrightbigg2\n.\n(3)\nHere, Λisthemomentumcut-offforthelinearWeylspec-\ntrum,|q|<Λ. SinceaΛ<1 and the characteristic fea-\ntures of the universal damping appear near |Ω| ∼vq, the\ndamping rates are generally comparable to the energy\nscaleJof the induced RKKY interactions. For example,\nthe RKKY energy scale in the magnetic Weyl semimetal\nNdAlSi13can be crudely estimated as J∼1 meV. Even\nif the damping rate is more than an order of magnitude\nbelow this value of J, it should be detectable with high\nresolutionneutron instruments (a spin echospectrometer\ncan achieve energy resolution below 10 µeV).\nIII. DISSIPATION BY WEYL ELECTRONS\nHere we calculate the Gaussian dissipative part of the\neffective action for local moments which arises due to\ntheir coupling to itinerant Weyl electrons. The non-\ndissipative part of this action, computed in Ref.45, cap-\ntures the induced RKKY interactions among the lo-\ncal moments: Heisenberg, Kitaev and Dzyaloshinskii-\nMoriya. All Gaussian terms δnaΓabδnbof the action ob-\ntain from a single two-point Feynman diagram which in-\nvolves momentum integration of a singular function; the\nprincipal part of this integral yields the interactions, and\nthe contribution of its pole singularity amounts to dissi-\npation. We will focus only on the latter, following the\nprocedure from Ref.45.\nThe essential dynamics of local moments ˆnicoupled to\nconduction electrons ψiis given by the Hamiltonian:\nH0=Hn+/summationdisplay\nkǫkψ†\nkψk+JK/summationdisplay\niˆniψ†\niσψi.(4)\nBoth the local moments and electrons live on a lattice\nwhose sites are labeled by i, but we will immediately\ntake the continuum limit. The basic two-spin correla-\ntions∝an}bracketle{tˆna\niˆnb\nj∝an}bracketri}htare contained in the second-order Feynman\ndiagram shown in Fig.5:\nΓab\nmn(q) =iJ2\nK\n2/integraldisplayd4k\n(2π)4tr/bracketleftBig\nGm/parenleftBig\nk−q\n2/parenrightBig\nσaGn/parenleftBig\nk+q\n2/parenrightBig\nσb/bracketrightBig\n(5)The Weyl electron Green’s functions\nGn(ω,k) =/bracketleftBig\nω−Hn(k)+isign(ǫn(k))0+/bracketrightBig−1\n(6)\nare treated as spinor matrices and refer to the low-energy\nelectronic states near any Weyl node nwhose wavevector\nin the first Brillouin zone is Qn; the wavevector kis a\n“small” displacement |k|<Λ fromQn, where Λ is the\nmomentum cut-off for the linear Weyl dispersion. These\nlow-energy electrons are described by the Hamiltonian\nHn(k+Qn) =vχnσk−µ , (7)\nwhereµis the chemical potential that determines the\nWeyl Fermi pocket character and size, vis the Fermi\nvelocity, and χn=±1 is the Weyl node chirality. We\nassume for simplicity that all Weyl nodes are spherically\nsymmetric, share the same node energy, chemical poten-\ntial and Fermi velocity, but have arbitrary wavevectors\nQnand chiralities χn=±1 (as long as the chiralities of\nall nodes in the first Brillouin zone add up to zero). By\nthis construction, the expression (5) is associated with a\npairm,nof Weyl nodes, and qis a “small” wavevector\nmeasured relative to Qm−Qn.\nWe will carry out all calculations with the formal as-\nsumption that no external or effective magnetic field is\nexerted on electrons. Realistically, however, we are inter-\nested in magnetic Weyl semimetals whose local moments\nmay carry a non-zero net magnetization ˆn0that presents\nitself as an effective magnetic field B=−JKˆn0to elec-\ntrons. This is of no concern because the correction of the\nspectrum (7) amounts merely to a shift of the wavevector\nk→k−B/vχn. Hence, an effective magnetic field only\nalters the locations Qnof the Weyl nodes in momentum\nspace, which are arbitrary in our formalism.\nThe full effective action matrix Γ for local moments\ntakes contributions from all Weyl node pairs:\nΓ(Q,Ω) =/summationdisplay\nm,nΓmn(Q−Qm+Qn,Ω).(8)\nIn this sense, it is possible to experimentally address a\nparticular pair of Weyl nodes, or a set of pairs, by prob-\ningthe momentumspaceinthe vicinityof Q∼Qm−Qn.\nThe dissipative part of Γ mnwill contain information\nabout the addressed Weyl nodes.\nA. Calculation of the dissipative terms in the\neffective spin Lagrangian\nThe calculationof(5) is lengthy, sowe will onlyoutline\nits key steps. The trace has been evaluated before45, and\nthe frequency integration yields:5\nΓab\nmn(q) =−J2\nK\n2/integraldisplayd3k\n(2π)3/bracketleftBigg\nXab(Ω,q;vχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle+Ω\n2−µ,k)\n2vχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/productdisplay\ns=±1θ/parenleftbig\nµ−vχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/parenrightbig\nΩ+vχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle−vsχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle+i0+F(sχn,χm)\n−Xab(Ω,q;−vχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle+Ω\n2−µ,k)\n2vχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/productdisplay\ns=±1θ/parenleftbig\nµ+vχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/parenrightbig\nΩ−vχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle−vsχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle+i0+F(sχn,−χm)\n+Xab(Ω,q;vχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle−Ω\n2−µ,k)\n2vχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/productdisplay\ns=±1θ/parenleftbig\nµ−vχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/parenrightbig\nvχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle−Ω−vsχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle−i0+F(χn,sχm)\n−Xab(Ω,q;−vχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle−Ω\n2−µ,k)\n2vχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/productdisplay\ns=±1θ/parenleftbig\nµ+vχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/parenrightbig\n−vχn/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle−Ω−vsχm/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle−i0+F(−χn,sχm)\nHere,θ(x) is the step function, and two more functions,\nXab(Ω,q;ω,k) andF(s+,s−) are introduced to simplify\nnotation. The function Xab(Ω,q;ω,k) obtains from the\nnumerator of the trace in (5). Introducing the Kronecker\nsymbolδaband the Levi-Civita symbol ǫabc, we have:\nXab(Ω,q;ω,k) =/bracketleftbigg\n(ω+µ)2−Ω2\n4/bracketrightbigg\nδab\n+v2χmχn/bracketleftbigg\n2/parenleftbigg\nkakb−qaqb\n4/parenrightbigg\n−δab/parenleftbigg\nkckc−qcqc\n4/parenrightbigg/bracketrightbigg\n+ivǫabc/bracketleftbigg\nχm/parenleftbigg\nω+Ω\n2+µ/parenrightbigg/parenleftbigg\nkc−qc\n2/parenrightbigg\n−χn/parenleftbigg\nω−Ω\n2+µ/parenrightbigg/parenleftbigg\nkc+qc\n2/parenrightbigg/bracketrightbigg\n. (9)\nThe function F(s+,s−) withs+,s−=±1 keeps track of\nthe infinitesimal imaginary terms in the denominators of\nGreen’s functions:\nF(s+,s−) = sign/parenleftBig\nvs+/vextendsingle/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/vextendsingle−µ/parenrightBig\n−sign/parenleftBig\nvs−/vextendsingle/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/vextendsingle/parenrightBig\n=θ/parenleftbigg\n|qk|−/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftBigµ\nv/parenrightBig2\n−k2−q2\n4/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenrightbigg\n×/bracketleftbigg/parenleftbigg\nsign(µ)+s++s−\n2/parenrightbigg\nsign(qk)+s+−s−\n2/bracketrightbigg\n+(s+−s−)θ/parenleftbigg\nk2+q2\n4−|qk|−/parenleftBigµ\nv/parenrightBig2/parenrightbigg\n.(10)\nAt this point, we use the relationship\n1\nx±i0+=P1\nx∓iπδ(x) (11)\nto isolate the dissipative processes that curb the x→0\nresonances. Dropping all terms that involve the principal\npartP, we get:\n/tildewideΓab\nmn(q) =iπJ2\nK\n8v2/summationdisplay\nsm,snsmsn/integraldisplayd3k\n(2π)3F′(snχn,smχm)\nχmχn/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle\n×Xab/parenleftbigg\nΩ,q;vsmχm/vextendsingle/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/vextendsingle+Ω\n2−µ,k/parenrightbigg\n(12)\n×δ/parenleftBig\nΩ+vsmχm/vextendsingle/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/vextendsingle−vsnχn/vextendsingle/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/vextendsingle/parenrightBig\n×/bracketleftBig\nθ/parenleftBig\nµ−vsmχm/vextendsingle/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/vextendsingle/parenrightBig\n−θ/parenleftBig\nµ−vsnχn/vextendsingle/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/vextendsingle/parenrightBig/bracketrightBigWe introduced F′= sign(F)(1−δF,0), and the sum goes\noversm,sn=±1. All chirality factors χm,χn=±1\nthat appear outside of Xabare clearly eliminated by the\nsummation over sm,sn, so it will be convenient do define\ns−=smχm=±1 ands+=snχn=±1. The Dirac\nδ-function in (12) imposes:\ns+/vextendsingle/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/vextendsingle−s−/vextendsingle/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/vextendsingle=Ω\nv. (13)\nThis pins the magnitude of the wavevector kto\nk=|Ω|\n2v/radicalBigg\nΩ2−v2q2\nΩ2−v2q2cos2θ, (14)\nassuming qk=qkcosθ, and further requires satisfying\none of these two conditions:\n|Ω|>vq ∧s±=±sign(Ω)\n|Ω|<vq|cosθ| ∧s+=s−= sign(Ωcos θ)\nThe wavevector k= (k,θ,φ) integration in (12) is now\nconveniently performed in the spherical coordinate sys-\ntemreferencedtotheexternalwavevector q. Theintegral\noverk=|k|is immediately solved due to the Dirac δ-\nfunction and we merely need to replace the occurrences\nof|k|with (14). The integral over φaffects only the\nquantities (9) leading to/integraltext\ndφXab= 2πv2/tildewideXabwith the\nfollowing non-zero components:\n/tildewideX/bardbl/bardbl=s+s−/vextendsingle/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/vextendsingle (15)\n+χmχn/bracketleftbigg\nk2(2cos2θ−1)−q2\n4/bracketrightbigg\n/tildewideX⊥⊥=s+s−/vextendsingle/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/vextendsingle\n+χmχn/parenleftbigg\n−k2cos2θ+q2\n4/parenrightbigg\n/tildewideX⊥⊥′=iǫ/bardbl⊥⊥′/bracketleftbigg\n−/parenleftBig\nχms+/vextendsingle/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/vextendsingle+χns−/vextendsingle/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/vextendsingle/parenrightBigq\n2\n+/parenleftBig\nχms+/vextendsingle/vextendsingle/vextendsinglek+q\n2/vextendsingle/vextendsingle/vextendsingle−χns−/vextendsingle/vextendsingle/vextendsinglek−q\n2/vextendsingle/vextendsingle/vextendsingle/parenrightBig\nkcosθ/bracketrightbigg\n.\nHere and onward, the upper spin indices denote direc-\ntions∝bardblparallel to q, and two mutually perpendicular6\ndirections ⊥,⊥′which are also perpendicular to q. Note\nthatǫ/bardbl⊥⊥′implements a chiral “right-hand-rule” rela-\ntionship between the three spin directions. The integral\noverθis finite and conveniently evaluated numerically.\nAt the end, we arrive at:\n/tildewideΓab\nmn(q) =iJ2\nKΩ2\n128πv3fab\nmn/parenleftbigg|Ω|\nvq,|Ω|\n2|µ|;sign(µ,Ω)/parenrightbigg\n(16)\nwhere the dimensionless functions f=α+βhave con-\ntributions from intra-band αand inter-band βelectronscattering. Note that the inter-band processes require\ntransferringan electron between the two states whoseen-\nergies have opposite signs, and thus can occur only when\n|Ω|>2|µ|. Defining\nλ=vq\n|Ω|, x=2|µ|\n|Ω|, κ=/radicalBigg\n1−λ2\n1−λ2ξ2(17)\nwith|ξ|=|cosθ|, we have:\nα⊥⊥\nmn=1/integraldisplay\n0dξθ/parenleftbig\n2κλξ−|x2−κ2−λ2|/parenrightbig\nκ2(18)\n×/bracketleftBigg/parenleftBigg\n1−χmχn−κ2ξ2+λ2\n/radicalbig\n(κ2+λ2)2−(2κλξ)2/parenrightBigg\nθ(1−λ)+/parenleftBigg\n1+χmχn−κ2ξ2+λ2\n/radicalbig\n(κ2+λ2)2−(2κλξ)2/parenrightBigg\nθ(λξ−1)/bracketrightBigg\nα/bardbl/bardbl\nmn=1/integraldisplay\n0dξθ/parenleftbig\n2κλξ−|x2−κ2−λ2|/parenrightbig\nκ2\n×/bracketleftBigg/parenleftBigg\n1−χmχnκ2(2ξ2−1)−λ2\n/radicalbig\n(κ2+λ2)2−(2κλξ)2/parenrightBigg\nθ(1−λ)+/parenleftBigg\n1+χmχnκ2(2ξ2−1)−λ2\n/radicalbig\n(κ2+λ2)2−(2κλξ)2/parenrightBigg\nθ(λξ−1)/bracketrightBigg\nα⊥⊥′\nmn=−iǫ/bardbl⊥⊥′1/integraldisplay\n0dξθ/parenleftbig\n2κλξ−|x2−κ2−λ2|/parenrightbig\nκ2\n×/summationdisplay\ns=±1(χm+χn)sign(µ)+s(χm−χn)sign(Ω)\n2/radicalbig\nκ2+λ2−2sκλξ(κξ−sλ)/bracketleftBig\nθ(1−λ)−sθ(λξ−1)/bracketrightBig\n,\nand\nβ⊥⊥\nmn=1/integraldisplay\n−1dξθ/parenleftbig\nκ2+λ2−2κλ|ξ|−x2/parenrightbig\nκ2\n1−χmχn−κ2ξ2+λ2\n/radicalBig\n(κ2+λ2)2−(2κλξ)2\nθ(1−λ) (19)\nβ/bardbl/bardbl\nmn=1/integraldisplay\n−1dξθ/parenleftbig\nκ2+λ2−2κλ|ξ|−x2/parenrightbig\nκ2\n1−χmχnκ2(2ξ2−1)−λ2\n/radicalBig\n(κ2+λ2)2−(2κλξ)2\nθ(1−λ)\nβ⊥⊥′\nmn=−iǫ/bardbl⊥⊥′1/integraldisplay\n−1dξθ/parenleftbig\nκ2+λ2−2κλ|ξ|−x2/parenrightbig\nκ2/summationdisplay\ns=±1s(χm−χn)sign(Ω)\n2/radicalbig\nκ2+λ2−2sκλ|ξ|(κ|ξ|−sλ)θ(1−λ).\nThe functions fabhave the same characteristics in all\nspin channels a,b∈ {⊥⊥,∝bardbl∝bardbl,⊥⊥′}. Their plots in Fig-\nures 1, 2, 4 illustrate that fabvanish for |Ω|<v|q|−2|µ|,\n2|µ|−v|q|>|Ω|>v|q|and|Ω|=v|q|. Thedissipationat|Ω|>max(2|µ|,v|q) is dominated by the collective mode\ndecay into “high energy” particle-hole pairs which are\nexcited across the Weyl node. Outside of this frequency-\nmomentum region, the decay occurs by generating “low7\nenergy” particle-hole pairs across the Fermi surface on\nthe Weyl node. This “low energy” channel is weaker,\nbut has several features that clearly reveal the relativis-\ntic properties of the Weyl spectrum. Fig. 4 shows how\ntheminimumsandmaximumsofacollectivemodedamp-\ning rate can be used to characterize the Fermi surface of\nWeyl electrons.\nB. Spin wave damping\nThe actual damping rate of collective excitations gen-\nerally obtains from a mixture of spin channels. Consider\nthe spin waves with wavevectors ∆ Q+qin the vicinity\nof the momentum-space separation ∆ Q=Qm−Qnbe-\ntween two particular Weyl nodes. Let −SΩab\n0(q) be the\nintrinsic part of the effective Lagrangian density δLeff\nfor the local moment fluctuations δn, excluding the spin\nBerry phase SΩδab(Sis the spin magnitude of local\nmoments). This can contain any exchange interactions\nof the localized electrons and crystal field anisotropies.\nThe Lagrangian density terms induced by the itinerant\nWeyl electrons are all contained in the Γabtensor (5).\nThe principal part of (5) yields a variety of induced\nRKKY interactions45, while its dissipative components\n/tildewideΓabare collected in (16). The presence of magnetic or-\nder in the ground state further affects the dynamics of\nspin waves because the small spin fluctuations δnof low-\nenergy modes must be orthogonal to the local spins ˆn.\nThis can be incorporated into the general analysis44, but\nwe will simplify the discussion here by considering only a\nferromagnetic ground state ˆn(r) =ˆn0. The spectrum of\ndamped spin waves is extracted from the Gaussian part\nof the Lagrangian density in momentum space\nδLeff= (δna)∗/bracketleftBig\nSΩδab−SΩab\n0(q)+a3Γab(Ω,q)/bracketrightBig\nδnb(20)\nThe factor of a unit-cell volume a3converts the energy\ndensity Γabto the energy per lattice unit-cell, and the\nfactor of1\n2in the Berry phase term Ω is appropriate for\nthe local moments with spin S=1\n2. Introducing\ngab= Ωab\n0−a3\nSΓab(21)\nto simplify notation, the spin wave modes obtain by di-\nagonalizing PMP, wherePab=δab−ˆna\n0ˆnb\n0projects-out\nthe high-energy amplitude fluctuations (keeps δn⊥ˆn0)\nand\nMab= Ωδab−g⊥⊥(δab−ˆqaˆqb)−g/bardbl/bardblˆqaˆqb−g⊥⊥′ǫabcˆqc\nis the matrix embedded in (20). An arbitrary choice of\nthe background magnetization ˆn0=ˆzreveals two polar-\nization modes δn= (δnx,δny) atq=qˆq\nδn±∝/parenleftBigg\ng/bardbl/bardbl−g⊥⊥\n2(ˆq2\nx−ˆq2\ny)±δǫ\n(g/bardbl/bardbl−g⊥⊥)ˆqxˆqy−g⊥⊥′ˆqz/parenrightBigg\n(22)with energies\nΩ±=g⊥⊥\n0+g/bardbl/bardbl−g⊥⊥\n2(1−ˆq2\nz)±δǫ(23)\nwhereδǫ=1\n2/radicalbig\n(g/bardbl/bardbl−g⊥⊥)2(1−ˆq2z)2−(2g⊥⊥′ˆqz)2.\nThese polarizations are generally elliptical, but become\ncircularδn∝(±i,1) with Ω ±=g⊥⊥∓ig⊥⊥′for the\nmodes that propagate along the magnetization direction\n(q∝bardblˆn0), and linear δn+∝ˆq,δn−∝ˆn0׈qwith\nΩ+=g/bardbl/bardbl, Ω−=g⊥⊥respectively for the modes that\npropagate in the plane perpendicular to the magneti-\nzation (q⊥ˆn0). The character and non-degeneracy\nof the two polarization modes is the hallmark of the\nRKKY interactions induced through the spin-orbit cou-\npling: Dzyaloshinskii-Moriya(DM) in the caseof circular\npolarizations, and Kitaev in the case of linear polariza-\ntions.\nThe equation (23) has to be solved self-consistently\nsince the components of the gabtensor on its right-hand\nside depend on frequency, but the revealed form of its\nsolutions ensures all of the spin wave properties that\nwe discuss. The two circular polarizations at the same\nwavevector q∝bardblˆn0carry opposite spin currents\nja\ni=−iqiǫabc(δnb)∗δnc∝ ∓|g⊥⊥′|2qiδaz,(24)\nso their energy difference Ω ±=g⊥⊥∓ig⊥⊥′due to the\nDM interaction implies spin-momentum locking. Note\nthat the DM interactions appears as gab\nDM∝ǫabc(iqc), so\nit does shift the spin wave energy. The dissipative com-\nponents/tildewidegab∝/tildewideΓabofgabimpart an imaginary part on\nthe pole frequency Ω, which corresponds to the damping\nrate. The signs of both /tildewideΓ⊥⊥,/tildewideΓ/bardbl/bardbl(f⊥⊥,f/bardbl/bardbl>0) in-\ndeed correspond to damping and not an instability, and\nthe chiral contributions are not large enough to overturn\nthis at any Ω. The chiral dissipative part extracted from\n(16) is real,/tildewidegab\nDM∝ǫabcqc, and hence introduces differ-\nent damping rates for the two circular spin waves. These\nqualitative conclusions hold for the elliptical modes as\nwell.\nC. The absence of uniform precession damping\nThe universal dependence of (16) on |Ω|/vqintroduces\nanon-analyticbehavioratΩ ,q→0inthedampingterms\n/tildewideLof the spin Lagrangian density. Therefore, one cannot\nstrictly expand /tildewideLin powers of Ω ,qto represent the dis-\nsipation as a result of local processes. /tildewideLcan be approx-\nimated by an expansion only in special limits. Suppose\nthe spin waves have dispersion |Ω|=uqat low ener-\ngies (in the vicinity of ∆ Q=Qm−Qn→0 for intra-\nnode scattering m=n). If the spin wave velocity uis\nsmaller than the Weyl electrons’ velocity v, then a suf-\nficiently large qpushes the spin waves into the regime\n|Ω|< vq−2|µ|where/tildewideΓab= 0 in (16) and the damping\nis absent (see Fig.2). Alternatively, if u≫v, then the8\nspin waves are in the regime |Ω| ≫vqand their damp-\ning at energies |Ω|>2|µ|is approximately characterized\nby the dominant local terms /tildewideΓ/bardbl/bardbl,/tildewideΓ⊥⊥∼i(AΩ2+Bq2)\nand a smaller chiral term /tildewideΓ⊥⊥′∼DqΩ. Together with\nthe non-dissipative Hermitian terms χ−1\n0, the electron-\ninduced part of the local moments’ effective Lagrangian\ndensity (20) contains\nΓab|Ω|≫vq− −−−− →1\n2/bracketleftBig\n(χ−1\n0)ab+i(AabΩ2+Babq2)+DǫabcqcΩ/bracketrightBig\n(25)\nwithAab=A⊥⊥(δab−qaqb/q2)+A/bardbl/bardblqaqb/q2andlikewise\nforBab. By construction (5), Γ ≡1\n2χ−1is the inverse\ntime-ordered correlation function\n∝an}bracketle{tδsa(q,Ω)δsb(q′,Ω′)∝an}bracketri}ht=iχab(q,Ω)δ(q+q′)δ(Ω+Ω′)\nfor the small fluctuations δsof the Weyl electron spins\naway from their equilibrium magnetization. We will con-\nsider only the simplest case of a collinear ferromagnet\nin the following analysis. The equilibrium state will be\ngiven by the uniform magnetization of local moments ˆn0\nand electrons ∝an}bracketle{ts0∝an}bracketri}ht ∝bardblˆn0.\nA semiclassical representation of the local moment dy-\nnamics is given by the field equation for ˆn. The pres-\nence of non-Hermitian damping terms in the effective ac-\ntion for local moments prevents us from deriving the field\nequation by considering the stationary action condition.\nInstead, we can use linear response theory to learn about\nthe semiclassical dynamics. The retarded electrons’ spin\ncorrelation function\nχR(q,Ω) =/braceleftBigg\nχ(q,Ω),Ω>0\nχ†(q,Ω),Ω<0(26)\nis readily obtained from (25)\n(χ−1\nR)ab|Ω|≫vq− −−−− →(χ−1\n0)ab(27)\n+sign(Ω)/bracketleftBig\ni(AabΩ2+Babq2)+DǫabcqcΩ/bracketrightBig\n,\nand then the response of electron spins to the local mo-\nment field is\n∝an}bracketle{tδsa(r,t)∝an}bracketri}ht=JK\na3/integraldisplay\ndt′d3r′χab\nR(r−r′,t−t′)δnb(r′,t′).\n(28)\nThis follows from the Kondo interaction JKin (4) be-\ntween the “perturbation” field nand the responding\nelectrons spin s=ψ†σψon a lattice site (the unit-cell\nvolumea3effectively converts the integration over coor-\ndinates to a summation over lattice sites). Note that\nχab\nR(q,Ω) = (χab\nR)∗(−q,−Ω) is established globally in\nmomentumspace(notnecessarilyintheimmediatevicin-\nity of the Weyl node wavevector ∆ Q)61, so that its in-\nverse Fourier transform χab\nR(r,t) is real. The thermody-\nnamic potential for local moments is simply\nF[ˆn] =JK∝an}bracketle{ts∝an}bracketri}htˆn. (29)The local moment dynamics is driven by an effective\n“magnetic” field in units of energy\nHeff(r,t) =−δF[ˆn]\nδˆn(r,t)=−JK∝an}bracketle{ts(r,t)∝an}bracketri}ht(30)\nTaking into account the Berry phase of local moments\nyields the usual semiclassical field equation\n∂ˆn\n∂t=ˆn×Heff. (31)\nwith\nHa\neff(r,t)≈ −JKˆna\n0−J2\nK\na3/integraldisplay\ndδtd3δrχab\nR(δr,δt) (32)\n×δˆnb(r+δr,t+δt)\n≈ −JKˆna\n0−J2\nK\na3/integraldisplay\ndδtd3δrχab\nR(δr,δt)\n×/bracketleftbigg\n1+δr∇+δt∂\n∂t+···/bracketrightbigg\nδˆnb(r,t)\nThis is seen to generate Gilbert damping which dissi-\npates the precession of uniform magnetization in typical\nferromagnets\n∂ˆn\n∂t=ˆn×Heff=···+ˆn×αG∂ˆn\n∂t(33)\nwith the damping tensor\nαab\nG=−J2\nK\na3/integraldisplay\ndδtd3δrχab\nR(δr,δt)δt (34)\n=iJ2\nK\na3/integraldisplaydΩ\n2π/integraldisplay\ndδte−iΩδt∂χab\nR(0,Ω)\n∂Ω\n=−J2\nK\na3∂Imχab\nR\n∂Ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n(q,Ω)=0.\nThe real part of χR(q,Ω) generally does not contribute\nbecause it is an even function of Ω at q= 0 (even though\nit diverges for gapless spin waves when Ω →0). In the\ncase of damping induced by Weyl electrons, the imagi-\nnary part of χRbecomes zero when 2 |µ|−vq>|Ω| ≥vq,\nfollowing the behavior of the time-ordered χ−1≡Γab\nthat was discussed earlier (see Fig.4). Therefore, χRis\nreal in the limit Ω ,q→0 and the decay of spin waves\ninto Stoner excitations of the Weyl electrons does not\ngenerate Gilbert damping.\nThe complete equation of motion for local moments\ncanbe extractedfrom(31)and(32), but thenon-analytic\nfrequency dependence of the dissipative terms in (27) in-\ntroduces (via its Fourier transform) non-local relation-\nships between the fields ˆn(t) at different times t. If one\nwere to ignore this issue, or approximate the non-local\neffect by couplings over small time intervals, then a local\nfield equation would be obtained from the expansion in-\ndicated in (32). We will not pursue this here any further.9\nIV. CONCLUSIONS AND DISCUSSION\nWe analyzed the dynamics of local magnetic moments\ncoupled to itinerant Weyl electrons, and focused on the\ndissipation of spin waves via the continuum of Stoner\nparticle-holeexcitations. Wedescribedthisdissipationat\nthe level of the effective Lagrangian of local moments, or\nequivalently the spin-spin correlation function (dynamic\nsusceptibility). For the spin waves at wavevector∆ Q+q\nand frequency Ω in the vicinity of the momentum differ-\nence∆Q=Qm−QnbetweentwoWeylnodes,thedamp-\ning rate is proportional to Ω2and a universal function of\n|Ω|/v|q|wherevis the Weyl electron (Fermi) velocity.\nThe presence of Fermi pockets with chemical potential µ\nintroduces additional dependence of the damping rate on\n|Ω/µ|. If the Weyl nodes are well-separated in momen-\ntum space, then there is no cross-talk between them in\nthe damping rates and the momentum-space locations of\nthe Weyl nodes can be discerned from the wavevectors\nat which the spin wave dissipation is locally maximized.\nThe Weyl-electron origin of dissipation can be experi-\nmentally verified by the universal relativistic properties\nof damping over a range of mode frequencies and mo-\nmenta, while various parameters of the Weyl spectrum\ncan be extracted from the momentum space locations\nof the characteristic damping features (e.g. local maxi-\nmums and points where damping vanishes). The damp-\ning rates involving Weyl electrons also generally exhibit\n“non-reciprocity”or chirality– the modes ofdifferent po-\nlarizationsthat propagateatthe samemomentum qhave\ndifferent lifetimes. We presented a procedure to obtain\nthe field equation for the semi-classical dynamics of the\nlocalmomentmagnetizationfield, andfoundthatthedis-\nsipation on Weyl electrons does not give rise to Gilbert\ndamping.\nOne important conclusion of this study is that the\nspin wave damping rate reveals the relativistic nature\nof Weyl electrons – both through its universal depen-\ndence on |Ω|/v|q|and the places in momentum space\nwhere it vanishes. We computed the damping rate asso-\nciated with Stoner excitations, but similar results should\nhold for zero-spin particle-hole excitations as well. Then,\notherkindsofcollectivemodescoupledtoWeylelectrons,\ne.g. the phonons of the crystal or a charge density wave,\nshouldexhibitsimilaruniversalityintheirdampingrates.\nThiswouldbeinterestingtoexploreinthefuturesincein-\nelastic neutron scattering is sensitive to phonons as well.\nThe developed theory is very general within its limi-\ntations. It makes no assumptions about the Weyl node\nlocations, so it applies to Diracsemimetals aswell (where\nthe opposite-chirality Weyl nodes coexist at the samewavevectors). It also makes no assumptions about the\nmagnetic order, so it holds for ferromagnets, antiferro-\nmagnets and paramagnets, with or without local spin\nanisotropy. In this regard, however, the damping rates\nof spin waves are affected by the nature of magnetic or-\nder; we demonstrated the calculations only in the fer-\nromagnetic (and implicitly also the paramagnetic) case.\nAnalytical progress was made by simplifying the model\nto spherically symmetric Weyl nodes that all live at the\nsame energy. This is the main limitation of the current\ntheory, although many implications of realistic model ex-\ntensions can be readily anticipated. Energy differences\nbetween the nodes are easily included by associating dif-\nferent chemical potentials to the nodes, while a small\nWeyl node anisotropy is expected to introduce a simi-\nlar anisotropy in the induced dynamics and dissipation\nof local moments. It is possible that type-II Weyl nodes\nfalloutsideofthistheory’sdomain, sotheirexplorationis\nleft for future study. We also did not consider corrections\ndue to finite temperature and disorder.\nTheusefulnessofthistheoryfortheexperimentalchar-\nacterization of magnetic Weyl semimetals is guarantied\nin principle, but depends on several factors in reality.\nThe needed level of detail is not easy to achieve in the\nmeasurements of spin wave spectra. It requires at least\nvery clean samples, low temperatures, as well as a suffi-\nciently high energy resolution and adequate statistics to\nresolvewithlowerrorbarsthe energy/momentumdepen-\ndence of the inelastic neutron scattering. These aspects\nof measurements can always be improved, but there are\nalso material-related constraints: phonons, for example,\nmust not coexist with spin waves at the same momenta\nand frequencies. Still, some regions of the first Brillouin\nzone should expose the electronic damping mechanism\nand enable the proposed experimental characterization\nof magnetic Weyl semimetals. On the purely theoretical\nfront, the present study was concerned with a basic but\nintricate and important aspect of interaction physics in\na topological system. It plays a role in piecing together\na broader picture of magnetic correlated topological ma-\nterials, which can host non-trivial anisotropic magnetic\norders13, chiral magnetic states and excitations44, and\npossibly even exotic spin liquids62.\nV. ACKNOWLEDGMENTS\nI am very grateful for insightfull discussions with\nJonathanGaudet and Collin Broholm. 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Nikoli´ c, Physical Review B 101, 115144 (2020)."    },    {        "title": "1609.01094v1.Coarsening_dynamics_of_topological_defects_in_thin_Permalloy_films.pdf",        "content": "Coarsening dynamics of topological defects in thin Permalloy films\nIlari Rissanen∗and Lasse Laurson\nCOMP Centre of Excellence and Helsinki Institute of Physics,\nDepartment of Applied Physics, Aalto University,\nP.O.Box 11100, FI-00076 Aalto, Espoo, Finland.\nWestudythedynamicsoftopologicaldefectsinthemagnetictextureofrectangularPermalloythin\nfilm elements during relaxation from random magnetization initial states. Our full micromagnetic\nsimulations reveal complex defect dynamics during relaxation towards the stable Landau closure\ndomain pattern, manifested as temporal power-law decay, with a system-size dependent cut-off time,\nofvariousquantities. Theseincludetheenergydensityofthesystem, andthenumberdensitiesofthe\ndifferent kinds of topological defects present in the system. The related power-law exponents assume\nnon-trivial values, and are found to be different for the different defect types. The exponents are\nrobust against a moderate increase in the Gilbert damping constant and introduction of quenched\nstructural disorder. We discuss details of the processes allowed by conservation of the winding\nnumber of the defects, underlying their complex coarsening dynamics.\nPACS numbers: 75.78.-n,76.60.Es,75.70.Kw\nI. INTRODUCTION\nThe topic of coarsening dynamics of topological de-\nfects in diverse systems ranging from liquid crystals1–4\nto biosystems5and cosmology6has attracted consider-\nable interest as it is related to properties of symmetry-\nbreaking phase transitions. As the system is quenched\nfrom a high-temperature disordered phase to a low-\ntemperature ordered phase, the symmetry of the disor-\ndered phase is broken and topological defects are gener-\nated, subsequentlyexhibitingslowcoarseningdynamics7.\nSuch phase-ordering kinetics is often characterized by a\npower-law temporal decay ρ(t)∼t−ηof the density ρof\nthe defects, with the value of the exponent ηdepending\non the characteristics of the system, and/or the defects8.\nIn ferromagnetic thin films dominated by shape\nanisotropy, elementary topological defects within the\nmagnetic texture, i.e. vortices, antivortices and edge\ndefects, may occur9–11. For instance, magnetic domain\nwalls can be envisaged as composite objects consisting of\ntwo or more such elementary defects, each characterized\nbytheirintegerorfractionalwindingnumbers9. Also,the\npresence of vortices is intimately related to magnetic flux\nclosure patterns minimizing the stray field energy of mi-\ncron or submicron magnetic particles12–14. While coars-\nening of e.g. the domain structures in Ising and Potts\ntype of models15,16as well as that of the defect struc-\nture in the XY model1,17have been extensively studied,\nless is known about the details of coarsening dynamics\nin soft (low-anisotropy) ferromagnetic thin films, involv-\ning the collective dynamics of vortices, antivortices and\nedge defects, when all the relevant effective field terms\n(exchange, and demagnetizing energies) are included in\nthe description.\nHere we study, by performing an extensive set of full\nmicromagnetic simulations of the magnetization dynam-\nics in Permalloy thin films, the relaxation process of\nsuch magnetic topological defects, starting from ran-\ndom magnetization initial states, mimicking the high-temperature disordered paramagnetic phase. Our zero-\ntemperature simulations, resembling a rapid quench to\nthe low-temperature ferromagnetic phase, show how de-\nfects emerge from the disordered initial states, and sub-\nsequently exhibit coarsening dynamics. The focus of our\nstudy is on the time period after a short initial transient\nsuch that the length of the magnetization vectors is ap-\nproximately constant, and the system can be modeled by\nthe Landau-Lifshitz-Gilbert equation. In the coarsening\nprocess, the densities of the various defect types, as well\nas the energy density of the system18, follow power-law\ntemporal decay towards the ground state with a flux-\nclosure Landau pattern. These power laws are charac-\nterized by non-trivial exponent values which are differ-\nent for the different defect types, and exhibit a cut-off\ntime scale growing with the system size. We also ad-\ndress the question of how the values of these exponents\nare affected by changes in the Gilbert damping constant\nαand introduction of random structural disorder within\nthe film, and discuss the role of the conservation of the\nwinding number on the possible annihilation reactions,\nunderlying the complex coarsening dynamics of the var-\nious defect populations.\nThe paper is organized as follows: In the following sec-\ntion (Section II), properties of the elementary topological\ndefects in soft ferromagnetic thin films are reviewed, and\ndetails of the micromagnetic simulations and data anal-\nysis are presented in Section III. In Section IV, we show\nour results on the defect coarsening dynamics and ana-\nlyze the possible annihilation reactions underlying such\ndynamics. Finally, Section V finishes the paper with dis-\ncussion and conclusions.\nII. TOPOLOGICAL DEFECTS IN\nMAGNETICALLY SOFT THIN FILMS\nIn the absence of an external magnetic field, the ori-\nentation of the spins in thin films of magnetically softarXiv:1609.01094v1  [cond-mat.mes-hall]  5 Sep 20162\nmaterial such as Permalloy is determined by the compe-\ntition of shape anisotropy and exchange interaction. For\nsmall films or nanodots (up to few tens of nanometers\ndepending on the film/dot thickness19), the exchange in-\nteraction energy dominates and the ground state is a sin-\ngle magnetic domain20. In larger films, up to a couple of\ntens of microns21, the ground state configuration consists\nof one or more elementary topological defects, depending\non the geometry of the film9. Square thin films can con-\ntain three types of stable elementary magnetic defects:\nvortices, antivortices and edge defects. Other structures,\nsuch as domain walls, are composed of these elementary\ndefects.\nMagnetic vortex is a point defect with core radius ap-\nproximately equal to the magnetic exchange length of\nthe material (between 5 and 6 nm in Permalloy22). The\ncore magnetization, often referred to as the polarization\nof the vortex, points out of the thin film plane, while\nthe surrounding magnetization rotates around the core\n(Fig. 1 a). Vortices are thus characterized by two quan-\ntities: the core polarization, and the rotation direction\nof the surrounding magnetization (clockwise or counter-\nclockwise).\nAn antivortex is another type of point defect with out-\nof-plane polarization, with a core radius similar to that\nof vortices. Unlike vortices, however, the magnetization\naround an antivortex does not rotate around the core of\nthe defect. Instead, the magnetization points into the\ncore from two opposite directions and out of the core\nfrom two perpendicular directions, with the rest of the\nsurrounding magnetization assuming orientation in be-\ntween these main directions (Fig. 1 b)10.\nVortices and antivortices are both bulk defects, i.e.\nthey form in the bulk of the film and tend to stay away\nfrom the edges unless driven there by the relaxation or\nan outside influence such as an external magnetic field.\nThe third type of defects, edge defects, are confined to\nthe edge and cannot move to the bulk9. The edge defects\nare also different from vortices and antivortices in that\nthe core magnetization of the defect does not necessarily\npoint out of plane. Edge defects can further be divided\ninto two types, henceforth referred to as the positive edge\ndefect (Fig. 1 c) and negative edge defect (Fig. 1 d), ac-\ncording to their winding numbers.\nThe topological defects can be characterized by the\nwinding number W, defined as a normalized line integral\nof the magnetization vector angle θover a closed loop\naround the defect10,W=1\n2π/contintegraltext\nSθ(φ)ds, whereφis the\nangle of the vector from the defect core to the point on\nthelinebeingintegratedover, and Stheintegrationpath.\nThe winding number (or topological charge) is +1for\nvortices and−1for antivortices. Though edge defects\ncannot be similarly circled around, they can be shown to\nhave fractional winding numbers of ±1/2.9\nThe total winding number of a thin film is a conserved\nquantity. In a film with nholes, the total winding num-\nber isW=/summationtextk\ni=1Wi= 1−n, whereWiis the winding\nnumber of defect iandkis the number of defects9. The\na) b)\nc) d)Figure 1. The elementary topological defects: a)vortex,b)\nantivortex, c)positive edge defect, d)negative edge defect.\nThe film is viewed from the direction of the z-axis, with the\narrows showing the magnetization direction in the xy-plane.\nThe vortex/antivortex cores pointing out of plane are denoted\nwith⊗. In the case of edge defects, the cores (not necessarily\nout of plane) are marked with #and the black bar represents\nthe edge of the sample.\ntotal number of defects can be quite high in large mag-\nnetically unrelaxed films, but the number will eventually\ndecay due to the collision-induced annihilations of the\ndefects. In a film with no holes, such as the ones simu-\nlated in this paper, the total winding number is equal to\n1. This corresponds to a few possible configurations, out\nof which the single vortex state ( flux-closure orLandau\npattern) is energetically most favorable13.\nIII. MICROMAGNETIC SIMULATIONS\nA. Simulation details\nDuring the relaxation of the magnetization from a ran-\ndomized initial state, the time evolution of the mag-\nnetic moments m=M/Msis described by the Landau-\nLifshitz-Gilbert (LLG) equation,\n∂m\n∂t=γHeff×m+αm×∂m\n∂t, (1)\nwhereγis the gyromagnetic ratio, Heffthe effective\nmagnetic field, Msthe saturation magnetization, and\nαthe Gilbert damping constant. Hefftakes into ac-\ncount four energy contributions, which are the afore-\nmentioned exchange energy, energy due to magnetocrys-\ntalline anisotropy, Zeeman energy (energy of an external\nfield) and the demagnetizing field energy. In the con-\ntext of this work, the Zeeman and anisotropy contribu-3\ntionsarenegligible, asnoexternalfieldsarebeingapplied\nand the magnetocrystalline anisotropy of Permalloy is in-\nsignificant.\nSimulations were performed with a GPU-based micro-\nmagnetic code Mumax3, using the adaptive Dormand-\nPrince method and finite differences for temporal and\nspatial discretization, respectively23. Simulations were\nrun for square samples of thickness 20 nm, with four\ndifferent linear film sizes Lof 512 nm, 1024 nm, 2048\nnm and 4096 nm. The dimensions of a single simula-\ntion cell were chosen to be 4 nm ×4 nm×20 nm, so\nthat the smallest film corresponds to 128 ×128 cells; the\nnumber of out-of-plane z-direction cells is 1. Here, typ-\nical parameter values of Permalloy13,18,24are used, i.e.,\nMs= 860·103A/m, and A= 13·10−12J/m. Un-\nless stated otherwise, we consider α= 0.02, which corre-\nsponds to slightly Nd-doped25or Pt-doped26Permalloy.\nWe also investigate the effect of αon the relaxation pro-\ncess, using values ranging from the typical 0.01for pure\nPermalloyupto 0.1representinghighlydopedPermalloy,\nand a couple of very high values, α= 0.5andα= 0.9.\nWhile the latter two α-values are clearly too high to re-\nalistically describe magnetization dynamics of Permal-\nloy, they allow to address the more general question of\nthe effect of damping on the defect coarsening dynam-\nics. Moreover, we also check the stability of our results\nwith respect to adding quenched structural disorder to\nthe system27–31, by performing a Voronoi tessellation to\ndividethefilmsintograins, mimickingthepolycrystalline\nstructure of the material30,31; we consider average grain\nsizes of 10nm,20nm and 40nm. Disorder is then imple-\nmented by either setting a random saturation magnetiza-\ntion in each grain (from a normal distribution with mean\nMsand standard deviation of 0.1Msor0.2Ms)27,30, or\ndecreasing the exchange coupling across the grain bound-\naries by 10 %,30 %or50 %30,31.\nAt the start of the simulation, the magnetization is\nrandomized in each cell, after which the system is let re-\nlax at zero temperature without external magnetic fields.\nFour example snapshots of the relaxation process are\nshown in Fig. 2. The relaxation process consists of ap-\nproximately three phases. In the beginning of the relax-\nation (usually from 0 to approx. 1 ns, though less with\nstronger damping), the system experiences large fluctua-\ntions in magnetization without well-defined magnetic de-\nfects or domains. After these fluctuations have settled,\nthe dynamics consist of defects moving and annihilating\nwith each other; the properties of this defect coarsen-\ning dynamics are the main focus of the paper. The final\nrelaxation stage consists of a single vortex experiencing\ndamped gyrational motion towards the center of the film.\nThe configuration of the resulting ground state displays\nthe Landau pattern: four large domains separated by di-\nagonal domain walls starting from the corners of the film\nand meeting at a 90◦angle in a vortex at the center12–14.\nOne should note that during the very early stages of\nthe relaxation starting from the disordered paramagnetic\nstate, the LLG equation does not fully describe the mag-\nFigure 2. The relaxation process in a square thin film with\nlateral size L= 1024 nm. The color wheel in the center\nshows the direction of the magnetization corresponding to\neach color. In this case, the final vortex (black dot at the\ncenterofthebottomrightpicture)isa −z-polarizedclockwise\nvortex.\nnetization dynamics since it assumes that the lengths of\nthe magnetic moments are conserved; the latter is not\nstrictlyspeakingthecaseduringthefirststageofquench-\ning the system across the phase boundary from the high-\ntemperature paramagnetic to the ferromagnetic phase.\nHowever, multiple studies32–34concerning the longitudi-\nnal relaxation of the magnetic moments point out that\nin low temperatures, the longitudinal relaxation is or-\nders of magnitude faster than the transverse relaxation,\nand takes place in the femto- and picosecond timescale.\nThus, afterthefirstfewpicoseconds, andespeciallyinthe\nannihilation-dominatedrelaxationregimewhichisthefo-\ncus of the present study, longitudinal relaxation is nonex-\nistent, and the LLG equation suffices to fully describe the\nmagnetization dynamics.\nB. Locating and characterizing defects\nHere, we describe the algorithm used to find the de-\nfects from the magnetization data. Finding bulk de-\nfects (vortices and antivortices) is relatively simple, as\nthey have strong z-directional magnetization at the core.\nThe cores of the defects are determined by comparing\nthez-magnetization with the nearest neighbors and find-\ning the local maximum or minimum, and comparing it\nto a threshold value of 0.5Ms. The type of the defect\nis then determined by performing a discretized version4\nX12 3\n7654 8\nX12 3\n7654 8\n1234\n5 1234\n5a) b)\nc) d)\nFigure 3. A schematic view of the different defects and the\n(ideal) surrounding magnetization in the nearest neighbors\nin thexy-plane, similarly as in Fig. 1. The numbers in the\ncorners of the cells show the cell traversal order when deter-\nmining the defect type.\nof the winding number integration: The nearest neigh-\nbors are looped through in a circle, and the rotation di-\nrection of the magnetization vector is monitored. Do-\ning a counterclockwise loop, the vector in two consec-\nutive cells would turn left in the case of a vortex and\nright in the case of an antivortex (Fig. 3 a,b). Ideally,\nthe angle between two consecutive neighbors in the xy-\nplane would be 45◦. Since each magnetization vector\nis normalized to Ms, the length of the cross product of\ntwo consecutive magnetization vectors in the vortex case\n(choosing counterclockwise turn as positive) would yield\n||mi×mi+1||=M2\nssin 45◦=M2\ns√\n2. The corresponding re-\nsult for an antivortex would be −M2\ns/√\n2. Summing the\nresults for each neighbor pair, this method would ideally\ngive 2√\n2M2\nsfor vortices and−2√\n2M2\nsfor antivortices.\nDue to nonidealities in rotation and the fact that the\nspins in the cells neighboring the core also tend to have\nnonzeroz-components, the sums can be smaller. Thus,\nthresholdvaluesforrecognizingdefectsaresetto M2\nsand\n-M2\nsfor vortices and antivortices, respectively.\nIn addition to the winding number, the vortices can\nhave clockwise or counterclockwise rotation. The rota-\ntion is determined as above, but considering the center-\nto-neighbor vector and the magnetization vector for each\nneighbor. This approach would ideally yield 4Ms(1+√\n2)\nfor counterclockwise rotation and −4Ms(1 +√\n2)for\nclockwise rotation, since the angle between the vectors\nwould be 90◦. Due to nonidealities, threshold values were\nset to 5Msand−5Ms, respectively.\nEdge defects are somewhat harder to detect, as they\nusually do not have polarized cores. Thus the defects are\n2.0\n1.6\n1.2\n0.8\n0.4\n00            0.4           0.8          1.2           1.6          2.0y [µm]\nx [µm]\nFigure 4. The various kinds elementary defects present in the\nrelaxation of the largest film: clockwise vortices (squares),\ncounterclockwise vortices (circles), antivortices ( +-signs) and\nedge defects (triangles). The color of a bulk defect shows its\npolarization (black for −zand white for +z). For the edge\ndefects, the color indicates whether the winding number of\nthe defect is positive (black) or negative (white).\n10-1110-1010-910-8\nt [s]-2×104-1×10401×104WtotL = 512 nm\nL = 1024 nm\nL = 2048 nm\nL = 4096 nm\n10-910-8\nt [s]02468Wtot\nFigure 5. The total average winding numbers for the four film\nsizes used. The inset shows that the winding number for the\nlargest sizes takes more time to converge into 1.\ndetermined only by performing a loop through the near-\nest neighbors as in the case of bulk defects (Fig. 3 c,d).\nThough the method performed quite well in finding the\nedge defects, the difficulty of singling out the core some-\ntimes resulted in multiple detections in the same region.\nThis problem was somewhat mitigated by introducing\nan area around the edge defects in which similar defects\nwould be ignored. Fig. 4 shows a snapshot of the relax-\nation with defects pinpointed by the detection algorithm.\nIn the simulations, the initial random fluctuations\ncause many false defect detections. This can be seen\nas the total winding number fluctuating in the beginning5\na)                               b)\nFigure 6. The two most commonly encountered metastable\nstates, shown for the system with L= 1024nm.a)A two\nedge defect, two vortex state which also displays significant\nbending of domain walls. b)A more complex state, showing\nan isolated vortex and an arc of other defects.\nbefore converging close to 1 (Fig. 5). The tendency of\nthe total winding number to be below 0 in the beginning\nis due to the initial fluctuations being more easily cate-\ngorized as antivortices, since they do not have a rotation\ndirectionthreshold. Thewindingnumbercanalsochange\nmomentarilyduringannihilationsduetothedetectional-\ngorithm having difficulties determining defects that are\nvery close to one another. Moreover, sometimes a dis-\nturbance (such as a spin wave from an annihilation) near\na defect could cause the defect to become momentarily\nunrecognizable to the algorithm. To lessen the fluctua-\ntion in defect amounts, a persistence time of 20 ps for\nthe already detected defects was introduced. During the\npersistence time, the algorithm considers a defect to exist\nin the location it was last detected even if it can’t find\nit at the present time. The persistence time reduces the\nnoise in the number of defects.\nIV. RESULTS\nDepending on the system size, the relaxation from ran-\ndommagnetizationtothesinglevortexgroundstatetook\nusually approximately from 5 ns to 40 ns. Hence the sim-\nulations were run for 20 ns for the two smallest film sizes,\n30 ns for the L= 2048nm film and 50 ns for the largest\nfilm. Usually the ground state was reached relatively\nquickly compared to the simulation time. Only with the\nlargest film size there were a couple of instances in which\nthe system had not relaxed to the single-vortex state or\na metastable state before the simulation time ran out,\nthough in these cases the system was still close to the\nrelaxed state with only 2−4defects left.\nMetastable states were encountered in 10 of the total\n80 simulations. With the smallest film, only one simula-\ntion ended up in a metastable state, whereas each larger\nsize had three simulations finished in a metastable state.\nOf these states, two different kinds were common: a sim-\nple one with two negative edge defects on two opposite\nsides and two vortices close to the center (Fig. 6 a) and a\n10-1110-1010-910-8\nt [s]10-510-410-310-210-1100(E(t) - E(tR))/ E(0)L= 512 nm\nL= 1024 nm\nL= 2048 nm\nL= 4096 nm\nt-1.22 Defect\nannihilationInitial\nfluctuation\n0 10 20 30 4050\nρd[1/µm2]0.00.51.01.5ρE[J/µm2]L= 512 nm\nL= 1024 nm\nL= 2048 nm\nL= 4096 nm×10-14Figure 7. The time evolution of the total energy of the films,\nconsisting of an “initial fluctuation” phase independent of the\nsystem size, and a defect annihilation/coarsening phase dis-\nplaying power law relaxation terminated by a cut-off time\nincreasing with the film size. The linear dependence of en-\nergy density and defect density during the coarsening phase\nis shown in the inset.\nmore complicated state with an antivortex, three vortices\nand two edge defects (Fig. 6 b), in which one vortex is\nisolated and the other defects are in an arc close to one\nof the edges.\nA. Time evolution of energy and defect densities\nIn Fig. 7, the time evolution of the energy towards the\nvalueE(tR)is shown for all four system sizes, averaged\nover 20 simulations for each size. Here tRis the time\nafter which the number of defects in the system does not\ndecrease, which corresponds to either the single vortex\nstate, a metastable state with more than one defect, or\nthe state the system had time to reach before the simu-\nlation time ran out.\nThe total energy of the system drops very little in the\nfirst 0.1 ns, and then starts decreasing in a slightly oscil-\nlating fashion in the initial fluctuation regime (0 – 1 ns).\nStable defects start to form at roughly 0.5 ns, but the\nenergy evolution is dominated by the global fluctuations\nin the system. As can be seen from the figure, during the\ninitial phase the time evolution of the energy, normalized\nby the initial value, is independent of the system size.\nThis results from the fact that during the initial fluctua-\ntions the magnetization is largely random, and thus the\nenergy contributions from the exchange interaction and\nthe stray fields are proportional to the system size.\nIn the ”defect annihilation” or coarsening phase, the\ntime evolution of the energy resembles a power law\nE(t)−E(tR)∝t−ηE, with an exponent ηE= 1.22±0.08.\nIn this phase, the total energy of the system consists\nmostly of the energy contained in the domain walls con-\nnecting the elementary defects and the stray fields cre-\nated by the (anti)vortex cores in which magnetization6\n10-1110-1010-910-810-7\nt[s]10-310-210-1100101102103ρd(t) -ρd(tR) [1/µm2]L= 512 nm\nL= 1024 nm\nL= 2048 nm\nL= 4096 nm\nt-1.42\n10-1110-1010-910-810-7\nt[s]10-310-210-1100101102103ρav(t) -ρav(tR) [1/µm2]L= 512 nm\nL= 1024 nm\nL= 2048 nm\nL= 4096 nm\nt-1.62\n10-1110-1010-910-810-7\nt[s]10-210-1100101102ρed(t) -ρed(tR) [1/µm]L= 512 nm\nL= 1024 nm\nL= 2048 nm\nL= 4096 nm\nt-0.8210-1110-1010-910-810-7\nt[s]10-310-210-1100101102103ρv(t) -ρv(tR) [1/µm2]L= 512 nm\nL= 1024 nm\nL= 2048 nm\nL= 4096 nm\nt-1.51a) b)\nc) d)\nFigure 8. The time evolution of the densities of a)all defects, b)vortices,c)antivortices and d)edge defects, with solid lines\ncorresponding to power law fits as guides to the eye. The power laws can be seen most clearly in the largest film sizes. In the\ncase of edge defects, the positive edge defects are noted with dashed symbols.\npoints out of plane. The largest system sizes are the\nslowest to reach the energy minimum and thus show the\nmost clear-cut power-law behavior. The inset of Fig. 7\nshows that during the coarsening phase the energy den-\nsityρE=E/L2of the system is linearly proportional\nto the density of the defects ρd=Nd/L2, whereNdis\nthe total number of all defects. This implies that the de-\nfects are the main contributors to the energy at this stage\nof the relaxation, a result previously obtained for the\nXY-model by Qian et al.17. The slope changes slightly\nwithsystemsize, likelyduetothefactthatwhiletheedge\ndefects also contribute to the total energy, their number\nscales with the lateral film size as 4Linstead of the L2-\nscaling of bulk defects. Thus the smaller the film, the\ngreater the relative amount of edge defects at the begin-\nning of the simulation. In smaller films the reduction\nof total number of defects involves more annihilations\nof edge defects, compared to vortex-antivortex annihila-\ntions that dominate in larger films. If the edge defects\nare energetically less expensive than bulk defects (which\nwould seem reasonable, considering that they don’t usu-\nally have large out-of-plane components and the overallchange in magnetization direction around the defect is\nless than that of bulk defects), these annihilations re-\nsult in smaller decrease in energy than vortex-antivortex\nannihilations. Hence, for smaller systems, the energy de-\ncreases more slowly as a function of the total number of\ndefects.\nIn the coarsening phase, also the densities of the dif-\nferent defect types decay as power laws, with different\nexponents for each type of defect. These exponent are\ndetermined only by the topological charge of the defect,\ngiven that considering separately the chirality and/or\ncore polarization (in the case of vortices and antivortices)\ndid not have an effect on the exponent. The total num-\nber density of all defects (counting vortices, antivortices\nand both types of edge defects) decays as a power law\nρd(t)−ρd(tR)∝t−ηdwith the exponent ηd= 1.42±0.06\n(Fig. 8 a). Here tRis again the time after which no anni-\nhilations take place. The exponent value is considerably\nhigher than the asymptotic value ( ηd= 1) found in sim-\nulations of the XY-model with linear damping and local\ninteractions1,17,35.\nFor the total amounts of vortices (summing both7\na) b)0.4 µm\n0.4 µm\nΔt = 0 ps Δt = 50 ps\nΔt = 100 ps Δt = 150 psΔt = 0 ps Δt = 150 ps\nΔt = 300 ps Δt = 450 ps\nFigure 9. The annihilation of a vortex and an antivortex with a)antiparallel b)parallel polarizations.\nclockwise and counterclockwise rotations and both\ncore polarizations) the time evolution is well-described\nby a power law ρv(t)−ρv(tR)∝t−ηv, with\nηv= 1.51±0.05(Fig. 8 b). The exponent of the time\nevolution of the antivortex density ρavis consis-\ntently found to be somewhat larger: We find a\npower law ρav(t)−ρav(tR)∝t−ηavwith an exponent\nηav= 1.62±0.09(Fig. 8 c). Since the typical relaxed\nstate achieved in the simulations is a single-vortex state,\nρav(tR)is usually 0, though few unrelaxed/metastable\nend states have ρav(tR)between 1-2.\nThedensityofnegativeedgedefectsishigherthanthat\nof the positive ones in all the simulations. Positive edge\ndefects were observed to be short-lived byproducts of an-\nnihilations of vortices and negative edge defects. This\nsupports the notion in Ref.9that negative edge defects\nare energetically preferable over positive edge defects for\nfilms withLt>L2\nex, whereLandtare the lateral length\nand thickness of the film, respectively, and Lexis the\nexchange length. Like vortices and antivortices, the den-\nsity of negative edge defects appears to show power-law\nbehaviorρned(t)−ρned(tR)∝t−ηned, with an exponent\nηned= 0.82±0.09(Fig. 8 d). One should note here that\nfor edge defects, ρned=Nned/Lis a line density instead\nof an area density. The number density of positive edge\ndefects decays close to zero soon after the initial fluctua-\ntions and there’s no visible power-law behavior.\nB. Defect dynamics during relaxation\nExamining the motion of defects during the relax-\nation/coarsening process reveals complex dynamical de-\nfect behavior, including various kinds of annihilations,\nvortex and antivortex emissions and core switching. Allof these events are restricted by the conservation of the\ntotal winding number.\nThe possible annihilation events are limited to four\ntypes: positive and negative edge defect annihilation,\nvortex-antivortex annihilation, vortex and 2×negative\nedge defect annihilation and antivortex and 2×positive\nedge defect annihilation. Out of these four annihilation\nprocesses, only two were primarily encountered in the\nsimulations: vortex-antivortex annihilation, and vortex\nand2×edge defect annihilation. In the former case, the\nparallelityorantiparallelityofthepolarizationsofthean-\nnihilating vortex/antivortex pair affects the nature of the\nannihilation process. This is related to the conservation\nof another topological quantity, the skyrmion number36.\nWhen the polarizations of the annihilating vortex and\nantivortex are parallel, the skyrmion number is con-\nserved, resulting in a continuous and relatively slow an-\nnihilation process. The vortex and antivortex approach\neach other until they’re indistinguishable and start ac-\ncelerating in a direction perpendicular to a line connect-\ning them. During the acceleration, the combined vortex-\nantivortex defect widens and diffuses continuously into\nthe surrounding magnetization. This process is depicted\nin Fig. 9 a. By contrast, if the polarizations are antipar-\nallel, the skyrmion number is not conserved, and a more\nabrupt annihilation (referred to as ”exchange explosion”\nby some authors10) takes place: the vortex and antivor-\ntex circle around one another in decaying orbits until\nmeeting at the center and explosively releasing circular\nspins waves (Fig. 9 b).\nThe steps of the annihilation process where a vortex\nannihilates with two negative edge defects are harder to\npinpoint. In a typical vortex-edge defect annihilation,\none of the edge defects changes sign and emits an an-\ntivortex, which annihilates with the approaching vortex.\nThe remaining edge defects, now having opposite signs,8\nΔt = 100 ps\nΔt = 0 ps\nΔt = 200 ps Δt = 300 ps0.4 µm\nFigure 10. Though somewhat difficult to see, during this edge\ndefect-vortex annihilation, the lower edge defect emits an an-\ntivortex with which the vortex actually annihilates.\nthenannihilatewitheachother. Thiskindofannihilation\nalso causes an emission of spin waves (Fig. 10). An edge\ndefect could also absorb or emit a vortex or an antivor-\ntex and change sign without a vortex/antivortex close by\nto annihilate with, since a +1/2edge defect emitting a\nvortex or absorbing an antivortex and changing into a\n−1/2defect conserves the winding number. Such emis-\nsions and absorptions were observed in the simulations,\nthough in most cases the emitted vortex/antivortex was\nshortly absorbed again accompanied with an emission of\nspin waves.\nThevelocitiesofthedefectsdonottypicallyexceedthe\ncore switching velocity of Permalloy ( 340±20m/s)37.\nHowever, sometimes an exception occurs in antiparal-\nlel vortex-antivortex annihilations. In this case the in-\ncreasing velocity of the vortex and/or antivortex causes\nthe formation of a dip particle, an antiparallelly polar-\nizedmagnetizationregionclosetothefast-movingcore38.\nJust before annihilation, the vortex/antivortex exceeds\nthe core switching velocity and the dip particle separates\ninto a vortex-antivortex pair. The consecutive annihi-\nlations of the two pairs then take place (Fig. 11). In\naddition to velocity, the environment of a vortex also af-\nfects the possibility of a core switch. Some core switches\nwere observed to happen even for relatively stationary\nvortices, usually after being excited by a spin wave orig-\ninating from a nearby annihilation.\nAnother core switching behavior was sometimes found\nat the corners of the film: a vortex could ”bounce”\n(shortly get absorbed and then again emitted by the edge\ndefect) between two edge defects on different edges of\nthe film while reversing polarization with each bounce\n(Fig. 12) and emitting spin waves. This kind of bouncing\nΔt = 150 ps\nΔt = 0 ps\nΔt = 300 ps Δt = 450 ps0.4 µmDip particle\nforming\nAVO-VG annihilation  \nAVG-VO annihilation  Figure 11. In this antiparallel annihilation, the negatively\npolarizedantivortex(blackdot)generatesadipparticlewhich\nthen splits into a positively polarized vortex-antivortex pair.\nThus two annihilations occur: an antiparallel annihilation of\nthe original antivortex and the generated vortex (AV O-VG),\nand a parallel annihilation of the generated antivortex and\nthe original vortex (AV G-VO).\nΔt = 150 ps\nΔt = 0 ps\nΔt = 300 ps Δt = 450 ps0.4 µm\nFigure 12. The core switching of a vortex due to a momen-\ntary absorption into an edge defect. Usually before and after\nthe absorption and emission of the bouncing defect, the edge\ndefect cores gain short-lived out-of-plane magnetization com-\nponents.\nalways ended up in both the vortex and the edge defects\nannihilating at the corner. Typically there were two or\nthree such core switches before the final annihilation.9\nC. Effects of damping and quenched disorder\nHere, we discuss briefly how the above results are af-\nfected by changes in the damping constant α, and when\nintroducing quenched disorder to the system. Fig. 13\nshows the time evolution of the total defect density ρd(t)\ninapuresystemfordifferentvaluesof αintherangefrom\n0.01 to 0.9; notice that while the higher values of αcon-\nsideredareclearlyunphysicalforPermalloy, theyallowto\naddress the question of how the defect coarsening process\nis modified when the overdamped limit (as often consid-\nered in coarse-grained models of defect coarsening, such\nas the XY-model) is approached. As indicated by the in-\nsetofFig. 13,thepowerlawexponent ηdevolvesfromthe\nlow-αvalueofηd≈1.4toalowervalueof ηd= 1.07±0.05\nfor the highest α-value considered. We note that ηdob-\ntainedhereinthelimitoflarge αisclosetothatobtained\nfor XY-model in earlier works1,17,35. The correspond-\ning exponents for the different defect types also exhibit\nsimilar evolution with α, with the values obtained for\nα= 0.9found to be ηv= 1.09±0.06,ηav= 1.13±0.06,\nηned= 0.72±0.09for vortices, antivortices and edge\ndefects, respectively (not shown). Qualitatively, with in-\ncreasingαfrom 0.01 towards 0.1, the initial fluctuations\ntend to settle down somewhat faster, and core switching\nevents are found to be less abundant. For the highest\nα-values considered (0.5 and 0.9), the system forms well-\ndefined defects almost instantaneously, with their subse-\nquent motionbeingquite sluggish. Also, no core switches\nnor ”bounces” of vortices from edge defects are observed.\nAsaresult,thedurationofthecoarseningphaseincreases\nsignificantly, with the largest system taking more than 70\nns to fully relax in some simulation runs.\nFinally, introducing random structural disorder due\nto the polycrystalline nature of Permalloy to the films\nwithα= 0.02has the effect that some of the simula-\ntion runs finish with more than one defect pinned by the\ndisorder. However, for the parameter values used in our\nsimulations for the grain size, exchange coupling reduc-\ntions across the grain boundary, and saturation magne-\ntization variations in different grains (see Section III),\nthe exponents of the power law relaxations remain the\nsame as in the corresponding pure system (not shown).\nWhen the exchange coupling between grains is weakened,\nthe defects prefer to move along the grain boundaries.\nAdditionally, core switches were observed to occasion-\nally occur when a vortex/antivortex crosses over a grain\nboundary. The probability for such core switches ap-\npears to increase with weaker inter-grain exchange cou-\npling strength. Varying the saturation magnetization in\nthe grains makes the movement of the defects somewhat\nchoppy, and increases the chance of defect pinning, but\notherwise the dynamics of the relaxation process remains\nsimilartothatinthenon-disorderedPermalloyfilmscon-\nsidered above.\n10-1010-910-810-7\nt [s]10-210-1100101102103ρd(t) - ρd(tR) [1/µm2]α= 0.01\nα= 0.1\nα= 0.5\nα= 0.9\n0.0 0.2 0.4 0.6 0.8 1.0α 1.01.11.21.31.41.5ηdFigure13. Mainfigureshowstheaveragetimeevolutionofthe\ntotal number density of defects ρdforL= 4096nm with four\ndifferentα-values. Forlarger α, thepowerlawcharacterofthe\nrelaxation (black lines indicate the the power-law fits used)\nstartsearlierduetothestronglydampedinitialmagnetization\nfluctuations. The inset shows the resulting ηd-exponent as a\nfunction of α.\nV. CONCLUSIONS\nIn this paper, we have investigated the magnetic relax-\nation starting from disordered initial states of Permalloy\nthin films of various sizes by extensive micromagnetic\nsimulations. We conclude that the resulting coarsen-\ning dynamics involve complex processes and display a\nmultitude of phenomena, such as defect annihilations,\ncore switching and vortex absorption/emission, many of\nwhichhavepreviouslybeenindividuallystudiedindetail.\nTogether these phenomena result in highly nontrivial dy-\nnamics for single defects which then give rise to interest-\ning time evolution of system-wide quantities such as the\ntotal energy density and the defect densities.\nIn the defect coarsening/annihilation phase, this com-\nplexityismanifestedinparticularasslowpower-lawtem-\nporal decay characterized by non-trivial exponents of\nquantities such as the energy density of the system, of\nthe form of ρ(t)−ρ(tR)∝t−ηE, withηE= 1.22±0.08for\nthe energy density time evolution. For the defect den-\nsities, different values of ηwere observed depending on\nthe defect type: For vortices, antivortices and negative\nedge defects we find ηv= 1.51±0.05,ηav= 1.62±0.09\nandηned= 0.82±0.09, respectively. The temporal de-\ncay of the total density of defects is characterized by the\nexponentηd= 1.42±0.06. These exponents show lit-\ntle change (within error bars) when using the Gilbert\ndamping constant αwithin the range of 0.01 - 0.1, and\nare found to be robust against adding quenched disorder\nof moderate strength. When αis increased further, the\nrelaxation exponents approach the asymptotic value for\nthe XY-model with local interactions ( ηd= 1)1. This\nshould be due to the large damping practically eliminat-\ning the precessional motion of the magnetic moments so10\nthat they align with the local effective field almost im-\nmediately; thus, the dynamics of the magnetic moments\nstarts to resemble that of the XY-model in the no-inertia\n(overdamped) limit. Our results thus suggest that the\nrelatively low damping of Permalloy has a key role in the\nemergence of the non-trivial values of the relaxation ex-\nponents, and that quenched disorder, present in any real\nsamples, is irrelevant for the relaxation exponent values.\nDue to the relatively small size of the films and as\na consequence the number of defects (about 500 in the\nlargest films at the initial stages of the coarsening phase),\nthe power-law relaxation phase of energy and defect den-\nsities was limited in time to roughly one or two or-\nders of magnitude. Thus, simulations and experiments\nwith larger films and, consequently, longer relaxation\ntimes, would be useful. For experimental investigation,\ntime-resolvedX-rayimagingtechniquesshouldhavegood\nenough spatial and temporal resolutions (25 - 30 nm and\n70 - 100 ps, respectively)39–41to observe the defects andtheir dynamics. Even though these resolutions are still\nlimited when compared to our simulations, the longer\nrelaxation times and larger inter-defect separations ex-\npected during the later stages of the relaxation process\nin larger films (e.g., with linear sizes in the range of\ntens of microns) should make it possible to experimen-\ntally observe the approximate time evolution of the vor-\ntex/antivortex number densities.\nACKNOWLEDGMENTS\nWe thank Mikko Alava for useful comments. We ac-\nknowledge the support of the Academy of Finland via an\nAcademy Research Fellowship (LL, projects no. 268302\nand 273474), and the Centres of Excellence Programme\n(2012-2017, project no. 251748). We acknowledge the\ncomputational resources provided by the Aalto Univer-\nsity School of Science Science-IT project.\n∗ilari.rissanen@aalto.fi\n1B. Yurke, A. N. Pargellis, T. Kovacs, and D. A. Huse,\nPhys. Rev. E 47, 1525 (1993).\n2T. Nagaya, H. Hotta, and H. Oriharaand Yoshi-\nhiro Ishibashi, J. Phys. Soc. 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Van Waeyenberge, Nat.\nPhys.5, 332 (2009)."    },    {        "title": "1605.06797v1.Low_Gilbert_damping_in_Co2FeSi_and_Fe2CoSi_films.pdf",        "content": "arXiv:1605.06797v1  [cond-mat.mtrl-sci]  22 May 2016Low Gilbert damping in Co 2FeSi and Fe 2CoSi films\nChristian Sterwerf,1,∗Soumalya Paul,2Behrouz Khodadadi,2Markus Meinert,1\nJan-Michael Schmalhorst,1Mathias Buchmeier,2Claudia K. A. Mewes,2Tim Mewes,2and G¨ unter Reiss1\n1Center for Spinelectronic Materials and Devices,\nPhysics Department, Bielefeld University, Germany\n2Department of Physics and Astronomy/MINT Center,\nThe University of Alabama, Tuscaloosa, AL 35487, USA\n(Dated: August 15, 2018)\nThin highly textured Fe 1+xCo2−xSi (0≤x≤1) films were prepared on MgO (001) substrates\nby magnetron co-sputtering. The magneto-optic Kerr effect ( MOKE) and ferromagnetic resonance\n(FMR) measurements were used to investigate the compositio n dependence of the magnetization,\nthe magnetic anisotropy, the gyromagnetic ratio and the rel axation of the films. The effective mag-\nnetization for the thin Fe 1+xCo2−xSi films, determined by FMR measurements, are consistent wit h\nthe Slater Pauling prediction. Both MOKE and FMR measuremen ts reveal a pronounced fourfold\nanisotropy distribution for all films. In addition we found a strong influence of the stoichiometry\non the anisotropy as the cubic anisotropy strongly increase s with increasing Fe concentration. The\ngyromagnetic ratio is only weakly dependent on the composit ion. We find low Gilbert damping pa-\nrameters for all films with values down to 0 .0012±0.00012 for Fe 1.75Co1.25Si. The effective damping\nparameter for Co 2FeSi is found to be 0 .0018±0.0004. We also find a pronounced anisotropic relax-\nation, which indicates significant contributions of two-ma gnon scattering processes that is strongest\nalong the easy axes of the films. This makes thin Fe 1+xCo2−xSi films ideal materials for the appli-\ncation in STT-MRAM devices.\nI. INTRODUCTION\nHalf-metallic ferromagnets have attracted great inter-\nest during the past few years because they promise to\nboost the performance of spintronic devices. High spin\npolarization at the Fermi level can generate high tun-\nnel magnetoresistance (TMR) ratios. A TMR effect can\nbe measured in a magnetic tunnel junction (MTJ) that\nconsists of two ferromagnetic films separated by a thin\ninsulator. The same structures can also be utilized to\nspin transfer torque induced magnetization switching [1],\nhoweverin this casea lowswitching currentdensity is de-\nsirable. Thus, low magnetic damping and a high spin po-\nlarization are frequently required for spin transfer torque\nbased devices [2]. A high spin polarization can be found\nin half-metals where one spin band structure is semicon-\nducting while the other spin band structure is metallic.\nCo- and Fe-based Heusler compounds are good candi-\ndates for materials with high Curie temperatures and\nhalf-metallic behavior.\nFull Heusler compounds have the formula X 2YZ, where\nX and Y are transition metals and Z is a main group\nelement. There are two different ordered structures: the\nL21structure and the X astructure with a different occu-\npation sequence. Both structures consist of a four-atom\nbasis and an fcc lattice. The prototype of the L2 1struc-\nture is Cu 2MnAl (space group Fm ¯3m) with the occupa-\ntion sequence X-Y-X-Z [3]. The prototypes for the X a\nstructure are Hg 2CuTi and Li 2AgSb with an occupation\nsequence Y-X-X-Z, with the two X-atoms at inequivalent\npositions in the lattice [4, 5]. In this work, we investigate\n∗csterwerf@physik.uni-bielefeld.dethe magnetic properties of a stoichiometric series rang-\ning from Co 2FeSi to Fe 2CoSi, where Co 2FeSi crystalizes\nin the L2 1structure and Fe 2CoSi in the X astructure, re-\nspectively. Both compounds should have a (pseudo-)gap\nin the minority states as predicted by first principle cal-\nculations. By substituting Co and Fe atoms the number\nof electrons varies and the Fermi level is expected to be\nshifted to lower energies when the Fe concentration is\nincreased. As we reported previously, magnetic tunnel\njunctions based on the Fe 1+xCo2−xSi films exhibit very\nhigh TMR ratios for all stoichiometries [6]. At 15K a\nmaximum TMR ratio of 262% was found for the inter-\nmediate stoichiometry Fe 1.75Co1.25Si, while the Co 2FeSi\nand Fe 2CoSi based MTJs showed a TMR ratio of 167%\nand 227%, respectively. One possible explanation for the\nhigh TMR ratio is that for Fe 1.75Co1.25Si the Fermi en-\nergy is shifted inside the pseudo-gap. In this work we\npresent results of the magnetic properties for the mag-\nnetization dynamics in particular including anisotropy\nand the Gilbert damping parameter of the Fe 1+xCo2−xSi\nfilms, as the intrinsic relaxation is are expected to be low\nfor half-metals [7].\nII. PREPARATION AND\nCHARACTERIZATION TECHNIQUES\nThin Fe 1+xCo2−xSi (x=0, 0 .25, 0.5, 0.75, 1) films were\nfabricated using co-sputtering in an UHV sputtering sys-\ntemwithabasepressureof1 ·10−9mbar. TheArpressure\nduringsputteringwas2 ·10−3mbar. Thefilmsweregrown\nby dc- and rf-magnetron sputtering from elemental tar-\ngets ontoMgO (001) substrates. Additional MgOand Cr\nseed layers were used to accommodate small lattice mis-2\nmatches and to promote coherent and epitaxial growth,\nas the Cr seed layer grows in 45◦direction on the MgO\nlayer, which has a lattice parameter of 4 .212˚A. The lat-\ntice mismatch between two unit cells of Cr (2 ×2.885˚A\nat 20◦C [8]) and one unit cell of Co 2FeSi (5.64˚A [9])\nor Fe2CoSi (5.645˚A [10]) is about 2%. The 5nm thick\nMgO and Cr films were in-situ annealed at 700◦C to ob-\ntain smooth surfaces. Fe 1+xCo2−xSi films with a thick-\nness of 20nm were deposited at room temperature and\nex-situ vacuum annealed at 500◦C. A 2nm thick MgO\ncapping layer was used to prevent oxidation of the films.\nTo determine the stoichiometry and to adjust the sput-\ntering powers, x-rayfluorescencemeasurementswere car-\nried out. To obtain information about the magnetization\ndynamics, in-plane ferromagneticresonance(FMR) mea-\nsurements were performed using a broadband coplanar\nwaveguide setup up to a maximum frequency of 40GHz.\nLeast square fits of the raw data using a first derivative\nof a Lorenzian line shape were done to precisely deter-\nmine the resonance field and the peak-to-peak linewidth\n∆H[11, 12]. For the FMR in-plane angle dependent\nmeasurements the samples were mounted on a rotating\nstage and the resonance spectra were measured at a fre-\nquencyof30GHz whilethe in-planeanglewaschangedin\n5◦steps. In addition quasistatic magnetization reversal\nmeasurements were carried out using the magneto-optic\nKerr effect (MOKE) in a vector MOKE setup with an s-\npolarized laser with a wavelength of 488nm. Anisotropy\nmeasurements were carried out using a rotating sample\nholder. The magnetic field was applied in the plane of\nthe films.\nIII. CRYSTALLOGRAPHIC PROPERTIES\nX-ray diffraction measurements were used to investi-\ngate the crystallographic properties of the Fe 1+xCo2−xSi\nfilms. Ordering parameters, determined from x-ray\ndiffraction, were already discussed in our previous work\n[6] and found to be high for Co 2FeSi and decrease when\ngoing to Fe 2CoSi. In order to test the films for crystal-\nlographic symmetry ϕscans are performed on the (220)\nplanes of the Fe 1+xCo2−xSi films. Figure 1 shows the\nresults together with the (220) plane of the MgO (001)\nsubstrate. The result shows that the (100) Heusler plane\nis rotated by 45◦with respect to the MgO (100) plane.\nThe fourfold symmetry of the ϕ-scans clearly verifies the\nhighly textured growth of all Fe 1+xCo2−xSi films of this\nstudy.\nIV. MAGNETIZATION DYNAMICS\nIn this section we present in-plane broadband FMR\nmeasurements for the Fe 1+xCo2−xSi samples to obtain\ninformation about the magnetic properties of the films.\nThe Landau-Lifshitz-Gilbert equation describes the dy-\nnamics of the magnetization vector /vectorMin the presencesqrt intensity (a.u.) \n360 315 270 225 180 135 90 45 0\n (°)x=0.75 x=0.5 x=0.25 x=0   MgO \nsubstrate \nx=1 \nFIG. 1. ϕ-scans of the (220) Fe 1+xCo2−xSi peak and (220)\nMgO substrate peak showing the fourfold symmetry of the\nfilms.\n40 \n30 \n20 \n10 \n0f (GHz) \n8 6 4 2 0\nH (kOe)  Fe2CoSi [100]\n Fe2CoSi [110]\nFIG. 2. Resonance frequency versus magnetic field (Kittel\nplot) along the in-plane magnetic hard [110] and the magneti c\neasy [100] axis for Fe 2CoSi. The experimental data are fitted\nusing a combined fit (equations (3 and 4)) to determine Meff\nandγ′.\nof an effective field /vectorHeff, which contains both dc and ac\nfields.\nIt is given by [13]:\nd/vectorM\ndt=−γ/vectorM×/vectorHeff+α\nM/parenleftBigg\n/vectorM×d/vectorM\ndt/parenrightBigg\n,(1)\nwhereγis the gyromagnetic ratio and the quantity pa-\nrameter αis the Gilbert damping parameter. Accord-\ning to the Landau-Lifshitz-Gilbert equation (1), the res-\nonance condition can be expressed in terms of the sec-\nond derivatives of the free-energy density Eby the Smit-\nBeljers formula [14]:\n/parenleftbiggf\nγ′/parenrightbigg2\n=1\n(Msinθ)2/bracketleftBigg\n∂2E\n∂θ2∂2E\n∂ϕ2−/parenleftbigg∂2E\n∂θ∂ϕ/parenrightbigg2/bracketrightBigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nθ0,ϕ0,(2)\nwhereγ′=γ/2π,θandϕare the polar and azimuthal\nangles of the magnetization /vectorMandθ0andϕ0the corre-3\nMeff ( B / f.u.) \n0 0.5 1 0.25 0.75\nx7\n6\n5\n4\n3\n2\n1\n07\n6\n5\n4\n3\n2\n1\n0\nMS ( B / f.u.) \nFIG. 3. Dependence of the fitted effective magnetic moment\nper formula unit for Fe 1+xCo2−xSi films with x=0, 0 .25, 0.5,\n0.75, 1shownontheleftaxis. Thedashedlineshows theinter-\npolated expected magnetic moments according to the Slater-\nPauling rule (right axis).\nsponding equilibrium values. Measurements of the mag-\nnetic field dependent resonance frequency were carried\nout in two different orientations of the sample: in [100]\nand [110] direction of the Fe 1+xCo2−xSi Heusler alloy, as\nthe [100] direction is the magnetic easy axis and the [110]\ndirection the magnetic hard axis, respectively. Figure 2\nshowsthe exemplaryKittel plots along[100]and [110]di-\nrections for the Fe 2CoSi sample. The experimental data\nwere fitted simultaneously using the Kittel equation for\nboth easy and hard configurations [15]:\nf=γ′/radicalbigg\n(Hres−ha−H4)(Hres−ha+H4\n2+4πMeff) (3)\nf=γ′/radicalbig\n(Hres−ea+H4)(Hres−ea+H4+4πMeff) (4)\nwhereMeff,γ′andH4are shared fit parameters. H4de-\nscribes the magnitude of the in-plane fourfold anisotropy\nfield.Hres−haandHres−eadenote the resonance field\nalong the magnetic hard and the magnetic easy axis,\nrespectively. The resulting fit parameters for the gyro-\nmagnetic ratio γ′are presented in Fig. 6 a) for all xin\nFe1+xCo2−xSi. Within the errorbarsitisnearlyconstant\nfor x≥0.25 and slightly smaller for Co 2FeSi. The fitted\neffective magnetization, which includes any perpendicu-\nlar anisotropy present in the films, is shown in Fig. 3\nfor the Fe 1+xCo2−xSi samples. The error bars originate\nfrom fitting of the Kittel equations and the determina-\ntion of the volume of the unit cell. For bulk Co 2FeSi\nand Fe 2CoSi the experimentally determined magnetiza-\ntionsare5 .95µB/f.u.[9]and4 .99µB/f.u.[10], respectively,\nwhich match the expected magnetizations according to\nthe Slater-Pauling rule (visualized by the dashed line\nin Fig. 3 on the right axis). The deviation from the\nexpected values might be attributed to residual atomic\ndisorder in the films or the presence of a perpendicular\nanisotropy caused by a small tetragonal distortion in the\n[001] direction.100\n80 \n60 \n40 \n20 \n0H (Oe)\n40 30 20 10 0\nf (GHz) Co2FeSi\n Fe1.25Co 1.75Si \n Fe1.5Co 1.5Si \n Fe1.75 Co1.25Si \n Fe2CoSi\nFIG. 4. Frequency dependent FMR linewidth for all sam-\nples measured along the magnetic hard axis [110] of the\nFe1+xCo2−xSi films.\nThe frequency dependence of the linewidth of the fer-\nromagnetic resonance absorption provides direct infor-\nmation about the magnetic relaxation. The frequency\ndependence of the linewidth [16, 17] can under certain\nconditions be characterized by an inhomogeneous resid-\nual linewidth at zero field ∆ H0and an intrinsic contri-\nbution [18]:\n∆H= ∆H0+2√\n3αeff\nγ′f. (5)\nFor correct determination of the effective damping pa-\nrameter it is necessary to measure the linewidth over a\nwide frequency range to determine the slope. It is not\nsufficient to measure ∆ Hat a fixed frequency, because\na non-zero extrinsic linewidth ∆ H0results in an over-\nestimated damping parameter αeff. Figure 4 shows the\npeak-to-peak linewidth ∆ Hfor all frequencies and all\nx. The measurements were performed in the direction\nof the magnetic hard axis of the Heusler films. The ex-\nperimental data were fitted by equation (5) to determine\nthe effective damping parameters. The slope at higher\nfrequencies was used to determine the damping parame-\nters. The inhomogeneous residual linewidth at zero field\n∆H0is presented in Fig. 6 b) for all stoichiometries. The\nerror margins result from the different slopes in the ∆ H\nvs.fcurves. The residual linewidth decreases as the\nFe concentration increases and reaches its lowest value of\n∆H0= 12Oe for Fe 2CoSi. McMichael et al.[19] found\nthat small grain size distributions can lead to low inho-\nmogeneous line broadening.\nThe effective Gilbert damping parameter αeffis shown\nin Fig. 6 c). All damping parameters have the same\norder of magnitude and vary between 0 .0012±0.00012\nto 0.0019±0.00013. The very upper limit of the er-\nror margins was calculated assuming that the linewidth\nmeasured at 40GHz is caused solely by Gilbert type\ndamping. Co 2FeSi exhibits a damping parameter of\n0.0018±0.0004, while Fe 2CoSi shows a slightly larger\nvalue of 0 .0019±0.00013. Kasatani et al.found damping4\n60 \n50 \n40 \n30 \n20 \n10 \n0) e O (  H\n40 30 20 10 0\nf (GHz)[100]\n[110]Fe 2CoSi\nmagnetic hard axis magnetic easy axis \nFIG. 5. FMR linewidth for Fe 2CoSi measured along both the\nmagnetic hard [110] and magnetic easy [100] axis.\nparameters from 0 .0023 to 0 .0061 for Co 2FeSi films and\n0.002for Fe 2CoSi [20]. In general, the Gilbert damping is\nexpected to be low in half-metallic materials, where spin-\nflip processes are suppressed [7, 21]. The small damping\nparameters of the metallic films show that a pseudo-gap\naspresentinthe Fe 1+xCo2−xSisystemissufficienttogive\nrise to a low Gilbert damping.\nFigure 5 shows the frequency dependent linewidth\nalong easy and hard axes for the Fe 2CoSi. The linewidth\nexhibits almost linearbehavior(the Gilbert model) along\nthe hard axis. We observed non-linear behavior in the\nlinewidth vs. frequency responsealongthe magneticeasy\naxis. ThisnonlineardependenceoftheFMRlinewidthon\nfrequencyisatypicalobservationwhentwomagnonscat-\ntering contributes significantly to the relaxation [22, 23].\nTwo-magnon scattering is an extrinsic relaxation mecha-\nnism and can be induced by means of different scattering\ncenters such as voids or pores [24], surface roughness [22]\nand grain size [25] or by network of misfit dislocations\nwhich causes scattering of the FMR mode (k=0) into\npropagating spin waves (k /negationslash=0).\nA. FMR in-plane rotation measurements\nTo obtain further information about the magnetic\nanisotropies and magnetic relaxation additional FMR\nmeasurements were carried out as a function of the\nin-plane angle of the applied field with respect to the\nFe1+xCo2−xSi[110]axis. Theoperatingfrequencyforthe\nrotation measurements was 30GHz. At this frequency\nthe resonancefields are high enough to saturate the mag-\nnetization along the easy and hard axes. All measure-\nments were performed at room temperature.\nA fourfold symmetry is observed in the in-plane angle\ndependence of the ferromagnetic resonance field for all\nsamples. Figure7a)exemplarilyshowstheferromagnetic\nresonancefield Hresversus the in-plane rotation angle for\nFe2CoSi. The dependence of the resonance field on the\nin-plane angle was simulated numerically using equationH0) e O (  2.94\n2.92\n2.90\n2.88\n2.86' (MHz/Oe) \n60 \n40 \n20 \n0K4e k (  rmc/g3)\n0 0.5 1 0.25 0.75\nxa) \nb) \nc) \nd) eff 60 \n50 \n40 \n30 \n20 \n10 \n0\n0.005\n0.004\n0.003\n0.002\n0.001\n0.000\nFIG. 6. a) Gyromagnetic ratio γ′, b)Extrinsic contribution to\nthe linewidth ∆ H0of the FMR spectra, c) effective Gilbert\ndamping parameter and d) cubic magnetic anisotropy con-\nstantK4for Fe 1+xCo2−xSi films with x= 0, 0.25, 0.5, 0.75\n,1.\n(2), assuming a cubic magnetic anisotropy contribution\nto the Gibbs free energy [26, 27]:\nEcubic=−1\n2K4/parenleftbig\nα4\n1+α4\n2+α4\n3/parenrightbig\n, (6)\nwhereK4is the cubic magnetic anisotropy constant and\nα1,α2,α3are the directional cosines with respect to\nthe cubic principal axes. The experimentally determined\nin-plane angle dependent Hresdata were fitted with the\nnumerical solution (red line in Fig. 7 a)) to determine\nthe cubic anisotropy constant. Figure 7 b) shows the\ncorresponding linewidth data, which also shows a clear\nfourfoldsymmetry. The linewidth exhibitsmaximaalong\nthe easy axes and minima along the hard axes of the\ncubic magnetic anisotropy. Randomly distributed crys-\ntalline defects oriented along the in-plane principal crys-\ntallographic axis [28] or a fourfold distribution in misfit\ndislocations [29] which induce the same symmetry on the\nstrength of two magnon scattering can explain the ob-\nserved anisotropic relaxation.\nThe magnetic fourfold symmetry matches the crystal-\nlographic symmetry of the highly textured Fe 1+xCo2−xSi\nfilms mentioned before. A polar plot of the MOKE\nsquareness versus the rotational angle for Fe 2CoSi is pre-\nsented in Fig. 8. This measurement confirms the cubic5\n5100520053005400\n5200\n5300\n5400045 90 \n135\n180\n225\n270315\n020 40 60 \n20 \n40 \n60 045 90 \n135\n180\n225\n270315Hres  (O e) \n[110] [110] \n[100][100] a) \nb) \nH (Oe) \nFIG. 7. Polar plots of a) the resonance fields H resand b)\nthe linewidth ∆ Has a function of the in-plane angle of the\napplied field with respect to the [110] axis of a 20nm thick\nFe2CoSi film measured at a microwave frequency of 30GHz.0.60.81\n0.8\n1045 90 \n135\n180\n225\n270315[110] [100]\nMR/M S\nFIG.8. Polar plotsofthesquarenessMR\nMSforFe 2CoSiobtained\nby MOKE measurements.\nanisotropy present in the films as seen in the FMR mea-\nsurement. The magnetic easy axis is located along the\n[100] crystallographic axis and the magnetic hard axis\nis located along the [110] crystallographic axis. A cubic\nanisotropy with the easy magnetic axis in the Heusler\n[100] direction is found for all samples. The cubic mag-\nnetic anisotropy constant K4obtained from the FMR\nmeasurements changes significantly in this series from\n55.8kerg\ncm3for Fe 2CoSi to 16 .6kerg\ncm3for Co 2FeSi, respec-\ntively. The cubic anisotropy constants for all stoichiome-\ntries are presented in Fig. 6 d). Hashimoto et al.found a\nsimilarcubicanisotropyconstantof18kerg\ncm3forcrystalline\nCo2FeSi with a film thickness of 18 .5nm [30]. Some films\nshowanadditionaluniaxialanisotropycomponent,which\ncan originate from miscut substrates.\nV. CONCLUSION\nIn summary we found very small damping parame-\ntersforthehalf-metallicFe 1+xCo2−xSifilmsvaryingfrom\n0.0012±0.00012 to 0 .0019±0.00013. Co 2FeSi exhibits\na damping parameter of 0 .0018±0.0004. Thus, the\nfilms are suitable for the use in STT-MRAMs. FMR\nandMOKEmeasurementsrevealafourfoldmagnetocrys-\ntalline anisotropy for all films in accordance with the\nfourfold crystalline symmetry in the highly textured\nfilms. The need for frequency dependent FMR measure-\nments was exemplified by the finding that the residual\nlinewidth changes both with composition and with the\nmeasurement direction.\nACKNOWLEDGMENTS\nThe authors gratefully acknowledge financial support\nfrom Bundesministerium f¨ ur Bildung und Forschung\n(BMBF) and Deutsche Forschungsgemeinschaft (DFG,\ncontract no. RE 1052/32-1) as well as support through\nthe MINT Center summer program. S. Paul, B. Kho-\ndadadi and T. Mewes would like to acknowledge sup-\nport by the NSF-CAREER Award No. 0952929, C.K.A.\nMewes would like to acknowledge support by the NSF-\nCAREER Award No. 1452670.6\n[1] L. Berger, Physical Review B 54, 9353 (1996).\n[2] J. C. Slonczewski, Journal of Magnetism and Magnetic\nMaterials 159, L1 (1996).\n[3] A. J. Bradley and J. W. Rodgers, Proceedings of the\nRoyal Society of London Series A 144, 340 (1934).\n[4] M. Puselj and Z. Ban, Croat. Chem. Acta 41, 79 (1969).\n[5] H. Pauly, A. Weiss, and H. Witte, Z. Metallkunde 59,\n47 (1968).\n[6] C. Sterwerf, M. Meinert, J.-M. Schmalhorst, and\nG. Reiss, IEEE Transactions on Magnetics 49, 4386\n(2013).\n[7] C. Liu, C. Mewes, M. Chshiev, and T. Mewes, Applied\nPhysics Letters 95, 022509 (2009).\n[8] M. E. Straumanis and C. C. Weng,\nActa Crystallographica 8, 367 (1955).\n[9] S. Wurmehl, G. Fecher, H. Kandpal, V. Ksenofontov,\nC. Felser, H.-J. Lin, and J. Morais, Physical Review B\n72, 184434 (2005).\n[10] H. Luo, Z. Zhu, L. Ma, S. Xu, H. Liu, J. Qu, Y. Li, and\nG. Wu, Journal of Physics D: Applied Physics 40, 7121\n(2007).\n[11] M. E. Schabes, H. Zhou, and H. N. Bertram, Journal of\nApplied Physics 87, 5666 (2000).\n[12] N. Pachauri, B. Khodadadi, and M. Althammer, Journal\nof Applied Physics 117, 233907 (2015).\n[13] B. Heinrich and J. F. Cochran, Advances in Physics 42,\n523 (1993).\n[14] H. Suhl, Physical Review 97, 555 (1955).\n[15] X. Liu, Y. Sasaki, and J. K. Furdyna, Physical Review\nB67, 205204 (2003).[16] B. Heinrich, J. F. Cochran, and R. Hasegawa, Journal\nof Applied Physics 57, 3690 (1985).\n[17] H. Lee, L. Wen, M. Pathak, and P. Janssen, Journal of\nPhysics D: Applied Physics 41, 215001 (2008).\n[18] C. Mewes and T. Mewes, Relaxation in Magnetic Ma-\nterials for Spintronics, in: Handbook of Nanomagnetism\n(Pan Stanford, 2015) p. 74.\n[19] R. D. McMichael, D. J. Twisselmann, and A. Kunz,\nPhys. Rev. Lett. 90, 227601 (2003).\n[20] Y. Kasatani, S. Yamada, H. Itoh, M. Miyao, K. Hamaya,\nand Y. Nozaki, Applied Physics Express 7, 123001\n(2014).\n[21] G. M. M¨ uller, J. Walowski, M. Djordjevic, and G. X.\nMiao, Nature Materials 8, 56 (2009).\n[22] H. Lee, Y. Wang, C. Mewes, and W. H. Butler, Applied\nPhysics Letters 95, 082502 (2009).\n[23] P. Landeros, R. E. Arias, and D. L. Mills, Physical Re-\nview B77, 214405 (2008).\n[24] M. J. Hurben and C. E. Patton, Journal of Applied\nPhysics83, 4344 (1998).\n[25] R. D. McMichael, M. D. Stiles, and P. J. Chen, Journal\nof Applied Physics 83, 7037 (1998).\n[26] M. Farle, Reports on Progress in Physics 61, 755 (1998).\n[27] B. Heinrich and J. Bland, eds., Radio Frequency Tech-\nniques, in: Ultrathin Magnetic Structures II (Springer,\n1994) p. 195.\n[28] I. Barsukov, P. Landeros, R. Meckenstock, J. Lindner,\nD. Spoddig, Z.-A. Li, B. Krumme, H. Wende, D. L. Mills,\nand M. Farle, Phys. Rev. B 85, 014420 (2012).\n[29] G. Woltersdorf and B. Heinrich, Physical Review B 69,\n184417 (2004).\n[30] M. Hashimoto, J. Herfort, H. P. Schonherr, and K. H.\nPloog, Applied Physics Letters 87, 102506 (2005)."    },    {        "title": "1010.1626v3.A_unified_first_principles_study_of_Gilbert_damping__spin_flip_diffusion_and_resistivity_in_transition_metal_alloys.pdf",        "content": "A uni\fed \frst-principles study of Gilbert damping, spin-\rip di\u000busion and resistivity\nin transition metal alloys\nAnton A. Starikov,1Paul J. Kelly,1Arne Brataas,2Yaroslav Tserkovnyak,3and Gerrit E. W. Bauer4\n1Faculty of Science and Technology and MESA+Institute for Nanotechnology,\nUniversity of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands\n2Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway\n3Department of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\n4Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands\n(Dated: October 25, 2018)\nUsing a formulation of \frst-principles scattering theory that includes disorder and spin-orbit\ncoupling on an equal footing, we calculate the resistivity \u001a, spin \rip di\u000busion length lsfand the\nGilbert damping parameter \u000bfor Ni 1\u0000xFexsubstitutional alloys as a function of x. For the tech-\nnologically important Ni 80Fe20alloy, permalloy, we calculate values of \u001a= 3:5\u00060:15\u0016Ohm-cm,\nlsf= 5:5\u00060:3 nm, and \u000b= 0:0046\u00060:0001 compared to experimental low-temperature values in\nthe range 4 :2\u00004:8\u0016Ohm-cm for \u001a, 5:0\u00006:0 nm forlsf, and 0:004\u00000:013 for\u000bindicating that the\ntheoretical formalism captures the most important contributions to these parameters.\nPACS numbers: 72.25.Rb, 71.70.Ej, 72.25.Ba, 75.40.Gb, 76.60.Es\nIntroduction. The drive to increase the density and\nspeed of magnetic forms of data storage has focussed at-\ntention on how magnetization changes in response to ex-\nternal \felds and currents, on shorter length- and time-\nscales [1]. The dynamics of a magnetization Min an ef-\nfective magnetic \feld He\u000bis commonly described using\nthe phenomenological Landau-Lifshitz-Gilbert equation\ndM\ndt=\u0000\rM\u0002He\u000b+M\u0002\"~G(M)\n\rM2sdM\ndt#\n;(1)\nwhereMs=jMjis the saturation magnetization, ~G(M)\nis the Gilbert damping parameter (that is in general a\ntensor) and the gyromagnetic ratio \r=g\u0016B=~is ex-\npressed in terms of the Bohr magneton \u0016Band the Land\u0013 e\ngfactor, which is approximately 2 for itinerant ferromag-\nnets. The time decay of a magnetization precession is\nfrequently expressed in terms of the dimensionless pa-\nrameter\u000bgiven by the diagonal element of ~G=\rMsfor\nan isotropic medium. If a non-equilibrium magnetization\nis generated in a disordered metal (for example by inject-\ning a current through an interface), its spatial decay is\ndescribed by the di\u000busion equation\n@2\u0001\u0016\n@z2=\u0001\u0016\nl2\nsf(2)\nin terms of the spin accumulation \u0001 \u0016, the di\u000berence be-\ntween the spin-dependent electrochemical potentials \u0016s\nfor up and down spins, and the spin-\rip di\u000busion length\nlsf[2, 3]. In spite of the great importance of \u000bandlsf,\nour understanding of the factors that contribute to their\nnumerical values is at best sketchy. For clean ferromag-\nnetic metals [4] and ordered alloys [5] however, recent\nprogress has been made in calculating the Gilbert damp-\ning using the Torque Correlation Model (TCM) [6] and\nthe relaxation time approximation in the framework ofthe Boltzmann equation. Estimating the relaxation time\nfor particular materials and scattering mechanisms is in\ngeneral a non-trivial task and application of the TCM to\nnon-periodic systems entails many additional complica-\ntions and has not yet been demonstrated. Thus, the the-\noretical study of Gilbert damping or spin-\rip scattering\nin disordered alloys and their calculation for particular\nmaterials with intrinsic disorder remain open questions.\nMethod. In this paper we calculate the resistivity \u001a,\nspin-\rip di\u000busion length lsfand Gilbert damping param-\neter\u000bfor substitutional Ni 1\u0000xFexalloys within a single\n\frst-principles framework. To do so, we have extended a\nscattering formalism [7] based upon the local spin den-\nsity approximation (LSDA) of density functional theory\n(DFT) so that spin-orbit coupling (SOC) and chemical\ndisorder are included on an equal footing. Relativistic\ne\u000bects are included by using the Pauli Hamiltonian.\nFor a disordered region of ferromagnetic (F) alloy sand-\nwiched between leads of non-magnetic (N) material, the\nscattering matrix Srelates incoming and outgoing states\nin terms of re\rection ( r) and transmission matrices ( t)\nat the Fermi energy. To calculate the scattering ma-\ntrix, we use a \\wave-function matching\" (WFM) scheme\n[7] implemented with a minimal basis of tight-binding\nlinearized mu\u000en-tin orbitals (TB-LMTOs) [8]. Atomic-\nsphere-approximation (ASA) potentials [8] are calculated\nself-consistently using a surface Green's function (SGF)\nmethod also implemented [9] with TB-LMTOs. Charge\nand spin densities for binary alloy AandBsites are calcu-\nlated using the coherent potential approximation (CPA)\n[10] generalized to layer structures [9]. For the transmis-\nsion matrix calculation, the resulting spherical potentials\nare assigned randomly to sites in large lateral supercells\n(SC) subject to maintenance of the appropriate concen-\ntration of the alloy [7]. Solving the transport problem\nusing lateral supercells makes it possible to go beyondarXiv:1010.1626v3  [cond-mat.mtrl-sci]  19 May 20112\ne\u000bective medium approximations such as the CPA. Be-\ncause we are interested in the properties of bulk alloys,\nthe leads can be chosen for convenience and we use Cu\nleads with a single scattering state for each value of crys-\ntal momentum, kk. The alloy lattice constants are de-\ntermined using Vegard's law and the lattice constants of\nthe leads are made to match. Though NiFe is fcc only\nfor the concentration range 0 \u0014x\u00140:6, we use the fcc\nstructure for all values of x.\nFor the self-consistent SGF calculations (without\nSOC), the two-dimensional (2D) Brillouin zone (BZ) cor-\nresponding to the 1 \u00021 interface unit cell was sampled\nwith a 120\u0002120 grid. Transport calculations including\nspin-orbit coupling were performed with a 32 \u000232 2D BZ\ngrid for a 5\u00025 lateral supercell, which is equivalent to\na 160\u0002160 grid in the 1 \u00021 2D BZ. The thickness of\nthe ferromagnetic layer ranged from 3 to 200 monolay-\ners of fcc alloy; for the largest thicknesses, the scattering\nregion contained more than 5000 atoms. For every thick-\nness of ferromagnetic alloy, we averaged over a number\nof random disorder con\fgurations; the sample to sample\nspread was small and typically only \fve con\fgurations\nwere necessary.\nResistivity. We calculate the electrical resistivity to\nillustrate our methodology. In the Landauer-B uttiker\nformalism, the conductance can be expressed in terms of\nthe transmission matrix tasG= (e2=h)Tr\b\ntty\t\n[11, 12].\nThe resistance of the complete system consisting of ideal\nleads sandwiching a layer of ferromagnetic alloy of thick-\nnessLisR(L) = 1=G(L) = 1=GSh+ 2Rif+Rb(L) where\nGSh=\u0000\n2e2=h\u0001\nNis the Sharvin conductance of each lead\nwithNconductance channels per spin, Rifis the interface\nresistance of a single N jF interface, and Rb(L) is the bulk\nresistance of a ferromagnetic layer of thickness L[7, 13].\nWhen the ferromagnetic slab is su\u000eciently thick, Ohmic\nbehaviour is recovered whereby Rb(L)\u0019\u001aLas shown in\nthe inset to Fig. 1 for permalloy (Py = Ni 80Fe20) and\nthe bulk resistivity \u001acan be extracted from the slope\nofR(L) [14]. For currents parallel and perpendicular to\nthe magnetization direction, the resistivities are di\u000berent\nand have to be calculated separately. The average resis-\ntivity is given by \u0016 \u001a= (\u001ak+ 2\u001a?)=3, and the anisotropic\nmagnetoresistance ratio (AMR) is ( \u001ak\u0000\u001a?)=\u0016\u001a.\nFor permalloy we \fnd values of \u0016 \u001a= 3:5\u00060:15\u0016Ohm-\ncm and AMR = 19 \u00061%, compared to experimental low-\ntemperature values in the range 4 :2\u00004:8\u0016Ohm-cm for\n\u0016\u001aand 18% for AMR [15]. The resistivity calculated as a\nfunction of xis compared to low temperature literature\nvalues [15] in Fig. 1. The plateau in the calculated values\naround the Py composition appears to be seen in the\nexperiments by Smit and Jaoul et al. [15]. The overall\nagreement with previous calculations is good [16]. In\nspite of the smallness of the SOC, the resistivity of Py\nis underestimated by more than a factor of four when it\nis omitted, underlining its importance for understanding\ntransport properties.\n0 20 40 60 80 1000123456ρ [µΩ ⋅ cm]\nFe concentration [%]With SOC\nWithout SOCCadeville\nMcGuire\nJaoul\nSmit\n0 10 20 30\n123R|| [fΩ ⋅ m2]L [nm]FIG. 1. Calculated resistivity as a function of the concen-\ntrationxfor fcc Ni 1\u0000xFexbinary alloys with (solid line) and\nwithout (dashed-dotted line) SOC. Low temperature experi-\nmental results are shown as symbols [15]. The composition\nNi80Fe20is indicated by a vertical dashed line. Inset: resis-\ntance of CujNi80Fe20jCu as a function of the thickness of the\nalloy layer. Dots indicate the calculated values averaged over\n\fve con\fgurations while the solid line is a linear \ft.\nThree sources of disorder which have not been taken\ninto account here will increase the calculated values of\n\u001a; short range potential \ructuations that go beyond the\nsingle site CPA, short range strain \ructuations re\recting\nthe di\u000bering volumes of Fe and Ni and spin disorder.\nThese will be the subject of a later study.\nGilbert Damping. Recently, Brataas et al. showed\nthat the energy loss due to Gilbert damping in an N jFjN\nscattering con\fguration can be expressed in terms of the\nscattering matrix S[17]. Using the Landau-Lifshitz-\nGilbert equation (1), the energy lost by the ferromagnetic\nslab is,\ndE\ndt=d\ndt(He\u000b\u0001M) =He\u000b\u0001dM\ndt=1\n\r2dm\ndt~G(M)dm\ndt\n(3)\nwhere m=M=Msis the unit vector of the magnetization\ndirection for the macrospin mode. By equating this en-\nergy loss to the energy \row into the leads [18] associated\nwith \\spin-pumping\" [19],\nIPump\nE =~\n4\u0019Tr\u001adS\ndtdSy\ndt\u001b\n=~\n4\u0019Tr\u001adS\ndmdm\ndtdSy\ndmdm\ndt\u001b\n;\n(4)\nthe elements of the tensor ~Gcan be expressed as\n~Gi;j(m) =\r2~\n4\u0019Re\u001a\nTr\u0014@S\n@mi@Sy\n@mj\u0015\u001b\n: (5)\nPhysically, energy is transferred slowly from the spin de-\ngrees of freedom to the electronic orbital degrees of free-\ndom from where it is rapidly lost to the phonon degrees\nof freedom. Our calculations focus on the role of elastic\nscattering in the rate-limiting \frst step.\nAssuming that the Gilbert damping is isotropic for cu-\nbic substitutional alloys and allowing for the enhance-\nment of the damping due to the F jN interfaces [19{21],3\n0 20 40 60 80 10002468101214α [x 10−3]\nFe concentration [%]Rantschler\nIngvarsson\nMizukami\nNakamuraPatton\nBailey\nBonin\nNibargerInaba\nLagae\nOogane0 5 10 15 20 2500.050.10.15G/(γ ⋅ µs A) [nm]\nL [nm]\nFIG. 2. Calculated zero temperature (solid line) and exper-\nimental room temperature (symbols) values of the Gilbert\ndamping parameter as a function of the concentration xfor\nfcc Ni 1\u0000xFexbinary alloys [21{23]. Inset: total damping of\nCujNi80Fe20jCu as a function of the thickness of the alloy\nlayer. Dots indicate the calculated values averaged over \fve\ncon\fgurations while the solid line is a linear \ft.\nthe total damping in the system with a ferromagnetic slab\nof thickness Lcan be written ~G(L) =~Gif+~Gb(L) where\nwe express the bulk damping in terms of the dimension-\nless Gilbert damping parameter ~Gb(L) =\u000b\rMs(L) =\n\u000b\r\u0016sAL, where\u0016sis the magnetization density and Ais\nthe cross section. The results of calculations for Ni 80Fe20\nare shown in the inset to Fig. 2, where the derivatives of\nthe scattering matrix in (5) were evaluated numerically\nby taking \fnite di\u000berences. The intercept at L= 0, ~Gif,\nallows us to extract the damping enhancement [20] but\nhere we focus on the bulk properties and leave consid-\neration of the material dependence of the interface en-\nhancement for later study. The value of \u000bdetermined\nfrom the slope of ~G(L)=(\r\u0016sA) is 0:0046\u00060:0001 that\nis at the lower end of the range of values 0 :004\u00000:013\nmeasured at room temperature for Py [21{23].\nFig. 2 shows the Gilbert damping parameter as a func-\ntion ofxfor Ni 1\u0000xFexbinary alloys in the fcc structure.\nFrom a large value for clean Ni, it decreases rapidly to a\nminimum at x\u00180:65 and then grows again as the limit\nof clean fccFe is approached. Part of the decrease in\n\u000bwith increasing xcan be explained by the increase in\nthe magnetic moment per atom as we progress from Ni\nto Fe. The large values of \u000bcalculated in the dilute al-\nloy limits can be understood in terms of conductivity-like\nenhancement at low temperatures [24] that has been ex-\nplained in terms of intraband scattering [4, 6]. The trend\nexhibited by the theoretical \u000b(x) is seen to be re\rected\nby experimental room temperature results. In spite of\na large spread in measured values, these seem to be sys-\ntematically larger than the calculated values. Part of this\ndiscrepancy can be attributed to an increase in \u000bwith\ntemperature [22, 25].\nSpin di\u000busion. When an unpolarized current is in-\njected from a normal metal into a ferromagnet, the polar-\nization will return to the value characteristic of the bulk\n0 5 10 15 20 25 3000.20.40.60.81\nz [nm]  \n1+β\n2\n1−β\n2p↑\np↓FIG. 3. Fractional spin-current densities for electrons injected\natz= 0 from Cu into Ni 80Fe20alloy. Calculated values (sym-\nbols) and \fts to Eq. (6) (solid lines).\nferromagnet su\u000eciently far from the injection point, pro-\nvided there are processes which allow spins to \rip. Fol-\nlowing Valet-Fert [3] and assuming there is no spin-\rip\nscattering in the N leads, we can express the fractional\nspin current densities p\"(#)=J\"(#)=Jas a function of\ndistancezfrom the interface as\np\"(#)(z) =1\n2\u0006\f\n2\u0014\n1\u0000exp(\u0000z=lsf)r\u0003\nif(\f\u0000\r+\rsech\u000e)\n\f(r\u0003\nif+lsf\u000e\u001a\u0003\nFtanh\u000e)\u0015\n;\n(6)\nwhereJis the total current through the device, J\"\nandJ#are the currents of majority and minority elec-\ntrons, respectively, lsfis the spin-di\u000busion length, \u001a\u0003\nF=\n(\u001a#+\u001a\")=4 is the bulk resistivity and \fis the bulk spin\nasymmetry ( \u001a#\u0000\u001a\")=(\u001a#+\u001a\"). The interface resistance\nr\u0003\nif= (r#\nif+r\"\nif)=4, the interface resistance asymmetry\n\r= (rif#\u0000r\"\nif)=(r#\nif+r\"\nif) and the interface spin-relaxation\nexpressed through the spin-\rip coe\u000ecient \u000e[26] must be\ntaken into consideration resulting in a \fnite polarization\nof current injected into the ferromagnet. The correspond-\ning expressions are plotted as solid lines in Fig. 3.\nTo calculate the spin-di\u000busion length we inject non-\npolarized states from one N lead and probe the transmis-\nsion probability into di\u000berent spin-channels in the other\nN lead for di\u000berent thicknesses of the ferromagnet. Fig. 3\nshows that the calculated values can be \ftted using ex-\npressions (6) if we assume that J\u001b=J=G\u001b=G, yielding\nvalues of the spin-\rip di\u000busion length lsf= 5:5\u00060:3 nm\nand bulk asymmetry parameter \f= 0:678\u00060:003 for\nNi80Fe20alloy compared to experimentally estimated val-\nues of 0:7\u00060:1 for\fand in the range 5 :0\u00006:0 nm for\nlsf[27].\nlsfand\fare shown as a function of concentration x\nin Fig. 4. The convex behaviour of \fis dominated by\nand tracks the large minority spin resistivity \u001a#whose\norigin is the large mismatch of the Ni and Fe minority\nspin band structures that leads to a \u0018x(1\u0000x) concen-\ntration dependence of \u001a#(x) [16]. The majority spin band\nstructures match well so \u001a\"is much smaller and changes\nrelatively weakly as a function of x. The increase of lsf\nin the clean metal limits corresponds to the increase of4\n0 20 40 60 80 100510152025l sf [nm]\nFe concentration [%]← lsfβ →\n0 20 40 60 80 1000.50.60.70.80.9\nβ\nFIG. 4. Spin-di\u000busion length (solid line) and polarization \fas\na function of the concentration xfor Ni 1\u0000xFexbinary alloys.\nthe electron momentum and spin-\rip scattering times in\nthe limit of weak disorder.\nIn summary, we have developed a uni\fed DFT-based\nscattering theoretical approach for calculating transport\nparameters of concentrated alloys that depend strongly\non spin-orbit coupling and disorder and have illustrated\nit with an application to NiFe alloys. Where comparison\nwith experiment can be made, the agreement is remark-\nably good o\u000bering the prospect of gaining insight into\nthe properties of a host of complex but technologically\nimportant magnetic materials.\nThis work is part of the research programs of \\Sticht-\ning voor Fundamenteel Onderzoek der Materie\" (FOM)\nand the use of supercomputer facilities was sponsored by\nthe \\Stichting Nationale Computer Faciliteiten\" (NCF),\nboth \fnancially supported by the \\Nederlandse Organ-\nisatie voor Wetenschappelijk Onderzoek\" (NWO). It was\nalso supported by \\NanoNed\", a nanotechnology pro-\ngramme of the Dutch Ministry of Economic A\u000bairs and\nby EC Contract No. IST-033749 DynaMax.\n[1] See the collection of articles in Ultrathin Magnetic Struc-\ntures I-IV , edited by J. A. C. Bland and B. Heinrich\n(Springer-Verlag, Berlin, 1994-2005).\n[2] P. C. van Son, H. van Kempen, and P. Wyder, Phys.\nRev. Lett., 58, 2271 (1987); 60, 378 (1988).\n[3] T. Valet and A. 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Benjamin Jung\reisch1,y\n1Department of Physics and Astronomy, University of Delaware, Newark, Delaware 19716, United States\n(Dated: February 20, 2023)\nWe demonstrate the in\ruence of damping and \feld-like torques in the magnon-photon coupling\nprocess by classically integrating the generalized Landau-Lifshitz-Gilbert equation with RLC equa-\ntion in which a phase correlation between dynamic magnetization and microwave current through\ncombined Amp\u0012 ere and Faraday e\u000bects are considered. We show that the gap between two hybridized\nmodes can be controlled in samples with damping parameter in the order of 10\u00003by changing the\ndirection of the dc current density Jif a certain threshold is reached. Our results suggest that an\nexperimental realization of the proposed magnon-photon coupling control mechanism is feasible in\nyttrium iron garnet/Pt hybrid structures.\nI. INTRODUCTION\nCoherent magnon-photon coupling in hybrid cavity-\nspintronics contributed to the advancement of magnon-\nbased quantum information and technologies [1{15]. The\ncollective excitations of an electron spin system in mag-\nnetically ordered media called magnons can couple to mi-\ncrowave photons via dipolar interaction, demonstrating\nlevel repulsion and Rabi oscillations [3]. Strongly cou-\npled magnon-photon systems have been explored to bring\nmany exotic e\u000bects into the limelight, some of which in-\nclude the manipulation of spin currents [16], and bidi-\nrectional microwave-to-optical transduction [17, 18]. In\naddition to the coherent magnon-photon coupling, there\nexists an exciting domain of dissipative magnon-photon\ncoupling where level attraction can be observed, which is\ncharacterized by a coalescence of the hybridized magnon-\nphoton modes [19{25].\nThe theoretical framework of magnon-photon coupling\nis given by the following dispersion relation [26] of the\nhybridized modes:\ne!\u0006=1\n2\u0014\n(e!m+e!c)\u0006q\n(e!m\u0000e!c)2+ 4g2\u0015\n;(1)\nwheree!m=!m\u0000i\u000b!mande!c=!c\u0000i\f!care the\ncomplex resonance frequencies of the magnon and pho-\nton (cavity) modes, respectively. gis the coupling be-\ntween the two modes. \u000band\fare the intrinsic damping\nrates of the magnon and photon modes, respectively. The\nreal and imaginary parts of e!\u0006represent the dispersion\nshape and the linewidth of the coupled modes, respec-\ntively. The second term of the square root in Eq. (1) not\nonly gives the strength of the coupling but also reveals\nthe nature of the coupling. Harder and co-workers [19]\ncarefully introduced a coupling term based on the cavity\nLenz e\u000bect to mitigate the Amp\u0012 ere e\u000bect. However, the\non-demand manipulation of the magnon-photon polari-\nton by spin torques has not been addressed so far.\n\u0003arai@udel.edu\nymbj@udel.eduIn this work, we examine the in\ruence of damping-\nand \feld-like torques in the magnon-photon coupling pro-\ncess. Our results indicate that the magnitude of the\nlevel repulsion (manifested by the frequency gap of the\nhybridized modes) and, hence, the magnon-photon cou-\npling strength can e\u000eciently be controlled by varying the\nmagnitude and the direction of dc current density Jfor\nrealistic parameters of the magnetic properties. By cou-\npling the generalized Landau-Lifshitz Gilbert equation\nwith the RLC equation of the cavity, we show that an\non-demand manipulation of the magnon-photon coupling\nstrength can be achieved for current densities of the order\nas small as 105A/cm2.\nThis article is structured in the following fashion. In\nsection II, we discuss the classical description to model\nour system, in which the ferromagnetic resonance of the\nmagnetic system is strongly coupled to photon resonator\nmode of the microwave cavity. In section III, we intro-\nduce the parameters used for the analysis followed by\na detailed discussion of our \fndings. In section IV, we\nsummarize our work.\nFIG. 1. The schematic of the experimental setup. A pat-\nterned YIG/platinum(Pt) bilayer is the sample under consid-\neration. The dc current is passed through the platinum layer.\nThe microwave current is passed through the cavity and an-\nalyzed using a Vector Network Analyzer (VNA). Here, the\nexternal magnetic \feld is applied along bzdirection.arXiv:2302.08910v1  [cond-mat.mes-hall]  17 Feb 20232\nFIG. 2. Magnon-photon coupling control for an intermediate value of the Gilbert damping parameter \u000band a continuous, low\ncurrent density J: The dispersion ( !\u0000!c) in (a-c) and the linewidth (\u0001 !) in (d-f) are plotted as a function of the \feld detuning\n(!m\u0000!c) for\u000b= 2:27\u000210\u00003. The hybridization of magnon and photon modes is compared for di\u000berent dc current densities\nJ: (a), (d) J = \u00005\u0002105A=cm2, (b), (e) J = 0 A =cm2and (c), (f) J =5 \u0002105A=cm2. The blue and red line represent the two\nhybridized modes. The inset in (b) shows that for larger \feld detuning ( !m\u0000!c), the uncoupled photon mode approaches !c\nmaking (!\u0000!c) approach zero.\nII. CLASSICAL DESCRIPTION\nThe magnetization dynamics in ferromagnetic systems\ncan be described by the generalized Landau-Lifshitz-\nGilbert equation [27{29] given by:\nd~M\ndt=\r~M\u0002~He\u000b\u0000\u000b\nMs \n~M\u0002d~M\ndt!\n+\n\raJ\nMs\u0010\n~M\u0002\u0010\n~M\u0002~ p\u0011\u0011\n\u0000\rbJ\u0010\n~M\u0002~ p\u0011\n;(2)\nwhere~Mis the magnetization vector, Msis the satu-\nration magnetization, ~He\u000bis the e\u000bective magnetic \feld\nincluding external \feld ~H, anisotropy, microwave, and\ndemagnetization \felds, \ris the gyromagnetic ratio, \u000bis\nthe Gilbert damping parameter, ~ pis the spin polarization\nunit vector. Furthermore, the terms proportional to aJ\nandbJare the damping-like torque and \feld-like torque,\nrespectively. The coe\u000ecients aJandbJare de\fned as\n[30]:\naJ=\u0011aJ~\n2eMsd;bJ=\u0011bJ~\n2eMsd; (3)where\u0011aand\u0011bare the damping-like torque e\u000eciency\nand \feld-like torque e\u000eciency, respectively. Jis the\ndc current density, whose polarity determines the direc-\ntions of \feld-like and damping-like torque terms through\nEqs. (2) and (3), ~is the reduced Planck's constant, e\nis the electron charge, and dis the thickness of the fer-\nromagnetic sample. We de\fne the magnetic \feld, mag-\nnetization, and spin polarization unit vectors as ~Ht=\nhx(t)bx+hy(t)by+Hbz,~M=mx(t)bx+my(t)by+Msbzand\n~ p=bz, whereHandMsare the dc magnetic \feld and\nsaturation magnetization, respectively, and hx;y(t) and\nmx;y(t) are the dynamic magnetic \feld and magnetiza-\ntion.\nIf we de\fne the dynamic components, h=hx+ihy\nandm=mx+imy, then Eq. (2) can be reduced to:\n(!\u0000~!m+\r~cJ)m+!sh= 0; (4)\nwhere ~!mis the complex ferromagnetic resonance fre-\nquency de\fned by ~ !m=!m\u0000i\u000b!(where!m'\rHis\nthe ferromagnetic resonance frequency), !s=\rMs, and\n~cJ=bJ\u0000iaJ. The e\u000bective RLC circuit for the cavity3\nFIG. 3. Magnon-photon coupling control for an intermediate value of the Gilbert damping parameter \u000band a pulsed, high\ncurrent density J: The dispersion ( !\u0000!c) in (a-c) and the linewidth (\u0001 !) in (d-f) are plotted as a function of the \feld detuning\n(!m\u0000!c) for\u000b= 2:5\u000210\u00003. The hybridization of magnon and photon modes is compared for di\u000berent dc current densities\nJ: (a), (d) J = \u00005\u0002106A=cm2, (b), (e) J = 0 A =cm2and (c), (f) J =5 \u0002106A=cm2. The blue and red line represent the two\nhybridized modes.\ncan be written as [19]:\nRjx;y(t) +1\nCZ\njx;y(t)dt+Ldjx;y(t)\ndt=V0x;y(t);(5)\nwhere R, L, and C represent the resistance, inductance,\nand capacitance, respectively. V0x;yis the voltage that\ndrives the microwave current. For j=jx+ijyandV0=\nV0x+iV0y, we have [19]\n\u0000\n!2\u0000!2\nc+i2!!c\f\u0001\nj=i!\nLV0; (6)\nwhere!c= 1=p\nLCis the cavity resonance frequency and\n\f= (R=2)p\nC=L is the intrinsic damping of the cavity-\nphoton mode.\nThe microwave magnetic \feld will exert a torque on\nthe magnetization through Amp\u0012 ere's law. The relation\ncan be expressed as:\nhx=KAjy;hy=\u0000KAjx; (7)\nwhereKAis the positive coupling term associated with\na phase relation between jx;yandhx;y. In a similar way,\nthe precessional magnetization will induce a voltage inthe RLC circuit through Faraday induction:\nVx=\u0000KFLdmy\ndt;Vy=KFLdmx\ndt; (8)\nwhereKFis the positive coupling term associated with\na phase relation between Vx;yandmx;y. Combining\nEqs. (4)-(8) gives us the coupled equations of the form:\n\u0012\n!2\u0000!2\nc+i2\f!c! i!2KF\n\u0000i!sKA!\u0000~!m+\r~cJ\u0013\u0012\nj\nm\u0013\n=\u0012\ni!!cj0\n0\u0013\n;(9)\nwherej0=V0p\nC=L. The hybridized eigenmodes are\ncalculated by solving the determinant of Eq. (9). This\nyields the following analytical form [see Supplemental\nMaterial (SM)]:\n~!\u0006=\u0010\n!c\n1+i\f+!m\u0000\u000e\n1+i\u000b\u0011\n\u0006r\u0010\n!c\n1+i\f\u0000!m\u0000\u000e\n1+i\u000b\u00112\n+2!c!sKFKA\n(1+i\u000b)(1+i\f)\n2;\n(10)\nwhere\u000e=\r~cJ. Here,\ris the gyromagnetic ratio and ~ cJ\nis a complex term associated with bJandaJde\fned by\n~cJ=bJ\u0000iaJ.4\nFIG. 4. Variation of coherent magnon-photon coupling (minimum frequency gap between two hybridized modes) for di\u000berent\nvalues of\u000bandJ. For (a), (b), and (c) \u000bis varied from 3 \u000210\u00003to 5\u000210\u00005andJ(continuous) is varied from \u00005\u0002105A=cm2\nto 5\u0002105A=cm2and for (d), (e), and (f) \u000bis varied from 4 \u000210\u00003to 5\u000210\u00005andJ(pulsed) is varied from \u00005\u0002106A=cm2to\n5\u0002106A=cm2. Based on our model, we can distinguish between \feld-like contribution (a) and (d), damping-like contribution\n(b) and (e), and a combination of \feld-like and damping-like contribution to the manipulation of the anticrossing gap (c) and\n(f). For (a) and (d) \u0011a= 0 and\u0011b= 0:05 (pure \feld-like torque e\u000bect), for (b) and (e) \u0011a= 0:2 and\u0011b= 0 (pure damping-like\ntorque e\u000bect), and for (c) and (f) \u0011a= 0:2 and\u0011b= 0:05 (combination of damping-like and \feld-like torque e\u000bects).\nIII. RESULTS AND DISCUSSION\nFor our model we choose the following realistic pa-\nrameters [18, 31{34]. The frequency of the cavity mode\nis selected at !c=2\u0019= 10 GHz with a cavity damping\n\f= 1\u000210\u00004(corresponding to quality factor Q\u00195000).\nThe reduced gyromagnetic ratio ( \r=2\u0019), damping-like\ntorque e\u000eciency ( \u0011a), and \feld-like torque e\u000eciency ( \u0011b)\nare taken as 2 :8\u0002106Hz=Oe, 0:2, and 0:05, respectively.\nFor a Pt/FM bilayer, the typical range of damping-like\ntorque e\u000eciency ( \u0011a) is 0.10 to 0.20 [32, 35, 36] and the\ntypical value of \feld-like torque e\u000eciency ( \u0011b) is\u00190.05\n[33, 37{39]. Due to its low Gilbert damping parame-\nter and high spin density, we choose yttrium iron garnet\n(YIG) as magnetic material. In particular, we consider\na YIG \flm with a thickness t= 2\u000210\u00005cm (smallest\nthickness available commercially) and saturation magne-\ntization,Ms= 144 emu =cm3[18]. For the calculation,\nthe termKFKAis taken as 5\u000210\u00006[19]. For YIG \flms,\ndepending upon the thickness and preparation method, \u000b\nvaries from order 10\u00003to 10\u00005[34, 38, 40{47]. Therefore,\nwe vary\u000bin our model from 3 \u000210\u00003to 5\u000210\u00005. Further-more, we vary Jfrom\u00005\u0002105A=cm2to 5\u0002105A=cm2.\nThe maximum value of chosen current density is at least\none order of magnitude smaller than what is used for\nmagnetic tunnel junctions (MTJs) [48, 49]. Note that, a\ncurrent density of this order of magnitude has previously\nbeen reported for YIG/Pt systems [50] to thermally con-\ntrol magnon-photon coupling in experiment. Reference\n[50] reports that such current density leads to a rise of the\nsystem temperature above 40\u000eC. Negative e\u000bects of heat-\ning on the magnetic properties can be drastically reduced\nby using a pulsed dc current through the Pt layer [51] in-\nstead of using a continuous current. For instance, using a\npulsed current with duty cycle of 50%, heating e\u000bects can\nbe mitigated while reaching reasonable high levels of cur-\nrent density between \u00005\u0002106A=cm2to 5\u0002106A=cm2\n. Such a high value of current density will create an\nOersted \feld and, hence, modify the resonance condi-\ntion. The generated Oersted \feld can be considered as a\ncontribution to the e\u000bective magnetic \feld presented in\nEq. (2). Hence, this \feld will modify the resonance posi-\ntion of the magnon modes in the following way: for one\npolarity of the current density (J), the resonance \feld5\nshifts up, while for the other, it shifts down. Experi-\nmentally, this a\u000bect can be compensated by tuning the\nbiasing magnetic \feld so the resonance frequency remains\nthe same. In the following analysis, we consider two sce-\nnarios: (1) a relatively low continuous current density\nand (2) a higher pulsed current density. The e\u000bects of\nboth conditions on the magnon-photon coupling process\nare compared below. The proposed experimental set up\nand measurement con\fguration is shown in Fig. 1.\nA. Dispersion and Linewidth\nIn Fig. 2 (intermediate value of Gilbert damping pa-\nrameter\u000band continuous, low value of current density\nJ) and Fig. 3 (intermediate value of \u000band pulsed, high\nvalue ofJ), the hybridized mode frequency ( !\u0000!c) and\nlinewidth (\u0001 !) are plotted as a function of the \feld de-\ntuning (!m\u0000!c).\nWe \frst focus on the former, shown in Fig. 2, top pan-\nels: For\u000b= 2:27\u000210\u00003and forJ= 0 A=cm2, we observe\na level attraction of the real part of the eigenvalues [Fig.\n2 (b)] in a small region. For J=\u00005\u0002105A=cm2, a\nsimilar behavior is found [Fig. 2(a)]. However, the be-\nhavior drastically changes for reversed current polarity:\nforJ= 5\u0002105(A=cm2), a gap (a prominent level repul-\nsion) is seen between the hybridized modes. This clearly\nshows that depending upon the strength and direction of\nthe dc current density Jone can tune the gap between\nthe hybridized modes, i.e., transitioning the system into\nthe strong coupling regime. Let us now consider Fig. 3,\ntop panels: A similar but enhanced behavior can be ob-\nserved for higher values of J(i.e.,jJj= 5\u0002106A/cm2)\nand\u000b= 2:5\u000210\u00003[Fig. 3]. As is obvious from Figs. 2\nand 3, there is a shift in the position where the coher-\nent coupling occurs. For negative and positive values of\nJ, the resonance shifts towards the negative and positive\nsides of the \feld detuning ( !m\u0000!c), respectively. This\nshift can be understood by the fact that di\u000berent mag-\nnitudes of \feld-like torques directly a\u000bect the resonance\ncondition as will be discussed in Sec. III B.\nNext, we discuss the lower panels of Figs. 2 and 3. The\nlinewidths of the two hybridized modes distinctly cross\neach other for J= 5\u0002105A=cm2andJ= 5\u0002106A=cm2\nas is expected for a broad coupling region [Fig. 2(f) and\nFig. 3(f)]. This e\u000bect is less distinct for the cases J=\n\u00005\u0002105A=cm2,J=\u00005\u0002106A=cm2andJ= 0 A=cm2.\nHowever, despite the lower number of region in the cross-\ning regime (the crossing is less spread), we emphasize that\nlevel crossings in the linewidths of the hybridized modes\nare are also observed here. We note that the coupling re-\ngion broadens as the current density increases from neg-\native values to positive values [Figs. 2(d,e,f) and Figs. 3\n(d,e,f)] and \fnally a distinct crossing of linewidths is ob-\nserved over a broad range [Fig. 2(f) and Fig. 3 (f)].\nFor special cases discussed in Sec. II of the SM, a\nlevel attraction [Fig. S1 (a,b)] in the real part and level\nrepulsion [Fig. S1 (d,e)] in the imaginary part of theeigenvalues are observed along with exceptional points\n(EPs) [52{55]. For more details on the observed EP we\nrefer to the SM.\nB. Anticrossing gap between hybridized modes\nFigure 4 shows the variation of the anticrossing gap\nbetween the hybridized modes for di\u000berent values of \u000b\nandJ. It is clear that the variation is nonlinear in nature.\nAs is evident from the \fgure, the gap between the two\nhybridized modes becomes smaller for a larger value of\n\u000b. On the other hand, the gap also depends on the dc\ncurrent density J: the value of \u000bfor which the gap is very\nsmall increases if we go from from negative to positive\nvalue of the dc current density. For a positive value of\nJ, we also observe the gap between the hybridized modes\nslowly increases as \u000bincreases and becomes maximum for\na particular value of \u000b, and then decreases if we further\nincrease the value of \u000b, as shown in the inset of Fig 4(f).\nFor the low \u000bregime, the anticrossing gap remains nearly\nthe same for di\u000berent orders of magnitude and directions\nof current density, as is shown in Figs. S2, S3 and S4\nof the SM. However, for the high \u000bregime, we observe\na level repulsion in the real part and a level crossing in\nthe imaginary part of the eigenvalues for di\u000berent orders\nof magnitude and directions of the current density, as is\nshown in Figs. S5, S6 and S7 of the SM. For a di\u000berent\ncoupling strength ( KFKA), we observe a similar trend.\nA positive current density is needed to increase the gap\nbetween the two hybridized modes as is shown in Fig. S8\n(SM).\nIn the following discussion, we chose Gilbert damping\nparameters of \u000b= 2:27\u000210\u00003and\u000b= 2:5\u000210\u00003for\ndi\u000berent orders of magnitude of Jwhere a very small\nanticrossing gap for zero current density is seen, as illus-\ntrated in Fig. 4. For large \u000b(= 4\u000210\u00003) and for very low\n\u000b(= 5\u000210\u00005), the hybridized mode frequency ( !\u0000!c)\nand the linewidth (\u0001 !) plotted as a function of the \feld\ndetuning (!m\u0000!c) are shown in the Fig. S1 and Fig.\nS2 of the SM. Figure 5 shows the variation of the magni-\ntude of the gap between the two hybridized modes with\nrespect to \feld detuning ( !m\u0000!) for\u000b= 2:27\u000210\u00003\nand\u000b= 2:5\u000210\u00003. For a pure \feld-like torque e\u000bect,\nthere is a horizontal shift [as shown in Figs. 5(a) and\n5(d)] of the gap between the hybridized modes towards\nthe positive value of \feld detuning as we go from negative\nto positive values of the current density J. However, for\na damping-like torque e\u000bect, there is a vertical shift [as\nshown in Figs. 5(b) and 5(e)] of the minimum gap (an-\nticrossing gap): the anticrossing gap increases if we go\nfrom negative to positive values of J. For the combined\n\feld-like and damping-like torques e\u000bect, there are both\nhorizontal and vertical shifts as can be seen in Figs. 5(c)\nand 5(f). However, the horizontal shift due to the e\u000bect of\n\feld-like torque, vertical shift due to damping-like torque\nand the combined shift due to both \feld and damping-\nlike torques are more pronounced for higher magnitudes6\nFIG. 5. Variation of the gap between two hybridized modes with respect to the \feld detunings ( !m\u0000!c) for\u000b= 2:27\u000210\u00003\nand continuous low Jfrom \u00005\u0002105A=cm2to 5\u0002105A=cm2[(a),(b) and (c)] and for \u000b= 2:5\u000210\u00003and pulsed high J\nfrom \u00005\u0002106A=cm2to 5\u0002106A=cm2[(d),(e) and (f)]. There is (a),(d) a horizontal shift in the location of the gap for\n\u0011a= 0 and\u0011b= 0:05 (pure \feld-like torque e\u000bect), (b), (e) vertical shift in the location of gap for \u0011a= 0:2 and\u0011b= 0\nand (pure damping-like torque e\u000bect), and (c),(f) horizontal and vertical shifts for \u0011a= 0:2 and\u0011b= 0:05 (combined e\u000bect of\ndamping-like and \feld-like torques).\nofJleading to an unusual behavior as is shown in panel\n(f).\nFinally, we note that introducing the \feld-like and\ndamping-like torques in the Landau-Lifshitz-Gilbert\nequation and coupling it with cavity mode through\ncombined Amp\u0012 ere and Faraday e\u000bects do not produce\nlevel attraction. The coupling term in our analysis\nis not a\u000bected by the \u000eterm [see Eq. (10)], which is\nthe parameter governed by spin torque. This means\nthat transitioning the system from strong coupling to\ndissipative coupling and vice versa cannot be achieved\nby spin-transfer torques.\nIV. SUMMARY\nBy coupling of the generalized LLG equation with the\nRLC equation of the cavity, we revealed the coupling be-\ntween magnon and photon modes under the in\ruence of\ndamping and \feld-like torques. Our results indicate that\nthe magnitude of the level repulsion (manifested by the\nfrequency gap of the hybridized modes) and, hence, themagnon-photon coupling strength can e\u000eciently be con-\ntrolled by varying the magnitude and the direction of dc\ncurrent density Jfor realistic parameters of the magnetic\nproperties. Our model suggests that an on-demand ma-\nnipulation of the magnon-photon coupling strength can\nbe achieved for current densities of the order as small\nas 105A/cm2and an intermediate Gilbert damping of\nthe order 10\u00003. Higher values of Jcan de\fnitely en-\nhance the e\u000bect of damping and \feld-like torques on the\nmagnon-photon coupling provided we use pulses of dc\ncurrent to reduce possible heating e\u000bects. 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An exact solution for the\ndomain wall velocity is provided, including the e\u000bect of non-equilibrium conduction electron spin-\ndensity, Gilbert damping, and the Rashba interaction parameter. We demonstrate explicitly that\nthe in\ruence of spin-orbit interaction can be qualitatively di\u000berent from the role of non-adiabatic\nspin-torque in the sense that the former is sensitive to the chirality of the domain wall whereas\nthe latter is not: the domain wall velocity shows a reentrant behavior upon changing the chirality\nof the domain wall. This could be used to experimentally distinguish between the spin-orbit and\nnon-adiabatic contribution to the wall speed. A quantitative estimate for the attainable domain\nwall velocity is given, based on an experimentally relevant set of parameters for the system.\nPACS numbers:\nI. INTRODUCTION\nThe study of domain-wall motion in ferromagnetic ma-\nterials has attracted much interest in recent years. Be-\nsides its allure from a fundamental physics viewpoint,\nelectric control of magnetic textures is attractive in terms\nof potential new applications such as magnetic memory.\nA key concept in the context of domain wall motion is\nthe so-called spin-transfer torque [1{3]: in essence, it con-\nsists of a transfer of a transverse spin-current component\nto the ferromagnetic order parameter which may occur\nin a non-collinear magnetization con\fguration. Active\ncontrol over domain-wall motion is a chief objective in\nterms of realizing the \"magnetic race-track\" technology\nput forward in [4]. Other ways to manipulate domain wall\nmotion include energy redistribution in the presence of\nan external magnetic \feld, by applying microwave radia-\ntion [5{9], and by using magnons [10{13]. Moreover, the\nconcept of spin-transfer torque has recently been studied\nin antiferromagnets [14{19] including the possibility to\nmove domain-walls.\nInterestingly, it turns out that spin-orbit interactions\ncan substantially modify both the spin-transfer torque\nand the resulting domain wall motion [20, 21, 25]. The\ncombined in\ruence of a magnetic exchange \feld together\nwith spin-orbit interaction, typically taken in the Rashba\nform, can be shown to give rise to a non-equilibrium\nspin density perpendicular to the injected current \row.\nIn turn, this gives rise to an e\u000bective magnetic \feld\nwhich causes magnetization dynamics and, for su\u000e-\nciently strong current density, magnetization switching.\nThe e\u000bective \"spin-orbit torque\" arising in this manner is\nqualitatively di\u000berent from the conventional spin-transfer\ntorque due to the di\u000berent mechanism at hand: it does\nnot require the presence of non-collinear magnetic ele-\nments and will cause magnetization dynamics in a single\nferromagnetic layer [28]. In addition, it is important to\nnote that whereas di\u000berent types of domain walls behave\nin the same way in the absence of spin-orbit interactions,\nthe exact magnetization texture plays a key role whenspin-orbit coupling is present due to the coupling between\nthe electron motion and the spin torque.\nIn light of the above, several experimental and theoreti-\ncal works has recently explored the in\ruence of spin-orbit\ninteractions on magnetization dynamics in various mag-\nnetic systems [22{30]. Since the addition of spin-orbit\nterms in the equations of motion for the magnetization\ntexture complicates their solution, the large majority of\nthese works have relied on numerical methods to solve\nthe Landau-Lifshitz-Gilbert (LLG) equation. In this pa-\nper, we utilize the Lagrangian formalism in order to write\ndown and solve analytically the equations of motion for\na domain wall within a collective-coordinate description.\nThe analytical nature of this approach allows us to iden-\ntify a transparent expression for the domain wall velocity\nand how it depends on parameters such as the spin-orbit\ninteraction, exchange \feld, and Gilbert damping of the\nsystem. Alternatively, one could have derived this result\nvia the LLG equation, but the present formalism makes\nit easier to accommodate non-equilibrium spin-density\nterms and higher order corrections due to the spin-orbit\ninteraction. We show from the analytical expression that\nthe presence of spin-orbit interactions renders the domain\nwall velocity to be chirality sensitive, in e\u000bect depend-\ning on whether the wall changes magnetization direction\nfrom positive to negative or vice versa along the direction\nof the current. We provide an estimate for the magnitude\nof the domain wall velocity using a set of experimentally\nrelevant parameters and show that the velocity behaves\nqualitatively di\u000berently depending on the chirality of the\ndomain wall and displays reentrant behavior. Our \fnding\nsuggests a way to experimentally distinguish between the\nnon-adiabatic and spin-orbit contribution to the domain-\nwall speed.\nII. THEORY\nWe consider \frst a ferromagnetic domain wall with an\neasy (hard) axis of magnetic anisotropy [36] along thearXiv:1302.4744v1  [cond-mat.mes-hall]  19 Feb 20132\nTilt angleφˆz\nˆy(easy)\nˆx(hard)\nTilt angleφˆz(easy)\nˆy(hard)\nCurrent direction(a)\n(b)Current direction\nFIG. 1: (Color online) Sketch of the magnetization textures\nconsidered in this paper. The domain wall nanowire extends\nalong thex-axis. (a) Hard axis along current direction ( x) and\neasy axis along z-direction. In equilibrium, the magnetization\nof the domain wall rotates in the yz-plane exactly as in Ref.\n[21] and similarly to Ref. [28]. Out of equilibrium, i.e. under\na current bias, a \fnite component may be acquired along the\nx-axis as indicated by the tilt angle \u001e. This geometry is the\nmain focus of this article, relevant for nanowires with perpen-\ndicular magnetic anisotropy. (b) Hard axis along y-direction\nand easy axis along z-direction. In equilibrium, the magne-\ntization rotates in the xz-plane. Out of equilibrium, a \fnite\ncomponent may be acquired along the y-axis as indicated by\nthe tilt angle \u001e.\ny-axis (x-axis). An electric current is injected along the\nx-axis [see Fig. 1a)], and the inversion symmetry is as-\nsumed broken in the ^ z-direction. This gives rise to a\nRashba spin-orbit coupling term, which in\ruences the\nmagnetization dynamics. More precisely, it can be shown\n[20] that the spin-orbit coupling generates an e\u000bective\nmagnetic \feld\nHSOC/\u000bR^z\u0002je (1)\nwhere\u000bRis the Rashba interaction strength and jeis\nthe current density vector. In order to treat the domain\nwall as rigid in a collective-coordinate framework, thus\nmaking other modes of deformation apart from the tilt\nangle\u001eirrelevant, the easy axis anisotropy energy Kis\nassumed larger than its hard axis equivalent K?[31], i.e.\njKj\u001djK?j. As is normally done, we do not take into\naccount the e\u000bect of the end-boundaries of the nanowire,\nassuming thus that the domain-wall center is located suf-\n\fciently far away from these during its propagation.\nThe starting point is the Lagrangian for such a Blochdomain-wall which was derived in Ref. [21]. For this type\nof domain wall, the magnetization rotates in the plane\nperpendicular to the extension of the wire (and current\ndirection), similarly to Ref. [28] (the only di\u000berence from\nRef. [28] is which of the perpendicular axes that is the\neasy one, in both cases the hard axis is along the current\ndirection). It reads:\nL= (\u001e_x\u0000sin2\u001e)\u00002\u0015xJ\u0001\n\u0016\u0000syx_\u001e\u0000\u001eJ\u0010\u0001\n\u0016+F(\u0015)\u0011\n(2)\nwith the de\fnition\nF(\u0015) =~2\u00152\nmL2\u00101\n\u0016+2\n\u0001\u0011\n: (3)\nAbove,\u001eis the tilt angle of the domain wall (see Fig.\n1) whilexis the normalized position of the center of\nthe domain wall ( x=X=L whereLis the wall thick-\nness). The quantities \u0015andJare the Rashba interac-\ntion strength and the current density normalized against\nmL= ~2andevc, respectively, with vcbeing the drift ve-\nlocity of electrons at the intrinsic threshold current for\n\u0001=\u0016= 1 without Rashba interactions. Moreover, \u0001 is\nthe exchange splitting, \u0016is the Fermi energy while m\nis the electron mass. Finally, syrepresents the constant\nnon-equilibrium spin density induced by the applied elec-\ntric \feld generating the current.\nIII. RESULTS AND DISCUSSION\nWe will model dissipation in this system by a Rayleigh\ndissipation function of the form [31]\nW=\u000b\n2( _x2+_\u001e2): (4)\nThe Lagrange equations are obtained via\nd\ndt@L\n@_q\u0000@L\n@q=\u0000@W\n@_q; q2fx;\u001eg: (5)\nwheretis dimensionless time-coordinate normalized\nagainstl=vc. De\fning for convenience\nc=\u0001\n\u0016+F(\u0015) (6)\nwe obtain the following Lagrange-equations, which were\nstudied numerically in [21]:\n_\u001e+\u000b_x=sy_\u001e\u00002\u0015J\u0001\n\u0016;_x\u0000\u000b_\u001e=\u0000sy_x+ sin 2\u001e+cJ:\n(7)\nThe role of the dissipative spin-transfer torque (also\nknown as non-adiabatic or \f-torque) will be addressed\nlater on - we will see that it plays a similar role as the\nspin-orbit coupling, but with one important di\u000berence.3\n0 200 400−150−100−50050100150200250300350\nCurrent J[m/s]Domain wall velocity /angbracketleftVDW/angbracketright[m/s]\n0 200 400−150−100−50050100150200250300350\nCurrent J[m/s]Domain wall velocity /angbracketleftVDW/angbracketright[m/s]\n  |Λ|= 0\n1.0e-14 eV ·m\n5.0e-14 eV ·m\n1.0e-13 eV ·m\n2.0e-13 eV ·m\n1.0e-12 eV ·m\n1.0e-11 eV ·m(b) (a)\nFIG. 2: (Color online) Plot of the terminal domain wall ve-\nlocity using \u000b= 0:005,L= 75 nm,m= 0:04me,\u0016= 0:05\neV, \u0001 = 0:02 eV, and vc= 150 m/s. (a) Positive chirality\n\u0015>0. (b) Negative chirality \u0015<0.\nBy combining the above equations, we are able to elimi-\nnate the _x-dependence and obtain a \frst-order di\u000beren-\ntial equation for the tilt angle:\n_\u001e=\u0000\u000b\n1 +\u000b2\n1+sy\u0000sy\u0010sin 2\u001e\n1 +sy+cJ\n1 +sy+ 2J\u0015\u0001\n\u000b\u0016\u0011\n(8)This equation may be cast into integral form as follows:\nZ\nd\u001e\u000b\u00001(1 +sy)(sy\u00001\u0000\u000b2=(1 +sy))\na+ sin 2\u001e=t:(9)\nWe de\fne the quantities\na=cJ+2\u0015\u0001J(1 +sy)\n\u000b\u0016;\nb=\u0000\u000b\u0000\u000b\u00001[1\u0000(sy)2]: (10)\nThe formal solution of this integral is obtained after some\nalgebraic manipulation:\ntan\u001e=\u0000a\u00001+a\u00001p\na2\u00001 tan(b\u00001tp\na2\u00001 +C0);\n(11)\nwhereC0is an integration constant to be determined\nfrom the initial conditions. In particular, \u001e(t= 0) = 0\nand\u001e(t= 0) =\u0019=2 yieldC0= atan(1=p\na2\u00001) and\nC0=\u0019=2, respectively.\nHaving obtained an explicit expression for the tilt\nangle, we are now in a position to identify the time-\ndependence of the domain-wall center x, thus also ob-\ntaining the domain-wall velocity _ x. To do so, we \frst\nobtain _\u001efrom Eq. (11) as:\n_\u001e=(a2\u00001)\nabcos\u00002(b\u00001tp\na2\u00001 +C0)\n1 + [\u0000a\u00001+a\u00001p\na2\u00001 tan(b\u00001tp\na2\u00001 +C0)]2: (12)\nSubstituting this into the \frst Lagrange-equation, one obtains an explicit expression for the domain-wall velocity vDW\nvDW=\u00002J\u0015\n\u000b\u0001\n\u0016\u0000(1\u0000sy)(a2\u00001)\nab\u000bcos\u00002(b\u00001tp\na2\u00001 +C0)\n1 + [\u0000a\u00001+a\u00001p\na2\u00001 tan(b\u00001tp\na2\u00001 +C0)]2: (13)\nEqs. (11) and (13) determine an exact analytical expression for the time-dependence of the domain wall tilt-angle\nand velocity, respectively, which we will analyze in more detail below.\nBy integration, Eq. (13) may be used to identify the\ntime-evolution of the domain-wall center:\nx=\u00002Jt\u0015\n\u000b\u0001\n\u0016\u0000(1\u0000sy)\n\u000bZ\ndt_\u001e: (14)\nThe integral over _\u001eis evaluated by making use of the\nformula\nZdycos\u00002(y+\r)\n[1 + (\u0011+\u0010tan(x+\r))2]=\u0010\u00001atan[\u0010tan(x+\r) +\u0011];\n(15)\nresulting in the following equation describing the instan-taneous position of the domain-wall center\nx\u0000x0=\u00002Jt\u0015\n\u000b\u0001\n\u0016\u00001\u0000s\r\n\u000b\u0002\ntan\u00001[a\u00001p\na2\u00001 tan(b\u00001tp\na2\u00001 +C0)\u0000a\u00001];\n(16)\nwithx0being an integration constant related to the ini-\ntial position of the domain-wall.\nIt is worth noting here that there exists a particularly\nsimple solution to the equation set Eq. (7) in the spe-\ncial case of a constant tilt angle, i.e. a domain-wall that\npreserves its shape and magnetization direction and thus4\nonly propagates. This amounts to setting _\u001e= 0, which\ndictates that the tilt angle must be constant:\nsin(2\u001e0) =\u0000cJ\u00002(1 +s\r)J\u0015\n\u000b\u0001\n\u0016; (17)\nand gives for the domain-wall velocity:\nvDW\u0011_x=\u00002J\u0015\n\u000b\u0001\n\u0016: (18)\nThis is identical to the \frst term in Eq. (13). Physically,\nthis implies a constant drift velocity of the domain-wall\nunder the application of a current. Interestingly, it is\nseen thatvDW= 0 in the absence of \u0015, meaning that the\nspin-orbit interaction is fully responsible for the domain-\nwall motion. In a more general scenario where the tilt\nangle is not restricted to being constant, i.e. allowing for\ndomain-wall deformation, the general expression for the\nvelocity is given by Eq. (13). As we shall discuss later,\nthis corresponds to the Walker breakdown threshold.\nThe necessity of spin-orbit coupling to drive the\ndomain-wall motion is a feature pertaining speci\fcally\nto the Bloch-domain wall with anisotropy axis chosen\nas described in the beginning of this section [see Fig.\n1a)]. In fact, for di\u000berent choices of anisotropy directions,\nspin-orbit coupling mainly contributes as a quantitative\ncorrection to the domain-wall velocity without being a\nprerequisite for its existence. To see this, one may con-\nsider instead a domain-wall where the magnetic easy and\nhard axes lie along the zandy-axes, respectively [see\nFig. 1b)]. In this case, the appropriate Lagrangian to\nconsider is [21]\nL= (\u001e_x\u0000sin2\u001e) + cos\u001e_\u001e\u0000\u0019sy\n2( _xcos\u001e\u0000sin\u001e_\u001e)\n\u0000szx_\u001e\u0000\u001eJ(\u0001\n\u0016\u0000~2\n2\u0001mL2) +\u0019\u0015sin\u001eJ\u0001=\u0016: (19)Using the same approach as above, one arrives at the\nfollowing set of integro-di\u000berential equations for the time-\nevolution of the domain-wall center and tilt angle:\n_\u001e+\u0019sy\n2sin\u001e_\u001e+sz_\u001e=\u0000\u000b_x (20)\nin addition to\nZ\nd\u001e\u000b\u00001[\u000b2+ (1 +\u0019sy\n2sin\u001e+sz)2]=[\u0019\u0015cos\u001eJ\u0001=\u0016\n\u0000sin 2\u001e\u0000J(\u0001\n\u0016\u0000~2\u00152\n2\u0001mL2)] =t\u0000t0: (21)\nUnlike the situation considered formerly, this equation\nset may be solved exactly analytically only under simpli-\nfying circumstances. For instance, by assuming that the\n\fnal term in the denominator of the integral equation\nabove dominates and moreover using s\r\u001c1,\r=fy;zg,\none identi\fes the domain-wall velocity in the strong-\ncurrent regime J\u001d1 as\nvDW=\u0000J\n1 +\u000b2\u0010\u0001\n\u0016\u0000~2\u00152\n2\u0001mL2\u0011\n: (22)\nAs seen, the presence of spin-orbit coupling in this case\nbrings about a minor correction to the \fnal velocity, es-\npecially for a strong ferromagnet where \u0001 =\u0016dominates\nthe expression in parantheses. The domain-wall is driven\ndirectly by the current Jwith a velocity that increases\nwith decreasing dissipation \u000b. The above result for the\ndomain wall velocity shows that the spin-orbit coupling\nhas a qualitatively di\u000berent e\u000bect on di\u000berent domain\nwall textures.\nIn the following, we focus on the more interesting case where the spin-orbit coupling in\ruences qualitatively the\ndomain-wall motion [Eqs. (11) and (13)] and investigate the precise dynamics using the derived analytical expressions.\nFor concreteness, we consider initial conditions such that the tilt angle of the domain-wall at t= 0 is zero, meaning\nthat we may write:\ntan\u001e=1\nahp\na2\u00001 tan\u0010tp\na2\u00001\nb+ atan\u00101p\na2\u00001\u0011\u0011\n\u00001i\n;\nvDW=\u00002J\u0015\n\u000b\u0001\n\u0016\u0000\u000b\u00001(1\u0000sy)(a2\u00001)\nab\u0002cos\u00002(b\u00001tp\na2\u00001 + atan(1=p\na2\u00001))\n1 +1\na2[p\na2\u00001 tan(b\u00001tp\na2\u00001 + atan(1=p\na2\u00001))\u00001]2: (23)\nIn the limit \u000b!1 , we \fnd that \u001e(t)!0 andvDW!0\nas expected. The analytical expression for vDWabove\nreveals that the velocity has non-monotonic behavior as\na function of time. In particular, there is a resonance\nconditiont=tresat which the velocity increases in mag-nitude:\ntres=(n+ 1=2)\u0019bp\na2\u00001\u0000b\na2\u00001; (24)\nassuming that a\u00151. In e\u000bect, vDWexhibits oscillations\nwhich persist even for its terminal behavior t!1 . It is5\ntherefore of interest to establish the average domain wall\nvelocity by averaging over one period T=\u0019b=p\na2\u00001:\nhvDWi=1\nTZT\n0dtvDW: (25)\nNote that for a<1, only the drift-term in vDWsurvives\nast!1 [since cos2(\u0006it)!1 in this limit]. Performingthe integral with this in mind, one \fnds for arbitrary a\nthat\nhvDWi=\u00002J\u0015\u0001\n\u000b\u0016\u00001\u0000sy\n\u000bbsignfagRefp\na2\u00001g;(26)\nwhich written out in terms of the original physical pa-\nrameters reads\nhvDWi=\u00002J\u0015\u0001\n\u000b\u0016+(1\u0000sy)signfJg\n\u000b2+ (1\u0000sy)\u0002signn2\u0015\u0001(1 +sy)\n\u000b\u0016+\u0001\n\u0016+F(\u0015)o\n\u0002Re(sh2\u0015\u0001(1 +sy)\n\u000b\u0016+\u0001\n\u0016+F(\u0015)i2\nJ2\u00001)\n(27)\nEq. (27) is the main result of this work and constitutes\na generally valid analytical expression for the terminal\ndomain wall velocity taking into account both Rashba\nspin-orbit coupling, the non-equilibrium spin density, and\nGilbert damping. For consistency, we have veri\fed that\na fully numerical solution of the equations of motion give\nidentical results as the above analytical expression for the\ndomain wall velocity.\nWe will proceed to analyze this velocity quantitatively\nfor a realistic set of parameters and investigate how it de-\npends in particular on the applied current and the mag-\nnitude of the spin-orbit coupling. Before doing so, one\nshould note that it follows from Eq. (27) that there exists\nboth a threshold current Jcfor which the second term in\nEq. (27) is non-zero:\njJcj=\f\f\f2\u0015\u0001(1 +sy)\n\u000b\u0016+\u0010\u0001\n\u0016+~2\u00152\nmL2\u0016+2~2\u00152\nmL2\u0001\u0011\f\f\f\u00001\n:\n(28)\nIn the limiting case of zero spin-orbit interaction, one ob-\ntainsjJcj= (\u0001=\u0016)\u00001in agreement with previous studies.\nThe presence of spin-orbit interaction is seen from the\nanalytical expression of jJcjto reduce the threshold cur-\nrent monotonically with increasing \u0015, consistently with\nthe numerical study in Ref. [21]. This monotonic be-\nhavior appears also when tuning the chemical potential\n\u0016: increasing \u0016lowers the polarization and increases the\nthreshold current.\nEq. (28) is in fact the Walker threshold value which\nseparates the regimes of domain wall motion with a \fxed\npro\fle, i.e. _\u001e= 0 and the regime with a domain wall\nrotating its spatial pro\fle as time increases, i.e. _\u001e6= 0.\nTo see this, one may revert to the original equations of\nmotion in Eq. (7). There exists a tilt angle \u001ewhich\nsatis\fes _\u001e= 0 if the following equation is satis\fed:\nsin 2\u001e=\u0000\u00102\u0015\u0001(1 +sy)\n\u000b\u0016J+cJ\u0011\n: (29)Since the left-hand side varies between \u00061, one may \fnd\na solution if the following equation holds:\njJj\f\f\f2\u0015\u0001(1 +sy)\n\u000b\u0016+c\f\f\f<1: (30)\nwhich is completely equivalent to Eq. (28) after rewrit-\ning. For larger currents J, there exists no time-\nindependent solution \u001eand domain wall distortion _\u001eis\nnow inevitable past the Walker breakdown.\nIt has previously been suggested that the spin-orbit\ninteraction and non-adiabatic spin-torque in\ruence the\nmagnetization dynamics in the same manner, since the\nlatter may be included by substituting \u0015!\f+\u0015[21].\nHowever, it was noted in Ref. [25] that the chirality of the\ndomain wall determines the e\u000bective sign of the spin-orbit\ncoupling\u0015in the equations of motion. Formally, this\ncorresponds to taking the domain wall pro\fle represented\nwith polar angles \u001eand\u0012asM=M0(\u0000sin\u0012sin\u001e^x+\ncos\u0012^y+sin\u0012cos\u001e^z) and performing the transformations\n\u001e!(\u0000\u001e) and cos\u0012!(\u0000cos\u0012). The parameter \u001e=\n\u001e(t) is the time-dependent tilt angle, whereas \u0012is de\fned\nby\nsin\u0012= sech[(~x\u0000X(t))=LDW]; (31)\nwhere ~xis the position along the magnetic wire, X(t)\nis the time-dependent center position of the domain-\nwall, whereas LDWis the domain wall width. This sug-\ngests that the role of spin-orbit coupling is chirality-\nsensitive and in this regard di\u000bers qualitatively from non-\nadiabaticity. Below, we will investigate this e\u000bect and\nshow that the chirality indeed gives rise to highly di\u000ber-\nent behavior for the domain wall velocity. The chirality\nis changed in our analytical expression Eq. (27) by let-\nting\u0015!(\u0000\u0015). We note that our conclusions remain\nunchanged even when including the \f-term, as long as it\nis small (which is typically the case).\nIn the remainder of this paper, we \fx the following pa-\nrameters:\u000b= 0:005,L= 75 nm,m= 0:04me,\u0016= 0:056\neV, \u0001 = 0:02 eV, and vc= 150 m/s to model an exper-\nimentally realistic semiconductor system [32], where me\nis the bare electron mass. We restrict our attention to a\nscenario where the electron-spin density satis\fes sy\u001c1.\nIn Eq. (27), note that Jand\u0015are normalized quanti-\nties. To make a quantitative estimate for the domain wall\nvelocity, we de\fne the non-normalized current J=Jvc\nand spin-orbit interaction \u0003 = ~\u0015=(mL) which have units\nm/s and eV\u0001m, respectively. Similarly, we restore the di-\nmension of the terminal domain wall velocity by de\fning\nhVDWi=vchvDWiwith units m/s.\nWe show in Fig. 2 a plot of the terminal domain wall\nvelocity as a function of the applied current for several\nvalues of the spin-orbit interactions. To illustrate the ef-\nfect of the chirality, we plot in (a) the case \u0003 >0 while\nin (b) \u0003<0. The latter results are consistent with the\nnumerical study of Ref. [21], and shows that the spin-\norbit interaction can greatly enhance the domain wall\nvelocity at low currents up to the Walker breakdown. In\nthe former case, however, the domain wall velocity be-\nhaves di\u000berently when changing the current J. The spin\ntorque induced by the e\u000bective Rashba-\feld is now di-\nrected opposite to the conventional current-driven spin\ntorque and there is a competition between the two. For\nnon-zero \u0003, the domain wall velocity starts by moving in\nthe direction opposite to the current, whereas it eventu-\nally changes sign with increasing J(above the threshold\ncurrent). Thus, experimentally observing a sign-reversal\nof the domain-wall velocity with applied current would\nbe an indication of precisely this chirality sensitive spin-\norbit coupling e\u000bect. We note that this sign-reversal of\nthe wall velocity is di\u000berent from the one predicted in\n[25], which originated from a Slonczewski-like spin-orbit\ntorque proportional to \fand when considering a di\u000ber-ent type of domain wall pro\fle. In our case, the \feld-\nlike spin-orbit torque is su\u000ecient to cause the velocity-\nreversal.\nIV. SUMMARY\nIn conclusion, we have used the Lagrangian formalism\nto derive an exact analytical expression for the domain\nwall velocity in a spin-orbit coupled ferromagnet. 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Suzuki, Nature Mater. 11, 39(2012)\n[35] W.-G. Wang, M. Li, S. Hageman, and C. L. Chien, Na-\nture Mater. 11, 64 (2012).\n[36] The anisotropy energy has the standard form EA=\n\u0000KM2\ny=2 +K?M2\nx=2, which is assumed to e\u000bectively\ninclude magnetostatic contributions of the same form as\nin C. Kittel, Phys. Rev. 73, 155 (1948)."    },    {        "title": "2105.07376v1.Anatomy_of_inertial_magnons_in_ferromagnets.pdf",        "content": "arXiv:2105.07376v1  [cond-mat.mes-hall]  16 May 2021Anatomy of inertial magnons in ferromagnetic nanostructur es\nAlexey M. Lomonosov1,∗Vasily V. Temnov2,3,†and Jean-Eric Wegrowe3‡\n1Prokhorov General Physics Institute of the Russian Academy of Sciences, 119991, Moscow, Russia\n2Institut des Mol´ ecules et Mat´ eriaux du Mans, UMR CNRS 6283 ,\nLe Mans Universit´ e, 72085 Le Mans, France and\n3LSI, Ecole Polytechnique, CEA/DRF/IRAMIS, CNRS,\nInstitut Polytechnique de Paris, F-91128, Palaiseau, Fran ce\n(Dated: May 18, 2021)\nWe analyze dispersion relations of magnons in ferromagneti c nanostructures with uniaxial\nanisotropy taking into account inertial terms, i.e. magnet ic nutation. Inertial effects are\nparametrized by damping-independent parameter β, which allows for an unambiguous discrimi-\nnation of inertial effects from Gilbert damping parameter α. The analysis of magnon dispersion\nrelation shows its two branches are modified by the inertial e ffect, albeit in different ways. The up-\nper nutation branch starts at ω= 1/β, the lower branch coincides with FMR in the long-wavelength\nlimit and deviates from the zero-inertia parabolic depende nce≃ωFMR+Dk2of the exchange\nmagnon. Taking a realistic experimental geometry of magnet ic thin films, nanowires and nanodiscs,\nmagnon eigenfrequencies, eigenvectors and Q-factors are found to depend on the shape anisotropy.\nThe possibility of phase-matched magneto-elastic excitat ion of nutation magnons is discussed and\nthe condition was found to depend on β, exchange stiffness Dand the acoustic velocity.\nPACS numbers: Valid PACS appear here\nI. INTRODUCTION\nAfter the first description of the dynamics of the mag-\nnetization by Landau and Lifshitz [1], Gilbert proposed\nan equation that contains a correction due to the preces-\nsional damping [2, 3]. Since then, the so-called Landau-\nLifshitz-Gilbert (LLG) equation is known to give an ex-\ncellent description of the dynamics of the magnetization,\nincluding ferromagnetic resonance (FMR) and magneto-\nstatic waves [4, 5], as well as the magnetization reversal\n[6, 7]. Ferromagnetic resonance and time-resolved mag-\nnetization measurementsallow its spatially homogeneous\nprecession ( k= 0) but also non-uniform modes of the\nmagnetizationprecession( k∝ne}ationslash= 0, where kisthe wavevec-\ntor of spin waves) to be measured [8–10]. During the last\ndecades, these techniques have been advanced in the con-\ntext of ultrafast demagnetization dynamics [11, 12] that\npaved the way for the description of new physics at the\nsub-picosecond regime. High-frequency resonant modes\nof exchange magnons have been measured with ultrafast\ntime-resolved optical techniques [8, 10, 13]. Therefore,\nthe validity of the LLG equations has been confirmed\ndown to the picosecond time scale and below.\nHowever, limitations of LLG equations has been es-\ntablished in the stochastic derivation performed by W.\nF. Brown in a famous paper published in 1963 [14]. This\nlimit is due to the hypothesis that the typical time scales\nof magnetization dynamics are much longer than those\nof other degrees of freedom forming the dissipative envi-\nronment. In analogy to the common description of the\ndiffusion process of a Brownianparticle, the inertial (mo-\nmentum) degrees of freedom are supposed to relax much\n∗lom@kapella.gpi.ru\n†vasily.temnov@univ-lemans.fr\n‡jean-eric.wegrowe@polytechnique.edufaster than its spatial coordinate. This means that the\ndegrees of freedom related to the linear momentum (in\nthe case of the usual diffusion equation), or to the an-\ngular momentum (in the case of the magnetization) are\nincluded into the heat bath. As a consequence, the iner-\ntial terms do not explicitly appear in the equations, but\nare considered to be part of the damping term [15].\nThe possibility of measuring the contribution to iner-\ntial degrees of freedom led to a generalizationof the LLG\nequation with an additional term, incorporating the sec-\nond time-derivative of magnetization:\n˙m=−γm×Heff+αm×˙m+βmרm,(1)\nwherem=M/Msis the unit magnetization vector\nthatgivesthedirectionofthemagnetizationateachpoint\n(andMsis the modulus of the magnetization, which is\nconstant), γ=γ0µ0is the gyromagnetic ratio, αstands\nfor the Gilbert damping. Inertial effects are character-\nized by the parameter β, which is introduced in a phe-\nnomenological way, i.e. independent on αandγ[16].\nThis generalized LLG equation has been derived in the\nframework of different and independent theoretical con-\ntexts [15, 17–32]; its solutions have been studied in a\nseries of publications [33–36]. The main consequence of\ninertia for the uniform magnetization (magnon with the\nwavevector k= 0) is the existenceofnutation oscillations\nthat are superimposed to the precession. This leads to\nan appearance of the second resonance peak at a higher\nfrequency in FMR spectra. The direct measurement of\nnutation has been reported recently [37, 38].\nThe goal of the present report is to study the conse-\nquences of these inertial effects on the exchange magnons\n(i.e.k∝ne}ationslash= 0 modes), in the perspective of experimental\nstudies. Magnons are defined as linear magnetic exci-\ntations propagating in ferromagnets at the micromag-\nnetic limit. This work completes the first description2\nFMR \nmagnon ys(t) s(t) \nNutation \n   magnon (a)                                                                   (b) \nxΨz\nk\nFIG. 1. (a) Inertial magnons propagating in ferromagnetic\nnanostructures with wavevector kalong the zdirection under\nan external magnetic field Hresult in complex magnetization\ndynamics. (b) They can be decomposed in FMR magnon\nand nutation magnon precessing in opposite directions on el -\nliptical trajectories at different frequencies, giving ris e to a\ncharacteristic flower-shaped trajectory.\npublished in 2015, Section IV of the remarkable work of\nToru Kikuchi and Gen Tatara [22], and independently\nreconsidered by Makhfudz et al. in 2020 [39].\nThe paper is organized as follows. Section II presents\nthe derivation of the linear magnetic excitations deduced\nfrom (1). Section III describes the dispersion relation\nin a simple case of zero dipolar field (spherical symme-\ntry). The first consequence of the inertia is that the dis-\npersion relation splits in two branches: the lower one\ns1exp(ikz−iω1t) (FMR magnons ) and the upper one\nfors2exp(ikz−iω2t) (nutation magnons ). The second\nconsequence is that the quality factor Qincreases with\nthek-vector. SectionIVgeneralizesthedescriptiontothe\ncaseofauniaxialanisotropyquantifiedbythe dimension-\nless (shape) anisotropy parameter ξ. In the anisotropic\ncase the trajectories of both FMR magnons and nuta-\ntion magnons become elliptical and rotating in opposite\ndirections at each point in space. For a given k-vector\nthe magnetization vector corresponding to a superposi-\ntion of both magnons draws a typical trochoidal trajec-\ntory (see Fig. 1). Section V discusses the conditions\nfor phase-matched excitation the nutation magnons by\nco-propagatinglongitudinalacoustic phonons, illustrated\nby the material parameters for Gd-doped Permalloy thin\nfilms [13].\nII. EXCHANGE MAGNONS IN FERROMAGNETIC\nTHIN FILMS WITH MAGNETIC INERTIA\nWe start with the LLG equation for unit magnetiza-\ntion vector mwith aneffective field Heff, which includes\nexchange interactions with stiffness D, an external field\nH= (Hx,0,Hz) and a demagnetizing field induced by\nthe shape anisotropy Hd=−MS/hatwideNm. The demagne-\ntization tensor /hatwideNdepends on the specific shape of the\nferromagnetic sample. Hereafter we assume the diagonal\nformof/hatwideNwithdiagonalelements Nx,NyandNz. Damp-ing of the magnetization dynamics is described by the\nconventional Gilbert term with parameter α. In addition\nto the conventional LLG equation we take into account\nthe inertial effect characterized by the independent pa-\nrameterβ. Then the inertial LLG equation (ILLG) takes\nthe form of Eq.(1) with Heff=H+D∆m+Hd.\nThe coordinate system was chosen such that the ex-\nternal field lies in the y= 0 plane, as is shown in Fig. 1.\nThe material is assumed to be magnetically isotropic, so\nthat the unperturbed magnetization vector also lies in\nthey= 0 plane. We seek for time- and space-dependent\nsolutions in the form m=m0+s(z,t) with spin-wave\nsolutions\ns(z,t) = (sx,sy,sz)exp(ikz−iωt) (2)\npropagating as plane waves with a real wave vector k\nalong the z-axis, see Fig. 1. Substitution m(z,t) into\nequation (1) and its linearization with respect to small\nperturbations sx,sy,szresults in a homogeneous system\nof three linear equations:\n/hatwideA\nsx\nsy\nsz\n= 0 (3)\nwhere the matrix Ais given by\n\n−iω A 12(ω,k) 0\nA21(ω,k)−iω A 23(ω,k)\n0A32(ω,k)−iω\n (4)\nwith coefficients Aij(ω,k) defined as:\nA12=mz(γDk2+γMSξyz−iαω−βω2)+γHz\nA21=−mz(γDk2+γMSξxz−iαω−βω2)+γHz\nA23=mx(γDk2+γMSξzx−iαω−βω2)+γHx(5)\nA32=−mx(γDk2+γMSξyx−iαω−βω2)−γHx\nwhere coefficients ξij=Ni−Njcharacterize the shape\nanisotropy. The condition for the nontrivial solution of\nthehomogeneoussystem(3) toexist, i.e. det A= 0, gives\nrise to the secular equation\nω2+A12(ω,k)A21(ω,k)+A23(ω,k)A32(ω,k) = 0 (6)\nwhich is used to calculate the spin wave dispersion re-\nlationω(k) for different shapes/symmetries, i.e.charac-\nterized by different types of the /hatwideNtensor.\nIII. INERTIAL EXCHANGE MAGNONS IN SAMPLES\nWITH SPHERICAL SYMMETRY\nExamples of such symmetry are infinite homogeneous\nisotropic ferromagnetic media, or any spherical body. In3\nthesecasesthedemagnetizationtensor /hatwideNisdiagonalwith\nall nonzero elements equal 1 /3, so that its contribution\nto the magnetization dynamics (1) and correspondingly\nto the wave matrix components(5) vanishes. The secular\nequation (6) takes a concise form:\n/parenleftbig\nγH+γDk2−βω2−iαω+ω/parenrightbig\n×(7)\n×/parenleftbig\nγH+γDk2−βω2−iαω−ω/parenrightbig\n= 0.\nDue to the symmetry of /hatwideN, equation (7), and hence all\nits roots, remains independent on the direction of Hand\nthe equilibrium magnetization m0with respect to the\nwave propagation direction along the z-axis. For each\npositive wavenumber k, the determinant (7) is solved\nforω. Given that the presumed solution has a form\n∼exp(ikz−iωt), positive ωdesignates the waves trav-\nelling in the positive direction. The two positive roots\ncorresponding to the first parenthesis in (7) have the fol-\nlowing forms:\nω1=1\n2β/parenleftig\n−1−iα+/radicalbig\n4γβ(Dk2+H)+(1+iα)2/parenrightig\n(8)\nω2=1\n2β/parenleftbigg\n1−iα+/radicalig\n4γβ(Dk2+H)+(1−iα)2/parenrightbigg\n(9)\nThefirstrootisthelowermagnonbranchorprecession,\nslightly modified by the inertial term and the second one\nexhibits the inertial magnon branch or nutation. It is\nconvenient to split these roots into real and imaginary\nparts:ω1,2=ω′\n1,2+iω′′\n1,2. Taylor series approximation of\nthose roots assuming the smallness of γβDk2,γβH,α ≪\n1 results in the following expressions for their real parts:\nω′\n1≈γ[Dk2+H−2βγHDk2+...] (10)\nω′\n2≈1\nβ+ω′\n1 (11)\nIn this approximation the nutation magnon branch\nis simply shifted by 1 /βwith respect to FMR magnon\nbranch. The validity of this approximation is illustrated\nin Fig. 2, where the exact roots given by (8) and (9) are\ndepicted with solid lines, whereas dashed lines represent\nthe power series approximation of (10) and (11).\nLower branch emerges from the Larmor’s frequency\nγHand grows parabolically with k. Effect of inertia re-\nduces the coefficient at the term quadratic in k. Upper\nbranch is simply displaced by +1 /βand has the simi-\nlar shape. Imaginary parts of the roots ω′′\n1,2represent\nattenuation of the corresponding magnetization dynam-\nics in time as ∝exp/parenleftbig\nω′′\n1,2t/parenrightbig\n, and therefore they must be\nnegative. In the frequency domain they characterize the\nwidth ∆f=|ω′′|/π(FWHM) of the Lorentzian spectral\nline.0.3 0.5 0.8 1.0 1.3 1.5 0.25 0.50 0.75 1.00 1.25 \nβ= 0.17ps \nβ= 0.27ps β= 0.17ps Frequency, THz\nwavenumber  k, rad/nm β= 0.27ps \n0.25 0.50 0.75 1.00 1.25 1.50 10 20 30 \nβ= 0β= 0\nα= 0.023 α= 0.023 \nα= 0.01 \nα= 0.01 Line Width, GHz\nwavenumber  k, rad/nm β= 0.27ps \nFIG. 2. (a) Dispersion of the inertial magnon (red, magenta)\nand conventional magnon (blue,green). Dashed curves shows\nthe approximations by (10) and (11). (b) corresponding line\nwidths. In order to indicate the effect of inertia on the pre-\ncession, dashed curves show the line width without inertia\nβ= 0.\nω′′\n1≈ −αγ/bracketleftbig\nDk2+H−6βγ2HDk2+.../bracketrightbig\n(12)\nω′′\n2≈ −α\nβ−ω′′\n1 (13)\nNote that in the limiting case of (12), (13) field and\nexchange stiffness have opposite effects on the attenua-\ntion of the two magnon branches: they increase the at-\ntenuation in the lower branch ω1and decrease it for the\ninertial branch ω1. In the other limiting case of large\nfield and large kattenuation of both branches tends to\nexp/parenleftig\n−α\n2βt/parenrightig\n. The damping for both branches appears\nto be naturally proportional to the Gilbert damping pa-\nrameterα. A conspicuous decrease of nutation linewidth\nω′′\n2(k= 0) with growing αreported by Cherkasski et\nal. [36] roots back to the parametrization of the nu-\ntation phenomenon in terms of a product ατ, whereτ\ndenotes the characteristic nutation lifetime. Within this\nparametrization, a variation of α, while keeping τcon-\nstant, leads to the simultaneous decrease of the nutation4\n0.2 0.4 0.6 0.810203040Q\nk, nm-11/2H=0 2 T5 T\n0.2 0.4102030in a thin film\nFIG. 3. Q-factor dependencies on kfor different values of the\nexternal field. Dashed line shows the quadratic in kapprox-\nimation for H= 0. Inset shows Q-factors for the ordinary\nmagnon (blue curve) and inertial magnon (red curve) in a\nthin film, H= 0\nfrequency 1 /β= 1/(ατ) rendering the analysis of damp-\ning extremely difficult. An alternative notation in terms\nofαandβ, introduced in this paper, resolves this prob-\nlem and allows for an independent investigation of iner-\ntial and damping effects.\nAnother parameter, which characterizes the resonant\nspectral line centered at frequency f0, is its quality factor\ndefined as Q=f0/∆f=ω′/(2ω′′). As can be seen from\nequations (8) and (9), Q-factors for both branches coin-\ncide within the accuracy of ∼(ωα)2. Dependence of the\nQ-factor on the wavenumber klooks counterintuitive in\nthat it essentially grows with k. Assuming for simplicity\nH= 0, for small kthe Q-factor can be approximated by\nexpansion of ω′andω′′in the power series in k, which\nresults in:\nQ(k)∼1\n2α1+γβDk2\n1−γβDk2+...∼1\n2α/parenleftbig\n1+2γβDk2/parenrightbig\n(14)\nExact values for the Q-factor in comparison to the\nestimate of (14) are shown in Fig. 3 for the external\nfield ranging from 0 to 5 T.\nField effect for small kcan be approximated as Q∼\n1/(2α)(1+2γβH).\nIV. INERTIAL EXCHANGE MAGNONS IN SAMPLES\nWITH CYLINDRICAL SYMMETRY\nExamplesofsuchbodies aredisks, wires, infinite plates\nand films. Axial symmetry about the z-axis retains the\ndiagonal form of /hatwideNwith the diagonal elements satisfying\nthefollowingconditions: Nx=NyandNx+Ny+Nz= 1.\nAs a result, components of the matrix /hatwideAgiven by (5)acquire terms proportional to γMS. Lack of symmetry\nmakes the magnon propagation dependent on the orien-\ntation of vectors m0andHwith respect to the z-axis.\nWe consider two limiting cases: collinear arrangement\nwithm0parallel to the axis of symmetry (Θ = Ψ = 0\nin Fig. 1); and orthogonal arrangement with m0paral-\nlel to the x-axis and Θ = Ψ = π/2. In the collinear\ncase the demagnetizing field acts simply against the ex-\nternal field, hence the secularequation remains similarto\n(7), but with field Hsubstituted with the reduced field\nH′=H−ξMS:\n/parenleftbig\nγH′+γDk2−βω2−iαω+ω/parenrightbig\n×(15)\n×/parenleftbig\nγH′+γDk2−βω2−iαω−ω/parenrightbig\n= 0\nwhereξ=Nz−Nx=Nz−Nycharacterizes the shape\neffect on demagnetizing, so that in an infinite wire ξ=\n−1/2, in the spherical symmetric (or unbounded) body\nξ= 0 and in the infinite film ξ= 1. Correspondingly the\nroots to (15) are similar to ones given in (8) and (9) with\nmodified field:\nω1=1\n2β/parenleftbigg\n−1−iα+/radicalig\n4γβ(Dk2+H′)+(1+iα)2/parenrightbigg\n(16)\nω2=1\n2β/parenleftbigg\n1−iα+/radicalig\n4γβ(Dk2+H′)+(1−iα)2/parenrightbigg\n(17)\nAt the low- klimit the lower branch roughly tends\nto the Larmor’s frequency γ(H−ξMS) and the upper\nbranch limit is 1 /β+γ(H−ξMS), which is similar to the\ncase of spherical symmetry, but with modified field. In\nthe orthogonal configuration with m0andHperpendic-\nular to the axis of symmetry and to the magnon propa-\ngation direction, roots of the determinant (4) generally\ncannot be found in an analytical form. Therefore we\nfirstconsideranapproximatesolutions,andthendescribe\nbrieflythe numericalgorithmforobtainingthedispersion\ncurves. By neglecting the Gilbert attenuation ( α= 0),\nthe approximate solutions to (4) for the in-plane mag-\nnetization and field can be found in a concise analytical\nform:\nω1,2=1\nβ√\n2{2γβ(Dk2+H)+γβξM s+1\n∓/radicalbig\n4γβ(Dk2+H)+(γβξM s+1)2}1/2(18)\nHere indices 1 and 2 denote the frequencies of the\nconventional and inertial magnons respectively, sign ‘-\n‘ prior to the square root in (18) corresponds to the\nlower branch ω1; sign ‘+’ denotes the inertial branch\nω2. Numerical procedure for building the dispersion\nrelations of the magnonic modes for nonzero αor ar-\nbitrary orientation of the external field H starts with\ncalculation of the stationary equilibrium magnetization5\nm0= (mx,my,mz). This can be done by solving (1)\nin its stationary form, i.e.with all time derivatives set\nzero. In a thin film, for example, quantities Hiandmj\nare related by MSmxmz+Hxmz−Hzmx= 0. Thus\nobtained stationary magnetization components are then\nsubstituted into (5) and (4). At some fixed small kthe\ndeterminant (4) as a function of complex-valued ωpos-\nsesses two minima, which correspond to the FMR and\nnutational branches. Their exact locations can be evalu-\natedbyanumericalroutinewhichminimizestheabsolute\nvalue of the determinant (4) in the vicinity of the guess\nvalues for those branches, for example given by equa-\ntions (19) and (20). Then we give ka small increment\nand repeat the extremum search using the ωs obtained\nat the previous step as guess values, and so on. As a\nresult, calculated values for ω1andω2follow the disper-\nsion curves of both branches. Note that for nonzero α\nrootsω1,2possess imaginary parts, which determine the\nline width and Q-factor for each mode. Let us consider\nthe magnetization behavior in a thin film in more detail.\nFor this geometry /hatwideNpossesses the only nonzero compo-\nnentNz= 1, and correspondingly ξ= 1. In the small- k\nlimit, the lower branch approaches the Kittel’s frequency\nωFMR=γ/radicalbig\nH(H+MS) from below as β,k→0:\nω1≈ωFMR−1\n2γωFMR(2H+MS)β+...(19)\nEffect of the demagnetizing field on the inertial branch\nis exhibited by an upward shift by1\n2γMS; whereas effect\nof inertia is opposite:\nω2≈1\nβ+γH+1\n2γMS−/parenleftbigg\nω2\nFMR+1\n8γ2M2\nS/parenrightbigg\nβ+...(20)\nFor the orthogonal configuration, when both m0and\nHlie in the film plane, we can estimate the trajectories\nof the magnetization dynamics of both modes for small\nkand small α. For each root given by (18) we solve\nthe homogeneous equation for perturbations s(3). Nor-\nmalization of the solutions can be chosen in an arbitrary\nway,here for simplicity we define sz= 1. In orthogonal\ngeometry sxcomponent is obviously negligible or equals\nzero, so the system reduces to two equations in syand\nsz. Results shown as a power series expansion for small\ninertiaβω≪1 for the precession:\nsp=\n0\n−i/radicalig\n1+ξMS\nDk2+H/parenleftig\n1+βγξM S\n2/parenrightig\n1\nexp(−iω1t)\n(21)\nand nutation:\nsn=\n0\ni(1−βγξM S/2)\n1\nexp(−iω2t) (22)0.1 0.2 0.3 0.4 0.5 0.61.01.21.41.61.82.0\n1 T0.2 Tellipticity sy’/sx’\nk, nm-1sx’ sy’0.1 T\nFMR\nnutation\nFIG. 4. Ratio of the polarization axes for the external field\nof 0.1T, 0.2T, and 1T. Handm0are parallel to the inward\nnormal to the figure plane.\nwith the parameter of anisotropy ξ= 1 for a thin film\nnormal to the z−axis and ξ= 1/2 for a thin wire spread\nalong the z−axis. Both perturbations exhibit elliptical\npolarization within the ( x′,y′) plane. Precession trajec-\ntory is deformed by the demagnetizing effect so that the\ny-axis of the ellipse is stretched with the/radicalbig\n1+MS/H\nfactor due to demagnetizing effect, and in addition on\naccount of inertial effect. On the contrary, the nutation\nellipse is squeezed along the y-axis proportionally to the\ninertial parameter β. Ellipticity of the lower branch de-\npends on the external field (21), whereas that of the up-\nper branch in this approximation shows no dependence\non the field. Signs of sycomponents are opposite for\nnutation and precession, this indicates that they are ro-\ntating in the opposite directions. Exact polarizationscan\nbe found numerically for a reasonable set of material pa-\nrameters and fields, as is shown in Fig. 4.6\nV. EXCITATION MECHANISMS OF INERTIAL\nEXCHANGE MAGNONS\nThe only experimental evidence of inertial effects in\nferromagnets has been reported for k= 0 nutation man-\ngons in Py-thin films resonantly excited with a mag-\nnetic field of an intense quasi-monochromatic THz pulse\n[37]. In order to excite k∝ne}ationslash= 0 exchange magnon modes\none would need to have either spatially localized and\ninstantaneous stimuli [10] or any other source of effec-\ntive magnetic field characterized by spectral and spatial\noverlap with investigated magnon modes. The letter can\nbe provided through ultrashort large-amplitude acoustic\npulses [40, 41] producing effective magneto-elastic fields\nrapidly varying in time and space [42]. Acoustic pulses\npropagating through a thin ferromagnetic sample at an\nacoustic velocity vare quantified by a linearized disper-\nsion relation ωac=vk. Crossing between acoustic and\nmagnon brunches, i.e. satisfying the phonon-magnon\nphase-matchingcondition, usually facilitatesthe acoustic\nexcitation of magnetization dynamics [43, 44]. A ques-\ntion arises under which conditions the crossing between\ndispersion curves for longitudinal phonons and inertial\nmagnons can occur. Whereas for realistic magnetic fields\nthe acoustic dispersion always intersects the lower FMR-\nbranch at a frequency close to FMR frequency [42], the\ncrossing of the upper nutation brunch is less obvious.\nFIG. 5. The magneto-acoustic phase matching condition for\nnutation magnons can be tuned vie the reduction of exchange\nstiffness in Gd-doped Py samples. Gd concentration x varies\nfrom 0 to 13%. The dashed line displays the acoustic disper-\nsion relation ωac/(2π). Magneto-elastic coupling with inertial\nmangon is efficient when the dashed line lies within the pink\ntinted area. Material parameters are taken from Ref. [13] an d\nβ=0.276 ps.\nIt is possible to quantify the criterion for magneto-\nelastic crossing with nutation magnons analytically. To\ndo that we note that for larger wavenumbers ksatisfying\nDk2≫H,MStheexchangetermplaysthedominantrole\nand the asymptotic behaviourfor both branchesbecomeslinear in k:\nω1,2≈ ∓1\n2β+k/radicaligg\nγD\nβ. (23)\nIt follows from (23) that the condition for the nutation\nmagnon branch to intersect the acoustical dispersion re-\nlationωac(k), requires the asymptotic slope of ω2(k) to\nbe smaller than the acoustic velocity v:\n/radicaligg\nγD\nβ< v. (24)\nThis expression shows that for a given βthe magneto-\nelastic crossing is facilitated by small exchange stiffness\nDand small acousticvelocity. This approximateanalysis\nbreaks down for acoustic frequencies in above-THz spec-\ntral range, where the acoustic dispersion starts deviating\nfrom its linear approximation.\nFigure 5 highlights the remarkable role of exchange\nstiffness to achieve the dispersion crossing between nuta-\ntionmagnonsandlongitudinalacousticphonons. Doping\nPy thin films with Gadolinium has been shown to gradu-\nally reduce the exchange stiffness upon Gd-concentration\nfrom 300 to 100 [meV ·˚A2] [13]. For a fixed value of in-\nertial parameter β=0.276 ps, nutation magnons for pure\nPy samples do not display any crossing with acoustic\nphonons within the displayed range of k-vectors but the\nGd-doped Py with 13% Gd concentration does. The nu-\ntation magnon-phonon crossing point occurs at 0.75 THz\nfrequency and k= 0.85 nm−1(magnon wavelength of\napproximately 5 nm), i.e. magnon parameters readily\naccessible in ultrafast magneto-optical experiments [10].\nVI. CONCLUSIONS\nIn this paper we have theoretically studied exchange\ninertial magnons in ferromagnetic samples of different\nshapes under the action of an external magnetic field.\nThe parametrizationof magnetization dynamics in terms\nof two independent parameters, the Gilbert damping α\nand the inertial time β, allows for unambiguous discrim-\nination between the inertial and damping effects as well\nas their impact on both branches of magnon dispersion.\nInertial effects are found to strongly effect not only the\nfrequencies (magnon eigenvalues) of both branches but\nalso result in a monotonous increase of the Q-factor as\na function of the external magnetic field and magnon k-\nvector. The two magnon branchesare found to precessin\nopposite directions along the elliptical trajectories with\nperpendicularlyorientedlongaxisoftheellipses(magnon\neigenvectors). Their ellipticity is found to depend on the\ncomponents of the demagnetizing tensor. An analyti-\ncal criterion for the existence of phase-matched magneto-\nelastic excitation of nutation magnons has been derived\nandillustratedforGd-dopedpermalloysampleswithtun-\nable exchange stiffness.7\nACKNOWLEDGMENTS\nFinancial support by Russian Basic Research Founda-\ntion (Grant No. 19-02-00682)is gratefully acknowledged.\n[1] L. D. Landau and L. M. Lifshitz, Physik. Zeits. Sowjetu-\nnion8, 153 (1935).\n[2] T. L. Gilbert, Ph. D. Thesis (1956).\n[3] T. L. Gilbert, IEEE transactions on magnetics 40, 3443\n(2004).\n[4] R. W. Damon and J. R. Eshbach, Journal of Physics and\nChemistry of Solids 19, 308 (1961).\n[5] M. Farle, Reports on progress in physics 61, 755 (1998).\n[6] L. Thevenard, J.-Y. Duquesne, E. Peronne, H. J.\nVon Bardeleben, H. Jaffres, S. Ruttala, J.-M. George,\nA. Lemaitre, and C. Gourdon, Physical Review B 87,\n144402 (2013).\n[7] V. S. Vlasov, A. M. Lomonosov, A. V. Golov, L. N. Ko-\ntov, V. Besse, A. Alekhin, D. A. Kuzmin, I. V. Bychkov,\nand V. V. 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B\n95, 060409(R) (2017)."    },    {        "title": "1412.2479v1.Magnetization_Dynamics_driven_by_Non_equilibrium_Spin_Orbit_Coupled_Electron_Gas.pdf",        "content": "arXiv:1412.2479v1  [cond-mat.mes-hall]  8 Dec 2014Magnetization Dynamics driven by Non-equilibrium Spin-Or bit Coupled Electron Gas\nYong Wang,1Wei-qiang Chen,2and Fu-Chun Zhang3,1,4\n1Department of Physics, The University of Hong Kong, Hong Kon g SAR, China\n2Department of Physics, South University of Science and Tech nology of China, China\n3Department of Physics, Zhejiang University, China\n4Collaborative Innovation Center of Advanced Microstructu res, Nanjing University, Nanjing, 210093, China\nThe dynamics of magnetization coupled to an electron gas via s-d exchange interaction is investi-\ngated by using density matrix technique. Our theory shows th at non-equilibrium spin accumulation\ninduces a spin torque and the electron bath leads to a damping of the magnetization. For the\ntwo-dimensional magnetization thin film coupled to the elec tron gas with Rashba spin-orbit cou-\npling, the result for the spin-orbit torques is consistent w ith the previous semi-classical theory. Our\ntheory predicts a damping of the magnetization, which is abs ent in the semi-classical theory. The\nmagnitude of the damping due to the electron bath is comparab le to the intrinsic Gilbert damping\nand may be important in describing the magnetization dynami cs of the system.\nI. INTRODUCTION\nIn study of spin transfer torque (STT), it has been\nproposed [1, 2] to manipulate magnetic order parameter\ndynamics by using non-equilibrium electron bath instead\nof external magnetic fields. The proposal has already led\nto commercial products in spintronics engineering. Re-\ncently, there has been much attention on the ”spin-orbit\ntorque”(SOT), which was first proposed in theory[3, 4],\nand later confirmed in experiments[5–8] (see Ref. 9 and\n10 for a comprehensive review). After applying an ex-\nternal electric field to the electron gas with spin-orbit in-\nteraction(SOI), a component ofthe accumulated electron\nspin density mis-aligned with the ferromagnetic ordering\ncan be created[3, 4], which then will induce a field-like\ntorque. The SOT opens the possibility of manipulat-\ning the magnetic order parameter in collinear magnetic\nstructures and may efficiently reduce the critical current\ndensity for magnetization switching[3, 4]. In the theoret-\nical side, a full quantum theory has been proposed and\ndeveloped to describe the dynamics of a single domain\nmagnetunderthecontinuousscatteringbyspin-polarized\nelectrons. The quantum STT theory recovers the results\nof the semiclassical STT theory, and has revealed more\ndetailsaboutthemagnetizationdynamicsintheSTT[11–\n13]. Therefore, it will be natural to apply a full quantum\ntheory to study the magnetization dynamics influenced\nby the SOI electron gas. This may be an extension of\nthe quantum STT theory to SOT. In the full quantum\ntheory, the quantum dynamics of the magnetization can\nbe described by the evolution of its density matrix under\nthe influence of the electron gas, which can be tuned by\nthe external electric field. This treatment will not only\ngive the mean-field effect on the magnetization dynamics\nby the electron bath, but also include the damping of the\nmagnetization due to the fluctuation of the electron spin.\nThe similarstrategyhas been exploited to investigatethe\nphoto-excited dynamics of the order parameter in Peierls\nchain[14].\nThis paper is organized as follows. In section II, we\napply density matrix technique to derive general formal-E\nFIG. 1. (Color online). Schematic diagram for the lattice of\nlocalized spins (orange) coupled to the conductions electr ons\n(blue) through s-d exchange interaction. An external elect ric\nfieldEcan be applied to tune the electron bath.\nism for the magnetization dynamics driven by the elec-\ntron bath through s-d exchange interaction. In section\nIII, we apply the general formalism to the special case\nwhere the spatially uniform magnetization is coupled to\na two dimensional electron gas with Rashba SOI, and\ncalculate the spin-orbit torque and the damping effect of\nthe electron bath. The main results are summarized and\ndiscussed in section IV.\nII. GENERAL FORMALISM\nWe apply density matrix technique to study dynamics\nof the magnetization driven by the electron bath via an\ns-d exchange interaction. The system is schematically\nillustratedinFig.1, wheretheelectronbathcanbe tuned\nbyanexternalelectricfield. TheHamiltonianofthetotal\nsystem is formally written as\nH=HM+He+Hsd. (1)\nHere,HMis the Hamiltonian for the magnetization sub-\nsystem in terms of the local spin operators /hatwideSi,µat sitei\nwith spin directions µ(=x,y,z);Heis the Hamiltonian\noftheelectronsubsystem; Hsddescribesthes-dexchange\ninteraction between the magnetization and the electron,2\nwhere\nHsd=J/summationdisplay\ni,µ/hatwideSi,µ/hatwideσi,µ. (2)\nHere,/hatwideσi,µrepresents the electron spin operator at site i\nwithout /planckover2pi1/2, andJis the exchange coupling strength.\nNote that we have not specified the forms of HMandHe\nyet, thus the results below will be quite general.\nThe effect of the s-d exchange interaction Hsdis\ntwofold. On one hand, the magnetization dynamics is\ndriven by the electron bath via Hsd; on the other hand,\nthe electron states are also affected by the magnetization\nconfiguration in turn due to Hsd. Since the time scale of\nthe electron dynamics is usually much faster than that\nof the magnetization dynamics, we may assume that the\nelectrons under the bias voltage establish a stationary\nnon-equilibrium distribution in a very short time inter-\nval, during which the change of the magnetization con-\nfiguration is negligible and the non-equilirium electron\nbath is approximated to be constant. The validity of\nthis assumption only holds if the spin-lattice interaction\nis stronger than the s-d exchange interaction to relax\nthe electron spin. Consider a short time interval [ t0,t],\nwhere the initial density matrices of the magnetization\nand the electron bath are ρM(t0) andρe(t0) respectively.\nThen the initial magnetization configuration at each site\nisSi,µ(t0) = Tr[/hatwideSi,µρM(t0)], and the initial electron den-\nsity matrix ρe(t0) is determined by the bath Hamiltonian\nHB=He+J/summationtext\ni,µSi,µ(t0)/hatwideσi,µand the open boundary\nconditions.\nIn order to investigate the magnetization dynamics\nduring the time interval [ t0,t] defined above, we rede-\nfine the local spin operators /hatwideSi,µ=Si,µ(t0) +/hatwidesi,µ, then\nthe Hamiltonian Hin Eq. (1) can be rewritten as\nH=HM+HB+Vsd, (3)\nwith the interaction term\nVsd=J/summationdisplay\ni,µ/hatwidesi,µ/hatwideσi,µ. (4)\nDuring this time interval, the electron density matrix ρe\nmay be approximated to be constant because of the neg-\nligible change of the magnetization, and this can be jus-\ntified in the limit t→t0. Assuming the total density\nmatrix as ρ(t) =ρM(t)⊗ρe(t0) and to the second or-\nder of interaction strength, the equation for the density\nmatrix/tildewideρM(t) in the interaction picture is[15]\nd\ndt/tildewideρM(t) =J\ni/planckover2pi1/summationdisplay\ni,µσi,µ(t)[/tildewidesi,µ(t),/tildewideρM(t0)]\n+(J\ni/planckover2pi1)2/summationdisplay\ni,µ;j,ν/integraldisplayt\nt0dτ{Ci,µ;j,ν(t,τ)[/tildewidesi,µ(t),/tildewidesj,ν(τ)/tildewideρM(τ)]\n−Cj,ν;i,µ(τ,t)[/tildewidesi,µ(t),/tildewideρM(τ)/tildewidesj,ν(τ)]}. (5)\nHere,/tildewider···denotes the operators in the interaction\npicture; the electron spin polarization is σi,µ(t) =Tre[/tildewideσi,µ(t)/tildewideρe(t0)]; the electron spin-spin correlation func-\ntion isCi,µ;j,ν(t,τ) = Tr e[/tildewideσi,µ(t)/tildewideσj,ν(τ)/tildewideρe(t0)], which\nis a function of t−τonly and satisfies the relation\nCi,µ;j,ν(t,τ) =C∗\nj,ν;i,µ(τ,t). In priciple, the solution of\nEq. (5) gives the density matrix of the magnetization in\nthe time interval [ t0,t] under the influence of the electron\nbath, and can be applied to study the physical qualities\nthat we are particularly interested in.\nBased on Eq. (5), the dynamical equation for Sl,λ(t) =\nTrM[/tildewideSl,λ(t)/tildewideρM(t)] is obtained as\nd\ndtSl,λ(t) =1\ni/planckover2pi1/an}b∇acketle{t[/hatwideSl,λ,HM]/an}b∇acket∇i}htt+J\n/planckover2pi1/summationdisplay\nµ,νǫλµνσl,µ(t)Sl,ν(t)\n+i(J\ni/planckover2pi1)2/summationdisplay\nj,µ,ν,ξǫλµν/integraldisplayt\nt0dτ{Cl,µ;j,ξ(t,τ)/an}b∇acketle{t/hatwideSl,ν(t)/hatwidesj,ξ(τ)/an}b∇acket∇i}htτ\n−Cj,ξ;l,µ(τ,t)/an}b∇acketle{t/hatwidesj,ξ(τ)/hatwideSl,ν(t)/an}b∇acket∇i}htτ}. (6)\nHere,/an}b∇acketle{t···/an}b∇acket∇i}htt≡TrM[···ρM(t)], and the spin commutation\nrelation [/hatwideSl,λ,/hatwideSi,µ] =iδli/summationtext\nνǫλµν/hatwideSl,νhas been exploited.\nThe first term in the r.h.s.of Eq. (6) gives the intrin-\nsic magnetization dynamics due to HM; the second term\nis the spin torque term due to the accumulation of the\nelectron spin density; the third term gives the damping\neffect of the electron bath. If the operator /hatwideSl,ν(t) in the\ndamping term is approximately replaced by its expecta-\ntion value Sl,ν(t), Eq. (6) becomes\nd\ndtSl,λ(t) =1\ni/planckover2pi1/an}b∇acketle{t[/hatwideSl,λ,HM]/an}b∇acket∇i}htt+J\n/planckover2pi1/summationdisplay\nµ,νǫλµνσl,µ(t)Sl,ν(t)\n+2J2\n/planckover2pi12/summationdisplay\nj,µ,ν,ξǫλµνSl,ν(t)/integraldisplayt\nt0dτKl,µ;j,ξ(t−τ)sj,ξ(τ),(7)\nwhereKl,µ;j,ξ(t−τ) is the imaginary part of Cl,µ;j,ξ(t,τ),\nandsj,ξ(τ) =/an}b∇acketle{t/hatwidesj,ξ/an}b∇acket∇i}htτ. We introduce the kernel func-\ntionγ(t) which satisfies the relation dγl,µ;j,ξ(t)/dt=\nKl,µ;j,ξ(t). The integral in the last term in Eq. (7) is\nrewrittenas/integraltextt\nt0dτγl,µ;j,ξ(t−τ)˙Sj,ξ(τ) afterintegratingby\nparts and neglecting the boundary terms in the limiting\ncaset→t0. It can be further simplified as Γ l,µ;j,ξ˙Sj,ξun-\nder the Markovian approximation ˙Sj,ξ(τ)≈˙Sj,ξ(t), with\nthe coefficient Γ l,µ;j,ξ=/integraltextδt\n0dτγl,µ;j,ξ(τ) forδt=t−t0.\nBased on the discussions above, Eq. (7) can be written\nin a compact form\nd\ndtSl(t) =1\ni/planckover2pi1/an}b∇acketle{t[/hatwideSl,HM]/an}b∇acket∇i}htt+γeBl(t)×Sl(t),(8)\nwhereγeis the gyromagnetic ratio; Blis the effective\nmagnetic field on the the local spin Sloriginating from\nthe electron bath. The µ-component of Blis expressed\nas\nBl,µ(t) =J\nγe/planckover2pi1σl,µ(t)+2J2\nγe/planckover2pi12/summationdisplay\nj,ξΓl,µ;j,ξ(t)˙Sj,ξ(t).(9)\nThe first term in (9) will give the torque term due to the\nelectron spin accumulation, which has been discussed ex-\ntensively in previous studies; the second term will give3\nthe damping effect of the electron bath on the magne-\ntization dynamics, which only emerges in the quantum\ntreatment. The non-local feature of the damping term\ncan be found here, which depends on the spatial correla-\ntion of Γ l,µ;j,ξ.\nSo far we have established a general dynamical equa-\ntion for the magnetization when it is coupled to the elec-\ntron bath via s-d exchange interaction. Here, both the\nHamiltonian for the magnetization subsystem HMand\nthe Hamiltonian for the electron subsystem Hehave not\nbeen specified yet. The treatment is similar to the previ-\nous work on the order parameter dynamics in the photo-\nexcited Peierls chain[14]. In the next section, we apply\nthis general formula to study the magnetization dynam-\nics of a two-dimensional ferromagnetic thin film under\nthe influence of an electron gas with Rashba SOI, i.e. a\nmodel system for “spin-orbit torque”.\nIII. SPIN-ORBIT TORQUE\nA. Electron Bath with Rashba SOI\nWe consider a special system studied by Manchon and\nZhang[3] for the spin-orbit torque. The two-dimensional\nmagnetization thin film in x-y plane consists of N=\nM×Nlattice sites with the lattice constant a, and we\nwill use the discrete notations in both real and reciprocal\nspace. The magnetization is assumed to be uniform due\nto strong exchange interaction. The lack of inversion\nsymmetry in z-direction induces the Rashba spin-orbit\ninteraction in the two-dimensional electron gas. In this\ncase, the Hamiltonian for the electron bath is given as[3]\nHB=/hatwidep2\n2m∗e+αR\n/planckover2pi1(/hatwidep×/hatwidez)·/hatwideσ+JS·/hatwideσ,(10)\nwhere/hatwidepis the electron momentum operator; m∗\neis the\neffective mass of electrons; αRis the Rashba interaction\nstrength; S=Siis the localized spin at each site. For\nS=S(sinθcosφ,sinθsinφ,cosθ), the energy dispersion\nrelation of the electron is\nEk,±=/planckover2pi12k2\n2m∗e±∆k. (11)\nHere, we have denoted the electron wavevector k=\nk(cosϕ,sinϕ), and\n∆k=/radicalBig\nJ2S2+α2\nRk2−2JSαRksinθsin(φ−ϕ).\nThe corresponding electron eigenstates |k,±/an}b∇acket∇i}htare\n|k,±/an}b∇acket∇i}ht=1√\nNeik·r/parenleftBigg\ncosΘk,±\n2e−iΦk\nsinΘk,±\n2/parenrightBigg\n,(12)where the angles Θ k,±and Φ kare determined by\ncosΘk,±\n2=/radicalbig\n∆2\nk−J2S2cos2θ/radicalbig\n2∆2\nk∓2JS∆kcosθ,\nsinΘk,±\n2=±∆k−JScosθ/radicalbig\n2∆2\nk∓2JS∆kcosθ,\ncosΦk=JSsinθcosφ+αRksinϕ/radicalbig\n∆2\nk−J2S2cos2θ,\nsinΦk=JSsinθsinφ−αRkcosϕ/radicalbig\n∆2\nk−J2S2cos2θ.\nThespinpolarizationvectorforthestate |k,±/an}b∇acket∇i}htisPk,±=\n(sinΘk,±cosΦk,sinΘk,±sinΦk,cosΦk,±).\nThe statistical properties of the electron bath are de-\ntermined by the probability distribution function fk,sfor\nthe state |k,s=±/an}b∇acket∇i}ht, which can be tuned by the external\nfield. If an electric field Eis applied, the non-equilibrium\ndistribution of the electron states will be established due\nto the random scattering potential by impurities[3]. The\ndistribution function fk,sis determined by the Boltz-\nmann equation\n−eE\n/planckover2pi1·∇kfk,s=Sc[fk,s]. (13)\nThe collision integral Sc[fk,s] describes the relaxation of\nthe occupied state |k,s/an}b∇acket∇i}htand can be treated by the relax-\nation time approximation, namely,\nSc[fk,s] =−fk,s−f0\nk,s\nτ. (14)\nHere,f0\nk,sis the equilibrium distribution function, and\nan isotropic relaxation time τhas been assumed[3]. To\nthe first orderofthe electric field, the solution of Eq. (13)\nisfk,s=f0\nk,s+gk,s, where the out of equilibrium part\ninduced by the external electric field is\ngk,s=∂f0\nk,s\n∂EeE·vk,sτ, (15)\nwith the electron velocity vk,s=1\n/planckover2pi1∇kEk,s. Such a treat-\nment of the non-equilirium electron distribution was also\nexploited in the previous semiclassical theory[3].\nB. Electron Spin Polarization and Torque\nWith the non-equilibrium distribution function fk,s\ngiven above, the electron spin polarization σl,µat sitel\nand the correlationfunction Cl,µ;j,ξ(t,τ) in Eq. (9) can be\ncalculated, and the torque and damping effect due to the\nelectron bath can be obtained. In the second quantiza-\ntion representation of the basis set {|k,s/an}b∇acket∇i}ht}, the operator\n/hatwideσl,µis expressed as\n/hatwideσl,µ=1\nN/summationdisplay\nk,s;k′,s′χµ\nk,s;k′,s′ei(k′−k)·rl/hatwidec†\nk,s/hatwideck′,s′,4\nwhere the matrix element\nχµ\nk,s;k′,s′= (cosΘk,s\n2eiΦk,sinΘk,s\n2)σµ/parenleftBigg\ncosΘk′,s′\n2e−iΦk′\nsinΘk′,s′\n2/parenrightBigg\n.\nThen the electron spin polarization σl,µis\nσl,µ=1\nN/summationdisplay\nk,sχµ\nk,s;k,sfk,s=1\nN/summationdisplay\nk,sPµ\nk,sfk,s.(16)\nFor the physically relevant case αRk≪JS, the approx-\nimate value of Pk,±to the first order ofαRk\nJSis\nPk,±=±\nSx+αR\nJSSxSykx+αR\nJS(1−S2\nx)ky\nSy−αR\nJS(1−S2\ny)kx−αR\nJSSxSyky\nSz+αR\nJSSySzkx−αR\nJSSxSzky\n.\nHere, the unit vector for the magnetization is denoted as\nS= (sinθcosφ,sinθsinφ,cosφ).\nFor the electric current density je=je(cosϑ,sinϑ,0),\nthe non-equilibrium spin polarization δσlwhich is per-\npendicular to Sis calculated to be (Appendix A)\nδσl=−αRm∗\nejea3\ne/planckover2pi1Ef\ncosϑSxSy+sinϑ(1−S2\nx)\n−cosϑ(1−S2\ny)−sinϑSxSy\ncosϑSySz−sinϑSxSz\n,\nwhereEfdenotes the Fermi energy. The torque Tlis\nthen obtained as\nTl=JSαRm∗\nejea3\ne/planckover2pi12Ef\ncosϑSz\nsinϑSz\n−cosϑSx−sinϑSy\n\n=JαRm∗\nea3\ne/planckover2pi12Ef(/hatwidez×je)×Sl.\nThis result reproduces the form of SOT obtained\nbefore[3], but the magnetization vector is not restricted\nin two-dimensional x-y plane in our derivations. It is\neasily understood from the effective Hamiltonian (10),\nwhere the non-equilibrium distribution of electron states\nwill produce an extra electron spin polarizationalong the\ndirection/hatwidez×je.\nC. Correlation Function and Damping\nWe now calculate the correlation function Cl,µ;j,ξ(t,τ),\nwhich gives the damping term for the magnetization\ndynamics due to the electron bath. Since /hatwideck,s(t) =\n/hatwideck,se−iEk,st//planckover2pi1, the correlation function Cl,µ;j,ξ(t,τ) is for-\nmally written as\nCl,µ;j,ξ(t,τ)\n=1\nN2/summationdisplay\nk,s;k′,s′/summationdisplay\nk′′,s′′;k′′′,s′′′ei(k′−k)·rlei(k′′′−k′′)·rj\n×ei(Ek,s−Ek′,s′)t//planckover2pi1ei(Ek′′,s′′−Ek′′′,s′′′)τ//planckover2pi1\n×χµ\nk,s;k′,s′χξ\nk′′,s′′;k′′′,s′′′/an}b∇acketle{t/hatwidec†\nk,s/hatwideck′,s′/hatwidec†\nk′′,s′′/hatwideck′′′,s′′′/an}b∇acket∇i}ht.(17)We see that Cl,µ;j,ξ(t,τ) is the function of rl−rjand\nt−τ, due to the space and time translation invari-\nance for the investigated system. For simplicity, we es-\ntimateCl,µ;j,ξ(t,τ) with several approximations. Firstly,\nwe assumethat the phase factor ei(k′−k)·(rl−rj)will cause\nthe cancellation of the summations over kandk′if\nrl/ne}ationslash=rj, thusCl,µ;j,ξ=Cµξδlj. Secondly, χµ\nk,s;k′,s′are\ncalculated to the zeroth order ofαRk\nJSfor the relevant\ncaseαRk≪JS, where the electron spin states are k-\nindependent, i.e.\nχ±±=±(sinθcosφ,sinθsinφ,cosθ),\nχ+−= (−cosθcosφ−isinφ,−cosθsinφ+icosφ,sinθ).\nFurthermore, we calculate the correlation function\n/an}b∇acketle{t/hatwidec†\nk,s/hatwideck′,s′/hatwidec†\nk′′,s′′/hatwideck′′′,s′′′/an}b∇acket∇i}htwith the electron bath at equi-\nlibrium, where the effect of the non-equilibrium electric\ncurrent induced by the external field will be neglected.\nThis enable us to apply the Wick contraction[16] to sim-\nplify the calculations. The negligence of the dependence\nof the damping coefficient on the Rashba SOI and the\nnon-equilibrium electric current is valid if the dynamical\nequation (8) is kept to the first order of these two factors.\nWith the above approximations, we get\nCµξ(t)\n=1\nN2/summationdisplay\nk,sχµ\nssχξ\nssfk,s+1\nN2/summationdisplay\nk,s;k′,s′χµ\nssχξ\ns′s′fk,sfk′,s′\n+1\nN2/summationdisplay\nk,s;k′,s′ei(Ek,s−Ek′,s′)t//planckover2pi1χµ\nss′χξ\ns′sfk,s(1−fk′,s′),\nwhere|k,s/an}b∇acket∇i}htand|k′,s′/an}b∇acket∇i}htare different states.\nSince the kernel function γl,µ;j,ξ(t) is given by the\nrelation dγl,µ;j,ξ(t)/dt=Kl,µ;j,ξ(t), where Kl,µ;j,ξ(t) =\nℑ[Cl,µ;j,ξ(t)], their Fourier transformations are related by\nγl,µ;j,ξ(ω) =i\nωKl,µ;j,ξ(ω). The Fourier transformation of\nKl,µ;j,ξ(t) is obtained as (Appendix B)\nKl,µ;j,ξ(ω)\n=δlj(m∗\nea2\n2π/planckover2pi12)2/planckover2pi1\n2i/summationdisplay\ns,s′[χµ\nss′χξ\ns′sgs(ω)−(χµ\nss′χξ\ns′s)∗gs(−ω)],\nwhere the function gs(ω) is defined as\ngs(ω) =\n\n0,/planckover2pi1ω <0;\n/planckover2pi1ω ,0</planckover2pi1ω < E f−sJS;\nEf−sJS , /planckover2pi1ω > E f−sJS.\nThen the damping kernel function γl,µ;j,ξ(t) can be\ncalculated by the inverse Fourier transformation from\nγl,µ;j,ξ(ω), which results in (Appendix B)\nγl,µ;j,ξ(t) =δlj(m∗\nea2\n2π/planckover2pi1)21\n2/summationdisplay\ns(δµξg−\ns(t)+is/summationdisplay\nνǫµξνSνg+\ns(t)).\n(18)\nHere,g±\ns(t) =/integraltext+∞\n−∞dωg±\ns(ω)e−iωtandg±\ns(ω) =\n1\n/planckover2pi1ω(gs(ω)±gs(−ω)), as schematically shown in Fig. 2.5\nThen the coefficient Γ l,µ;j,ξin Eq. (9) is obtained as\nΓl,µ;j,ξ=δlj(m∗\nea2\n2π/planckover2pi1)2(Γ(1)δµξ+Γ(2)/summationdisplay\nνǫµξνSν),(19)\nwith Γ(1)=1\n2/summationtext\ns/integraltextδt\n0dτg−\ns(τ) and Γ(2)=\ni\n2/summationtext\nss/integraltextδt\n0dτg+\ns(τ). Then the damping part in Eq. (8)\ncan be explicitly written as\nDl= 2(Jm∗\nea2\n2π/planckover2pi12)2(Γ(1)˙Sl×Sl+SΓ(2)˙Sl),(20)\nwhich is independent of the Rashba constant and the\nelectric current due to our approximations above.\n0−1−0.500.51\nhωgs+(ω)(a)\nEf+JSs = −s = +\nEf−JS\n000.51\nhωgs−(ω)(b)\ns = +\ns = −Ef−JSEf+JS\nFIG. 2. (Color online). Schematic diagram for g±\ns(ω). Blue\nline fors= +, and red line for s=−. Notice that g+\nsis an\nodd function of ωandg−\ns(ω) is an even function of ω, and\nthey approach to 0 when |ω| → ∞.\nThe first term in (20) will give the damping effect\nwhich drivesthe local spin towardsthe direction with the\nlower energy; while the second term in (20) will give a\nrenormalized factor in Eq. (8). Assuming that J∼1 eV,\nm∗\ne∼me,a∼1˚A, one gets the rough estimation of the\nmagnitudeorderforthe factor(Jm∗\nea2\n2π/planckover2pi12)2∼10−3, thusthe\ndamping effect due to the electron bath is comparable to\nthe intrinsic Gilbert damping of some ferromagnetic ma-\nterials. This damping effect can become important to\nunderstand the dissipative features of the magnetization\ndynamics driven by spin-orbit torque.\nIV. CONCLUSION\nIn conclusion, we have applied density matrix tech-\nnique to formulate the magnetization dynamics of a sys-\ntem consisting of local magnetic moments influenced by\nan electron gas through s-d exchange interaction. In\nthis approach, the magnetic subsystem is treated as an\nopen quantum system and the electron gas acts as a non-\nequilibrium bath tuned by the external electricfield. The\nspin torque due to the non-equilibrium electron spin ac-\ncumulation and the damping effect of the electron bath\nhave been taken into account simultaneously. We ap-\nply the developed formula to the model system for spin-\norbit torque, where the two-dimensional magnetization\nfilm is coupled to the Rashba electron gas through s-dexchange interaction. We have calculated the spin-orbit\ntorque and the results are consistent with the previous\nstudy. However, our method does not require the mag-\nnetization direction to be in the two-dimensional plane\nas in the previous study. Our approach enables us to ob-\ntain the damping effect due to the electron bath, which\nis a new feature absent in the semiclassical theory. The\ndamping caused by the electron bath is estimated to be\ncomparableto the intrinsic Gilbert damping, and may be\nimportanttodescribethemagnetizationdynamicsdriven\nby spin-orbit torque. In brief, this work has extended\nthe previous semiclassical theory for spin-orbit torque\nto a more complete description. Further applications of\nthis approach are expected to understand and to manip-\nulate the magnetization dynamics through electron gas\nin other complex cases.\nACKNOWLEDGMENTS\nThis work was supported in part by the Hong Kong’s\nUniversity Grant Council via grant AoE/P-04/08. This\nwork is also partially supported by National Basic Re-\nsearch Program of China (No. 2014CB921203), NSFC\ngrant (No.11274269), and NSFC grant (No.11204186).\nAppendix A: Electron Spin Polarization\nWe first assume that the electric field is applied along\nx-direction, then\nδσl=1\nN/summationdisplay\nk,sgk,sPk,s=1\nN/summationdisplay\nk(gk,+−gk,−)kxαR\nJSΣx,\nwhereΣx= (SxSy,−(1−S2\ny),SySz). The corresponding\nelectric current density is\nje=−e\nNa3/summationdisplay\nk,sgk,s(vk,s)x≈ −e/planckover2pi1\nm∗e1\nN/summationdisplay\nk,sgk,skx,\nand the spin current density is\njs=/planckover2pi1\n2Na3/summationdisplay\nk,sgk,s(vk,s)xPk,s\n≈/planckover2pi12\n2m∗e1\nNa3/summationdisplay\nk(gk,+−gk,−)kxS.\nThus a rough relation is obtained as\nδσl=−αRm∗\nejea3\ne/planckover2pi1EfΣx,\nwhere the relation js≈ −/planckover2pi1JS\n2eEfjeShas been used here.\nSimilarly, if the electric field is applied along the y-\ndirection, the non-equilibrium spin polarization will be\nδσl=−αRm∗\nejea3\ne/planckover2pi1EfΣy,6\nwithΣy= (1−S2\nx,−SxSy,−SxSz). Therefore, for the\nelectric current density je=je(cosϑ,sinϑ,0), we get\nδσl=−αRm∗\nejea3\ne/planckover2pi1Ef\ncosϑSxSy+sinϑ(1−S2\nx)\n−cosϑ(1−S2\ny)−sinϑSxSy\ncosϑSySz−sinϑSxSz\n.\nAppendix B: Correlation Function and Damping\nKernel\nThe imaginary part of Cµξ(t) is given as\nKµξ(t)\n=ℑ[Cµξ(t)]\n= (m∗\nea2\n2π/planckover2pi12)2/summationdisplay\ns,s′ℑ[χµ\nss′χξ\ns′s/integraldisplayEf\nsJSdǫ/integraldisplay∞\nEfdǫ′ei\n/planckover2pi1(ǫ−ǫ′)t]\n= (m∗\nea2\n2π/planckover2pi12)2/summationdisplay\ns,s′/integraldisplayEf\nsJSdǫ/integraldisplay∞\nEfdǫ′[−i\n2χµ\nss′χξ\ns′sei\n/planckover2pi1(ǫ−ǫ′)t+h.c.].\nHere,fk,sis approximated as the zero-temperature\nFermi distribution function, and the relation1\nN/summationtext\nk→\na2\n(2π)2/integraltext\nd2k=m∗\nea2\n2π/planckover2pi12/integraltext\ndǫhas been used. Its Fourier trans-formation Kµξ(ω) is then\nKµξ(ω)\n=1\n2π/integraldisplay+∞\n−∞dtKµξ(t)eiωt\n= (m∗\nea2\n2π/planckover2pi12)2/summationdisplay\ns,s′/integraldisplayEf\nsJSdǫ/integraldisplay∞\nEfdǫ′\n×[−i\n2χµ\nss′χξ\ns′sδ(ω+ǫ−ǫ′\n/planckover2pi1)+i\n2(χµ\nss′ξξ\ns′s)∗δ(ω+ǫ′−ǫ\n/planckover2pi1)]\n=−(m∗\nea2\n2π/planckover2pi12)2i/planckover2pi1\n2/summationdisplay\ns,s′[χµ\nss′χξ\ns′sgs(ω)−(χµ\nss′χξ\ns′s)∗gs(−ω)],\nwhere the function g(ω) is defined as\ngs(ω) =\n\n0,/planckover2pi1ω <0;\n/planckover2pi1ω ,0</planckover2pi1ω < E f−sJS;\nEf−sJS , /planckover2pi1ω > E f−sJS.\nTherefore,\nγl,µ;j,ξ(ω)\n=δlj(m∗\nea2\n2π/planckover2pi1)21\n2/summationdisplay\ns,s′[ℜ(χµ\nss′χξ\ns′s)g−\ns(ω)+iℑ(χµ\nss′χξ\ns′s)g+\ns(ω)],\nwhereg±\ns(ω) =1\n/planckover2pi1ω(gs(ω)±gs(−ω)), andγl,µ;j,ξ(t) is cal-\nculated as\nγl,µ;j,ξ(t)\n=/integraldisplay+∞\n−∞dωγl,µ;j,ξ(ω)e−iωt\n=δlj(m∗\nea2\n2π/planckover2pi1)21\n2/summationdisplay\ns,s′[ℜ(χµ\nss′χξ\ns′s)g−\ns(t)+iℑ(χµ\nss′χξ\ns′s)g+\ns(t)]\n=δlj(m∗\nea2\n2π/planckover2pi1)21\n2/summationdisplay\ns(δµξg−\ns(t)+is/summationdisplay\nνǫµξνSνg+\ns(t))\n≈δljδµξ(m∗\nea2\n2π/planckover2pi1)2g−(t),\nwhereg±\ns(t) =/integraltext+∞\n−∞dωg±\ns(ω)e−iωtand we have used the\nexpressions\nχµ\n+,+χξ\n+,+=χµ\n−,−χξ\n−,−=SµSξ.\nχµ\n+,−χξ\n−,+= (χµ\n−,+χξ\n+,−)∗=δµξ−SµSξ+i/summationdisplay\nνǫµξνSν.\n[1] J. 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Hals, Nat. Nanotechnol. 9, 86\n(2014).\n[11] Y. Wang and L. J. Sham, Phys. Rev. B 85, 092403\n(2012).\n[12] Y. Wang and L. J. Sham, Phys. Rev. B 87, 174433\n(2013).[13] T. Tay and L. J. Sham, Phys. Rev. B 87, 174407 (2013).\n[14] Y. Wang, W.Q. Chen, and F.C. Zhang, Phys. Rev. B 90,\n205110 (2014).\n[15] K. Blum, Density Matrix Theory and Applications\n(Springer-Verlag, Berlin Heidelberg, 2012).\n[16] A. L. Fetter and J. D. Walecka, Quantum Theory of\nMany-Particle Systems , (McGraw-Hill, New York, 1971)."    },    {        "title": "0706.4270v2.Coherent_Magnetization_Precession_in_GaMnAs_induced_by_Ultrafast_Optical_Excitation.pdf",        "content": "1 Coherent Magnetization Precession in GaMnAs induced  \nby Ultrafast  Optical Excitation  \n \nJ. Qi, Y. Xu, N. Tolk  \nDepartment of Physics and Astronomy, Vanderbilt University, Nashville, TN, 37235  \nX. Liu, J. K. Furdyna  \nDepartment of Physics, University of Notre Dam e, Notre Dame, IN 46556  \nI. E. Perakis  \nDepartment of Physics, University of Crete, Heraklion, Greece  \n \nWe use femtosecond optical pulses to induce, control and monitor magnetization \nprecession in ferromagnetic Ga 0.965 Mn0.035 As. At temperatures below ~40 K we  \nobserve coherent oscillations of the local Mn spins, triggered by an ultrafast \nphotoinduced reorientation of the in -plane easy axis. The amplitude saturation of the \noscillations above a certain pump intensity indicates that the easy axis remains \nunchanged  above ~T C/2. We find that the observed magnetization precession damping \n(Gilbert damping) is strongly dependent on pump laser intensity, but largely \nindependent on ambient temperature. We provide a physical interpretation of the \nobserved light -induced col lective Mn -spin relaxation and precession.  \n \n \nThe magnetic semiconductor GaMnAs has received considerable attention in \nrecent years, largely because of its potential role in the development of spin -based \ndevices1,2. In this itinerant ferromagnet, the collec tive magnetic order arises from the \ninteraction between mobile valence band holes and localized Mn spins. Therefore, the \nmagnetic properties are sensitive to external excitations that change the carrier density \nand distribution. Ultrafast pump -probe magnet o-optical spectroscopy is an ideal \ntechnique for controlling and characterizing the magnetization dynamics in the \nmagnetic materials, and has been applied to the GaMnAs system by several groups3,4. \nAlthough optically induced precessional motion of magne tization has been studied in 2 other magnetic systems5, magnetization precession in ferromagnetic GaMnAs has \nbeen observed only recently4 and has yet to be adequately understood.  \n \nIn this paper, we  report comprehensive temperature and photoexcitation intens ity \ndependent measurements of photoinduced magnetization precession in Ga 1-xMnxAs (x \n= 0.035) with no externally imposed magnetic field. By comparing and contrasting \nthe temperature and intensity dependence of the precession frequency, damping, and \namplitu de, we identify the importance of light -induced nonlinear effects and obtain \nnew information on the relevant physica l mechanisms. Our measurements of the \nphotoinduced magnetization show coherent oscillations, arising from the precession \nof collective Mn sp ins. Amplitude of the magnetization precession saturates above \ncertain pump intensity is a strong indication that direction of the magnetic easy axis \nremains unchanged at temperatures above about half the Curie temperature (T C). The \nprecession  is explained  by invoking an ultrafast change in the orientation of the \nin-plane easy axis, due to an impulsive change in the magnetic anisotropy induced by \nthe laser pulse. We also find that the Gilbert damping coefficient, which characterizes \nthe Mn -spin relaxation, depends only weakly on the ambient temperature but changes \ndramatically with pump intensity . Our results suggest a general model for \nphotoinduced precessional motion and relaxation of magnetization in the GaMnAs \nsystem under compressive strain.  \n \nTime -resolved magneto -optical Kerr effect (MOKE) measurements were \nperformed on a 300 nm thick ferromagnetic Ga 1-xMnxAs (x = 0.035) sample, with \nbackground hole density p ≈1020 cm-3 and T C ≈70 K. The sample was grown by low \ntemperature molecular beam epitaxy  on a G aAs(001) substrate, and was therefore \nunder compressive strain. The pump -probe experiment employed a Coherent MIRA \n900 Ti:Sapphire laser, which produced ~150 -fs-wide pulses in the 720 nm (1.719 eV) \nto 890 nm (1.393 eV) wavelength range with a repetition ra te of 76 MHz. The pump \nbeam was incident normal to the sample, while the probe was at an angle of about 30o \nto the surface normal. The polarization of the pump beam could be adjusted to be 3 linear, right -circular ( σ+), or left -circular ( σ-) polarization. Th e probe beam was \nlinearly polarized. This configuration produced a combination of polar and \nlongitudinal MOKE, with the former dominating6. The temporal Kerr rotation signal \nwas detected using a balanced photodiode bridge, in combination with a lock -in \namplifier. Both pump and probe beams were focused onto the sample with a spot \ndiameter of about 100 µm, with an intensity ratio of 15:1. The pump light typically \nhad a pulse energy of 0.065 nJ, and a fluence of 0.85 µJ/cm2. \n \nFigure 1(a) shows our time -resolve d Kerr rotation (TRKR) measurements at \ntemperature of 20 K. The amplitude of the temporal Kerr rotation  signal was found to \nbe symmetric with respect to right or left -circularly polarized photo -excitation. In \nparticular, we observe a superimposed oscillato ry behavior at temperatures less than \n~40 K, indicating magnetization precession. It is important to note that these \noscillations were observed not only with σ+ and σ- polarized but also with linearly \npolarized pump light . The phase difference among the os cillations for σ+ or σ- pump \nexcitation is less than ~5o. This negligible phase difference implies that the oscillatory \nbehavior is not due to the non -thermal circular polarization -dependent carrier spin \ndynamics5. After the initial few picoseconds, where equilibration between the \nelectronic and lattice systems occurs, the oscillations can be fitted well by the \nfollowing equation (see Fig. 1(b)):  \n) cos()/ exp( )( 0j w t q + − = t t AtK           (1),  \nwhere A0, w, t, and j are the amplitude, precession frequency, decay time, and initial \nphase of  the oscillation, respectively. Some fitted parameters are shown in figure 2 as \na function of pump intensity and temperature.  \n \nOn a sub -picosecond time scale, the photoexcited electrons/holes scatter  and  \nthermalize with the Fermi sea via electron -electron  interactions. Following this initial \nnon-thermal temporal regime, the properties of the GaMnAs system can be \ncharacterized approximately by time -dependent carrier and lattice temperatures heT/ 4 and lT. Subsequently, the carrier system transfers its energy to the lattice within a \nfew picoseconds via the electron -phonon interaction. This leads to a \nquasi -equilibration of heT/and lT, which then relax back to the equilibrium \ntempera ture via a slow (ns) thermal diffusion process. Mn precession was also \nobserved in Ref. 4 and was attributed to the change of uniaxial anisotropy due to the \nincrease in hole concentration7. In our experiment, for a typical pump intensity of \n0.065 nJ/pulse,  we estimate that the relative increase in hole concentration is ~0.1%. \nThe resulting transient increase in local temperature and hole concentration leads to \nan impulsive change  in the in -plane magnetic anisotropies and in the easy axis \norientation . As a r esult, the effective magnetic field experienced by the Mn spins  \nchanges, thereby triggering the observed precession.  \n \nIt is known that the magnetic anisotropy parameters (i.e., uniaxial anisotropy \nconstant K1u and cubic anisotropy constant K1c), which dete rmine the direction of the \neasy axis in the GaMnAs system, are functions of temperature and hole \nconcentration4,7,8. Thus when the GaMnAs system is excited by an optical pulse, a \ntransient change in local hole concentration Δp and local temperature ΔT, reflecting \nvariation of both the carrier temperature )(/t TheΔ  and the lattice temperature ()lTtΔ, \ncan lead to changes in the magnetic anisotropy parameters. Below the Curie \ntemperature, the direction of the in -plane magnetic eas y axes (given by the angle f) \ndepends on the interplay between K1u and -K1c. After the optical excitation, the new \nangle of the in -plane easy axes is given by 100\n100((),())\n()((),())u\ncKTTtppt\ntKTTtpptff +Δ+Δ=+Δ+Δ , \nwhere T0 and p0 are the initial (ambient) temperature and hole conce ntration7,8. \nTherefore, the in -plane easy axes may quickly assume a new direction following \nphotoexcitation if ΔT(t) and Δp(t) are sufficiently strong. This transient change in the \nmagnetic easy axis, due to the change in the minimum of the magnetic free e nergy as \nfunction of Mn spin induced by the photoexcitation, triggers a precessional motion of \nthe magnetization around the new effective magnetic field.  5  \nWithin the mean field treatment of the p-d magnetic exchange interaction,  the \nMn spins precess aroun d the effective magnetic field Mn\neffH, which is determined by \nthe sum of the anisotropy field Mn\nanisH  and the hole -spin mean field JNholem. The \ndynamics of the hole magnetization  m is determined by its precession around the  \nmean field JNMnM due to the Mn spins , and by its rapid relaxation due to the strong \nspin-orbit interaction in the valence band with a rate ΓSO of the order of tens of \nfemtoseconds9,10. Here, m (M) is the hole (Mn) magnetization , J is the exchange \nconstant , and Nhole (NMn) is the number of holes (Mn -spins)2. For small fluctuations of \nthe magnetization orientation around the easy axis, the magnetization  dynamics  can \nbe described by the Landau -Lifshitz -Gilbert (LLG) equation2, which is appropriate to \napply to  our experimental data at low -pump intensities (e.g., 0.065 nJ/pulse). In the \nadiabatic limit, where the hole spins precess and relax much faster than the Mn spins, \nwe can eliminate the hole spins by transforming to the rotating frame11. In this way \nwe obt ain an effective LLG equation for the Mn magnetization M, whose precession \nis governed by the anisotropy field MnanisH  and  the effective Gilbe rt damping \ncoefficient including  the damping a0 due to e.g. spin -lattice interactions and the \ncontribution due to the p-d exchange interaction, which depends on the hole \nconcentration, the ratio of hole spin relaxation energy over exchange interaction \nenergy, and the ratio m/M of the collective hole and Mn spins[9,10].  \n \n    The LLG equation predict s an oscillatory behavior of the magnetization due to \nthe precession of the local Mn moments around the magnetic anisotropy field \nMnanisH (T0+??(t), p0+?p(t)). The precession frequency is proportional to this anisotropy \nfield, which is given by the gradient of magnetic free energy and is proportional to the \nanisotropy constants  K1u and K1c.2 The magnitude of MnanisH decreases as the ambient \ntemperature T0 or the transient temperature DT increases, primarily due to the  \ndecrease in K1c8. This leads to the decrease of the precession  frequency as the ambient 6 temperature or the pump intensity increases, and is consistent with the be havior \nobserved in Fig. 2. It should be pointed out that, in the Fourier transform of each \ntemporal signal of the oscillations, only one oscillatory mode was observed (see also \nin Fig. 1(b) and Fig. 1(c)). This indicates that only a single uniform -precessi on \nmagnon was excited in our experiment, and the scattering among uniform -precession \nmagnons can be neglected when interpreting damping of the Mn spin precession.  \n \nAs can be seen in Fig. 2,  the amplitude of the oscillations increas es as the \nambient tempera ture T0 decreas es or as the pump intensity increases. This result is in \naccord with the fact that the relative change DT/T0 and Δp/p0, which determines the \nmagnitude of f(t) and the photo -induced tilt in the easy axis, increase as T0 decreases \nor as the pu mp intensity increases. It is important to note that in our experiment the \namplitude of the oscillations saturates as the pump intensity exceeds about 4 I0 \n(I0=0.065nJ/pulse) at T0=10 K. Thus, the observed saturation may indicate that the \nmagnetic easy axi s is stabilized at pump intensity larger than 4 I0. We estimated that \nthe increase of local hole concentration Δp/p0 is about 0.4%, and the local temperature \nincrease ΔT /T0 is about 160% using the value of specific heat of 1mJ/g/K for GaAs12 \nfor pump inte nsity ~4 I0. This results in the transient local temperature T0+ΔT close to \nTC/2. Because the magnetic easy axis is already along the [110] direction for T0+ΔT \nclose to or higher than T C/28, our observed phenomenon is in agreement with the \nprevious reported  results.  \n \nFinally we turn to the oscillation damping, which is intimately related to \ncollective localized -spin lifetimes, and consequently to spintronic device development. \nFigure 3 shows the fitted Gilbert damping coefficient a obtained by using the LLG \nequation as a function of pump intensity and ambient temperature, respectively. It can \nbe seen that in Figure 3(a) a changes weakly with the ambient temperature and has an \naverage value ~0.135 for a fixed pump intensity of 0.065 nJ/pulse. However, Figure \n3(b) shows that a increases nonlinearly as pump intensity increases. To interpret these \nresults, we note that, as discussed above, the p-d kinetic -exchange coupling between 7 the local Mn moments and the itinerant carrier spins contributes significantly to \nGilbert damping10. In particular, a increases with increasing ratio m/M. The ratio \nm/M is known to increase nonlinearly with increasing temperature13 and should \ntherefore depend nonlinearly on the photoexcitation. The Gilbert damping  coefficient \ndue to the e xchange interaction also increases as hole density p and hole spin \nrelaxation rate SO\nMnJMNΓ\n increase. Here, ΓSO and Δp/p (<0.01) are relatively small. \nThus we can conclude that the damping coefficient due to the p-d exchange \ninteraction should increase with increasing ambient temperature ( T0) or increasing \npump intensity  (ΔT and Δp). On the other hand, we also note that damping may arise \nfrom an extrinsic inhomogeneous MnanisH  broadening attributed to a local temperatu re \ngradient due to inhomogeneities in the laser beam intensity profile and the detailed \nstructure of the sample. This extrinsic damping effect is expected to decrease as the \nambient temperature increases or the pump intensity decreases.  Thus, the data in F ig. \n3(a), which shows a only weakly dependent on ambient temperature, may result from \nthe competition between these two mechanisms, both of which, however, predict the \nresult in Fig. 3(b) that the damping coefficient increases nonlinearly with the increase  \nof pump intensity. It is important to note that the LLG equation is valid only at low \npump intensities. At high pump intensities, an alternative theoretical approach must be \nintroduced[14]. So our new experimental results in the time domain are not access ible \nwith static FMR experiments, and provide new information on the physical factors \nthat contribute to the damping effect.  \n \nIn conclusion, we have studied the photoinduced magnetization dynamics in \nGa0.965 Mn0.035 As by time -resolved MOKE with zero externa l magnetic fields. At \ntemperatures below ~40 K, a precessional motion associated with correlated local Mn  \nmoments was observed. This precession is attributed to an ultrafast reorientation of \nthe in -plane magnetic easy axis from an impulsive change in the m agnetic anisotropy \ndue to photoexcitation. Our results indicate that the magnetic easy axis does not 8 change at temperatures above about T C/2. We find the Gilbert damping coefficient is \nindependent of ambient temperature but depends nonlinearly on the pump intensity, \nWe attribute this nonlinearity to the hole -Mn spin exchange interaction and the \nextrinsic anisotropy field broadening due to temperature gradients in the sample. Our \nresults show that ultrafast optical excitation provides a way to control the am plitude, \nprecession frequency and damping of the oscillations arising from coherent localized \nMn spins in the GaMnAs system.  \n \nThis work was supported by ARO Grant W911NF -05-1-0436 (VU), NSF Grant \nDMR06 -03752 (ND), and by the EU STREP program HYSWITCH (Cret e).  \n \n1 H. Ohno, Science 281, 951 (1998)  \n2 J. Jungwirth, J.Sinova, J.Masek, J.Kucera, and A.H. MacDonald, Rev. Mod. Phys. \n78, 809 (2006)  \n3 J. Wang, C. Sun, Y. Hashimoto, J. Kono, G.A. Kh odaparast, L. Cywinski, L.J. Sham, \nG.D. Sanders, C.J. Stanton  and H. Munekata , J. Phys.:Condens. Matter 18, R501 \n(2006)  \n4 A. Oiwa, H. Takechi, and H. Munekata, Journal of Superconductivity 18, 9 (2005)  \n5 A.V. Kimel, A. Kirilyuk, F. Hansteen, R.V. Pisarev, and Th. Rasing , \nJ.Phys.:Condens. Matter 19, 043201(2007)  \n6 V. A. Kotov and A.  K. Zvezdin. Modern Magnetooptics and Magneto Optical \nMaterials . (Institute of Physics, London, 1997)  \n7 T. Dietl, H. Ohno, and F. Matsukura, Phys. Rev. B 63, 195205 (2001)  \n8 U.Wel p, V.K.Vlasko -Vlasov, X.Liu, J.K.Furdyna, and T.Wojtowicz, Phys. Rev. Lett. \n90, 167206 (2003)  \n9 J. Chovan, E. G. Kavousanaki, and I. E. Perakis, Phys. Rev. Lett. 96, 057402 (2006) \nand unpublished.  \n10J.Sinova, T.Jungwirth, X.Liu, Y.Sasaki, J.K.Furdiyna, W.A .Atkinson, and \nA.H.MacDonald, Phys.  Rev. B 69, 085209  (2004)  \n11 Y. Tserkovnyak, G. A. Fiete, and B. I. Halperin, Appl. Phys. Lett. 84, 5234 (2004)  9 12 J. S. Blakemore, J.  Appl.  Phys. 53, R123  (1982)  \n13 S. Das Sarma, E. H. Hwang, and A. Kaminski, Phys. Rev. B 67, 155201(2003)  \n14 H. Suhl, IEEE Trans.Magn. 34, 1834(1998)  10 0 200 400 600 800-0.4-0.20.00.20.4FFT (a.u.)Kerr rotation (mdeg)\nLinear\nσ- σ+T=20K Kerr rotation (mdeg)\nDelay time (ps)0 200 400 600 800-0.010.000.01(b)\n   \nDelay time (ps)\n(a)\n0 5 10 15 20 25 30024(c)\n  \nFrequency (GHz)\n \nFigure 1. (a) Kerr rotation measurements for Ga 1-xMnxAs (x = 0.035) excited by \nlinearly -polarized and circularly -polarized light ( σ+ and σ-) at a temperature of 20 K. \nThe photon energy was 1 .56 eV. Oscillations due to magnetization precession are \nsuperimposed on the decay curves. (b) Oscillation data  (open circles) extracted from \n(a). The solid line is the fitted result. (c) Fourier transformation profile for the \noscillation data in (b).  \n10 20 30 40102030\nPump Intensity( I0) Temperature (K)\n \n ω (GHz)204060\n  \n A0 (µdeg) I=I0\n50100150200\nT0=10K  \n \n \n246810142128 \n \n \n \nFigure 2 Amplitude A0 and angular frequency w as a function of temperature T0 at \nconstant pump intensity I=I0; and as a function of pump intensity (in units of I0) at T0 \n= 10 K. I0 = 0.065 nJ/pulse.  11 0.5 1.0 1.5 2.00.100.150.200.25Damping Coefficient α\n  \nPump Intensity (I0)(b)\nT0=10K10 20 30 400.090.120.150.18\n  Damping Coefficient α\nTemperature (K)(a)\nI=I0\n \nFigure 3 (a) Gilbert damping coefficient a as a function of temperature (T 0) at a \nconstant pump intensity I=I0. I0=0.065 nJ/pulse; (b) Gilbert damping coefficient a as a \nfunction of pump intensity in units of I0 at T0= 10 K.  "    },    {        "title": "1411.2857v1.Capturing_of_a_Magnetic_Skyrmion_with_a_Hole.pdf",        "content": "Capturing of a Magnetic Skyrmion with a Hole\nJan M uller1\u0003and Achim Rosch1\n1Institut f ur Theoretische Physik, Universit at zu K oln, D-50937 Cologne, Germany\n(Dated: October 5, 2018)\nMagnetic whirls in chiral magnets, so-called skyrmions, can be manipulated by ultrasmall current\ndensities. Here we study both analytically and numerically the interactions of a single skyrmion in\ntwo dimensions with a small hole in the magnetic layer. Results from micromagnetic simulations are\nin good agreement with e\u000bective equations of motion obtained from a generalization of the Thiele\napproach. Skyrmion-defect interactions are described by an e\u000bective potential with both repulsive\nand attractive components. For small current densities a previously pinned skyrmion stays pinned\nwhereas an unpinned skyrmion moves around the impurities and never gets captured. For higher\ncurrent densities, jc1< j < jc2, however, single holes are able to capture moving skyrmions. The\nmaximal cross section is proportional to the skyrmion radius and top\u000b, where\u000bis the Gilbert\ndamping. For j > jc2all skyrmions are depinned. Small changes of the magnetic \feld strongly\nchange the pinning properties, one can even reach a regime without pinning, jc2= 0. We also show\nthat a small density of holes can e\u000bectively accelerate the motion of the skyrmion and introduce a\nHall e\u000bect for the skyrmion.\nPACS numbers: 12.39.Dc,75.76.+j,74.25.Wx,75.75.-c\nI. INTRODUCTION\nTopologically stable magnetic whirls, so-called\nskyrmions, have recently gained a lot of attention both\ndue to their interesting physical properties and their\npotential for applications. A single skyrmion is shown\nin Fig. 1. A skyrmion is a smooth magnetic con\fgu-\nration where the spin direction winds once around the\nunitsphere. This implies that the spin con\fguration is\ntopological protected and can unwind only by creating\nsingular spin con\fgurations1,2. In bulk chiral magnets,\nlattices of skyrmions are stabilized by Dzyaloshinskii-\nMoriya interactions and thermal \ructuations in a\nsmall temperature and \feld regime3. In \flms of chiral\nmagnets they occur as a stable phase4in a wide range\nof temperatures in the presence of a stabilizing \feld.\nSingle skyrmions are metastable in an even broader\nregime of parameters4. They have been observed in a\nwide range of materials, including insulators5, doped\nsemiconductors4,6and metals3,7,8, with sizes ranging\nfrom a few nanometers up to micrometers and from\ncryogenic temperatures almost up to room temperature8.\nIn bilayer PdFe \flms on Ir substrates, single nanoscale\nskyrmions have been written using the current through\na magnetic tip9.\nDue to their e\u000ecient coupling to electrons by Berry\nphases and the smoothness of the magnetic texture,\nskyrmions can be manipulated by extremely small elec-\ntric current densities10{14. Therefore the potential exists\nto realize new types of memory or logic devices based\non skyrmions2,15. Several studies have therefore investi-\ngated the dynamics of skyrmions in nanostructures and\ntheir creation at defects15,16.\nIn this paper, we investigate how a single defect a\u000bects\nthe dynamics of a single skyrmion in a magnetic \flm, see\ne.g. Fig. 1. As an example of a defect we consider a\nvacancy, i.e., a single missing spin, or more general a\nFIG. 1. Snapshot of a micromagnetic simulation of skyrmion\ndriven by a current ( D= 0:3J=a,\u0016B= 0:09J=a2,vs=\n0:001aJex, and\u000b=\f= 0:4) in the presence of a single\nvacancy: a missing spin (grey sphere). The trajectory of\nthe skyrmion center is indicated by a red line.\nhole in the magnetic \flm with radius small compared to\nthe skyrmion radius. This problem is of interest for at\nleast two reasons. First, this is perhaps the most simple\nexample of a nanostructure which can interact with the\nskyrmion. As we will show, one can use such defects to\ncontrol the capturing and release of skyrmions via the\nmagnetic \feld and the applied current density. Second,\ndefects are always present in real materials. As long as\nthe typical distance of defects is small compared to the\nskyrmion radius, the e\u000bects of a \fnite density of defects\ncan be computed from the solution of the single-defect\nproblem. The in\ruence of a \fnite defect density on a\nlattice of skyrmions has been studied using micromag-\nnetic simulations by Iwasaki, Mochizuki and Nagaosa,\nRef. 12. Interestingly, they observed in their simulations\nthat skyrmions move e\u000eciently around defects. While a\ndi\u000berent type of defect (enhanced easy axis anisotropy)\nwas considered by them, a similar phenomenon will also\nbe of importance for our study.\nBesides the use of micromagnetic simulations, the main\ntheoretical tool will be the analysis of e\u000bective equations\nof motion for the center of the skyrmion. Thiele17pi-arXiv:1411.2857v1  [cond-mat.str-el]  11 Nov 20142\noneered the approach to project the e\u000bective equations\nof motion on the translational mode of a magnetic tex-\nture. This approach has also been successfully applied\nto skyrmions and skyrmion lattices12,16,18,19. Here, we\ncombine this approach with microscopic evaluations of an\ne\u000bective potential describing the defect-skyrmion interac-\ntion. The resulting e\u000bective equation of motion for the\nskyrmion accurately reproduces the results of the micro-\nmagnetic simulation and allows for an analytical analysis\nof the skyrmion dynamics.\nIn the following, we will \frst introduce the model, de-\nrive the e\u000bective potential and resulting equations of mo-\ntion, and use them to investigate how skyrmions are cap-\ntured, released and de\rected by a single defect. Finally,\nwe study the e\u000bects of a \fnite, but low density of defects.\nII. THE MODEL\nTo describe the magnetization of the system we use\nclassical Heisenberg spins M(r) with uniform length\nkM(r)k= 1 on a square lattice. The corresponding free\nenergy functional in the continuum reads\nF[M]=Z\nd2rJ\n2[rM(r)]2+DM(r)\u0001[r\u0002M(r)]\u0000\u0016B\u0001M(r),\n(1)\nincluding the ferromagnetic coupling J, Dzyaloshinskii-\nMoriya interaction Dand the Zeeman interaction with\nthe magnetic \feld B= (0;0;B).\u0016is the magnetiza-\ntion per area. For a single spin 1 =2 per square unit cell\nwith lattice constant a0andgfactorg= 2 one has, for\nexample,\u0016=\u0016B=a2\n0.\nOn a square lattice we use the following discretized\nversion\nF[M] =\u0000JX\nrMr\u0001\u0000\nMr+aex+Mr+aey\u0001\n\u0000DaX\nr\u0000\nMr\u0002Mr+aex\u0001ex+Mr\u0002Mr+aey\u0001ey\u0001\n\u0000B\u0016a2\u0001X\nrMr, (2)\nwhere exandeyare unit vectors in the xandydirection,\nrespectively. The lattice constant aand the interaction\nstrengthJare set to 1 in the following. If not otherwise\nstated, we use D= 0:3J=aand\u0016B= 0:09J=a2. For these\nparameters the ground state is ferromagnetic. Hence the\nsingle skyrmion is a topologically protected, metastable\nexcitation. A vacancy at position Rdis created by setting\nthe magnetization Mat this site to zero.\nThe micromagnetic dynamics of each spin in the pres-\nence of an electric current density jare described by\nthe Landau-Lifshitz-Gilbert (LLG) equation20{22. In the\ncontinuum case the LLG equation reads\n[@t+(vs\u0001r)]M=\u0000\rM\u0002Be\u000b\n+\u000bM\u0002\u0014\n@tM+\f\n\u000b(vs\u0001r)M\u0015\n, (3)wherevsis the drift velocity of spin currents which\nis directly proportional to the current density jand\n\r=g\u0016B=~is the gyromagnetic ratio. Note that we set\nvs;\u000band\fto a constant value, not taking into account\nthat depending on the microscopic realization of the de-\nfect, they might be modi\fed in proximity of the defect.\nAt least for defects small compared to the skyrmion ra-\ndius and su\u000eciently small currents, this approximation\nis justi\fed as the forces on the skyrmions add up from\nall parts of the skyrmion (see below). The (very weak)\ne\u000bects of changes to the current pattern around a notch\nin a nanowire have been studied in Ref. 16. The e\u000bec-\ntive magnetic \feld is given by Be\u000b=\u0000\u000eF[M]\n\u0016\u000eM.\u000band\f\nare phenomenological damping terms. Note that \u000b=\f\nis a special point as in this case the magnetic texture\ndrifts with the current as long as no defects are present,\nM(r;t) =M(r\u0000vst). In our lattice model we rewrite\nEq. (3) using @iM(r) =1\n2a(Mr+aei\u0000Mr\u0000aei).\nIII. EFFECTIVE DYNAMICS OF SKYRMIONS\nA. Generalized Thiele approach\nThe LLG equation describes the movement of every\nmagnetic moment in the system. As we do not want to\ndescribe every spin but the movement of the skyrmion\ncenter, which is a collective movement of spins, we ap-\nply the so-called Thiele approach17. Originally, this ap-\nproach is based on the approximation that the skyrmion\nis a completely rigid object. While this approximation\nfails in the presence of a local defect, we will show that\none can nevertheless use this approach if one performs a\nmicroscopic calculation of the potential V(r) describing\nthe forces between skyrmion and defect.\nOur goal is to derive an equation of motion for the\ncenter Rof the skyrmion ( Ris de\fned below), which\ntakes into account deformations of the skyrmion. If the\nmotion of the skyrmion is su\u000eciently slow, we expect\nthat for each \fxed Rthe skyrmion con\fguration is in a\nlocal minimum of the energy. We therefore approximate\nM(r;t)\u0019M0(R(t)\u0000Rd;r\u0000R(t) ). (4)\nThe magnetic con\fguration M0depends on the distance\nof skyrmion center R(t) and defect position Rdand is\ndetermined from the condition that\nV(R\u0000Rd) =F[M0(R\u0000Rd;r\u0000R)]\u0000F0\n= min\nR\u0000Rd\fxedF[M(r)]\u0000F0 (5)\nis at a local minimum for \fxed distance of skyrmion and\ndefect, R\u0000Rd.V(R\u0000Rd) is the e\u000bective potential\ndescribing the skyrmion-defect interaction. The o\u000bset\nF0is chosen such that V(R!1 ) = 0. Note that the\nstandard Thiele approach neglects the deformation of the\nskyrmion, i.e., the dependence of MonR\u0000Rd.\nTo calculate M0andF[M0] numerically, we have used\ntwo di\u000berent methods. In the case that Ris located3\n0510152025-0.4-0.3-0.2-0.10.00.1\ndistance rapotential VHrLJ\n05101520-0.050.000.05\nFIG. 2. Potential V(R\u0000Rd) of the skyrmion-hole inter-\naction as a function of distance shown for D= 0:3J=a\nand various magnetic \felds from \u0016B= 0:05J=a2(red) to\n\u0016B= 0:12J=a2(blue) in steps of \u0001 \u0016B= 0:01J=a2. Inset:\nRaw data used to calculate the smoothened potential shown\nin the main \fgure. The dark red (light green) data has been\nobtained for \u0016B= 0:09J=a2using the \frst and second algo-\nrithm described in the text. The spread in each curve arises\nas on the square lattice the potential does not only depend\non the distance from the defect but also has a tiny angular\ndependence. For comparison, we also show the estimate for\nthe potential which is obtained when deformations of the\nskyrmion are ignored (dashed line).\non one of the lattice sites, we \fx the position Rof\nthe skyrmion by setting the magnetization at r=R\nto (0;0;\u00001), opposite to the ferromagnetic background.\nThis approach is similar to the method used in Ref. 14\nto numerically calculate the potential of the skyrmion-\nskyrmion interaction. In a second approach, we \frst\ncompute the skyrmion con\fguration Mc(r\u0000R) in the\nclean system without a defect. To determine the en-\nergy minimum in the presence of the defect for \fxed R,\nwe minimize F[M(r)] with the boundary condition that\nM(r) =Mc(r\u0000R) forjr\u0000Rdj>r0. It turns out that this\nprocedure rapidly converges with r0andr0= 4:5agives\naccurate results in the considered parameter range. The\nresults forV(R\u0000Rd) determined from the two methods\nare almost identical, see inset of Fig. 2.\nIn Fig. 2 the resulting potentials V(jR\u0000Rdj) are\nshown. In the continuum model, Eq. (1), the e\u000bective\npotential depends only on the distance of skyrmion and\ndefect,jR\u0000Rdj, whereas in the lattice there is a small\nangular dependence (raw data is shown in the inset of\nFig. 2). For simplicity, we average over this angular\ndependence. We \ft an exponential law for very large\njR\u0000Rdjand interpolate the curve by a polynomial oth-\nerwise. The shape of the potential not only quantita-\ntively but also qualitatively depends on the strenght of\nthe magnetic \feld, which will be important for the fol-\nlowing discussion.\nTo derive an e\u000bective equation of motion for R(t), we\nproceed as follows17. First, both sides of the LLG equa-\ntion are multiplied by\u0016\n\rM\u0002such that\u0016Be\u000b=\u0000\u000eF[M]\n\u000eMis isolated (using that Be\u000bcan be chosen to be perpen-\ndicular to M). Second, Mis replaced by M0de\fned in\nEq. (4). Third, to project onto the translational mode\nin direction ithe resulting equation is multiplied bydM0\ndRiand integrated over space.\nThe resulting di\u000berential equation for R(t), the gen-\neralized Thiele equation, can be written in the following\nform\n\u0000dV\ndR=GR\u0002\u0010\n_R\u0000vs\u0011\n+\u000eGR\u0001vs\n+DR\u0001\u0010\n\u000b_R\u0000\fvs\u0011\n+\f\u000eDR\u0001vs, (6)\nwhere the potential V, the gyrocoupling GRand the ma-\ntrices\u000eGR,DRand\u000eDRare functions of the distance\nfrom the defect, R\u0000Rd.Vis de\fned in Eq. (5), the\nother terms are determined from\n(GR)i=s\u000fijkZ\nd2r1\n2M0\u0001\u0012dM0\ndRj\u0002dM0\ndRk\u0013\n(7)\n(DR)ij=sZ\nd2rdM0\ndRi\u0001dM0\ndRj(8)\n(\u000eGR)ij=sZ\nd2rM0\u0001\u0012dM0\ndRj\u0002\u0012dM0\ndRi+dM0\ndri\u0013\u0013\n(9)\n(\u000eDR)ij=sZ\nd2rdM0\ndRi\u0001\u0012dM0\ndRj+dM0\ndrj\u0013\n, (10)\nwhere we included the spin density\ns=\u0016\n\r, (11)\ne.g.,s=~=(2a2\n0) for a single spin 1/2 in a unit cell of\nlengtha0. Note that some of the derivatives are with\nrespect to the skyrmion position Rand further that the\ncombinationdM0\ndRi+dM0\ndri=\u0000dM0\ndRd;idescribes the change\nof the skyrmion con\fguration when only the position of\nthe defect changes.\nIf the deformation of the skyrmion (and therefore the\nderivativesd\ndRd;i) are ignored, then the correction terms\n\u000eGRand\u000eDRvanish and one can replaced\ndRiby\u0000d\ndrito\nrecover the Thiele equation in the standard form. Within\nthis approximation, the gyrocoupling GR=Gis in the\ncontinuum limit topologically quantized to a multiple of\n4\u0019M(Mis the magnetization per unit cell set to 1 within\nour conventions). This is, however, notthe case if the\ndependence of M0onR\u0000Rdis taken into account.\nThe most important e\u000bect of the deformation is that\nthey strongly modify the e\u000bective potential V(R\u0000Rd),\nas is shown in the inset of Fig. 2. Taking into account\nthe adjustment of the magnetic texture to the defect is\nimportant as it gives rise to corrections of order 1.\nChanges of the gyrocoupling and dissipative tensor\nare, in general, also of importance when nanostructures\nlead to a signi\fcant deformation of the magnetic texture.\nThey are, however, not important for the situation con-\nsidered in our paper. We study the case where the radius\nadof the defect is much smaller than the radius of the4\nskyrmion,as. In this case the deformations a\u000bect only a\nsmall part of the skyrmion and give therefore only small\ncorrections of order ( ad=as)2\u001c1 on the right-hand side\nof the generalized Thiele equation (6). This is shown in\nFig. 3, wherejGRj,Dr\nRandDt\nRare shown as a func-\ntion of the distance from the defect, jR\u0000Rdj. Here\nDr\nR= ^er\u0001DR\u0001^erandDt\nR= ^e\u001e\u0001DR\u0001^e\u001edescribe the\ndissipative tensor projected on the radial and tangential\ndirection, respectively, with ^ er= (R\u0000Rd)=jR\u0000Rdj\nand ^e\u001e= ^z\u0002^er. Far from the defect, one recovers\nthe results predicted by the standard Thiele approach\nwithDr\nR=Dt\nRandjGRj= 4\u0019in the continuum limit,\nwhereas there are small deviations of a few percent when\nthe distance of the defect is of the order of the skyrmion\nradius. Similarly, the corrections arising from \u000eDRand\n\u000eGRare also small. For the following analysis, we will\ntherefore neglect the modi\fcation of the right-hand side\nof the Thiele equation (6) using\n\u0000dV\ndR=G\u0002\u0010\n_R\u0000vs\u0011\n+D\u0001\u0010\n\u000b_R\u0000\fvs\u0011\n, (12)\nwith space-independent G= lim R!1GRandD=\nlimR!1DRwhile the modi\fed potential is fully taken\ninto account.\nDRr/4π\nDRt/4π\n|GR|/4π\n0 2 4 6 8 10 12 141.001.051.101.15\ndistance r/a(DRr,DRt,|GR|) /4π\nFIG. 3.jGRj,Dr\nR, andDt\nRshown as a function of the\ndistance from the defect, r=jR\u0000Rdj, forD= 0:3J=aand\nmagnetic \feld \u0016B= 0:09J=a2.\nB. Comparison of the generalized Thiele approach\nand micromagnetic simulations\nUsing the numerically determined potential, see\nFig. 2, one can directly calculate the trajectories of the\nskyrmions using the Thiele equation, Eq. (12). In Fig. 4\nthe trajectories are shown for two values of the damping\nconstants, i.e., \u000b=\f= 0:4 and 0:04. The properties\nof these solutions will be discussed in Sec. IV. Here we\ncompare them to full micromagnetic simulations of the\nsystem. To track the center of the skyrmion R, we used\nR\u0019P\ni(Mz\n0\u0000Mz\ni)ri=P\ni(Mz\n0\u0000Mz\ni) summing only over\nsites withMz\ni<Mz\n0=\u00000:3.\n-1001020-10-5051015\n-1001020-10-5051015FIG. 4. Comparison of trajectories of the moving skyrmion\nobtained from the full micromagnetic simulation (colored)\nto the results of the e\u000bective potential approach (black).\nThe vacancy is placed in the origin. For better orientation,\nthe green (red) circle indicates the minimum (maximum) of\nthe e\u000bective potential. Parameters are D= 0:3J=a,\u0016B=\n0:09J=a2,vs= 0:001aJex, and\u000b=\f= 0:04 (left),\u000b=\n\f= 0:4 (right).\nComparing the results from the micromagnetic simula-\ntions and the simpli\fed Thiele equation, we \fnd that the\ntwo approaches agree quantitatively with high precision.\nTiny deviations arise partially from using the simpli\fed\nThiele equation (12) instead of (6) neglecting, e.g., the\nspatial variations of GRandDRshown in Fig. 3. The\nother source of error is that the motion of the skyrmion\nleads to internal excitations and emission of spin waves\nnot captured by the generalized Thiele approach. These\ne\u000bects are expected to be larger for smaller \u000bbut even\nfor\u000b= 0:04 they give only small quantitative corrections\nin the considered parameter regime. Most importantly,\nthese corrections do not a\u000bect the physics of capturing\nand depinning discussed in the following paragraphs.\nC. Scaling of the e\u000bective potential and\ndimensionless units\nMost results presented in this paper have been ob-\ntained for a \fxed value of the Dzyaloshinskii-Moriya in-\nteraction,D= 0:3J=a, and for a defect described by\na single missing spin in a two-dimensional lattice. As\nskyrmions are smooth objects, one can use simple scaling\nrelations to determine the e\u000bective potential and there-\nfore also the equation of motion and the skyrmion tra-\njectories for other parameters.\nWithin the continuum theory, the free energy, Eq. (1),\nin the presence of a defect of radius adis invariant under\nthe transformation r!r0=r=\u0015,ad!a0\nd=ad=\u0015,\nJ!J0=J,D!D0=\u0015D, and B!B0=\u00152B. The\nshape of the skyrmion is thereby determined by the scale\ninvariant dimensionless parameter\n\u0010=J\u0016B\nD2, (13)\nwith\u00100=\u0010(called\u00142=Q2in Ref. 23).\nThe scaling invariance implies that the e\u000bective poten-5\ntial in the continuum limit can be written as\nVe\u000b(r) =JV\u0012\n\u0010;ad\u0016B\nD;r\u0016B\nD\u0013\n\u0019a2\ndJ\u00162B2\nD2V\u0010\u0012r\u0016B\nD\u0013\n,\n(14)\nwhereVandV\u0010are dimensionless scaling functions with\ndimensionless arguments and V\u0010(\u0018) =1\n2d2V(\u0010;\r;\u0018 )\nd\r2\f\f\f\n\r=0.\nFor the last expression, we assumed that adis much\nsmaller than the skyrmion radius and used that the term\nlinear inadvanishes.\nStrictly speaking, the above derived equation is fully\nvalid in the limit that the lattice constant ais much\nsmaller than both the skyrmion radius and the defect\nradiusad. But even for a defect given by a single missing\nspin (ad=a), one can use that the skyrmion radius as\nis much larger than a. Therefore the dependence of the\ne\u000bective potential on J;D and\u0016Bcan be described as\nVe\u000b(r)\u0019a2J\u00162B2\nD2~V\u0010\u0012r\u0016B\nD\u0013\n. (15)\nNote that ~V\u0010(\u0018) will in general di\u000ber from V\u0010(\u0018). This\nprediction is con\frmed in Fig. 5 which shows ~V\u0010obtained\nfor a single missing spin and various skyrmion sizes ob-\ntained by changing DandBsuch that\u0010= 1 remains\n\fxed.\n1r0=0.3a\n1r0=0.25a\n1r0=0.2a\n1r0=0.15a\n0123456-0.4-0.20.00.20.40.6\ndistance rr0potential V\nz\nFIG. 5. Rescaled potentials, ~V\u0010=Ve\u000b=E0withE0=\na2J\u00162B2\nD2, see Eq. (15), as a function of the rescaled distance\nr=r0withr0=D=\u0016B . Parameters are a=ad= 1 and\n\u0010= 1.\nThe scaling properties for the skyrmion trajectories\ncan simply be obtained by realizing that both Gand\nDare invariant under the scaling transformation as can\ndirectly be seen from Eqs. (7) and (8). Therefore the\n(generalized) Thiele equations (6) or (12) are invariant if\none uses for \fxed defect radius adthe scaling r!r=\u0015,\nD!\u0015D,B!\u00152Bsuch thatdVe\u000b=dr!\u00153dVe\u000b=dr\nand simultaneously rescales the drift velocity vs!\u00153vs\nand the time t!\u00154t.\nAn important consequence of this analysis is that the\ntypical drift velocity vsof electrons or the corresponding\ncritical current density, jc, to depin a skyrmion from adefect scales with the third power of the inverse skyrmion\nradiusas\u0018D=\u0016B\njc\u0018vs\u0018as\u00003. (16)\nThis is part of the reason why skyrmions can be manip-\nulated by ultrasmall current densities11.\nIt is also an interesting question how the potentials\nchange when instead of a single magnetic layer NL>1\nlayers are considered. For a line defect where all spins\nare removed in a line perpendicular to the surface and\nforNL\u001d1 one can use that away from the surface the\nmagnetic con\fguration is translationally invariant in zdi-\nrection. Therefore, the e\u000bective potential is simply given\nby multiplying Ve\u000bbyNL. As also the gyrocoupling and\ndamping matrix scale linearly in NL, the equation of mo-\ntion for the skyrmion center remains unmodi\fed as long\nas the phase with a single skyrmion in a ferromagnetic\nbackground remains stable. Increasing NLallows to elim-\ninate all e\u000bects of thermal \ructuations. The situation is\nmore complicated when only a few layers NLare con-\nsidered. As the properties of the surface and the inner\nlayers are di\u000berent, Ve\u000bcannot simply be computed from\nthe single-layer result.\nFor the presentation of our results, it is useful to \fnd\nthe minimal set of dimensionless parameters needed to\nparametrize our results. Here it is useful to note, that\nthe dependence on \fin the e\u000bective Thiele equations can\nbe eliminated by parametrizing the e\u000bect of the current\nby the drift velocitiy vdof the skyrmion in the absence\nof any defect. It can be obtained from the equation G\u0002\nvs+\fDvs=G\u0002vd+\u000bDvd. Further we also de\fne the\ndimensionless drift velocity vby\nvd=1\nG2+\u000b2D2\u0000\n(\u000b\u0000\f)DG\u0002vs+ (G2+\u000b\fD2)vs\u0001\nv=vdsD3\na2J\u00163B3. (17)\nIn units where all length scales are measured in units\nofD=\u0016B and all times in units ofsD4\na2J\u00164B4the e\u000bective\nThiele equation (12) takes the form\n\u0000d~V\u0010(R)\ndR=\u00004\u0019^z\u0002\u0010\n_R\u0000v\u0011\n+\u000bD\u0010\u0010\n_R\u0000v\u0011\n, (18)\nwithD\u0010=D=s. Originally, the continuum theory was\nparametrized by J,D,\u0016B,\u000b,\f,vs, and the size of the\ndefect. For a point-like defect, we \fnd that the three di-\nmensionless variables \u0010,v, and\u000bare su\u000ecient to describe\nall regimes.\nIV. SKYRMION DEPINNING, CAPTURING\nAND DEFLECTION\nA. Phase diagram\nWhen studying the qualitative behavior of the\nskyrmions when a current is slowly switched on, it is use-\nful to distinguish two initial states, an initially localized6\nP2 P1CF\n0.6 0.7 0.8 0.9 1.0 1.1 1.20.000.020.040.060.080.100.12\nmagnetic field B/B 0current density j/j a\n0.00.51.01.52.02.53.03.5\nσc/σ0\nFIG. 6. Phase diagram as function of the magnetic \feld\nB=B 0=\u0010and current density j=ja=v\u00103for\u000b= 0:1. Here\nwe use the combination v\u00103=vdJ2s\na2D3as it is independent\nof the magnetic \feld. The colored area encodes the value\nfor the capturing cross section \u001bc=\u001b0with the characteristic\nlength\u001b0=D=\u0016B , see Sec. IV C, which is a measure for\nthe e\u000eciency of capturing.\nskyrmion and a skyrmion approaching the defect from\nfar away.\nIf the skyrmion is initially localized close to the de-\nfect and if the potential has a local minimum, it will re-\nmain there for small current densities and gets depinned\nfor larger current densities. Similarly, a skyrmion ap-\nproaching the defect from far away can either get cap-\ntured (green trajectories in Fig. 4) by the defect or is\njust de\rected (blue trajectories).\nAn overview over these possibilities is given in the\nphase diagram, Fig. 6. The solid lines mark the depin-\nning transition. Below these lines, in the regimes denoted\nby P1, P2 and C, an initially localized skyrmion remains\nlocalized close to the defect when the current is switched\non slowly. In P1 the e\u000bective potential has a local min-\nimum atr= 0 while in P2 and C it has a minimum\nat \fnite skyrmion-defect distance. In the free phase, F,\npinning is not possible and all skyrmions move freely. At\nlow magnetic \felds we \fnd this phase even for zero cur-\nrent density. Note that we consider only \u0010 > 0:56 as\nat this point23the circular symmetric skyrmion becomes\nunstable towards the formation of a bimeron24.\nAn unexpected result is that in the pinning regimes P1\nand P2 a skyrmion approaching the defect from far apart\nisnotcaptured. Instead of getting trapped, it moves\naround the defect and is only de\rected. This is a con-\nsequence of the fact that for long distances the defect-\nskyrmion potential is repulsive. Capturing of approach-\na-6-4-2 024-20246\nb-6-4-2 024-20246\nc-6 -4 -2 0 2 4-20246\nd-6-4-2 024-20246\ne-6-4-2 024-20246\nf-6 -4 -2 0 2 4-20246\ng-6-4-2 024-20246\nh\n-6-4-2 024-20246FIG. 7. Trajectories (black) of the single skyrmion obtained\nfrom the e\u000bective potential approach. The coordinates rare\nde\fned relative to the position of the vacancy in dimension-\nless units r=\u001b0=r\u0016B=D . Parameters are \u0010= 1,\u000b= 0:1 and\ndrift velocities from left to right, top to bottom are v= 0:004,\nv= 0:009,v= 0:015,v= 0:026,v= 0:039,v= 0:060,\nv= 0:091, andv= 0:107. The corresponding drift velocities\nvare also marked in Fig. 8. The orange curve is the separa-\ntrix; the orange area is the capturing area. Red arrows mark\noutgoing \row and green arrows mark ingoing \row at a \fxed\npoint. The green (red) circle indicates the potential minimum\n(maximum).\ning skyrmions is only possible in the region C.\nB. Fixed points and separatices\nFor a quantitative analysis of the qualitatively di\u000berent\ntrajectories and for the construction of the phase diagram\nshown in Fig. 6, an analysis of the stable and unstable\n\fxed points of the Thiele equation (18) is useful.\nIn the continuum limit, the e\u000bective potential depends7\nonly on the relative distance rof skyrmion and defect,\nV(r) =V(r). If we now look for \fxed points of the Thiele\nequation, _R= 0, we \fnd that all \fxed points are on the\nline in the direction ^ eofG\u0002vd+\u000bDvd. At the \fxed\npoint one has\nj~V0\n\u0010(rFP)j=v\r, (19)\nwhere\r=\u0010\n(4\u0019)2+\u000b2D2\n\u0010\u00111\n2. There can be 0 ;2;4 or 6\n\fxed points. To classify the \fxed points, one linearizes\nthe equation of motion around them to obtain a matrix\nequation of the type _R=M\u000eR. It is useful to distin-\nguish 5 di\u000berent types of \fxed points characterized by the\neigenvalues, \u00151;2, of the 2\u00022 matrixM. The eigenvalues\nare either both real or are a pair of complex conjugate\nnumbers. If the real part of an eigenvalue is positive (neg-\native) it describes repulsion (attraction). A \fnite imagi-\nnary part gives an oscillatory behavior around the \fxed\npoint on top of the repulsion or attraction. We therefore\ndistinguish attractive ( \u00151;2<0), repulsive ( \u00151;2>0),\nsemide\fnite ( \u00151>0>\u00152), as well as oscillating attrac-\ntive (Re\u00151= Re\u00152<0;Im\u00151=\u0000Im\u001526= 0) and oscil-\nlating repulsive \fxed points (Re \u00151= Re\u00152>0;Im\u00151=\n\u0000Im\u001526= 0).\nGiven the potential exhibits a local minimum, for suf-\n\fciently small drift velocities, v<vc2, always one attrac-\ntive \fxed point exists. Therefore, for v<vc2, an initially\npinned skyrmion will remain pinned when the current\nand hence the drift velocity is increased slowly (a fast\nincrease is discussed below). In Fig. 7, the trajectories\nof skyrmion centers are shown. All skyrmions starting in\nthe orange-shaded region \fnally end up in the attractive\n\fxed point.\nThis capturing region can either be a bounded re-\ngion or extended to in\fnity. Only in the latter case, a\nskyrmion approaching from far apart can be captured\nby the defect. For small drift velocities, v < vc1, and\na local minimum of the potential, we obtain always\na bounded capturing region where all skyrmions move\naround the defect without being trapped. Only in the\nregimevc1< v < vc2capturing of mobile skyrmions is\npossible.\nOn each separatrix (orange lines in Fig. 7), we \fnd\nat least one semide\fnite \fxed point, where the `critical'\ntrajectories, which de\fne the separatrix, end. In Fig. 7\nthe semide\fnite \fxed points are marked by ingoing green\nand outgoing red arrows. To construct the separatrix nu-\nmerically, it is useful to investigate the time-reversed ver-\nsion of the equation of motion. After time reversal, the\ncritical trajectories start at the metastable \fxed point.\nUsing a small perturbation in the direction of the at-\ntractive eigenvector as a starting point, one can obtain\ndirectly the separatrix by computing the time-reversed\ntrajectories. This procedure is numerically stable as af-\nter time-reversal the repulsive direction of the \fxed point\nbecomes attractive. The time-reversed trajectories either\nreach in\fnity or end at the repulsive direction of a \fxed\npoint. Using this method, we compute for a given drift\nh g f e d cbaα=0.4\nα=0.1\nα=0.04\nα=0.01\n0.00 0.02 0.04 0.06 0.08 0.100123\ndrift velocity vσc/σ0FIG. 8. Cross section for capuring a skyrmion \u001bc=\u001b0as a\nfunction ofvwith the characteristic length \u001b0=D=\u0016B shown\nfor\u0010= 1 and various values of \u000b. Only for a \fnite drift\nvelocity range, vc1< v < vc2, the skyrmion gets trapped by\nthe defect. The vertical lines denote boundaries of various\nregimes (a-h) for \u000b= 0:1. In Fig. 7 typical trajectories for\nthese regimes are shown.\nvelocity the separatrix and identify the orange shaded\ncapturing region. At v=vc1, there is one trajectory\ndirectly connecting the two semide\fnite \fxed points.\nForvclose tovc2the orange-shaded capturing region\nshrinks in width and is displaced. This has the e\u000bect\nthat the way the current is switched on profoundly af-\nfects the pinning properties of skyrmions. For an ad-\nabatic switching on of a current with v<vc2, a skyrmion\nwhich was pinned at v= 0 follows the stable \fxed point\nand hence remains pinned. If, in contrast, the current is\nsuddenly switched on, then the initially pinnied skyrmion\nmay end up outside of the capturing region and thus gets\ndepinned. At v= 0 the position of the trapped skyrmion\nis determined by the location of the minimum of the ef-\nfective potential, shown as a green dashed line in Fig. 7.\nThis line is only partially within the capturing region for\nlarge current densities. Therefore a corresponding frac-\ntion of initially localized skyrmions, Fig. 7e and f, or even\nallskyrmions, Fig. 7g, will be released after a sudden\nswitching on of the current.\nC. Capturing cross section \u001bc\nTo quantify the e\u000eciency for capturing a mobile\nskyrmion by a defect, it is useful to de\fne the 'capturing\ncross-section` \u001bc. As for other scattering experiments, the\ncross section is obtained by dividing the capturing rate\nby the incoming \rux of particles. In the two-dimensional\nsituation discussed here, \u001bcis directly given by the width\nof the orange-shaded capturing region, compare Fig. 7,\nforx!\u00001 . In Fig. 8, \u001bcis shown as a function of\nthe drift velocity vfor a \fxed value of the magnetic \feld\n(\fxed\u0010). As discussed above, mobile skyrmions are only\ntrapped for vc1<v<vc2. The change of \u001bcas a function\nof both \feld and current density (drift velocity) is shown\nin Fig. 6.\nThev-dependence of \u001bcshows a sequence of kinks.8\nThe origin of each of these kinks can be traced back\nto a change of the topological structure de\fned by the\n\fxed points and the separatrices connecting and encir-\ncling them. In Fig. 7 examples of such \fxed point con-\n\fgurations are shown.\nTo trap a skyrmion, damping is needed. Indeed, Fig. 8\nshows that \u001bcgets smaller and smaller for \u000b!0, while\nthe two critical drift velocities vc1andvc2remain \fnite\nin this limit. To prove analytically that the skyrmions\nare not captured for \u000b= 0, one can, for example, rewrite\nEq. (12) in the form _R=\u0000G\u0002dVtot\ndR=jGj2using the\npotentialVtot(R) =V(R)\u0000(G\u0002vs)\u0001R. This implies\nthat skyrmions \row along equipotential lines of Vtotand\na mobile skyrmion thus never gets trapped.\nTaking into account that \u001bcvanishes for \u000b!0 it is\nperhaps surprising that the maximal \u001bcin Fig. 8 appears\nto be of the order of the skyrmion radius even for \u000b=\n0:01. The capturing cross section is maximal for rather\nsmall drift velocities slightly above vc1. A quantitative\nanalysis shows that for \u000b!0 the maximal cross section\nscales with bothp\u000band the typical length scale \u001b0=\nD=(\u0016B) of the order of the skyrmion radius\n\u001bmax\nc\u0019p\u000bc\u0010\u001b0, (20)\nwhere the dimensionless constant c\u0010depends on \u0010with\nc1\u001912. For larger currents (i.e., the regimes f and g in\nFig. 8) we \fnd instead \u001bc/\u000band therefore only tiny\ncapturing cross sections for small \u000b.\nForv!vc1andv!vc2the capturing cross section\n\u001bcvanishes.\nTo analyze the behavior of \u001bcforvclose to the lower\ncritical drift velocity, v&vc1, we use that the proper-\nties of this critical point are governed by the semide\fnite\n\fxed point at the top of Fig. 7a-c and that the sepa-\nratrix can be computed by analyzing the time-reversed\nequation of motion as described above. At v=vc1\nthe time-reversed trajectory coming from the semide\f-\nnite \fxed point at the bottom ends in this \fxed point\nat the top with eigenvalues \u00151>0> \u0015 2and eigenvec-\ntorsb1andb2. Forv=vc1+\u000ev, this trajectory ap-\nproaches the \fxed point from the b1direction and leaves\ninto theb2direction. Close to the \fxed point, one ob-\ntains\u000eR(t) =b1x1(t) +b2x2(t) withxi(t) =xi(0)e\u0000\u0015it.\nThe linearized equation of motion is only valid for small\nx1(t);x2(t)< x 0, wherex0is a cuto\u000b scale. We choose\ntwo timest1andt2such thatx1(t1) =x0andx2(t2) =x0\nin a way that the linearization is valid for t1< t < t 2.\nHeret1(t2) describes a point on the trajectory when\napproaching (leaving) the \fxed point. With these de\f-\nnitions we obtain x1(t2) =x0\u0010\nx0\nx2(t1)\u0011\u00151=\u00152\n. Using that\nthe cross section is approximately proportional to x1(t2)\nand thatx2(t1) depends linearly on v\u0000vc1, we obtain\n\u001bc\u0018(v\u0000vc1)j\u00151=\u00152j. (21)\nWe have checked numerically, that this result is valid\nclose tovc1. Forv!vc2in contrast, we \fnd that thedecay of the capturing cross section can be described by\n\u001bc\u0018(vc2\u0000v)2. (22)\nV. SKYRMIONS AND WEAK DISORDER\nFinally, we will discuss the case of a skyrmion moving\nthrough a weakly disordered medium. The distance of\ndefects is assumed to be much larger than the skyrmion\nradius,nd\u001c(\u0016B=D )2, wherendis the density of defects.\nIn this limit it is interesting to investigate, how the de-\nfects in\ruence the skyrmion Hall e\u000bect and the skyrmion\nmobility.\nIn the absence of any defects, the skyrmions move on\na straight line in a direction set by v. This direction is\nset by the direction of the external current and the dissi-\npation constants, see Eq. 17. When a skyrmion scatters\nfrom a defect, it therefore cannot change its direction.\nThe only net-e\u000bect of scattering is a displacement \u0001 k\nand \u0001?, parallel and perpendicular to v, respectively.\nA parallel displacement \u0001 kimplies that the skyrmion is\ndelayed, \u0001k>0, or has moved faster when passing the\ndefect, \u0001k<0. Therefore \u0001 kleads to changes of the mo-\nbility of the skyrmion. \u0001 ?in contrast, describes a 'side\njump` of the skyrmion due to the defect. Similar to the\nside-jump mechanism of electron scattering25, this leads\nto a contribution to the skyrmion Hall e\u000bect.\n\u0001kand \u0001?are functions of the impact parameter b,\ndescribing the o\u000bset of the incoming skyrmion trajectory\nrelative to the defect position. This dependence is shown\nin Fig. 9 for v < vc1(top \fgure) and v > vc2(bottom\n\fgure). When a skyrmion travels a long distance L, it\nhits several randomly distributed defects with impact pa-\nrameterbi. To calculate the total shift of a skyrmion one\ncan therefore average over all defect positions\n\u0001k=?\nL=1\nLX\ni\u0001k=?(bi)\u00191\nLZ\nnd\u0001k=?(b(r))d2r\n=nd\u0001I\nk=?\n\u0001I\nk=?=Z\ndb\u0001k=?(b). (23)\nThe o\u000bset integrals \u0001I\n?and \u0001I\nkparametrize how e\u000e-\nciently a defect can lead to a displacement of the trajec-\ntory.\nTo linear order in the density of defects nd, the average\nvelocity and the mobility thus change by\n\u0001v\nv\u0019nd\u0001I\nk (24)\nand the average direction of motion of the skyrmions is\nrotated by the angle\n'\u0019nd\u0001I\n?. (25)\nIn Fig. 10 the o\u000bset integrals are shown for v < vc1\nandv>vc2. Forvc1<v<vc2a single skyrmion always\ngets trapped for L! 1 and therefore the discussion9\nv=7.410-3\nv=4.110-3\nv=7.410-4\n-5 0 5-10-50510\nstarting offset bs0DÞ s0,DÈÈs0\nv=0.276\nv=0.158\nv=0.112\n-4-2024-10123456\nstarting offset bs0DÞ s0,DÈÈs0\nFIG. 9. \u0001k(dashed) and \u0001 ?(solid) shown as functions of\nthe impact parameter bfor drift velocities below (top) and\nabove (bottom) the capturing regime. Parameters used here\nare\u0010= 1,\u001b0= 1=0:3a,\u000b= 0:1, and drift velocities as given\nin the \fgures.\ngiven above can only be applied for \fnite systems (not\ndiscussed here).\nFor large drift velocities, v!1 , one can calculate\n\u0001I\nk=?by using that in this limit the potential only induces\nsmall corrections to straight skyrmion trajectories which\ncan be treated perturbatively. In this limit we obtain\n\u0001I\n?\u00192\u0019\u000bGD\nv2(G2+ (\u000bD)2)2Z1\n0drr(V0(r))2+O\u00121\nv3\u0013\n\u0001I\nk\u0019\u000bD\nG\u0001I\n?+O\u00121\nv3\u0013\n. (26)\nNumerically, we \fnd that this formula although derived\nforv! 1 works accurately for \u0001I\n?for allv > vc2\nwhereas for \u0001I\nkcorrections to this formula are of order 1\nforv&vc2, see lower part of Fig. 10.\nBoth o\u000bset integrals are strongly enhanced for v<vc1\nwhen the skyrmion moves around the defect instead of\npassing it. For the parameters shown in Fig. 10, they\nare about two orders of magnitude larger for v.vv1\ncompared to the case v&vc2and also become much\nlarger than the area of the skyrmion. The main reason is\nthat in this regime the skyrmions move around the defect\nDÞIs02\nlog2fit\nlog2fit\nDÈÈIs02\n0.000 0.002 0.004 0.006 vc1-120-100-80-60-40-20\ndrift velocity vDÞIs02,DÈÈIs02\nDÞIs02\nDÞ PTIs02\nDÈÈPTIs02\nDÈÈIs02\nvc20.20.30.40.50.60.70.8-0.8-0.6-0.4-0.20.00.2\ndrift velocity vDÞIs02,DÈÈIs02FIG. 10. O\u000bset integrals shown as functions of the drift ve-\nlocity below (top) and above (bottom) the capturing regime,\ni.e.,v<vc1= 0:0083 andv>vc2= 0:1044. Parameters used\nhere are\u0010= 1,\u001b0= 1=0:3a, and\u000b= 0:1. The logarithmic \fts\nin the upper plot are related to Eq. (27). The perturbative\napproximation in the lower plot is given in Eq. (26).\nin a distance/ln 1=vdue to the exponential tails of the\nskyrmion-defect potential. This sets the relevant length\nscale independent of the damping and hence the e\u000bects\nare not any more suppressed by \u000b. We therefore obtain\n\u0001I\nk;\u0001I\n?/ln21=vforv!0, (27)\nas shown in the upper part of Fig. 10. Note that thermal\n\ructuations, not considered in this study, are expected\nto cut o\u000b the divergency.\nA counter-intuitive result is that for v < vc1, \u0001I\nkis\nnegative implying that defects accelerate the motion of\nskyrmions. This is possible because the speed of the\nskyrmion can grow when the angle \u001ebetween vand _R\ngrows. A simple, analytically solvable limit is the motion\nof the skyrmion parallel to a wall. From the balance of\nforces parallel to the wall ( dV=dRk= 0), one obtains\nusing Eq. (12) that _Rk=vsGcos\u001e\u0000\fDsin\u001e\n\u000bDwhere\u001eis the\nangle between drift velocity and the wall normal. For\nsmall\u000b\u0018\f, obstacles can therefore speed up skyrmion\nmotion by a maximal factor of order 1 =\u000b\u001d1. While\nthe path of the skyrmion which moves around a defect\nincreases, the increased velocity typically overcompen-10\nsates this longer path for v<vc1as shown in Fig. 10 for\n\u000b= 0:1.\nA way to measure \u000b\u0000\fin a weakly damped skyrmion\nsystem experimentally is to determine (in the absence of\ndefects) the skyrmion Hall angle, '0\u0019D(\u000b\u0000\f)\nG, which\nis the angle between the direction of the electric current\nand the skyrmion \row direction. This angle is changed\nby defects as described by Eq. (25). Especially for small\n\u000band\fandv<vc1one can easily reach a regime where\nthe impurity induced contribution to the skyrmion Hall\nangle dominates, '&'0. An alternative point of view is\nthat the defects lead to a strong renormalization of \u000b\u0000\f,\nwith\n(\u000b\u0000\f)e\u000b\u0019\u000b\u0000\f+G\nDnd\u0001I\n?. (28)\nVI. CONCLUSION AND QUANTITATIVE\nESTIMATES FOR SKYRMION DEVICES\nWe found that the Thiele ansatz combined with an\ne\u000bective potential is an e\u000ecient and reliable tool to de-\nscribe the interaction of a skyrmion with a vacancy. Us-\ning this ansatz we determined the phase diagram which\nshows the three phases appearing in this interaction: The\npinning phase where a skyrmion freely moves around va-\ncancies and never gets captured but stays pinned if it\nwas pinned initially, the capturing phase in which a mov-\ning skyrmion can get captured within a certain captur-\ning cross section and the freephase for current densities\nj >jmaxwhere a skyrmion can never be pinned.\nTc\u0015 a m n\nMnSi 29 K 180 \u0017A 4:6\u0017A 0:4\u0016B 3:8\u00011028m\u00003\nFeGe 280 K 700 \u0017A 4:7\u0017A 1\u0016B 2:4\u00011028m\u00003\nTABLE I. Input parameters26{29for the quantitative esti-\nmates.Tc\u0018Jis the transition temperature30,athe lat-\ntice constant for a unit cell containing 4 Mn (or Fe) ions,\n\u0015\u00192\u0019J=D the pitch in the helical phase, m\u0019s\u0016Ba2=(2~) is\nthe magnetization per Mn (or Fe) ion and hence \u0016= 4m=a2,\nandnthe charge density.\nOur results can be used to obtain estimates of neces-\nsary current densities and the depth of the pinning po-\ntential and can, hopefully, be used as a starting point to\ndesign simple skyrmion devices. As an example, we try\nto give estimates of the relevant parameters for MnSi,\nthe perhaps best studied skyrmion material, and for\nFeGe, the skyrmion system with the largest transition\ntemperature8up to now. Input parameters for these es-\ntimates are shown in table I.\nIn table II, we show typical parameters characterizing\na defect with the size of one unit cell for a single-layer of\nMnSi or FeGe for \u0010= 1. Note that the actual numbers\nwill depend on the microscopics of the induced defect andshould therefore be viewed only as order-of-magnitude\nestimates.\nB0E0=kBv0j0 jc2\nMnSi 0.7 T 0.7 K 9m\ns5\u00011010A\nm2 6\u0001109A\nm2\nFeGe 0.2 T 0.5 K 8m\ns3\u0001109A\nm2 3\u0001108A\nm2\nTABLE II. Estimates of typical parameters for the pinning\nof a skyrmion by a single-site defect in a single layer of the\nmaterials MnSi and FeGe at \u0010= 1, i.e., for B=B0=D2\n\u0016J.\nE0=J(a\u0016B)2\nD2is the strength of the pinning potential de-\n\fned by the prefactor in Eq. (15). The typical velocity\nv0=a2J(\u0016B)3\nsD3and the typical current density j0=nev0are\nde\fned such that v=vd=v0=j=j0, whilejc2= 0:11j0is the\ncritical current density for \u0010= 1 atT= 0. Note that the\nparameters depend strongly on the size of the defect and the\nlayer thickness, see text.\nA main result of these estimates is that a single-site va-\ncancy in a monolayer of these materials will not be able\nto pin a skyrmion due to the presence of thermal \ruc-\ntuations,E0\u001ckBT. This shows that indeed skyrmions\nare very insensitive to defects. To build a device with a\nnanostructure which is capable to pin a skyrmion, one\ntherefore needs to consider both larger defects and also\n\flms with a larger number of layers, NL\u001d1, using that\nE0/a2NL, see Sec. III C. The critical current density\nfor depinning, jc2, is independent of NLbut also scales\nwith the area of the defect. For example, using a hole\nwith a diameter of 10 nm for a FeGe \flm with a thick-\nness of 50 nm in a magentic \feld of 0 :2 T, we obtain as\nan order-of-magnitude estimate\nE0=kB\u001920:000 K; jc2\u00191011A=m2, (29)\nclearly su\u000ecient for thermal stability.\nAs we have shown, the shape of the e\u000bective impurity-\nskyrmion potential depends quantitatively and qualita-\ntively on the strength of the magnetic \feld. Changing,\nfor example, the magnetic \feld from 0 :2 T to 0:13 T is\nsu\u000ecient to avoid allpinning, see Fig. 6. By controlling\nboth the magnetic \feld and the current density one can\nvary in a \rexible way not only the capability of a de-\nfect to hold a skyrmion but also its ability to capture a\nskyrmion moving close by, see Fig. 8. We believe that\nthis \rexibility will allow to control skyrmions e\u000eciently\nin devices based on holes and similar nanostructures.\nVII. ACKNOWLEDGEMENTS\nWe would like to thank C. 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Bailey1,b)\nMaterials Science and Engineering, Department of Applied Physics and\nApplied Mathematics, Columbia University, New York, New York 10027,\nUSA\nWe present measurements of interfacial Gilbert damping due to the spin pumping\ne\u000bect in Ni 81Fe19/W heterostructures. Measurements were compared for heterostruc-\ntures in which the crystallographic phase of W, either \u000b(bcc)-W or \f(A15)-W, was en-\nriched through deposition conditions and characterized using X-ray di\u000braction (XRD)\nand high-resolution cross-sectional transmission electron microscopy (HR-XTEM).\nSingle phase Ni 81Fe19/\u000b-W heterostructures could be realized, but heterostructures\nwith\f-W were realized as mixed \u000b-\fphase. The spin mixing conductances (SMC)\nfor W at interfaces with Ni 81Fe19were found to be signi\fcantly lower than those\nfor similarly heavy metals such as Pd and Pt, but comparable to those for Ta, and\nindependent of enrichment in the \fphase.\na)Electronic mail: wc2476@columbia.edu\nb)Electronic mail: web54@columbia.edu\n1arXiv:1904.05950v2  [cond-mat.mtrl-sci]  4 Sep 2019I. INTRODUCTION\nThe heavy metals Ta, W and Pt have drawn attention as charge-to-spin-current-\nconverters using spin Hall and related e\u000bects1{4. Beta phase W, \f-W, with the topologically\nclose-packed A15 structure5, possesses a \\giant\" spin Hall angle of \u0012SH\u00190.3{0.43,6. The\nspin transport properties of \f-W, such as the spin Hall angle \u0012SHand spin di\u000busion length\n\u0015SD, have been characterized by di\u000berent methods3,6{8. In these studies, the metastable\n\f-W layers were deposited directly on the substrate, were only stable for small W thickness,\nand were presumably stabilized through residual water vapor or oxygen on the substrate\nsurface; thicker W \flms typically revert to the stable (bcc) \u000bphase.\nRecently, some of us9{11have optimized a di\u000berent method to stabilize the metastable-\n\f-phase, using the introduction of N 2gas12while sputtering at low power. Relatively thick\n(over 100 nm) monophase \f-W \flms could be stabilized this way, when deposited on glass\nsubstrates. This technique has allowed deposition of majority \fphase W for 14 nm W\n\flms on CoFeB, as CoFeB/W(14 nm), and of minority \fphase for 14 nm W \flms on\nNi and Ni 81Fe19(\\Py\"), as Ni/W(14 nm) and Py/W(14 nm). In the present work, we\nhave prepared both monophase Py/ \u000b-W (here Py/\\ \u000b\"-W) and mixed phase Py/( \u000b+\f)-\nW (here Py/\\ \f\"-W) heterostructures using our optimized sputtering technique to enrich\nthe fraction of \f-W. Crystallographic phases of W were characterized by X-ray di\u000braction\n(XRD), and high-resolution cross-sectional transmission electron microscopy (HR-XTEM);\nsecondary structural information was provided by electrical resistivity measurements at room\ntemperature. We note that our measurements cannot distinguish between purely metallic,\nA15\f-W and A15 W oxide or nitride (e.g. W 3O); the identity of \f-W as a purely metallic\nphase or a compound is a longstanding controversy12,13.\nIn ferromagnet (FM)/normal metal (NM) heterostructures, pure (chargeless) spin cur-\nrents can be injected from the FM into the NM by exciting ferromagnetic resonance (FMR)\nin the FM layer, \\pumping\" out spin current14,15. If the spin current is absorbed in the\nNM layer, the in\ruence of \\spin pumping\" can be observed through the increase in the\nlinewidth of the resonance, proportional to frequency !as Gilbert damping, due to the\nloss of angular momentum from the precessing spin system14,15. The e\u000eciency of the spin\npumping e\u000bect for a given interface is characterized through the spin mixing conductance\n(SMC)g\"#\nFM=NM. The SMC is also an important parameter for the interpretation of inverse\n2spin Hall e\u000bect (ISHE) measurements4,16, in which the spin Hall angle \u0012SHis measured by\npumping chargeless spin current into the NM by FMR and measuring spin-to-charge current\nconversion through the generated charge current. Measurements of spin mixing conductance\nfor Py/\u000b-W and Py/ \f-W have not been reported previously, although some measurements\nhave been reported for W oxide8. For these measurements, the simplest way to isolate the\ncontribution of the FM/NM interface to the damping, and thus the spin pumping e\u000bect and\nspin mixing conductance g\"#\nFM=NM, is to deposit the FM on the bottom and the NM on top,\nso that comparison structures without the NM layer have nearly identical microstructure.\nThe ability to deposit enriched \f-W on Py rather than on an insulating substrate is thus im-\nportant for the measurement of spin mixing conductance of Py/ \f-W. In this manuscript, we\nreport measurements of spin mixing conductances for Py/\\ \u000b\"-W and Py/\\ \f\"-W interfaces\nusing variable-frequency, swept-\feld FMR, as in our previous work17{19.\nII. SAMPLE PREPARATION\nUltrahigh vacuum (UHV) magnetron sputtering was used to deposit substrate/Ta(5\nnm)/Cu(5 nm)/Ni 81Fe19(Py)/W/Cu(5 nm)/Ta(5 nm) heterostructures on both oxidized\nSi and glass substrates at room temperature, with base pressure better than 2 \u000210\u00008Torr.\nThe samples consist of two thickness series in \\ \u000b\"-W and \\\f\"-W for a total of four se-\nries. In the \frst thickness series, the thickness of Py ( tPy= 5 nm) was \fxed and the\nthickness of W was varied, with tW= 2, 5, 10, 30 nm, for both \\ \u000b\"-optimized and \\ \f\"-\noptimized conditions. This thickness series was used for resistivity measurements, X-ray\ndi\u000braction (XRD) ( tW= 10, 30 nm), high-resolution cross-sectional transmission electron\nmicroscope (HR-XTEM) ( tW= 30 nm) and FMR characterization. In the second thickness\nseries, the thickness of W ( tW= 10 nm) was \fxed and the thickness of Py was varied, with\ntPy= 3, 5, 10, 20 nm, also for both \\ \u000b\" and \\\f' conditions. This thickness series was used\nonly for FMR characterization. The same stacks without W layers, Py(3, 5, 10, 20 nm),\nwere deposited as reference samples for FMR measurements. One heterostructure with re-\nverse depostion order, \\ \u000b\"-W(10 nm)/Py(5 nm), was deposited in the absence of N 2gas\nand characterized by XRD and FMR; this was not possible for \\ \f\"-W because the \fphase\ncannot be stabilized on Cu underlayers10.\nThe W layers in all samples were deposited with 10 W power, nearly constant deposition\n3rate (<0:1\u0017A/s), and Ar pressure of 3 \u000210\u00003Torr. Nitrogen gas, with 1 :2\u000210\u00005Torr\npressure measured by a residual gas analyzer, was introduced to promote the growth of \f\nphase W10.\nIII. STRUCTURAL CHARACTERIZATION\nCrystalline phases of W in the Py/W heterostructures were characterized primarily by\nXRD (Section A), with supporting measurements by HR-XTEM (Section B), and \fnally with\nsome indirect evidence in room-temperature electrical resistivity measurements (Section C).\nOur basic \fndings are that \flms deposited without N 2, optimized for \\ \u000b\"-W, are nearly\nsingle-phase \u000bin Py/\\\u000b\"-W, while in the Py/\\ \f\"-W optimized heterostructures, deposited\nin the presence of N 2, the W layers are mixed \u000b+\fphase, with a roughly 50%{50% mixture\nof\u000b-W and\f-W averaged over a 10 nm \flm. The phase composition within the \frst 5 nm\nof the interface may have a slightly greater fraction of \u000b-W, but\f-W could be positively\nidenti\fed here as well.\nA. X-ray di\u000braction\nBoth symmetric ( \u0012-2\u0012) and grazing-incidence, \fxed sample angle X-ray di\u000braction (XRD)\nscans were carried out on Py(5 nm)/W(10 nm) and Py(5 nm)/W(30 nm) heterostructures\ndeposited on glass substrates. The scans are compared for \\ \u000b\"-W and \\\f\"-W depositions.\nScans were recorded using Cu K\u000bradiation and a commercial di\u000bractometer.\nThe symmetric ( \u0012-2\u0012) scans, with scattering vector perpendicular to \flm planes, are\npresented \frst. We point out some obvious features of the symmetric XRD spectra, shown\nin Figures 1a) and 1b). For the Py/\\ \u000b\"-W(30 nm) \flm in Fig. 1a), all peaks can be indexed\nto the close-packed planes, Cu(111)/Py(111) (fcc) and \u000b-W(110) (bcc). The small peak\nat 2\u0012= 36\u000ecan be indexed to the re\rection of a small amount of Cu K\fradiation from\n\u000b-W(110). Moving to the thinner \u000bphase \flm in Fig. 1b), Py/\\ \u000b\"-W(10 nm), it is still\nthe case that all re\rections can be indexed to the close-packed Cu(111)/Py(111) and \u000b-\nW(110) planes. However, there is greater structure in these re\rections, presumably due to\n\fnite-size oscillations (Laue satellites), expected to be more evident in thinner \flms. Nearly\nidentical spectra are recorded for the 10 nm \\ \u000b\"-W \flms regardless of deposition order:\n4Py(5 nm)/\\ \u000b\"-W(10 nm) and \\ \u000b\"-W(10 nm)/Py(5 nm) \flms scatter X-rays very similarly,\nas shown in Fig. 1b). We should note that Cu deposited on Ta has strong f111gtexture in\nour \flms. Py (Ni 81Fe19) deposited on Cu also has strong f111gtexture; growth of Py on Cu\nand vice-versa is found to be largely coherent within grains. Both layers are fcc with similar\nlattice parameters: aCu\u00193:61\u0017A for Cu10,20andaPy\u00193:55\u0017A for Py10,21, with a small\nmis\ft strain of \u000f=jaCu\u0000aPyj=aCu\u00192%. The XRD peaks for (111)-re\rections in bulk\nphases, broadened by \fnite-size e\u000bects ( FWHM\u00191:7\u000efor 5 nm \flms, using the Scherrer\nequation22,23), are very close to each other, at 44 :2\u000e(Py) and 43 :4\u000e(Cu) respectively, so we\nexpect (and have observed) one averaged peak for Cu and Py.\nThe nominal \\ \f\"-W \flms (red lines) clearly show the presence of the \fphase through\nthe unique\f-W(200) re\rection at 2 \u0012'36\u000e. This unique re\rection is very strong in the \\ \f\"-\nW(30 nm) heterostructure (Fig. 1a) but weaker as a proportion of the total intensity in the\nthinner \\\f\"-W(10 nm) heterostructure (Fig. 1b). In Fig. 1a), experimental \f-W(200) and\n\f-W(210) re\rections have intensities in a ratio similar to the theoretical scattering intensity\nratios for randomly-oriented \fgrains. This is not the case for the thinner \\ \f\"-W(10 nm)\nheterostructure in Fig. 1b); here the unique \f-W(200) peak is less intense than expected.\nWe interpret the relative weakness of \f(200) as the presence of a large fraction of \u000bgrains\nin the nominal Py/\\ \f\"-W(10 nm) heterostructure.\nIn order to quantify the amount of \u000b-W in the nominal \\ \f\"-W \flm, we have carried out\ngrazing incidence measurements of Py(5 nm)/W(10 nm) samples (20\u000e\u00142\u0012\u0014100\u000e) on the\nsame di\u000bractometer, as illustrated in Fig. 1c). The samples were measured at a \fxed source\nposition of 5\u000ewith 0:1\u000estep size, 0:25\u000e\fxed slit and the 15 mm beam mask. From the TEM\nmeasurements in Fig. 2b), we \fnd that the deposited \\ \u000b\"-W \flms havef110gtexture, i.e.,\nthe hexagonal arrangement (60\u000eangles) of thef011g\u000b-W re\rections away from the surface\nnormal. Thus with the grazing incidence geometry, in which the scattering vector does not\nremain perpendicular to the \flm plane, the relative intensities of the peaks will not match\ntheoretical calculations (vertical lines) based on randomly-oriented, untextured \flms. For\nexample, the \u000b-W(200) peak (blue, \u001858\u000ein 2\u0012) almost vanishes in the XRD scan here, due\nto thef011g\u000b-W texture.\nHere we focus on the \u000b-W(211) peaks near 2 \u0012= 72\u000e, observed in both Py/\\ \u000b\"-W and\nPy/\\\f\"-W samples. As shown in the Fig. 1c) inset, the \u000b-W(211) peaks (60\u000e\u00142\u0012\u001485\u000e),\nwere \ftted as the sum of Lorentzian peak and identical background, assumed quadratic in\n52\u0012, for both Py/\\ \u000b\"-W and Py/\\ \f\"-W samples. First we \ft the \u000b-W(211) peak (blue)\nin the Py/\\ \u000b\"-W sample to the summed function to determine the Lorentzian peak and\nquadratic background parameters. Next, we use this \ftted background in the \ft to the\n\u000b-W(211) peak (red) in the Py/\\ \f\"-W sample. The two \ftted \u000b-W(211) peaks are shown\nas blue (for Py/\\ \u000b\"-W) and red (for Py/\\ \f\"-W) dashed lines in the Fig. 1c) inset. The \fts\nreproduce the experimental data well in the \ftted region. The integrated \u000b-W(211) peak\n(i.e., the 2\u0012-integrated area between the measured data and the \ftted background) for the\nPy/\\\f\"-W sample has roughly half the intensity of the integrated peak for the Py/\\ \u000b\"-W\nsample. Assuming that the nominal \u000b-W is 100% \u000bphase and that the \u000bgrains in mixed\nphase \\\f\"-W have similar f110gtexture, as is supported by the HR-XTEM measurements\nin Figures 2 and 3, we conclude that the Py/\\ \f\"-W(10 nm) \flm is roughly 50% \u000b-W and\n50%\f-W.\nB. Transmission electron microscopy\nThe phases of the nominal Py(5 nm)/\\ \u000b\"-W(30 nm) and the nominal Py(5 nm)/\\ \f\"-\nW(30 nm) samples deposited on oxidized Si substrates were characterized in high-resolution\ncross-sectional imaging, selected-area di\u000braction, and focused-beam nanodi\u000braction, by\ntransmission electron microscopy (for details see the endnote1).\nFig. 2 shows a cross-sectional image and di\u000braction pattern for the nominal Py/\\ \u000b\"-\nW(30 nm) heterostructure. First, one can see from the mass contrast between W and the 3d\ntransition metal elements (Ni, Fe, Cu) that the Py/W and W/Cu interfaces are relatively\n\rat and sharp on the scale of the image resolution of \u00183 nm, presumably broadened\nby topographic variation through the thickness of the TEM foil. Second, based on (less\npronounced) di\u000braction contrast parallel to the interface, the grains appear to be columnar,\nin many cases extending through the \flm thickness, with an average (lateral) grain diameter\nof 10{20 nm. The selected-area di\u000braction (SAD) pattern can be indexed according to unique\n(111)Py//(011) \u000b-W \fber texture, as shown by the hexagonal arrangement (60\u000eangles) of\nthef011gre\rections in \u000b-W, and the arrangement of f111gre\rections in Py,\u001870:5\u000eaway\nfrom the (vertical) \fber axis. The calculated di\u000braction spots based on f111gPy//f011g\u000b-\nW \fber texture with 1-fold rotational symmetry about the \flm-normal axis are shown in\nFig. 2 b), inset; good agreement is found.\n6Cross-sectional images and di\u000braction patterns for the Py/\\ \f\"-W(30 nm) heterostructure\nare shown in Fig. 3. Here again, in Fig. 3 a), the mass contrast shows similarly well-\nde\fned interfaces, but the topographic variations have a shorter wavelength, due presumably\nto smaller, more equiaxed grains in the mixed-phase \\ \f\"-W. Circles indicate areas where\nconvergent nanobeam electron di\u000braction (CBED) patterns were taken. The di\u000braction\npatterns over these small regions can be indexed to single phases: fcc Ni 81Fe19(Py) in\ngreen, bcc\u000b-W in blue, and A15 \f-W in red.\nThe CBED patterns in Fig. 3 a) con\frm that the nominal \\ \f\"-W \flm is mixed-phase \u000b-W\nand\f-W. The critical question for distinguishing the spin mixing conductances of \u000b-W and\n\f-W in Py/W is the identity of the W phase located within the \frst several nanometers of\nthe interface with Py: the pumped spin current is ejected through the interface and absorbed\nover this region; see the x-axis of Fig. 6. We have addressed this question locally using high\nresolution imaging (see Fig. 3 b) and over a larger area using frequency analysis (see Fig. 3\nc) of the image, roughly equivalent to SAD. In Fig. 3 b), a 10 nm area (red box) shows what\nappears to be a single-crystal region with (1 \u001611)[110]Py//(011)[1 \u001611]\u000b-W//(002)[200] \f-W,\nindicating that the \fcrystals may nucleate on top of the \u000bcrystals; however, this is contrary\nto our previous observations10and not distinguishable in the image from the superposition\nof grains through the foil, with nucleation of \fat the Py/W interface. The discrete spatial\nFourier transform (FT) of this region shows that the four vertically/horizontally circled \f-\nWf002gspots are similar in intensity to the six \u000b-Wf011gspots, supporting a similar \f-W\ncontent in this region. Carrying out a spatial FT of the full selected region within 5 nm of\nthe interface (dotted box) in Fig. 3 a), we can con\frm that \f-W is indeed present adjacent\nto the interface, as indicated by the \f-Wf002gFT spots in Fig. 3 c), although these appear\nto be somewhat less intense than the \u000b-Wf011gspots.\nC. Resistivity\nFour-point probe van der Pauw resistivity measurements were performed at room tem-\nperature on the \frst thickness series of samples ( tPy= 5 nm \fxed, variable tW) deposited\non 25\u000225 mm square glass substrates, i.e., glass substrate/Ta(5 nm)/Cu(5 nm)/Py(5\nnm)/W(tW)/Cu(5 nm)/Ta(5 nm). Two point probes for current and two point probes\nfor voltage were placed at the four corners of the square coupons. For square samples, the\n7voltage-to-current ratios were converted to resistance per square using the known geomet-\nrical factor \u0019=ln 2\u00194:5324. To isolate the W resistances, we plot the thickness-dependent\nsheet conductance and \ft according to:\n1\nRtotal=Gtotal=G0+4:53\n\u001aWtW (1)\nwhereRtotal(Gtotal) is the total resistance (conductance) of the sample, \u001aWandtWare\nthe resistivity and the thickness of the W layer, and G0is the parallel conductance of other\nlayers in the stack.\nWe have veri\fed Ohmic response by \ftting the proportional dependence of voltage V\non current Iover the range 2 mA \u0014I\u001410 mA for each sample. Fig. 4 summarizes the\ntotal conductance Gtotal= 1=Rtotalas a function of W thickness tWfor all Py(5 nm)/W( tW)\nheterostructures. Solid lines represent linear \fts for the W resistivity \u001aW, assumed constant\nas a function of W thickness for \\ \u000b\"-W and \\\f\"-W samples. The extracted resistivity for\n\\\u000b\" phase W \u001a\u000bis found to be\u001835\u0016\ncm and for \\ \f\" phase W \u001a\f\u0018148\u0016\ncm. The\nresistivity for \\ \f\"-W more than four times greater than that for \\ \u000b\"-W, is due in large part\nto the much smaller grain size for \f-W and is typically observed in prior studies25. Here the\nresistivity for \\ \u000b-W\" is larger by a factor of 2{3 than \flms deposited at room temperature\nand postannealed in previous work26, also attributable to a smaller grain size in these \flms\ndeposited at ambient temperature. The resistivity measurements for these thin \flms might\nbe taken as indirect evidence for the presence of the \fphase in the nominal \\ \f\"-W layers.\nIV. FERROMAGNETIC RESONANCE MEASUREMENTS\nThe four thickness series of Py( tPy)/W(tW) \flms, for \\ \u000b\"-W and \\\f\"-W, as described\nin Section II were characterized using variable-frequency \feld-swept FMR using a coplanar\nwaveguide (CPW) with center conductor width of 300 \u0016m. The bias magnetic \feld was\napplied in the \flm plane ( pc-FMR, or parallel condition). For details, see e.g., our prior\nwork in Ref. [20].\nFig. 5 summarizes half-power FMR linewidth \u0001 H1=2as a function of frequency !=2\u0019\nfor Py(5 nm), Py(5 nm)/\\ \u000b\"-W(10 nm) and Py(5 nm)/\\ \f\"-W(10 nm) samples. The mea-\nsurements were taken at frequencies from 3 GHz to above 20 GHz. Solid lines are linear\nregression of the variable-frequency FMR linewidth \u0001 H1=2= \u0001H0+ 2\u000b!=\r , where \u0001H1=2\n8is the full-width at half-maximum, \u0001 H0is the inhomogeneous broadening, \u000bis the Gilbert\ndamping,!is the resonance frequency and \ris the gyromagnetic ratio. Good linear \fts\nwere obtained with resonance frequency !=2\u0019for experimental linewidths \u0001 H1=2(!) of all\nthe samples measured.\nFor the \frst sample thickness series Py(5 nm)/W( tW), we plot damping parameters \u000b\nextracted from the linear \fts, as a function of W thickness in Fig. 6. Standard deviation\nerrors in the \ft for \u000bare\u00182\u000210\u00004. The Gilbert damping \u000bsaturates quickly as a function\noftWfor both \\\u000b\"-W and \\\f\"-W, with almost all of the e\u000bect realized with the \frst 2 nm\nof W. Loosely speaking, this fast saturation implies a short spin di\u000busion length \u0015SD\u00142\nnm, so the identity of the W phase ( \u000bor\f) over this length scale near the interface is\nmost relevant. The averaged damping, \u000bPy=\\\u000b\"\u0000Wand\u000bPy=\\\f\"\u0000W, are shown as horizontal\ndashed lines in the \fgure. \u000bPy=\\\u000b\"\u0000Wis slightly smaller than \u000bPy=\\\f\"\u0000W, but this may be\nwithin experimental error. Due to spin pumping, the damping is enhanced with the addition\nof W layers \u0001 \u000b=\u000bPy=W\u0000\u000bPy, normalized to the Gilbert damping \u000bPyof the reference\nsample without W layers. The e\u000bective SMC g\"#\neffat the Py/W interfaces can be calculated\nfollowing:\n\u0001\u000b=\r\u0016hg\"#\neff\n(4\u0019MS)tPy(2)\nwhere\ris the gyromagnetic ratio, \u0016 his the reduced Planck constant, and 4 \u0019MS\u001910\nkG is the saturation inductance of Py. In this series of samples, the e\u000bective SMC at the\nPy/\\\u000b\"-W interface g\"#\nPy=\\\u000b\"\u0000W\u00197:2\u00060:3 nm\u00002and the e\u000bective SMC at the Py/\\ \f\"-\nW interface g\"#\nPy=\\\f\"\u0000W\u00197:4\u00060:2 nm\u00002. These values are signi\fcantly lower than those\nreported in Ref.8for CoFeB/W (20{30 nm\u00002), as measured by spin-torque FMR.\nFor the second sample thickness series Py( tPy)/W(10 nm), we plot the extracted Gilbert\ndamping\u000band damping enhancement \u0001 \u000b=\u000bPy=W\u0000\u000bPyas a function of Py thickness\nin Fig. 7. The enhanced damping is normalized to the Gilbert damping \u000bPyof reference\nsamples with the same Py thickness tPy. The result is in good agreement with the inverse\nthickness dependence of contributed damping predicted from Equation 2. The experimental\ndata is \ftted with Equation 2 to extract the e\u000bective SMC. In this series of samples, the\ne\u000bective SMC at the Py/\\ \u000b\"-W interface g\"#\nPy=\\\u000b\"\u0000W\u00196:7\u00060:1 nm\u00002and the e\u000bective SMC\nat the Py/\\ \f\"-W interface g\"#\nPy=\\\f\"\u0000W\u00197:4\u00060:3 nm\u00002.\nPrevious studies on W have shown that the formation of \u000b-W is preferred, for thicker\n9W layers (e.g. 10 nm)3,26. We also prepared the sample \\ \u000b\"-W(10 nm)/Py(5 nm) with\nreverse deposition order, with the same seed and cap layers, on an oxidized Si substrate.\nHere the top surface of the 10 nm thick \u000b-W layer is pure \u000bphase, as shown by XRD in Fig.\n1 a). We performed the same FMR measurement on the reverse order sample; its Gilbert\ndamping enhancement \u0001 \u000bis plotted as the green dot in Fig. 7. This point almost overlaps\nwith the measurement for the normal order sample Py(5 nm)/\\ \u000b\"-W(10 nm), indirectly\nsupporting the conclusion that the phase of the Py/\\ \u000b\"-W interface is similar to the phase\nof the \\\u000b\"-W/Py interface, i.e., almost 100% \u000bphase W. Note that it was not possible to\ndeposit a reverse-order \fphase sample because no \fphase W could be stabilized on Cu\nusing our technique10.\nThe FMR measurements of spin mixing conductance g\"#for Py/\\\u000b\"-W and Py/\\ \f\"-W\nare new in this study. We \fnd that the value is similar to that measured for Ta27(g\"#\u0018\n10 nm\u00002) regardless of the enriched phase. First-principles-based calculations including\nrelativistic e\u000bects28forg\"#at Py/NM interfaces have shown that Ta, next to W in the\nperiodic table, is a good spin sink due to its large spin-orbit coupling (SOC), but has a\nrelatively small g\"#\u00188{9 nm\u00002. The e\u000ecient absorption of spin current can be connected\nwith a large SOC from the large atomic number, and the low SMC can be connected to\nrelatively poor band matching across the Py/W interface, compared with that for Py/Cu\nor Py/Pt28. The conclusion for Ta is consistent with our experimental results for the Py/W\nsystem, i.e., the rapid saturation of Gilbert damping within the \frst 2 nm of W, indicating\nW is also a good spin sink, with a similarly low g\"#\u00187 nm\u00002.\nV. DISCUSSION\nWe have found very little di\u000berence between the spin scattering properties (spin mixing\nconductance and spin di\u000busion length) of \u000b-W and mixed phase ( \u000b+\f)-W. The simplest\ninterpretation is that both spin mixing conductances and spin di\u000busion lengths are nearly\nequal for the two phases. However, despite our development of an optimized technique9{11\nto stabilize the \fphase, our control over the amounts of deposited \u000band\fphases is less\nthan complete, particularly near the Py/W interface.\nThe \\\u000b\"-structure we deposited, Py/ \u000b-W, is nearly\u0018100%\u000bphase. We observed no\nstrong\f-W peaks in the XRD scans, and neither crystalline structure nor di\u000braction patterns\n10for the\fphase in HR-XTEM characterization. According to our previous work10,26,29,\nwe know that ionically and covalently bonded substrates/underlayers are favorable for the\nformation of some \f-W, whereas metallic underlayers promote \u000b, so on Py even at a thickness\nof 2 nm, the nominally \u000b-W \flm is fully \u000bif deposited in the absence of nitrogen.\nIn the thinnest \\ \f\"-structure which we can characterize by XRD, Py/\\ \f\"-W(10 nm),\nwe identify a roughly 50%-50% mixture of \u000band\fphases. If this balance persists at the\ninterface as well, the SMC cannot di\u000ber by more than 10-20% for the two phases. While\nthe measurement of the 5 nm region near the interface seems to show somewhat less than\n50%\fphase, there is still a substantial population of \f-W in this region, and it would seem\nthat a strong di\u000berence in SMC for \u000b-W and\f-W should be resolvable if present. Given\nthat the measured values are very similar, we conclude that the \u000band\fphases do not di\u000ber\nstrongly in this spin transport study.\nOne might ask why the spin mixing conductance, in contrast to the spin Hall angle3, does\nnot di\u000ber much for the two phases of W. The spin mixing conductance (SMC) g\"#\nFM=NMis a\nproperty of the FM/NM interface, rather than a bulk property of the NM layer. The SMC\nmay be approximated (in a single-band, free-electron model) as g\"#\u0019\u0014k2\nFA=4\u00192, wherekF\nis the Fermi wave number for the NM, \u0014represents the number of scattering channels in\nunits of one channel per interface atom, and A is the total surface area of the interface30.\nDespite the possibility that bulk \f-W has a stronger e\u000bective spin-orbit coupling and spin\nHall e\u000bect due to its A15 structure, \f-W could have similar numbers of conducting channels\nper atom at the FM/NM interface as \u000b-W, which could lead to the similar values of SMC\nmeasured here.\nAnother possibility is that the spin di\u000busion length \u0015SDmay vary along the W layer\nthickness, due to nonuniformly distributed \u000b-W and\f-W phases in \\ \f\"-W samples. If this\nis true, \ftting a single spin di\u000busion length for spin pumping into very thin W layers will\nbe problematic31. However, because we have observed a very rapid saturation of Gilbert\ndamping over the \frst 2 nm of W for both \\ \u000b\"-W (almost pure \u000bphase) and \\ \f\"-W (mixed\nphase) in Fig. 6, we can only assign an upper bound for \u0015SD, similarly short in the two\nphases.\n11VI. CONCLUSIONS\nIn summary, we report measurements of spin mixing conductances of Py/W \flms with\ncontrolled amounts of \u000band\fphase W, measured by Gilbert damping through ferromag-\nnetic resonance (FMR). We \fnd no strong di\u000berences in the spin mixing conductances of\nPy/\u000b-W and Py/ \f-W, measured as g\"#= 6.7{7.4 nm\u00002, although control of the \fphase is\nseen to be more di\u000ecult near the interface with Py. Our experimental results also indicate\nthat W, no matter of which phase, is a good spin sink, but with relatively small spin mixing\nconductance in Ni 81Fe19(Py)/W, similar to Ta in Py/Ta.\nVII. ACKNOWLEDGEMENTS\nThe authors thank Daniel Paley of Columbia Nano Initiative for the grazing incidence\nXRD scans and Kadir Sentosun of Columbia University for the satellite peak calculations.\nThis work is supported by the US NSF-DMR-1411160.\nNOTES\n1Focused ion-beam (FIB) and FEI Helios NanoLab 660 were used to prepare foils for TEM studies. To\nprotect the heterostructures against the ion-beam damage during sample preparation, amorphous Platinum\n(1.5\u0016mthick) was sputtered on the surface of the wafers by electron and ion beam, respectively. TEM and\nhigh-resolution cross-sectional TEM (HR-XTEM) analyses were performed by image Cs-corrected FEI Titan\nThemis 200 at an accelerating voltage of 200 kV. Nano-beam electron di\u000braction pattern (DP) technique and\nFourier transform (FT) analysis of the HRTEM have been utilized to identify the nature of each phase at\nthe scale of 1{2 nm wide. The nano-beam DPs were obtained by FEI Talos TEM operating at 200 kV. 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Toney, Journal of Vacuum Science & Technology A:\nVacuum, Surfaces, and Films 29, 051512 (2011).\n27S. Mizukami, Y. Ando, and T. Miyazaki, Journal of Magnetism and Magnetic Materials\n226-230 , 1640 (2001), proceedings of the International Conference on Magnetism (ICM\n2000).\n28Y. Liu, Z. Yuan, R. J. H. Wesselink, A. A. Starikov, and P. J. Kelly, Phys. Rev. Lett.\n113, 207202 (2014).\n29D. Choi, X. Liu, P. K. Schelling, K. R. Co\u000bey, and K. Barmak, Journal of Applied Physics\n115, 104308 (2014).\n30Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. Lett. 88, 117601 (2002).\n31E. Montoya, P. Omelchenko, C. Coutts, N. R. Lee-Hone, R. H ubner, D. Broun, B. Heinrich,\nand E. Girt, Phys. Rev. B 94, 054416 (2016).\n14FIGURES\n15FIG. 1. X-ray di\u000braction (XRD) measurements for Py(5 nm)/\\ \u000b\"-W(tW) (blue) and Py(5\nnm)/\\\f\"-W(tW) (red) deposited on glass substrates. (a) tW= 30 nm; (b) tW= 10 nm. Solid\nvertical lines show the calculated re\rections and intensities for \u000b-W and\f-W peaks. (c) Grazing-\nincidence XRD measurements for Py(5 nm)/\\ \u000b\"-W(10 nm) and Py(5 nm)/\\ \f\"-W(10 nm) samples.\nThe inset shows the \u000b(211) re\rections observed in both samples. The blue and the red dashed\nlines refer to the \fts for Py/\\ \u000b\"-W and Py/\\ \f\"-W, respectively. The black dashed line refers to\nthe identical quadratic background.\n16FIG. 2. (a) High-resolution cross-sectional transmission electron microscopy (HR-XTEM) image\nof SiO 2/Ta(5 nm)/Cu(5 nm)/Py(5 nm)/\\ \u000b\"-W(30 nm)/Cu(5 nm)/Ta(5 nm) heterostructure. The\n\u000b-W grains are columnar with lateral radius of 10{20 nm, with larger grain size in the growth\ndirection. (b) Selected-area di\u000braction (SAD) pattern of the heterostructure, showing the preferred\ntexture of\u000b-W grains on Py layer, f111gPy//f011g\u000b-W (see calculated pattern in the inset). No\nsign of\f-W was detected in this heterostructure.\n17FIG. 3. (a) HR-XTEM image of SiO 2/Ta(5 nm)/Cu(5 nm)/Py(5 nm)/\\ \f\"-W(30 nm)/Cu(5\nnm)/Ta(5 nm), showing mixed-phase \u000b-W and\f-W. Convergent nanobeam electron di\u000braction\n(CBED) patterns, bottom, reveal the co-existence of separated \u000b-W,\f-W, and fcc Py. (b) Close-\nup of one region near the Py/W interface in (a), with discrete spatial Fourier Transform (FT). The\nFT is consistent with a single-crystal pattern of (1 \u001611)[110]Py//(011)[1 \u001611]\u000b-W//(002)[200] \f-W, as\nshown in the calculated pattern (bottom right). (c) FT of interface region (dotted box), showing\nco-existence of \u000b-W and\f-W in the \frst 5 nm W adjacent to the Py/W interface.\n18FIG. 4. The total conductance Gtotal= 1=Rtotalas a function of W thickness. Blue dots refer\nto Py(5 nm)/\\ \u000b\"-W(tW) samples and red dots refer to Py(5 nm)/\\ \f\"-W(tW) samples. The solid\nlines are linear \fts.\n19FIG. 5. Half-power FMR linewidth \u0001 H1=2spectra of reference sample Py(5 nm) (black), Py(5\nnm)/\\\u000b\"-W(10 nm) (blue) and Py(5 nm)/\\ \f\"-W(10 nm) (red) samples. The solid lines are linear\n\fts.\n20FIG. 6. Gilbert damping \u000bof the reference sample Py(5 nm) (black), Py(5 nm)/\\ \u000b\"-W(tW) (blue)\nand Py(5 nm)/\\ \f\"-W(tW) (red) samples. The blue and red dash lines refer to averaged enhanced\ndamping for Py(5 nm)/\\ \u000b\"-W(tW) and Py(5 nm)/\\ \f\"-W(tW), respectively.\n21FIG. 7. Damping enhancement \u0001 \u000b=\u000bPy=W\u0000\u000bPyof Py(tPy)/\\\u000b\"-W(10 nm) (blue), Py( tPy)/\\\f\"-\nW(10 nm) (red) and \\ \u000b\"-W(10 nm)/Py(5 nm) (green) samples, normalized to the Gilbert damping\nof reference samples \u000bPywith the same Py thickness. Solid lines refer to \ftting with Equation 2.\nInset: Gilbert damping \u000bof the reference sample Py( tPy) (black), Py( tPy)/\\\u000b\"-W(10 nm) (blue),\nPy(tPy)/\\\f\"-W(10 nm) (red) and \\ \u000b\"-W(10 nm)/Py(5 nm) (green) samples.\n22"    },    {        "title": "2401.00486v1.Molecular_Hybridization_Induced_Antidamping_and_Sizable_Enhanced_Spin_to_Charge_Conversion_in_Co20Fe60B20__β__W_C60_Heterostructures.pdf",        "content": "Molecular Hybridization Induced Antidamping and Sizable Enhanced Spin-to-Charge\nConversion in Co 20Fe60B20/β-W/C 60Heterostructures\nAntarjami Sahoo 1, Aritra Mukhopadhyaya 2, Swayang Priya Mahanta 1, Md. Ehesan Ali 2, Subhankar Bedanta 1,3\n1 Laboratory for Nanomagnetism and Magnetic Materials (LNMM),\nSchool of Physical Sciences, National Institute of Science Education and Research (NISER),\nAn OCC of Homi Bhabha National Institute (HBNI), Jatni 752050, Odisha, India\n2 Institute of Nano Science and Technology, Knowledge City, Sector-81, Mohali, Punjab 140306, India and\n3 Center for Interdisciplinary Sciences (CIS), National Institute of Science Education and Research (NISER),\nAn OCC of Homi Bhabha National Institute (HBNI), Jatni, Odisha 752050, India\nMd. Ehesan Ali∗and Subhankar Bedanta†\nDevelopment of power efficient spintronics devices has been the compelling need in the post-CMOS\ntechnology era. The effective tunability of spin-orbit-coupling (SOC) in bulk and at the interfaces of\nhybrid materials stacking is a prerequisite for scaling down the dimension and power consumption of\nthese devices. In this work, we demonstrate the strong chemisorption of C 60molecules when grown\non the high SOC β-W layer. The parent CFB/ β-W bilayer exhibits large spin-to-charge intercon-\nversion efficiency, which can be ascribed to the interfacial SOC observed at the Ferromagnet/Heavy\nmetal interface. Further, the adsorption of C 60molecules on β-W reduces the effective Gilbert\ndamping by ∼15% in the CFB/ β-W/C 60heterostructures. The anti-damping is accompanied by a\ngigantic ∼115% enhancement in the spin-pumping induced output voltage owing to the molecular\nhybridization. The non-collinear Density Functional Theory calculations confirm the long-range en-\nhancement of SOC of β-W upon the chemisorption of C 60molecules, which in turn can also enhance\nthe SOC at the CFB/ β-W interface in CFB/ β-W/C 60heterostructures. The combined amplifica-\ntion of bulk as well interfacial SOC upon molecular hybridization stabilizes the anti-damping and\nenhanced spin-to-charge conversion, which can pave the way for the fabrication of power efficient\nspintronics devices.\nI. INTRODUCTION\nSpintronic logic and memory devices have proven to\nbe one of the most suitable research domains to meet\nthe ultra-low power consumption demand in the post-\nComplementary Metal Oxide Semiconductor (CMOS)\ntechnology era. Especially, with the advent of artificial\nintelligence and the Internet of Things (IoT), the further\nscaling down of CMOS technology can reach its physi-\ncal limits in size, speed, and static energy consumption.\nThe conceptualized spin orbit torque magnetic random\naccess memory (SOT-MRAM) devices which take the ad-\nvantage of spin Hall effect (SHE) can bring down the\nenergy consumption to femto Joule from the pico Joule\nscale [1, 2]. The SHE based magnetization switching\nmechanism in SOT-MRAMs also offers much improved\nendurance owing to the separation in data writing and\nreading paths. Though these potentials of SOT-MRAMs\nhave attracted major foundries, several challenges need\nto be addressed before the commercialization of SOT-\nMRAMs [1, 2]. The increase of writing efficiency to re-\nduce power consumption is one of those aspects which\nrequires significant consideration. In this context, the\nspin Hall angle, θSH(JS⁄JC) of the nonmagnetic layer\npresent in the SOT-MRAMs, where J Cand J Sare the\ncharge and spin current densities, respectively, plays a\n∗ehesan.ali@inst.ac.in\n†sbedanta@niser.ac.incritical role in determining the writing efficiency [3]. The\nefficient charge to spin interconversion can lead to the\nfaster switching of magnetization of the adjacent mag-\nnetic layer via SHE. Hence, various types of heavy metals\n(HMs), like Pt, Ta, W, Ir etc. have been investigated in\nthe past two decades to reduce the power consumption\nof future spintronic devices [4–6]. On a similar note, the\nRashba-Edelstein effect (REE) occurring at the interfaces\nwith spatial inversion symmetry breaking and high spin\norbit coupling (SOC) has also the potential for the man-\nifestation of efficient charge to spin interconversion[7–9].\nHence, the combination of SHE and REE can be the most\nsuitable alternative for the development of power efficient\nspintronics application.\nAmong all the heavy metals, highly resistive ( ρβ−W∼\n100−300µΩ cm) metastable β-W possesses the largest\nθSH∼-0.3 to -0.4 [10–14], which makes it a strong can-\ndidate for SOT-MRAM devices. Usually, additional re-\nactive gases, like O 2, N2, and F are employed to stabilize\nthe A15 crystal structure of β-W [11] and consequently, a\nlarger θSHis realized. For example, Demasius et al., have\nbeen able to achieve θSH∼-0.5 by incorporating the oxy-\ngen into the tungsten thin films [12]. Interface engineer-\ning also acts as a powerful tool for enhancing the writing\nefficiency in β-W based SOT-MRAM devices [15–17]. For\ninstance, the presence of an interfacial atomically thin\nα-W layer in CoFeB/ α-W/β-W trilayer suppresses the\nspin backflow current, resulting in a 45% increase in the\nspin mixing conductance [15]. Further, the REE evolved\nat the W/Pt interface owing to the charge accumulationarXiv:2401.00486v1  [cond-mat.mtrl-sci]  31 Dec 20232\ngenerates an additional spin orbit field on the adjacent\nferromagnet (FM) NiFe (Py) layer [18]. The coexistence\nof SHE and REE has also been reported in CoFeB/ β-Ta\nand NiFe/Pt bilayers, where the interfacial SOC arising\nat the FM/HM interface plays a vital role in the spin-\nto-charge interconversion phenomena [19, 20]. More in-\nterestingly, a recent theoretical work has predicted the\ninterfacial SOC mediated spin Hall angle of Pt can be 25\ntimes larger than the bulk value in NiFe/Pt heterostruc-\nture [21]. The interfacial SOC mediated spin accumula-\ntion has also been reported to occur at the Rashba-like\nβ-Ta/Py interface without flowing the DC current [22].\nThe spin pumping induced by the ferromagnetic reso-\nnance results in non-equilibrium spin accumulation at the\ninterface which consequently reduces the effective Gilbert\ndamping of the β-Ta/Py bilayer. The reduction in ef-\nfective damping, also termed as antidamping, is similar\nto the interfacial Rashba like SOT, observed in various\nHM/FM heterostructures [22]. The anti-damping phe-\nnomena without the requirement of DC current depends\non several factors, like SOC of HM, strength of built in\nelectric field at the interface, interface quality etc. Hence,\nthe interface engineering via tuning the interfacial SOC\ninβ-W based HM/FM heterostructures can be the path\nforward for developing power efficient SOT-MRAM de-\nvices.\nTill the date, most of the interface engineering re-\nsearch have been focused on employing an additional\nmetallic or oxide layer in the HM/FM system for the\nenhancement of spin-to-charge interconversion efficiency.\nWhereas, the organic semiconductors (OSCs) can also\nbe incorporated in the HM/FM system to fabricate hy-\nbrid power efficient spintronic devices owing to their\nstrong interfacial hybridization and charge transfer na-\nture at metal/OSC interface [23]. Recently, the SOC\nof Pt has been found to be enhanced due to the on-\nsurface physical adsorption of C 60(fullerene) molecules\nin YIG/Pt/C 60trilayer [24]. However, the θSHof Pt is\nusually found to be smaller compared to β-W and it is\nimportant to investigate the magnetization dynamics and\nspin to charge conversion phenomena in FM/ β-W/C 60\nheterostructures. Hence, in this article, we report the\neffect of molecular hybridization at β-W/C 60interface\non magnetization dynamics and spin-to-charge conver-\nsion phenomena in Co 20Fe60B20(CFB)/ β-W/C 60het-\nerostructures. The molecular hybridization reduces the\neffective Gilbert damping and also enhances the spin-to-\ncharge conversion efficiency owing to the enhanced SOC\nofβ-W and consequent strengthening of possible Rashba-\nlike interaction at the CFB/ β-W interface. The strong\nchemisorption at the β-W/C 60interface and evolution of\nenhanced SOC of β-W upon the molecular hybridization\nhave also been confirmed by the first principle density\nfunctional theory (DFT) based calculations.II. EXPERIMENTAL AND COMPUTATIONAL\nMETHODS\nFour different types of heterostructures with CFB (7\nnm)/ β-W (2.5, 5 nm) (Figure 1 (a)) and CFB (7 nm)/ β-\nW (2.5, 5 nm)/C 60(13 nm) (Figure 1 (b)) stackings were\nfabricated on Si/SiO 2(300 nm) substrates for the inves-\ntigation of magnetization dynamics and spin pumping\nphenomena. In addition, the CFB (7 nm)/ β-W (10, 13\nnm) heterostructures were also fabricated to reaffirm the\nstabilization of β-W. The heterostructure stackings and\ntheir nomenclatures are mentioned in Table I. The CFB\nandβ-W layers were grown by DC magnetron sputter-\ning, while the Effusion cell equipped in a separate cham-\nber (Manufactured by EXCEL Instruments, India) was\nused for the growth of the C 60over layers in the CFWC\nseries. While preparing the CFWC1 and CFWC2, the\nsamples were transferred in-situ into the chamber with\nEffusion cell in a vacuum of ∼10−8mbar for the de-\nposition of C 60. Before the fabrication of heterostruc-\ntures, thin films of CFB, β-W, and C 60were prepared for\nthickness calibration and study of magnetic and electrical\nproperties. The base pressure of the sputtering chamber\nand chamber with Effusion cells were usually maintained\nat∼4×10−8mbar and ∼6×10−9mbar, respectively.\nThe structural characterizations of individual thin films\nand heterostructures were performed by x-ray diffrac-\ntion (XRD), x-ray reflectivity (XRR), and Raman spec-\ntrometer. The magneto-optic Kerr effect (MOKE) based\nmicroscope and superconducting quantum interference\ndevice based vibrating sample magnetometer (SQUID-\nVSM) were employed for the static magnetization char-\nacterization and magnetic domain imaging. The mag-\nnetization dynamics was investigated by a lock-in based\nferromagnetic resonance (FMR) spectrometer manufac-\ntured by NanOsc, Sweden. The heterostructures were\nkept in a flip-chip manner on the co-planner waveguide\n(CPW). The FMR spectra were recorded in the 4-17 GHz\nrange for all the samples. The FMR spectrometer set-up\nis also equipped with an additional nano voltmeter using\nwhich spin-to-charge conversion phenomena of all the de-\nvices were measured via inverse spin Hall effect (ISHE)\nwith 5-22 dBm RF power. The contacts were given at\nthe two opposite ends of 3 mm ×2 mm devices using\nsilver paste to measure the ISHE induced voltage drop\nacross the samples. The details of the ISHE measure-\nment set-up are mentioned elsewhere [25, 26].\nDensity functional theory (DFT)-based electronic\nstructure calculations were performed in the Vienna Ab-\ninitio simulation package (VASP) [27, 28] to understand\nthe interface’s chemical bonding and surface reconstruc-\ntions. The plane wave basis sets expand the valance\nelectronic states, and the core electrons are treated\nwith the pseudopotentials. The core-valance interac-\ntions are considered with the Projected Augmented Wave\nmethod. The exchange-correlation potentials are treated\nwith Perdew, Bruke and Ernzerof (PBE) [29] functional\nwhich inherits the Generalized Gradient Approximation3\nTABLE I. Details of the heterostructures and their nomenclatures\nSl. No. Stacking Nomenclature\n1 Si/SiO 2(300 nm)/CFB (7 nm)/ β-W (2.5 nm) CFW1\n2 Si/SiO 2(300 nm)/CFB (7 nm)/ β-W (5 nm) CFW2\n3 Si/SiO 2(300 nm)/CFB (7 nm)/ β-W (2.5 nm)/C 60(13 nm) CFWC1\n4 Si/SiO 2(300 nm)/CFB (7 nm)/ β-W (5 nm)/C 60(13 nm) CFWC2\n(GGA). This functional produces a reliable understand-\ning of similar kinds of interfaces. The convergences in the\nself-consistent field iterations were ensured with a plane-\nwave cutoff energy of 500 eV and a tolerance of 10−6\neV/cycle. A D3 dispersion correction term, devised by\nGrimme, accounts for the long-range interaction terms\nwas employed in the calculations. The optimized unit\ncell parameter obtained from the aforementioned meth-\nods for the cubic A15 crystal of the β-W is 5.014 ˚A,\nwhich resembles the experimental parameter of 5.036 ˚A.\nA 5×2×1 repetition is used to construct the (210) surface\nunit cell of the β-W to model the surface supercell. The\nlower two atomic layers were fixed at the bulk, and the\nremaining three layers were allowed to relax during the\ngeometry optimization. The surface layer of β-W con-\ntains the C 60molecules. To understand the effect of the\nspin-orbit coupling interactions, we have performed the\nnon-collinear DFT calculations as implemented in VASP.\nThe E SOC calculated from these calculations quantifies\nthe strength of the SOC term in the Hamiltonian.\nIII. RESULTS AND DISCUSSION\nThe grazing incidence x-ray diffraction (GIXRD) was\nperformed for all the heterostructures. The XRD pat-\nterns of CFB/ β-W heterostructures with different thick-\nnesses of β-W are shown in Figure 1 (c). The presence of\n(200), (210) and (211) Bragg’s peaks of W at 35.5◦, 39.8◦\nand 43.5◦indicate the stabilization of metastable βphase\nof W (A-15 crystal structure) [5, 30]. In addition, we\nhave also observed the (320) and (321) Bragg’s peaks of\nW, which further suggests the growth of polycrystalline\nβ-W. The relative intensity of (320) and (321) Bragg’s\npeaks of W is lower compared to (200), (210) and (211)\nBragg’s peaks, consistent with previous reports [30]. The\nBragg’s peaks are more prominent for heterostructures\nwith thicker W layers as diffraction intensity increases\nwith the increase in W thickness. The XRD patterns for\nCFWC1 and CFWC2 are similar to that for CFW1 and\nCFW2, respectively, as the thickness of β-W are same.\nHere, we have not used the reactive gases like, O 2and\nN2for the growth of β-W unlike some previous report\n[11]. The resistivity of W films with thicknesses 2.5, 5,\n10 nm were measured by standard four probe methods.\nThe resistivity decreases with increase in thickness of W\nand were found in between ∼300-100 µΩ-cm, further\nconfirming the growth of βphase of W [5, 30]. We donot also observe the (110), (200), (210) Bragg’s peaks for\nthe bcc α-W in the XRD patterns and the α-W would\nhave also exhibited one order less resistivity compared to\nwhat we have observed [30]. The stabilization of pure β\nphase of W is quite important for future SOT device fab-\nrication and hence, we can expect a high spin-to-charge\nconversion efficiency in our CFB/ β-W heterostructures\nowing to high SOC of β-W [10, 30].\nThe XRR measurements were performed for all the\nsamples in both the CFW and CFWC series to confirm\nthe desired thickness of individual layers and to investi-\ngate the interface quality. Figure S1 (Supporting Infor-\nmation) shows the XRR patterns of all the heterostruc-\ntures considered for the present study. The experimental\ndata were fitted using GenX software and the simulated\npatterns are also shown in Figure S1 (red curves). The\npresence of Kiessig oscillations for all the films infer the\nabsence of a high degree of interfacial disorder and dis-\nlocations. The relative peak positions and intensity of\nthe simulated patterns agree quite well with the experi-\nmentally observed low angel XRR data. The fit provides\nthe anticipated thickness of individual layer in each sam-\nple as mentioned in Table I. The interface roughness for\nall the heterostructures were found in between 0.2-0.5\nnm, further inferring the high quality growth of both the\nseries of samples. Figure S2 (Supporting Information)\ndisplays the Raman spectra of 13 nm C 60film grown on\nSi/SiO 2(300 nm) substrate with the same growth con-\ndition as in the heterostructures. The presence of A g(2)\nand H g(8) Raman modes of C 60around ∼1460 cm−1\nand 1566 cm−1, respectively confirms the growth of C 60\nfilm [31, 32]. In addition, the Raman mode around ∼\n495 cm−1corresponding to A g(1) mode of C 60is also ob-\nserved in the Raman spectrum. The anticipated thick-\nness of C 60in the C 60thin film, CFWC1 and CFWC2\nhas also been confirmed from the XRR measurements.\nThe Raman spectrum of our C 60film grown by effusion\ncells are quite similar to those prepared by different so-\nlution methods in HCl or N 2atmosphere [31, 32]. The\nsaturation magnetization and the magnetic domain im-\nages of all the heterostructures are found to be similar\n(see Supporting Information) as the bottom CFB layer\nis same for all the heterostructures.\nThe magnetization relaxation and propagation of spin\nangular momentum in the CFB thin film and the het-\nerostructures in both the CFW and CFWC series were\nstudied to explore the effect of high resistive β-W and\nβ-W/C 60bilayer by in-plane FMR technique. The het-4\nFIG. 1. Schematics of (a) Si/SiO 2/CFB/ β-W and (b) Si/SiO 2/CFB/ β-W/C 60heterostructures, (c) GIXRD patterns of\nSi/SiO 2/CFB/ β-W heterostructures with various thicknesses of β-W.\nerostructures are placed in a flip-chip manner on CPW\nas shown in the schematics in Figure S4 (a) (Support-\ning Information). Figure S4 (c) shows the typical FMR\nspectra of CFW1 and CFWC1 heterostructures measuredin the 4-17 GHz range. All the FMR spectra were fit-\nted to the derivative of symmetric and antisymmetric\nLorentzian function to evaluate the resonance field ( Hres)\nand linewidth (∆ H) [33]:\nFMRSignal =K14(∆H)(H−Hres)\n[(∆H)2+ 4(H−Hres)2]2−K2(∆H)2−4(H−Hres)2\n[(∆H)2+ 4(H−Hres)2]2+Offset, (1)\nwhere K 1and K 2are the antisymmetric and symmetric\nabsorption coefficients, respectively. The extracted Hres\nand ∆ Hvalues at different resonance frequencies ( f) of\nall the heterostructures are shown in Figure 2 (a-b). The\nfvsHresof different samples in the CFW and CFWC\nseries are plotted in Figure 2 (a). The fvsHresplots\nare fitted by using equation 2 [33]:\nf=γ\n2πq\n(HK+Hres)(HK+Hres+ 4πMeff),(2)\nwhere\n4πMeff= 4πMS+2KS\nMStFM\nand H K, KS, and t FMare the anisotropy field, perpen-\ndicular surface anisotropy constant, and the thicknessof FM, respectively. Here, γis the gyromagnetic ra-\ntio and 4 πMeffrepresents the effective magnetization.\nThe 4 πMeffextracted from the fitting gives similar val-\nues as compared with the saturation magnetization value\n(4πMS) calculated from the SQUID-VSM. Further, the\neffective Gilbert damping constant ( αeff) and hence, the\nmagnetization relaxation mechanism are studied from\nthe resonance frequency dependent FMR linewidth be-\nhavior. The ∆ Hvsfplots are shown in Figure 2 (b).\nThe linear dependency of ∆ Honfindicates the mag-\nnetic damping is mainly governed by intrinsic mechanism\nvia electron-magnon scattering rather than the extrinsic\ntwo magnon scattering. The ∆ Hvsfplots are fitted by5\nFIG. 2. (a) Frequency ( f) versus resonance field ( Hres) and (b) linewidth (∆ H) versus frequency ( f) behaviour for various\nheterostructures. The solid lines are the best fits to equation 2 and 3.\nthe following linear equation [33] to evaluate the αeff.\n∆H= ∆H0+4παeff\nγf, (3)\nwhere the ∆ H0is the inhomogeneous linewidth broad-\nening. The αeffvalues for all the heterostructures and\nCFB thin film obtained from the fitting are shown in Ta-\nble II. The αeffvalue for CFW series ( ∼0.0075 ±0.0001\nfor CFW1 and ∼0.0080 ±0.0001 for CFW2) are found\nto be larger compared to that of the CFB thin film\n(∼0.0059 ±0.0001). The enhancement of αeffindicates\nthe possible evolution of spin pumping mechanism in the\nCFB/ β-W bilayers. Interestingly, the αeffdecreases to\n∼0.0065 ±0.0001 upon the deposition of C 60molecules\non CFB/ β-W bilayers in CFWC series. The signifi-\ncant change in αefffor the CFB/ β-W/C 60heterostruc-\ntures compared to CFB/ β-W bilayers infers the modi-\nfication of physical properties of β-W layer in CFB/ β-\nW/C 60. The deposition of C 60molecules can lead to the\nmetal/molecule hybridization at the β-W/C 60interface,\nwhich in turn can alter the properties of β-W.\nThe DFT based first principle calculations were per-\nformed to elucidate further the molecular hybridization\nat the β-W/C 60interface and its consequences on the\nmagnetization dynamics of CFB/ β-W/C 60heterostruc-\ntures. The extended simulation supercell for the C 60on\nβ-W(210) are shown in Figure 3 (a). The C 60molecule\nis observed as strongly chemisorbed onto the β-W (210)\nsurface with an adsorption energy of -253.5 kcal/mol.\nThe adsorption energy is quite high as compared to the\nother substrates. For example, the adsorption energy for\nCo/C 60was found to be -90 kcal/mol [34] while for the\nPt/C 60interface it is reported to be -115 kcal/mol [35].\nThe chemisorption in case of β-W/C 60is quite strong\nand induces distortion to the spherical shape of the ad-\nsorbed C 60. The distance between two carbon atoms\nfrom two opposite hexagons of adsorbed C 60is shorter\nalong one direction compared to the other measured in\nthe plane (left panel of Figure 3 (a)). The diameter of C 60molecules decreases by 0.3 ˚Awhen it is measured perpen-\ndicular to the β-W (210) surface (right panel of Figure\n3 (a)). This distortion can be attributed to the W-C\nbond formation due to the strong chemisorption at the\nβ-W/C 60interface. This chemisorption strongly alters\nthe electronic structure of the β-W and C 60molecule\n(Figure 3 (b)). The pzorbital, which accommodates the\nπ-electrons of the C 60, hybridizes with the d-orbitals of\ntheβ-W atom and forms the hybridised interfacial states.\nThe out-of-plane d-orbitals ( dxz,dyzanddz2orbitals) are\nstrongly hybridized with the pzorbital of the carbon\natom over a large energy window near the Fermi energy\nlevel (Figure 3 and Figure S5 (Supporting Information)).\nThe sharp peaks observed in the DOS of free C 60layer\ngets significantly broadened, flattened, and shifted for β-\nW/C 60stacking. The strong metallo-organic hybridiza-\ntion also modifies the PDOS of various d-orbitals of β-\nW. The various d-orbitals become flattened and spread\nover larger energy spectrum around the Fermi level upon\nmolecular hybridization. The formation of the W-C bond\nalso costs a transfer of 3.25e−from the interfacial layer of\ntheβ-W to C 60molecule (Figure 3 (c)). This is relatively\nhigher compared to the previously reported the 0.25e−\ntransfer from Pt (111) and 3e−transfer from Cu (111) to\nthe adjacent C 60molecule, inferring the metallo-organic\nhybridization is quite stronger in case of β-W/C 60in-\nterface [35].Hence, the molecular hybridization of β-W\nis expected to alter its physical properties with greater\neffect and can be considered as an important tool to op-\ntimize the spintronics device performances.\nThe modified electronic structure was found to carry\na long-range effect on the strength of the spin-orbit cou-\npling. The E SOC of bare 2.5 nm β-W and 2.5 nm β-\nW covered with C 60molecules, and the variation of the\nESOC(∆E SOC) due to β-W/C 60hybridization are shown\nin Figure 4. The interfacial W atoms involved in the hy-\nbridization with C 60show a decrease in the E SOC. The\nrest of the W atoms from the surface layer exhibit an\nincrease in the E SOC. The lower atomic layers of W6\nFIG. 3. (a) The extended simulation supercell for the C 60onβ-W(210) substrate. The left panel shows the top view of the\nsurface supercell (along the z-axis), and the right panel shows the side view of the same. The pink balls of larger size and cyan\nballs of smaller size represent the tungsten and carbon atoms, respectively. The yellow bonds highlight the part of the C 60\nwhich takes part in the interface formation. The double-headed dotted arrows quantify the diameter of the C 60spheres in two\ndirections. (b-c) The modification in the electronic structure due to chemisorption of the C 60molecule on the β-W. (b) The\natom projected orbital resolved partial density of states of β-W(210), C 60, and β-W(210)/C 60, and (c) The electron density\nredistribution due to chemisorption. The red and green iso-surfaces depict electron density depletion and accumulation of the\nelectron density at the interface, respectively. The bi-coloured arrow depicts the direction of the electron transfer process.\nalso show an increment in the E SOC. The W layer, far-\nthest from the β-W/C 60interface (nearer to the CFB/ β-\nW interface), exhibits the most increased E SOC. Hence,\nthe hybridization at the β-W/C 60interface increases the\noverall spin-orbit coupling strength of the β-W layer.\nMore importantly, the SOC at the CFB/ β-W interface\nis enhanced for CFB/ β-W/C 60stacking compared to the\nCFB/ β-W bilayer. The enhanced bulk SOC of β-W and\nthe interfacial SOC at CFB/ β-W interface can facilitate\nan efficient spin to charge conversion in CFB/ β-W/C 60\nheterostructures.\nThe decrease in damping, usually know as anti-\ndamping, has been observed previously in FM/HM bilay-ers [22, 26, 30]. In those systems, the effective damping\nvalues become lower than the αeffof the FM layer and\nthis phenomenon has been attributed to the formation\nof Rashba like interfacial states [22, 30]. Similar type of\nevolution of Rashba like states at the CFB/ β-W inter-\nface can be expected due to structural inversion asym-\nmetry and large SOC of β-W. The spin accumulation\nat the CFB/ β-W interface can lead to evolution of the\nnon-equilibrium spin states. The non-equilibrium spin\nstates along with the enhanced SOC at CFB/ β-W inter-\nface due to molecular hybridization as confirmed from the\nDFT calculations can generate an additional charge cur-\nrent due to IREE and can also induce the antidamping7\nFIG. 4. The effect of the chemisorption of the C 60molecule at the β-W(210) surface on the E SOCof various atomic sites. The\npercentage change in the E SOC (∆E SOC) is calculated in terms of the change in the E SOC of the bare β-W(210) substrate.\nLayer 5 is the interfacial layer that interacts with the C 60, and layer 1 is the opposite to the β-W/C 60interface layer.\ntorque on the magnetization of FM layer. The antidamp-\ning torque can make the magnetization precession rela-\ntively slower and thus decreasing the αeffof the CFB/ β-\nW/C 60heterostructures compared to the CFB/ β-W bi-\nlayer. The control of Gilbert damping of FMs by inter-\nfacing with adjacent non-magnetic metal/organic bilay-\ners can also provide an alternative to the search for low\ndamping magnetic materials. Especially, the low cost and\nabundant availability of carbon based organic molecules\ncan be commercially beneficial in optimizing the mag-\nnetic damping for spintronic applications. Further, the\nGilbert damping modulation can also control the effec-\ntive spin mixing conductance ( g(↑↓)\neff) of the heterostruc-\ntures which also plays a vital role for efficient spin current\ntransport across the interface. Hence, the g(↑↓)\neffof all the\nheterostructures was calculated from the damping con-\nstant measurement by equation 4 [33]:\ng(↑↓)\neff=4πMstCFB\ngµB(αCFB/NM −αCFB), (4)\nwhere g, µBandtCFB are the Land´ e g factor (2.1),\nBohr’s magnetron, and thickness of CFB layer, respec-\ntively. αCFB/NM is the damping constant of bilayer ortri-layers and αCFB is the damping constant of the ref-\nerence CFB thin film. The g(↑↓)\nefffor CFW1 and CFW2\n(Table II) are relatively higher compared to the previ-\nous reports on FM/ β-W bilayers. Especially, the g(↑↓)\neffof\nCFW2 is one order higher than that reported for Py/ β-W\nbilayer (1.63 ×1018m−2) [30], and 2 order higher com-\npared to that of the YIG/ β-W (5.98 ×1017m−2) [14].\nThis indicates the absence of any significant amount of\nspin back flow from β-W layer and high SOC strength\nof parent β-W layer in our system. However, the g(↑↓)\neff\nvalues decrease for the CFWC1 and CFWC2 tri-layers\nowing to anti-damping phenomena.\nThe ISHE measurements were performed for all the\nheterostructures in CFW and CFWC series to gain more\ninsights about the effect of molecular hybridization in\nCFB/ β-W/C 60on the magnetization dynamics and spin\nto charge conversion efficiency. Figure 5 shows the typi-\ncal field dependent DC voltage ( Vdc) measured across the\nCFB (7 nm)/ β-W (5 nm)/C 60(13 nm) heterostructure\nunder FMR conditions. In order to separate the symmet-\nric (VSY M) and asymmetric ( VASY M ) components, the\nVdcvsHplots were fitted with the following Lorentzian\nfunction:\nVdc=VSY M(∆H)2\n(∆H)2+ (H−Hres)2+VASY M(∆H)(H−Hres)\n(∆H)2+ (H−Hres)2(5)\nThe extracted field dependent VSY M andVASY M are also plotted in Figure 5. Similar type of field depen-8\nFIG. 5. VMEAS ,VSY M andVASY M versus Hfor CFB (7)/ β-W(5)/C 60(13) [CFWC2] heterostructure with ϕ∼(a) 180◦and\n(b) 0◦measured at 15 dBm RF power. The red curve is Lorentzian fit with equation 5 to VdcvsHplot.\nTABLE II. Effective Gilbert damping, spin mixing conduc-\ntance, and symmetric component of measured DC voltage for\ndifferent heterostructures.\nHeterostructures αeff(±0.0001) g(↑↓)\neff(1019m−2)VSY M(µV)\nCFB 0.0059 - -\nCFW1 0.0075 0.87 1.08\nCFW2 0.0080 1.13 1.25\nCFWC1 0.0064 0.27 2.32\nCFWC2 0.0065 0.32 1.78\ndent VMEAS ,VSY M, and VASY M are also observed for\nother samples in both CFW and CFWC series. The\nVSY M is mainly contributed by the spin pumping voltage\n(VISHE ) and the spin rectification effects arising from\nthe anisotropic magnetoresistance (AMR) [ VAMR] [33].\nWhereas, the asymmetric component of the measured\nvoltage arises solely due to anomalous Hall effect and\nAMR [33]. The sign of VSY M is reversed when ϕ(angel\nbetween the perpendicular direction to the applied mag-\nnetic field ( H) and direction of voltage measurement) is\nchanged from 0◦to 180◦(Figure 5), confirming the pres-\nence of ISHE in our heterostructures. The field depen-\ndent VSY M for all the four heterostructures are plotted\nin Figure 6 (a-b). Interestingly, the VSY M value at the\nresonance field for CFB/ β-W/C 60trilayers is found to be\nincreased compared to that for CFB/ β-W bilayers. Theincrement is ∼115% for β-W thickness 2.5 nm, while it\nbecomes ∼20% for β-W thickness 5 nm. The gigantic\nenhancement of VSY M for CFB (7)/ β-W(2.5)/C 60(13)\ninfers the modification of SOC of β-W when capped with\norganic C 60molecules and the presence of an additional\nspin to charge conversion effect in the heterostructures.\nThe power dependent spin-to-charge conversion measure-\nments were also performed to further confirm the en-\nhancement of VSY M. The spin pumping induced voltage\nincreases linearly with the RF power as shown in Fig-\nure 6 (c) for both CFW1 and CFWC1. The VSY M at\ndifferent RF power is found to be increased for CFWC1\ncompared to CFW1, which further confirms the molec-\nular hybridization induced enhanced spin-to-charge con-\nversion. As the thickness, magnetic properties of bot-\ntom CFB layer is same for all the heterostructures, the\ncontribution of VAMR is expected to be same for CFB\n(7 nm)/ β-W(2.5 nm)/C 60(13 nm) and CFB (7 nm)/ β-\nW(2.5 nm). Hence, the sizable increase in the measured\nvoltage can be attributed to the enhanced SOC of β-W\ndue to molecular hybridization and additional charge cur-\nrent flowing at the CFB/ β-W interface due to IREE as\nshown in the Figure 6 (d). In order to understand the en-\nhanced spin-to-charge conversion phenomena further, we\nalso calculated the θSHof the heterostructures by using\nequations 6 and 7 [14, 33]:\nJs=g(↑↓)\neffγ2h2\nrfℏ[γ4πMs+p\n(γ4πMs)2+ 4ω2]\n8πα2\neff[(γ4πMs)2+ 4ω2]×(2e\nℏ), (6)9\nFIG. 6. VSY M versus applied magnetic field with ϕ∼180◦for (a) CFB (7)/ β-W(2.5) [CFW1] and CFB (7)/ β-W(2.5)/C 60\n(13) [CFWC1] and (b) CFB (7)/ β-W(5) [CFW2] and CFB (7)/ β-W(5)/C 60(13) [CFWC2] heterostructures measured at 15\ndBm RF power, (c) Power dependent VSY M for CFW1 and CFWC1 (The solid line is the linear fit), (d) Schematic showing\nthe spin-to-charge conversion phenomena in CFB/ β-W/C 60heterostructures.\nVISHE =wyLρNM\ntNMθSHλNMtanh(tNM\n2λNM)Js, (7)\nwhere the ρNMis the resistivity of the β-W measured by\nfour-probe technique and Lis the length of sample. The\nRF field ( hrf) and the width of the CPW transmission\nline ( wy) in our measurements are 0.5 Oe (at 15 dBm RF\npower) and 200 µm, respectively. The λNMfor the β-W\nhas been taken as ∼3 nm from the literature [36]. Angel\ndependent ISHE measurements were performed to sepa-\nrate the AMR contribution from the VSY M. The contri-\nbution of VAMR was found to be one order smaller com-\npared to V ISHE . For example, the VAMR and V ISHE for\nCFW2 heterostructure are found to be ∼0.15µV and ∼\n1.25µV, respectively (See Supporting Information). The\nρNMfor 5 nm β-W is found to be 250 µΩ cm. Hence, the\nθSHfor CFB (7 nm)/ β-W (5 nm) bilayer estimated using\nequations 6 and 7 is found to be ∼-0.6±0.01. A similar\ntype of calculation for CFB (7 nm)/ β-W (2.5 nm) bilayer\nestimates the θSHto be∼-0.67±0.01. The observed θSHvalue is larger compared to that reported in the literature\n[10–12]. The high SOC of our β-W and higher spin mix-\ning conductance could be responsible for this enhanced\nθSH. Further, the interfacial SOC at CFB/ β-W interface\ncan also induce an additive spin-to-charge conversion ef-\nfect, contributing to the enhancement of θSH. Such type\nof interfacial SOC mediated enhanced spin-to-charge con-\nversion has been reported previously for NiFe/Pt and\nCFB/ β-Ta [19, 20]. Here, it is important to note that\nit is difficult to disentangle the IREE and ISHE effect in\nthese type of FM/HM systems. On the other hand, the\ng(↑↓)\nefffor CFWC1 and CFWC2 decreases by 70 % due to\nthe anti-damping phenomena and hence, the reduction\ninJsaccording to equation 6. However, the VISHE for\nthe CFWC1 and CFWC2 are found to be larger than\nCFW1 and CFW2, respectively (Figure 6). This leads to\ntheθSHvalue >1, calculated using the equation 6 and10\n7 for CFB/ β-W/C 60heterostructures. This type of gi-\ngantic enhancement of θSHcannot be explained by mere\nbulk ISHE in β-W. The enhanced θSHcan be partly at-\ntributed to the enhanced bulk SOC of β-W upon molec-\nular hybridization as predicted by the DFT calculations.\nFurther, our DFT calculations also predict the enhance-\nment of SOC of β-W layer closer to the CFB/ β-W inter-\nface due to the molecular hybridization in the CFB/ β-\nW/C 60heterostructures. The larger interfacial SOC and\ninversion symmetry breaking at the CFB/ β-W interface\nmakes the scenario favorable for realizing an enhanced\ninterfacial charge current due to the IREE as depicted in\nFigure 6 (d). Hence, the combination of bulk and interfa-\ncial SOC enhancement owing to the strong chemisorption\nof C 60onβ-W can attribute to the sizable increase in the\nθSHin CFB/ β-W/C 60heterostructures.\nThe enhanced output DC voltage due to the spin\npumping upon the C 60deposition on β-W is also con-\nsistent with the reduced effective damping value as dis-\ncussed earlier. The enhanced SOC of β-W and the struc-\ntural inversion asymmetry at the CFB/ β-W interface\ncan stabilize the Rashba like states at FM/HM inter-\nface [19, 20]. The IREE mediated spin to charge con-\nversion has received considerable interest after it was\ndiscovered at the Ag/Bi interface [7]. Till the date,\nmost of the IREE effects have been experimentally re-\nalized at the all inorganic metal/metal, metal/oxide or\noxide/oxide interfaces [9]. Our experiments and theoret-\nical calculations show that the molecular hybridization\nat the HM/OSC interface can also help in strengthen-\ning the Rashba spin-orbit coupling at the FM/HM in-\nterface. The Rashba interaction leads to the spin split-\nting of bands, whose magnitude is dependent on the SOC\nstrength at the interface. Upon the molecular hybridiza-\ntion, the SOC strength of β-W is further enhanced. This\ncould have lead for a larger Rashba coefficient αRand\nhence, a relatively larger IREE at the FM/HM inter-\nface. The simultaneous observation of ISHE and IREE\nby engineering the HM interface with OSC can help in\nreducing the power consumption of future SOT-MRAM\ndevices. As the CFB/ β-W stacking is employed for fab-\nrication of spin Hall nano oscillators (SHNOs) [37], theincorporation organic molecules can also significantly en-\nhance their efficiency. Hence, the HM/C 60interface can\nreduce the power consumption for data storage as well as\nfacilitate in performing efficient spin logic operations.\nIV. CONCLUSION\nIn conclusion, we present that a strong interfacial SOC\ncan lead to the larger spin Hall angle in CFB/ β-W bi-\nlayer. The thermally evaporated organic C 60molecules\non CFB/ β-W bilayer leads to a strong chemisorption at\ntheβ-W/C 60interface. The experimental and theoreti-\ncal calculations confirm that the molecular hybridization\nenhances the bulk as well as interfacial SOC in CFB/ β-\nW/C 60heterostructures. The strengthening of techno-\nlogically important SOC manifests an anti-damping phe-\nnomena and gigantic ∼115% increase in spin-pumping\ninduced output voltage for CFB/ β-W/C 60stacking. The\ncontrol of magnetization dynamics and output efficiency\nin spintronics devices by the molecular hybridization can\nbe a viable alternative to the other interface engineering\nand surface alloying techniques. The stabilization of the\nanti-damping and enhanced spin-to-charge conversion by\ntuning the bulk as well interfacial SOC via employing\nthe cost effective, abundant organic molecule can pave\nthe way for the fabrication of next generation power effi-\ncient spintronics devices.\nV. ACKNOWLEDGEMENT\nWe acknowledge the Department of Atomic En-\nergy (DAE), the Department of Science and Technol-\nogy (DST) of the Government of India, and SERB\nproject CRG/2021/001245. A.S. acknowledges the DST-\nNational Postdoctoral Fellowship in Nano Science and\nTechnology. 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In\ruence of the disorder are studied in detail by use of\nlarge supercells with random atomic arrangement. It is found that disorder a\u000bects the magnon\nspectrum in vastly di\u000berent ways depending on the system. Speci\fcally, it is more pronounced in\nFe-Ni alloys compared to Fe-Co alloys. In particular, the magnon spectrum at room temperature\nin Permalloy (Fe 20Ni80) is found to be rather di\u000buse in a large energy interval while in Fe 75Co25it\nforms sharp branches. Fe-Co alloys are very interesting from a technological point of view due to\nthe combination of large Curie temperatures and very low calculated Gilbert damping of \u00180.0007\nat room temperature for Co concentrations around 20{30%.\nI. INTRODUCTION\nThere has been a growing interest in disordered\nmagnetic materials in the last few decades in the\nform of transition metal alloys and diluted magnetic\nsemiconductors1{12. A central motivation for many stud-\nies is the potential of these materials in spintronics and\nmagnonics applications. Magnon excitations are com-\nmonly studied experimentally using inelastic neutron\nscattering suitable for bulk systems such as Co13or spin\npolarized electron loss spectroscopy (SPEELS) for low\ndimensional magnets such as Co 8/Cu00114. Theoreti-\ncally, the simplest approach for calculating magnon spec-\ntrum for elements and compounds is through linear spin\nwave theory of the Heisenberg Hamiltonian. However,\nfor accurate studies of alloys, both the treatment of dis-\norder and thermal e\u000bects needs to be handled reliable.\nMagnons in disordered magnets, either random alloys or\ndiluted, are more complicated than for ordered systems\nfor a number of reasons. Due to broken translational\nsymmetry, perfect magnon modes with in\fnite life time\nas in ordered magnets are absent but at certain condi-\ntions, one may still expect well de\fned magnon modes\nbut with a \fnite lifetime due to disorder. The damping\nand formation of these modes are of great interest both\ntheoretically and for applications.\nPrevious studies of magnon properties of disordered\nmagnets have been focused on diluted magnets and the\ne\u000bect of dilution on the magnon spectrum and spin\nsti\u000bness15{18. The main \fndings from these studies are\nthat the region with well de\fned magnon modes are de-\ncreasing with dilution and properties are strongly dimen-\nsionality dependent. Surprisingly, there are only very few\npublished studies of magnons in random alloys with full\nconcentration of magnetic elements19{21, such as Fe-Co\nalloys22,23. The aim for the present study is to introduce\na simple methodology for theoretical studies of magnons\nin disordered materials. We are using this methodologyto investigate magnon and other \fnite temperature prop-\nerties, i.e. spin sti\u000bness, Cure temperatures and Gilbert\ndamping for bulk transition metal alloys that hopefully\nwill stimulate experiments in the new generation of neu-\ntron scattering facilities currently in construction.\nThe paper is organized as follows: In Section II we\nintroduce the methodology and give the details of the\ncalculations, in Section III we present our \fndings and\nin Section IV we give a summary and provide an outlook.\nII. FORMALISM\nA. Spin excitations in solids\nA magnetic solid at \fnite temperature displays two dif-\nferent kinds of magnetic excitations, namely spin wave\nexcitations (magnons) and electron-hole pair excitations\n(Stoner). The magnon excitations are responsible for\ntransversal \ructuations while Stoner excitations cause\nlongitudinal changes of the moments. At low temper-\natures and in particular for bulk materials, as in this\nstudy, the magnon excitations dominate and as a \frst\napproximation the Stoner excitations can be neglected.\nHowever, it is worth noting that they may play an im-\nportant role at high temperatures and also for certain\nmaterials with induced magnetic moments. Longitudinal\n\ructuations can however be modelled in a more advanced\ntreatment24.\nThe low energy spin excitation in a form of a magnon\nis characterized by the wave vector qwithin the Brillouin\nzone and for a cubic, ordered, material the magnon en-\nergyE(q) = ~!(q)\u0019Dq2, whereDis the spin wave\nsti\u000bness25\nD=2\n3MX\njJ0jR2\noj; (1)arXiv:1810.08487v1  [cond-mat.mtrl-sci]  19 Oct 20182\nandMis the magnetization, Jijis the exchange inter-\nactions between magnetic moments mat sitesiandj\nconnected with position vector R. In the case of disor-\nder, the spin wave sti\u000bness Dis obtained in by averaging\nover allNatoms in the system as\nD=2\n3M1\nNX\nnX\njJnjR2\nnj; (2)\nB. Atomistic spin dynamics\nThe dynamics of a magnetic material at \fnite tem-\nperature and thus the magnetic excitations, is conve-\nniently modelled through atomistic spin dynamics (ASD)\nsimulations26. Within ASD, the temporal evolution of\nthe atomic moments mat \fnite temperature is governed\nby Langevin dynamics, through coupled stochastic dif-\nferential equations, the Landau-Lifshitz-Gilbert (LLG)\nequations, here written in the Landau-Lifshitz form,\ndmi\ndt=\u0000\r\n(1 +\u000b2)mi\u0002[Bi+bi(t)] (3)\n\u0000\r\u000b\nm(1 +\u000b2)mi\u0002fmi\u0002[Bi+bi(t)]g;\nwhere\ris the electron gyromagnetic ratio and \u000bis the\nGilbert damping parameter. The latter can either be\ntaken from experiments using ferromagnetic resonance\n(FMR) or calculated from \frst-principles. The e\u000bective\ninteraction \feld Biexperienced by each atomic moment\niis given by\nBi=\u0000@H\n@mi: (4)\nwhereHis the spin Hamiltonian governing the interac-\ntions between the magnetic moments. We are employing\nthe semi-classical Heisenberg model,\nH=\u0000P\nijJijmi\u0001mj, where the exchange interactions\nare parametrized from \frst-principles calculations. The\ne\u000bective interaction \feld is complemented with a stochas-\ntic \feld bithat is modeled with uncorrelated white noise\nwith a temperature dependent variance26.\nC. Magnon dispersion\nWe are employing two di\u000berent complementary meth-\nods for calculating the magnon spectrum, 1) the adia-\nbatic magnon spectrum (AMS) valid for the ground state\nand 2) from ASD simulations through the dynamical\nstructure factor at \fnite temperatures and damping.\n1. Adiabatic magnon spectrum\nThe adiabatic magnon spectrum is directly connected\nto the real-space exchange interactions JijthroughFourier transformation27,28. LetJ\u000b\f(q) denote the\nFourier transform of the exchange interaction between\nchemical type \u000band\fwith a wave-vector qlying in the\nBrillouin zone (BZ). J\u000b\f(q) is calculated as\nJ\u000b\f(q) =X\nj6=0J\u000b\f\n0jexp(iq\u0001R0j): (5)\nIn the spirit of virtual crystal approximation (VCA), it\nis tempting to perform a chemical average of the Fourier\ntransformed exchange interactions, i.e.\n~J(q) =J11(q)x2\n1+J12(q)x1x2+J21(q)x1x2+J22(q)x2\n2\n(6)\nin the case of binary alloy and where x1andx2are the\nconcentration of each chemical type. In such a case,\nthe \"e\u000bective\" magnon energy ( ~=1) for each wavevec-\ntorqcan then be adapted to the expression valid for one\natom/cell of ordered systems27,28\n~!(q) =4\n~M\u0010\n~J(0)\u0000~J(q)\u0011\n; (7)\nwhere ~Mis the saturation magnetization. However,\nthis treatment of the disorder is over-simpli\fed and\ndoes not reproduce experimentally observed excitations.\nAnalogous to multi-sublattice ordered systems, where N\nmagnon branches appear in the spectrum ( Nis the num-\nber of sublattices), chemically disordered systems con-\ntainingKchemical components will exhibit Kmagnon\nbranches. More speci\fcally, in the case of a binary al-\nloy (K=2), the magnon energies at each wave-vector q\nwill be given by the eigenvalues of the following 2 \u00022\ndynamical matrix\n!(q) = 4Eig (J11(0)\u0000J11(q))x1+J12(0)x2\nM1\u0000J12(q)x2\nM1\n\u0000J21(q)x1\nM2(J22(0)\u0000J22(q))x2+J21(0)x1\nM2!\n:\n(8)\n2. Dynamical structure factor\nThe magnon dispersion at \fnite temperatures are di-\nrectly accessible in ASD through the dynamical structure\nfactorS(q;!)29{31. The key ingredient is the measure-\nment of the time and space correlation function\nC\u0016\u0017(r;t) =1\nNX\ni;jwhere\nri\u0000rj=rhm\u0016\ni(t)m\u0017\nj(0)i\u0000hm\u0016\ni(t)ihm\u0017\nj(0)i:\n(9)\nThe correlation function de\fned in Eqn. (9) describes\nhow the magnetic order evolves both in space ( \u0016;\u0017de-\nnotes carteisian components) and over time. The per-\nhaps most valuable application of C(r;t) is obtained by3\na Fourier transform over space and time to give the dy-\nnamical structure factor\nS\u0016\u0017(q;!) =1p\n2\u0019NX\nreiq\u0001rZ1\n\u00001ei!tC\u0016\u0017(r;t)dt:(10)\nThe magnon energies are determined by the peak val-\nues ofS(q;!) at wavevector q. In contrast to the adi-\nabatic treatment, temperature e\u000bects from the Gilbert\ndamping processes are included that give rise to inten-\nsity variation of the available energies. In the present\nstudy, we have not included longitudinal \ructuations of\nthe magnetic moment. Such \ructuations give rise to\nStoner excitations and an additional damping mechanism\nfor magnons so-called Landau damping.\nD. Details of calculation\nAll \frst-principles calculations in this study was\nperformed using a multiple-scattering (Korringa-Kohn-\nRostoker, KKR) implementation of the density func-\ntional theory (DFT) as implemented in the SPR-KKR\nsoftware32,33. The generalized gradient approxima-\ntion (GGA) using the Perdew-Burke-Enzerhof (PBE)\nparametrization was used as exchange-correlation for\nthe volume relaxation while all other calculations em-\nployed the local spin density approximation (LDA). The\ncalculations are fully relativistic employing the atomic\nsphere approximation with a basis set consisting of spdf-\norbitals. The coherent potential approximation (CPA)\nwas employed for treating the disorder. In order to study\nthe magnetic excitations and \fnite temperature proper-\nties, the total energies from the electronic structure cal-\nculations are mapped onto an e\u000bective Heisenberg Hamil-\ntonian generalized to random alloys.\nThe magnetic exchange interactions were obtained\nfrom the magnetic force theorem using the formalism\nof Lichtenstein, Katsnelson, Antropov and Gubanov\n(LKAG)34,35. Gilbert damping was calculated using the\nlinear response formalism of the torque-torque correla-\ntion method as described in Ref.[36]. The alloy-analogy\nmodel within CPA37was employed for the \fnite tempera-\nture damping where both atomic displacements and spin\n\ructuations from Monte Carlo data were included. The\natomistic simulations, either the Monte Carlo or atom-\nistic spin dynamics, were performed using the UppASD\nsoftware26,38. Here the disorder is instead treated by\nusing a large supercell in which each site is chemically\nrandomly occupied according to the concentration. We\nare using large supercells consisting of between 110592\natoms (for the calculation of the spin sti\u000bness and AMS)\nand 512000 atoms (for the calculation of the dynamical\nstructure factor), such that most of the local environ-\nment con\fgurations from a central atom exist within the\nsupercell. The spin sti\u000bness was calculated for each in-\ndividual atom in the supercell and the \fnal result was\nobtained by performing an average over all atoms.III. RESULTS\nA. Electronic band structure\nFIG. 1. Electronic band structure in terms of the Bloch spec-\ntral function of a) ordered Fe-Co compound in the B2 struc-\nture and b) Fe 50Co50random alloy in the bcc structure.\nBefore describing in details how the magnetic proper-\nties are a\u000bected by chemical disorder, we \frst look into\nthe electronic band structure. For ordered elements and\ncompounds, the electron bands are well de\fned with no\nassociated broadening as function of energy and wave\nvector within the LDA/PBE treatment. This corre-\nsponds to electrons having in\fnite lifetime. A typical\nelectron band structure of an ordered Fe-Co compound is\ndisplayed in Fig. 1a). However, if the system has chemical\ndisorder (or if the \fnite lifetime of the quasi-particles are\ntaken into account) the bands become \"fuzzy\" and obtain\na \fnite broadening, with a line width inversely propor-\ntional to the lifetime. The broadening is however not\nuniform over the considered energy range. For random\nalloys, the electron band structure is conveniently ob-\ntained through the Bloch spectral function within CPA,\nas demonstrated in Fig. 1b) where the electron band\nstructure of Fe 50Co50random alloy in the body centered\ncubic (bcc) lattice is displayed. For this particular system\nand concentration, the disorder is most visible around\nthe Fermi level . This a\u000bects many properties such as4\nthe Gilbert damping (see Section III D).\nB. Spin sti\u000bness\nFIG. 2. Calculated spin sti\u000bness Din (meV \u0017A2) for the ran-\ndom alloys Fe 1\u0000xNix, Co 1\u0000xNixand Fe 1\u0000xCox.\nThe calculated values of the spin sti\u000bness Dare shown\nin Fig. 2 for Fe-Ni, Co-Ni and Fe-Co random alloys. For\nthe Fe-Ni alloys, Dis monotonously increasing with the\nNi concentration. This has a rather simple explanation.\nPure Fe in the face-centered cubic (fcc) lattice at the\nhere considered volumes possess rather complicated non-\ncollinear magnetic structures39which translates into a\nvanishing spin sti\u000bness. Overall, the magnetic properties\nin Fe-Ni alloys are rather sensitive to volume changes.\nEven at Invar concentration (Fe 65Ni35) it is possible to\nkill the ferromagnetic order by applying pressure, and in\nthis way obtain a vanishing spin sti\u000bness.\nThe Co-Ni alloys behave di\u000berently. Here the val-\nues of the spin sti\u000bness are rather constant throughout\nthe whole concentration range and therefore the magnon\nproperties are not expected to change much. This is per-\nhaps not so unexpected since both elemental Co and Ni\nare stable in the fcc lattice.\nThe spin sti\u000bness of the Fe-Co alloys in the bcc lattice\nshows a more interesting behaviour. At low concentra-\ntions of Co ( x <0:2), the spin sti\u000bness is similar as for\nelemental Fe while it increases for higher concentrations\nof Co. At the phase boundary around x= 0:7, the spin\nsti\u000bness is approximately twice as large as that of Fe.\nThis suggests that the are ample possibilities for tuning\nthe magnetic properties in this system.\nC. Curie temperatures\nOur computed Curie temperatures are shown in Fig. 3.\nTwo di\u000berent approaches have been used, the mean \feld\napproximation (MFA) and the random phase approxima-\ntion (RPA). In principle, MFA corresponds to the arith-\nmetric average of the exchange interactions and RPA tothe harmonic average. It can be shown40,41that for fer-\nromagnetic interactions TMFA\nc> Tc> TRPA\nc, whereTc\nis the \"true\" value (which can be obtained from Monte\nCarlo). The two di\u000berent methods (MFA and RPA) then\nset the upper and lower bounds of Tc. Of the three con-\nsidered alloy systems, the Fe-Ni system displays the low-\nest values of Tcwhile Fe-Co the highest with values peak-\ning around 1500 K for Co concentrations around 0.5.\nFIG. 3. Calculated Curie temperatures for the random alloys\na) Fe 1\u0000xNix), b) Co 1\u0000xNixand c) Fe 1\u0000xCox. MFA denotes\nvalues from mean \feld approximation and RPA from random\nphase approximation.\nD. Gilbert damping\nFIG. 4. Calculated Gilbert damping at T= 300 K for the ran-\ndom alloys Fe 1\u0000xNix(red), Co 1\u0000xNix(green) and Fe 1\u0000xCox\n(black).\nGilbert damping in magnetic materials determines the\nrate of dissipative energy processes with the surround-\nings. Very often a low damping is wanted in order to min-\nimize energy losses but equally important is the ability to\ntune the damping. This can be achieved by, e.g., impurity\ndoping42or by varying the alloy composition. The latter\nis pursued here. For both fcc alloy systems considered5\nin this study, i.e. Fe-Ni and Co-Ni, the Gilbert damping\nincreases with Ni concentration. The Gilbert damping\nfor Fe-Ni is consistently lower than the one seen in Co-\nNi. For elemental Ni (o\u000b scale), we obtain \u000b= 0:013,\nwhich is in the same range as reported previously36,43.\nWorth noting is that the damping is one order of mag-\nnitude smaller in elemental Co and Fe. What is perhaps\nmost remarkable however is the very low damping found\nin certain Fe-Co alloys, in which it is even lower than\nfor elemental Fe. This behaviour is due to variation of\nthe density of states and was explained in detail in pre-\nvious studies36,44. The experimental values reported in\nRef.[44] are in good agreement with our calculated values\npresented here.\nE. Magnon properties\nFIG. 5. Magnon spectrum of permalloy (Fe 20Ni80). The\nthin red line denotes e\u000bective adiabatic spectrum, Eq.(7), and\nthick black lines full adiabatic treatment, Eq.(8). Blue (green)\npoints denote peak position at each wavevector of the dynami-\ncal structure factor from atomistic spin dynamics calculations\natT= 10 K ( T= 300 K) using the calculated Gilbert damp-\ning.\nOverall, we \fnd that the main features of the magnon\nspectra are quite similar in all systems we have consid-\nered here. We therefore choose in this section to present\nresults only for two systems of particular technological\ninterest: i) permalloy (Fe 20Ni80) in the fcc lattice and ii)\nFe75Co25in the bcc lattice chosen due to its large mag-\nnetic moment and low damping.\nIn Fig. 5, the calculated magnon spectrum of Py is dis-\nplayed using a variety of di\u000berent tools as described in\nSection. II C. Both the thin red line and the bold black\nlines are from adiabatic calculations, Eqs. (7) and (8), re-\nspectively. Between the two, the spectrum in black lines\nis expected to hold which is clear from comparison with\ndynamical structure factor from atomistic spin dynam-\nics calculations as indicated in squares for two di\u000berent\ntemperatures, namely T= 10 K and T= 300 K. It is\nFIG. 6. Magnon density of states of permalloy (Fe 20Ni80)\nfrom AMS and atomistic spin dynamics simulations at T=\n10 K and T= 300 K.\nimportant to remember that AMS only re\rects the ex-\nchange interactions and the chemical disorder of the sys-\ntem. Temperature e\u000bects in the form of transversal \ruc-\ntuations and damping are however included in the ASD\nsimulations where the calculated Gilbert damping at T=\n300 K was employed. The curvature around the \u0000 point\nis the spin sti\u000bness and by inspection it is clear that the\nspectrum softens drastically at room temperature com-\npared to the low temperature data. This temperature\ndependence of the sti\u000bness was also analyzed in a re-\ncent study20. At higher energies, due to a combination\nof thermal \ructuations, disorder and damping processes\nthe spectrum broadens which is much clearly shown in\nFig. 6 where the magnon density of states (MDOS) is\ndisplayed.\nFIG. 7. Magnon spectrum of Fe 75Co25. The thin red line\ndenotes e\u000bective adiabatic spectrum, Eq. (7), and thick black\nlines full adiabatic treatment, Eq. (8). Blue (red) points de-\nnote peak position at each wavevector of the dynamical struc-\nture factor from atomistic spin dynamics calculations at T=\n0 K (T= 300 K) using the calculated Gilbert damping.\nThe MDOS obtained from AMS has two distinct peaks6\nwhich we can denote \"acoustic\" and \"optical\" branch in\nanalogy to phonons. The two branches are separated by\na small gap. However, even at low temperature ( T= 10\nK), the MDOS as obtained by ASD simulations is broad-\nened enough such that the two branches overlap. The\npeak positions are however almost identical. Although\none need to keep in mind that AMS is using a simpli\fed\ntreatment of disorder, namely VCA, while ASD simula-\ntions are treating the disorder much more accurately by\nusing a large random supercell. Increasing the temper-\nature to 300 K softens the spectrum almost uniformly.\nThis \fnding has been used to describe the low tempera-\nture dependence of MDOS with a quasiharmonic approx-\nimation in Refs. [45 and 46].\nThe calculated magnon spectrum for Fe 75Co25, dis-\nplayed in Fig. 7, is quite di\u000berent from the one of Py.\nFirst of all, given its much higher Curie temperature the\ndi\u000berence of the spectrum between T= 10 K and T=\n300 K is minimal. Secondly, since Fe and Co atoms are\nrather similar chemically, both in terms of magnetic mo-\nments, 2.5\u0016Band 1.8\u0016B, respectively and the exchange\ninteractions are of similar magnitude, the spectrum has\nmuch less disorder broadening.\nF. Magnon lifetimes, ordered versus disordered\nTo further quantify the e\u000bects of disorder on magnon\nproperties, in this section we compare ordered system\nwith disordered system having the same composition.\nMore speci\fcally, we compare Fe 50Co50which exists both\nin ordered structure, B2, or as a disordered random\nalloy in bcc structure. Both the magnetic moments\n(Ms\u00192:2\u0016B) and Curie temperatures ( \u00191400 K) are\nrather similar between the two structures. The calculated\nGilbert damping at room temperature is however lower\nfor the ordered B2 structure, 0.0007 vs 0.0011 for the ran-\ndom alloy. It is worth noting that the damping for the B2\nis remarkable low for a metallic compound. In Fig. 8, the\nmagnon spectrum from ASD simulations at T= 300 K is\nshown for the both compounds, together with the AMS\nspectrum as reference. Due to the lack of disorder in the\nB2 structure, the magnon states are very well de\fned\nthroughout the whole Brillouin zone. It is immediately\nclear that the disorder of the random alloy broadens the\nmagnon states a\u000becting the magnon lifetimes, similar as\nfound for the electron bands in Fig. 1. However, even for\nthe random alloy there are relatively well de\fned magnon\nstates throughout the Brillouin zone, in contrast to the\nFe-Ni alloys where the magnon states away from the \u0000-\npoint are very di\u000buse.\nIn Fig. 9, the magnon DOS is displayed for the two\ncompounds. As also clear from the spectrum, in the B2\nstructure the magnon states are divided in two distinct\nbranches, \"acoustic\" and \"optical\" with small tempera-\nture dependence of the peak positions. The width of the\npeak is inversely related to the magnon lifetime. From\ninspection, the width for the B2 structure at T= 300 K isslightly larger than at T= 10 K and thus giving shorter\nmagnon lifetimes. However, the magnon lifetimes will\nalso have a wave-vector dependence and a more involved\nanalysis is needed. In the random alloy, the \"acoustic\"\nmagnon branch is roughly located at the same energies\nas in the B2 structure, while the \"optical\" branch is sig-\nni\fcantly broadened in comparison.\nThe most elaborate way to determine magnon life-\ntimes theoretically is through time dependent density\nfunctional theory and linear response47. Due to the com-\nplexity of such calculations, it has so far only been ap-\nplied to elemental systems and not in alloys. Here, we\ntherefore use an alternative simpli\fed method to obtain\nthe wave vector dependent magnon lifetimes \u001c(q). By\n\ftting the dynamical structure factor for each wave vec-\ntor with a Lorentzian and determine the full width half\nmaximum (FWHM), \u0001( q), the corresponding magnon\nlifetime is obtained through the relation \u001c(q) =2~\n\u0001(q).\nIt is worth stressing that this approach only takes into\naccount decay through Gilbert damping mechanism and\nnot via Landau damping corresponding to electron-hole\npair excitations within the Stoner continuum. However,\nfor bulk materials as in this study it should be a rea-\nsonable good approximation, at least for the \"acoustic\"\nmagnon branch.\nFIG. 8. Magnon spectrum from atomistic spin dynamics sim-\nulations at T= 300 K of Fe 50Co50in the ordered B2 structure\n(top) and as a random alloy (bottom). The white line is the\ncorresponding spectrum as obtained from AMS.7\nFIG. 9. Magnon density of states of Fe 50Co50in ordered B2\nstructure (top) and random alloy (bottom) from AMS and\natomistic spin dynamics simulations at T= 10 K and T=\n300 K.\nIn Fig. 10, the calculated magnon lifetimes in speci\fc\ndirections of the Brillouin zone are displayed for both the\nordered B2 structure and as random alloy. Due to the\nsensitivity of the \ftting, the calculated lifetimes for each\nwave vector have an associated error bar that is of the\norder of the variation between neighbouring wave vectors\nvalues. Nevertheless, it is clear that on average the or-\ndered B2 structure has longer magnon lifetimes compared\nto the random alloy. For the speci\fc directions in Fig. 10,\nthe average magnon lifetime in B2 is approximately three\ntimes larger than for the random alloy (0.6 ps vs 0.2 ps).\nThis behaviour is in line with what is normally expected\nfrom disordered systems, i.e. that increased disorder in-\ncreases the scattering rates which e\u000bectively gives shorter\nquasi-particle lifetimes. A direct comparison can be made\nwith the broadening of the electron bands in the spectral\nfunctions shown in Fig. 1.\nIV. SUMMARY\nWe have presented magnon and \fnite temperature\nproperties of random alloys using a combination of elec-\ntronic structure calculations and atomistic spin dynamics\nsimulations. Disorder is seen to have a pronounced e\u000bect\non the magnon properties causing additional scattering\nand damping of magnon modes. However, the degree of\nmagnon scattering and damping depends sensitively on\nthe chemical composition of the alloy and also on the\nrelative concentration of the constituent atomic species,\nprompting for material speci\fc studies. For example,the magnon spectrum of permalloy (Fe 20Ni80) is much\nmore a\u000bected by disorder causing di\u000buse spectra in most\nof the Brillouin zone than that of Fe 75Co25where well-\nde\fned magnon states exist everywhere. Similarly, we\ncompared the magnon properties of Fe 50Co50, both as\nordered structure and as random alloy. We found a dis-\ntinct di\u000berence in the magnon density of states between\nthe two as well as observing shorter magnon lifetimes in\nthe random alloy. We hope that the present study will\nmotivate new experiments in the next generation of neu-\ntron scattering facilities. 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B\n84, 174418 (2011)."    },    {        "title": "2111.03233v1.Giant_oscillatory_Gilbert_damping_in_superconductor_ferromagnet_superconductor_junctions.pdf",        "content": "Giant oscillatory Gilbert damping in \nsuperconductor/ferromagnet/superconductor junctions  \n \nAuthors  \nYunyan Yao1,2†, Ranran Cai1,2†, Tao Yu3, Yang Ma1,2, Wenyu Xing1,2, Yuan Ji1,2, Xin -Cheng \nXie1,2,4,5, See -Hun Yang6*, and Wei Han1,2* \nAffiliations   \n1International Center for Quantum Materials, School of Physics, Peking University, Beijing \n100871, China.  \n2Collaborative Innovation Center of Quantum Matter, Beijing 100871, China.  \n3Max Planck Institute for the Structure and Dynamics of Matte r, 22761 Hamburg, Germany  \n4CAS Center for Excellence in Topological Quantum Computation, University of Chinese \nAcademy of Sciences, Beijing 100190, P. R. China  \n5Beijing Academy of Quantum Information Sciences, Beijing 100193, P. R. China  \n6IBM Research - Almaden, San Jose, California 95120, USA   \n†These authors contributed equally to the work  \n*Correspondence to: weihan@pku.edu.cn (W.H.); seeyang@us.ibm.com (S.H.Y.) . \n \n \nAbstract   \nInterfaces between materials with differently ordered phases present unique opportunities for \nexotic physical properties, especially the interplay between ferromagnetism and superconductivity \nin the ferromagnet/superconductor heterostructures. The investig ation of zero- and π-junctions has \nbeen of particular interest for both fundamental physical science and emerging technologies.  Here, \nwe report the experimental observation of giant oscillatory Gilbert damping in the superconducting \nNb/NiFe/Nb junctions wi th respect to the NiFe thickness. This observation suggests an \nunconventional spin pumping and relaxation via zero-energy Andreev bound states that exist only \nin the Nb/NiFe/Nb π-junctions, but not in the Nb/NiFe/Nb zero-junctions. Our findings could be \nimportant for further exploring the exotic physical properties of ferromagnet/superconductor \nheterostructures, and potential applications of ferromagnet π-junctions in quantum computing, \nsuch as half -quantum flux qubits.  2 \n  \nOne sentence summary: Giant oscillat ory Gilbert damping is observed in \nsuperconductor/ferromagnet/superconductor junctions with varying the ferromagnet thickness.  \n \n \nIntroduction  \nThe interplay between ferromagnetism and superconductivity  has induced many exotic and \nexciting physical properties in ferromagnet (FM)/superconductor (SC) heterostructures (1-3). Of \nparticular interest is the unconventional π-phase ground state SC/FM/SC junction  that might be \nrealized for cert ain FM thicknesses arising from the quantum intermixing of the wave functions \nbetween spin -singlet Cooper pairs in SC and spin -polarized electrons in FM (1, 3, 4). At the FM/SC \ninterface, a Cooper pair moving into the FM will ha ve a finite center -of-mass momentum, resulting \nin the oscillation of the  real part of  superconducting order parameter  (Re {Ψ})  with respect to the \nFM thickness  (Fig. 1A ) (1, 5, 6). Depending on the FM thicknesses, the Cooper pair wavefunctions \nin the two superconductors on either side of the FM can have a phase difference from zero or π, \nforming so -called zero-junctions with positive Josephson coupling (Fig. 1B ) or π-junctions with  \nthe negative Josephson coupling  (Fig. 1C) . The FM π-junctions can be used for quantum \ncomputing applications (7, 8), as half quantum flux qubits (9). Due to the scientific and technical \nimportance, the research on the FM π-junctions has been active for the last tw o decades (6, 10-13). \nPrevious experimental studies have demonstrated the switching between zero- and π-junctions in \nSC/FM/SC structures by varying the temperature and the FM thickness (11, 14-17). These reports \nmainly focus on the electrical properties of the FM zero- and π-junctions. Recently, dynamic spin \ninjection into SCs has attracted considerable interest  both the exper imentally  (18-21) and \ntheoretically (22-26). However, the spin -dependent properties in FM zero- and π-junctions have \nnot been explored yet. The investigation of the spin -dependent properties requires the spin current \nprobes, such as the dynamical spin pumping (27). Furthemore, for the application of the FM π-\njunctio ns in quantum computing technologies (9), the magnetization/spin dynamic properties are \nextremely important to be studi ed. \nHere, we report the experimental observation of giant oscillatory Gilbert damping in the \nsuperconducting Nb/NiFe/Nb junctions with respect to the NiFe thickness, which can be 3 \n qualitatively explained by the different spin pumping efficiency via the Andr eev bound states \n(ABS) of Nb/NiFe/Nb zero- and π-junctions. Using a minimal model based on the ABS, we show \nthat an unconventional spin pumping into the zero -energy ABS  penetrat ed into SCs  could occur s \nonly for the π-junctions, which can lead to the oscillatory Gilbert damping as a function of the \nNiFe thick ness. \nResults  \nFigures 1D and 1E  show the schematic s of the spin pumping, magnetization dynamics, and \nenhanced Gilbert damping in the  SC/FM/SC  zero- and π-junction s. Spin pumping refers to the \nspin-polarized current injection to non -magnetic materials from a FM with precessing \nmagnetization around its ferromagnetic resonance (FMR) conditions (28, 29). In FM and its \nheterostructures, the Gilbert damping (\n) characterizes the magnetization dynamics , as described \nby the Landau -Lifshitz -Gilbert formula with an additiona l Slonczewski -torque term (30-32): \n 𝑑𝒎\n𝑑𝑡=−𝛾𝒎×𝑯𝒆𝒇𝒇+𝛼𝒎×𝑑𝒎\n𝑑𝑡+𝛾\n𝑀𝑠𝑉(ℏ\n4𝜋𝑔↑↓𝒎×𝑑𝒎\n𝑑𝑡) (1) \nwhere 𝒎=𝑴/|𝑴| is the magnetization unit vector,  𝛾 is the gyromagnetic ratio, 𝑯𝒆𝒇𝒇 is the total \neffective magnetic field, 𝑀𝑠=|𝑴| is the saturation magnetization, and 𝑔↑↓ is the interface spin  \nmixing conductance. The pumped spin current from FM into SCs can be expressed by Js =\nℏ\n4𝜋𝑔↑↓𝒎×𝑑𝒎\n𝑑𝑡 (29). The spin pumping into the SCs  give rise  to an enhanced Gilbert damping \nconstant  that is proportional to the spin pumping current  (αsp~ J s) (29). Fig. 1E illustrates the \npumped spin current mediated by the zero -energy ABS inside the superconducting gap  in π-\njunctions , which will be discussed later in details.  While for a zero-junction, the pumped spin \ncurrent is mediated by the ABS near the superconducting gap (Fig. 1D). The ABS can be formed \nwithin the FM layer and then extended into the interface of SCs with the superconducting coherent \nlength scale (33, 34).   \nThe SC/FM/SC junctions consist of a NiFe (Ni 80Fe20) layer (thickness: ~ 5 - 20 nm) \nsandwiched by two Nb layers (thickness: 100 nm) grown by magnetron sputt ering (see Methods \nand f ig. S1). To maximize the integrity of samples for a systematic study, more than tens of \nsamples are grown in each run via rotation mask technique in a sputtering system, which is the \nsame as in the previous study of the oscillatory exchange coupling in mag netic multilayer 4 \n structures (35). The Gilbert damping  and spin pumping  are measured by the ferromagnetic \nresonance  (FMR)  technique (see Methods for details) .  \nAbove the TC of Nb, spin pumping in the Nb/NiFe/Nb junctions leads to the spin accumulation \nin Nb near the interface, which can be described by the spin -dependent chemical potentials, as \nillustrated in Fig. 2 A. The Gilbert damping of NiFe in the Nb/NiFe/Nb junctions is determined \nfrom the microwave frequen cy-dependent FMR spectra ( fig. S2). A typical FMR curve with the \nLorentzian fitting is shown in Fig. 2 B, from which the  half linewidth (ΔH) can be obtained. The \nGilbert damping can be extracted from the best linear -fitting curve of ΔH vs. f (Fig. 2 C). Figure \n2D shows the NiFe thickness dependence of the Gilbert damping in the Nb/NiFe/Nb junctions \nmeasured at T = 10, 15, and 20 K, respectively. Interestingly,  an oscillating feature of the Gilbert \ndamping is observed as a function of 𝑑NiFe in the region of 𝑑NiFe < ~15 nm. This oscillating \nbehavior can be attributed to the quantum -interference effect of angular momentum transfer \nbetween the local precessing magnetic moment and conduction electrons in thin NiFe that was \ntheoretically predicted by Mills (36), but has not been experimentally reported yet. Above TC, the \ncontinuous energy bands of Nb, similar to the normal met al in the Mills theory, overlap with both \nspin-up and spin -down bands of NiFe at the interface, thus allowing the conducting electrons in \nNiFe to flip between the spin -down and spin -up states. As illustrated in the inset of Fig. 2 D, one \nspin-down electron scatters with the local magnetic moment and then flips to the spin -up \npolarization, giving rise to the angular momentum transfer between the spin -polarized electrons \nand the magnetic moment. Besides the change of angular momentum, the momentum of the \nelect ron also changes ( ∆𝑘), due to different Fermi vectors for spin -up (𝑘𝐹↑) and spin -down ( 𝑘𝐹↓) \nelectrons with exchange splitting (Fig. 2 A). When the NiFe layer is thin enough to become \ncomparable with 1\n∆𝑘, quantum -interference effect of the spin -polariz ed electrons shows up, which \ngives rise to the oscillating spin -transfer torque to the NiFe. When the NiFe thickness is \n2𝑛𝜋/[𝑘F↑−𝑘F↓] (n is an integer), the matching of the quantum levels between the spin -up and \nspin-down electrons in NiFe induces smaller Gilbert damping. On the other hand, when the NiFe \nthickness is (2n+1)𝜋/[𝑘F↑−𝑘F↓], a larger Gilbert damping is induced. Consequently, th e \nGilbert damping in the Nb/NiFe/Nb structures oscillates with a period of 2𝜋/[𝑘F↑−𝑘F↓] \n(Supplementary Materials S1) . Experimentally, an oscillating period ( λ) of ~ 1.8 nm is identified 5 \n (see the red dashed arrow in Fig. 2 D). At T = 50 K, the oscillating f eature disappears since the \nquantum -interference effect is smeared by thermal excitations ( fig. S3) . \nNext, we investigate the spin pumping and spin transfer torque of the Nb/NiFe/Nb junctions \nin the superconducting states below TC with a superconducting gap (Fig. 3 A). TC in the \nNb/NiFe/Nb junctions is obtained from typical four -probe resistance measurement as a function \nof the temperature. A typical temperature -dependent resistance curve measured on the Nb/NiFe \n(12 nm)/Nb junction is shown in Fig. 3b, indicating the TC of ~ 8.6 K. As dNiFe changes, TC of the \nNb/NiFe/Nb junctions exhibits little variat ion between ~ 8.4 and ~ 8.9 K ( fig. S4). Similar to the \nnormal states of Nb, the Gilbert damping below TC is also obtained from the be st linear -fitting \nresult of the  half linewidth vs. frequency ( fig. S5). During the FMR measurement, TC varies a little \n(< 1 K) ( Fig. S 6). As the temperature decreases, 𝛼 decreases abruptly from ~ 0.012 to ~ 0.0036 \nacross the TC (fig. 3C), which indicates the decrease of spin current injected into Nb due to the \nformation of superconducting gap below TC. This observation is consistent with previous reports \non spin pumping into SCs where the spin current is mediated by Bogoliubov quasiparticles (18, \n19, 37). As the tem perature decreases far below the TC, the quasiparticle population dramatically \ndecreases, leading to reduced spin pumping and Gilbert damping.  \nRemarkably, the oscillating amplitude of the Gilbert damping of the Nb/NiFe/Nb junctions as \na function of the NiF e thickness is dramatically enhanced as the temperature decreases into the \nsuperconducting states of Nb (Fig. 3 D). At T = 4 K, the oscillating magnitude of the Gilbert \ndamping constant is ~ 0.005 for the first three oscillations, which is comparable to the  background \nvalue of ~ 0.006. The obtained Gilbert damping values are not affected by thermal cycles, and the \nlarge oscillating feature has been confirmed on a different set of samples. Such a giant oscillation \nof the Gilbert damping cannot be explained by  spin pumping of Bogoliubov  quasiparticle -\nmediated spin current in SCs. Since as the temperature decreases, the population of the Bogoliubov  \nquasiparticles monotonically and rapidly decreases with an increase of the SC gap, which would \nlead to lower Gilber t damping and also smaller oscillation compared to the normal states. Note \nthat the oscillating period of the Gilbert damping at T = 4 K is the same as that at T = 10 K that is \nsupposed to be 2𝜋/[𝑘F↑−𝑘F↓] due to the quantum interference effect . Such oscillating period of \n2𝜋/[𝑘F↑−𝑘F↓] is the also same as that of the zero- and π-phase ground states transitions in FM \nJosephson devices, which is equal to the coherence length in NiFe film of 2𝜋/[𝑘F↑−𝑘F↓] in the 6 \n ballistic regime (1, 11, 17), and √ℏ𝐷𝑑𝑖𝑓𝑓 ∕𝐸𝑒𝑥  in the diffusive regime ( 𝐷𝑑𝑖𝑓𝑓 is the diffusion \ncoefficient, and 𝐸𝑒𝑥 is the exchange energy ). The observed oscillating period of ~ 1.8 nm in our \nstudy is similar to the zero - 𝜋 oscillating period measured in the NiFe Josephson junctions in the \ndiffusive regime reported previously (11, 17).  \nThe Gilbert damping difference  (∆𝛼) between the zero- and 𝜋-junctions is extracted as a \nfunction of NiFe thickness, as shown in Fig. 4 A. We assume the larger Gi lbert damping for the 𝜋-\njunctions and smaller value s for the zero-junctions, which will be discuss ed later in details . The \nthickness -dependent  Gilbert damping of  the zero- and 𝜋-junctions  are expected to both behave as \nα ~ 1 ⁄ dNiFe (29). Hence, we can treat them separately, as illustrated by the guide lines in the inse t \nof Fig. 4A, and ∆𝛼 is obtained by subtracting the fitted 1/d curve for the expected  zero-junctions  \n(black  dashed line) . Clearly, there is a pronounced oscillating feature of ∆𝛼 for the Nb/NiFe/Nb \njunctions with NiFe thickness from ~ 5 nm to ~ 11 nm. When the NiFe thickness is above ~ 11 \nnm, the oscillating feature of the Gilbert damping is largel y suppressed compared to thinner NiFe \njunctions. This feature might be associated wi th the strong Josephson coupling for thin NiFe \njunctions and the exponential decaying of the Josephson coupling as the NiFe thickness increases \n(11, 17). To confirm this, the Jose phson junctions are fabricated using the shadow mask technique, \nand a Josephson coupling is observed from the Nb/NiFe (5 nm and 10 nm)/Nb junctions \n(Supplementary Materials and fig. S7). \nDiscussion  \nLet us discuss the physical mechanism that induces the giant oscillating Gilbert damping in the \nfollowing. Apart from the spin pumping via ABS  discussed above (Fig. 1 D and 1E ), the spin \ncurrent in SCs can also be mediated by Bogoliubov quasiparticles  (fig. S8A) (18, 19, 22, 23, 38), \nspin-triplet pairs  (fig. S 8B) (3). Regarding Bogoliubov  quasiparticles, they populate around the \nedge of superconducting gap at elevated temperatures close to TC (39). As shown both theoretical \nand experimental studies, the enhanced Gilbert damping in the SC/FM/SC heterostructures \nhappens around TC (18, 19, 22, 23, 38). As the temperature decreases down to 0.5 TC, the \nBogoliubov quasiparticles are mostly frozen out, for which the spin pumping is forbidden that will \nno longer contribute to the enhanced Gilbert damping. Hence, the Bogoliubov quasi particles are \nvery unlikely to account for our experimental results. Regarding  the spin-triplet pairs, it has been \nshown in previous studies that the spin -triplet current under FMR conditions and spin triplet 7 \n correlations would be different for zero- and 𝜋-junctions (4, 38, 40, 41), which might result in \ndifferent Gilbert damping theoretically. However, in our study, there are not spin sinks adjacent to \nthe Nb layers , thus not allowing the spin -triplet Cooper pairs to be relaxed in the Nb. This is \ndiffe rent from previous report on the Pt/SC/FM/SC/Pt heterostructures (20), where the Pt is used \nas the spin sink. Experimentally, as the temperature below TC, the Gilbert damping exhibits a \nmonotonic decrease for the Nb/NiFe/Nb heterostructures (Fig. 3 C), which is different from the \nenhanced Gilbert damping due to spin -triplet pairs  (20). Furthermore, no Josephson current in the \nNb/NiFe/Nb heterostructures is observed i n Nb/NiFe (30 nm)/Nb junction ( fig. S7), which  \nindicates the absence of long -range spin -triplet Josephson coupling. Both these experimental \nresults indicate that the contribution from the spin -triplet pairs is not significant to the enhanced \nGilbert damping in the superconducting Nb/NiFe/Nb junctions . \nTo our best understanding, the most reasonable mechanism is the spin pumping via the ABS, \nwhich can qualitatively  describe our experimental observation. Previous studies have \ndemonstrated that the energy of ABS inside the superconducting gap depends on the  \nsuperconducting -phase (42, 43). For the FMR measurement under  open -circuit ed conditions , the \ninversion symmetry of the current -phase  (𝜑) relationships is preserved (43-45). For 𝜋-junctions, \nthere is a 𝜋-phase shift in the current -phase relationship curves compared to zero-junctions , i.e., \nthe properties of 𝜑 = 0 of a 𝜋-junction is the same as those of 𝜑 = 𝜋 of a zero-junction. Since this \n𝜋-phase shift is already taken into account by the FM exchange field , the ABS energy of the 𝜋-\njunctions  can be obtained at 𝜑 = 0 in the ground states , which is similar to that of 𝜑 = 𝜋 of zero-\njunctions .  \nFor 𝜋-junctions, ABS is located around the  zero-energy inside of the superconducting gap (Fig. \n1D). The ABS could penetrate into the superconducting Nb films with scale of superconducting \ncoherent length (~ 30 nm), which is evanescent to dissipate the spin angular moment um (25, 26, \n44). As shown in Fig. 4 B, the transfer efficiency of spin angular momentum via the zero -energy \nABS can lead to an enhanced Gilbert damping . Whileas, for  zero-junctions, the distribution of the \nABS is near the edge of the superconducting gap (Fig. 1C) , thus, the spin pumping effic iency is \nsuppressed due to the reduced population of the ABS at low temperatures  (Fig. 4 C). Furthermore , \nthe oscil latory energy levels of the ABS  between the zero- and 𝜋-junctions is also consistent with \nthe density of states (DOS)  oscillating in supercon ductors  between the zero- and 𝜋-junctions (1, 6, 8 \n 38, 44, 46). In consequence, as the NiFe thickness increas es, the oscillatory spin pumping \nefficiency via ABS  at the FM/SC interface  (or DOS in SC s) gives rise to the oscillatory Gilbert \ndamping.  We have proposed a simplified model for the case of ideal transparency of electrons \n(Supplementary Materials S2 and f ig. S9 ). For the less transparency cases, i.e., in diffusive regime , \n(42, 43), the energy level s of the ABS in  𝜋-junctions  locates away from zero -energy, but they are \nstill much smaller than those  of the ABS in zero-junctions. Actually, the similar oscillating \nbehaviors of ABS (or DOS) can be preserved in the diffusive regime  (6, 46). Hence, an oscillating \nspin pumping efficiency would also be expected in the diffusive regime, which could lead to the \noscillating Gilbert damping  observed in our experiment . To fully understand the experimental \nobservation of the oscillatory Gilbert damping and the detailed spin relaxation process in the \ndiffusive regime, further theoretical studies are needed.  \nFurthe rmore, the control samples of bilayer Nb/NiFe heterostructur es do not exhibit the large \noscillatory feature for the Gilbert damping as the NiFe thickness varies  at T = 4 K  (fig. S10), which \nfurther presents the important role of phase difference across NiFe  in the large the oscillatory \nGilbert damping  observed in t he trilayer  Nb/NiFe/Nb heterostructures .  \nIn conclusion, giant oscillatory Gilbert damping is observed in the superconducting \nNb/NiFe/Nb junctions with respect to the NiFe thickness. To our best knowledge, neither the \nBogoliubov quasiparticles, nor the spin -triplet pairs are relevant to this observation.  The most \npossible explanation for such giant oscillatory Gilbert damping could be related to the different \nABS energy levels and the DOS at the NiFe/SC interface in zero- and π- junctions. To full y \nunderstand these results, further theoretical studies are needed. Looking forward, our experimental \nresults  might pave the way for controlling the magnetization dynamics  by the superconducting \nphase  in a FM Josephson junction  in the SQUID setup, and could be important potential \napplications of ferromagnet π-junctions in quantum computing, such as half -quantum flux qubits.  \n \nMaterials and Methods  \nMaterials growth  \nThe SC/FM/SC heterostructures consisting of Nb (100 nm) and Ni 80Fe20 (NiFe ; ~ 5 - 20 nm) were \ngrown on thermally oxidized Si substrates in a d.c. magnetron sputtering system with a base \npressure of ∼1× 10−8 torr. To systematically vary the NiFe thickness that is crucial for the quantum -9 \n size effect, we adopted the rotating multi -platter technique that allows us to grow dozens of \nNb/NiFe/Nb samples in each run  (35). The thickness of the Nb layer is fixed to be ~100 nm that is \nmuch larger than the spin diffusion length of Nb  (20, 47). After the growth, a thin Al 2O3 layer (~ \n10 nm) was deposited in situ  as a capping layer to avoid sample degradation against air/water \nexposure. The crystalline properties of Nb/NiFe/Nb heterostructures were chara cterized by X -ray \ndiffraction (fig. S1A ) and high -resolution cross -section al tra nsmission electron microscopy (f ig. \nS1B) using a 200 -kV JEOL 2010F field -emission microscope. The NiFe thickness is determined \nby the growth rate that is calibrated by TEM measurement, where the uncertainty of the NiFe \nthickness is obtained to be  smaller than ~ 0.8  nm (f ig. S1B).  The resistivity of the NiFe layers \n(thickness: 5 - 20 nm) is ranging from 60 to 35 μΩ ·cm, which corresponds to the  mean free path \nbetween 2.3 and  3.9 nm.  \n \nFerromagnetic resonance measurement.   \nThe spin pumping of Nb/NiFe/Nb heterostructures was characterized via FMR  using the coplanar \nwave guide technique connected with a vector network analyzer (VNA; Agilent E5071C) in the \nvariable temperature insert of a Physical Properties Measurement System (PPMS; Quantum \nDesign)  (19). The FMR spectra were characterized by measuring the amplitudes of forward \ncomplex transmission coefficients (S 21) as the in -plane magnetic field decreases from 4000 to 0 \nOe under the microwave power of 1 mW. The typical FMR results measured on the Nb/NiFe (12 \nnm)/Nb het erostructures are shown in the fig. S2A  (T = 10 K) and fig. S 5A (T = 4 K). Weaker \nFMR signals are observed in the superconducting states compared to the normal states.  \nThe half linewidth ( ∆𝐻) can be obtained by the Lorentz fitting of the magnetic field -dependent \nFMR sign al following the relationship ( figs. S2B  and S4 B): \n 𝑆21∝𝑆0(∆𝐻)2\n(∆𝐻)2+(𝑯−𝑯𝒓𝒆𝒔)2 (3) \nwhere 𝑆0 is the coefficient for the transmitted microwave power,  𝑯 is the external in -plane \nmagnetic field, and 𝑯𝒓𝒆𝒔 is the resonance magnetic field.  The Gilbert damping constant (α) can be \nobtained from the slope of the best linear -fitting results of the ∆𝐻 vs. the microwave frequency ( f) \n(48-51): 10 \n  ∆𝐻=∆𝐻0+(2𝜋𝛼\n𝛾)𝑓 (4) \nwhere ∆𝐻0 is the zero -frequency line broadening that is related to the inhomogeneous properties, \nand  𝛾 is the gyromagnetic ratio. From the best linearly fits of the ∆𝐻 vs. f results measured on the \ntypical Nb/Py ( 12 nm)/Nb sample (red lines in f igs. S2 C and S 5C), 𝛼 is determined to be 0.012 \nand 0.0054 at T = 10 and 4 K, respectively. A larger zero -frequency line broadening ∆𝐻0 is \nobserved for the superconducting state compared to the normal state of Nb/Py/Nb heterostructures, \nwhich could be attributed to Meissn er screening effect and the formation of trapped magnetic \nfluxes in Nb  (51). The thickness dependent ∆𝐻0 is shown in f ig. S1 1C, and no obviously \noscillatory behaviors  are observed . \nThe effective magnetization and the gyromagn etic ratio can be fitted via the in -plane Kittel \nformula (51): \n 𝑓𝑟𝑒𝑠=𝛾\n2𝜋√(𝐻𝑟𝑒𝑠+ℎ)(𝐻𝑟𝑒𝑠+ℎ+4𝜋𝑀𝑒𝑓𝑓),                                      (5)  \nwhere 𝑓𝑟𝑒𝑠 and 𝐻𝑟𝑒𝑠 are the resonant microwave frequency and magnetic field respectively , \n4𝜋𝑀𝑒𝑓𝑓 is the effective saturated magnetization , and ℎ is the shifted magnetic field induced by \nsuperconducting proximity effect. The thickness -dependent gyromagnetic ratio and eff ective \nmagnetization can be found in fig. S1 1A and fig. S1 1B. Both parameters do not exhibit  any \noscillatory features  as the Gilbert damping does (Fig . 4A), which demonstrates that the oscillatory \nGilbert damping is not caused by any unintentional experimental error . \n \nSuperconducting transition temperature measurement.  \nThe superconducting transition temperature ( TC) of the Nb/NiFe/Nb heterostructures was \ndetermined via the zero -resistance temperature measured by four -probe method in a P PMS using \nstandard a.c. lock -in technique at a low frequency of 7 Hz. The TC of Nb (100 nm)/NiFe/Nb (100 \nnm) heterostructures exhibits little variation as a f unction of the NiFe thickness ( fig. S 4). It is \nnoticed that the FMR measurement can affect the TC a little (< 1 K), as shown in fig. S6.  \n 11 \n Supplementary Materials  \nSupplementary Materials and Methods  \nfig. S1 . The crystalline properties of the Nb/NiFe/Nb heterostructures.  \nfig. S2. Gilbert dampin g measurement of Nb/NiFe/Nb heterostructures at T = 10 K.  \nfig. S3. NiFe thickness dependence of Gilbert damping at T = 50 K.  \nfig. S4. NiFe thickness dependence of TC for the Nb/NiFe/Nb heterostructures.  \nfig. S5. Measurement of the Gilbert damping of Nb/ NiFe/Nb heterostructures at T = 4 K.  \nfig. S6. The effect of FMR measurement on the TC of Nb/NiFe/Nb heterostructures.  \nfig. S7. The measurement of Josephson coupling in Nb/NiFe/Nb junctions.  \nfig. S 8. Illustration of magnetization dynamics and spin pumping i n the SC/FM/SC \nheterostructures due to Bogoliubov quasiparticles and equal spin -triplet Cooper pairs.  \nfig. S9. Calculation of the enhanced Gilbert damping due to spin pumping via the ABS at T = 4 K.  \nfig. S10. Gilbert damping of control sample of bilayer Nb /NiFe junctions.  \nfig. S1 1. Thickness dependen ce of  gyromagnetic ratio, effective magnetization and \ninhomogeneous half -linewidth.  \n \n \nReferences and Notes:  \n1. A. I. Buzdin, Proximity effects in superconductor -ferromagnet heterostructures. Rev. 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T.Y. is \nfinancially supported by DFG Emmy Noether program (SE 2558/2 -1).  \n \nAuthor contributions:  W.H. conceived and supervised the project. Y.Y. and R.C. \nperformed the ferromagnetic resonance measurements. Y.Y. and Y.M. performed X -ray \ndiffraction measurements. T.Y. performed the theor etical calculations. S.H.Y. synthesized \nthe Nb/NiFe/Nb heterostructures. Y.Y. and W.H. wrote the manuscript with the \ncontribution from all authors. All the authors discussed the results.  \n \nCompeting interests:  The authors declare no competing interests.  \n \nData Availability: All data needed to evaluate the conclusions in the paper are present in \nthe paper and/or the Supplementary Materials.  \n  16 \n  \nFig. 1. Illustration of magnetization dynamics and spin pumping in the SC/FM/SC \nheterostructures.  (A) The oscillatory real part of the superconducting order parameter (Re {Ψ}, \ngreen curve ) penetrated into FM leads to the zero-state and  𝜋-state. (B) The symmetric order \nparameter in the zero-junction s. (C) The anti -symmetric order parameter in the 𝜋-junction s. (D-E) \nSpin pumping via t he ABS  in SCs  in the zero- and 𝜋-junction s. M and 𝛼FM are the magnetization \nand Gilbert damping of the FM layer itself, and 𝛼sp is the enhanced Gilbert damping, which arises \nfrom the spin dissipation in SC layers during the spin pumping process.  \n \n17 \n  \nFig. 2. Oscillatory Gilbert damping of the Nb/NiFe/Nb heterostructures above TC. (A) The \nillustration of spin pumping into the normal states of Nb layers and the electronic band structure \nof NiFe with different spin-up and spin -down Fermi vectors (𝑘F↑ and 𝑘F↓) due to the exchange \nsplitting (2 𝐸𝑒𝑥). The spin pumping gives rise to the spin accumulation in the Nb layers, indicated \nby the spin -split chemical potential ( 𝜇↑ and 𝜇↓). (B) A typical FMR  curve measured with f = 12 \nGHz (black circles) and the Lorentzian fitting curve (red line) measured on Nb/Py (12 nm)/Nb . \nΔH is the half line width at the half maximum of FMR signal. (C) The determination of the Gilbert \n18 \n damping from ΔH vs. f. The red line represents the best linear -fitting curve. (D) The oscillatory \nGilbert damping as a function of NiFe thickness ( dNiFe) measured at T = 10, 15, and 20 K, \nrespectively. The experimental oscillating period ( 𝜆) is marked by the red dashed arrow. The inset: \nIllustration of the quantum -interference effect o f the angular momentum transfer between the local \nmagnetic moment and the spin -polarized electrons . When the NiFe thickness decreases to scale of \n1\n∆𝑘, the quantum -interference effect starts to be significant in the angular momentum transfer  and \nspin pumping into the Nb layers .  \n  \n \n \n \n \n \n \n \n \n \n \n 19 \n            \nFig. 3. Giant oscillatory Gilbert damping in Nb/NiFe/Nb heterostructures below TC. (A) The \nillustration of electronic band structures of Nb in the normal and superconducting states. (B) The \ndetermination of TC via zero -resistance temperature measured on the typical Nb/NiFe (12 nm)/Nb \nheterostructures. (C) Temperature dependence of Gilbert d amping of the typical Nb/NiFe (12 \nnm)/Nb heterostructures. (D) The oscillatory Gilbert damping as a function of the NiFe thickness \nin the Nb/NiFe/Nb heterostructures measured at T = 10, 7, 5, and 4 K, respectively. The oscillating \nfeature below TC (T = 4 and 5 K) is dramatically enhanced compared to that above TC (T = 10 K).  \n  \n20 \n  \nFig. 4. Physical mechanism of the giant oscillatory Gilbert dam ping in Nb/NiFe/Nb junctions. \n(A) The NiFe thickness dependence of the Gilbert damping difference ( ∆α) between the  \nNb/NiFe/Nb π- and zero-junctions  at T = 4 K . Inset: The Gilbert damping of zero- and π-junctions. \nThe solid balls represent the experimental data, the blue and black dash lines are the guide lines \nfor π- and zero-junctions, respectively. For bo th guide lines, the damping is expected to behave as \nα ~ 1 ⁄ dNiFe. (B-C) Illustration of the spin pumping via the ABS and the enhanced Gilbert damping \nfor Nb/NiFe/Nb π- and zero-junctions, respectively.  The red thick -arrows indicate the pumping \nand relaxation  of the spin current in SCs . \n21 \n Supplementary Materials for  \n \n \nGiant oscillatory Gilbert damping in \nsuperconductor/ferromagnet/superconductor junctions  \n \nAuthors  \nYunyan Yao1,2†, Ranran Cai1,2†, Tao Yu3, Yang Ma1,2, Wenyu Xing1,2, Yuan Ji1,2, Xin -Cheng \nXie1,2,4,5, See -Hun Yang6*, and Wei Han1,2* \n \n \nThis SM file includes : \n⚫ Supplementary Materials and Methods  \n⚫ fig. S1. The crystalline properties of the Nb/NiFe/Nb heterostructures.  \n⚫ fig. S2. Gilbert damping measurement of Nb/NiFe/Nb heterostructures at T = 10 K.  \n⚫ fig. S3. NiFe thickness dependence of  Gilbert damping at T = 50 K . \n⚫ fig. S4. NiFe thickness dependence of TC for the Nb/NiFe/Nb heterostructures.  \n⚫ fig. S5. Measurement of the Gilbert damping of Nb/NiFe/Nb heterostructures at T = 4 K.  \n⚫ fig. S6. The effect of FMR measurement on the TC of Nb/NiFe/N b heterostructures.  \n⚫ fig. S7. The measurement of Josephson coupling in Nb/NiFe/Nb junctions.  \n⚫ fig. S8. Illustration of magnetization dynamics and spin pumping in the SC/FM/SC \nheterostructures due to Bogoliubov quasiparticles and equal spin -triplet Cooper pai rs.  \n⚫ fig. S9. Calculation of the enhanced Gilbert damping due to spin pumping via the ABS at \nT = 4 K.  \n⚫ fig. S10. Gilbert damping of control sample of bilayer Nb/NiFe junctions.  \n⚫ fig. S11. Thickness dependence of gyromagnetic ratio, effective magnetization an d \ninhomogeneous half -linewidth . \n \n  22 \n Supplementary Materials and Methods  \nSection 1: Model of oscillating Gilbert damping above TC.  \nThe oscillatory Gilbert damping in normal metal (NM)/ferromagnet (FM)/NM \nheterostructures arising from quantum interference effect is analyzed based on previous theory by \nMills (36). Within the linear response theory, the enhanced Gilbert damping is related to the \ndynamical spin susceptibility ( 𝜒−+(𝛺)) of conduction electrons in a FM,  \n 𝛼sp=𝐽2𝑀𝑠𝑉\n2𝑁2ℏ3𝛾Λ2 (S1) \nwhere Λ2=Im(𝑑𝜒−+(𝛺)\n𝑑𝛺|\nΩ=0). Using one dimensional model,  we obtain   \n Λ2=1\n𝜋2∫𝑑𝑥𝑑 𝑥′Im[𝐺↑(𝑥,𝑥′,𝜖F)]Im[𝐺↓(𝑥,𝑥′,𝜖F)]\nFM (S2) \nwhere 𝐺𝜎(𝑥,𝑥′,𝜖F) is the Green’s function for conduction electrons with 𝜎-spin at the Fermi \nenergy ( 𝜖F). In a FM, 𝐺𝜎(𝑥,𝑥′,𝜖F) is related to the exchange energy.  \n [−ℏ2\n2𝑚𝑑\n𝑑𝑥2−𝜖±𝐸ex]𝐺𝜎(𝑥,𝑥′,𝜖F)=𝛿(𝑥−𝑥′) (S3) \nFor the FM film with a thickness dFM in − dFM/2 < x < dFM/2, the Green’s function satisfies the \nrelation  \n 𝐺𝜎(𝑥,𝑥′,𝜖F)=𝐺𝜎(𝑥′,𝑥,𝜖F)=𝐺𝜎(−𝑥,−𝑥′,𝜖F) (S4) \nHence, the imaginary part of the Green’s function could be expressed by  \n Im[𝐺𝜎(𝑥,𝑥′,𝜖F)]=−π{𝑁F𝜎cos [𝑘F𝜎(𝑥−𝑥′)]+𝑁F𝜎′cos [𝑘F𝜎(𝑥+𝑥′)]}     (S5) \nwhere 𝑘F𝜎=√2𝑚\nℏ2(𝜖F∓𝐸ex) is the Fermi wave -vector in the FM, 𝑁F𝜎 and 𝑁F𝜎′ are equivalent to \nthe density of states and the modulation amplitude of the local density of states, respectively. For \nthe same position of x, the local density of states is equal to  \n 𝑁𝜎(𝑥,𝜖F)=𝑁F𝜎+𝑁F𝜎′cos [2𝑘F𝜎𝑥] (S6) \nSince 𝐸ex is much smaller compared to 𝜖F, the spatial modulation of the local density of states is \nnegligible. The combination of  equations (S2) and (S5) leads to  \nΛ2=∫ 𝑑𝑥𝑑 𝑥′𝑑FM 2⁄\n−𝑑FM 2⁄{𝑁F↑cos[𝑘F↑(𝑥−𝑥′)]}∗{𝑁F↓cos[𝑘F↓(𝑥−𝑥′)]}        (S7) 23 \n                =2𝑁F↑𝑁F↓{1\n(𝑘F↑−𝑘F↓)2sin2[𝑘F↑−𝑘F↓\n2𝑑FM]+1\n(𝑘F↑+𝑘F↓)2sin2[𝑘F↑+𝑘F↓\n2𝑑FM]} \nClearly,  the enhanced  Gilbert  damping  is expected  to oscillate  as a function  of the FM \nthickness  with two periods  of  2𝜋/[𝑘F↑−𝑘F↓] and 2𝜋/[𝑘F↑+𝑘F↓]. For real FM materials,  such as \nNiFe  with (𝑘F↑+𝑘F↓)≫(𝑘F↑−𝑘F↓), the second  term in the equation  (S7) could  be negligible,  \nleaving  only one oscillating  period  of 2𝜋/[𝑘F↑−𝑘F↓]. When  the FM thickness  is equal  to \n2𝑛𝜋/[𝑘F↑−𝑘F↓], a lower  Gilbert  damping  is obtained.  On the other  hands  with FM thickness  of \n(2𝑛+1)𝜋/[𝑘F↑−𝑘F↓], a larger  Gilbert  damping  is obtained.   \nSection  2: Calculation  of the enhanced  Gilbert  damping  in Nb/NiFe/Nb  by spin pumping  via \nAndreev  bound  states  (ABS).   \nAs the reciprocal process of the spin transfer torque, conventional spin pumping is achieved \nby the magnetization torques provided by the driven quasiparticle carriers (29, 52-54), which are \nthe electrons in the normal metals. In SC/FM heterostructures, however, the quasiparticle carriers \ncan be either Bogoliubov quasiparticles or ABS (42), which lie above and within the \nsuperconducting gaps, respectively. Therefore, it is desirable to formulate and estim ate the \ncontribution to the spin pumping via the ABS (55), when the temperature is much smaller than the \nsuperconducting critical temperature.   \nWithout loss of generality, we start the analysis from a left -propagating electron of energy 𝜀 \nand spin 𝜎 = {↑, ↓} = {+, −,}. When the Zeeman splitting J is much smaller than the Fermi energy \nEF, it has momentum  \n 𝑘𝜎=𝑘𝐹+(𝜀+𝜎𝐽)/(ℏ𝜈𝐹),                                                             (S8) \nwhere 𝜈𝐹 is the Fermi velocity of the electron. When one electron goes from the FM to the SCs, it \nis reflected as a hole by the Andreev reflection at the right FM/SC interface; this hole has a phase \nshift χ=−arccos (𝜀/Δ) with respect to the electron (56), where Δ is the superconducting gap . \nSimilarly, when a hole goes from the metal to the superconductor at the left FM/SC interface, an \nelectron can be reflected. With a proper energy, the Andreev reflections can form a closed path, as \na result of which the ABS forms. This requires that the phase accumulated in the reflections \nsatisfies the Sommerfeld quantization condition, i.e. in the ballistic regime,  24 \n     𝜀𝐿\nℏ𝜈𝐹+𝜎𝐽𝑑NiFe\nℏ𝜈𝐹−arccos (𝜀\n∆)=𝑛𝜋+𝜑\n2,                                 (S9) \nwhere 𝜑 is the phase difference between the two superconductors , dNiFe is the thickness of the FM \nlayer and n is an integer. Since ℏ𝜈𝐹/∆  ≥ 100 nm with 𝜈𝐹 = 2.2 ×  105 m/s and ∆ = 1 meV at T = 4 \nK in our experiment (17, 57, 58), dNiFe < 19 nm << ℏ𝜈𝐹/∆ such that the first term in Eq. (S9) can \nbe safely disregarded. For the FMR measurements with open -circuited configuration, the junctions \nalways stay in the ground states (43-45). For 𝜋-junctions, there is a 𝜋-phase shift in the current -\nphase relationship curves compared to zero-junctions , i.e., the properties of 𝜑 = 0 of a 𝜋-junction \nis the same as those of 𝜑 = 𝜋 of a zero-junction. Since this 𝜋-phase shift is already taken into \naccount by the FM exchange field , the ABS energy of the 𝜋-junctions  can be obtained at 𝜑 = 0 in \nthe ground states , which is similar to that of 𝜑 = 𝜋 of zero-junctions . Hence, the energy of the ABS \ncan be described by   ε0=±∆cos (𝐽𝑑NiFe\nℏ𝜈𝐹) for ideal case with perfect transparency of \nelectrons/holes.  \nIn reality, the interfacial scattering and transport conditions (ballistic or diffusive regimes) of \nFM could affect th e energy of the ABS. Following previous studies (42, 59), a transmission \ncoefficient ( D) could be introduced to describe this issue , which is close to unity in the ballistic \nregime but can also be large in the  diffusive regime  with an ideal transparency at the interface (43, \n60, 61). In this work, we focus on the ideal cases with perfect transparency of electrons/holes . The \nenergy of the ABS oscillates from the ed ge of the superconducting gap to the zero -energy with \nrespect to the FM thickness (fig. S9A).   \nThe pumped spin current reads (29, 52-54), \n𝐉𝑠(𝑡)=ℏ\n4𝜋𝑔eff↑↓𝒎×𝑑𝒎\n𝑑𝑡,                                                          (S10)  \nwhere m is the magnetization unit vector, and we define the effective mixing spin conductivity \n𝑔eff↑↓ at the finite temperature via the zero -temperature one 𝑔↑↓ by (53, 55)  \n  𝑔eff↑↓=𝑛0∫𝑑𝜀𝑑𝑓(𝜀)\n𝑑𝜀Re[𝑔↑↓(𝜀)].                                              (S11)  \nHere, n0 is the number of the conduction channel that roughly corresponds the conduction electron \ndensity at the interface and 𝑓(𝜀)=1/{exp [𝜀/(𝑘𝐵𝑇)]+1} is the Fermi -Dirac distribution of \nelectron at the temperature T. Importantly, in the ballistic limit Re[𝑔↑↓(𝜀)]=1 when  𝜀=𝜀0; it 25 \n has width ∆𝜀 depending on the FM thickness dNiFe in the ballistic regime or the mean free path 𝑙𝑚 \nin the diffusive regime . By the uncertainty principle, ∆𝜀∆𝑡=2𝜋ℏ, where ∆𝑡=𝑙𝑚/𝜈𝐹 is the \npropagation time of the electron in the junction, leading to  ∆𝜀 ~ 2𝜋ℏ𝜈𝐹/𝑙𝑚. By further \nconsidering the degeneracy due to spin (× 2) and the existence of two interfaces (× 2), we thus can \nestimate  \n    𝑔eff↑↓ ~ 8𝜋𝑛0ℏ𝜈𝐹\n𝑙𝑚𝑑𝑓(𝜀0)\n𝑑𝜀.                                                         (S12)  \nThe pumped spin current carries the angular momentum away from the precessing magnetization \nand hence cause an enhanced Gilbert damping, which is described by  \n𝛿𝛼=2𝛾ℏ2𝜈𝐹\n𝑀𝑠𝑙𝑚𝑑NiFe𝑑𝑓(𝜀0)\n𝑑𝜀 ,                                                      (S13)  \nwhere 𝛾  is electron gyromagnetic ratio and 𝑀𝑠 is the saturated magnetization of the ferromagnet.  \n      We are now ready to estimate the contribution of ABS to  the Gilbert damping at T = 4 K with \nvarying transmission coefficient. We take 𝑛0 =0.5 ×  1016 m−2 following Ref. 44 , 𝑙𝑚~ 3 nm,  𝜈𝐹= \n2.2 ×  105 m/s, J = 400 meV and 𝜇0𝑀𝑠≈1 𝑇 from previous experimental results (17). With \nsuperconducting gaps ∆ ≈ 1 meV at T = 4 K for Nb (57, 58), Fig.S8A plots the normalized energy \nof ABS by the superconducting gap at T = 4 K as a function of dNiFe for the ideal transparency case . \nThe oscillation of the Gilbert damping can be resolved by using the FM exchange field -induced \nphase shift of 𝐽𝑑NiFe\nℏ𝜈𝐹 (fig. S9B) . For simplicity, we have disregarded the possible thickness \ndependence of the superconducting gaps and magnetizations. To  be noted, our theoretical \nestimation is based on a simplified model that assumes D = 1. For the diffusive regime or the case \nof non -perfect transparency of electrons at the interface  (42, 43), similar oscillating behaviors of \nABS (or DOS) in the SCs can also be preserved. For example, the oscillating ABS (or DOS) in the \nSCs have been shown to exist in the diffusive regime theoretically (6, 46), and indeed, the zero to \n𝜋 transitions have been experimentally observed in both the ballistic and diffusiv e regimes from \nthe supercurrent measurements (11, 17). To fully understand the experimental observation of the \noscillatory Gilbert damping in the diffusive regime, further theoreti cal studies are needed.  \n \n \n \n 26 \n Section 3: Measurement of the Josephson coupling in Nb/NiFe/Nb.  \nThe Nb/NiFe/Nb  Josephson devices are fabricated using the shadow mask techniques during \nthe films growth. As shown in figs. S7A and S7B, the Josephson devices have a junction area ( A) \nof ~ 80 μm  × 80 μm, and the other areas are electrically isolated by a 100 nm AlO x layer. The \nJosephson current is measured by standard a.c. lock -in technique. The normalized differential \nresistances (dV/dI) measured on the Nb/NiFe (5 nm)/Nb junction at various temperatures are \nshown in fig. S7C. The critical current ( Ic) is defined as poi nt where the differential resistance \nincreases above the value for the zero -bias current. The normal resistance (R n) is determined to be \nthe saturated value of the normal states of the Josephson coupling measurement. The measured \narea-resistance product (R nA) of ~  5×10−10 Ω𝑚2 is higher than that reported in metallic \nJosephson junction (17, 61), and comparable to that of FM Josephson junction with a thin tunnel \nbarrier (62). This behavior indicates that there is more likely a thin NiFeO x layer (indicated by Fig. \nS8B) in the junction formed during the AlO x growth step in the presence of oxygen gas. As the \ntemperature increases, I c and the characteristic voltage (I cRn) decrease (figs. S7C and S7D). Clear \nJosephson currents are observed on the Nb/NiFe (5 nm)/Nb junction and Nb/NiFe (10 nm)/Nb \njunction (figs. S7E and S78F). And the estimated SC gap energy is ~ 0.9 meV at T = 2 K (1, 43), \nwhich is comparable to the value of ~1.36 meV at T = 0 K estimated from TC of ~ 8.5 K (fig. S4). \nOn the other hand, no Josephson current could be observed in the Nb/NiFe (30 nm)/Nb junction \n(figs. S7E and S7F). The absence of Jo sephson current in Nb/NiFe (30 nm)/Nb junction indicates \nthat there is no long -range spin -triplet Josephson coupling in the Nb/NiFe/Nb heterostructures in \nour experiment.  \n  27 \n  \n \n \nfig. S1. The crystalline properties of the Nb/NiFe/Nb heterostructures. (A) The θ -2θ X -ray \ndiffraction results measured on the typical Nb/NiFe (12 nm)/Nb sample, where Nb (110) and NiFe \n(111) peaks are observed. ( B) High -resolution transmission electron micrographs mea sured on the \ntypical Nb/NiFe (12 nm)/Nb sample. The dashed lines show the interfaces between Nb and NiFe \nlayers. The red bars indicate the deviation of NiFe at the interface.  \n \n \n \n \n \n \n  \n28 \n  \n \n \nfig. S2. Gilbert damping measurement of Nb/NiFe /Nb heterostructures at T = 10 K. (A) The \ntypical FMR  spectra as a function of magnetic field with microwave frequency ( f) of 10, 12, 14, \n16, and 18 GHz, respectively. (B) The typical FMR  spectrum measured with f = 12 GHz (black \ncircles) and the Lorentz fi tting curve (red line). ΔH is the half linewidth of the FMR signal. (C) \nThe determination of the Gilbert damping from ΔH vs. f. The red line indicates the best linear -\nfitting curve. These results are obtained on the typical  Nb/Py (12 nm)/Nb sample.  \n  \n29 \n  \n \n \n \nfig. S3. NiFe thickness dependence of  Gilbert damping at T = 50 K .  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n30 \n  \nfig. S4. NiFe thickness dependence of TC for the Nb/NiFe/Nb heterostructures. The TC is \ndetermined from the zero -resistance temperature via four -probe resistance measurement.  \n \n31 \n  \nfig. S5. Measurement of the Gilbert damping of Nb/NiFe/Nb heterostructures at T = 4 K. (A) \nThe typical FMR  spectra as a function of magnetic field with microwave  frequency ( f) of 10, 12, \n14, 16, and 18 GHz, respectively. (B) The typical FMR  spectrum measured with f = 12 GHz (black \ncircles) and the Lorentz fitting curve (red line). ΔH is the half linewidth of the FMR signal. (C) \nThe determination of the Gilbert damping from ΔH vs. f. The red line indicates the best linear -\nfitting curve. These results are obtained on the typical  Nb/Py (12 nm)/Nb sample.  \n  \n32 \n  \n \nfig. S6. The effect of FMR measurement on the TC of Nb/NiFe/Nb heterostructures. The four -\nprobe resistances vs. temperature are probed from the typical Nb/NiFe  (12 nm)/Nb sample \nwith/without the presence of the in -plane magnetic field and microwave power.   \n33 \n  \n \nfig. S7. The measurement of Josephson coupling in Nb/NiFe/Nb junctions.  (A) The optical \nimage of a typical Nb/NiFe/Nb Josephson device and schematic of the electrical measurement \ngeometry.  (B) The cross -section of the Josephson devices with a junction area of ~ 80 μm  × 80 \nμm. At the junction, a thin oxide layer of NiFeO x is mostly liked formed on the top surface of NiFe \nduring the growth of A lOx in the presence of oxygen. ( C) The normalized differential resistance \n(dV/dI) as a function of the bias  current  measured on the Nb/NiFe (5 nm)/Nb junction from T = 2 \nto 6 K.  (D) The temperature dependence of the characteristic voltage (I cRn) of the Nb/NiFe (5 \nnm)/Nb Josephson  junction . (E) The normalized differential resistance as a function of the bias  \ncurrent  of the Nb/NiFe/Nb junctions ( dNiFe = 5, 10 and 30 nm) at T = 2 K. ( F) The NiFe thickness \ndependence of the characteristic voltages a t T = 2 K.  \n \n34 \n  \n \nfig. S8. Illustration of magnetization dynamics and spin pumping in the SC/FM/SC \nheterostructures due to Bogoliubov quasiparticles (A) and equal spin -triplet Cooper pairs \n(B). The dark and light balls represent the electron -like and hole-like quasiparticles respectively. \nThe red and blue arrows indicate the spin up and spin down respectively.  \n  \n35 \n   \n \nfig. S9.  Calculation of the enhanced Gilbert damping due to spin pumping via the ABS at T \n= 4 K. (A) The normalized energy of ABS by the superconducting gap at T = 4 K as a function of \ndNiFe for the ideal transparency case. ( B) The enhanced Gilbert damping via ABS as a function of \ndNiFe.  \n  \n36 \n  \n \nfig. S10. Gilbert damping of control sample of bilayer Nb/Ni Fe junctions. (A)  Gilbert damping \nof bilayer Nb/NiFe junctions at T = 4, 5, and 10  K. (B) Comparison of the Gilbert damping of \nbilayer Nb/NiFe and trilayer Nb/NiFe/Nb junctions at T = 4 K.  \n \n \n \n \n \n \n \n \n37 \n  \nfig. S11.  Thickness dependence of gyromagnetic ratio (and g factor) (A), effective \nmagnetization (B) and inhomogeneous half -linewidth (C) . The blue, black, green and red \ndotted -lines represent to temperature of T = 10, 7, 5 and 4 K, respectively.  \n \n \n"    },    {        "title": "2302.02556v3.Global_solutions_of_the_Landau__Lifshitz__Baryakhtar_equation.pdf",        "content": "arXiv:2302.02556v3  [math.AP]  29 Jun 2023GLOBAL SOLUTIONS OF THE LANDAU–LIFSHITZ–BARYAKHTAR\nEQUATION\nAGUS L. SOENJAYA AND THANH TRAN\nAbstract. The Landau–Lifshitz–Baryakhtar (LLBar) equation is a generalisa tion of the\nLandau–Lifshitz–Gilbertand the Landau–Lifshitz–Blochequations which takesinto account\ncontributions from nonlocal damping and is valid at moderate temper ature below the Curie\ntemperature. Therefore, it is used to explain some discrepancies b etween the experimen-\ntal observations and the known theories in various problems on mag nonics and magnetic\ndomain-wall dynamics. In this paper, the existence and uniqueness of global weak, strong,\nand regular solutions to LLBar equation are proven. H¨ older contin uity of the solution is\nalso discussed.\n1.Introduction\nThe theory of micromagnetism deals with the study of magnetic phen omena occurring\nin ferromagnetic materials at sub-micrometre length scales. A bett er understanding of the\nmagnetisation dynamics at elevated temperature would contribute to the development of\nultrahigh-density storage technology based on heat-assisted ma gnetic recording; see [ 11,29,\n30,32] and the references therein.\nA widely-studied equation which describes the evolution of magnetic s pin field in ferro-\nmagnetic material is the Landau–Lifshitz–Gilbert (LLG) equation [ 18,24]. According to this\ntheory, the magnetisation of a magnetic body Ω ⊂Rd,d∈ {1,2,3}, denoted by u(t,x)∈R3\nfort >0 andx∈Ω, is described by\n∂u\n∂t=−γu×Heff−λu×(u×Heff), (1.1)\nwhereγ >0 andλ >0 are the gyromagnetic ratio and a phenomenological damping param e-\nter, respectively, and Heffistheeffectivefield(consistingoftheexchangefield, demagnetising\nfield, external magnetic field and others). It is known that far belo w the Curie temperature,\nthe magnetisation of a ferromagnetic material preserves its magn itude. This property is\nreflected in equation ( 1.1) (by taking the dot product of both sides of the equation with u).\nMathematically, the LLG equation has been extensively studied eithe r on bounded or\nunbounded domains where various existence, uniqueness and regu larity properties were dis-\ncussed. A non-exhaustive list includes [ 1,9,10,12,16,20,21,22,33]. Since then, various\ngeneralisations and improvements to the LLG equation have been ma de in the physical and\nmathematical literatures. Awidely-used physical model formicrom agnetism abovetheCurie\ntemperature is the Landau–Lifshitz–Bloch (LLB) equation [ 17]. This equation interpolates\nbetween the LLG equation at low temperatures and the Ginzburg–L andau theory of phase\ntransitions, and is known not to preserve the magnitude of the mag netisation. Mathemati-\ncally, the existence and regularity properties for LLB equation hav e been studied [ 25,27].\nDate: June 30, 2023.\n12 AGUS L. SOENJAYA AND THANH TRAN\nTheLLGandLLBequations, nevertheless, cannotaccountforso meexperimentaldataand\nmicroscopic calculations. These include the nonlocal damping in magne tic metals and crys-\ntals [14,36], or the higher-than-expected spin wave decrement for short-w ave magnons [ 5].\nTheLandau–Lifshitz–Baryakhtar (LLBar)equationproposedby Baryakhtar [ 3,4,5]isbased\nonOnsager’srelationsandgeneralisestheLLGandLLBequations[ 14,15,35]. Thisequation\nhas also been implemented on several commonly-used micromagnetic simulation software,\nsuch asMuMax [2,26] andFidimag [34,35]. Moreover, various micromagnetic simulations\nprovide evidence that the LLBar equation agrees with some of the o bserved experimental\nfindings in micromagnetics, especially those related to ultrafast mag netisation at an elevated\ntemperature; see [ 2,14,28,34,35,36] and the references therein.\nThe LLBar equation in its most general form [ 5,35] reads\n∂u\n∂t=−γu×Heff+Λr·Heff−Λe,ij∂2Heff\n∂xi∂xj,\nwhereurepresents themagnetisation vector, ΛrandΛedenote therelaxation tensor and the\nexchange tensor, respectively. Here, Einstein’s summation notat ion is used. For a polycrys-\ntalline, amorphous soft magnetic materials and magnetic metals at mo derate temperature\n(where nonlocal damping and longitudinal relaxationare significant) , this equation simplifies\n[14,35] to\n∂u\n∂t=−γu×Heff+λrHeff−λe∆Heff.\nwhere the positive scalars γ,λr, andλeare the electron gyromagnetic ratio, relativistic\ndamping constant, and exchange damping constant, respectively . The effective field Heffis\ngiven by\nHeff= ∆u+1\n2χ(1−|u|2)u+lower order terms ,\nwithχ >0 being the magnetic susceptibility of the material.\nIftheexchange interactionisdominant (asisthecaseforordinary ferromagneticmaterial),\nthenu: [0,T]×Ω→R3solves the following problem:\n∂u\n∂t−β1∆u+β2∆2u=β3(1−|u|2)u−β4u×∆u+β5∆(|u|2u) in (0,T)×Ω,(1.2a)\nu(0,·) =u0 in Ω, (1.2b)\n∂u\n∂n=∂(∆u)\n∂n=0 on (0,T)×∂Ω,(1.2c)\nwhereβ1=λr−λe/(2χ) is a real constant (which may be positive or negative), while\nβ2,...,β 5are positive constants. Here, ∂Ω is the boundary of Ω with exterior unit normal\nvector denoted by n.\nTypically, the constant β1will be positive since λe/(2χ) is much smaller than λr. How-\never, in certain situations occurring in spintronics or magnonics whe re the wavelength of\nthe magnons is approaching the exchange length of the ferromagn etic material, λecan be\nsignificant [ 14]. Therefore, we allow β1to take positive or negative values in ( 1.2).\nTo the best of our knowledge, mathematical analysis of the LLBar e quation does not\nexist in the literature. In this paper, we prove the existence, uniqu eness, and regularity\nof a weak and strong solution to problem ( 1.2) in one, two and three spatial dimensions\n(see Theorem 2.2), by using the Faedo–Galerkin approximation and compactness met hod.GLOBAL SOLUTIONS OF LLBAR EQUATION 3\nWe also prove H¨ older continuity properties of the solution (Theore m2.3). This gives a\nmathematical foundation for the rigorous theory of LLBar equat ion which is not currently\navailable in the literature.\nAnother advantage of studying the LLBar equation is for a given init ial datau0, the weak\nsolution to the LLBar equation generally has better regularity comp ared to that of the LLG\nor the LLB equation. Moreover, it is known that the existence of glo bal solutions to the\nLLG equation in 2-D is only guaranteed for sufficiently small initial data [9,16], whereas for\ngeneral initial data, solutions in 2-D could blow-up in finite time [ 23]. As we show in this\npaper, the solution to the LLBar equation exists globally.\nWe note that a related model of magnetisation dynamics in the frame work of frustrated\nmagnets(whichtakesintoaccountlocalandnonlocalinteractions )hasrecentlybeenexplored\nin [13]. The model is based on the LLG equation involving the bilaplacian opera tor, where\nthe magnitude of the magnetisation is conserved in that case.\nThepaper is organisedasfollows. InSection 2, we introduce some notationsandformulate\nthe main results. In Section 3, we establish some a priori estimates that are needed for the\nproof of the main theorems. Section 4is devoted to the proof of the main results. Finally,\nwe collect in the appendix some essential mathematical facts that a re used throughout the\npaper.\n2.Formulation of the main results\n2.1.Notation. We begin by defining some notations used in this paper. The function\nspaceLp:=Lp(Ω;R3) denotes the space of p-th integrable functions taking values in R3and\nWk,p:=Wk,p(Ω;R3) denotes the Sobolev space of functions on Ω ⊂Rdtaking values in R3.\nAlso, we write Hk:=Wk,2. Here, Ω ⊂Rdford= 1,2,3 is an open domain with smooth\nboundary. The operator ∆ denotes the Neumann Laplacian. The pa rtial derivative ∂/∂xi\nwill be written by ∂ifor short.\nIfXisanormedvector space, thespaces Lp(0,T,X)andWk,p(0,T,X)denoterespectively\nthe usual Lebesgue and Sobolev spaces of functions on (0 ,T) taking values in X. The\nspaceC([0,T],X) denotes the space of continuous function on [0 ,T] taking values in X.\nThroughout this paper, we denote the scalar product in a Hilbert sp aceHby/an}b∇acketle{t·,·/an}b∇acket∇i}htHand\nits corresponding norm by /ba∇dbl · /ba∇dblH. We will not distinguish between the scalar product of\nL2vector-valued functions taking values in R3and the scalar product of L2matrix-valued\nfunctions taking values in R3×3, and still denote them by /an}b∇acketle{t·,·/an}b∇acket∇i}htL2.\nThe following frequently-used notations are collected here for the reader’s convenience.\nFirstly, for any vector z∈R3and matrices A,B∈R3×d, we define\nz·A:=/bracketleftbig\nz·A(1)···z·A(d)/bracketrightbig\n∈R1×d,A·B:=d/summationdisplay\nj=1A(j)·B(j)∈R,\nz×A:=/bracketleftbig\nz×A(1)···z×A(d)/bracketrightbig\n∈R3×d,A×B:=d/summationdisplay\nj=1A(j)×B(j)∈R3,(2.1)\nwhereA(j)andB(j)denote the jthcolumn of AandB, respectively.4 AGUS L. SOENJAYA AND THANH TRAN\nNext, for any vector-valued function v= (v1,v2,v3) : Ω⊂Rd→R3, we define\n\n\n∇v: Ω→R3×dby∇v:=/bracketleftbig∂1v···∂dv/bracketrightbig\n=\n∂1v1···∂dv1\n∂1v2···∂dv2\n∂1v3···∂dv3\n,\n∂v\n∂n:∂Ω→R3×1by∂v\n∂n:= (∇v)n=/bracketleftbigg\n∂v1\n∂n∂v2\n∂n∂v3\n∂n/bracketrightbigg⊤\n,\n∆v: Ω→R3×1by ∆v:=/bracketleftbig∆v1∆v2∆v3/bracketrightbig⊤,\n∆∇v: Ω→R3×dby ∆∇v:=\n∆∂1v1···∆∂dv1\n∆∂1v2···∆∂dv2\n∆∂1v3···∆∂dv3\n=∇∆v.(2.2)\nAs a consequence, if uandvsatisfy suitable assumptions and ∂u/∂n= 0 (where nis the\noutward normal vector to ∂D), then\n−/an}b∇acketle{t∆u,v/an}b∇acket∇i}htL2=−3/summationdisplay\ni=1/an}b∇acketle{t∆ui,vi/an}b∇acket∇i}htL2=3/summationdisplay\ni=1/an}b∇acketle{t∇ui,∇vi/an}b∇acket∇i}htL2=/an}b∇acketle{t∇u,∇v/an}b∇acket∇i}htL2,\n/an}b∇acketle{tu×∆u,v/an}b∇acket∇i}htL2=/an}b∇acketle{t∆u,v×u/an}b∇acket∇i}htL2=−/an}b∇acketle{tu×∇u,∇v/an}b∇acket∇i}htL2.(2.3)\nFinally, throughout this paper, the constant Cin the estimate denotes a generic constant\nwhichtakesdifferent valuesatdifferent occurrences. Ifthedepe ndence of Consomevariable,\ne.g.T, is highlighted, we often write C(T). The notation A/lessorsimilarBmeansA≤CBwhere the\nspecific form of the constant Cis not important to clarify.\n2.2.Main results. In the following, we define the notion of weak solutions to ( 1.2). We\nfirst multiply ( 1.2a) (dot product) with a test function φ, integrate over Ω, and (formally)\nuse integration by parts, noting ( 1.2c), to obtain\n/angbracketleftBig∂u(t)\n∂t,φ/angbracketrightBig\nL2+β1/an}b∇acketle{t∇u(t),∇φ/an}b∇acket∇i}htL2+β2/an}b∇acketle{t∆u(t),∆φ/an}b∇acket∇i}htL2\n=β3/angbracketleftbig\n(1−|u(t)|2)u(t),φ/angbracketrightbig\nL2+β4/an}b∇acketle{tu(t)×∇u(t),∇φ/an}b∇acket∇i}htL2−β5/angbracketleftbig\n∇/parenleftbig\n|u(t)|2u(t)/parenrightbig\n,∇φ/angbracketrightbig\nL2.\n(2.4)\nWe next find sufficient conditions for the terms on the right-hand sid e to be well-defined for\nd= 1,2,3. Ifu∈H1, thenu∈L4so that|u|2u∈L4/3. Therefore, the term /an}b∇acketle{t|u|2u,φ/an}b∇acket∇i}htL2is\nwell defined if u∈H1andφ∈H1. Moreover, if u∈H2, thenu∈L∞. Thus, the second\nterm/an}b∇acketle{tu(t)×∇u(t),∇φ/an}b∇acket∇i}htL2and the third term\n/angbracketleftbig\n∇/parenleftbig\n|u(t)|2u(t)/parenrightbig\n,∇φ/angbracketrightbig\nL2=d/summationdisplay\ni=13/summationdisplay\nj=1/angbracketleftbig\n∂i/parenleftbig\n|v|2vj/parenrightbig\n,∂iφj/angbracketrightbig\nL2\non the right-hand side of the above equation are also well defined if u∈H2andφ∈H1.\nThis motivates the following definition of solutions to problem ( 1.2).\nDefinition 2.1. GivenT >0 andu0∈L2(Ω), a function u: [0,T]→H2is aweak solution\nto the problem ( 1.2) ifubelongs to C([0,T];L2)∩L2(0,T;H2) and satisfies\n/an}b∇acketle{tu(t),φ/an}b∇acket∇i}htL2+β1/integraldisplayt\n0/an}b∇acketle{t∇u(s),∇φ/an}b∇acket∇i}htL2ds+β2/integraldisplayt\n0/an}b∇acketle{t∆u(s),∆φ/an}b∇acket∇i}htL2dsGLOBAL SOLUTIONS OF LLBAR EQUATION 5\n=/an}b∇acketle{tu0,φ/an}b∇acket∇i}htL2+β3/integraldisplayt\n0/angbracketleftbig\n(1−|u(s)|2)u(s),φ/angbracketrightbig\nL2ds\n+β4/integraldisplayt\n0/an}b∇acketle{tu(s)×∇u(s),∇φ/an}b∇acket∇i}htL2ds−β5/integraldisplayt\n0/angbracketleftbig\n∇/parenleftbig\n|u(s)|2u(s)/parenrightbig\n,∇φ/angbracketrightbig\nL2ds,(2.5)\nfor allφ∈H2andt∈[0,T].\nA weak solution u: [0,T]→H2is called a strong solution if it belongs to C([0,T];H2)∩\nL2(0,T;H4). In this case, it satisfies ( 1.2) almost everywhere in [0 ,T]×Ω.\nWenowstatethemaintheoremsofthepaper, theproofsofwhichw ill begiveninSection 4\nand Section 5. The first theorem gives the existence, uniqueness, and regularit y of the\nsolution.\nTheorem 2.2. Let Ω⊂Rd,d= 1,2,3, be a bounded domain with Cr+1,1-boundary and let\nu0∈Hr,r∈ {0,1,2,3}, be a given initial data. For any T >0, there exists a global weak\nsolution to ( 1.2) such that\nu∈C([0,T];Hr)∩L2(0,T;Hr+2). (2.6)\nFurthermore, this solution depends continuously on the Hr-norm of the initial data, which\nimplies uniqueness. More precisely, if uandvare solutions corresponding to the initial\ndatau0andv0, respectively, then the following estimates hold\n/ba∇dblu(t)−v(t)/ba∇dbl2\nL2/lessorsimilar/ba∇dblu0−v0/ba∇dbl2\nL2exp/parenleftbigg/integraldisplayt\n0/parenleftBig\n1+/ba∇dblu(s)/ba∇dbl4\nL∞+/ba∇dblv(s)/ba∇dbl4\nL∞/parenrightBig\nds/parenrightbigg\n(2.7)\nand\n/ba∇dblu(t)−v(t)/ba∇dblHr/lessorsimilar/ba∇dblu0−v0/ba∇dblHr, (2.8)\nwhere the constant depends on T. In particular, if the solution uto problem ( 1.2) satisfies\n/integraldisplayT\n0/ba∇dblu(s)/ba∇dbl4\nL∞ds <∞, (2.9)\nthen it is unique.\nMoreover, if r= 2,3 then the solution is a strong solution in the sense of Definition 2.1.\nThe next theorem shows that the strong solution is H¨ older continu ous in time.\nTheorem 2.3. LetT >0 andube the unique strong solution of ( 1.2) with initial data\nu0∈H2. Then\nu∈C0,α(0,T;L2)∩C0,β(0,T;L∞),\nwhereα∈/parenleftbig\n0,1\n2/bracketrightbig\nandβ∈/parenleftbig\n0,1\n2−d\n8/bracketrightbig\n.\nRemark2.4.It can be seen that assumption ( 2.9) holds in many different cases.\n(i) Ifr= 0, assumption ( 2.9) holds for d= 1 ord= 2. Indeed, in case d= 1, we\nhave by using the Gagliardo–Nirenberg inequality (Theorem 6.2) and the fact that\nu∈L2(0,T;H2),\n/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl4\nL∞ds/lessorsimilar/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl2\nL2/ba∇dblu(s)/ba∇dbl2\nH1ds≤/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl2\nH1ds/lessorsimilar1,\nand in case d= 2,\n/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl4\nL∞ds/lessorsimilar/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl2\nL2/ba∇dblu(s)/ba∇dbl2\nH2ds≤/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl2\nH2ds/lessorsimilar1.6 AGUS L. SOENJAYA AND THANH TRAN\n(ii) Ifr∈ {1,2,3}, assumption ( 2.9) holds for d= 1,2,3. In this case, with d= 3, we\nhave\n/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl4\nL∞ds/lessorsimilar/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl2\nH1/ba∇dblu(s)/ba∇dbl2\nH2ds≤/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl2\nH2ds/lessorsimilar1.\n3.Faedo–Galerkin Approximation\nLet{ei}∞\ni=1denote an orthonormal basis of L2consisting of eigenvectors for −∆ such that\n−∆ei=λieiin Ω and∂ei\n∂n=0on∂Ω,\nwhereλi>0 are the eigenvalues of −∆, associated with ei. By elliptic regularity results, ei\nis smooth up to the boundary, and we also have\n∆2ei=λ2\nieiin Ω and∂ei\n∂n=∂∆ei\n∂n=0on∂Ω.\nLetVn:= span{e1,...,en}and Π n:L2→Vnbe the orthogonal projection defined by\n/an}b∇acketle{tΠnv,φ/an}b∇acket∇i}htL2=/an}b∇acketle{tv,φ/an}b∇acket∇i}htL2for allφ∈Vnand allv∈L2.\nNote that Π nis self-adjoint and satisfies\n/ba∇dblΠnv/ba∇dblL2≤ /ba∇dblv/ba∇dblL2for allv∈L2. (3.1)\nTo prove the existence of a weak solution to ( 1.2), we will use the Faedo–Galerkin method.\nWe first prove the following two lemmas.\nLemma 3.1. For any vector-valued function v: Ω→R3, we have\n∇(|v|2v) = 2v(v·∇v)+|v|2∇v, (3.2)\n∂/parenleftbig\n|v|2v/parenrightbig\n∂n= 2v/parenleftBig\nv·∂v\n∂n/parenrightBig\n+|v|2∂v\n∂n, (3.3)\n∆(|v|2v) = 2|∇v|2v+2(v·∆v)v+4∇v(v·∇v)⊤+|v|2∆v, (3.4)\nprovided that the partial derivatives are well defined.\nProof.Recall the notations introduced in ( 2.1) and (2.2). Also note that\n∇(|v|2) = 2(v·∇v)⊤and ∆(|v|2) = 2|∇v|2+2v·∆v.\nHence, it follows from the product rule that\n∇(|v|2v) =v/parenleftbig\n∇(|v|2)/parenrightbig⊤+|v|2∇v= 2v(v·∇v)+|v|2∇v,\nproving ( 3.2). Identity ( 3.3) then follows from ( 3.2) and the definition of normal derivatives.\nFinally, the product rule gives\n∆(|v|2v) = ∆(|v|2)v+2∇v∇(|v|2)+|v|2∆v\n= 2|∇v|2v+2(v·∆v)v+4∇v(v·∇v)⊤+|v|2∆v,\nproving ( 3.4). /squareGLOBAL SOLUTIONS OF LLBAR EQUATION 7\nLemma 3.2. For each n∈Nandv∈Vn, define\nF1\nn(v) = ∆v,\nF2\nn(v) = ∆2v,\nF3\nn(v) = Πn(|v|2v),\nF4\nn(v) = Πn(v×∆v),\nF5\nn(v) = Πn∆(|v|2v).\nThenFj\nn,j= 1,...,5, are well-defined mappings from VnintoVn. Moreover, F1\nnandF2\nn\nare globally Lipschitz while F3\nn,F4\nn, andF5\nnare locally Lipschitz.\nProof.For anyv∈Vn, sincev=/summationtextn\ni=1/an}b∇acketle{tv,ei/an}b∇acket∇i}htL2ei, we have\n−∆v=n/summationdisplay\ni=1λi/an}b∇acketle{tv,ei/an}b∇acket∇i}htL2ei∈Vnand ∆2v=n/summationdisplay\ni=1λ2\ni/an}b∇acketle{tv,ei/an}b∇acket∇i}htL2ei∈Vn.\nTherefore, F1\nnandF2\nnmapVnintoVn. Moreover, if the boundary of Ω is sufficiently smooth,\nthen the eigenfunctions ei,i∈N, are smooth functions, and so is v∈Vn. This implies that\n|v|2v, ∆(|v|2v), andv×∆vall belong to L2(Ω), so that F3\nn,F4\nn, andF5\nnare well defined.\nWe now prove the Lipschitz property of these mappings. Using the t riangle inequality, the\northonormality of {ei}and H¨ older’s inequality, for any v,w∈Vnand forj= 1,2, we have\n/vextenddouble/vextenddoubleFj\nn(v)−Fj\nn(w)/vextenddouble/vextenddouble2\nL2=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddoublen/summationdisplay\ni=1λj\ni/an}b∇acketle{tv−w,ei/an}b∇acket∇i}htL2ei/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2\n=n/summationdisplay\ni=1λ2j\ni|/an}b∇acketle{tv−w,ei/an}b∇acket∇i}htL2|2≤/parenleftBign/summationdisplay\ni=1λ2j\ni/parenrightBig\n/ba∇dblv−w/ba∇dbl2\nL2.\nHence,F1\nnandF2\nnare globally Lipschitz.\nNext, it follows from ( 3.1) that\n/ba∇dblF3\nn(v)−F3\nn(w)/ba∇dblL2≤ /ba∇dbl|v|2v−|w|2w/ba∇dblL2\n≤ /ba∇dbl|v|2(v−w)/ba∇dblL2+/ba∇dbl(v−w)·(v+w)w/ba∇dblL2\n≤/parenleftbig\n/ba∇dblv/ba∇dbl2\nL∞+/ba∇dblv+w/ba∇dblL∞/ba∇dblw/ba∇dblL∞/parenrightbig\n/ba∇dblv−w/ba∇dblL2,\nwhere we used the fact that all norms are equivalent in the finite dime nsional subspace Vn.\nThis shows that F3\nnis locally Lipschitz.\nSimilarly, it follows from ( 3.1) that\n/ba∇dblF4\nn(v)−F4\nn(w)/ba∇dblL2≤ /ba∇dblv×∆v−w×∆w/ba∇dblL2\n≤ /ba∇dblv×(∆v−∆w)/ba∇dblL2+/ba∇dbl(v−w)×∆w/ba∇dblL2\n≤ /ba∇dblv/ba∇dblL∞/ba∇dblF2\nn(v)−F2\nn(w)/ba∇dblL2+/ba∇dblv−w/ba∇dblL2/ba∇dbl∆w/ba∇dblL∞.\nSinceF2\nnis Lipschitz, we deduce that F4\nnis locally Lipschitz.\nFinally, note that if v∈Vn, then∂v/∂n= 0. Thus ( 3.3) implies ∂/parenleftbig\n|v|2v/parenrightbig\n/∂n= 0, which\nallows us to use integration by parts to obtain\n/angbracketleftbig\n∆/parenleftbig\n|v|2v/parenrightbig\n,w/angbracketrightbig\n=−/angbracketleftbig\n∇/parenleftbig\n|v|2v/parenrightbig\n,∇w/angbracketrightbig\n=/angbracketleftbig\n|v|2v,∆w/angbracketrightbig\n∀v,w∈Vn.8 AGUS L. SOENJAYA AND THANH TRAN\nTherefore, for any v,w∈Vn, we can use the definition of Π nand integration by parts again\nto have\n/angbracketleftbig\nΠn∆/parenleftbig\n|v|2v/parenrightbig\n,w/angbracketrightbig\n=/angbracketleftbig\n∆/parenleftbig\n|v|2v/parenrightbig\n,w/angbracketrightbig\n=/angbracketleftbig\n|v|2v,∆w/angbracketrightbig\n=/angbracketleftbig\nΠn/parenleftbig\n|v|2v/parenrightbig\n,∆w/angbracketrightbig\n=/angbracketleftbig\n∆Πn/parenleftbig\n|v|2v/parenrightbig\n,w/angbracketrightbig\n.\nThis means ∆ and Π ncommute, so that F5\nn(v) =F1\nn◦F3\nn(v). SinceF1\nnis Lipschitz and F3\nnis\nlocally Lipschitz, we have that F5\nnis locally Lipschitz as well. This completes the proof. /square\nThe Faedo–Galerkin method seeks to approximate the solution to ( 1.2) byun(t)∈Vn\nsatisfying the equation\n\n\n∂un\n∂t−β1∆un+β2∆2un−β3Πn((1−|un|2)un)\n+β4Πn(un×∆un)−β5Πn(∆(|un|2un)) =0in (0,T)×Ω,\nun(0) =u0n in Ω,(3.5)\nwhereu0n∈Vnis an approximation of u0such that if u0∈Hr, then\n/ba∇dblu0n/ba∇dblHr/lessorsimilar/ba∇dblu0/ba∇dblHr (3.6)\nLemma3.2assures us that all the terms in ( 3.5) are well defined. Moreover, the existence\nof solutions to the above ordinary differential equation in Vnis guaranteed by this lemma\nand the Cauchy–Lipschitz theorem.\nWe now prove some a priori estimates for the solution of ( 3.5). First we need the following\nresults.\nLemma 3.3. Let Ω⊂Rdbe an open bounded domain with C∞-boundary and ǫ >0 be\ngiven. Then there exists a positive constant Csuch that the following inequalities hold:\n(i) for any v∈L2(Ω) such that ∆ v∈L2(Ω) satisfying∂v\n∂n= 0 on∂Ω,\n/ba∇dblv/ba∇dbl2\nH2≤C/parenleftbig\n/ba∇dblv/ba∇dbl2\nL2+/ba∇dbl∆v/ba∇dbl2\nL2/parenrightbig\n, (3.7)\n/ba∇dbl∇v/ba∇dbl2\nL2≤C/ba∇dblv/ba∇dbl2\nL2+ε/ba∇dbl∆v/ba∇dbl2\nL2, (3.8)\n(ii) for any v∈H1(Ω) such that ∆ v∈H1satisfying∂v\n∂n= 0 on∂Ω,\n/ba∇dbl∆v/ba∇dbl2\nL2≤ /ba∇dbl∇v/ba∇dblL2/ba∇dbl∇∆v/ba∇dblL2, (3.9)\n/ba∇dblv/ba∇dbl2\nH3≤C/parenleftbig\n/ba∇dblv/ba∇dbl2\nL2+/ba∇dbl∇v/ba∇dbl2\nL2+/ba∇dbl∇∆v/ba∇dbl2\nL2/parenrightbig\n, (3.10)\n(iii) for any v∈H2(Ω) such that ∆2v∈L2(Ω) satisfying∂v\n∂n=∂∆v\n∂n= 0 on∂Ω,\n/ba∇dbl∇∆v/ba∇dbl2\nL2≤ /ba∇dbl∆v/ba∇dblL2/vextenddouble/vextenddouble∆2v/vextenddouble/vextenddouble\nL2, (3.11)\n/ba∇dblv/ba∇dbl2\nH4≤C/parenleftBig\n/ba∇dblv/ba∇dbl2\nL2+/ba∇dbl∆v/ba∇dbl2\nL2+/vextenddouble/vextenddouble∆2v/vextenddouble/vextenddouble2\nL2/parenrightBig\n, (3.12)\n(iv) for any v∈H3(Ω) such that ∆2v∈H1(Ω) satisfying∂v\n∂n=∂∆v\n∂n= 0 on∂Ω,\n/ba∇dblv/ba∇dbl2\nH5≤C/parenleftBig\n/ba∇dblv/ba∇dbl2\nL2+/ba∇dbl∇v/ba∇dbl2\nL2+/ba∇dbl∇∆v/ba∇dbl2\nL2+/vextenddouble/vextenddouble∇∆2v/vextenddouble/vextenddouble2\nL2/parenrightBig\n. (3.13)GLOBAL SOLUTIONS OF LLBAR EQUATION 9\nProof.Inequality ( 3.7) follows from the standard elliptic regularity result with Neumann\nboundary data [ 19, Corollary 2.2.2.6]. Next, using integration by parts, H¨ older’s inequa lity,\nand Young’s inequality, we obtain\n/ba∇dbl∇v/ba∇dbl2\nL2=/an}b∇acketle{t∇v,∇v/an}b∇acket∇i}htL2=−/an}b∇acketle{tv,∆v/an}b∇acket∇i}ht ≤ /ba∇dblv/ba∇dblL2/ba∇dbl∆v/ba∇dblL2≤C/ba∇dblv/ba∇dbl2\nL2+ε/ba∇dbl∆v/ba∇dbl2\nL2,\nproving ( 3.8).\nSimilarly, we have\n/ba∇dbl∆v/ba∇dbl2\nL2=−/an}b∇acketle{t∇∆v,∇v/an}b∇acket∇i}ht ≤ /ba∇dbl∇∆v/ba∇dblL2/ba∇dbl∇v/ba∇dblL2\nand\n/ba∇dbl∇∆v/ba∇dbl2\nL2=−/angbracketleftbig\n∆v,∆2v/angbracketrightbig\n≤ /ba∇dbl∆v/ba∇dblL2/vextenddouble/vextenddouble∆2v/vextenddouble/vextenddouble\nL2,\nproving ( 3.9) and (3.11).\nNext, applying the higher-order regularity result [ 19, Remark 2.5.1.2] to the elliptic oper-\nator (−∆+I) yields\n/ba∇dblv/ba∇dbl2\nH3≤C/parenleftbig\n/ba∇dblv/ba∇dbl2\nH1+/ba∇dbl∆v/ba∇dbl2\nH1/parenrightbig\n≤C/parenleftbig\n/ba∇dblv/ba∇dbl2\nL2+/ba∇dbl∇v/ba∇dbl2\nL2+/ba∇dbl∇∆v/ba∇dbl2\nL2/parenrightbig\n,\nproving ( 3.10). Furthermore, by the same elliptic regularity result,\n/ba∇dblv/ba∇dbl2\nH4≤C/parenleftbig\n/ba∇dblv/ba∇dbl2\nL2+/ba∇dbl∆v/ba∇dbl2\nH2/parenrightbig\n≤C/parenleftbig\n/ba∇dblv/ba∇dbl2\nL2+/ba∇dbl∆v/ba∇dbl2\nL2+/vextenddouble/vextenddouble∆2v/vextenddouble/vextenddouble2\nL2/parenrightbig\n,\nwhere we applied ( 3.7) to ∆v, noting that the assumptions are satisfied. This proves ( 3.12).\nSimilarly, we have\n/ba∇dblv/ba∇dbl2\nH5≤C/parenleftbig\n/ba∇dbl∆v/ba∇dbl2\nH3+/ba∇dblv/ba∇dbl2\nH3/parenrightbig\n≤C/parenleftbig\n/ba∇dblv/ba∇dbl2\nL2+/ba∇dbl∇v/ba∇dbl2\nL2+/ba∇dbl∇∆v/ba∇dbl2\nL2+/vextenddouble/vextenddouble∇∆2v/vextenddouble/vextenddouble2\nL2/parenrightbig\n,\nwhere we applied ( 3.10) to ∆v, thus proving ( 3.13). /square\nRemark 3.4.Elliptic regularity result in H2(3.7) holds more generally for a domain with\nC1,1-boundary[ 19, Remark 2.2.2.6]or a convex polygonal domain[ 19, Theorem 4.3.1.4]. The\nHr-regularity results also hold for a domain with Cr−1,1-boundary (see [ 19, Remark 2.5.1.2]).\nLemma 3.5. Letk= (k1,···,kd) be a multi-index and define the operator\nDk:=∂|k|\n∂k1x1···∂kdxd\nwhere|k|=k1+···+kd. Then the following inequalities hold:\n(i) for any u,v∈Hs(Ω), where s > d/2,\n/ba∇dbl|u||v|/ba∇dblHs≤C/ba∇dblu/ba∇dblHs/ba∇dblv/ba∇dblHs. (3.14)\n(ii) for any u,v∈H2∩H|k|,\n/vextenddouble/vextenddoubleDk(|u|2u−|v|2v)/vextenddouble/vextenddouble\nL2/lessorsimilar\n\n/parenleftbig\n/ba∇dblu/ba∇dbl2\nL∞+/ba∇dblv/ba∇dbl2\nL∞/parenrightbig\n/ba∇dblu−v/ba∇dblL2 if|k|= 0,\n/parenleftbig\n/ba∇dblu/ba∇dblH1+/ba∇dblv/ba∇dblH1/parenrightbig/parenleftbig\n/ba∇dblu/ba∇dblH2+/ba∇dblv/ba∇dblH2/parenrightbig\n/ba∇dblu−v/ba∇dblH1if|k|= 1,\n/parenleftbig\n/ba∇dblu/ba∇dbl2\nH|k|+/ba∇dblv/ba∇dbl2\nH|k|/parenrightbig\n/ba∇dblu−v/ba∇dblH|k| if|k| ≥2.\n(3.15)\n(iii) for any u,v∈H|k|∩L∞,\n/vextenddouble/vextenddoubleu×Dku−v×Dkv/vextenddouble/vextenddouble\nL2/lessorsimilar/ba∇dblu/ba∇dblL∞/vextenddouble/vextenddoubleDk(u−v)/vextenddouble/vextenddouble\nL2+/vextenddouble/vextenddouble/parenleftbig\nu−v/parenrightbig\nDkv/vextenddouble/vextenddouble\nL2.(3.16)10 AGUS L. SOENJAYA AND THANH TRAN\nProof.Firstly, (3.14)followsfrom /ba∇dbl|v||w|/ba∇dblHs/lessorsimilar/ba∇dblv/ba∇dblL∞/ba∇dblw/ba∇dblHs+/ba∇dblv/ba∇dblHs/ba∇dblw/ba∇dblL∞andtheSobolev\nembedding. Next, we prove ( 3.15). For the case |k|= 0, by H¨ older’s inequality we have/vextenddouble/vextenddouble|u|2u−|v|2v/vextenddouble/vextenddouble\nL2≤/vextenddouble/vextenddouble|u|2(u−v)/vextenddouble/vextenddouble\nL2+/ba∇dbl|u+v||u−v||v/ba∇dblL2\n≤/parenleftbig\n/ba∇dblu/ba∇dbl2\nL∞+/ba∇dblv/ba∇dblL∞/ba∇dblu+v/ba∇dblL∞/parenrightbig\n/ba∇dblu−v/ba∇dblL2\n/lessorsimilar/parenleftbig\n/ba∇dblu/ba∇dbl2\nL∞+/ba∇dblv/ba∇dbl2\nL∞/parenrightbig\n/ba∇dblu−v/ba∇dblL2.\nFor the case |k|= 1, note that/vextendsingle/vextendsingleDk(|u|2u−|v|2v)/vextendsingle/vextendsingle≤/vextendsingle/vextendsingleDk(|u|2)(u−v)/vextendsingle/vextendsingle+/vextendsingle/vextendsingle|u|2Dk(u−v)/vextendsingle/vextendsingle+/vextendsingle/vextendsingleDk(|u|2−|v|2)v/vextendsingle/vextendsingle\n+/vextendsingle/vextendsingle(|u|2−|v|2)Dkv/vextendsingle/vextendsingle\n≤2|u|/vextendsingle/vextendsingleDk(u)/vextendsingle/vextendsingle/vextendsingle/vextendsingleu−v/vextendsingle/vextendsingle+|u|2/vextendsingle/vextendsingleDk(u−v)/vextendsingle/vextendsingle+/vextendsingle/vextendsingleDku+Dkv/vextendsingle/vextendsingle|u−v||v|\n+|u+v|/vextendsingle/vextendsingleDku−Dkv/vextendsingle/vextendsingle|v|+|u+v||u+v|/vextendsingle/vextendsingleDkv/vextendsingle/vextendsingle.\nUsing this, we have/vextenddouble/vextenddoubleDk(|u|2u−|v|2v)/vextenddouble/vextenddouble\nL2\n≤2/vextenddouble/vextenddouble|u|/vextendsingle/vextendsingleDk(u)/vextendsingle/vextendsingle|u−v|/vextenddouble/vextenddouble\nL2+/vextenddouble/vextenddouble|u|2Dk(u−v)/vextenddouble/vextenddouble\nL2+/vextenddouble/vextenddouble|Dku+Dkv| |u−v| |v|/vextenddouble/vextenddouble\nL2\n+/vextenddouble/vextenddouble|u+v| |Dku−Dkv| |v|/vextenddouble/vextenddouble\nL2+/vextenddouble/vextenddouble|u+v| |u−v| |Dkv|/vextenddouble/vextenddouble\nL2\n/lessorsimilar/parenleftbig\n/ba∇dblu/ba∇dblL6/vextenddouble/vextenddoubleDku/vextenddouble/vextenddouble\nL6+/ba∇dblu/ba∇dblL6/vextenddouble/vextenddoubleDkv/vextenddouble/vextenddouble\nL6+/ba∇dblv/ba∇dblL6/vextenddouble/vextenddoubleDku/vextenddouble/vextenddouble\nL6+/ba∇dblv/ba∇dblL6/vextenddouble/vextenddoubleDkv/vextenddouble/vextenddouble\nL6/parenrightbig\n/ba∇dblu−v/ba∇dblL2\n+/parenleftbig\n/ba∇dblu/ba∇dbl2\nL∞+/ba∇dblv/ba∇dbl2\nL∞/parenrightbig/vextenddouble/vextenddoubleDku−Dkv/vextenddouble/vextenddouble\nL2\n/lessorsimilar/parenleftbig\n/ba∇dblu/ba∇dblH1/vextenddouble/vextenddoubleDku/vextenddouble/vextenddouble\nH1+/ba∇dblu/ba∇dblH1/vextenddouble/vextenddoubleDkv/vextenddouble/vextenddouble\nH1+/ba∇dblv/ba∇dblH1/vextenddouble/vextenddoubleDku/vextenddouble/vextenddouble\nH1+/ba∇dblv/ba∇dblH1/vextenddouble/vextenddoubleDkv/vextenddouble/vextenddouble\nH1/parenrightbig\n/ba∇dblu−v/ba∇dblH1\n+/parenleftbig\n/ba∇dblu/ba∇dblH1/ba∇dblu/ba∇dblH2+/ba∇dblv/ba∇dblH1/ba∇dblv/ba∇dblH2/parenrightbig\n/ba∇dblu−v/ba∇dblH1\n/lessorsimilar/parenleftbig\n/ba∇dblu/ba∇dblH1/ba∇dblu/ba∇dblH2+/ba∇dblu/ba∇dblH1/ba∇dblv/ba∇dblH2+/ba∇dblv/ba∇dblH1/ba∇dblu/ba∇dblH2+/ba∇dblv/ba∇dblH1/ba∇dblv/ba∇dblH2/parenrightbig\n/ba∇dblu−v/ba∇dblH1\n=/parenleftbig\n/ba∇dblu/ba∇dblH1+/ba∇dblv/ba∇dblH1/parenrightbig/parenleftbig\n/ba∇dblu/ba∇dblH2+/ba∇dblv/ba∇dblH2/parenrightbig\n/ba∇dblu−v/ba∇dblH1,\nwhere in the third step we used Gagliardo–Nirenberg’s inequality /ba∇dblu/ba∇dbl2\nL∞/lessorsimilar/ba∇dblu/ba∇dblH1/ba∇dblv/ba∇dblH2\n(valid for d∈ {1,2,3}) and the Sobolev embedding H1⊂L6.\nFor the case |k| ≥2, we use H¨ older’s inequality and ( 3.14) to obtain\n/vextenddouble/vextenddoubleDk(|u|2u−|v|2v)/vextenddouble/vextenddouble\nL2≤/vextenddouble/vextenddoubleDk(|u|2u−|v|2v)/vextenddouble/vextenddouble\nL2≤/vextenddouble/vextenddouble|u|2u−|v|2v/vextenddouble/vextenddouble\nH|k|\n≤/parenleftbig\n/ba∇dblu/ba∇dbl2\nH|k|+/ba∇dblv/ba∇dblH|k|/ba∇dblu+v/ba∇dblH|k|/parenrightbig\n/ba∇dblu−v/ba∇dblH|k|\n/lessorsimilar/parenleftbig\n/ba∇dblu/ba∇dbl2\nH|k|+/ba∇dblv/ba∇dbl2\nH|k|/parenrightbig\n/ba∇dblu−v/ba∇dblH|k|.\nThis completes the proof of ( 3.15). The proof of ( 3.16) is obvious and is omitted. /square\nWe now use the above lemmas to derive a priori estimates on the Galer kin solution un.\nProposition 3.6. LetT >0 be arbitrary and assume that u0∈L2. For each n∈Nand\nallt∈[0,T],\n/ba∇dblun(t)/ba∇dbl2\nL2+/integraldisplayt\n0/ba∇dbl∇un(s)/ba∇dbl2\nL2ds+/integraldisplayt\n0/ba∇dbl∆un(s)/ba∇dbl2\nL2ds+/integraldisplayt\n0/ba∇dblun(s)/ba∇dbl4\nL4ds\n+/integraldisplayt\n0/ba∇dbl|un(s)||∇un(s)|/ba∇dbl2\nL2ds+/integraldisplayt\n0/ba∇dblun(s)·∇un(s)/ba∇dbl2\nL2ds/lessorsimilar/ba∇dblu0/ba∇dbl2\nL2,(3.17)\nwhere the constant depends on T, but is independent of n.GLOBAL SOLUTIONS OF LLBAR EQUATION 11\nProof.Taking the inner product of ( 3.5) withun(t), integrating by parts with respect to x\n(noting ( 3.3)) and using the identity ( 3.2), we obtain, for any ǫ >0,\n1\n2d\ndt/ba∇dblun/ba∇dbl2\nL2+β2/ba∇dbl∆un/ba∇dbl2\nL2+β3/ba∇dblun/ba∇dbl4\nL4+2β5/ba∇dblun·∇un/ba∇dbl2\nL2+β5/ba∇dbl|un||∇un|/ba∇dbl2\nL2\n≤β3/ba∇dblun/ba∇dbl2\nL2+|β1|/ba∇dbl∇un/ba∇dbl2\nL2≤C/ba∇dblun/ba∇dbl2\nL2+ǫ/ba∇dbl∆un/ba∇dbl2\nL2,\nwhere in the last step we used ( 3.8). Rearranging the above equation, choosing sufficiently\nsmallǫ, and integrating over (0 ,t), we deduce\n/ba∇dblun(t)/ba∇dbl2\nL2+/integraldisplayt\n0/ba∇dbl∆un(s)/ba∇dbl2\nL2ds+/integraldisplayt\n0/ba∇dblun(s)/ba∇dbl4\nL4ds+/integraldisplayt\n0/ba∇dbl|un(s)||∇un(s)|/ba∇dbl2\nL2ds\n+/integraldisplayt\n0/ba∇dblun(s)·∇un(s)/ba∇dbl2\nL2ds/lessorsimilar/ba∇dblun(0)/ba∇dbl2\nL2+/integraldisplayt\n0/ba∇dblun(s)/ba∇dbl2\nL2ds.\nInvoking Gronwall’s inequality yields the required estimate for all the t erms on the left-\nhand side of ( 3.17), except/integraltextt\n0/ba∇dbl∇un(s)/ba∇dbl2\nL2ds. The bound for this term follows from ( 3.8),\ncompleting the proof of the proposition. /square\nProposition 3.7. Under the assumption of Proposition 3.6, we have\n/ba∇dbl∂tun/ba∇dblL2(0,T;H−2)/lessorsimilar/ba∇dblu0/ba∇dbl2\nL2,\nwhere the constant depends on Tbut is independent of n.\nProof.Taking the inner product of ( 3.5) withϕ∈H2such that /ba∇dblϕ/ba∇dblH2≤1 and integrating\nby parts with respect to x, we have\n/an}b∇acketle{t∂tun,ϕ/an}b∇acket∇i}htL2=−β1/an}b∇acketle{t∇un,∇ϕ/an}b∇acket∇i}htL2−β2/an}b∇acketle{t∆un,∆ϕ/an}b∇acket∇i}htL2+β3/an}b∇acketle{tun,ϕ/an}b∇acket∇i}htL2−β3/angbracketleftbig\n|un|2un,ϕ/angbracketrightbig\nL2\n+β4/an}b∇acketle{tun×∇un,∇ϕ/an}b∇acket∇i}htL2+β5/angbracketleftbig\n∇(|un|2un),∇ϕ/angbracketrightbig\nL2.\nIt follows from this equation and H¨ older’s inequality that/vextendsingle/vextendsingle/an}b∇acketle{t∂tun,ϕ/an}b∇acket∇i}htL2/vextendsingle/vextendsingle/lessorsimilar/ba∇dblun/ba∇dblH1/ba∇dblϕ/ba∇dblH1+/ba∇dblun/ba∇dblH2/ba∇dblϕ/ba∇dblH2+/ba∇dblun/ba∇dblL2/ba∇dblϕ/ba∇dblL2+/ba∇dblun/ba∇dblL2/ba∇dblun/ba∇dbl2\nL6/ba∇dblϕ/ba∇dblL6\n+/ba∇dblun/ba∇dblL4/ba∇dbl∇un/ba∇dblL2/ba∇dbl∇ϕ/ba∇dblL4+/ba∇dblun/ba∇dblL2/ba∇dblun/ba∇dblL6/ba∇dbl∇un/ba∇dblL6/ba∇dbl∇ϕ/ba∇dblL6,\nwhere we used the Sobolev embedding H1⊂L6. Therefore, integrating over (0 ,t) and noting\nthat/ba∇dblϕ/ba∇dblH2≤1, we obtain\n/integraldisplayt\n0/vextendsingle/vextendsingle/an}b∇acketle{t∂sun(s),ϕ/an}b∇acket∇i}htL2/vextendsingle/vextendsingle2ds/lessorsimilar/integraldisplayt\n0/ba∇dblun/ba∇dbl2\nH2ds/lessorsimilar1\nby Proposition 3.6. Taking supremum over the set {ϕ∈H2:/ba∇dblϕ/ba∇dblH2≤1}and noting that\n/ba∇dbl∂tun/ba∇dblH−2≤sup\nϕ∈H2/vextendsingle/vextendsingle/an}b∇acketle{t∂tun,ϕ/an}b∇acket∇i}htL2/vextendsingle/vextendsingle\n/ba∇dblϕ/ba∇dblH2,\nwe obtain the required estimate. /square\nProposition 3.8. LetT >0 be arbitrary and assume that u0∈H1. Then there exists\nT∗>0 such that for n∈Nandt∈[0,T∗], we have\n/ba∇dbl∇un(t)/ba∇dbl2\nL2+/integraldisplayt\n0/ba∇dbl∇∆un(s)/ba∇dbl2\nL2ds+/integraldisplayt\n0/ba∇dbl|un(s)||∆un(s)|/ba∇dbl2\nL2ds+/integraldisplayt\n0/ba∇dblun(s)·∆un(s)/ba∇dbl2\nL2ds\n/lessorsimilar/ba∇dblun(0)/ba∇dbl2\nH1/lessorsimilar/ba∇dblu0/ba∇dbl2\nH1.12 AGUS L. SOENJAYA AND THANH TRAN\nThe constant depends on T∗, but is independent of n. Here,\n/braceleftBigg\nT∗=Tford= 1,2,\nT∗≤Tford= 3,\nwhereT∗=T∗(/ba∇dblu0/ba∇dblH1).\nProof.Taking the inner product of ( 3.5) with−∆un(t) andintegrating by parts with respect\ntox, we have\n1\n2d\ndt/ba∇dbl∇un/ba∇dbl2\nL2+β1/ba∇dbl∆un/ba∇dbl2\nL2+β2/ba∇dbl∇∆un/ba∇dbl2\nL2−β3/ba∇dbl∇un/ba∇dbl2\nL2\n+β3/angbracketleftbig\n∇(|un|2un),∇un/angbracketrightbig\nL2+β5/angbracketleftbig\n∆(|un|2un),∆un/angbracketrightbig\nL2= 0.(3.18)\nIt follows from ( 3.2) and (3.4) that\n/angbracketleftbig\n∇(|un|2un),∇un/angbracketrightbig\nL2= 2/ba∇dblun·∇un/ba∇dbl2\nL2+/ba∇dbl|un||∇un|/ba∇dbl2\nL2\nand\n/angbracketleftbig\n∆(|un|2un),∆un/angbracketrightbig\nL2= 2/angbracketleftbig\n|∇un|2un,∆un/angbracketrightbig\nL2+2/ba∇dblun·∆un/ba∇dbl2\nL2\n+4/angbracketleftbig\n∇un(un·∇un)⊤,∆un/angbracketrightbig\nL2+/ba∇dbl|un||∆un|/ba∇dbl2\nL2.\nTherefore, after rearranging the terms in ( 3.18), we obtain\n1\n2d\ndt/ba∇dbl∇un/ba∇dbl2\nL2+β2/ba∇dbl∇∆un/ba∇dbl2\nL2+2β3/ba∇dblun·∇un/ba∇dbl2\nL2+β3/ba∇dbl|un||∇un|/ba∇dbl2\nL2\n+2β5/ba∇dblun·∆un/ba∇dbl2\nL2+β5/ba∇dbl|un||∆un|/ba∇dbl2\nL2\n=−β1/ba∇dbl∆un/ba∇dbl2\nL2+β3/ba∇dbl∇un/ba∇dbl2\nL2−2β5/angbracketleftbig\n|∇un|2un,∆un/angbracketrightbig\nL2−4β5/angbracketleftbig\n∇un(un·∇un)⊤,∆un/angbracketrightbig\nL2\n≤ |β1|/ba∇dbl∆un/ba∇dbl2\nL2+β3/ba∇dbl∇un/ba∇dbl2\nL2+6β5/integraldisplay\nΩ|un||∆un||∇un|2dx. (3.19)\nUsing H¨ older’s inequality and Young’s inequality, we can estimate the la st term on the\nright-hand side by\n6β5/integraldisplay\nΩ|un||∆un||∇un|2dx≤6β5/ba∇dbl|un||∆un|/ba∇dblL2/ba∇dbl∇un/ba∇dbl2\nL4\n≤ǫ/ba∇dbl|un||∆un|/ba∇dbl2\nL2+C/ba∇dbl∇un/ba∇dbl4\nL4,\nwhereε >0 is sufficiently small. This inequality together with ( 3.19) yields\nd\ndt/ba∇dbl∇un/ba∇dbl2\nL2+/ba∇dbl∇∆un/ba∇dbl2\nL2+/ba∇dblun·∇un/ba∇dbl2\nL2+/ba∇dbl|un||∇un|/ba∇dbl2\nL2+/ba∇dblun·∆un/ba∇dbl2\nL2+/ba∇dbl|un||∆un|/ba∇dbl2\nL2\n/lessorsimilar/ba∇dbl∇un/ba∇dbl2\nL2+/ba∇dbl∆un/ba∇dbl2\nL2+/ba∇dbl∇un/ba∇dbl4\nL4. (3.20)\nWeestimate thelast term onthe right-handside of ( 3.20) byinvoking Gagliardo–Nirenberg’s\ninequality ( 6.2).\nCase 1:d= 1. Applying inequality ( 6.2) withv=un,q= 4,r= 1,s1= 0, and s2= 3 gives\n/ba∇dbl∇un/ba∇dbl4\nL4/lessorsimilar/ba∇dblun/ba∇dbl7/3\nL2/ba∇dblun/ba∇dbl5/3\nH3/lessorsimilar/ba∇dblun/ba∇dbl5/3\nH3GLOBAL SOLUTIONS OF LLBAR EQUATION 13\nwhere in the last step we used Proposition 3.6and the assumption that un(0) =u0n∈Vn\nwhich approximates u0. Young’s inequality implies, for any ǫ >0,\n/ba∇dbl∇un/ba∇dbl4\nL4≤C+ǫ/ba∇dblun/ba∇dbl2\nH3/lessorsimilar1+/ba∇dbl∇un/ba∇dbl2\nL2+ǫ/ba∇dbl∇∆un/ba∇dbl2\nL2,\nwhere in the last step we used ( 3.10). Therefore, by choosing ǫ >0 sufficiently small, we\ndeduce from ( 3.20)\nd\ndt/ba∇dbl∇un(t)/ba∇dbl2\nL2/lessorsimilar1+/ba∇dbl∇un(t)/ba∇dbl2\nL2+/ba∇dbl∆un(t)/ba∇dbl2\nL2.\nIntegrating over (0 ,t) and using Proposition 3.6, we obtain\n/ba∇dbl∇un(t)/ba∇dbl2\nL2≤ /ba∇dbl∇un(0)/ba∇dbl2\nL2+C+C/integraldisplayt\n0/ba∇dbl∇un(s)/ba∇dbl2\nL2ds.\nGronwall’s inequality yields the required estimate.\nCase 2:d= 2. Applying inequality ( 6.2) withv=∇un,q= 4,r= 0,s1= 0, and s2= 1\ngives\n/ba∇dbl∇un/ba∇dbl4\nL4/lessorsimilar/ba∇dbl∇un/ba∇dbl2\nL2/ba∇dbl∇un/ba∇dbl2\nH1/lessorsimilar/parenleftbig\n1+/ba∇dbl∆un/ba∇dbl2\nL2/parenrightbig\n/ba∇dbl∇un/ba∇dbl2\nL2,\nwhere in the last step we used ( 3.7) and Proposition 3.6. Therefore, inequality ( 3.20) gives\nd\ndt/ba∇dbl∇un(t)/ba∇dbl2\nL2/lessorsimilar/ba∇dbl∆un(t)/ba∇dbl2\nL2+/parenleftbig\n1+/ba∇dbl∆un(t)/ba∇dbl2\nL2/parenrightbig\n/ba∇dbl∇un(t)/ba∇dbl2\nL2.\nIntegrating over (0 ,t), using Gronwall’s inequality and Proposition 3.6, we deduce\n/ba∇dbl∇un(t)/ba∇dbl2\nL2/lessorsimilar1+exp/parenleftBig/integraldisplayT\n0/parenleftbig\n1+/ba∇dbl∆un(t)/ba∇dbl2\nL2/parenrightbig\ndt/parenrightBig\n/lessorsimilar1,\nproving the result for d= 2.\nCase 3:d= 3. Applying inequality ( 6.2) withv=∇un,q= 4,r= 0,s1= 0,s2= 2, and\nusing (3.10), we infer\n/ba∇dbl∇un/ba∇dbl4\nL4/lessorsimilar/ba∇dbl∇un/ba∇dbl5/2\nL2/ba∇dbl∇un/ba∇dbl3/2\nH2/lessorsimilar/ba∇dbl∇un/ba∇dbl5/2\nL2/parenleftbig\n1+/ba∇dbl∇un/ba∇dbl2\nL2+/ba∇dbl∇∆un/ba∇dbl2\nL2/parenrightbig3/4\n/lessorsimilar/ba∇dbl∇un/ba∇dbl5/2\nL2+/ba∇dbl∇un/ba∇dbl4\nL2+/ba∇dbl∇un/ba∇dbl5/2\nL2/ba∇dbl∇∆un/ba∇dbl3/2\nL2.\nYoung’s inequality yields, for ǫ >0 sufficiently small,\n/ba∇dbl∇un/ba∇dbl4\nL4≤C/parenleftBig\n/ba∇dbl∇un/ba∇dbl5/2\nL2+/ba∇dbl∇un/ba∇dbl4\nL2+/ba∇dbl∇un/ba∇dbl10\nL2/parenrightBig\n+ǫ/ba∇dbl∇∆un/ba∇dbl2\nL2.\nInserting this estimate into ( 3.20) and rearranging the terms, we deduce\nd\ndt/ba∇dbl∇un/ba∇dbl2\nL2/lessorsimilar/ba∇dbl∇un/ba∇dbl2\nL2+/ba∇dbl∆un/ba∇dbl2\nL2+/ba∇dbl∇un/ba∇dbl5/2\nL2+/ba∇dbl∇un/ba∇dbl4\nL2+/ba∇dbl∇un/ba∇dbl10\nL2\n/lessorsimilar/ba∇dbl∇un/ba∇dbl2\nL2+/ba∇dbl∆un/ba∇dbl2\nL2+/ba∇dbl∇un/ba∇dbl10\nL2.\nIntegrating over (0 ,t) and using Proposition 3.6give\n/ba∇dbl∇un(t)/ba∇dbl2\nL2≤ /ba∇dbl∇un(0)/ba∇dbl2\nL2+C+C/integraldisplayt\n0/ba∇dbl∇un(s)/ba∇dbl10\nL2ds.\nBy using Gronwall–Bihari’s inequality (Theorem 6.1) withf(x) =x5(so thatF(x) =x−4/4\nandF−1(x) =x−1/4/√\n2), and noting ( 3.6), we obtain the required estimate for any t∈14 AGUS L. SOENJAYA AND THANH TRAN\n[0,T∗], where T∗=/parenleftbig\n/ba∇dbl∇un(0)/ba∇dbl2\nL2+C/parenrightbig−4/4. This completes the proof of the proposition.\n/square\nRemark3.9.Sinceun(0) =u0napproximates u0, we have T∗≈ /ba∇dbl∇u0/ba∇dbl−8\nL2.\nProposition 3.10. Under the assumption of Proposition 3.8, we have\n/ba∇dbl∂tun/ba∇dbl2\nL2(0,T∗;H−1)/lessorsimilar/ba∇dblu0/ba∇dbl2\nH1,\nwhere the constant depends on T∗but is independent of n.\nProof.Taking the inner product of ( 3.5) withϕ∈H1such that /ba∇dblϕ/ba∇dblH1≤1 and integrating\nby parts with respect to x, we have\n/an}b∇acketle{t∂tun,ϕ/an}b∇acket∇i}htL2=−β1/an}b∇acketle{t∇un,∇ϕ/an}b∇acket∇i}htL2+β2/an}b∇acketle{t∇∆un,∇ϕ/an}b∇acket∇i}htL2+β3/an}b∇acketle{tun,ϕ/an}b∇acket∇i}htL2−β3/angbracketleftbig\n|un|2un,ϕ/angbracketrightbig\nL2\n+β4/an}b∇acketle{tun×∇un,∇ϕ/an}b∇acket∇i}htL2+β5/angbracketleftbig\n∇(|un|2un),∇ϕ/angbracketrightbig\nL2.\nIt follows from this equation and H¨ older’s inequality that\n/vextendsingle/vextendsingle/an}b∇acketle{t∂tun,ϕ/an}b∇acket∇i}htL2/vextendsingle/vextendsingle/lessorsimilar/ba∇dblun/ba∇dblH1/ba∇dblϕ/ba∇dblH1+/ba∇dblun/ba∇dblH3/ba∇dblϕ/ba∇dblH1+/ba∇dblun/ba∇dblL2/ba∇dblϕ/ba∇dblL2+/ba∇dblun/ba∇dblL2/ba∇dblun/ba∇dbl2\nL6/ba∇dblϕ/ba∇dblL6\n+/ba∇dblun/ba∇dblL4/ba∇dbl∇un/ba∇dblL4/ba∇dbl∇ϕ/ba∇dblL2+/ba∇dblun/ba∇dblL6/ba∇dblun/ba∇dblL6/ba∇dbl∇un/ba∇dblL6/ba∇dbl∇ϕ/ba∇dblL2.\nTherefore, integrating over (0 ,t), using the Sobolev embedding H1⊂L6, and noting that\n/ba∇dblϕ/ba∇dblH1≤1, we obtain\n/integraldisplayt\n0/vextendsingle/vextendsingle/an}b∇acketle{t∂sun(s),ϕ/an}b∇acket∇i}htL2/vextendsingle/vextendsingle2ds/lessorsimilar/integraldisplayt\n0/ba∇dblun/ba∇dbl2\nH3ds/lessorsimilar1\nby Proposition 3.6. Taking supremum over the set {ϕ∈H1:/ba∇dblϕ/ba∇dblH1≤1}and noting that\n/ba∇dbl∂tun/ba∇dblH−1≤sup\nϕ∈H1/vextendsingle/vextendsingle/an}b∇acketle{t∂tun,ϕ/an}b∇acket∇i}htL2/vextendsingle/vextendsingle\n/ba∇dblϕ/ba∇dblH1,\nwe obtain the required estimate. /square\nProposition 3.11. Assume that u0∈H2. LetT >0 be arbitrary and T∗be defined as in\nProposition 3.8. For each n∈Nand allt∈[0,T∗],\n/integraldisplayt\n0/ba∇dbl∂sun(s)/ba∇dbl2\nL2ds+/ba∇dbl∆un(t)/ba∇dbl2\nL2+/ba∇dblun(t)/ba∇dbl4\nL4+/ba∇dblun(t)·∇un(t)/ba∇dbl2\nL2+/ba∇dbl|un(t)| |∇un(t)|/ba∇dbl2\nL2\n/lessorsimilar/ba∇dblun(0)/ba∇dbl2\nH2/lessorsimilar/ba∇dblu0/ba∇dbl2\nH2,\nwhere the constant depends on T∗, but is independent of n. Here,∂sun:=∂un\n∂s.\nProof.Taking the inner product of ( 3.5) with∂tunand integrating by parts with respect\ntox, we obtain\n/ba∇dbl∂tun/ba∇dbl2\nL2+β1\n2d\ndt/ba∇dbl∇un/ba∇dbl2\nL2+β2\n2d\ndt/ba∇dbl∆un/ba∇dbl2\nL2+β3\n4d\ndt/ba∇dblun/ba∇dbl4\nL4+β4/an}b∇acketle{tun×∆un,∂tun/an}b∇acket∇i}htL2\n+β5/angbracketleftbig\n∇(|un|2un),∂t∇un/angbracketrightbig\nL2=β3\n2d\ndt/ba∇dblun/ba∇dbl2\nL2.\nFor the last term on the left-hand side, it follows from ( 3.2) that\nβ5/angbracketleftbig\n∇(|un|2un),∂t∇un/angbracketrightbig\nL2= 2β5/angbracketleftbig\nun/parenleftbig\nun·∇un/parenrightbig\n,∂t∇un/angbracketrightbig\nL2+β5/angbracketleftbig\n|un|2∇un,∂t∇un/angbracketrightbig\nL2GLOBAL SOLUTIONS OF LLBAR EQUATION 15\n=β5d\ndt/ba∇dblun·∇un/ba∇dbl2\nL2−β5/an}b∇acketle{tun·∇un,∂tun·∇un/an}b∇acket∇i}htL2\n+β5\n2/angbracketleftbig\n|un|2,∂t/parenleftbig\n|∇un|2/parenrightbig/angbracketrightbig\nL2\n=β5d\ndt/ba∇dblun·∇un/ba∇dbl2\nL2−β5/an}b∇acketle{tun·∇un,∂tun·∇un/an}b∇acket∇i}htL2\n+β5\n2d\ndt/ba∇dbl|un| |∇un|/ba∇dbl2\nL2−β5/angbracketleftbig\n|∇un|2,un·∂tun/angbracketrightbig\nL2.\nTherefore,\n/ba∇dbl∂tun/ba∇dbl2\nL2+β1\n2d\ndt/ba∇dbl∇un/ba∇dbl2\nL2+β2\n2d\ndt/ba∇dbl∆un/ba∇dbl2\nL2+β3\n4d\ndt/ba∇dblun/ba∇dbl4\nL4\n+β5d\ndt/ba∇dblun·∇un/ba∇dbl2\nL2+β5\n2d\ndt/ba∇dbl|un| |∇un|/ba∇dbl2\nL2\n=β3\n2d\ndt/ba∇dblun/ba∇dbl2\nL2−β4/an}b∇acketle{tun×∆un,∂tun/an}b∇acket∇i}htL2\n+β5/an}b∇acketle{tun·∇un,∂tun·∇un/an}b∇acket∇i}htL2+β5/angbracketleftbig\n|∇un|2,un·∂tun/angbracketrightbig\nL2\n≤β3\n2d\ndt/ba∇dblun/ba∇dbl2\nL2+β4/ba∇dblun/ba∇dblL∞/ba∇dbl∆un/ba∇dblL2/ba∇dbl∂tun/ba∇dblL2+2β5/ba∇dblun/ba∇dblL∞/ba∇dbl∇un/ba∇dbl2\nL4/ba∇dbl∂tun/ba∇dblL2\n≤β3\n2d\ndt/ba∇dblun/ba∇dbl2\nL2+C/ba∇dblun/ba∇dbl2\nL∞/ba∇dbl∆un/ba∇dbl2\nL2+C/ba∇dblun/ba∇dbl2\nL∞/ba∇dbl∇un/ba∇dbl4\nL4+ǫ/ba∇dbl∂tun/ba∇dbl2\nL2,\nfor anyǫ >0, where in the last step we used Young’s inequality. Rearranging the inequality,\nwe obtain\n/ba∇dbl∂tun/ba∇dbl2\nL2+d\ndt/ba∇dbl∇un/ba∇dbl2\nL2+d\ndt/ba∇dbl∆un/ba∇dbl2\nL2+d\ndt/ba∇dblun/ba∇dbl4\nL4+d\ndt/ba∇dblun·∇un/ba∇dbl2\nL2+d\ndt/ba∇dbl|un| |∇un|/ba∇dbl2\nL2\n/lessorsimilard\ndt/ba∇dblun/ba∇dbl2\nL2+/ba∇dblun/ba∇dbl2\nL∞/ba∇dbl∆un/ba∇dbl2\nL2+/ba∇dblun/ba∇dbl2\nL∞/ba∇dbl∇un/ba∇dbl4\nL4. (3.21)\nWe now estimate the last two terms on the right-hand side of ( 3.21).\nCase 1:d= 1. It follows from the Sobolev embedding, Proposition 3.6, and Proposition 3.8\nthat\n/ba∇dblun(t)/ba∇dbl2\nL∞/lessorsimilar/ba∇dblun(t)/ba∇dbl2\nH1/lessorsimilar1, t∈[0,T].\nMoreover, the Gagliardo–Nirenberg inequality (Theorem 6.2withv=un,q= 4,r= 1,\ns1= 1, and s2= 2) together with ( 3.7) implies\n/ba∇dbl∇un(t)/ba∇dbl4\nL4/lessorsimilar/ba∇dblun(t)/ba∇dbl3\nH1/ba∇dblun(t)/ba∇dblH2/lessorsimilar1+/ba∇dbl∆un(t)/ba∇dbl2\nL2, t∈[0,T].\nTherefore, inequality ( 3.21) yields the required result, after integrating over (0 ,t) and using\nProposition 3.6.\nCase 2:d= 2. The Gagliardo–Nirenberg inequality (respectively with v=un,q=∞,\nr=s1= 0,s2= 2, and with v=∇un,q= 4,r=s1= 0,s2= 1) implies\n/ba∇dblun(t)/ba∇dbl2\nL∞/lessorsimilar/ba∇dblun(t)/ba∇dblL2/ba∇dblun(t)/ba∇dblH2/lessorsimilar/ba∇dblun(t)/ba∇dblH2/lessorsimilar1+/ba∇dbl∆un(t)/ba∇dblL2, t∈[0,T],\n/ba∇dbl∇un(t)/ba∇dbl4\nL4/lessorsimilar/ba∇dbl∇un(t)/ba∇dbl2\nL2/ba∇dbl∇un(t)/ba∇dbl2\nH1/lessorsimilar/ba∇dblun(t)/ba∇dbl2\nH2/lessorsimilar1+/ba∇dbl∆un(t)/ba∇dbl2\nL2, t∈[0,T],16 AGUS L. SOENJAYA AND THANH TRAN\nwhere we also used ( 3.7) and Proposition 3.8. Inserting these estimates into ( 3.21) and\nintegrating over (0 ,t) yield\n/integraldisplayt\n0/ba∇dbl∂sun(s)/ba∇dbl2\nL2ds+/ba∇dbl∆un(t)/ba∇dbl2\nL2+/ba∇dblun(t)/ba∇dbl4\nL4+/ba∇dblun(t)·∇un(t)/ba∇dbl2\nL2+/ba∇dbl|un(t)| |∇un(t)|/ba∇dbl2\nL2\n/lessorsimilar1+/integraldisplayt\n0/parenleftBig\n/ba∇dbl∆un(s)/ba∇dblL2+/ba∇dbl∆un(s)/ba∇dbl2\nL2+/ba∇dbl∆un(s)/ba∇dbl3\nL2/parenrightBig\nds\n/lessorsimilar1+/integraldisplayt\n0/parenleftBig\n/ba∇dbl∆un(s)/ba∇dbl2\nL2+/ba∇dbl∆un(s)/ba∇dbl3\nL2/parenrightBig\nds, (3.22)\nwhere in the last step we used Young’s inequality for the term /ba∇dbl∆un(s)/ba∇dblL2. For the last\nterm on the right-hand side, we use ( 3.9) to obtain\n/ba∇dbl∆un/ba∇dbl3\nL2≤ /ba∇dbl∇un/ba∇dbl3/2\nL2/ba∇dbl∇∆un/ba∇dbl3/2\nL2/lessorsimilar/ba∇dbl∇∆un/ba∇dbl3/2\nL2/lessorsimilar1+/ba∇dbl∇∆un/ba∇dbl2\nL2,\nwhere in the penultimate step we used Proposition 3.8, and in the last step we used Young’s\ninequality. Therefore the right-hand side of ( 3.22) is bounded independent of ndue to\nProposition 3.8, proving the proposition for this case.\nCase 3:d= 3. The Gagliardo–Nirenberg inequality (respectively with v=un,q=∞,\nr= 0,s1= 1,s2= 2, and with v=∇un,q= 4,r=s1= 0,s2= 1) implies\n/ba∇dblun(t)/ba∇dbl2\nL∞/lessorsimilar/ba∇dblun(t)/ba∇dblH1/ba∇dblun(t)/ba∇dblH2/lessorsimilar/ba∇dblun(t)/ba∇dblH2/lessorsimilar1+/ba∇dbl∆un(t)/ba∇dblL2,\n/ba∇dbl∇un(t)/ba∇dbl4\nL4/lessorsimilar/ba∇dbl∇un(t)/ba∇dblL2/ba∇dbl∇un(t)/ba∇dbl3\nH1/lessorsimilar/ba∇dblun(t)/ba∇dbl3\nH2/lessorsimilar1+/ba∇dbl∆un(t)/ba∇dbl3\nL2,\nfor allt∈[0,T∗] whereT∗is given in Proposition 3.8. Inserting these estimates into ( 3.21),\nintegrating over (0 ,t), and using ( 3.6) yield\n/integraldisplayt\n0/ba∇dbl∂sun(s)/ba∇dbl2\nL2ds+/ba∇dbl∆un(t)/ba∇dbl2\nL2+/ba∇dblun(t)/ba∇dbl4\nL4+/ba∇dblun(t)·∇un(t)/ba∇dbl2\nL2+/ba∇dbl|un(t)| |∇un(t)|/ba∇dbl2\nL2\n/lessorsimilar1+/integraldisplayt\n0/parenleftBig\n/ba∇dbl∆un(s)/ba∇dblL2+/ba∇dbl∆un(s)/ba∇dbl2\nL2+/ba∇dbl∆un(s)/ba∇dbl3\nL2+/ba∇dbl∆un(s)/ba∇dbl4\nL2/parenrightBig\nds\n/lessorsimilar1+/integraldisplayt\n0/parenleftBig\n/ba∇dbl∆un(s)/ba∇dbl2\nL2+/ba∇dbl∆un(s)/ba∇dbl4\nL2/parenrightBig\nds\n/lessorsimilar1+/integraldisplayt\n0/parenleftBig\n/ba∇dbl∆un(s)/ba∇dbl2\nL2+/ba∇dbl∇∆un(s)/ba∇dbl2\nL2/parenrightBig\nds\n/lessorsimilar1,\nwhere in the last step we used Proposition 3.8, completing the proof of the proposition. /square\nProposition 3.12. Assume that u0∈H2. LetT >0 be arbitrary and T∗be defined as in\nProposition 3.8. For each n∈Nand allt∈[0,T∗],\n/integraldisplayt\n0/vextenddouble/vextenddouble∆2un(s)/vextenddouble/vextenddouble2\nL2ds/lessorsimilar/ba∇dblun(0)/ba∇dbl2\nH2/lessorsimilar/ba∇dblu0/ba∇dbl2\nH2,\nwhere the constant depends on T∗but is independent of n.\nProof.Taking the inner product of ( 3.5) with ∆2unand integrating by parts with respect\ntox, we obtain\n1\n2d\ndt/ba∇dbl∆un/ba∇dbl2\nL2+β1/ba∇dbl∇∆un/ba∇dbl2\nL2+β2/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble2\nL2GLOBAL SOLUTIONS OF LLBAR EQUATION 17\n=β3/ba∇dbl∆un/ba∇dbl2\nL2−β3/angbracketleftbig\n|un|2un,∆2un/angbracketrightbig\nL2\n−β4/angbracketleftbig\nun×∆un,∆2un/angbracketrightbig\nL2−β5/angbracketleftbig\n∆(|un|2un),∆2un/angbracketrightbig\nL2. (3.23)\nEach term on the right-hand side can be estimated as follows. For th e first term, by Young’s\ninequality, Sobolev embedding and Proposition 3.8,\n/vextendsingle/vextendsingle/angbracketleftbig\n|un|2un,∆2un/angbracketrightbig\nL2/vextendsingle/vextendsingle≤C/vextenddouble/vextenddouble|un|2un/vextenddouble/vextenddouble2\nL2+ǫ/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble2\nL2\n=C/ba∇dblun/ba∇dbl6\nL6+ǫ/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble2\nL2\n≤C/ba∇dblun/ba∇dbl6\nH1+ǫ/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble2\nL2\n/lessorsimilar1+ǫ/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble2\nL2\nfor anyǫ >0. For the second term, by H¨ older’s inequality, Young’s inequality, S obolev\nembedding, and Proposition 3.11, we have\n/vextendsingle/vextendsingle/angbracketleftbig\nun×∆un,∆2un/angbracketrightbig\nL2/vextendsingle/vextendsingle≤ /ba∇dblun/ba∇dblL∞/ba∇dbl∆un/ba∇dblL2/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble\nL2\n≤C(/ba∇dblun/ba∇dbl2\nH2/ba∇dbl∆un/ba∇dbl2\nL2)+ǫ/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble2\nL2\n/lessorsimilar1+ǫ/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble2\nL2.\nFinally, by H¨ older’s and Young’s inequalities, we have\n/vextendsingle/vextendsingle/angbracketleftbig\n∆(|un|2un),∆2un/angbracketrightbig/vextendsingle/vextendsingle≤/vextenddouble/vextenddouble∆(|un|2un)/vextenddouble/vextenddouble\nL2/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble\nL2\n≤C/vextenddouble/vextenddouble∆(|un|2un)/vextenddouble/vextenddouble2\nL2+ǫ/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble2\nL2. (3.24)\nFor the first term on the right-hand side, it follows from ( 3.4), H¨ older’s inequality, and\nSobolev embedding that\n/vextenddouble/vextenddouble∆(|un|2un)/vextenddouble/vextenddouble2\nL2/lessorsimilar/ba∇dbl∇un/ba∇dbl4\nL6/ba∇dblun/ba∇dbl2\nL6+/ba∇dbl∆un/ba∇dbl2\nL6/ba∇dblun/ba∇dbl4\nL6\n≤ /ba∇dbl∇un/ba∇dbl4\nH1/ba∇dblun/ba∇dbl2\nH1+/ba∇dbl∆un/ba∇dbl2\nH1/ba∇dblun/ba∇dbl4\nH1\n/lessorsimilar/ba∇dblun/ba∇dbl6\nH2+/ba∇dbl∆un/ba∇dbl2\nH1/ba∇dblun/ba∇dbl4\nH1\n/lessorsimilar1+/ba∇dbl∇∆un/ba∇dbl2\nL2, (3.25)\nwhere in the last step we also used Proposition 3.11. Altogether, we deduce from ( 3.23) after\nintegrating over (0 ,t) that\n/ba∇dbl∆un(t)/ba∇dbl2\nL2+/integraldisplayt\n0/vextenddouble/vextenddouble∆2un(s)/vextenddouble/vextenddouble2\nL2ds/lessorsimilar1+/integraldisplayt\n0/ba∇dbl∇∆un(s)/ba∇dbl2\nL2ds/lessorsimilar1,\nwhere in the last step we used Proposition 3.8. This completes the proof of the proposition.\n/square\nProposition 3.13. Assume that u0∈H3. LetT >0 be arbitrary and T∗be defined as in\nProposition 3.8. For each n∈Nand allt∈[0,T∗],\n/ba∇dbl∇∆un(t)/ba∇dbl2\nL2+/integraldisplayt\n0/ba∇dbl∇∆2un(s)/ba∇dbl2\nL2ds/lessorsimilar/ba∇dblun(0)/ba∇dbl2\nH3/lessorsimilar/ba∇dblu0/ba∇dbl2\nH3,\nwhere the constant depends on T∗but is independent of n.18 AGUS L. SOENJAYA AND THANH TRAN\nProof.Taking the inner product of ( 3.5) with ∆3unand integrating by parts with respect\ntox, we have\n1\n2d\ndt/ba∇dbl∇∆un/ba∇dbl2\nL2+β1/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble2\nL2+β2/vextenddouble/vextenddouble∇∆2un/vextenddouble/vextenddouble2\nL2−β3/ba∇dbl∇∆un/ba∇dbl2\nL2\n=β3/angbracketleftbig\n∆/parenleftbig\n|un|2un/parenrightbig\n,∆2un/angbracketrightbig\nL2−2β4/angbracketleftbig\n∇un×∇∆un,∆2un/angbracketrightbig\nL2\n−β5/angbracketleftbig\n∇∆(|un|2un),∇∆2un/angbracketrightbig\nL2. (3.26)\nEach term on the right-hand side can be estimated as follows. For th e first term, by ( 3.24)\nand (3.25) we have\n/vextendsingle/vextendsingle/angbracketleftbig\n∆/parenleftbig\n|un|2un/parenrightbig\n,∆2un/angbracketrightbig\nL2/vextendsingle/vextendsingle/lessorsimilar1+/ba∇dbl∇∆un/ba∇dbl2\nL2+ǫ/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble2\nL2. (3.27)\nFor the second term, Sobolev embedding and H¨ older’s inequality give\n/vextendsingle/vextendsingle/angbracketleftbig\n∇un×∇∆un,∆2un/angbracketrightbig\nL2/vextendsingle/vextendsingle≤ /ba∇dbl∇un/ba∇dblL3/ba∇dbl∇∆un/ba∇dblL6/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble\nL2\n≤ /ba∇dblun/ba∇dblH2/ba∇dblun/ba∇dblH4/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble\nL2\n/lessorsimilar1+/vextenddouble/vextenddouble∆2un/vextenddouble/vextenddouble2\nL2, (3.28)\nwhere in the last step we used ( 3.7), (3.12), and Proposition 3.11. For the last term on the\nright-hand side of ( 3.26), by H¨ older’s and Young’s inequalities, and ( 3.2), we deduce\n/vextendsingle/vextendsingle/angbracketleftbig\n∇∆(|un|2un),∇∆2un/angbracketrightbig\nL2/vextendsingle/vextendsingle≤/vextenddouble/vextenddouble∇(|un|2un)/vextenddouble/vextenddouble\nH2/vextenddouble/vextenddouble∇∆2un/vextenddouble/vextenddouble\nL2\n≤C/vextenddouble/vextenddoubleun/parenleftbig\nun·∇un/parenrightbig/vextenddouble/vextenddouble2\nH2+C/vextenddouble/vextenddouble|un|2|∇un|/vextenddouble/vextenddouble2\nH2+ǫ/vextenddouble/vextenddouble∇∆2un/vextenddouble/vextenddouble2\nL2\n≤C/ba∇dblun/ba∇dbl4\nH2/ba∇dbl∇un/ba∇dbl2\nH2+ǫ/vextenddouble/vextenddouble∇∆2un/vextenddouble/vextenddouble2\nL2\n/lessorsimilar1+/ba∇dbl∇∆un/ba∇dbl2\nL2+ǫ/vextenddouble/vextenddouble∇∆2un/vextenddouble/vextenddouble2\nL2 (3.29)\nfor anyǫ >0, where in the penultimate step we used ( 3.14) and in the last step we used\n(3.10), Proposition 3.8, and Proposition 3.11. Inserting the estimates ( 3.27), (3.28), and\n(3.29) into (3.26) and integrating over (0 ,t) yield\n/ba∇dbl∇∆un(t)/ba∇dbl2\nL2+/integraldisplayt\n0/vextenddouble/vextenddouble∇∆2un(s)/vextenddouble/vextenddouble2\nL2ds/lessorsimilar1+/integraldisplayt\n0/ba∇dbl∇∆un(s)/ba∇dbl2\nL2ds+/integraldisplayt\n0/vextenddouble/vextenddouble∆2un(s)/vextenddouble/vextenddouble2\nL2ds\n/lessorsimilar1,\nwhere in the last step we used Proposition 3.8and3.12. This completes the proof. /square\nProposition 3.14. Under the assumption of Proposition 3.13, we have\n/ba∇dbl∂tun/ba∇dbl2\nL2(0,T∗;H1)/lessorsimilar/ba∇dblu0/ba∇dbl2\nH3,\nwhere the constant depends on T∗, but is independent of n.\nProof.Taking the inner product of ( 3.5) with−∆∂tunand integrating by parts with respect\ntox, we have\n/ba∇dbl∇∂tun/ba∇dbl2\nL2+β1d\ndt/ba∇dbl∆un/ba∇dbl2\nL2+β2d\ndt/ba∇dbl∇∆un/ba∇dbl2\nL2\n=β3d\ndt/ba∇dbl∇un/ba∇dbl2\nL2−β3/angbracketleftbig\n∇/parenleftbig\n|un|2un/parenrightbig\n,∇∂tun/angbracketrightbig\nL2−β4/an}b∇acketle{t∇un×∆un,∇∂tun/an}b∇acket∇i}htL2\n−β4/an}b∇acketle{tun×∇∆un,∇∂tun/an}b∇acket∇i}htL2+β5/angbracketleftbig\n∇∆(|un|2un),∇∂tun/angbracketrightbig\nL2. (3.30)GLOBAL SOLUTIONS OF LLBAR EQUATION 19\nEach inner product on the right-hand side can be estimated as follow s. For the first in-\nner product, by H¨ older’s inequality, Proposition 3.6,3.8and3.11, and Sobolev embedding\nH1⊂L6, we have\nβ3/vextendsingle/vextendsingle/angbracketleftbig\n∇/parenleftbig\n|un|2un/parenrightbig\n,∇∂tun/angbracketrightbig\nL2/vextendsingle/vextendsingle/lessorsimilar/ba∇dbl∇un/ba∇dblL6/ba∇dblun/ba∇dbl2\nL6/ba∇dbl∇∂tun/ba∇dblL2\n≤ /ba∇dblun/ba∇dblH2/ba∇dblun/ba∇dbl2\nH1/ba∇dbl∇∂tun/ba∇dblL2\n/lessorsimilar1+ǫ/ba∇dbl∇∂tun/ba∇dbl2\nL2, (3.31)\nfor anyǫ >0, where in the last step we used Young’s inequality. For the second in ner\nproduct, H¨ older’s inequality, Proposition 3.6,3.8,3.13, and Sobolev embedding H2⊂L∞\ngive\nβ4/vextendsingle/vextendsingle/an}b∇acketle{t∇un×∆un,∇∂tun/an}b∇acket∇i}htL2/vextendsingle/vextendsingle/lessorsimilar/ba∇dbl∇un/ba∇dblL∞/ba∇dbl∆un/ba∇dblL2/ba∇dbl∇∂tun/ba∇dblL2\n≤ /ba∇dbl∇un/ba∇dblH2/ba∇dbl∆un/ba∇dblL2/ba∇dbl∇∂tun/ba∇dblL2\n/lessorsimilar1+ǫ/ba∇dbl∇∂tun/ba∇dbl2\nL2, (3.32)\nfor anyǫ >0. Similarly, for the third inner product, we have\nβ4/vextendsingle/vextendsingle/an}b∇acketle{tun×∇∆un,∇∂tun/an}b∇acket∇i}htL2/vextendsingle/vextendsingle/lessorsimilar/ba∇dblun/ba∇dblL∞/ba∇dbl∇∆un/ba∇dblL2/ba∇dbl∇∂tun/ba∇dblL2\n≤ /ba∇dblun/ba∇dblH2/ba∇dbl∇∆un/ba∇dblL2/ba∇dbl∇∂tun/ba∇dblL2\n/lessorsimilar1+ǫ/ba∇dbl∇∂tun/ba∇dbl2\nL2. (3.33)\nFor the last inner product on ( 3.30), by H¨ older’s and Young’s inequality, we obtain\nβ5/vextendsingle/vextendsingle/angbracketleftbig\n∇∆(|un|2un),∇∂tun/angbracketrightbig\nL2/vextendsingle/vextendsingle≤/vextenddouble/vextenddouble|un|2un/vextenddouble/vextenddouble\nH3/ba∇dbl∇∂tun/ba∇dblL2\n≤C/ba∇dblun/ba∇dbl3\nH3/ba∇dbl∇∂tun/ba∇dblL2\n/lessorsimilar1+ǫ/ba∇dbl∇∂tun/ba∇dbl2\nL2, (3.34)\nfor anyǫ >0, where in the penultimate step we used ( 3.14), and in the last step we used\n(3.10), Proposition 3.6,3.8,3.13, and Young’s inequality. Inserting the estimates ( 3.31),\n(3.32), (3.33) and (3.34) into (3.30), integrating over (0 ,t), and choosing ǫsufficiently small\nyield\n/integraldisplayt\n0/ba∇dbl∇∂sun(s)/ba∇dbl2\nL2ds/lessorsimilar1+/ba∇dbl∇un/ba∇dbl2\nL2/lessorsimilar1,\nwhere in the last step we used Proposition 3.8. This completes the proof. /square\nAs a consequence of Proposition 3.6–Proposition 3.14, we have the following result.\nCorollary 3.15. For anyT >0, letT∗be defined by Proposition 3.8. Assume that the\ninitial data u0satisfiesu0∈Hrforr∈ {0,1,2,3}. Assume further that\n/ba∇dblu0n−u0/ba∇dblHr→0 asn→ ∞,\nwhereu0,nis defined in ( 3.5). Then\n/ba∇dblun/ba∇dblL∞(0,T;Hr)+/ba∇dblun/ba∇dblL4(0,T;L4)+/ba∇dblun/ba∇dblL2(0,T;Hr+2)+/ba∇dbl∂tun/ba∇dblL2(0,T;Hr−2)/lessorsimilar1,(3.35)\nwhere\nT=/braceleftBigg\nTifr= 0,\nT∗ifr >0,(3.36)20 AGUS L. SOENJAYA AND THANH TRAN\nProof.First we recall from ( 3.6) that the given assumption yields /ba∇dblun(0)/ba∇dblHr/lessorsimilar/ba∇dblu0/ba∇dblHr/lessorsimilar1.\nTherefore, Proposition 3.6and Proposition 3.7imply (3.35) whenr= 0, while Proposi-\ntion3.8, Proposition 3.11, Proposition 3.10and inequality ( 3.10) give the result when r= 1.\nNext, Proposition 3.6, Proposition 3.8, Proposition 3.11, Proposition 3.12and inequal-\nity (3.10) give the required estimate for the case r= 2. Finally, Proposition 3.6, Proposi-\ntion3.8, Proposition 3.11, Proposition 3.12, Proposition 3.13, Proposition 3.14and inequal-\nity (3.13) give the result for the case r= 3, completing the proof of the corollary. /square\n4.Proof of Theorem 2.2\nLetr∈ {0,1,2,3}. It follows from ( 3.35) and the Banach-Alaoglu theorem that there\nexists a subsequence of {un}, which is still denoted by {un}, such that\n\n\nun⇀uweakly* in L∞(0,T;Hr),\nun⇀uweakly in L2(0,T;Hr+2),\nun⇀uweakly in L4(0,T;L4),\n∂tun⇀ ∂tuweakly in L2(0,T;Hr−2),(4.1)\nwhereTwas defined in ( 3.36). By the Aubin–Lions–Simon lemma (Theorem 6.3), a further\nsubsequence then satisfies\nun→ustrongly in L2(0,T;H1). (4.2)\nThe next proposition shows the convergence of the nonlinear term s in (3.5).\nProposition 4.1. LetT >0 be arbitrary. Let {φn}be a sequence in Vnsuch that φn→φ\ninH2. For all t∈[0,T], we have\nlim\nn→∞/integraldisplayt\n0/angbracketleftbig\nΠn/parenleftbig\n(1−|un(s)|2)un(s)/parenrightbig\n,φn/angbracketrightbig\nL2ds=/integraldisplayt\n0/angbracketleftbig/parenleftbig\n(1−|u(s)|2)u(s)/parenrightbig\n,φ/angbracketrightbig\nL2ds,(4.3)\nlim\nn→∞/integraldisplayt\n0/an}b∇acketle{tΠn(un(s)×∆un(s)),φn/an}b∇acket∇i}htL2ds=−/integraldisplayt\n0/an}b∇acketle{t(u(s)×∇u(s)),∇φ/an}b∇acket∇i}htL2ds,(4.4)\nlim\nn→∞/integraldisplayt\n0/angbracketleftbig\nΠn/parenleftbig\n∆/parenleftbig\n|un(s)|2un(s)/parenrightbig/parenrightbig\n,φn/angbracketrightbig\nL2ds=/integraldisplayt\n0/angbracketleftbig\n∇/parenleftbig\n|u(s)|2u(s)/parenrightbig\n,∇φ/angbracketrightbig\nL2ds.(4.5)\nProof.By the definition of Π n, in order to prove ( 4.3) it suffices to show\nlim\nn→∞/integraldisplayt\n0/angbracketleftbig\n|un(s)|2un(s),φn/angbracketrightbig\nL2ds=/integraldisplayt\n0/angbracketleftbig\n|u(s)|2u(s),φ/angbracketrightbig\nL2ds. (4.6)\nTo this end, note that H¨ older’s inequality implies/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/angbracketleftbig\n|un(s)|2un(s),φn/angbracketrightbig\nL2ds−/integraldisplayt\n0/angbracketleftbig\n|u(s)|2u(s),φ/angbracketrightbig\nL2ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/angbracketleftbig\n|un(s)|2un(s),φn−φ/angbracketrightbig\nL2ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/angbracketleftbig\n|un(s)|2(un(s)−u(s)),φ/angbracketrightbig\nL2ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/angbracketleftbig\n(|un(s)|2−|u(s)|2)u(s),φ/angbracketrightbig\nL2ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤ /ba∇dblφn−φ/ba∇dblL6/integraldisplayt\n0/ba∇dblun(s)/ba∇dbl2\nL6/ba∇dblun(s)/ba∇dblL2ds+/ba∇dblφ/ba∇dblL∞/integraldisplayt\n0/ba∇dblun(s)/ba∇dbl2\nL4/ba∇dblun(s)−u(s)/ba∇dblL2dsGLOBAL SOLUTIONS OF LLBAR EQUATION 21\n+/ba∇dblφ/ba∇dblL∞/integraldisplayt\n0/ba∇dblun(s)−u(s)/ba∇dblL2/ba∇dblun(s)+u(s)/ba∇dblL4/ba∇dblu(s)/ba∇dblL4ds\n≤ /ba∇dblφn−φ/ba∇dblH1/ba∇dblun/ba∇dblL2(0,T;H1)+/ba∇dblφ/ba∇dblH2/ba∇dblun/ba∇dblL4(0,T;L4)/ba∇dblun−u/ba∇dblL2(0,T;L2)\n+/ba∇dblφ/ba∇dblH2/ba∇dblun−u/ba∇dblL2(0,T;L2)/ba∇dblun+u/ba∇dblL4(0,T;L4)/ba∇dblu/ba∇dblL4(0,T;L4).\nBy using the Sobolev embedding H1⊂L6andH2⊂L∞, (3.35), and (4.2) we deduce ( 4.6).\nSimilarly, to show ( 4.4), it suffices to show\nlim\nn→∞/integraldisplayt\n0/an}b∇acketle{t(un(s)×∇un(s)),∇φn/an}b∇acket∇i}htL2ds=/integraldisplayt\n0/an}b∇acketle{t(u(s)×∇u(s)),∇φ/an}b∇acket∇i}htL2ds.\nTo this end, note that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/an}b∇acketle{tun(s)×∇un(s),∇φn/an}b∇acket∇i}htL2ds−/integraldisplayt\n0/an}b∇acketle{tu(s)×∇u(s),∇φ/an}b∇acket∇i}htL2ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/an}b∇acketle{tun(s)×∇un(s),∇φn−∇φ/an}b∇acket∇i}htL2ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/an}b∇acketle{t(un(s)−u(s))×∇un(s),∇φ/an}b∇acket∇i}htL2ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/an}b∇acketle{tu(s)×(∇un(s)−∇u(s)),∇φ/an}b∇acket∇i}htL2ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/lessorsimilar/ba∇dblun/ba∇dblL2(0,T;L4)/ba∇dbl∇un/ba∇dblL2(0,T;L4)/ba∇dbl∇φn−∇φ/ba∇dblL2\n+/ba∇dblun−u/ba∇dblL2(0,T;L4)/ba∇dbl∇un/ba∇dblL2(0,T;L4)/ba∇dbl∇φ/ba∇dblL2\n+/ba∇dblu/ba∇dblL2(0,T;L4)/ba∇dbl∇un−∇u/ba∇dblL2(0,T;L4)/ba∇dbl∇φ/ba∇dblL2.\nBy using the Sobolev embedding H1⊂L4, (3.35), and (4.2), we deduce the required conver-\ngence.\nFor the last convergence ( 4.5), it suffices to show\nlim\nn→∞/integraldisplayt\n0/angbracketleftbig\n∇/parenleftbig\n|un(s)|2un(s)/parenrightbig\n,∇φn/angbracketrightbig\nL2ds=/integraldisplayt\n0/angbracketleftbig\n∇/parenleftbig\n|u(s)|2u(s)/parenrightbig\n,∇φ/angbracketrightbig\nL2ds.\nTo this end, note that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/angbracketleftbig\n∇(|un(s)|2un(s)),∇φn/angbracketrightbig\nL2ds−/integraldisplayt\n0/angbracketleftbig\n∇(|u(s)|2u(s)),∇φ/angbracketrightbig\nL2ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/angbracketleftbig\n∇(|un(s)|2un(s)),∇φn−∇φ/angbracketrightbig\nL2ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/angbracketleftbig\n∇(|un(s)|2un(s))−∇(|un(s)|2u(s)),∇φ/angbracketrightbig\nL2ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0/angbracketleftbig\n∇(|un(s)|2u(s))−∇(|u(s)|2u(s)),∇φ/angbracketrightbig\nL2ds/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nThe arguments follow along the line of the previous convergence sta tements and are omitted.\n/square\nWe are now ready to prove Theorem 2.2.22 AGUS L. SOENJAYA AND THANH TRAN\nProof that usatisfies ( 2.5) and (2.6): For any φ∈H2, take a sequence {φn}inVnsuch\nthatφn→φinH2. It follows from ( 3.5) that\n/an}b∇acketle{tun(t),φn/an}b∇acket∇i}htL2+β1/integraldisplayt\n0/an}b∇acketle{t∇un(s),∇φn/an}b∇acket∇i}htL2ds+β2/integraldisplayt\n0/an}b∇acketle{t∆un(s),∆φn/an}b∇acket∇i}htL2ds\n=/an}b∇acketle{tu0n,φn/an}b∇acket∇i}htL2+β3/integraldisplayt\n0/angbracketleftbig\nΠn/parenleftbig\n(1−|un(s)|2)un(s)/parenrightbig\n,φn/angbracketrightbig\nL2ds\n+β4/integraldisplayt\n0/an}b∇acketle{tΠn(un(s)×∆un(s)),φn/an}b∇acket∇i}htL2ds−β5/integraldisplayt\n0/angbracketleftbig\nΠn/parenleftbig\n∆/parenleftbig\n|un(s)|2un(s)/parenrightbig/parenrightbig\n,φn/angbracketrightbig\nL2ds.\nHence, letting n→ ∞and using Proposition 4.1we deduce ( 2.5). Noting ( 4.1), we have\nu∈L∞(0,T;Hr)∩L2(0,T;Hr+2)∩L4(0,T;L4) and ∂tu∈L2(0,T;Hr−2).\nTherefore, applying Theorem 6.4, and noting that [ Hr−2,Hr+2]1/2≡Hr, we obtain ( 2.6).\nProof of ( 2.7): Letuandvbe weak solutions to ( 2.4) with initial data u0andv0∈L2,\nrespectively. Let w=u−v. Then, for all φ∈H2,\n/an}b∇acketle{t∂tw,φ/an}b∇acket∇i}htL2+β1/an}b∇acketle{t∇w,∇φ/an}b∇acket∇i}htL2+β2/an}b∇acketle{t∆w,∆φ/an}b∇acket∇i}htL2\n=β3/an}b∇acketle{tw,φ/an}b∇acket∇i}htL2−β3/angbracketleftbig\n|u|2u−|v|2v,φ/angbracketrightbig\nL2+β4/an}b∇acketle{tu×∇u−v×∇v,∇φ/an}b∇acket∇i}htL2\n−β5/angbracketleftbig\n∇(|u|2u−|v|2v),∇φ/angbracketrightbig\nL2. (4.7)\nBy using integration by parts for the terms with coefficient β4andβ5, we obtain\n/an}b∇acketle{t∂tw,φ/an}b∇acket∇i}htL2+β1/an}b∇acketle{t∇w,∇φ/an}b∇acket∇i}htL2+β2/an}b∇acketle{t∆w,∆φ/an}b∇acket∇i}htL2\n=β3/an}b∇acketle{tw,φ/an}b∇acket∇i}htL2−β3/angbracketleftbig\n|u|2u−|v|2v,φ/angbracketrightbig\nL2−β4/an}b∇acketle{tu×∆u−v×∆v,φ/an}b∇acket∇i}htL2\n+β5/angbracketleftbig\n|u|2u−|v|2v,∆φ/angbracketrightbig\nL2. (4.8)\nBoth equations above will be used at our convenience. Letting φ=win (4.8), we have\n1\n2d\ndt/ba∇dblw/ba∇dbl2\nL2+β2/ba∇dbl∆w/ba∇dbl2\nL2≤ |β1|/ba∇dbl∇w/ba∇dbl2\nL2+β3/ba∇dblw/ba∇dbl2\nL2+β3/vextendsingle/vextendsingle/vextendsingle/angbracketleftbig\n|u|2u−|v|2v,w/angbracketrightbig\nL2/vextendsingle/vextendsingle/vextendsingle\n+β4/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{tu×∆u−v×∆v,w/an}b∇acket∇i}htL2/vextendsingle/vextendsingle/vextendsingle\n+β5/vextendsingle/vextendsingle/vextendsingle/angbracketleftbig\n|u|2u−|v|2v,∆w/angbracketrightbig\nL2/vextendsingle/vextendsingle/vextendsingle. (4.9)\nWewillnowestimatetheinnerproductsontheright-handside. Fort hefirstinnerproduct,\napplying ( 3.15) yields\nβ3/vextendsingle/vextendsingle/angbracketleftbig\n|u|2u−|v|2v,w/angbracketrightbig\nL2/vextendsingle/vextendsingle/lessorsimilar/parenleftbig\n/ba∇dblu/ba∇dbl2\nL∞+/ba∇dblv/ba∇dbl2\nL∞/parenrightbig\n/ba∇dblw/ba∇dbl2\nL2. (4.10)\nFor the second inner product on the right-hand side of ( 4.9), we have by using H¨ older’s\ninequality and Young’s inequality\nβ4/vextendsingle/vextendsingle/an}b∇acketle{tu×∆u−v×∆v,w/an}b∇acket∇i}htL2/vextendsingle/vextendsingle=β4/vextendsingle/vextendsingle/an}b∇acketle{tu×∆w+w×∆v,w/an}b∇acket∇i}htL2/vextendsingle/vextendsingle=β4/vextendsingle/vextendsingle/an}b∇acketle{tu×∆w,w/an}b∇acket∇i}htL2/vextendsingle/vextendsingle\n/lessorsimilar/ba∇dblu/ba∇dblL∞/ba∇dbl∆w/ba∇dblL2/ba∇dblw/ba∇dblL2\n≤C/ba∇dblu/ba∇dbl2\nL∞/ba∇dblw/ba∇dbl2\nL2+ǫ/ba∇dbl∆w/ba∇dbl2\nL2, (4.11)GLOBAL SOLUTIONS OF LLBAR EQUATION 23\nfor anyǫ >0. For the last inner product in ( 4.9), applying ( 3.15), then using Young’s\ninequality yield\nβ5/vextendsingle/vextendsingle/vextendsingle/angbracketleftbig\n|u|2u−|v|2v,∆w/angbracketrightbig\nL2/vextendsingle/vextendsingle/vextendsingle≤C/parenleftbig\n/ba∇dblu/ba∇dbl4\nL∞+/ba∇dblv/ba∇dbl4\nL∞/parenrightbig\n/ba∇dblw/ba∇dbl2\nL2+ǫ/ba∇dbl∆w/ba∇dbl2\nL2 (4.12)\nfor anyǫ >0. Inserting ( 4.10), (4.11) and (4.12) into (4.9), and choosing ǫsufficiently small,\nwe obtain\nd\ndt/ba∇dblw/ba∇dbl2\nL2+/ba∇dbl∆w/ba∇dbl2\nL2/lessorsimilar/ba∇dbl∇w/ba∇dbl2\nL2+/parenleftbig\n/ba∇dblu/ba∇dbl2\nL∞+/ba∇dblv/ba∇dbl2\nL∞+/ba∇dblu/ba∇dbl4\nL∞+/ba∇dblv/ba∇dbl4\nL∞/parenrightbig\n/ba∇dblw/ba∇dbl2\nL2+ǫ/ba∇dbl∆w/ba∇dbl2\nL2\n/lessorsimilar/parenleftbig\n1+/ba∇dblu/ba∇dbl2\nL∞+/ba∇dblv/ba∇dbl2\nL∞+/ba∇dblu/ba∇dbl4\nL∞+/ba∇dblv/ba∇dbl4\nL∞/parenrightbig\n/ba∇dblw/ba∇dbl2\nL2+ǫ/ba∇dbl∆w/ba∇dbl2\nL2\n/lessorsimilar/parenleftbig\n1+/ba∇dblu/ba∇dbl4\nL∞+/ba∇dblv/ba∇dbl4\nL∞/parenrightbig\n/ba∇dblw/ba∇dbl2\nL2+ǫ/ba∇dbl∆w/ba∇dbl2\nL2, (4.13)\nwhere we used ( 3.8) and Young’s inequality. Choosing ǫsufficiently small, rearranging the\nabove equation, integrating over (0 ,t), and using Gronwall’s inequality, we infer ( 2.7). Under\nassumption ( 2.9), uniqueness then follows.\nProof of ( 2.8):\nThe case r= 1: Taking φ=−∆win (4.7), and integrating by parts (for the terms with\ncoefficient β1andβ2) we have (after rearranging the terms)\n1\n2d\ndt/ba∇dbl∇w/ba∇dbl2\nL2+β2/ba∇dbl∇∆w/ba∇dbl2\nL2\n≤ |β1|/ba∇dbl∆w/ba∇dbl2\nL2+β3/ba∇dbl∇w/ba∇dbl2\nL2+β3/vextendsingle/vextendsingle/angbracketleftbig\n|u|2u−|v|2v,∆w/angbracketrightbig\nL2/vextendsingle/vextendsingle\n+β4/vextendsingle/vextendsingle/an}b∇acketle{tu×∇u−v×∇v,∇∆w/an}b∇acket∇i}htL2/vextendsingle/vextendsingle+β5/vextendsingle/vextendsingle/angbracketleftbig\n∇(|u|2u−|v|2v),∇∆w/angbracketrightbig\nL2/vextendsingle/vextendsingle.(4.14)\nWe will estimate the inner products on the right-hand side. For the fi rst inner product, it\nfollows successively from ( 4.12), (3.9), and Young’s inequality that\nβ3/vextendsingle/vextendsingle/angbracketleftbig\n|u|2u−|v|2v,∆w/angbracketrightbig\nL2/vextendsingle/vextendsingle/lessorsimilar/parenleftbig\n/ba∇dblu/ba∇dbl4\nL∞+/ba∇dblv/ba∇dbl4\nL∞/parenrightbig\n/ba∇dblw/ba∇dbl2\nL2+/ba∇dbl∇w/ba∇dbl2\nL2+ǫ/ba∇dbl∇∆w/ba∇dbl2\nL2,(4.15)\nfor anyǫ >0. For the second inner product, applying ( 3.16) then using Young’s inequality\nyield\nβ4/vextendsingle/vextendsingle/an}b∇acketle{tu×∇u−v×∇v,∇∆w/an}b∇acket∇i}htL2/vextendsingle/vextendsingle/lessorsimilar/ba∇dblu/ba∇dbl2\nL∞/ba∇dbl∇w/ba∇dbl2\nL2+/ba∇dblw/ba∇dbl2\nL2/ba∇dbl∇v/ba∇dbl2\nL∞+ǫ/ba∇dbl∇∆w/ba∇dbl2\nL2,\n(4.16)\nfor anyǫ >0. For the third inner product on the right-hand side of ( 4.14), applying ( 3.15)\nthen using Young’s inequality give\nβ5/vextendsingle/vextendsingle/angbracketleftbig\n∇(|u|2u−|v|2v),∇∆w/angbracketrightbig\nL2/vextendsingle/vextendsingle/lessorsimilar/parenleftbig\n/ba∇dblu/ba∇dbl2\nH1+/ba∇dblv/ba∇dbl2\nH1/parenrightbig/parenleftbig\n/ba∇dblu/ba∇dbl2\nH2+/ba∇dblv/ba∇dbl2\nH2/parenrightbig\n/ba∇dblw/ba∇dbl2\nH1\n+ǫ/ba∇dbl∇∆w/ba∇dbl2\nL2, (4.17)\nfor anyǫ >0. Inserting ( 4.15), (4.16) and (4.17) into (4.14), integrating over (0 ,t), and\nchoosing ǫsufficiently small, we obtain\n/ba∇dbl∇w(t)/ba∇dbl2\nL2+/integraldisplayt\n0/ba∇dbl∇∆w(s)/ba∇dbl2\nL2ds\n/lessorsimilar/ba∇dbl∇w(0)/ba∇dbl2\nL2+/integraldisplayt\n0α(s)/ba∇dblw(s)/ba∇dbl2\nL2ds+/integraldisplayt\n0α(s)/ba∇dbl∇w(s)/ba∇dbl2\nL2ds\n/lessorsimilar/ba∇dbl∇w(0)/ba∇dbl2\nL2+/ba∇dblw(0)/ba∇dbl2\nL2/integraldisplayt\n0α(s)exp/parenleftbigg/integraldisplayt\n0α(τ)dτ/parenrightbigg\nds+/integraldisplayt\n0α(s)/ba∇dbl∇w(s)/ba∇dbl2\nL2ds,24 AGUS L. SOENJAYA AND THANH TRAN\nwhere\nα(s) :=/parenleftbig\n/ba∇dblu(s)/ba∇dbl2\nH1+/ba∇dblv(s)/ba∇dbl2\nH1/parenrightbig/parenleftbig\n/ba∇dblu(s)/ba∇dbl2\nH2+/ba∇dblv(s)/ba∇dbl2\nH2/parenrightbig\n+/ba∇dblu(s)/ba∇dbl2\nH3+/ba∇dblv(s)/ba∇dbl2\nH3\nand where in the last step we used ( 2.7) which was proved above in this proof. Note that for\nu0,v0∈H1, we have u,v∈L∞(0,T;H1)∩L2(0,T;H3), and thus (by Gagliardo–Nirenberg\ninequality)/integraldisplayt\n0α(s)ds/lessorsimilar/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl2\nH3+/ba∇dblv(s)/ba∇dbl2\nH3ds/lessorsimilar1.\nGronwall’s inequality then yields\n/ba∇dbl∇w(t)/ba∇dbl2\nL2≤C/ba∇dblw(0)/ba∇dbl2\nH1,\nwhich implies ( 2.8).\nThe case r= 2: Taking φ=∂twin (4.8) and using integration by parts for the term with\ncoefficient β5give\n/ba∇dbl∂tw/ba∇dbl2\nL2+β2d\ndt/ba∇dbl∆w/ba∇dbl2\nL2≤β1d\ndt/ba∇dbl∇w/ba∇dbl2\nL2+β3d\ndt/ba∇dblw/ba∇dbl2\nL2+β3/vextendsingle/vextendsingle/vextendsingle/angbracketleftbig\n|u|2u−|v|2v,∂tw/angbracketrightbig\nL2/vextendsingle/vextendsingle/vextendsingle\n+β4/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{tu×∆u−v×∆v,∂tw/an}b∇acket∇i}htL2/vextendsingle/vextendsingle/vextendsingle\n+β5/vextendsingle/vextendsingle/vextendsingle/angbracketleftbig\n∆(|u|2u−|v|2v),∂tw/angbracketrightbig\nL2/vextendsingle/vextendsingle/vextendsingle. (4.18)\nNote that when using integration by parts, the integrals on the bou ndary vanish due to the\nboundary property of wand (3.3). Each inner product on the right-hand side of ( 4.18) can\nbe estimated as follows. For the first inner product, similarly to ( 4.12) we have\nβ3/vextendsingle/vextendsingle/angbracketleftbig\n|u|2u−|v|2v,∂tw/angbracketrightbig\nL2/vextendsingle/vextendsingle≤C/parenleftbig\n/ba∇dblu/ba∇dbl4\nL∞+/ba∇dblv/ba∇dbl4\nL∞/parenrightbig\n/ba∇dblw/ba∇dbl2\nL2+ǫ/ba∇dbl∂tw/ba∇dbl2\nL2 (4.19)\nfor anyǫ >0. For the second inner product, applying ( 3.16) and Young’s inequality yield\nβ4/vextendsingle/vextendsingle/an}b∇acketle{tu×∆u−v×∆v,∂tw/an}b∇acket∇i}htL2/vextendsingle/vextendsingle≤C/parenleftbig\n/ba∇dblu/ba∇dbl2\nH2+/ba∇dblv/ba∇dbl2\nH2/parenrightbig\n/ba∇dblw/ba∇dbl2\nH2+ǫ/ba∇dbl∂tw/ba∇dbl2\nL2.(4.20)\nFor the last inner product in ( 4.18), by (3.15) and Young’s inequality, we have\nβ5/vextendsingle/vextendsingle/angbracketleftbig\n∆(|u|2u−|v|2v),∂tw/angbracketrightbig\nL2/vextendsingle/vextendsingle≤C/parenleftbig\n/ba∇dblu/ba∇dbl4\nH2+/ba∇dblv/ba∇dbl4\nH2/parenrightbig\n/ba∇dblw/ba∇dbl2\nH2+ǫ/ba∇dbl∂tw/ba∇dbl2\nL2.(4.21)\nInserting ( 4.19), (4.20) and (4.21) into (4.18), integrating over (0 ,t), and choosing ǫ >0\nsufficiently small, we obtain\n/integraldisplayt\n0/ba∇dbl∂tw(s)/ba∇dbl2\nL2ds+/ba∇dbl∆w(t)/ba∇dbl2\nL2/lessorsimilar/ba∇dblw(0)/ba∇dbl2\nH2+/integraldisplayt\n0β(s)/ba∇dblw(s)/ba∇dbl2\nL2ds+/integraldisplayt\n0β(s)/ba∇dbl∆w(s)/ba∇dbl2\nL2ds,\nwhere\nβ(s) := 1+/ba∇dblu(s)/ba∇dbl4\nH2+/ba∇dblv(s)/ba∇dbl4\nH2.\nGronwall’s inequality then yields\n/ba∇dbl∆w(t)/ba∇dbl2\nL2≤C/ba∇dblw(0)/ba∇dbl2\nH2.\nThis, together with ( 2.7) shown for r= 0 and ( 2.8) forr= 1, gives the required inequal-\nity (2.8) forr= 2.GLOBAL SOLUTIONS OF LLBAR EQUATION 25\nThe case r= 3: We now take φ=−∆∂twin (4.8). Using integration by parts for all\nterms, we have\n/ba∇dbl∂t∇w/ba∇dbl2\nL2+β1d\ndt/ba∇dbl∆w/ba∇dbl2\nL2+β2d\ndt/ba∇dbl∇∆w/ba∇dbl2\nL2\n=β3d\ndt/ba∇dbl∇w/ba∇dbl2\nL2−β3/angbracketleftbig\n∇(|u|2u−|v|2v),∇∂tw/angbracketrightbig\nL2\n+β4/an}b∇acketle{t∇u×∆u−∇v×∆v,∇∂tw/an}b∇acket∇i}htL2+β4/an}b∇acketle{tu×∇∆u−v×∇∆v,∇∂tw/an}b∇acket∇i}htL2\n−β5/angbracketleftbig\n∇∆(|u|2u−|v|2v),∇∂tw/angbracketrightbig\nL2. (4.22)\nEach inner product on the right-hand side of ( 4.22) can be estimated as follows. For the\nfirst inner product, similarly to ( 4.17) we have\nβ3/vextendsingle/vextendsingle/angbracketleftbig\n∇(|u|2u−|v|2v),∇∂tw/angbracketrightbig\nL2/vextendsingle/vextendsingle/lessorsimilar/parenleftbig\n/ba∇dblu/ba∇dbl2\nH1+/ba∇dblv/ba∇dbl2\nH1/parenrightbig/parenleftbig\n/ba∇dblu/ba∇dbl2\nH2+/ba∇dblv/ba∇dbl2\nH2/parenrightbig\n/ba∇dblw/ba∇dbl2\nH1\n+ǫ/ba∇dbl∇∂tw/ba∇dbl2\nL2 (4.23)\nfor anyǫ >0. For the second inner product, applying H¨ older’s and Young’s ineq uality yields\nβ4/vextendsingle/vextendsingle/an}b∇acketle{t∇u×∆u−∇v×∆v,∇∂tw/an}b∇acket∇i}htL2/vextendsingle/vextendsingle\n/lessorsimilar/ba∇dbl∇u×∆u−∇v×∆v/ba∇dbl2\nL2+ǫ/ba∇dbl∇∂tw/ba∇dbl2\nL2\n/lessorsimilar/ba∇dbl∇u/ba∇dbl2\nL∞/ba∇dbl∆w/ba∇dbl2\nL2+/ba∇dbl∇w/ba∇dbl2\nL∞/ba∇dbl∆v/ba∇dbl2\nL2+ǫ/ba∇dbl∇∂tw/ba∇dbl2\nL2\n/lessorsimilar/parenleftbig\n/ba∇dblu/ba∇dbl2\nH3+/ba∇dblv/ba∇dbl2\nH2/parenrightbig\n/ba∇dblw/ba∇dbl2\nH3+ǫ/ba∇dbl∇∂tw/ba∇dbl2\nL2 (4.24)\nfor anyǫ >0, where in the last step we used the Sobolev embedding H2⊂L∞. For the\nthird inner product we have by H¨ older’s inequality, Young’s inequality , and (3.16)\nβ4/vextendsingle/vextendsingle/an}b∇acketle{tu×∇∆u−v×∇∆v,∇∂tw/an}b∇acket∇i}htL2/vextendsingle/vextendsingle\n/lessorsimilar/ba∇dblu/ba∇dbl2\nL∞/ba∇dbl∇∆w/ba∇dbl2\nL2+/ba∇dblw/ba∇dbl2\nH2/ba∇dbl∇∆v/ba∇dbl2\nL2+ǫ/ba∇dbl∇∂tw/ba∇dbl2\nL2\n/lessorsimilar/parenleftbig\n/ba∇dblu/ba∇dbl2\nH2+/ba∇dblv/ba∇dbl2\nH3/parenrightbig\n/ba∇dblw/ba∇dbl2\nH3+ǫ/ba∇dbl∇∂tw/ba∇dbl2\nL2 (4.25)\nfor anyǫ >0. For the last inner product in ( 4.22), by using H¨ older’s and Young’s inequality,\nand (3.15), we have\nβ5/vextendsingle/vextendsingle/angbracketleftbig\n∇∆(|u|2u−|v|2v),∇∂tw/angbracketrightbig\nL2/vextendsingle/vextendsingle≤C/parenleftbig\n/ba∇dblu/ba∇dbl4\nH3+/ba∇dblv/ba∇dbl4\nH3/parenrightbig\n/ba∇dblw/ba∇dbl2\nH3+ǫ/ba∇dbl∇∂tw/ba∇dbl2\nL2.(4.26)\nDefine\nγ(s) := 1+/ba∇dblu(s)/ba∇dbl4\nH3+/ba∇dblv(s)/ba∇dbl4\nH3.\nInserting ( 4.23), (4.24), (4.25) and (4.26) into (4.22), integrating over (0 ,t), and choosing\nǫ >0 sufficiently small, we obtain\n/ba∇dbl∇∆w(t)/ba∇dbl2\nL2/lessorsimilar/ba∇dblw(0)/ba∇dbl2\nH3+/integraldisplayt\n0γ(s)/parenleftbig\n/ba∇dblw(s)/ba∇dbl2\nL2ds+/ba∇dbl∇w(s)/ba∇dbl2\nL2ds+/ba∇dbl∇∆w(s)/ba∇dbl2\nL2/parenrightbig\nds\n/lessorsimilar/ba∇dblw(0)/ba∇dbl2\nH3+/integraldisplayt\n0γ(s)/ba∇dbl∇∆w(s)/ba∇dbl2\nL2ds,\nwhere we used ( 3.10), (2.7) and (2.8) withr= 1. Gronwall’s inequality yields\n/ba∇dbl∇∆w(t)/ba∇dbl2\nL2≤C/ba∇dblw(0)/ba∇dbl2\nH3\nand the required estimate ( 2.8) forr= 3 then follows.26 AGUS L. SOENJAYA AND THANH TRAN\nExtension from [0 ,T∗] to [0,T] ford= 3: Recall that T∗≤Tford= 3. We will now show\nthat in this case, we also have T∗=T. First, it follows from ( 3.35) that, for r= 1,2,3,\nu∈L∞(0,T∗;Hr)∩L2(0,T∗;Hr+2). (4.27)\nAssume that the following estimate holds (which will be shown in Propos ition4.2later).\n/ba∇dbl∇u(t)/ba∇dbl2\nL2+/ba∇dblu(t)/ba∇dbl4\nL4+/integraldisplayt\n0/ba∇dbl∇∆u(s)/ba∇dbl2\nL2ds\n+/integraldisplayt\n0/vextenddouble/vextenddouble∇(|u(s)|2u(s))/vextenddouble/vextenddouble2\nL2ds+/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl6\nL6ds/lessorsimilar/ba∇dblu0/ba∇dbl2\nH1, t∈[0,T∗].(4.28)\nThen we can repeat the arguments leading to the proof of ( 3.35) withunreplaced by u\nto obtain similar estimates for u, with constant depending on T. Proposition 4.2and (3.35)\nimply that this weak solution uoriginally defined on [0 ,T∗] belongs to C([0,T∗];Hr)∩\nL2(0,T∗;Hr+2), and that u(t,x) remains bounded in this norm as t→T∗from the left.\nTherefore, the technique of continuation of solutions can be applie d and thus the solution u\nexists on the whole interval [0 ,T] for any T >0.\nIt remains to prove ( 4.28).\nProposition 4.2. LetT >0 be arbitrary and T∗be defined as in Proposition 3.8. Letu\nbe the unique weak solution of ( 1.2). Then ( 4.28) holds with a constant depending on T.\nProof.We aim to choose φ=α|u(t)|2u(t) in (2.5), for some positive constant α. Hence,\nwe first consider the nonlinear terms in the resulting equation with th at choice of φ. For the\nterm with coefficient β1, we use ( 3.2) to have\n/angbracketleftbig\n∇u(s),∇/parenleftbig\n|u(s)|2u(s)/parenrightbig/angbracketrightbig\nL2= 2/ba∇dblu(s)·∇u(s)/ba∇dbl2\nL2+/ba∇dbl|u(s)||∇u(s)|/ba∇dbl2\nL2.(4.29)\nFor the term with coefficient β2, we use integration by parts to have\n/angbracketleftbig\n∆u(s),∆/parenleftbig\n|u(s)|2u(s)/parenrightbig/angbracketrightbig\nL2=−/angbracketleftbig\n∇∆u(s),∇/parenleftbig\n|u(s)|2u(s)/parenrightbig/angbracketrightbig\nL2. (4.30)\nFor the terms involving β3andβ5, it is straightforward to have\n/angbracketleftbig/parenleftbig\n1−|u(s)|2/parenrightbig\nu(s),|u(s)|2u(s)/angbracketrightbig\nL2=/ba∇dblu(s)/ba∇dbl4\nL4−/ba∇dblu(s)/ba∇dbl6\nL6,\n/angbracketleftbig\n∇/parenleftbig\n|u(s)|2u(s)/parenrightbig\n,∇/parenleftbig\n|u(s)|2u(s)/parenrightbig/angbracketrightbig\nL2=/vextenddouble/vextenddouble∇/parenleftbig\n|u(s)|2u(s)/parenrightbig/vextenddouble/vextenddouble2\nL2.\nThe term involving β4vanishes. Altogether, we deduce from choosing φ= 4α|u(t)|2u(t)\nin (2.5) that\nα/ba∇dblu(t)/ba∇dbl4\nL4+8αβ1/integraldisplayt\n0/ba∇dblu(s)·∇u(s)/ba∇dbl2\nL2ds+4αβ1/integraldisplayt\n0/ba∇dbl|u(s)||∇u(s)|/ba∇dbl2\nL2ds\n−4αβ2/integraldisplayt\n0/angbracketleftbig\n∇∆u(s),∇(|u(s)|2u(s))/angbracketrightbig\nL2ds\n= 4α/angbracketleftbig\nu0,|u(t)|2u(t)/angbracketrightbig\nL2+4αβ3/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl4\nL4ds−4αβ3/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl6\nL6ds\n−4αβ5/integraldisplayt\n0/vextenddouble/vextenddouble∇(|u(s)|2u(s))/vextenddouble/vextenddouble2\nL2ds. (4.31)GLOBAL SOLUTIONS OF LLBAR EQUATION 27\nNext, we choose φ=−2∆u(t) in (2.5) and use integration by parts, noting ( 2.3) so that\nthe term involving β4vanishes. We then have, noting ( 4.29),\n/ba∇dbl∇u(s)/ba∇dbl2\nL2+2β1/integraldisplayt\n0/ba∇dbl∆u(s)/ba∇dbl2\nL2ds+2β2/integraldisplayt\n0/ba∇dbl∇∆u(s)/ba∇dbl2\nL2ds\n= 2/an}b∇acketle{t∇u0,∇u(t)/an}b∇acket∇i}htL2+2β3/integraldisplayt\n0/ba∇dbl∇u(s)/ba∇dbl2\nL2ds\n−4β3/integraldisplayt\n0/ba∇dblu(s)·∇u(s)/ba∇dbl2\nL2ds−2β3/integraldisplayt\n0/ba∇dbl|u(s)||∇u(s)|/ba∇dbl2\nL2ds\n+2β5/integraldisplayt\n0/angbracketleftbig\n∇∆u(s),∇(|u(s)|2u(s))/angbracketrightbig\nds. (4.32)\nAdding ( 4.31) and (4.32) gives\nα/ba∇dblu(t)/ba∇dbl4\nL4+/ba∇dbl∇u(t)/ba∇dbl2\nL2+2β2/integraldisplayt\n0/ba∇dbl∇∆u(s)/ba∇dbl2\nL2ds+4αβ3/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl6\nL6ds\n−(4αβ2+2β5)/integraldisplayt\n0/angbracketleftbig\n∇∆u(s),∇(|u(s)|2u(s))/angbracketrightbig\nL2ds+4αβ5/integraldisplayt\n0/vextenddouble/vextenddouble∇(|u(s)|2u(s))/vextenddouble/vextenddouble2\nL2ds\n=α/angbracketleftbig\nu0,|u(t)|2u(t)/angbracketrightbig\nL2+/an}b∇acketle{t∇u0,∇u(t)/an}b∇acket∇i}htL2\n−2β1/integraldisplayt\n0/ba∇dbl∆u(s)/ba∇dbl2\nL2ds+4αβ3/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl4\nL4ds+2β3/integraldisplayt\n0/ba∇dbl∇u(s)/ba∇dbl2\nL2ds\n−(8αβ1+4β3)/integraldisplayt\n0/ba∇dblu(s)·∇u(s)/ba∇dbl2\nL2ds−(4αβ1+2β3)/integraldisplayt\n0/ba∇dbl|u(s)||∇u(s)|/ba∇dbl2\nL2ds.(4.33)\nNote that if α=β5/2β2, then the third, fifth, and sixth terms on the left-hand side add up\nto\n2β2/ba∇dbl∇∆u(s)/ba∇dbl2\nL2−(4αβ2+2β5)/angbracketleftbig\n∇∆u(s),∇(|u(s)|2u(s))/angbracketrightbig\n+4αβ5/vextenddouble/vextenddouble∇(|u(s)|2u(s))/vextenddouble/vextenddouble2\nL2\n=/vextenddouble/vextenddouble/vextenddouble/radicalbig\n2β2∇∆u(s)−/radicalbig\n4αβ5∇(|u(s)|2u(s))/vextenddouble/vextenddouble/vextenddouble2\nL2≥0.\nHence, with this value of αand the use of Young’s inequality, ( 4.33) becomes\nα/ba∇dblu(t)/ba∇dbl4\nL4+/ba∇dbl∇u(t)/ba∇dbl2\nL2+4αβ3/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl6\nL6ds+/integraldisplayt\n0/vextenddouble/vextenddouble/vextenddouble/radicalbig\n2β2∇∆u−/radicalbig\n4αβ5∇(|u|2u)/vextenddouble/vextenddouble/vextenddouble2\nL2ds\n≤C/ba∇dblu0/ba∇dbl4\nL4+ǫ/ba∇dblu(t)/ba∇dbl4\nL4+C/ba∇dbl∇u0/ba∇dbl2\nL2+ǫ/ba∇dbl∇u(t)/ba∇dbl2\nL2\n+2|β1|/integraldisplayt\n0/ba∇dbl∆u(s)/ba∇dbl2\nL2ds+4αβ3/integraldisplayt\n0/ba∇dblu(s)/ba∇dbl4\nL4ds+2β3/integraldisplayt\n0/ba∇dbl∇u(s)/ba∇dbl2\nL2ds\n+(8α|β1|+4β3)/integraldisplayt\n0/ba∇dblu(s)·∇u(s)/ba∇dbl2\nL2ds+(4α|β1|+2β3)/integraldisplayt\n0/ba∇dbl|u(s)||∇u(s)|/ba∇dbl2\nL2ds\n≤C/ba∇dblu0/ba∇dbl2\nH1+ǫ/ba∇dblu(t)/ba∇dbl4\nL4+ǫ/ba∇dbl∇u(t)/ba∇dbl2\nL2,\nwhere in the last step we used the Sobolev embedding H1⊂L4(foru0) and Proposition 3.6\nto boundall the integrals onthe right-handside. Choosing ǫ >0 sufficiently small, we obtain\nthe required estimate. /square28 AGUS L. SOENJAYA AND THANH TRAN\n5.Proof of Theorem 2.3\nProof.For any Banach space X, sinceC0,α2([0,T];X)⊂C0,α1([0,T];X) for 0< α1< α2, it\nsuffices to prove the theorem for α= 1/2 andβ= 1/2−d/8.\nLetT >0 andτ,t∈[0,T] be such that τ < t. Performing integration by parts on ( 2.5)\n(and noting the regularity of the solution ugiven by Theorem 2.2), we have for any φ∈H2,\n/an}b∇acketle{tu(t)−u(τ),φ/an}b∇acket∇i}htL2−β1/integraldisplayt\nτ/an}b∇acketle{t∆u(s),φ/an}b∇acket∇i}htL2ds+β2/integraldisplayt\nτ/angbracketleftbig\n∆2u(s),φ/angbracketrightbig\nL2ds\n=β3/integraldisplayt\nτ/angbracketleftbig\n(1−|u(s)|2)u(s),φ/angbracketrightbig\nL2ds−β4/integraldisplayt\nτ/an}b∇acketle{tu(s)×∆u(s),φ/an}b∇acket∇i}htL2ds\n+β5/integraldisplayt\nτ/angbracketleftbig\n∆(|u(s)|2u(s)),φ/angbracketrightbig\nL2ds.\nTherefore, by H¨ older’s inequality,\n/vextendsingle/vextendsingle/an}b∇acketle{tu(t)−u(τ),φ/an}b∇acket∇i}htL2/vextendsingle/vextendsingle≤ |β1|/ba∇dblφ/ba∇dblL2/integraldisplayt\nτ/ba∇dbl∆u(s)/ba∇dblL2ds+β2/ba∇dblφ/ba∇dblL2/integraldisplayt\nτ/vextenddouble/vextenddouble∆2u(s)/vextenddouble/vextenddouble\nL2ds\n+β3/ba∇dblφ/ba∇dblL2/integraldisplayt\nτ/ba∇dblu(s)/ba∇dblL2ds+β3/ba∇dblφ/ba∇dblL2/integraldisplayt\nτ/ba∇dblu(s)/ba∇dbl3\nL6ds\n+β4/ba∇dblφ/ba∇dblL2/integraldisplayt\nτ/ba∇dblu(s)×∆u(s)/ba∇dblL2ds\n+β5/ba∇dblφ/ba∇dblL2/integraldisplayt\nτ/vextenddouble/vextenddouble∆(|u(s)|2u(s))/vextenddouble/vextenddouble\nL2ds.\nTakingφ=u(t)−u(τ), we obtain\n/ba∇dblu(t)−u(τ)/ba∇dblL2/lessorsimilar/integraldisplayt\nτ/ba∇dbl∆u(s)/ba∇dblL2ds+/integraldisplayt\nτ/vextenddouble/vextenddouble∆2u(s)/vextenddouble/vextenddouble\nL2ds+/integraldisplayt\nτ/ba∇dblu(s)/ba∇dblL2ds\n+/integraldisplayt\nτ/ba∇dblu(s)/ba∇dbl3\nL6ds+/integraldisplayt\nτ/ba∇dblu(s)×∆u(s)/ba∇dblL2ds\n+/integraldisplayt\nτ/vextenddouble/vextenddouble∆(|u(s)|2u(s))/vextenddouble/vextenddouble\nL2ds. (5.1)\nWe will now estimate each term on the right-hand side of ( 5.1). For the linear terms, by\nH¨ older’s inequality and Corollary 3.15,\n/integraldisplayt\nτ/ba∇dbl∆u(s)/ba∇dblL2ds≤ |t−τ|1\n2/ba∇dbl∆u/ba∇dblL2(0,T;L2)/lessorsimilar|t−τ|1\n2,\n/integraldisplayt\nτ/vextenddouble/vextenddouble∆2u(s)/vextenddouble/vextenddouble\nL2ds≤ |t−τ|1\n2/vextenddouble/vextenddouble∆2u/vextenddouble/vextenddouble\nL2(0,T;L2)/lessorsimilar|t−τ|1\n2,\n/integraldisplayt\nτ/ba∇dblu(s)/ba∇dblL2ds≤ |t−τ|1\n2/ba∇dblu/ba∇dblL2(0,T;L2)/lessorsimilar|t−τ|1\n2.\nFor the nonlinear terms on the right-hand side of ( 5.1), by H¨ older’s inequality, Corollary 3.15\nand the Sobolev embedding,\n/integraldisplayt\nτ/ba∇dblu(s)/ba∇dbl3\nL6ds≤ |t−τ|1\n2/ba∇dblu/ba∇dbl3\nL6(0,T;L6)/lessorsimilar|t−τ|1\n2/ba∇dblu/ba∇dbl3\nL∞(0,T;H1)GLOBAL SOLUTIONS OF LLBAR EQUATION 29\n/lessorsimilar|t−τ|1\n2,\n/integraldisplayt\nτ/ba∇dblu(s)×∆u(s)/ba∇dblL2ds≤/integraldisplayt\nτ/ba∇dblu(s)/ba∇dblL∞/ba∇dbl∆u(s)/ba∇dblL2ds\n≤ /ba∇dbl∆u/ba∇dblL∞(0,T;L2)|t−τ|1\n2/ba∇dblu/ba∇dblL2(0,T;L∞)\n≤ |t−τ|1\n2/ba∇dblu/ba∇dblL2(0,T;H2)/ba∇dbl∆u/ba∇dblL∞(0,T;L2)\n/lessorsimilar|t−τ|1\n2,\n/integraldisplayt\nτ/vextenddouble/vextenddouble∆(|u(s)|2u(s))/vextenddouble/vextenddouble\nL2ds≤/integraldisplayt\nτ/vextenddouble/vextenddouble|u(s)|2u(s)/vextenddouble/vextenddouble\nH2ds≤/integraldisplayt\nτ/ba∇dblu(s)/ba∇dbl3\nH2ds\n≤ |t−τ|1\n2/ba∇dblu/ba∇dbl3\nL6(0,T;H2)≤ |t−τ|1\n2/ba∇dblu/ba∇dbl3\nL∞(0,T;H2)\n/lessorsimilar|t−τ|1\n2,\nwhere for the last nonlinear term, we also used ( 3.14). Altogether, we derive from ( 5.1) that\nu∈C0,α(0,T;L2) forα∈(0,1/2].\nFinally, by the Gagliardo–Nirenberg inequality (Theorem 6.2withv=u(t)−u(τ),r= 0,\nq=∞,s1= 0,s2= 2),\n/ba∇dblu(t)−u(τ)/ba∇dblL∞/lessorsimilar/ba∇dblu(t)−u(τ)/ba∇dbl1−d\n4\nL2/ba∇dblu(t)−u(τ)/ba∇dbld\n4\nH2\n/lessorsimilar/ba∇dblu(t)−u(τ)/ba∇dbl1−d\n4\nL2/ba∇dblu/ba∇dbld\n4\nC([0,T];H2)\n/lessorsimilar|t−τ|1\n2−d\n8,\nwhere in the penultimate step we used Theorem 2.2and in the last step we used the previous\npart of this theorem. /square\n6.Appendix\nWe collect in this section a few results which were extensively used in th is paper.\nTheorem 6.1 (Gronwall–Bihari’s inequality [ 6,7]).Letfbe a non-decreasing continuous\nfunction which is non-negative on [0 ,∞) such that/integraltext∞\n11/f(x)dx <∞. LetFbe the anti-\nderivative of −1/fwhich vanishes at ∞. Lety: [0,∞)→[0,∞) be a continuous function\nand letgbe a locally integrable non-negative function on [0 ,∞). Suppose that there exists\ny0>0 such that for all t≥0,\ny(t)≤y0+/integraldisplayt\n0g(s)ds+/integraldisplayt\n0f(y(s))ds.\nLetT∗be the unique solution of the equation\nT∗=F/parenleftbigg\ny0+/integraldisplayT∗\n0g(s)ds/parenrightbigg\n.\nThen for any T′∈(0,T∗), we have\nsup\n0≤t≤T′y(t)≤F−1/parenleftBigg\nF/parenleftBig\ny0+/integraldisplayT′\n0g(s)ds/parenrightBig\n−T′/parenrightBigg\n. (6.1)\nNote that the expression on the right-hand side of ( 6.1) tends to ∞asT′→T∗.30 AGUS L. SOENJAYA AND THANH TRAN\nThe following theorem is a special case of a more general result in [ 8].\nTheorem 6.2 (Gagliardo–Nirenberg inequalities) .Let Ω be a bounded domain of Rdwith\nLipschitz boundary, and let v: Ω→R3. Then\n/ba∇dblv/ba∇dblWr,q≤C/ba∇dblv/ba∇dblθ\nHs1/ba∇dblv/ba∇dbl1−θ\nHs2 (6.2)\nfor allv∈Hs2(Ω), where s1,s2,rare non-negative real numbers satisfying\n0≤s1< s2, θ∈(0,1),0≤r < θs 1+(1−θ)s2,\nandq∈(2,∞] satisfies\n1\nq=1\n2+(s2−s1)θ\nd−s2−r\nd.\nMoreover, when 2 < q <∞, we have\nθ=2q(s2−r)−d(q−2)\n2q(s2−s1).\nTheorem 6.3 (Aubin–Lions–Simon lemma [ 31]).LetX0֒→X ֒→X1be three Banach\nspaces such that the inclusion X0֒→Xis compact and the inclusion X ֒→X1is continuous.\nFor 1≤p,q≤ ∞, let\nWp,q:={v∈Lp(0,T;X0) :vt∈Lq(0,T;X1)}.\n(1) Ifp <∞, thenWp,qis compactly embedded into Lp(0,T;X).\n(2) Ifp=∞andq= 1, then Wp,qis compactly embedded into C([0,T];X).\nTheorem 6.4 (Theorem II.5.14 in [ 7]).LetVandWbe Hilbert spaces. Then the space\n{v∈L2(0,T;V) :∂tv∈L2(0,T;W)}\nis continuously embedded into C([0,T];[V,W]1/2). 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Phe nomenological description of the\nnonlocal magnetization relaxation in magnonics, spintronics, and do main-wall dynamics. Phys. Rev. B ,\n92(2015), 054430.\n[36] T. Weindler, H. G. Bauer, R. Islinger, B. Boehm, J.-Y. Chauleau, and C. H. Back. Magnetic damping:\nDomain wall dynamics versus local ferromagnetic resonance. Phys. Rev. Lett. ,113(2014), 237204.\nSchool of Mathematics and Statistics, The University of New South Wales, Sydney 2052,\nAustralia\nEmail address :a.soenjaya@unsw.edu.au\nSchool of Mathematics and Statistics, The University of New South Wales, Sydney 2052,\nAustralia\nEmail address :thanh.tran@unsw.edu.au"    },    {        "title": "1606.09326v2.Skyrmion_dynamics_in_a_chiral_magnet_driven_by_periodically_varying_spin_currents.pdf",        "content": "arXiv:1606.09326v2  [cond-mat.mes-hall]  6 Dec 2016Skyrmion dynamics in a chiral magnet driven by periodically\nvarying spin currents\nRui Zhu*and Yin-Yan Zhang\nDepartment of Physics, South China University of Technolog y,\nGuangzhou 510641, People’s Republic of China\nAbstract\nIn this work, we investigated the spin dynamics in a slab of ch iral magnets induced by an\nalternating (ac) spin current. Periodic trajectories of th e skyrmion in real space are discovered\nunder the ac current as a result of the Magnus and viscous forc es, which originate from the Gilbert\ndamping, the spin transfer torque, and the β-nonadiabatic torque effects. The results are obtained\nby numerically solving the Landau-Lifshitz-Gilbert equat ion and can be explained by the Thiele\nequation characterizing the skyrmion core motion.\nPACS numbers: 75.78.-n, 72.25.-b, 71.70.-d\n*Corresponding author. Electronic address: rzhu@scut.edu.cn\n1I. INTRODUCTION\nThe skyrmion spin texture is a kind of topologically-nontrivial magnet ic vortex formed\nmost typically in the bulk chiral magnets (CMs) and magnetic thin films1–3. In CMs it is\nbelieved that the spin-orbit coupling induced Dzyaloshinskii-Moriya int eraction (DMI) gov-\nerns the spin twisting1. Recently the magnetic skyrmion structure attracts intensive fo cus,\nboth in the fundamental theoretic aspect and in its potential applic ation in the information\ntechnology4–7. In the magnetic skyrmion state the emergent electrodynamic effe ct originates\nfrom its nontrivial spin topology and gives rise to the topological Hall effect and a remark-\nable current-driven spin transfer torque effect1,8–14. The so-called skyrmionics makes use\nof the skyrmion as a memory unit favored by its topologically protect ed long lifetime and\nultralow driving current, which is five or six orders smaller than that f or driving a magnetic\ndomain wall1,15.\nAlthough the current-driven spin dynamics in the CMs with DMI has be en intensively\nstudied recently, less work of an alternating current (ac) driven s kyrmion dynamics was\nreported. The skyrmion-motion-induced ac current generation h as been predicted, which\nshares the reversed effect of our consideration16. In this work, we investigated the ac-\nspin-current driven skyrmion dynamics with the DMI, Gilbert damping17, adiabatic and\nnonadiabatic spin torques, and different current profiles taken int o account. Our proposition\nis inspired by the following several aspects. Firstly, it is both theore tically and technically\ninteresting to know the behavior of a skyrmion when an external ac current is applied.\nSecondly, ahigh-speedlow-powermodulationofaskyrmionisfavora bleforpotentialmemory\nprocessing. Lastly but not least, we noticed the mathematical sign ificance of the solution of\nthe Landau-Lifshitz-Gilbert (LLG) equation of a collinear magnet wit h periodically varying\nspin-currents applied, in which chaos is observed18,19.\nThe topological property of a spin texture can be described by the surface integral of the\nsolid angle of the unitary spin-field vector n(r). The skyrmion number is so defined as S=\n1\n4π/integraltext\nn·/parenleftBig\n∂n\n∂x×∂n\n∂y/parenrightBig\nd2rcounting how many times the spin field wraps the unit sphere. More\nspecific topological properties of a skyrmion can be considered by a nalyzing its radial and\nwhirling symmetric pattern1,20–22. In the continuum field theory, as a result of topological\nprotection, the skyrmion cannot be generated from a topologically trivial magnetic state\nsuch as a ferromagnet or a helimagnet by variation without a topolog ically nontrivial force\n2such as a spatially nontrivial spin current11,12, geometrical constriction13, domain wall pair\nsource5,6, the edge spin configuration22, etc., and vise versa. It is predicted by simulation\nthat the skyrmion can be generated from a quasi-ferromagnetic a nd helimagnetic state by\nexternal Lorentzian and radial spin current12and that transformation is possible between\ndifferent topologically-nontrivial states such as that between the domain-wall pair and the\nskyrmion5,6. The local current flowing from the scanning tunneling microscope t o generate\nthe skyrmion in experiment can be approximated by a radial spin curr ent, which imbues\nnontrivial topology into the helimagnet11. Also an artificial magnetic skyrmion can be\ntailored by an external magnetic field with nontrivial geometric distr ibution23. When the\nboundary geometry of the material is tailored such as by a notch in a long plate, a skyrmion\ncan be generated by a collinear spin current13. In this case, the nontrivial constriction\ntopology contributes to the formation of the skyrmion. The unifor m current can move and\nrotate a skyrmion without changing its topology15,24. In this work we will show that these\ntopological behaviors of the skyrmion are retained in the spin dynam ics driving by an ac\nspin current.\nAlmost all kinds of ferromagnetic and vortex spin dynamics can be de scribed by the LLG\nequation. The behavior of the LLG equation is of importance in both t he physical and\nmathematical sciences18. It has been shown by previous works that the spin torque effect\ndriven by a periodic varying spin current can be described as well by t he LLG equation\nwith the original time-independent current replaced by the time-de pendent current in the\nspin torque term18,19. Although chaotic behaviors are predicted in the spatially-uniform a c\nspin-current driven collinear ferromagnetic spin structure18,19, which is well described by the\nsingle-spin LLG equation, no similar phenomenon is reported in a spatia lly-nonuniform spin\nlattice, the latter of which can be attributed to the relaxation proc esses of the inter-site\nscattering. Even if some sort of chaotic behavior occurs after a lo ng time of evolution, it\nis workable to restore the original state by applying a magnetic field a fter some time. The\ninfluence of it on the skyrmionics exploitation is not large. In this work , we use a matrix-\nbased fourth-order Runge-Kutta method to solve the LLG equat ion with both the adiabatic\nand nonadiabatic spin torques taken into account. Analytical solut ion of the generalized\nThiele equation1,13,24reproduces our numerical results.\n3II. THEORETIC FORMALISM\nWe consider a thin slab of CM modulated by a constant magnetic field an d an ac spin\ncurrent. The strong DMI makes the material a skyrmion-host. In the continuum approxi-\nmation, the Hamiltonian of the localized magnetic spin in a CM can be desc ribed as1,12,13\nH=−J/summationtext\nrMr·/parenleftbig\nMr+ex+Mr+ey/parenrightbig\n−D/summationtext\nr/parenleftbig\nMr×Mr+ex·ex+Mr×Mr+ey·ey/parenrightbig\n−B·/summationtext\nrMr,(1)\nwithJandDthe ferromagnetic and Dzyaloshinskii-Moriya (DM) exchange energ ies, re-\nspectively. The dimensionless local magnetic moments Mrare defined as Mr≡ −Sr//planckover2pi1,\nwhereSris the local spin at rand/planckover2pi1is the plank constant divided by 2 π. We assume that\nthe length of the vector |Mr|=Mis fixed, therefore Mr=Mn(r) withn(r) the unitary\nspin field vector. The unit-cell dimension is taken to be unity. An exte rnal magnetic field\nBis applied perpendicular to the slab plane to stabilize the skyrmion confi guration. The\nBohr magneton µBis absorbed into Bto have it in the unit of energy. The typical DMI\nD= 0.18Jis used throughout this work13. This DM exchange strength corresponds to the\ncritical magnetic fields Bc1= 0.0075Jbetween the helical and skyrmion-crystal phases and\nBc2= 0.0252Jbetween the skyrmion-crystal and ferromagnetic phases, resp ectively. We\nadoptB= (0,0,0.01J) in our numerical considerations with J= 1 meV.\nTheextendedformoftheLLGequationthattakesintoaccountth eDMIandtheadiabatic\nand nonadiabatic spin torque effects can be expressed in the followin g formula1,12,13,25\ndMr\ndt=−γMr×Beff\nr+α\nMMr×dMr\ndt+pa3\n2eM[j(r,t)·∇]Mr\n−pa3β\n2eM2{Mr×[j(r,t)·∇]Mr}.(2)\nBy assuming that the energy of a magnet with the local magnetizatio nMrin a spatially\nvarying magnetic field Beff\nrhas the form of H=−γ/planckover2pi1/summationtext\nrMr·Beff\nr, we have\nBeff\nr=−1\n/planckover2pi1γ∂H\n∂Mr, (3)\nand therefore the first term in the right hand side of Eq. (2) is\n−γMr×Beff\nr=−J\n/planckover2pi1Mr×/parenleftbig\nMr+ex+Mr+ey+Mr−ex+Mr−ey/parenrightbig\n−1\n/planckover2pi1(Mr×B)\n−D\n/planckover2pi1Mr×\n/parenleftbig\nMr−ey,z−Mr+ey,z/parenrightbig\nex+(Mr+ex,z−Mr−ex,z)ey\n+/parenleftbig\nMr+ey,x−Mr+ex,y−Mr−ey,x+Mr−ex,y/parenrightbig\nez\n.(4)\n4The second to the last terms of Eq. (2) sequentially correspond to the effect of the Gilbert\ndamping, the time-dependent spin current j(t) =jesin(ωt)-induced adiabatic and nonadia-\nbatic spin torques, respectively. pmeasures the polarization of the conduction electrons, eis\nthe positive electron charge, and ais the average in-plane lattice constant of the CM. In our\nconsiderations the frequency of the ac spin current ωis small enough in comparison of the\nmagnetization evolution rate. Therefore, the spin torques can be satisfactorily described by\nusing the time-dependent current in thestandard torque expres sion, which has been justified\nby previous studies18,19. Here, the unit of time is set to be t0=/planckover2pi1/J≈6.6×10−13s. A\nphenomenologically expectedvalueof α= 0.1isusedinafterwardsnumerical considerations.\nBy looking deep into Eqs. (2) and (4), we can make some predictions o f the behavior of\nthe local magnetization. We know that the effect of the magnetic fie ld together with the\nGilbert term is to precess the magnetic spin into the direction of the e xternal field. The first\nterm in the right hand side of Eq. (4) is that the effective magnetic fie ld is in the direction\nof neighboring spins. Therefore the evolution tends to form a ferr omagnet. This contributes\nto the centripetal force of the magnetization in the direction of Mr×Mr′, which results in\nthe precession of one around the other. The effective field in the DM term in Eq. (4) is\n−∇×Mrwith unitary lattice constants. The integral counterpart of the c url is/contintegraltext\nMr′·dl.\nWhentheneighboringspinsformaring, theenergyisthelowest, hen cegeneratingaspiraling\nforce to the CM. It helps our understanding if we analogize all the ot her terms in the right\nhand side of Eq. (2) to the effect of a magnetic field. The local “magn etic field” of the\nphenomenological Gilbert damping force is proportional to −dMr/dtin the standard linear-\nresponse damping form, proportional to the velocity and pointing o ppositely to it. While\nthe local spin is precessing, the direction of Mr×dMr/dtpoints to the precession axis of\nMradding a force swaying to that axis. The last two terms are the effec t of the current-\ninduced spin torques. For convenience of interpretation, we discu ss the case that j(r,t) is\nspatially-uniform and along the x-direction. Then [ j(r,t)·∇]Mr=jx(t)∂Mr/∂x. In the\ncase of the adiabatic torque, this term adds a velocity to Mrmaking it sway to the direction\nofMr+ex, andMr+extoMr+2ex, and etc. if jx(t) is positive. Therefore, the complete spin\ntexture moves along the direction of the external spin current like a relay race no matter\nit is a skyrmion or a domain wall. In a periodic magnetic structure such a s a ferromagnet\nand a helimagnet the “relay race” goes back to itself and hence no sp in structure movement\noccurs. Following this physical picture, the local “magnetic field” of the nonadiabatic spin\n5torque is along the direction of jx(t)∂Mr/∂x. It exerts a velocity perpendicular to that\noriginates from the adiabatic spin torque. Its result is the motion of the spin texture in the\ndirection perpendicular to the spin current. We have already analyz ed the mechanisms of\ntheLLGequation termby term. However, they affectsthe system collaboratively. While the\nadiabatic spin torque moves the skyrmion along the spin current, th e Gilbert damping force\ncontributes a velocity in the direction of Mr×dMr/dtandtherefore the effect is a transverse\nmotion of the skyrmion, which is the so-called Hall-like motion12. Also it is noticeable that\nthe transverse velocity resulting from the Gilbert damping and the n onadiabatic spin torque\nis opposite to each other. In real situations, both αandβare much less than 1. The\nadiabatic spin torque makes the main contribution to the motion of th e skyrmion. And\nwhen the two transverse force is equal, the motion of the skyrmion is straightly along the\ndirection of the spin current. Therefore, periodic trajectories o f the skyrmion in real space\ncan be predicted under the influence of a spatially uniform ac spin cur rent.\nThe previous discussions are well expressed in the Thiele equation de scribing the motion\nof the center of mass of a skyrmion as1,13,24,26.\nG×[−j(t)−vd]+κ[−βj(t)−αvd]−∇U(r) =0, (5)\nwherevd=dR/dt=/parenleftBig\n˙X,˙Y/parenrightBig\nwithR= (X,Y) the center of mass coordinates, κis a\ndimensionless constant of the order of unity, and G= 2πSezis the gyrovector with ezin the\ndirection perpendicular to the CM plane. The minus sign before j(t) is because of that the\ndirection of the motion of conduction electrons is opposite to that o f the current. The Thiele\nequation (5) describes5coaction of the Magnus force Fg=G×[−j(t)−vd], the viscous\nforceFv=κ[−βj(t)−αvd], and the confining force Fp=−∇U(r). In our considerations,\nperiodic boundary conditions are used to justify an infinite two-dime nsional model. The\napplied magnetic field is spatially uniform and the impurity effect is neglec ted. Therefore\n∇U≈0. The analytical result of Eq. (5) assuming S=−1 andj(t) =jesin(ωt)excan be\nobtained as \n\nX=αβκ2+4π2\n(α2κ2+4π2)ωjecos(ωt),\nY=2πκ(β−α)\n(α2κ2+4π2)ωjecos(ωt).(6)\nSince the spin current is time dependent, FgandFvinstantaneously change their direc-\ntion with the motion of the skyrmion core and simultaneously react on the motion of the\nskyrmion, giving rise to the trigonometric trajectory of the skyrm ion shown in Eq. (6),\n6which agrees with the simulation results. Because the skyrmion vort ex moves in the relay\nfashion under the effect of the spin torque shown by the LLG equat ion, there is a π/2 phase\nlag between its core motion and the sinusoidally varying spin current.\nIII. NUMERICAL RESULTS AND INTERPRETATIONS\nBy multiplying ˜ α−1with\n˜α= 1−α\n0−(Mr)z(Mr)y\n(Mr)z0−(Mr)x\n−(Mr)y(Mr)x0\n, (7)\nfrom the left to Eq. (2), the matrix-based Runge-Kutta method is developed. In Figs. 1 to\n3, numerical results of our simulations are given. We set M= 1,p= 0.5, anda= 4˚A. The\nintegral step h= 0.1t0is used and its convergence is justified by comparison with the result s\nofh= 0.01t0. WithD= 0.18J, the natural helimagnet wavevector Q= 2π/λ=D/J\nwith the diameter of the skyrmion λ=D/J≈35 in the unit of a. A 30×30 square\nlattice is considered which approximately sustains a single skyrmion. P eriodic boundary\ncondition is used to allow the considered patch to fit into an infinite plan e. While part of the\nskyrmion moves out of the slab, complementary part enters from t he outside as the natural\nground state of a CM is the skyrmion crystal. We use the theoretica lly perfect skyrmion\nprofilen(r) = [cosΦ( ϕ)sinΘ(r),sinΦ(ϕ)sinΘ(r),cosΘ(r)] with Θ( r) =π(1−r/λ) and\nΦ(ϕ) =ϕin the polar coordinates as the initial state and it would change into a n atural\nskyrmion in less than one current period. The skyrmion number for t his state S=−1. The\nspatially-uniform ac spin current is applied in the x-direction as j(t) =jesin(ωt)exwithin\nthe CM plane.\nVariationofthe skyrmion number intimedriven bythe acspincurrent is shown inFig. 1.\nItcanbeseen thatcosinusoidal variationof Soriginatesfromthesinusoidal j(t)withexactly\nthe same period. Fig. 2 shows the snapshots of the spin profile at th e bottoms and peeks of\nthecosinusoidal variationof SandFig. 3shows thetrajectoriesof thecenter oftheskyrmion\n(see Ref. 27 for Supplementary Movie). The skyrmion number is a de monstration of the\nmotion pattern of the skyrmion. While the skyrmion moves to one side of the CM slab, only\npart of a skyrmion is within the view and hence the skyrmion number is r educed. Previous\nauthors have found that the velocity of the skyrmion increases line arly with the increase of\n7the current amplitude and that thedynamical threshold current t o move a skyrmion isin the\nsameorderofthatneededforadomainwall1. Herewehavereobtainedthetwo points. Itcan\nbeseeninFig. 1(a)thatthepeakhight oftheskyrmionnumber incre ases withtheamplitude\nof the current density and it becomes almost invisible when jeis as small as 1010Am−2. In\nFig. 1(b), the evolutions of Sfor different ac periods are shown. The frequency of the ac\ncurrent is in the order of GHz, which is sufficiently adiabatic as the rat e of the spin dynamics\nis in the order of 10−12s. We can see that the periodic pattern of Sis better kept with larger\namplitudes for smaller ac frequencies. It shows that the phenomen on is a good adiabatic\none. Within our numerical capacity, it can be predicted that strong cosinusoidal variation\ncan occur at MHz or smaller ac frequencies, which promises experime ntal realization.\nThe variation of Sis the result of the motion of the skyrmion. The periodic translation o f\nskyrmion is the result of the coaction of the instantaneous Magnus and viscous forces. The\nspin current gives rise to the drift velocity of the spin texture. As a combined result of the\nGilbert damping, the DMI, the adiabatic and nonadiabatic spin torque s, the skyrmion Hall\neffect, namely, the transverse motion of the skyrmion perpendicu lar to the spin current, is\nobserved in topologically-nontrivial spin textures. As shown in Fig. 2 , in spite of its motion,\nthe topological properties of the skyrmion are conserved becaus e the initial skyrmion state\nand the natural skyrmion ground state share similar topology and n o topology-breaking\nsource such as an in-plane magnetic field is present. When part of th e skyrmion moves out\nof the CM slab, only the remaining part contributes to the skyrmion n umber and hence Sis\ndecreased. The cosinusoidal variation of Sdirectly reflects the oscillating trajectory of the\nskyrmion Shown in Figs. 2 and 3. We can see that the skyrmion change s from the initial\nartificial skyrmion state into the natural skyrmion state with S=−1 conserved, as shown\nin Fig. 2(a) and (b). At the times of integer periods the skyrmion is at the center of the CM\nslab and at the times of half-integer periods the skyrmion moves to t he left side as shown\nin Fig. 2 (c) to (f).\nAs predicted by the Thiele equation, the trajectory of the skyrmio n follows a cosinusoidal\npattern expressed in Eq. (6). It is interesting that the trajecto ry of the skyrmion results\nfrom the competition between the drift motion of the skyrmion and t he skyrmion Hall effect\nunder the influence of the adiabatic and nonadiabatic spin torque eff ects. The adiabatic spin\ntorque effect exerts a force to align the spin at each site to its + x-direction neighbor while\nj(t) is in the exdirection, which results in the motion of the spin pattern to the + xdirection\n8in a relay fashion. The Gilbert damping effect and the nonadiabatic spin torque add a\ntransverse velocity to the moving skyrmion perpendicular to its orig inal velocity. These two\nforces are in opposite directions when αandβare both positive. Therefore the transverse\nmotion is determined by the sign and relative strength of these two e ffects. From Eq. (6)\nwe can see that when β−α >0 the skyrmion’s y-direction motion is in a cosinusoidal form\nand when β−α <0 it is in a negative cosinusoidal form. For the x-direction motion of the\nskyrmion, the direction is the same in the two cases and the magnitud e is slightly smaller for\nthe latter because |4π2| ≫ |αβκ2|holds for all physical parameter settings. And physically\nit is because the x-direction motion of the skyrmion is mainly determined by the adiabatic\nspin torque, which is the prerequisite for any motion of the skyrmion .\nOur simulation results of the skyrmion trajectories for β= 0.5α,α, and 2αwith fixed\nα= 0.1 are shown in Fig. 3. Good agreement with the prediction by the Thiele equation\nis obtained. In the three cases, Xevolves cosinusoidally with the initial position ( X,Y) =\n(15,15) at the center of the CM slab. For β= 0.5α,Yevolves minus-cosinusoidally; for\nβ=α,Yis constant at 15; for β= 2α,Yevolves cosinusoidally. As the difference between\nβandαis small in Fig. 3 (a) and (c), the cosinusoidal pattern shrinks into a s tep jump.\nBesides the oscillation, a tiny linear velocity of the skyrmion can be see n in Fig. 3 (a) and\n(c). And the directions of this velocity are different in the two cases . We attribute this\nlinear velocity to the whirling of the skyrmion from the artificial initial p rofile to the natural\nprofile sustained by the real CM. Because at this whirling step, the G ilbert damping and\nthe adiabatic and nonadiabatic torques are already in effect, the init ial linear velocities are\ndifferent in the two cases.\nIV. CONCLUSIONS\nIn this work, we have investigated the dynamics of the skyrmion in a C M driven by\nperiodically varying spin currents by replacing the static current in t he LLG equation by\nan adiabatic time-dependent current. Oscillating trajectories of t he skyrmion are found\nby numerical simulations, which are in good agreement with the analyt ical solution of the\nThiele equation. In the paper, physical behaviors of the general L LG equation with the\nGilbert damping and the adiabatic and nonadiabatic spin torques coex istent are elucidated.\nEspecially, the effect of the nonadiabatic spin torque is interpreted both physically and\n9numerically.\nV. AUTHOR CONTRIBUTION STATEMENT\nR.Z. wrote the program and the paper. Y.Y.Z. made the simulation.\nVI. 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Lett. 30, 230 (1972).\n27Supplementary Movie.\n12/s48 /s53/s48/s48 /s49/s48/s48/s48 /s49/s53/s48/s48 /s50/s48/s48/s48/s45/s49/s46/s48/s48/s45/s48/s46/s57/s53/s45/s48/s46/s57/s48/s45/s48/s46/s56/s53/s45/s49/s46/s48/s48/s45/s48/s46/s57/s53/s45/s48/s46/s57/s48/s45/s48/s46/s56/s53/s45/s48/s46/s56/s48/s45/s48/s46/s55/s53\n/s116/s83/s32/s106\n/s101/s61/s48\n/s32/s84/s61/s52/s48/s48\n/s32/s84/s61/s53/s48/s48\n/s32/s84/s61/s54/s48/s48/s40/s98/s41/s83/s32/s106\n/s101/s61/s49/s48/s49/s48\n/s65/s109/s45/s50\n/s32/s106\n/s101/s61/s49/s48/s49/s49\n/s65/s109/s45/s50\n/s32/s106\n/s101/s61/s50 /s49/s48/s49/s49\n/s65/s109/s45/s50/s40/s97/s41/s32\nFIG. 1: Variation of the skyrmion number S in time (a) for differ ent current amplitudes and (b)\nfor different ac current frequencies. The time tand ac spin current period Tare in the unit of\nt0≈6.6×10−13s.β= 0.05. In panel (a), T= 500. In panel (b), je= 1011Am−2.\n13(0,−0.91763)\n  \n(a)\n−1−0.500.51(750,−0.77054)\n(b)(1000,−0.97276)\n(c)\n(1250,−0.76819)\n(d)(1500,−0.97019)\n(e)(1750,−0.76716)\n(f)\nFIG. 2: Snapshots of the dynamical spin configurations at the bottoms and peaks of the skyrmion\nnumber shown in Fig. 1. The in-plane components of the magnet ic moments are represented by\narrows and their z-components are represented by the color plot. The paramete rs areje= 2×1011\nAm−2,T= 500, and β= 0.05. On the top of each panel are the ( t,S) values.\n14/s57/s49/s48/s49/s49/s49/s50/s49/s51/s49/s52/s49/s53/s49/s54\n/s57/s49/s48/s49/s49/s49/s50/s49/s51/s49/s52/s49/s53\n/s48 /s53/s48/s48 /s49/s48/s48/s48 /s49/s53/s48/s48 /s50/s48/s48/s48/s57/s49/s48/s49/s49/s49/s50/s49/s51/s49/s52/s49/s53\n/s32/s88\n/s32/s89/s40/s97/s41/s32 /s61/s48/s46/s48/s53\n/s40/s98/s41/s32 /s61/s48/s46/s49\n/s116/s40/s99/s41/s32 /s61/s48/s46/s50\nFIG. 3: Variation of the skyrmion center coordinates ( X,Y) in time (a) for β= 0.05, (b) for\nβ= 0.1, and (c) for β= 0.2. Other parameters are the same as Fig. 2.\n15"    },    {        "title": "1911.02775v2.Quantum_Oscillations_of_Gilbert_Damping_in_Ferromagnetic_Graphene_Bilayer_Systems.pdf",        "content": "arXiv:1911.02775v2  [cond-mat.mes-hall]  15 Apr 2020Quantum Oscillations of Gilbert Damping in Ferromagnetic/ Graphene Bilayer\nSystems\nYuya Ominato1and Mamoru Matsuo1,2\n1Kavli Institute for Theoretical Sciences, University of Ch inese Academy of Sciences, Beijing 100190, China and\n2CAS Center for Excellence in Topological Quantum Computati on,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n(Dated: April 16, 2020)\nWe study the spin dynamics of a ferromagnetic insulator on wh ich graphene is placed. We show\nthat the Gilbert damping is enhanced by the proximity exchan ge coupling at the interface. The\nmodulation of the Gilbert damping constant is proportional to the product of the spin-up and\nspin-down densities of states of graphene. Consequently, t he Gilbert damping constant in a strong\nmagnetic field oscillates as a function of the external magne tic field that originates from the Landau\nlevel structure of graphene. We find that a measurement of the oscillation period enables the\nstrength of the exchange coupling constant to be determined . The results theoretically demonstrate\nthat the ferromagnetic resonance measurements may be used t o detect the spin resolved electronic\nstructure of the adjacent materials, which is critically im portant for future spin device evaluations.\nIntroduction .—Graphene spintronics is an emergent\nfield aiming at exploiting exotic spin-dependent proper-\ntiesofgrapheneforspintronicsdevices[1]. Althoughpris-\ntinegrapheneisanon-magneticmaterial,therehavebeen\neffortstointroducemagnetismintographenetofindspin-\ndependent phenomena and to exploit its spin degrees of\nfreedom. Placing graphene on a magnetic substrate is\na reasonable way, which leads to magnetic proximity ef-\nfect and lifting of spin degeneracy [2, 3]. Subsequently,\nmagnetization was induced in graphene and spin depen-\ndent phenomena, such as the anomalous Hall effect [4, 5]\nand non-local spin transport [6, 7], were observed. In all\nthese experiments, a spin-dependent current was gener-\nated by an electric field. There is an alternative way to\ngenerate a spin current called spin pumping [8–12]. The\nproximity exchange coupling describes spin transfer at\nthe magnetic interface and a spin current is injected us-\ning ferromagnetic resonance (FMR) from ferromagnetic\nmaterials into the adjacent materials. The generation of\na spin current is experimentally detectable through both\nthe inverse spin Hall effect and modulation of the FMR,\nwhich were experimentally confirmed at magnetic inter-\nfaces between graphene and several magnetic materials\n[13–18].\nThe theory of spin transport phenomena at magnetic\ninterfaces has been formulated based on the Schwinger-\nKeldysh formalism [19], which is applicable to magnetic\ninterfaces composed of a variety of systems, such as a\nparamagnetic metal and a ferromagnetic insulator (FI)\n[20–23], a superconductor and FI [24, 25], and two FIs\n[26, 27]. The modulation of FMR has been investigated\nin several papers. The modulation of Gilbert damping\nwasfound to be proportionalto the imaginarypartofthe\ndynamical spin susceptibility [21, 23–25, 28, 29], which\nmeans that one can detect spin excitations and electronic\nproperties of adjacent materials through the FMR mea-\nsurements. This implies that the FMR measurements\nof FI/graphene bilayer systems allow us to access thespin-dependent properties of graphene in quantum Hall\nregime [30, 31]. However, the modulation of FMR at the\nmagnetic interface between a FI and graphene has not\nbeen investigated and the effect of Landau quantization\non the FMR signal is unclear.\nIn this work, we study the modified magnetization dy-\nnamics of a FI adjacent to graphene. Figure 1 (a) shows\naschematicofthe system. Microwavesareirradiatedand\nthe precession of localized spins is induced. Figure 1 (b)\nand (c) shows the electronic structure of graphene on the\nFI under aperpendicular magneticfield. The spin degen-\neracy is lifted by the exchange coupling at the interface\nand spin-split Landau levels are formed. The densities of\nstates for spin-up and spin-down are shown in the right\npanel; Landau level broadening is included. We find that\nthe modulation of Gilbert damping is proportionalto the\nproduct of the densities of states for spin-up and spin-\ndown, so that the FMR measurements may be used as\na probe of the spin-resolved densities of states. Owing\nto the peak structure of the density of states, the mod-\nulation of Gilbert damping exhibits peak structure and\nan oscillation as a function of Fermi level and magnetic\nfield, which reflects the Landau level structure. One may\ndetermine the exchange coupling constant by analyzing\nthe period of the oscillation.\nModel Hamiltonian .—The totalHamiltonian H(t)con-\nsists of three terms,\nH(t) =HFI(t)+HGr+Hex. (1)\nThe first term HFI(t) describes the bulk FI\nHFI(t) =/summationdisplay\nk/planckover2pi1ωkb†\nkbk−h+\nac(t)b†\nk=0−h−\nac(t)bk=0,(2)\nwhereb†\nkandbkdenote the creation and annihilation\noperators of magnons with momentum k. We assume a\nparabolic dispersion /planckover2pi1ωk=Dk2−/planckover2pi1γB, withγ(<0) the\nelectron gyromagnetic ratio. The coupling between the2\nMicrowaveB(a) System (b) spin splitting \nExchange \ncoupling\nB01230\n-1 \n-2 \n-3 \nDOSE\nup down(c) spin-split Landau level \nkx kyE\nE\nB123\n0-1 \n-2 \n-3 0\nkx kyE\nFIG. 1. (Color online) Schematic picture of the FMR measurem ent and the energy spectrum of graphene in a strong perpen-\ndicular magnetic field. (a) Graphene on a ferromagnetic insu lator substrate. The magnetic field perpendicular to graphe ne\nis applied and the microwave is irradiated to the FI. (b) The s pin degeneracy is lifted by the exchange coupling. (c) The\nperpendicular magnetic field leads to the spin-split Landau level structure. The density of states has a peak structure a nd the\nlevel broadening originating from disorder is included.\nmicrowave and magnons is given by\nh±\nac(t) =/planckover2pi1γhac\n2√\n2SNe∓iΩt, (3)\nwherehacand Ω are the amplitude and frequency of the\nmicrowave radiation, respectively, and Sis the magni-\ntude of the localized spin in the FI. The above Hamilto-\nnian is derived from a ferromagnetic Heisenberg model\nusing the Holstein-Primakoff transformation and the\nspin-wave approximation ( Sz\nk=S−b†\nkbk,S+\nk=√\n2Sbk,\nS−\n−k=√\n2Sb†\nk, whereSkis the Fourier transform of the\nlocalized spin in the FI).\nThe second term HGrdescribes the electronic states\naround the Kpoint in graphene under a perpendicular\nmagnetic field,\nHGr=/summationdisplay\nnXsεnc†\nnXscnXs, (4)\nwherec†\nnXsandcnXsdenote the creation and annihi-\nlation operators of electrons with Landau level index\nn= 0,±1,±2,···, guiding center X, and spin up s= +\nand spin down s=−. The eigenenergy is given by\nεn= sgn(n)√\n2e/planckover2pi1v2/radicalbig\n|n|B, (5)\nwherevis the velocity and the sign function is defined\nas\nsgn(n) :=\n\n1 (n >0)\n0 (n= 0)\n−1 (n <0). (6)\nIn the following, we neglect the Zeeman coupling be-\ntween the electron spin and the magnetic field because\nit is much smaller than the Landau-level separation and\nthe exchange coupling introduced below. In graphene,\nthere are two inequivalent valleys labelled KandK′. Inthis paper, we assume that the intervalley scattering is\nnegligible. This assumption is valid for an atomically flat\ninterface,whichisreasonablegiventherecentexperimen-\ntal setups [4, 17, 18]. Consequently, the valley degree\nof freedom just doubles the modulation of the Gilbert\ndamping.\nThe third term Hexis the exchange coupling at the\ninterface consisting of two terms\nHex=HZ+HT, (7)\nwhereHZdenotes the out-of-plane component of the\nexchange coupling and leads to the spin splitting in\ngraphene,\nHZ=−JS/summationdisplay\nnX/parenleftBig\nc†\nnX+cnX+−c†\nnX−cnX−/parenrightBig\n,(8)\nwithJthe exchange coupling constant. The z-\ncomponent of the localized spin is approximated as\n∝angbracketleftSz\nk∝angbracketright ≈S. The out-of-plane component HZis modeled\nas a uniform Zeeman-like coupling, although in general,\nHZcontains the effect of surface roughness, which gives\noff-diagonal terms. The Hamiltonian HTdenotes the in-\nplane component of the exchange coupling and describes\nspin transfer between the FI and graphene,\nHT=−/summationdisplay\nnX/summationdisplay\nn′X′/summationdisplay\nk/parenleftBig\nJnX,n′X′,ks+\nnX+,n′X′−S−\nk+h.c./parenrightBig\n,\n(9)\nwhereJnX,n′X′,kis the matrix element for the spin trans-\nfer processes and s+\nnX+,n′X′−is the spin-flip operator for\nthe electron spin in graphene.\nModulation of Gilbert Damping .—To discuss the\nGilbert damping, we calculated the time-dependent sta-\ntistical averageof the localized spin under the microwave\nirradiation. The first-order perturbation calculation\ngives the deviation from the thermal average,\nδ∝angbracketleftS+\nk=0(t)∝angbracketright=−h+\nac(t)GR\nk=0(Ω). (10)3\nThe retarded Green’s function is written as\nGR\nk(ω) =2S//planckover2pi1\nω−ωk+iαGω−(2S//planckover2pi1)ΣR\nk(ω),(11)\nwhere we have introduced the phenomenological dimen-\nsionlessdampingparameter αG, calledthe Gilbert damp-\ning constant, which originates from the magnon-phonon\nand magnon-magnon coupling, etc [32–34]. In this pa-\nper, we focus on the modulation of the Gilbert damping\nstemming from the spin transfer processes at the inter-\nface. The self-energy from the spin transfer processes at\nthe interface within second-order perturbation is given\nby\nΣR\nk(ω) =/summationdisplay\nnX/summationdisplay\nn′X′|JnX,n′X′,k|2χR\nn+,n′−(ω).(12)\nThe spin susceptibility is given by\nχR\nn+,n′−(ω) =fn+−fn′−\nεn+−εn′−+/planckover2pi1ω+i0,(13)\nwherefns= 1//parenleftbig\ne(εns−µ)/kBT+1/parenrightbig\nis the Fermi distribu-\ntion function and εns=εn−JSsis the spin-split Landau\nlevel. From the self-energy expression, one sees that the\nmodulation of the Gilbert damping reflects the property\nof the spin susceptibility of graphene. The modulation\nof the Gilbert damping under the microwave irradiation\nis given by [21, 23–25, 28, 29]\nδαK\nG=−2SImΣR\nk=0(ω)\n/planckover2pi1ω, (14)\nwhere the superscript Ksignifies the contribution from\ntheKvalley.\nTo further the calculation, we assume that the ma-\ntrix element JnX,n′X′,k=0is approximated by a constant\nJ0, including detail properties of the interface, that is,\nJnX,n′X′,k=0≈J0. Withthisassumption,theself-energy\nbecomes\nImΣR\nk=0(ω) =−|J0|2π/planckover2pi1ω/integraldisplay\ndε/parenleftbigg\n−∂f(ε)\n∂ε/parenrightbigg\nD+(ε)D−(ε),\n(15)\nwhereDs(ε) is the density of states for spin s=±\nDs(ε) =A\n2πℓ2\nB/summationdisplay\nn1\nπΓ\n(ε−εns)2+Γ2,(16)\nwith magnetic length ℓB=/radicalbig\n/planckover2pi1/(eB) and area of the\ninterface A. Here, we have introduced a constant Γ de-\nscribing level broadening arising from surface roughness\nand impurity scattering. This is the simplest approx-\nimation to include the disorder effect. The density of\nstates shows peaks at the Landau level, which is promi-\nnent when its separation exceeds the level broadening.\nLandau level (JS = 20  meV) δα G [δα 0   10-2 ]\n0.6\nB [T]1.0μ [meV] \n0.4 0.240 \n20 \n-20\n-400\n0.8s=-, n=0\ns=+, n=0123\n-1 \n-2 \n-3 Γ = 1  meV\nkBT = 1  meV (= 11  K)\n6\n03\n0.6\nB [T]1.0μ [meV] \n0.4 0.240 \n20 \n-20\n-400\n0.8\nFIG. 2. (Color online) Modulation of the Gilbert damping\nconstant δαGand spin-split Landau levels as a function of\nthe Fermi level µand the magnetic field B. The spin splitting\nJSis set to 20meV. In the left panel, δαGhas peaks at the\ncrossing points of spin-up and spin-down Landau levels. In\nthe right panel, the blue and red curves identify the spin-up\nand spin-down Landau levels, respectively.\nFinally, the modulation of the Gilbert damping constant\nδαGis derived as\nδαG= 2πgvS|J0|2/integraldisplay\ndε/parenleftbigg\n−∂f(ε)\n∂ε/parenrightbigg\nD+(ε)D−(ε),(17)\nwheregv= 2 denotes the valley degree of freedom.\nFrom this expression, one sees that the modulation of\nthe Gilbert damping is proportional to the product of\nthe densities of states for spin-up and spin-down. There-\nfore, combined with the density of states measurement,\nfor example, a capacitance measurement [35], the FMR\nmeasurement is used to detect the spin-resolved densities\nof states.\nFigure 2 shows the spin-split Landau levels and the\nmodulation of the Gilbert damping δαGas a function of\nthe Fermi level µand the magnetic field B. We use δα0\nas a unit of δαG\nδα0= 2πgvS|J0|2/parenleftbiggA\n2πℓ2\nB1\nmeV/parenrightbigg2\n.(18)\nWe note that δα0(∝B2) depends on the magnetic field.\nBoth the level broadening Γ and the thermal broadening\nkBTare set to 1meV, and JSis set to 20meV [2–4].\nδαGreflects the Landau level structure and has peaks at\ncrossing points of spin-up and spin-down Landau levels.\nThe peakpositions aredetermined bysolving εn+=εn′−\nand the inverse of the magnetic field at the peaks is given\nby\n1\nB=2e/planckover2pi1v2\n(2JS)2/parenleftBig/radicalbig\n|n|−/radicalbig\n|n′|/parenrightBig2\n. (19)\nThe peak structurebecomes prominent when the Landau\nlevel separation exceeds both level and thermal broaden-\ning.4\nΓ = 1 meV\nkBT = 1 meV (= 11 K)\n5 meV (= 57 K)\n10 meV (= 115 K)kBT = 1 meV (= 11 K)\nΓ = 1 meV \n2 meV \n4 meV μ = JS = 20 meV μ = JS = 20 meV (a) (b)\nΔ(1/B)\n3\n1/B [1/T]4δα G [δα 0   10 -2 ]\n2 18\n6\n4\n2\n0\n5 3\n1/B [1/T]4δα G [δα 0   10 -2 ]\n2 18\n6\n4\n2\n0\n5Δ(1/B)\nFIG. 3. (Color online) Quantum oscillation of the modulatio n\nof the Gilbert damping constant δαGas a function of the\ninverse of the magnetic field 1 /B. The Fermi level µand the\nmagnitude of the spin splitting JSare set to 20meV. (a)\nΓ = 1meV and δαGis plotted at several temperatures. (b)\nkBT= 1meV and δαGis plotted for several Γ’s. The period of\nthe oscillation ∆(1 /B) is indicated by double-headed arrows.\nFigure 3 shows the modulation of the Gilbert damping\nδαGas a function of the inverse of the magnetic field\n1/Bwith the Fermi level set to µ= 20meV, where the\nspin-down zeroth Landau level resides. δαGshows peak\nstructure and a periodic oscillatorybehavior. The period\nof the oscillation ∆(1 /B) is derived from Eq. (19) and is\nwritten as\n∆/parenleftbigg1\nB/parenrightbigg\n=2e/planckover2pi1v2\n(2JS)2. (20)\nThe above relation means that the magnitude of the spin\nsplitting JSis detectable by measuring the period of the\noscillation ∆(1 /B). For the peak structure to be clear,\nboth leveland thermalbroadeningmust to be sufficiently\nsmaller than the Landau level separation; otherwise, the\npeak structure smears out.\nDiscussion .—To observe the oscillation of Gilbert\ndamping, at least two conditions must be satisfied. First,\nthe well-separated landau levels have to be realized in\nthe magnetic field where the FMR measurements is fea-\nsible. Second, the FMR modulation caused by the ad-\njacent graphene have to be detectable. The graphene\nLandau levels are observed in recent experiments at 2T\n[36], andrecentbroadbandferromagneticresonancespec-\ntrometer enables the generation of microwaves with fre-\nquencies ≤40GHz and FMR measurements in a mag-\nnetic field ≤2T [37]. The modulation of the FMR\nlinewidth in Permalloy/Graphene [14, 16], yttrium iron\ngarnet/Graphene [17, 18] have been reported by sev-\neral experimental groups, although all of them were per-\nformed at room temperature. Therefore, the above two\nconditionsareexperimentallyfeasibleandourtheoretical\npredictions can be tested in an appropriate experimental\nsetup.Conclusion .—We have studied the modulation of the\nGilbert damping δαGin a ferromagnetic insulator on\nwhich graphene is placed. The exchange coupling at\nthe interface and the perpendicular magnetic field lead\nto the spin-split Landau levels in graphene. We showed\nthatδαGis proportional to the product of the densities\nof states for spin-up and spin-down electrons. Therefore,\nthe spin-resolved densities of states can be detected by\nmeasuring δαGand the total density of states. When the\nFermi level is fixed at a Landau level, δαGoscillates as a\nfunction of the inverse of the magnetic field. The period\nof the oscillation provides information on the magnitude\nof the spin splitting. Our main message is that the FMR\nmeasurement is a probe of spin-resolved electronic struc-\nture. In addition to spin current generation, one may use\ntheFMRmeasurementstodetectthe electronicstructure\nof adjacent materials.\nAcnowledgement WethankJ.Fujimoto, T.Kato,R.\nOhshima, and M. Shiraishi for helpful discussions. 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Nikoli´ c\nDepartment of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA\nWe analyze a quantum-classical hybrid system of steadily precessing slow classical localized mag-\nnetic moments, forming a head-to-head domain wall, embedded into an open quantum system of\nfast nonequilibrium electrons. The electrons reside within a metallic wire connected to macroscopic\nreservoirs. The model captures the essence of dynamical noncollinear and noncoplanar magnetic\ntextures in spintronics, while making it possible to obtain the exact time-dependent nonequilib-\nrium density matrix of electronic system and split it into four contributions. The Fermi surface\ncontribution generates dissipative (or damping-like in spintronics terminology) spin torque on the\nmoments, and one of the two Fermi sea contributions generates geometric torque dominating in the\nadiabatic regime. When the coupling to the reservoirs is reduced, the geometric torque is the only\nnonzero contribution. Locally it has both nondissipative (or field-like in spintronics terminology)\nand damping-like components, but with the sum of latter being zero, which act as the counter-\nparts of geometric magnetism force and electronic friction in nonadiabatic molecular dynamics.\nSuch current-independent geometric torque is absent from widely used micromagnetics or atomistic\nspin dynamics modeling of magnetization dynamics based on the Landau-Lifshitz-Gilbert equation,\nwhere previous analysis of Fermi surface-type torque has severely underestimated its magnitude.\nOne of the most fruitful applications of geometric (or\nBerry) phase [1] concepts is encountered in quantum-\nclassical hybrid systems where separation of time scales\nmakes it possible to consider fast quantum degrees of free-\ndom interacting with the slow classical ones [2, 3]. The\namply studied example of this kind are fast electrons in-\nteracting [4, 5] with slow nuclei in molecular dynamics\n(MD) [6–9] problems of physics, chemistry and biology.\nThe parameters driving adiabatic evolution of quantum\nsubsystem, with characteristic frequency smaller that its\nlevel spacing, are nuclear coordinates elevated to the\nstatus of dynamical variables. The electronic system\nthen develops geometric phase in states evolving out of\nan instantaneous energy eigenstate, while also acquiring\nshifts in the energy levels. Conversely, nuclei experience\nforces due to back-action from electrons. The simplest\nforce is the adiabatic Born-Oppenheimer (BO) force [4, 5]\nwhich depends only on the coordinates of the nuclei, and\nit is associated with electronic adiabatic potential sur-\nfaces [6, 7]. Even small violation of BO approximation\nleads to additional forces—the first nonadiabatic correc-\ntion generates forces linear in the velocity of the nuclei,\nand being Lorentz-like they are dubbed [2, 10] “geomet-\nric magnetism.” The “magnetism” is not a not a real\nmagnetic field, but an emergent geometrical property of\nthe Hilbert space [11], and akin to the true Lorentz force,\nthe emergent geometric force is nondissipative .\nAdditional forces appear upon making the quantum\nsystem open by coupling it to a thermal bath [10, 12]\n(usually modeled as an infinite set of harmonic oscilla-\ntors [13]) or to macroscopic reservoirs of particles [14].\nIn the latter case, one can also introduce chemical po-\ntential difference between the reservoirs to drive particle\nflux (i.e., current) through the quantum system which is,\nthereby, pushed out of equilibrium [14–16, 18, 19]. In\nFIG. 1. (a) Schematic view of a two-terminal system where\na single classical LMM, precessing steadily with frequency ω\nand cone angle θ, interacts with an open quantum system of\nconduction electron spins. The electrons hop along 1D infinite\ntight-binding chain which terminates into the left and right\nmacroscopic reservoirs kept at the same chemical potential µ.\nPanel (c) depicts 7 LMMs, M1–M7forming a head-to-head\nBloch domain wall, which precess with the same frequency\nbut are noncollinear andnoncoplanar . Both (a) and (c) can\nbe mapped in the rotating frame to a time-independent four-\nterminal system in (b) with an effective bias voltage ~ω/e\nbetween the left or right pair of leads.\nboth equilibrium and nonequilibrium cases, the energy\nspectrum of the quantum system is transformed into a\ncontinuous one, and frictional forces [8–10, 14–19] linear\nin the velocity of the nuclei become possible. Also, due\nto continuous spectrum, adiabaticity criterion has to be\nreplaced by a different one [14]. Stochastic forces also ap-\npear, both in equilibrium and in nonequilibrium, where\nin the former case [10, 12] they are due to fluctuations at\nfinite temperature while in the latter case they includearXiv:2005.14153v2  [cond-mat.mes-hall]  14 Jun 20202\nadditional contribution from nonequilibrium noise [14–\n16]. Finally, specific to nonequilibrium is the emergence\nof nonconservative forces [14–16, 18, 19]. The derivation\nof all of these forces is achieved by computing nonadia-\nbatic corrections to the density matrix (DM) [10, 12, 14–\n16, 18, 19]. This yields a non-Markovian stochastic\nLangevin equation, with nonlocal-in-time kernel describ-\ning memory effects [20], as the most general [16, 19] equa-\ntion for nuclei in nonadiabatic MD.\nThe analogous problem exists in spintronics, where the\nfast quantum system is comprised of conduction electron\nspins and slow classical system is comprised of localized-\non-atoms spins and associated localized magnetic mo-\nments (LMMs) described by unit vectors Mi(t). The\ndynamics of LMMs is accounted by the Landau-Lifshitz-\nGilbert (LLG) type of equation [21]\n∂Mi\ndt=−gM×Beff\ni+λMi×∂Mi\n∂t\n+g\nµM/parenleftBig\nTi/bracketleftBig\nISα\next/bracketrightBig\n+Ti[∂Mi/∂t]/parenrightBig\n. (1)\nThis includes phenomenological Gilbert damping, whose\nparameter λcan be measured or independently calcu-\nlated [22] by using electronic Hamiltonian with spin-orbit\ncoupling and impurities. It can also include Slonczewski\nspin-transfer torque (STT) term Ti/bracketleftBig\nISα\next/bracketrightBig\ndue to exter-\nnally supplied spin current ISα\next. The STT is a phe-\nnomenon [28] in which spin angular momentum of con-\nduction electrons is transferred to local magnetization\nnot aligned with electronic spin-polarization. Finally,\nsome analyses [23–25] also consider current-independent\ntorque Ti[∂Mi/∂t] as a back-action of electrons pushed\nout of equilibrium by time-dependent Mi(t). Neverthe-\nless, such effects have been deemed negligible [23, 26] or\neasily absorbed into Eq. (1) by renormalizing gandλ[23].\nHeregis the gyromagnetic ratio; Beff\ni=−1\nµM∂H/∂Mi\nis the effective magnetic field as the sum of external field,\nfield due to interaction with other LMMs and magnetic\nanisotropy field in the classical Hamiltonian Hof LMMs;\nandµMis the magnitude of LMM [21].\nThe STT vector, T=TFL+TDL, can be decomposed\n[Fig. 1(a)] into: ( i) even under time-reversal or field-like\n(FL) torque, which affects precession of LMM around\nBeff\ni; and ( ii) odd under time-reversal or damping-like\n(DL) torque, which either enhances the Gilbert damp-\ning by pushing LMM toward Beff\nior competes with\nGilbert term as “antidamping.” For example, negative\nvalues ofTDL=TDL·eDLin Figs. 2 and 3, where\neDL= (Mi×∂Mi/∂t)|Mi×∂Mi/∂t|−1, means that TDL\nvector points away from the axis of precession which is\nantidamping action. Similarly, TFL=TFL·eFL, where\neFL= (∂Mi/∂t)|∂Mi/∂t|−1, is plotted in Figs. 2 and 3.\nThe current-driven STT Ti/bracketleftBig\nISα\next/bracketrightBig\nacts as the coun-\nterpart of nonconservative force in nonadiabatic MD.\nThe Gilbert damping plus current-independent torque\nFIG. 2. The FL and DL components [Fig. (1)] of three spin\ntorques contributions in Eq. (4) exerted by nonequilibrium\nspin density of electrons onto a single localized precessing\nmagnetic moment in the setup of Fig. 1(a) as a function of\ncoupling to the leads. Black dotted line is the sum of the three\ntorques. In panels (a) and (c) Jsd= 0.1γ, while in panels\n(b) and (d) Jsd= 20γensures perfectly adiabatic regime [32],\nJsd/~ω/greatermuch1, for the chosen precession frequency ~ω= 0.001γ.\nTi[∂Mi/∂t] appear as the counterpart of electronic\nfriction [8, 9, 14–19], but Gilbert damping requires\nagents [22] other than electrons alone considered in nona-\ndiabatic MD. Thus, the geometric torque and damping,\nas counterparts of geometric magnetism force [2] and fric-\ntion [10], are absent from standard modeling of classi-\ncal magnetization dynamics. Geometric torque has been\nadded ad hoc into the LLG equation applied to spe-\ncific problems, such as spin waves within bulk magnetic\nmaterials [29–31]. A recent study [32] of a single clas-\nsical LMM embedded into a closed (i.e., finite length\none-dimensional wire) electronic quantum system finds\nthat nonequilibrium electronic spin density always gener-\nates geometric torque, even in perfectly adiabatic regime\nwhere electron-spin/LMM interaction is orders of mag-\nnitude larger than the characteristic frequency of LMM\ndynamics. It acts as a purely FL torque causing anoma-\nlous frequency of precession that is higher than the Lar-\nmor frequency. By retracing the same steps [14, 15] in\nthe derivation of the stochastic Langevin equation for\nelectron-nuclei system connected to macroscopic reser-\nvoirs, Ref. [33] derived the stochastic LLG equation [34–\n37] for a single LMM embedded into an open electronic\nsystem out of equilibrium. The novelty in this derivation\nis damping, present even in the absence of traditional\nspin-flip relaxation mechanisms [23, 25], while the same3\nFIG. 3. Spatial profile of FL and DL components of Tgeo\ni,Tsea\niandTsurf\nispin torques on precessing LMMs depicted in Fig. 1(c)\nfor closed or open electronic quantum system and for two different values of Jsd. Insets on the top of each row mark positions\nand static configuration of LMMs within the Bloch DW, with their x-component depicted by the colorbar next to panel (a).\nconclusion about geometric torque changing only the pre-\ncession frequency of LMM has been reached (in some\nregimes, geometric phase can also affect the stochastic\ntorque [38]). However, single LMM is a rather special\ncase, which is illustrated in Fig. 1(a) and revisited in\nFig. 2, and the most intriguing situations in spintronics\ninvolve dynamics of noncollinear textures of LMMs. This\nis exemplified by current- or magnetic-field driven dy-\nnamics of domain walls (DWs) and skyrmions [25, 37, 39–\n43] where a much richer panoply of back-action effects\nfrom fast electronic system can be expected.\nIn this Letter, we analyze an exactly solvable\nmodel of seven steadily precessing LMMs, M1(t)–M7(t)\n[Fig. 1(c)], which are noncollinear and noncoplanar and\nembedded into a one-dimensional (1D) infinite wire host-\ning conduction electrons. The model can be viewed as\na segment of dynamical noncollinear magnetic texture,\nand it directly describes magnetic field-driven [43] head-\nto-head Bloch DW [44] but without allowing it to prop-\nagate [41, 43]. Its simplicity makes it exactly solvable—\nwe fins the exact time-dependent DM via the nonequi-\nlibrium Green function (NEGF) formalism [45] and an-\nalyze its contributions in different regimes of the ratio\nJsd/~ωofsdexchange interaction Jsd[23] between elec-\ntron spin and LMM and frequency of precession ω. In\nboth Figs. 1(a) and 1(c), the electronic subsystem is an\nopen quantum system and, although no bias voltage is\napplied between the macroscopic reservoirs, it is pushed\ninto the nonequilibrium state by the dynamics of LMMs.\nFor example, electronic quantum Hamiltonian becomes\ntime-dependent due to M1(t) [Fig. 1(a)] or M1(t)–\nM7(t) [Fig. 1(c)], which leads to pumping [25, 27, 46]\n[Fig. 4(b),(c)] of spin current locally within the DW re-\ngion, as well as into the leads [Fig. 4(a)]. Pumping ofcharge current will also occur if the left-right symmetry\nof the device is broken statically [27] or dynamically [47].\nThe electrons are modeled on an infinite tight-binding\n(TB) clean chain with Hamiltonian in the lab frame\nˆHlab(t) =−γ/summationdisplay\n/angbracketleftij/angbracketrightˆc†\niˆcj−Jsd/summationdisplay\niˆc†\niˆ\u001bˆci·Mi(t). (2)\nHere ˆc†\ni= (ˆc†\ni↑,ˆc†\ni↓) and ˆc†\niσ(ˆciσ) creates (annihilates) an\nelectron of spin σ=↑,↓at sitei. The nearest-neighbor\nhoppingγ= 1 eV sets the unit of energy. The active re-\ngion in Figs. 1(a) or 1(c) consists of one or seven sites,\nrespectively, while the rest of infinite TB chain is taken\ninto account as the left (L) and the right (R) semi-infinite\nleads described by the same Hamiltonian in Eq. (2), but\nwithJsd= 0. The hopping between the leads and the\nactive region is denoted by γc. The leads terminate at\ninfinity into the macroscopic particle reservoirs with iden-\ntical chemical potentials µL=µR=EFdue to assumed\nabsence of bias voltage, and EF= 0 is chosen as the\nFermi energy. In contrast to traditional analysis in spin-\ntronics [23, 25], but akin to Refs. [32, 33], Hamiltonian in\nEq. (2) does not contain any spin-orbit or impurity terms\nas generators of spin-flip relaxation.\nThe spatial profile of Bloch DW is given by\nMx\ni=−sech[(hDW−zi)/W] tanh[(ZDW−zi)],My\ni=\nsech2[(ZDW−zi)/W] andMz\ni= tanh[(ZDW−zi)/W],\nwhereZDW= 4 andW= 0.9. Instead of solving LLG\nequations [Eq. (1)] for M1(t)–M7(t), we impose a so-\nlution where LMMs precess steadily around the z-axis:\nMx\ni(t) = sinθicos(ωt+φi);My\ni(t) = sinθisin(ωt+\nφi); andMz\ni(t) = cosθi. Using a unitary transfor-\nmation into the rotating frame (RF), the Hamiltonian\nin Eq. (2) becomes time-independent [25, 27], ˆHRF=4\nˆU†(t)ˆHlab(t)ˆU(t)−i~ˆU†∂ˆU/∂t =ˆHlab(t= 0)−~ωˆσα/2,\nwith LMMs frozen at t= 0 configuration from the lab.\nThe unitary operator is ˆU(t) = exp(−iωtˆσα/2) forα-axis\nof rotation. In the RF, the original two-terminal Lan-\ndauer setup for quantum transport in Figs. 1(a) and\n1(c) is mapped, due to ~ωˆσα/2 term, onto an effective\nfour-terminal setup [27] [illustrated for single LMM in\nFig. 1(b)]. Each of its four leads is an effective half-metal\nferromagnet which accepts only one spin species, ↑or↓\nalong theα-axis, and effective dc bias voltage ~ω/eacts\nbetween L or R pair of leads.\nIn the RF, the presence of the leads and macro-\nscopic reservoirs can be taken into account exactly us-\ning steady-state NEGFs [45] which depend on time\ndifferencet−t/primeand energy Eupon Fourier trans-\nform. Using the retarded, ˆG(E), and the lesser, ˆG<(E),\nGreen functions (GFs), we find the exact nonequilib-\nrium DM of electrons in the RF, ˆ ρRF=1\n2πi\u0001\ndEˆG<(E).\nHere the two GFs are related by the Keldysh equa-\ntion, ˆG<(E) = ˆG(E)ˆΣ<(E)ˆG†(E), where ˆΣ<(E) is\nthe lesser self-energy [45] due to semi-infinite leads and\nˆG(E) = [E−ˆHRF−ˆΣ(E,~ω)]−1with ˆΣ(E,~ω) =/summationtext\np=L,R,σ=↑,↓ˆΣσ\np(E−Qσ\nα~ω) being the sum of retarded\nself-energies for each of the four leads p,σin RF. We\nuse shorthand notation Q↑\np=−1/2 andQ↓\np= +1/2.\nSince typical frequency of magnetization dynamics is\n~ω/lessmuchEF, we can expand [48] ˆ ρRFin small ~ω/EF\nand then transform it back to the lab frame, ˆ ρlab(t) =\nˆU(t)ˆρRFˆU†(t) to obtain ˆ ρlab(t) = ˆρad\nt+ ˆρgeo(t)+ ˆρsea(t)+\nˆρsurf(t) where:\nˆρad\nt=−1\nπˆU+∞\u0002\n−∞dEImˆG0f(E)ˆU†, (3a)\nˆρgeo(t) =1\nπˆU+∞\u0002\n−∞dEIm/bracketleftbigg\nˆG0/parenleftbigg\ni~ˆU†∂ˆU\n∂t/parenrightbigg\nˆG0/bracketrightbigg\nf(E)ˆU†,(3b)\nˆρsea(t) =−~ω\n2πˆU/summationdisplay\np+∞\u0002\n−∞dEIm/bracketleftbigg\nˆG0/parenleftbigg∂ˆΣ↑\np\n∂E−∂ˆΣ↓\np\n∂E/parenrightbigg\nˆG0/bracketrightbigg\n×f(E)ˆU†, (3c)\nˆρsurf(t) =~ω\n4πˆU/summationdisplay\np+∞\u0002\n−∞dEˆG0(ˆΓ↑\np−ˆΓ↓\np)ˆG†\n0∂f\n∂EˆU†.(3d)\nWe confirm by numerically exact calculations [39] that\nthus obtained ˆ ρlab(t) is identical to ~G<(t,t)/icomputed\nin the lab frame. Here ˆG0(E) = [E−ˆHRF−ˆΣ(E,0)]−1\nisˆG(E) with ~ω= 0; ˆΓσ\np(E) =i[ˆΣσ\np(E)−ˆΣσ\np(E)†] is\nthe level broadening matrix due the leads; and fσ\np(E) =\nf(E−[EF+Qσ\nα~ω]) is the the Fermi function of macro-\nscopic reservoir p,σin the RF.\nThe total nonequilibrium spin density, /angbracketleftˆsi/angbracketright(t) =\nTr[ˆρlab(t)|i/angbracketright/angbracketlefti|⊗ˆ\u001b] =/angbracketleftˆsi/angbracketrightad\nt+/angbracketleftˆsi/angbracketrightgeo(t) +/angbracketleftˆsi/angbracketrightsea(t) +\n/angbracketleftˆsi/angbracketrightsurf(t), has the corresponding four contributions fromDM contributions in Eq. (3). Here /angbracketleftˆsi/angbracketrightad\ntis the equilib-\nrium expectation value at an instantaneous time twhich\ndefines ‘adiabatic spin density’ [23, 25, 30–32]. It is com-\nputed using ˆ ρad\ntas the grand canonical equilibrium DM\nexpressed via the frozen (adiabatic) retarded GF [14, 15,\n33], ˆGt(E) = [E−ˆHt−ˆΣ]−1, for instantaneous configu-\nration of Mi(t) while assuming ∂Mi/∂t= 0 [subscript\ntsignifies parametric dependence on time through slow\nvariation of Mi(t)]. The other three contributions—from\nˆρgeo(t) and ˆρsea(t) governed by the Fermi sea and ˆ ρsurf(t)\ngoverned by the Fermi surface electronic states—contain\nfirst nonadiabatic correction [14, 15, 33] proportional to\nvelocity∂Mi/∂t, as well as higher order terms due to\nˆρlab(t) being exact. These three contributions define STT\nout of equilibrium [23, 39, 48]\nTi=Jsd/angbracketleftˆsi/angbracketright(t)×Mi(t) =Tgeo\ni+Tsea\ni+Tsurf\ni.(4)\nEach term Tgeo\ni,Tsea\ni,Tsurf\nican be additionally sepa-\nrated into its own DL and FL components [Fig. 1(a)], as\nplotted in Figs. 2 and 3. Note that Tsea\niis insignificant\nin both Figs. 2 and 3, so we focus on Tgeo\niandTsurf\ni.\nTo gain transparent physical interpretation of Tgeo\niand\nTsurf\ni, we first consider the simplest case [32, 33]—a single\nM1(t) in setup of Fig. 1(a). The STT contributions as a\nfunction of the coupling γcto the leads (i.e., reservoirs)\nare shown in Fig. 2. We use two different values for Jsd,\nwhere large ratio of Jsd= 20 eV and ~ω= 0.001 eV is\nperfect adiabatic limit [30–32]. Nevertheless, even in this\nlimit and for γc→0 we find Tgeo\n1/negationslash= 0 in Fig. 2(b) as the\nonly nonzero and purely FL torque. This is also found\nin closed system of Ref. [32] where Tgeo\n1was expressed\nin terms of the spin Berry curvature. As the quantum\nsystem becomes open for γc>0,Tgeo\n1is slightly reduced\nwhile Tsurf\n1emerges with small FL [Fig. 2(b)] and large\nDL [Fig. 2(d)] components. The DL torque Tsurf,DL\n1\npoints toward the z-axis and, therefore, enhances the\nGilbert damping. In the wide-band approximation [49],\nthe self-energy ˆΣ(E) =−iΓˆI2is energy-independent for\nEwithin the bandwidth of the lead, which allows us to\nobtain analytical expression (at zero temperature)\nTgeo\n1(t) =~ω\n2π/bracketleftbigg\nπ−2 tan−1/parenleftbiggΓ\nJsd/parenrightbigg/bracketrightbigg\nsinθeφ(t).(5)\nHere eφ(t) =−sinωtex+ cosωtey. Thus, in per-\nfect adiabatic limit, Jsd/~ω→∞ , or in closed system,\nΓ→0,Tgeo\n1is independent of microscopic parameters\nas expected from its geometric nature [29]. The always\npresent Tgeo\ni/negationslash= 0 means that electron spin is never along\n‘adiabatic direction’ /angbracketleftˆsi/angbracketrightad\nt.\nSwitching to DW [Fig. 1(c)] embedded into a closed\nquantum system ( γc= 0) shows in Fig. 3(a)–(d) that\nonlyTgeo\ni/negationslash= 0, which also acquires DL component lo-\ncally with damping or antidamping action depending on\nthe position of LMM. Upon opening the quantum sys-\ntem (γc=γ), Fig. 3(e)–(h) shows emergence of ad-\nditional Tsurf\ni/negationslash= 0 which, however, becomes negligible5\nFIG. 4. (a) The z-component of total DL torques which act\non DW in Fig. 1(c) as a function of Jsdforγc=γ. Cir-\ncles show that sum of spin currents pumped into the leads\nmatches/parenleftbig/summationtext\niTsurf,DL\ni/parenrightbig\nz≡ISz\nL+ISz\nR. Panel (b) and (c),\nwhich correspond to Fig. 3(g), show spatial profile of lo-\ncal spin currents ISz\ni→jpumped between sites iandjfor\nJsd= 0.1γ, with their sum being identically zeroin panel (c).\nDashed black line in panels (a) and (b) is pumped local spin\ncurrent by SMF [24, 26], ISz\nSMF(x) =gµB~G0\n4e2[∂M(x,t)/∂t×\n∂M(x,t)/∂x]z, whereG0=G↑+G↓is the total conductivity.\n[Fig. 3(f),(h)] in the perfectly adiabatic limit Jsd/~ω/greatermuch1.\nAt first sight, Tgeo,DL\ni/negationslash= 0 violates Berry and Robbins\noriginal analysis [2] according to which an isolated quan-\ntum system, with discrete energy spectrum, cannot exert\nfriction onto the classical system. This apparent contra-\ndiction is resolved in Fig. 4(a) where we show that total/summationtext\niTgeo,DL\ni≡0 is always zero. Conversely, Fig. 4(a) con-\nfirms that total/parenleftBig/summationtextTsurf,DL\ni/parenrightBig\nz≡ISz\nL+ISz\nRis identical\nto net spin current pumped into the leads via which the\nconduction electrons carry away excess angular momen-\ntum of precessing LMMs [46]. Such identity underlies\nphysical picture where spin current generated by time-\ndependent magnetization becomes DL torque [24, 46].\nNote that pumped spin current ISz\ni→jdue to ˆρgeoor ˆρsea\nin Fig. 4(c) can be nonzero locally, but they sum to zero.\nThe nonuniform pumped spin current due to spatially\nand time varying magnetization has prompted propos-\nals [24, 26] to amend the LLG equation by adding the\ncorresponding DL torque M×D·∂M/∂twith 3×3 damp-\ning tensorDwhose spatial dependence is given by the so-\ncalled spin-motive force (SMF) formula. However, SMF\ncorrection was estimated to be small [26] in the absence\nof spin-orbit coupling in the band structure. We confirm\nits smallness in Fig. 4(a),(b) for our DW case, but this\nactually reveals that SMF formula produces incorrectly\nan order of magnitude smaller torque than obtained from\nour exact ˆρsurf(t). Due to possibly complex [40] time and\nspatial dependence of Tsurf\niandTgeo\ni, the accurate path\nto incorporate them is offered by self-consistent coupling\nof electronic DM and LLG calculations, as proposed in\nRefs. [39, 42, 50] and in full analogy to how electronic\nfriction is included in nonadiabatic MD [7–9, 14–19].This research was supported in part by the U.S. Na-\ntional Science Foundation (NSF) under Grant No. CHE\n1566074.\n[1] M. V. Berry, Quantal phase factors accompanying adia-\nbatic changes, Proc. R. Soc. Lond. A 392, 45 (1984).\n[2] M. V. Berry and J. M. Robbins, Chaotic classical and\nhalf-classical adiabatic reactions: geometric magnetism\nand deterministic friction, Proc. R. Soc. Lond. A 442,\n659 (1993).\n[3] Q. Zhang and B. 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(cond-mat/0702020) argued  in favor of th e Land au-Lifshitz dampin g term in  the \nmicromagnetic equations of  motion over that of  the more commo nly accepted  Gilbert d amping for m. Much of \ntheir argument r evolved  around  spin-torque dr iven domai n wall motion in n arrow magnetic wires, since th e \npresence of spin -torques can mor e acut ely draw a distinct ion b etween the two form s of dam ping. In this ar ticle, \nthe author  uses simple argum ents and exam ples to offer  an alterna tive po int of  view favoring  Gilb ert.  \n \nI. PREL IMINARIES \n \n       The Gilbert1 (G) or La ndau-Li fshitz2 (LL) equations of  \nmotion for unit magnetization vect or  \nare formally descri bed by the gene ric form sMt t /),( ),(ˆ rM rm≡\n \n) derivativeal (variationˆ/ /1)] (ˆ[ /ˆ\neffeff totdamp tot\nNC\nm HH H HH Hm m\n∂∂−≡+≡+×γ−=\nE Mdt d\ns      (1) \n \nwhere  the satu ration  magnetizatio n, and  sM γ is th e \ngyromagnetic ratio (tak en here to  be a po sitive constant). \nThe total (p hysical) field  by h as con tributions fro m the \nusual \"effecti ve field\" term , pl us t hat of a   \n\"nonconservative-field\"  that is supp osed not to be \nderivable from the -gradient of the (internal) free-energy \ndensity functional . Although also non conservative \nby defi nition, the \"dam ping-field\"  is p rimarily a \nmath ematica l vehicle for  describing a physical d amping \ntorque , and i s properly treated separat ely. \nFor most of the rem ainder of th is article, an y sp atial \ndepe ndence of  will be im plicitly unde rstood.  effH\nNCH\nmˆ\n)ˆ(mE\ndampH\ndampˆHm×sM\n),(ˆtrm\n       As was described by Brown,3 the Gilb ert eq uations of \nmotion m ay be de rived using st andard techni ques of \nLagra ngian mechanics .4 In particular, a phenomenological \ndamping of the motion in included via the use of a R ayleigh \ndissipation function : )/ˆ( dtdmℜ\ndt d dt d M/dtdM\nGG\nss\n/ˆ)/ ( )/ˆ(/ /1ˆ )2/ (\nG\ndamp2\nm m Hm\nγα−= ∂∂ℜ−≡γα=ℜ\n     (2) \n \nwhere dimensionless  is th e Gilb ert damping parameter. \nBy d efinition,Gα\n4 2\ndampˆ / /ˆ 2G/dtd M dtds G m m H γα=⋅−=ℜ  \nis th e in stantaneous rate o f energy lost fro m the \nmagnetizatio n syste m to its thermal environment (e.g ., to \nthe lattice) d ue to the viscous  \"friction\" re present ed by the \ndamping field .   dt d/ˆG\ndampm H−∝\n        The Lag rangian method is well su ited to include \nnonconservative fields , which can be  generally \ndefined using the principles of virtual work:0NC≠H\n3,4 \n \n)ˆ ( /1 ) ˆ() ˆ( ˆ\nNC NCNC NC NC\nmN H HmHm m H\n× =⇔×=⇒δ⋅×=δ⋅ =δ\ns ss s\nM MNM M W θ       (3) \n \nThe latter exp ressio n is useful in cases (e.g.,  spin-torques) \nwhere the torque density funct ional  is specified. \nTreatin g  as fi xed, the  (virtual) dis placem ent )ˆ(mN\nsM mˆδ is of \nthe fo rm m m ˆ ˆ×δ=δθ , and only the orthogonal compone nts \nof the torque mNmN ˆ ˆ××↔  are physically signi ficant.  \n       Combining (1) an d (2) gives the Gilbert equations: \n \n)/ˆ ˆ( ) ˆ( /ˆtot dtd dt dG mm Hm m ×α+×γ−=           (4) \n \nAs is well known, the G eq uations o f (4) may be rearra nged \ninto their equivalent (and perhaps m ore common) f orm:  \n \n)] ˆ(ˆ ˆ[\n1/ˆtot tot2\nGHmm Hm m ××α+×\nα+γ−=G dt d      (5) \n \n       With re gard to the LL e quations, the form  of  is \nnot uniquely  defined in problems where LL\ndampH\n0NC≠H , whic h \nhave only c ome to the forefront with the  recent interest in \nspin-torque phen omena. Two d efinitions  conside red are \n \n) ˆ(eff damp LLLLHm H ×α≡ ,                       (6a) \n                        (6b) ) ˆ(tot damp LLLLHm H ×α≡\n \nThe fi rst de finition of (6a) is the historical/conventional \nform  of LL, and is that em ployed by Stiles et al.5 Howe ver, \nin this a uthor's view, the re is no a-priori reason, other than \nhistorical,  to not replace  as in (6b). Doing so \nyields a form  of LL that reta ins it \"usual\" e quivalence (i.e., \nto first o rder in tot eff H H→\nα) to G w hether or not 0NC≠H , as is \nseen by com paring (5) and (6b). The form o f (6b) treats \nboth  and  on an equal footin g. effHNCH       Nonet heless, to facilita te a com parative discussi on with \nthe analysis of Stiles et al. ,5  (6a) will he ncefort h be used to \ndefine what will be re ferred to below as the LL eq uatio ns \nof motion: \n \n)] ˆ(ˆ ˆ[ /ˆeff tot LL Hmm Hm m ××α+×γ−=dt d        (7) \n \nIn cases of pre sent interest where  , the difference \nbetwee n G in (4) (or (5)) and the f orm of LL give n in ( 7) \nare first orde r in the dam ping param eter, and  thus o f a more \nfundam ental nature. T hese differences a re the subj ect of the  \nremainder of t his article. 0NC≠H\n \nII. SPIN-TOR QUE  EXAM PLES \n \n       Two distinct situations where spin-torque effects have \ngarnere d substantial intere st are those  of CPP-GMR \nnanopillars, and spin-torque driven dom ain wall motion i n \nnanowires as was conside red in R ef. 5. The spi n-torque \nfunctio n  is taken t o have a \npredominant \"adiabatic\" c omponent , alon g with a \nsmall \"nona diabatic\" com pone nt   described \nphenomenolog ically  by the relation )ˆ( )ˆ( )ˆ(nad ad ST m NmNmN + =\n)ˆ(admN\n)ˆ(nadm N\nad nadˆNm N ×β−≡ , \nwith . In t he case of a narrow nanowire along the -\naxis, with m agnetization  and electron curre nt density  \n, the torque function  and associate d \nfield (see ( 2)) are descri bed by1<<β xˆ\n)(ˆxm\nx J ˆe eJ= )ˆ(STmN\n)ˆ(STmH5 \n \n)/ˆ /ˆ ˆ() 2/ ()/ˆ()2/ ()ˆ(\nSTad\ndxd dxd eM PJdxde PJ\ns ee\nm mm Hm mN\nβ+× −==\nhh   (8) \n \nwhere P is the  spin-polarization of t he electron curre nt.  \n       To check if  is conse rvative, one ca n \"discretize\"  \nthe spatial derivatives app earing in (8) in the form  STH\nx dxdi iixx∆ −→−+=2/)ˆ ˆ( /ˆ1 1m m m , whe re )(ˆ ˆi i xmm≡  \nand , not unlik e the com mon m icrom agnetics  \napproxim ation. For a c onservative H-field where \n, the set of  Cartesian tensorsi i x xx−≡∆+1\ni i Em H ∂∂∝ / 33×6 \nj i j iuv\nji E H mm mH ∂∂∂∝∂∂≡ /2/t\n will be sym metric, i.e., \nvu\nijuv\nji H Htt\n= , under sim ultaneo us reversal of s patial indices \n and vect or in dices ji, z yxvu or,, ,= . For the adiabatic \nterm in (6), it can be readily  shown that the  uv\njiHt\n are in \ngene ral asymmetric , i.e., always antisym metric in vect or \nindices (du e to cross pr oduct) , but asy mmetr ic in  spatial \nindices , being antisym metric he re only for  \nlocally uniform  magnetization . The  \nnonadiabatic term  yields an -inde pendent 1 ,±=iji\ni im m ˆ ˆ1=±\nmˆuv\njiHt\n that is  \nalways antisym metric, i.e., symmetric in ve ctor i ndices, but antisym metric in spatial indic es . The concl usion \nhere t hat  is in ge neral nonconservative a grees with \nthat f ound in Ref. 5, by way of a rathe r diffe rent argument.  1 ,±=iji\nSTH\n       Anot her well known example is a nanopillar stack wit h \nonly two fe rrom agnetic (FM ) layers, the \" refere nce\" layer \nhaving a m agnetization  rigidly fixe d in time, and  a \ndynamically  varia ble \"free \" lay er refˆm\n)(ˆ)( ˆfree t tm m= . As  \ndescri bed by  Sloncze wski,7 the (adiaba tic) spin-t orque \ndensity function a nd field is given by: )ˆ(STmH\n \n]ˆ )ˆ ˆ[() 4/ ()ˆ ˆ(]ˆ ˆˆ[) 4/ ()ˆ ˆ(\nref reffree refref free ref ad\nST\nm m mm m Hm mm m m N\nβ+××⋅−=×× ⋅−=\ntMe PJ gte PJ g\ns ee\nhh\n   (9) \n \nwhere  is the free layer thickness, and freet )ˆ ˆ(refm m⋅ g  is a \nfunctio n of order unity, the de tails of which are not relevant  \nto the present discu ssion. From  the the -tenso r, or by  \nsimple inspection, t he adiabatic  term  in (7) is \nmanifestly  nonc onservative . However, app roxim ating uvHt\nm m ˆ ˆref×\n)ˆ ˆ(refm m⋅ g ~ consta nt, the conse rvative nonadiabatic ter m \nresem bles a magnetic field d escribe d by the -gradient  of \nan Zeem an-like ene rgy function mˆ\nm m ˆ ˆref nad ⋅∝ E . The  \nremaining discussion will restrict attention to \nnonconservative contrib utions. \n \nIII. STATIO NARY SOLU TIONS OF G AND LL \n \n       With , stationary  (i.e, ST NC H H→ 0 /ˆ=dt dm ) \nsolutions  of G-equatio ns (4) satisfy  the c onditions  that 0ˆm\n \nST STST\n0 eff 0 0G\ndamp eff 0\nˆ ˆ 0 ˆ0 /ˆ ;0) ( ˆ\nHm Hm Hmm H H H m\n×−=×⇒≠×=∝ =+× dt d      (10) \n \nThe clear and physically intuitiv e interpreta tion of (10) is \nthat stationary  state  satisfies a condition of zero \nphysical tor que, 0ˆm\n0 ˆtot 0=×Hm , includin g bot h \nconservative ( ) an d nonconservative s pin-torque \n( ) fields.  Being visco us in nature, the G dam ping \ntorque  inde pendently vanishes..  effH\nSTH\n0 /ˆ ˆG\ndamp 0 ≡∝× dt dm Hm\n       Previous measurem ents6 of the angular depe ndence of \nspin-torque critical curre nts  in CPP-GMR \nnanopillar syste ms by this  author and colleagues  \ndemonstrated t he existence of such stationary states with \nnon-collinear )ˆ ˆ(refcritm m⋅eJ\n0 ˆ ˆ0 ref≠×m m  and crit0e eJ J<< . In t his \nsituation, it follows from (9) an d (10) that the stationa ry \nstate  satisfies 0ˆm 0 ˆ ˆeff 0 0 ST ≠×=×− Hm Hm . It is  \nnoted that the last result i mplies that  is not a (therm al) 0ˆmequilibrium  state which m inimizes the free energy , \ni.e., )ˆ(mE\n0 ) ˆ()ˆ ()ˆ/(ˆ/eff 0 0 ≠δ⋅×∝×δ⋅∂∂=δδ θ θ Hm m m m E E  \nfor arbitra ry .  θδ\n       In the present described circum stance of  stationary   \nwith , the LL equatio ns of (7) differ from G \nin a fundam ental respect. Setting  in (7) yield s  0ˆm\n0 ˆST 0≠×Hm\n0 /ˆ=dt dm\n \n) ˆ( ˆ ) ( ˆeff 0 0 eff 0 LL ST Hm m H H m ××α−=+×         (11) \n \nLike (10), (11) im plies  that 0 ˆeff 0≠×Hm  whe n \n. However, (11)  also  imply a static , nonzero \nphysical tor que , alon g with a static,  \nnonzero damping tor que  (see (6a) ) to \ncancel it out . In sim ple mechanical term s, the latte r \namounts to non-visco us \"static-frictio n\". It has n o anal ogue \nwith G in a ny circum stance, or with LL in conventional  \nsituations with  and  equilibrium \nfor which LL  dam ping wa s origi nally  develo ped as a  \nphenomenolog ical dam ping f orm. It furth er contra dicts th e \nviscous (o r -depe ndent) nature o f the damping \nmechanism s desc ribed by  physical (rathe r than  \nphenomenolog ical) base d theoretical m odels0 ˆST 0≠×Hm\n0 ˆtot 0≠×Hm\n0 ˆLL\ndamp 0 ≠×Hm\n0ST NC =↔H H ↔0ˆm\ndtd/ˆm\n,8,9. \n       The above arguments ignored therm al fluctuatio ns of \n. However, thermal fluctuations mˆ10 scale approxim ately a s \n, while ( 10) or (11) are scale-inva riant \nwith 2\neff 0 ) /( Hm⋅ kT\nH.  In t he simple CPP nanopillar exam ple of (9), one \ncan (conce ptually at least)  continually increase both eJ \nand a n applied  field  contri bution to  to scale up appHeffH\nST 0ˆHm×  and eff 0ˆHm×  while approxim ately keeping \na fixe d statio nary state  (satis fying 0ˆm 0 ˆ ˆref 0≠×mm  with \nfixed ).  However, unique to LL eq uatio ns (11) based \non (6a) is t he additional requi rement that the static dam ping \nmechanism  be able to produce an refˆm\neff dampˆLLHm H ×∝  \nwhich sim ilarly scales (without li mit). This author finds thi s \na physically unreasonable proposition.  \n \nIV. ENERGY A CCOUN TING  \n \nIf one ignores/forgets t he Lagrangian formulation3 of the \nGilbert e quatio ns (4), one may derive t he followin g energy \nrelationships, substitutin g the right side of (4) for evaluating \nvecto r products of form : dtd/ˆmH⋅\n \n)/ˆ ˆ( ) ˆ()/ˆ ˆ( ) ˆ(/ˆ )/ˆˆ/ /( /1\neff effeff effeff\nNCNC\ndtddtddtd dtd E dtdE Ms\nmm H Hm Hmm H Hm Hm H mm\n×⋅α−×⋅γ−=×⋅α−×⋅γ=⋅−≡⋅∂∂=\n   (12a) \n )/ˆ ˆ( ) ˆ(/ˆ / /1\nNC NCNC NC\neff dtddtd dt dWMs\nmm H Hm Hm H\n×⋅α+×⋅γ−=⋅≡   (12b) \n \n)/ˆ ˆ() () ˆ(/ˆ /ˆ\nNC efftot2\ndt ddt d dt d\nmm H HHm m m\n×⋅+γ=×γ−⋅=          (12c) \n \nSubtractin g (12b) from (12a), and usin g (12c) one  finds \n \ndt d M dt dWdt d M dt dW dtdE\nss\n/ˆ //ˆ / / / :G\nG\nNCG NC\ndamp2\nm Hm\n⋅ + =γα− =\n         (13) \n \nThe re sult o f (13) is essentially a state ment of energy \nconservation. Nam ely, that the rate of change of the internal \nfree e nergy (density) of the magnetic sy stem  is give by  the \nwork done on the system  by the (exte rnal) no nconservative \nforces/fields , minus the loss of energy (t o the lattice) \ndue t o dam ping. The G damping term  in ( 13) is ( not \nsurprisi ngly) t he sam e as expected from  (2). It is a strictly  \nlossy, negative-definite  contributio n to .  NCH\ndtdE/\n       Over a finite interval of motion from  time  to , the \nchange 1t2t\n)ˆ( )ˆ(1 2 m m E EE −=∆  is, from (12b ) and (13):  \n \n∫⋅γα− =∆2\n1G NCˆ)/ˆ / )ˆ( (t\ntsdtddt d dt MEmm m H       (14) \n \nSince  is nonc onservative, t he work NCHNCW∆  is pat h-\ndepe ndent, and so use of (14) requires indepe ndent \nknowledg e of the solution  of (4). Sin ce \n itself depe nds on ) (ˆ2 1 ttt≤≤m\n)(ˆtmGα, the  term's contribution  \nto (14) also can vary with . Regardless, NCH\nGα 0>∆E  can \nonly result in the case of a positive  amount of work \n done by .  ∫⋅ =∆2\n1NC NC )/ˆ (t\nts dtdt d M W m HNCH\n       Working out the results  analogous to (12 a,b) for the LL \nequatio ns of (6a) and (7), one finds \n \n) ˆ() ˆ( /ˆ/ˆ / / :LL\ntot eff dampdamp\nLLLL\nNC\nHm Hm m Hm H\n×⋅×αγ−=⋅⋅ + =\ndtddtd M dt dW dtdEs\n      (15) \n \nThe form of (15) is the sam e as the latter result in (13). \nHowever, unlike G, the LL damping term in (15) is not \nmanifestly negative-definite, except when 0NC=H .  \n       The results of (13)-(15) apply equally to situations \nwhere one inte grates over the spatial distribution of  \nto evaluate the total syste m free energy, rat her tha n (local ) \nfree e nergy density . Total time derivatives  may be \nreplace d by partial deri vative s  whe re appropriate. ),(ˆtrm\ndtd/\nt∂∂/      Dropping terms of order  (and sim plifying notation \n), (7) is easily transfor med to a Gilbert-like form : 2\nLLα\nα→αLL\n \n)ˆˆ( )] ˆ( [ˆˆ:LLNC totdtd\ndtd mm Hm Hmm×α+×α−×γ−=  (16) \n \nwhic h differs from G in  (4) by the term  ) ˆ(NCHm×α  \nwhich is first order in both  and . For the \"wire \nproblem\" described by  (8), the equation s of m otion bec ome αNCH\n \n)ˆ ˆ(ˆ ˆˆ ˆ:LL)ˆ ˆ(ˆ ˆˆ ˆ\neffeff :G\ndxdv\ndtd\ndxdv\ndtddxdv\ndtd\ndxdv\ndtd\nm mm Hmm mm mm Hmm m\nαβ+α+×α+×γ−=+αβ+×α+×γ−=+\n (17) \n \nwhere , and terms of or der eM PJ vs e2/γ=h βα are \ndropped for LL. A s noted previously,5,9,11 (17) permits \n\"translational\" solutions ) (ˆ)(ˆeq vtx x,t −=m m  whe n α=β  \n(G) or  (LL), with  the static , equilibrium \n(minimum E) solution of . Evaluatin g \n by takin g  from (8), and \n with , one finds \nthat 0=β )(ˆeqxm\n0) ˆ(eff eq =×H m\ndtd M dt dWs /ˆ /ST ST m H⋅ =STH\n) (ˆ /ˆeq vtx v dtd −′−→m m dqd q /ˆ )(ˆ m m≡′\n2\neq2) (ˆ)/ ( /ST vtx Mv dt dWs −′γβ= m . In transl ational \ncases where  is exactly  collinear  to , only \nthe nonadiabatic term  does work on t he -system.  dtd/ˆm dx d/ˆm\nmˆ\n       Interestingly, the energy interpretation of these \ntranslational solutions is very diffe rent fo r G or LL. For G, \nthe positive rate of work  when dt dW /ST α=β  exactly \nbalances t he negative damping l oss as  given in (13), the \nlatter alway s nonzero and scaling as . For LL by  \ncontrast, the wo rk done by  vanishes when 2v\nSTH 0=β , \nmatching t he damping l oss whic h, from (6a)  or (15), is \nalways zero since  regardle ss of 0) ˆ(eff eq =×H m v. If \n is a sharp domain wall, )(ˆeqxm ) (ˆ /ˆeq vtx v dtd −′−=m m  \nrepresents, from a spatially local  perspective at a fixed  point \nx, an abrupt, irreversi ble, non-equilibrium  reor ientation of \n at/near tim e  whe n the wall core passes by. The  \nprediction of LL/(6a) that t his magnetization re versal c ould \ntake place locally (at arbitrarily large v), with the com plete \nabsence of the spin-orbit couple d, ele ctron scatteri ng \nprocessesmˆ vxt /≈\n8 that lead to spin-latti ce dam ping/r elaxation in all \nother known circum stances (e. g., external field-driven \ndomain wall motion) is, in the vi ew of this author, a rather \ndubious, nonphysical aspect of (6a) when 0ST≠H . \n       Stiles et al.5 repo rt that micromagnetic com putation s \nusing G in the case  sho w (non-translational)  \ntime/distance l imited dom ain wall displacement, resulting  in a net positive  increase \n0=βE∆. They claim  that 1) \"spi n \ntrans fer to rques do not cha nge the ene rgy of the sy stem\", \nand that 2) \"Gilbert dam ping to rque is the only  torque th at \nchanges the e nergy\". Acce pting  as accurate, it is \nthis aut hor's view that t he elementary physics/ mathematics \nleading t o (13) and (14) demonstrably  prove t hat bot h of \nthese claim s must be incorrect  (err or in the first per haps \nleading t o the misinterpretation of the second). On a related \npoint, the res ults of (1 3) and (1 5) shows that excludi ng \nwork  or 0>∆E\ndt dW /ST STW∆ , only LL- damping may possibly  \nlead to a positive contri bution to  or dtdE/ E∆ when \n0ST≠H , in a pparent contradictio n to the claim  in R ef. 5 \nthat LL damping \"uniquely and  irre versibly  reduces \nmagnetic free energy whe n spin-transfer torque is prese nt\". \n \nACK NOWLEDGM ENTS \n \nThe aut hor would like to ackno wledge em ail discussions on \nthese or related topics with W. Sa slow and R. Duine, as \nwell as an  extende d series of friendly discussio ns with  \nMark Stiles. Obviously, the latter have not (as of yet) \nachieve d a mutually agree d viewpoint on thi s subj ect.  \n \nREFERENCES \n \n1 T. L. Gilber t, Armour Research Report, M ay 1956; IEEE Tran s. \nMagn., 40, 3343  (2004).  \n2 L. Landau  and E. Lifshit z, Phys. Z. Sow jet 8, 153 (1935). \n3 W. F. Brown, Micromagnetics  (Krieger , New Y ork 1978). \n4 H. Gol dstein, Classical Me chanics , (Addison  Wesley, Reading \nMassachusetts, 1 950). \n5M. D. Stiles, W. M. Saslow, M. J. Donahue,  and A . Zangwill, \narXiv:cond-m at/0702020. \n6 N. Smith, J. A. Katine, J. R. Childress, and M. J. Carey, IEEE \nTrans. M agn. 41, 2935 (2005) ; N. Sm ith, J Appl. Ph ys. 99, \n08Q703 (2006). \n7 J. C. Slonczewski, J.  Magn. Magn. Mater. 159, L1 (1996); J. \nMagn. Magn . Mater. 247, 324 (2 002) \n8 V. Kambersky, Can. J. Phys. 48, 2906 (1970) ; V. Kam bersky and \nC. E. Patton , Phys. Rev . B 11, 2668 (1975). \n9 R. Duine, A. S. Nunez, J.  Sinova, and A. H. MacDonald, \narXiv:cond-m at/0703414. \n10 N. Sm ith, J. A ppl. Ph ys. 90, 57 68 (2001). \n11 S. E Barnes and S . Maekaw a, Phys. R ev. Lett. 95, 10720 4 \n(2005). \n "    },    {        "title": "1405.2267v1.Current_induced_magnetization_dynamics_in_two_magnetic_insulators_separated_by_a_normal_metal.pdf",        "content": "Current-induced magnetization dynamics in two magnetic insulators separated by a\nnormal metal\nHans Skarsv\u0017 ag1, Gerrit E. W. Bauer2;3and Arne Brataas1\n1Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\n2Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan\n3Kavli Institute of NanoScience, Delft University of Technology, 2628 CJ Delft, The Netherlands\n(Dated: March 14, 2022)\nWe study the dynamics of spin valves consisting of two layers of magnetic insulators separated\nby a normal metal in the macrospin model. A current through the spacer generates a spin Hall\ncurrent that can actuate the magnetization via the spin-transfer torque. We derive expressions\nfor the e\u000bective Gilbert damping and the critical currents for the onset of magnetization dynamics\nincluding the e\u000bects of spin pumping that can be tested by ferromagnetic resonance experiments.\nThe current generates an amplitude asymmetry between the in-phase and out-of-phase modes. We\nbrie\ry discuss superlattices of metals and magnetic insulators.\nPACS numbers: 76.50.+g,75.30.Ds,75.70.-i,75.76.+j\nI. INTRODUCTION\nElectric currents induce spin-transfer torques in het-\nerogeneous or textured magnetic systems.1In this con-\ntext, magnetic insulators such as yttrium iron garnet\n(YIG) combined with normal metal contacts exhibit-\ning spin-orbit interactions, such as Pt, have recently\nattracted considerable interest, both experimentally2{8\nand theoretically.9{15Since the discovery of non-local\nexchange coupling and giant magnetoresistance in spin\nvalves, i.e., a normal metal sandwiched between two fer-\nromagnetic metals, these systems have been known to\ndisplay rich physics. Some of these e\u000bects, such as the dy-\nnamic exchange interaction,19should also arise when the\nmagnetic layers are insulators. The spin Hall magnetore-\nsistance (SMR) is predicted to be enhanced in such spin\nvalves,10although experimental realizations have not yet\nbeen reported. Here, we consider multilayer structures\nwith ferromagnetic but electrically insulating (FI) layers\nand normal metal (N) spacers. In-plane electric currents\napplied to N generate perpendicular spin currents via the\nspin Hall e\u000bect (SHE). When these spin currents are ab-\nsorbed at the NjFI interfaces, the ensuing spin-transfer\ntorques can induce magnetization dynamics and switch-\ning. We consider ground state con\fgurations in which\nthe magnetizations are parallel or antiparallel to each\nother. For thin magnetic layers, even small torques can\ne\u000bectively modify the (Gilbert) damping, which can be\nobserved as changes in the line width of the ferromagnetic\nresonance (FMR) spectra. We employ the macrospin\nmodel for the magnetization vectors that is applicable\nfor su\u000eciently strong and homogeneous magnetic \felds,\nwhile extensions are possible.13{15Our results include\nthe observation of e\u000bective (anti)damping resulting from\nin-plane charge currents in FI jNjFI trilayers, magnetic\nstability analysis in the current-magnetic \feld parame-\nter space and a brief analysis of the dynamics for cur-\nrents above the critical value. We also consider current-\ninduced e\u000bects in superlattices. Our paper is organized as\nfollows. In Section II, we present our model for a FI jNjFIspin valve including the SHE spin current generation\nand spin pumping, modeled as additional torques in the\nLandau-Lifshitz-Gilbert equation. We proceed to formu-\nlate the linearized magnetization dynamics and the spin\naccumulation in N in Section III. In Section IV, we calcu-\nlate the eigenmodes and the current-controlled e\u000bective\nGilbert damping and determine the critical currents at\nwhich the magnetic precession becomes unstable. We dis-\ncuss the current-induced dynamics of \u0001\u0001\u0001jFIjNjFIjNj\u0001\u0001\u0001\nsuperlattices in Section V. Finally, we summarize our\nconclusions and provide an outlook in Section VI.\nII. MODEL\nFI1jNjFI2 denotes the heterostructure composed of a\nnormal metal (N) layer sandwiched between two layers of\nferromagnetic insulators (FIs) (see Fig. 1). We denote the\nthicknesses of FI1, N and FI2 by d1,dNandd2, respec-\ntively. We adopt a macrospin model of spatially constant\nmagnetization Miin each layer. The magnetization dy-\nnamics of the two layers are described by the coupled\nLandau-Lifshitz-Gilbert-Slonczewski (LLGS) equations:\n_Mi=\u0000\rMi\u0002\u0012\nHe\u000b;i+J\ndiMS;iMj\u0013\n+\u000biMi\u0002_Mi\n+\u001cDSP\ni+\u001cISP\ni+\u001cSH\ni; (1)\nwhere Miis the unit vector in the direction of the magne-\ntization in the left/right layer with indices i= 1;2;MS;i\nis the saturation magnetization; \ris the gyromagnetic\nratio;\u000biis the Gilbert damping constant; Jis the in-\nterlayer dipolar and exchange energy areal density, with\nj= 1(2) when i= 2(1); and He\u000b;iis an e\u000bective mag-\nnetic \feld:\nHe\u000b;i=Hext+Han;i(Mi) (2)\nconsisting of the external magnetic \feld Hextas well as\nthe anisotropy \felds Han;ifor the left/right layer. We dis-\ntinguish direct (DSP) and indirect spin pumping (ISP).arXiv:1405.2267v1  [cond-mat.mes-hall]  9 May 20142\nDSP generates the spin angular momentum current jDSP\n1(2)\nthrough the interfaces of FI1(2). A positive spin current\ncorresponds to a spin \row toward the FI from which it\noriginates. The DSP spin current is expressed as\njDSP\ni=~\neg?;iMi\u0002_Mi; (3)\nwhereg?;iis the real part of the spin-mixing conduc-\ntance of the NjFI1(2) interface per unit area for i= 1(2),\nrespectively, and \u0000eis the electron charge. This angu-\nlar momentum loss causes a damping torque (here and\nbelow in CGS units):\n\u001cDSP\ni=\r~2g?;i\n2e2MS;idiMi\u0002_Mi: (4)\nIn ballistic systems, the spin current emitted by the\nneighboring layer is directly absorbed and generates an\nindirect spin torque on the opposing layer:19\n\u001cISP\ni;ball=\u0000\r~2g?;i\n2e2MS;idiMi\u0002_Mi: (5)\nIn the presence of an interface or bulk disorder, the trans-\nport is di\u000buse, and the ISP is\n\u001cISP\ni=\u0000\r~\n2e2MS;idig?;iMi\u0002(Mi\u0002\u0016SP(zi));(6)\nwhere \u0016SP(zi) is the spin pumping contribution to the\nspin accumulation (di\u000berence in chemical potentials) at\nthe interface in units of energy, with zi\u0011 \u0007dN=2 for\ni= 1;2.\u0016SPis the solution of the spin di\u000busion equation\nin N as discussed below.\nDue to the SHE, an in-plane DC charge current pro-\nduces a transverse spin current that interacts with the\nFIjN interfaces. Focusing on the di\u000busive regime, the\nareal density of charge current jcas well as the spin jSH\nk\ncurrent in the k-direction, where jSH\nk=\f\fjSH\nk\f\fis the spin\npolarization unit vector, can be written in terms of a\nsymmetric linear response matrix:10\n0\nBB@jc\njSH\nx\njSH\ny\njSH\nz1\nCCA=\u001b0\nB@1 \u0002 SH^x\u0002\u0002SH^y\u0002\u0002SH^z\u0002\n\u0002SH^x\u0002 1 0 0\n\u0002SH^y\u0002 0 1 0\n\u0002SH^z\u0002 0 0 11\nCA\n0\nBB@\u0000r\u0016c=e\n\u0000r\u0016SH\nx=(2e)\n\u0000r\u0016SH\ny=(2e)\n\u0000r\u0016SH\nz=(2e)1\nCCA; (7)\nwhere \u0002SHis the spin Hall angle, \u001bis the electrical\nconductivity and \u0016cis the charge chemical potential.\n\u0016SH= (\u0016SH\nx;\u0016SH\ny;\u0016SH\nz) is the spin accumulation induced\nby re\rection of the spin currents at the interfaces. The\nspin transfer torques \u001cSH\niat the FI interfaces ( i= 1;2)\nare then expressed as\n\u001cSH\ni=\u0000\r~\n2e2MS;idig?;iMi\u0002\u0000\nMi\u0002\u0016SH(zi)\u0001\n:(8)The polarization of \u0016SHand thereby \u001cSH\nican be con-\ntrolled by the charge current direction. In the following\nsections, we assume that the shape anisotropy and ex-\nchange coupling favor parallel or antiparallel equilibrium\norientations of M1andM2. For small current levels, the\ntorques normal to the magnetization induce tilts from\ntheir equilibrium directions and, at su\u000eciently large cur-\nrents, trigger complicated dynamics, while torques di-\nrected along the equilibrium magnetization modify the\ne\u000bective damping and induce magnetization reversal.\nHere, we focus on the latter con\fguration, in which the\nspin accumulation in N is collinear to the equilibrium\nmagnetizations.\nIn the following equations, we take the thickness, satu-\nration magnetization, Gilbert damping and spin-mixing\nconductance to be equal in the two layers FI1 and FI2,\nwith an out-of-plane hard axis and an in-plane internal\n\feld:\nHe\u000b;1=!H\n\r^x\u0000!M\n\r(M1)z^z; (9a)\nHe\u000b;2=s!H\n\r^x\u0000!M\n\r(M2)z^z; (9b)\nwith!H=\r(Hext+ (Han;i)x) and!M= 4\u0019\rMS. Pure\ndipolar interlayer coupling with J <0 favors an antipar-\nallel ground state con\fguration, while the exchange cou-\npling oscillates as a function of dN.\nm\n1m1\nM\n1m\n2m2M2\n FI1                    NM              FI2\nx\n                      z\nyz=-d1-dN /2 z=-dN/2 z=dN/2z=dN/2+d2\njc jSH(SHE spin current direction)\nj1SP ~ M1 x m1j2SP ~ M2 x m2\nFIG. 1: (Color online) Spin valve of ferromagnetic insulators\n(FIs) sandwiching a normal metal (N). The equilibrium mag-\nnetizations M1andM2are collinear, i.e., parallel or antipar-\nallel. A spin-Hall-induced spin current \rows in the z-direction\nand is polarized along x.\nIII. SPIN-TRANSFER TORQUES\nThe spin-pumping and spin-transfer torques \u001cDSP\niand\n\u001cISP\ni(Eqs. (4) and (6)) cause dynamic coupling between\nthe two magnetizations. To leading order, these torques\ncan be treated separately. We now derive expressions3\nfor disordered systems that support spin accumulations\n\u0016X(z) (X = SH;SP) governed by the spin-di\u000busion equa-\ntion:\n_\u0016X=D@2\nz\u0016X\u0000\u0016X\n\u001csf: (10)\nHere,Dis the di\u000busion constant, and \u001csfis the spin-\n\rip relaxation time. The di\u000buse spin current in the\nz-direction related to this spin accumulation follows\nEq. (7):\njX=\u0000\u001b\n2e@\u0016X\n@z; (11)\nwhere\u001bis the conductivity of N.\nA. Spin-pumping-induced torques\nThe total spin current into an FI is the sum of the\nspin-transfer and spin-pumping currents. Disregarding\ninterface spin-\rip scattering, the boundary conditions for\nthe left/right layer are\n\u00001\neg?Mi\u0002\u0000\nMi\u0002\u0016SP(zi)\u0001\n+jDSP\ni=\u0007jSP(zi):(12)\nThe -(+) sign on the right-hand side is due to the oppo-\nsite \row direction of the spin currents at the left (right)\ninterface. We expand the magnetization direction around\nthe equilibrium con\fguration as\nM1=^ x+m1; (13a)\nM2=s^ x+m2; (13b)\nas long asjmij\u001cjMijormi\u0001Mi=O\u0010\njmij2\u0011\n. The\nparameters= 1 when the equilibrium con\fguration is\nparallel;s=\u00001 when it is antiparallel. The FMR fre-\nquency is usually much smaller than the di\u000buse electron\ntraversal rate D=d2\nNand spin-\rip relaxation 1 =\u001csfrate;\nthus, retardation of the spin \row may be disregarded. In\nthe steady state, the left-hand side of Eq. (10) vanishes.\nWe solve Eq. (10) for the adiabatic magnetization dy-\nnamics with boundary conditions Eq. (12) to obtain the\nspin accumulation:\n\u0016SP=\u0000~\n2^ x\u0002[(_ m1+s_ m2)\u00001(z)\n\u0000(_ m1\u0000s_ m2)\u00002(z)]; (14)\nwherelsf=pD\u001csfis the spin-di\u000busion length and\n\u00001(z)\u0011cosh (z=lsf)\ncosh (z=lsf) +\u001bsinh (z=lsf)=2g?lsf;(15a)\n\u00002(z)\u0011sinh (z=lsf)\nsinh (z=lsf) +\u001bcosh (z=lsf)=2g?lsf:(15b)\nThe torques are\n\u001cISP\ni=\r~\n2e2MSdg?\u0016SP(zi): (16)Because the spin accumulation is generated by the dy-\nnamics of both ferromagnets, we obtain spin-pumping-\ninduced dynamic coupling that is quenched when dN\u001d\nlsf. In the limit of vanishing spin-\rip scattering, the spin\naccumulation is spatially constant and is expressed as\n\u0016SPdN\u001clsf! \u0000~\n2^ x\u0002(_ m1+s_ m2): (17)\nThe corresponding di\u000busive torque is then a simple av-\nerage of the contributions from the two spin-pumping\ncurrents, in contrast to the ballistic torque that depends\nonly on the magnetization on the opposite side.\nB. Current-induced torques\nA charge current in the y-direction causes a spin Hall\ncurrent in the z-direction that is polarized along the x-\ndirection (see Fig. 1). At the interfaces, the current in-\nduces a spin-accumulation \u0016SHthat satis\fes the di\u000busion\nEq. (10) and drives a spin current (dropping the index z\nfrom now on):\njSH=\u0000\u001b\n2e@\u0016SH\n@z\u0000jSH\n0^x; (18)\nwherejSH\n0= \u0002 SHjc:Angular momentum conservation at\nthe left/right boundaries leads to\n\u00001\neg?Mi\u0002\u0000\nMi\u0002\u0016SH(zi)\u0001\n=\u0007jSH(zi): (19)\nWhen Mik\u0016SH, the spin Hall current is completely\nre\rected and the spin current at the interface van-\nishes, while the absorption and torque are maximal when\nMi?\u0016SH. Spin currents and torques at the interface scale\nfavor mifor small magnetization amplitudes. Let us de-\n\fne a time-independent \u0016SH\n0for collinear magnetizations\nand spin current polarization. For small dynamic mag-\nnetizations, then\n\u0016SH=\u0016SH\n0+\u000e\u0016SH; (20)\nwhere\u000e\u0016SH\u0018mi. We will show that the spin-Hall in-\nduced spin accumulation leads to a (anti)damping torque\nin the trilayer, while it gives a contribution to the real\npart of the frequency for superlattices (see Sec. V).\nSolving the di\u000busion Eq. (10) with boundary condi-\ntions, Eq. (19) yields\n\u0016SH\n0=\u00002elsf\n\u001bjSH\n0sinh(z=lsf)\ncosh(dN=2lsf)^ x: (21)\nThe dynamic correction\n\u000e\u0016SH=\u00001\n22elsf\n\u001bjSH\n0tanh(dN=2lsf)\n[(m1+sm2)\u00002(z)\u0000(m1\u0000sm2)\u00001(z)] (22)4\nleads to SHE torques [Eq. (8)]:\n\u001cSH\ni=\u0000\r~\n2e2MSdg?\u0002\nmi(\u0016SH\n0\u0001^x)\u0000\u000e\u0016SH(zi)\u0003\n:(23)\nEq. (1) then reduces to four coupled linear \frst-order\npartial di\u000berential equations for mi.\nIV. EIGENMODES AND CRITICAL\nCURRENTS\nAfter linearizing Eq. (1) and Fourier transforming to\nthe frequency domain _Mi!i!^mi, Eq. (1) becomes\nMv= 0; (24)\nwhere vT= ( ^m1;y;^m1;z;^m2;y;^m2;z) andMis a 4\u00024\nfrequency-dependent matrix that can be decomposed as\nM=M0+JMJ+(\u000b+\u000b0)Md+\u000b0MSP+jSH\n0MSH;(25)\nwith\nM0=0\nB@\u0000i!\u0000~!H\u0000!M 0 0\n~!H\u0000i! 0 0\n0 0\u0000i!\u0000s~!H\u0000s!M\n0 0 s~!H\u0000i!1\nCA;(26a)\nMd=0\nB@0\u0000i! 0 0\ni! 0 0 0\n0 0 0\u0000is!\n0 0is! 01\nCA; (26b)\nMJ=0\nB@0 0 0 !x\n0 0\u0000!x0\n0s!x0 0\n\u0000s!x0 0 01\nCA; (26c)\nMISP=0\nB@0i!F00is!G0\n\u0000i!F00\u0000is!G00\n0i!G00is!F0\n\u0000i!G00\u0000is!F001\nCA; (26d)\nMSH=0\nB@\u0000F0\u0000sG 0\n0\u0000F 0\u0000sG\nG 0sF 0\n0G 0sF1\nCA: (26e)\nHere,M0describes dissipationless precession in the e\u000bec-\ntive magnetic \felds, and Mdarises from Gilbert damping\nand the direct e\u000bect of spin pumping with a renormalized\ndamping coe\u000ecient ~ \u000b=\u000b+\u000b0and\n\u000b0=\r~2\n2e2Msdg?; (27)\nMJrepresents interlayer exchange coupling, MISPrep-\nresents spin-pumping-induced spin transfer, and MSH\nrepresents the spin transfer caused by the spin Hall cur-\nrent. The external and possible in-plane anisotropy \felds\nare modi\fed by the interlayer coupling, !H!~!H=\n!H+!x, where!x=\rJ=(Msd). The matrix elementsF0,G0,FandGare generalized susceptibilities extracted\nfrom Eqs. (16) and (23):\nF0=1\n\u000b0@(\u001cST\n1)y\n@_m1;z; (28a)\nG0=1\n\u000b0@(\u001cST\n1)y\n@(s_m2;z); (28b)\nF=\u00001\njSH\n0@(\u001cST\n1)y\n@m1;y; (28c)\nG=1\njSH\n0@(\u001cST\n1)y\n@(sm2;y): (28d)\nThe explicit expressions given in Appendix A are simpli-\n\fed for very thick and thin N spacers.\nThin N layer: WhendN\u001clsf, the interlayer coupling\nG0due to spin pumping approaches F0;, the intralayer\ncoupling:\nG0!F0!1\n2; (29)\nwhich implies that the incoming and outgoing spin cur-\nrents are the same. This outcome represents the limit of\nstrong dynamic coupling in which the additional Gilbert\ndamping due to spin pumping vanishes when the mag-\nnetization motion is synchronized.16In this regime, the\nSHE becomes ine\u000bective because FandGscale asdN=lsf.\nF=G!2 becauseFcontains a contribution from both\nthe static as well as the dynamic spin accumulation.\nThick N layer: In the thick \flm limit, dN\u001dlsf, the\ninterlayer coupling vanishes as G!0 andG0!0, while\nF0!1\n1 +\u001b\n2g?lsf; (30a)\nF!\r~\n2eMSd1\n1 +\u001b\n2g?lsf: (30b)\nIntroducing the spin conductance Gsf\u0011A\u001b=2lsfG?=\nAg?andRtot= (G?+Gsf)\u00001, the total resistance of the\ninterface and the spin active region of N, F0!RtotG?,\nrepresents the back\row of pumped spins. The same holds\nfor the part of Fthat originates from the dynamic part\nof\u0016SH, while the static part approaches a constant value\nwhendNbecomes large (see Appendix A). In this limit,\nthe system reduces to two decoupled FI jN bilayers.\nThe eigenmodes of the coupled system are the solutions\nof det [M(!n)] = 0 with complex eigenfrequencies !n.\nThe SHE spin current induces spin accumulations with\nopposite polarizations at the two interfaces. In the paral-\nlel case, the torques acting on the two FIs are exerted in\nopposite directions. The torques then stabilize one mag-\nnetization, but destabilize the other. When the eigenfre-\nquencies acquire a negative imaginary part, their ampli-\ntude grows exponentially in time. We de\fne the thresh-\nold current jSH\n0;thrby the value at which Im[ !n\u0010\njSH\n0;thr\u0011\n] =\n0:Because the total damping has to be overcome at the5\nthreshold,jSH\n0;thr\u0018~\u000b. We treat the damping and ex-\nchange coupling perturbatively, thereby assuming ~ \u000b\u001c1\nand!x\u001c!0;where!0=p\n~!H(~!H+!M) is the FMR\nfrequency. The spin Hall angle is usually much smaller\nthan unity; thus, jSH\n0is treated as a perturbation for cur-\nrents up to the order of the threshold current, implying\nthat\f\fIm[!n\u0000\njSH\n0\u0001\n]\f\f\u001c\f\fRe[!n\u0000\njSH\n0\u0001\n]\f\f.\nThe exchange coupling !x=\rJ=(Msd) for YIGjPtj\nYIG should be weaker than that of the well-studied\nmetallic magnetic monolayers, where it is known to be-\ncome very small for d&3 nm.17In the following sections,\nwe assume that !x\u001c!Mmay be treated as a perturba-\ntion.\nTo treat the damping, spin pumping, spin-Hall-\ninduced torques and static exchange perturbatively, we\nintroduce the smallness parameter \u000fand let\u000b!\u000f\u000b,\n\u000b0!\u000f\u000b0,jSH\n0!\u000fjSH\n0,!x!\u000f!x. In the following sec-\ntions, a \frst-order perturbation is applied by linearizing\nin\u000fand subsequently setting \u000f= 1.\nWe transformMby the matrixUthat diagonalizes\nM0with eigenvalues ( !0;!0;\u0000!0;\u0000!0). We then ex-\ntract the part corresponding to the real eigenfrequencies,\nwhich yields the following equation:\n\f\f\f\f(D)11(D)12\n(D)21(D)22\f\f\f\f= 0; (31)\nwhereD=U\u00001MU. We thus reduce the fourth-order\nsecular equation in !to a second-order expression. To\nthe \frst order, we \fnd for the parallel ( s= 1) case,\n!P= ~!0+i\u000bP\ne\u000b\n2(2~!H+!M); (32)\nwhere we introduced a current-controlled e\u000bective\nGilbert damping:\n\u000bP\ne\u000b=\u000b+\u000b0(1\u0000F0)\n\u0006s\u0012\n\u000b0G0\u0000i!x\n!0\u00132\n+4(F2\u0000G2)\u0000\njSH\n0\u00012\n(2~!H+!M)2:(33)\nThe imaginary part of the square root in Eq. (33) causes a\n\frst-order real frequency shift that we may disregard, i.e.,\nRe\u0002\n!P\u0003\n\u0019~!0\u0019!0. We thus \fnd two modes with nearly\nthe same frequencies but di\u000berent e\u000bective broadenings.\nThe critical current jSH;P\n0;thris now determined by requir-\ning that\u000bP\ne\u000bvanish, leading to\njSH;P\n0;thr=\u0006q\n(\u000b+\u000b0(1\u0000F0))2\u0000(\u000b0G0)2\n2p\nF2\u0000G2\ns\n1 +\u0012!x=!0\n\u000b+\u000b0(1\u0000F0)\u00132\n(2~!H+!M);(34)\nwhile the critical charge current is jP\nc;thr=jS;P\n0;thr=\u0002SH.\nSpin pumping and spin \rip dissipate energy, leading to a\nhigher threshold current, which is re\rected by 1 \u0000F0\u0015\nG0. The reactive part of the SHE-induced torque ( G)suppresses the e\u000bect of the applied current and thereby\nincreases the critical current as well. The static exchange\ncouples M1andM2, hence increasing jSH;P\n0;thr. The criti-\ncal spin current decreases monotonically with increasing\ndN=lsf, implying that the spin valve (with parallel mag-\nnetization) has a larger threshold current than the FI jN\nbilayer (with thick dN).\nAnalogous to the parallel case, we \fnd two eigenmodes\nfor the antiparallel case ( s=\u00001), with eigenfrequencies\n!AP=!0+\u0012\n\u0006\u0000!x\n2!0+i\u000bAP\ne\u000b\n2\u0013\n(2~!H+!M) (35)\nand corresponding e\u000bective Gilbert damping parameters\n\u000bAP\ne\u000b=\u000b+\u000b0(1\u0000F0)\n\u0006\u000b0G0!M\n2~!H+!M+2\n2~!H+!MFjSH\n0;(36)\nwhich depend on the magnetic con\fguration because the\ndynamic exchange coupling di\u000bers, while the resonance\nfrequency is a\u000bected by the static coupling. In the AP\ncon\fgurations, the spin Hall current acts with the same\nsign on both layers due to the increase/decrease in damp-\ning on both sides depending on the applied current direc-\ntion. The corresponding threshold current is expressed\nas\njSH;AP\n0;thr=\u0000(\u000b+\u000b0(1\u0000F0)) (2~!H+!M)\u0000\u000b0G0!M\n2F;\n(37)\nwithjSH;AP\nc;thr=jSH;AP\n0;thr=\u0002SH. Again, the threshold for\ncurrent-induced excitation is increased by the spin pump-\ning.\nm2M0\nIncreasing current FI1 FI2 Nm2\nm1M0m1\nFIG. 2: (Color online) The acoustic mode for the parallel case\nfor di\u000berent applied currents, ranging from zero to just below\nthe critical current. For large currents the oscillations of the\ntwo FIs become out of phase.\nTo zeroth order in the smallness parameter \u000f, we \fnd\nthat the eigenvectors for the parallel con\fguration take6\nthe form vP= (u;\fu)T, where uis the 2-component\nvector\nu=\u0012\nip\n1 +!M=~!H\n1\u0013\n: (38)\nThe imbalance in the amplitudes of both layers is param-\neterized by\n\f=2jSH\n0F\u0007r\n4(F2\u0000G2)(jSH\n0)2+\u0010\n\u000b0G0\u0000i!x\n!0\u00112\n(2~!H+!M)2\n\u00002jSH\n0G+\u0010\n\u000b0G0\u0000i!x\n!0\u0011\n(2~!H+!M);\n(39)\nwhere\u0007corresponds to the \u0006in Eq. (32). For the sym-\nmetric case, the applied current favors out-of-phase os-\ncillations. It can be demonstrated that in the limit of\nlarge currents and low spin-memory loss, the correspond-\ning amplitude di\u000berence is \f=\u00001, withjSH\n0= 0, and\nan interlayer coupling dominated by either dynamic or\nstatic exchange \f=\u00071, which correspond to an optical\nand an acoustic mode, respectively. We use the labels\n\\acoustic\" and \\optic\"even though the phase di\u000berence\nis not precisely 0 or \u0019due to the static exchange inter-\naction. Note that \f(\u0000jSH\n0) = 1=\f(jSH\n0) is required by\nsymmetry; inverting the current direction is equivalent\nto interchanging FI1 and FI2. For !x= 0,\f(jSH\n0) is a\npole or node depending on the current direction for the\nacoustic mode in which the magnetization in one layer\nvanishes. Above this current, \fchange signs, and both\nmodes have a phase di\u000berence of \u0019. The critical cur-\nrent lies above the current corresponding to the node at\nwhich the acoustic mode becomes unstable. The ballistic\nmodel also supports acoustic and optical modes,19with\nthe optical mode being more e\u000eciently damped.\nIn the antiparallel case, acoustic and optical modes can\nare characterized by amplitudes\nvAP\nA=0\nBB@i!0\n~!H\n1\ni!0\n~!H\n\u000011\nCCA;vAP\nO=0\nBB@i!0\n~!H\n1\n\u0000i!0\n~!H\n11\nCCA; (40)\nwhere the optical (acoustic) mode corresponds to the +(-\n) sign in Eq. (35). The labels optical and acoustic are\nkept because of the di\u000berence in e\u000bective damping; a 180\u000e\nrotation about the yaxis of FI2 map these modes to the\ncorresponding modes for the parallel case.\nWhen the composition of the spin valve is slightly\nasymmetric, the dynamics of the two layers can still be\nsynchronized by the static and dynamic coupling. How-\never, at some critical detuning \u0001 !=!2\u0000!1, this tech-\nnique no longer works, as illustrated by the eigenfrequen-\ncies for the asymmetric spin valve in Fig. 4. Here, we\nemploy YIGjPtjYIG parameters but tune the FMR fre-\nquency of the right YIG layer. In practice, the tuning\ncan be achieved by varying the direction of the applied\nmagnetic \feld.16When the FMR frequencies of the two\nlayers are su\u000eciently close, the precessional motions in\nFI1 FI2 NM0\nFI1 FI2 N(a)\n(b)m1M0m1\nm2m2\nm1M0m1\nM0\nm2\nm2FIG. 3: (Color online) The eigenmodes of the antiparallel con-\n\fguration. (a)/(b) corresponds to the acoustic/optical mode\nof Eq. (40). For the acoustic/optical mode the in-plane/out-\nof-plane component is equal in the two layers, and opposite\nfor the out-of-plane/in-plane component.\nTABLE I: Physical parameters used in the numerical calcu-\nlations\nConstant Value Units\ng?a3:4\u00011015cm\u00002e2=h\n\u001bb5:4\u00011017s\u00001\n4\u0019MSc1750 G\nHint 0:2\u00014\u0019MS G\n\u000bc3\u000110\u00004\nlsf 10 nm\nd1;dN;d210, 5, 10 nm\na) Ref. [20], b) Ref. [21], c) Ref. [22]\nthe two layers lock to each other. The asymmetry in-\ntroduced by higher currents is observed to suppress the\nsynchronization.\nThe non-linear large-angle precession that occurs for\ncurrents above the threshold is not amenable to analyti-\ncal treatments; however, numerical calculations can pro-\nvide some insights. Because the dissipation of YIG is\nvery low the number of oscillations required to achieve\na noticable change in the precession angle is very large.\nTo speed up the calculations and make the results more\nreadable we rescale both g?and\u000bby a factor 0 :005=\u000b, in\nthis way the e\u000bective damping is rescaled. Fig. IV shows\nthe components of the magnetization in the two layers as\na function of time when a large current is switched on for\nan initially parallel magnetization along xwith a slight\ncanting ofMi;y= 0:01 fori= 1;2. We apply a current\njSH\n0=jP;SH\n0;thr= 110% at t= 0. For 5 T.t <40T, the\nprecession is out of phase, and the amplitude gradually\nincreases. At t= 40T, the applied current is ramped up\ntojSH\n0=jP;SH\n0;thr= 130%. At t\u001860T, the precession angle is\nno longer small, and our previous perturbative treatment\nbreaks down. However, we can understand that the right\nlayer precesses with a large angle, while the left layer7\nFIG. 4: (Color online) The lowest resonance frequencies of a\nparallel FI1jNjFI2 spin valve as a function of the detuning of\nthe FMR frequencies of the individual layers and for di\u000berent\ncurrentsjSH\n0=jSH\n0;thr= 0;10%;50% for the solid, black dashed\nand grey dashed lines, respectively. At zero applied current\nthe two layers lock when detuning is small. The current sup-\npresses synchronization almost completely when reaching the\nthreshold value.The inset shows the corresponding broaden-\nings.\nstays close to the initial equilibrium from the opposite\ndirection of the interface spin accumulations \u0016SH\n0.\nV. SUPERLATTICES\nA periodic stack of FIs coupled through Ns supports\nspin wave excitations propagating in the perpendicular\ndirection. The coupling between layers is described by\nEq. (24); however, each FI is coupled through the N lay-\ners to two neighboring layers. The primitive unit cell\nof the superlattice with collinear magnetization is the\nFIjN bilayer for the parallel con\fguration (two bilayers\nin the antiparallel con\fguration). For equivalent satu-\nration magnetizations in all FI layers, we can write for\ni2Z\nMi=si^x+mi; (41)\nwheresi= 1 for the parallel and si= (\u00001)ifor the\nantiparallel ground state. We can then linearize the ex-\npression with respect to the small parameters mi. An\nin-plane charge current causes accumulations of opposite\nsign in each N layer. The long-wavelength excitations of\nthe superlattice magnetization can be treated in the con-\ntinuum limit. Denoting the total thickness of a unit cell\nb=dN+dFI, we \fnd for the parallel case ( si= 1)\n@tm=^x\u0002[!Hm+!Mmz^z+ (\u000b+ 2\u000b0(1\u0000F0\u0000G0))@tm\n\u0000\u000b0G02@t;zzm\u0000!xb2@zzm+ 2jSH\n0Gb@z^x\u0002m\u0003\n:(42)\nFIG. 5: (Color online) Magnetization dynamics for the par-\nallel con\fguration and currents above the threshold. (a)/(b)\nthe magnetization in the left/right layer as a function of time\nin units of T= 2\u0019=! 0. The e\u000bective damping is rescaled by\nlettingg?!g?0:005=\u000band\u000b!\u000b0:005=\u000b. The numeri-\ncal calculation was carried out by a 4th order Runge-Kutta\nmethod with a step size \u0001 t=T=50.\nForm=m0ei(!t\u0000kzz), the linearized dispersion relation\nis\n!=p\n!H(!H+!M) +1\n2(2!H+!M)!x\n!0b2k2\nz\u00002jSH\n0Gbkz\n+i1\n2(2!H+!M)(\u000b+ 2\u000b0(1\u0000F0\u0000G0) +\u000b0G0k2\nzb2):(43)\nThe applied current thus adds a term that is linear in\nkzto the real part of the frequency. The direct e\u000bect\nof the SHE now vanishes because the torques on both\nsides of any FI cancel. However, when m06=0, a net\nspin current \rows normal to the stack, which a\u000bects the\ndispersion. In the ferromagnetic layers, this phenomenon\nis equivalent to a pure strain \feld on the magnetization\nand is therefore non-dissipative. While generating jSH\n0\ncauses Ohmic losses, the magnetization dynamics in this\nlimit do not add to the energy dissipation, explaining\nthe contribution to Re[ !]:In this regime, there are no\nexternal current-induced contributions or instabilities.\nAntiferromagnetic superlattices appear to be di\u000ecult\nto realize experimentally because a staggered external\nmagnetic \feld would be required. The unit cell is dou-\nbled as is the number of variables in the equation of mo-\ntion. Determining the coupling coe\u000ecients from Eq. (26)8\nis straightforward but cumbersome and is not presented\nhere. Naively, one could expect that the SHE-induced\ntorque would act very di\u000berently in the antiferromag-\nnetic case. The SHE acts in a symmetric manner on the\nFI(\")jNjFI(#) system, stabilizing or destabilizing both\nlayers simultaneously. However, similarly to the ferro-\nmagnetic superlattice, the direct SHE vanishes also in\nthe antiferromagnetic superlattice. Each FI is in contact\nwith an N, with spin accumulations of opposite sign on\nthe left and right side of the interfaces, which leads to\nthe same cancellation of the direct SHE-induced torque\npresented for the ferromagnetic superlattice.\nWe can also envision a multilayer in which individual\nmetallic layers can be contacted separately and indepen-\ndently. NjFIjN structures have been predicted to display\na magnon drag e\u000bect through the ferromagnetic \flm,23\ni.e. a current in one layer induces an emf in the other one.\nA drag e\u000bect does also exists in our macrospin model: if\nwe induce dynamics by a current in one layer by the spin\nHall e\u000bect, the spin pumping and inverse spin Hall e\u000bect\ngenerates a current in the other layer, but only above a\ncurrent threshhold.\nWith separate contacts to the layers one may drive op-\nposite currents through neighboring \flms. In that case,\nthe spin currents absorbed by a ferromagnetic layer is\nrelatively twice as large as in the FI jN bilayer, thereby\nreducing the critical currents for the parallel con\fgura-\ntion, but of opposite sign for neighboring magnetic lay-\ners. A staggered current distribution in the superlattice\ndestabilizes the ferromagnetic con\fguration, but it can\nstabilize an antiferromagnetic one even in the absence of\nstatic exchange coupling. This leads to intricate dynam-\nics when competing with an applied magnetic \feld.VI. CONCLUSIONS\nWe study current-induced magnetization dynamics in\nspin valves and superlattices consisting of insulating mag-\nnets separated by metallic spacers with spin Hall ef-\nfect. The current-induced torques experienced by the two\nmagnetic layers in an FI( \")jNjFI(\") spin valve caused by\nthe spin Hall e\u000bect are opposite in sign. A charge current\nin N normal to the magnetization this leads to a damp-\ning and an antidamping, stabilizing one and destabilizing\nthe other magnetization. We calculate the magnetiza-\ntion dynamics when the two layers are exchange coupled\nand in the presence of the dynamic exchange coupling in-\nduced by spin pumping. In an antiparallel con\fguration\nFI(\")jNjFI(#) the interlayer couplings play a minor role\nin the current-induced e\u000bects. The threshold currents at\nwhich self-oscillation occur are higher for parallel than\nantiparallel spin valves. We predict interesting current-\ninduced e\u000bects for superlattices and multilayers in which\nthe metallic spacer layers can be individually contacted.\nAcknowledgments\nH.S. and A. B. acknowledge support from the Re-\nsearch Council of Norway, project number 216700. This\nwork was supported by KAKENHI (Grants-in-Aid for\nScienti\fc Research) Nos. 25247056 and 25220910, FOM\n(Stichting voor Fundamenteel Onderzoek der Materie),\nthe ICC-IMR, the EU-RTN Spinicur, EU-FET grant\nInSpin 612759 and DFG Priority Program 1538 \"Spin-\nCaloric Transport\" (GO 944/4).\n1A. Brataas, A. D. Kent, and H. Ohno, Nature Mat., 373\n(2012).\n2H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida, Y.\nKajiwara, 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., 110206601 (2013).\n3Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida,\nM. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K.\nTakanashi, S. Maekawa, and E. 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Physics for Scientists and Engineers with\nModern Physics (4th ed.), (2009) [1984]\n22A. A. Serga, A. V. Chumak and B. Hillebrands, J. Phys.\nD: App. Phys. 43, 264002 (2010)\n23Steven S.-L. Zhang and S. Zhang, Phys. Rev. Lett. 109,\n096603 (2012).\nAppendix A: Matrix Elements\nHere, we derive the response coe\u000ecients F,G,F0and\nG0that determine the torques, depending on the prop-\nerties of the normal metal. Let us \frst discuss the coef-\n\fcients related to the torques induced by the SHE. The\nfunctionsFandGare extracted from the derivatives of\nEq. (23) with respect to the transverse components of the\ndynamic magnetizations mi.Fgoverns the SHE-induced\ntorque in one layer due to displaced magnetization in the\nsame layer and can be computed as\n@(\u001cSH\n1)y\n@m1;y=@(\u001cSH\n1)z\n@m1;z=\u0000@(\u001cSH\n2)y\n@(sm2;y)=\u0000@(\u001cSH\n2)z\n@(sm2;z)=\u0000FjSH\n0:\n(A1)\nThus,\nF=\r~\n2e2MSdg?2elsf\n\u001btanh(dN=2lsf)\n\u0014\n1\u00001\n2\u00001(dN=2)\u00001\n2\u00002(dN=2)\u0015\n:(A2)\nSimilarly, we can identify G, which governs the cross-\ncorrelation of the SHE-induced torque in one layer arising\nfrom a displaced magnetization in the other layer from\n@(\u001cSH\n1)y\n@(sm2;y)=@(\u001cSH\n1)z\n@(sm2;z)=\u0000@(\u001cSH\n2)y\n@m1;y=\u0000@(\u001cSH\n2)z\n@m1;z=GjSH\n0;\n(A3)Thus\nG=\r~\n2e2MSdg?2elsf\n\u001btanh(dN=2lsf)\n1\n2[\u00001(dN=2)\u0000\u00002(dN=2)]: (A4)\nTorques generated by spin pumping contain terms of the\nform^ x\u0002miand couple the y- andz-components of the\nmagnetization dynamics. We \fnd\n@(\u001cST\n1)y\n@_m1;z=\u0000@(\u001cST\n1)z\n@_m1;y=@(\u001cST\n2)y\n@(s_m2;z)=\u0000@(\u001cST\n2)z\n@(s_m2;y)=F0\u000b0;\n(A5)\nwhere\n2F0= \u00001(dN=2) + \u0000 2(dN=2): (A6)\nSimilarly,\n@(\u001cST\n1)y\n@(s_m2;z)=\u0000@(\u001cST\n1)z\n@(s_m2;y)=@(\u001cST\n2)y\n@_m1;z=\u0000@(\u001cST\n2)z\n@_m1;y=G0\u000b0;\n(A7)\nwhere\n2G0= \u00001(dN=2)\u0000\u00002(dN=2): (A8)\nWe \fnally note that some of the coe\u000ecients are related:\nG\nG0\u000b0=1\n~2elsf\n\u001btanh(dN=2lsf): (A9)"    },    {        "title": "1508.07118v3.The_inviscid_limit_for_the_Landau_Lifshitz_Gilbert_equation_in_the_critical_Besov_space.pdf",        "content": "arXiv:1508.07118v3  [math.AP]  25 Aug 2016THE INVISCID LIMIT FOR THE LANDAU-LIFSHITZ-GILBERT\nEQUATION IN THE CRITICAL BESOV SPACE\nZIHUA GUO AND CHUNYAN HUANG\nAbstract. We prove that in dimensions three and higher the Landau-Lifshitz-\nGilbert equation with small initial data in the critical Besov space is glob ally well-\nposed in a uniform way with respect to the Gilbert damping parameter . Then we\nshow that the global solution converges to that of the Schr¨ oding er maps in the\nnatural space as the Gilbert damping term vanishes. The proof is ba sed on some\nstudies on the derivative Ginzburg-Landau equations.\n1.Introduction\nInthis paperwe study theCauchy problemfortheLandau-Lifshitz -Gilbert (LLG)\nequation\n∂ts=as×∆s−εs×(s×∆s), s(x,0) =s0(x), (1.1)\nwheres(x,t) :Rn×R→S2⊂R3,×denotes the wedge product in R3,a∈R\nandε >0 is the Gilbert damping parameter. The equation (1.1) is one of the\nequations of ferromagnetic spin chain, which was proposed by Land au-Lifshitz [19]\nin studying the dispersive theory of magnetisation of ferromagnet s. Later on, such\nequations were also found in the condensed matter physics. The LL G equation has\nbeen studied extensively, see [17, 7] for an introduction on the equ ation.\nFormally, if a= 0, then (1.1) reduces to the heat flow equations for harmonic\nmaps\n∂ts=−εs×(s×∆s), s(x,0) =s0(x), (1.2)\nand ifε= 0, then (1.1) reduces to the Schr¨ odinger maps\n∂ts=as×∆s, s(x,0) =s0(x). (1.3)\nBoth special cases have been objects of intense research. The p urpose of this paper\nis to study the inviscid limit of (1.1), namely, to prove rigorously that t he solutions\nof (1.1) converges to the solutions of (1.3) as ε→0 under optimal conditions on\nthe initial data.\nThe inviscid limit is an important topic in mathematical physics, and has b een\nstudied in various settings, e.g. for hyperbolic-dissipative equation s such as Navier-\nStokes equation to Euler equation (see [11] and references ther ein), for dispersive-\ndissipative equations such as KdV-Burgers equation to KdV equatio n (see [9]) and\nGinzburg-Landau equation to Schr¨ odinger equations (see [23, 12 ]). The LLG equa-\ntion (1.1) is an equation with both dispersive and dissipative effects. T his can be\n2010Mathematics Subject Classification. 35Q55.\nKey words and phrases. Landau-Lifshitz-Gilbert equation, Schr¨ odinger maps, Inviscid limit ,\nCritical Besov Space.\n12 Z. GUO AND C. HUANG\nseen from the stereographic projection transform. It was know n that (see [18]) let\nu=P(s) =s1+is2\n1+s3, (1.4)\nwheres= (s1,s2,s3) is a solution to (1.1), then usolves the following complex\nderivative Ginzburg-Landau type equation\n(i∂t+∆−iε∆)u=2a¯u\n1+|u|2n/summationdisplay\nj=1(∂xju)2−2iε¯u\n1+|u|2n/summationdisplay\nj=1(∂xju)2\nu(x,0) =u0.(1.5)\nOn the other hand, the projection transform has an inverse\nP−1(u) =/parenleftbiggu+ ¯u\n1+|u|2,−i(u−¯u)\n1+|u|2,1−|u|2\n1+|u|2/parenrightbigg\n. (1.6)\nTherefore, (1.1) is equivalent to (1.5) assuming PandP−1is well-defined, and we\nwill focus on (1.5). The previous works [13, 14, 1, 2, 3] on the Schr¨ odinger maps\n(ε= 0) were also based on this transform. Note that (1.5) is invariant u nder the\nfollowing scaling transform: for λ >0\nu(x,t)→u(λx,λ2t), u0(x)→u0(λx).\nThus the critical Besov space is ˙Bn/2\n2,1in the sense of scaling.\nTo study the inviscid limit, the crucial task is to obtain uniform well-pos edness\nwith respect to the inviscid parameter. Energy method was used in [1 1]. For\ndispersive-dissipative equations, one needs to exploit the dispersiv e effect uniformly.\nStrichartz estimates and energy estimates were used in [23] for Gin zburg-Landau\nequations, andBourgainspacewasused in[9]forKdV-Burgersequ ations. In[24,10]\nthe inviscid limit for the derivative Ginzburg-Landau equations were s tudied by us-\ning the Strichartz estimates, local smoothing estimates and maxima l function esti-\nmates. However, these results requires high regularities when app lied to equation\n(1.5). In this paper we will use Bourgain-type space and exploit the n ull structure\nthat are inspired by the latest development for the Schr¨ odinger m aps (ε= 0) (see\n[4, 3, 1, 2, 13, 14, 8]) to study (1.5) with small initial data in the critica l Besov\nspace. In [2] and [14] it was proved independently that global well-p osedness for\n(1.3) holds for small data in the critical Besov space. We will extend t heir results\nto (1.1) uniformly with respect to ε. We exploit the Bourgain space in a differ-\nent way from both [2] and [14]. One of the novelties is the use of X0,1-structure\nthat results in many simplifications even for the Schr¨ odinger maps. The presence\nof dissipative term brings many technical difficulties, e.g. the lack of s ymmetry\nin time and incompatibility with Xs,bstructure. We need to overcome these dif-\nficulties when extending the linear estimates for the Schr¨ odinger e quation to the\nSchr¨ odinger-dissipative equation uniformly with respect to ε.\nBy scaling we may assume a=±1. From now on, we assume a= 1 since the\nother case a=−1 is similar. For Q∈S2, the space ˙Bs\nQis defined by\n˙Bs\nQ=˙Bs\nQ(Rn;S2) ={f:Rn→R3;f−Q∈˙Bs\n2,1,|f(x)| ≡1 a.e. in Rn},\nwhere˙Bs\n2,1is the standard Besov space. It was known the critical space is ˙Bn/2\nQ.\nThe main result of this paper isLANDAU-LIFSHITZ EQUATION 3\nTheorem 1.1. Assumen≥3. The LLG equation (1.1)is globally well-posed for\nsmall data s0∈˙Bn/2\nQ(Rn;S2),Q∈S2in a uniform way with respect to ε∈(0,1].\nMoreover, for any T >0, the solution converges to that of Schr¨ odinger map (1.3)\ninC([−T,T] :˙Bn/2\nQ)asε→0.\nAs we consider the inviscid limit in the strongest topology (same space as the\ninitial data), no convergence rate is expected. This can be seen fr om linear solutions\nfor (1.5). However, if assuming initial data has higher regularity, on e can have\nconvergence rate O(εT) (see (5.3) below).\n2.Definitions and Notations\nForx,y∈R,x/lessorsimilarymeans that there exists a constant Csuch that x≤Cy, and\nx∼ymeans that x/lessorsimilaryandy/lessorsimilarx. We use F(f),ˆfto denote the space-time Fourier\ntransform of f, andFxi,tfto denote the Fourier transform with respect to xi,t.\nLetη:R→[0,1] be an even, non-negative, radially decreasing smooth function\nsuch that: a) ηis compactly supported in {ξ:|ξ| ≤8/5}; b)η≡1 for|ξ| ≤5/4. For\nk∈Zletχk(ξ) =η(ξ/2k)−η(ξ/2k−1),χ≤k(ξ) =η(ξ/2k),/tildewideχk(ξ) =/summationtext9n\nl=−9nχk+l(ξ),\nand then define the Littlewood-Paley projectors Pk,P≤k,P≥konL2(Rn) by\n/hatwidestPku(ξ) =χk(|ξ|)/hatwideu(ξ),/hatwideP≤ku(ξ) =χ≤k(|ξ|)/hatwideu(ξ),\nandP≥k=I−P≤k−1,P[k1,k2]=/summationtextk2\nj=k1Pj. We also define /tildewidePku=F−1/tildewideχk(|ξ|)/hatwideu(ξ)\nLetSn−1be the unit sphere in Rn. Fore∈Sn−1, define /hatwidePk,eu(ξ) =/tildewideχk(|ξ·\ne|)χk(|ξ|)/hatwideu(ξ). Since for |ξ| ∼2kwe have\n5n/summationdisplay\nl=−5nχk+l(ξ1)+···+5n/summationdisplay\nl=−5nχk+l(ξn)∼1,\nthen let\nβj\nk(ξ) =/summationtext5n\nl=−5nχk+l(ξj)\n/summationtextn\nj=1/summationtext5n\nl=−5nχk+l(ξj)·1/summationdisplay\nl=−1χk+l(|ξ|), j= 1,···,n.\nDefine the operator Θj\nkonL2(Rn) by/hatwidestΘj\nkf(ξ) =βj\nk(ξ)ˆf(ξ), 1≤j≤n. Lete1=\n(1,0,···,0),···,en= (0,···,0,1). Then we have\nPk=n/summationdisplay\nj=1Pk,ejΘj\nk. (2.1)\nFor anyk∈Z, we define the modulation projectors Qk,Q≤k,Q≥konL2(Rn×R) by\n/hatwidestQku(ξ,τ) =χk(τ+|ξ|2)/hatwideu(ξ,τ),/hatwideQ≤ku(ξ,τ) =χ≤k(τ+|ξ|2)/hatwideu(ξ,τ),\nandQ≥k=I−Q≤k−1,Q[k1,k2]=/summationtextk2\nj=k1Qj.\nFor anye∈Sn−1, we can decompose Rn=λe⊕He, whereHeis the hyperplane\nwith normal vector e, endowed with the induced measure. For 1 ≤p,q <∞, we\ndefineLp,q\nethe anisotropic Lebesgue space by\n/ba∇dblf/ba∇dblLp,q\ne=/parenleftBigg/integraldisplay\nR/parenleftbigg/integraldisplay\nHe×R|f(λe+y,t)|qdydt/parenrightbiggp/q\ndλ/parenrightBigg1/p4 Z. GUO AND C. HUANG\nwith the usual definition if p=∞orq=∞. We write Lp,q\nej=Lp\nxjLq\n¯xj,t. We use\n˙Bs\np,qto denote the homogeneous Besov spaces on Rnwhich is the completion of the\nSchwartz functions under the norm\n/ba∇dblf/ba∇dbl˙Bsp,q= (/summationdisplay\nk∈Z2qsk/ba∇dblPkf/ba∇dblq\nLp)1/q.\nTo exploit the null-structure we also need the Bourgain-type space associated to\ntheSchr¨ odinger equation. Inthis paperwe use themodulation-ho mogeneousversion\nas in [2, 8]. We define X0,b,qto be the completion of the space of Schwartz functions\nwith the norm\n/ba∇dblf/ba∇dblX0,b,q= (/summationdisplay\nk∈Z2kbq/ba∇dblQkf/ba∇dblq\nL2\nt,x)1/q. (2.2)\nIfq= 2 we simply write X0,b=X0,b,2. By the Plancherel’s equality we have\n/ba∇dblf/ba∇dblX0,1=/ba∇dbl(i∂t+ ∆)f/ba∇dblL2\nt,x. SinceX0,b,qis not closed under conjugation, we also\ndefine the space ¯X0,b,qby the norm /ba∇dblf/ba∇dbl¯X0,b,q=/ba∇dbl¯f/ba∇dblX0,b,q, and similarly write ¯X0,b=\n¯X0,b,2. It’s easy to see that X0,b,qfunction is unique modulo solutions of the homo-\ngeneous Schr¨ odinger equation. For a more detailed description of theX0,b,pspaces\nwe refer the readers to [21] and [20]. We use X0,b,p\n+to denote the space restricted to\nthe interval [0 ,∞):\n/ba∇dblf/ba∇dblX0,b,p\n+= inf\n˜f:˜f=font∈[0,∞)/ba∇dbl˜f/ba∇dblX0,b,p.\nIn particular, we have\n/ba∇dblf/ba∇dblX0,1\n+∼ /ba∇dbl˜f/ba∇dblX0,1 (2.3)\nwhere˜f=f(t,x)1t≥0+f(−t,x)1t<0.\nLetL=∂t−i∆ and¯L=∂t+i∆. We define the main dyadic function space. If\nf(x,t)∈L2(Rn×R+) has spatial frequency localized in {|ξ| ∼2k}, define\n/ba∇dblf/ba∇dblFk=/ba∇dblf/ba∇dblX0,1/2,∞\n++/ba∇dblf/ba∇dblL∞\ntL2x+/ba∇dblf/ba∇dbl\nL2\ntL2n\nn−2\nx\n+2−(n−1)k/2sup\ne∈Sn−1/ba∇dblf/ba∇dblL2,∞\ne+2k/2sup\n|j−k|≤20sup\ne∈Sn−1/ba∇dblPj,ef/ba∇dblL∞,2\ne,\n/ba∇dblf/ba∇dblYk=/ba∇dblf/ba∇dblL∞\ntL2x+/ba∇dblf/ba∇dbl\nL2\ntL2n\nn−2\nx+2−(n−1)k/2sup\ne∈Sn−1/ba∇dblf/ba∇dblL2,∞\ne\n+2−kinf\nf=f1+f2(/ba∇dblLf1/ba∇dblL2\nt,x+/ba∇dbl¯Lf2/ba∇dblL2\nt,x),\n/ba∇dblf/ba∇dblZk=2−k/ba∇dblLf/ba∇dblL2\nt,x\n/ba∇dblf/ba∇dblNk= inf\nf=f1+f2+f3(/ba∇dblf1/ba∇dblL1\ntL2x+2−k/2sup\ne∈Sn−1/ba∇dblf2/ba∇dblL1,2\ne+/ba∇dblf3/ba∇dblX0,−1/2,1\n+)+2−k/ba∇dblf/ba∇dblL2\nt,x.\nThen we define the space Fs,Ys,Zs,Nswith the following norm\n/ba∇dblu/ba∇dblFs=/summationdisplay\nk∈Z2ks/ba∇dblPku/ba∇dblFk,/ba∇dblu/ba∇dblYs=/summationdisplay\nk∈Z2ks/ba∇dblPku/ba∇dblYk,\n/ba∇dblu/ba∇dblZs=/summationdisplay\nk∈Z2ks/ba∇dblPku/ba∇dblZk,/ba∇dblu/ba∇dblNs=/summationdisplay\nk∈Z2ks/ba∇dblPku/ba∇dblNk.\nObviously, Fk∩Zk⊂Yk, and thus Fs∩Zs⊂Ys. In the end of this section, we\npresent a standard extension lemma (See Lemma 5.4 in [22]) giving the r elation\nbetween Xs,band other space-time norm.LANDAU-LIFSHITZ EQUATION 5\nLemma 2.1. Letk∈ZandBbe a space-time norm satisfying with some C(k)\n/ba∇dbleit0eit∆Pkf/ba∇dblB≤C(k)/ba∇dblPkf/ba∇dbl2\nfor anyt0∈Randf∈L2. Then\n/ba∇dblPku/ba∇dblB/lessorsimilarC(k)/ba∇dblPku/ba∇dblX0,1/2,1.\n3.Uniform linear estimates\nIn this section we prove some uniform linear estimates for the equat ion (1.5) with\nrespect to the dissipative parameter. First we recall the known line ar estimates for\nthe Schr¨ odinger equation, see [16] and [14].\nLemma 3.1. Assumen≥3. For any k∈Zwe have\n/ba∇dbleit∆Pkf/ba∇dbl\nL2\ntL2n\nn−2\nx∩L∞\ntL2x/lessorsimilar/ba∇dblPkf/ba∇dbl2, (3.1)\nsup\ne∈Sn−1/ba∇dbleit∆Pkf/ba∇dblL2,∞\ne/lessorsimilar2(n−1)k\n2/ba∇dblPkf/ba∇dbl2, (3.2)\nsup\ne∈Sn−1/ba∇dbleit∆Pk,ef/ba∇dblL∞,2\ne/lessorsimilar2−k\n2/ba∇dblPkf/ba∇dbl2. (3.3)\nLemma 3.2. Assumen≥3,u,Fsolves the equation: for ε >0\nut−i∆u−ε∆u=F(x,t), u(x,0) =u0.\nThen for b∈[1,∞]\n/ba∇dblPku/ba∇dblX0,1/2,b\n+/lessorsimilar/ba∇dblPku0/ba∇dblL2+/ba∇dblPkF/ba∇dblX0,−1/2,b\n+, (3.4)\n/ba∇dblPku/ba∇dblZk/lessorsimilarε1/2/ba∇dblPku0/ba∇dblL2+2−k/ba∇dblPkF/ba∇dblL2\nt,x, (3.5)\nwhere the implicit constant is independent of ε.\nProof.First we show the second inequality. We have\nu=eit∆+εt∆u0+/integraldisplayt\n0ei(t−s)∆+ε(t−s)∆F(s)ds (3.6)\nand thus\n/ba∇dblPku/ba∇dblZk/lessorsimilarε2k/ba∇dblPku/ba∇dblL2\nt,x+2−k/ba∇dblPkF/ba∇dblL2\nt,x\n/lessorsimilarε1/2/ba∇dblPku0/ba∇dblL2+2−k/ba∇dblPkF/ba∇dblL2\nt,x.\nNow we show the first inequality. We only prove the case b=∞since the other\ncases aresimilar. First we assume F= 0. Then u=eit∆+εt∆u0. Let ˜u=eit∆+ε|t|∆u0,\nthen ˜uis an extension of u. Then\n/ba∇dblPku/ba∇dblX0,1/2,∞\n+/lessorsimilarsup\nj2j/2/ba∇dblFt(e−it|ξ|2−ε|t|·|ξ|2)(τ)ˆu0(ξ)χk(ξ)χj(τ+|ξ|2)/ba∇dblL2\nτ,ξ\n/lessorsimilarsup\nj2j/2/ba∇dbl(ε|ξ|2)−1/parenleftbigg\n1+|τ+|ξ|2|2\n(ε|ξ|2)2/parenrightbigg−1\nˆu0(ξ)χk(ξ)χj(τ+|ξ|2)/ba∇dblL2\nτ,ξ\n/lessorsimilar/ba∇dblPku0/ba∇dbl2.\nNext we assume u0= 0. Fix an extension ˜FofFsuch that\n/ba∇dbl˜F/ba∇dblX0,−1/2,∞≤2/ba∇dblF/ba∇dblX0,−1/2,∞\n+.6 Z. GUO AND C. HUANG\nThen define ˜ u=F−11\nτ+|ξ|2+iε|ξ|2F˜F. We see ˜ uis an extension of uand then\n/ba∇dblPku/ba∇dblX0,1/2,∞\n+/lessorsimilar/ba∇dbl˜u/ba∇dblX0,1/2,∞\n/lessorsimilarsup\nj2j/2/ba∇dblχk(ξ)χj(τ+|ξ|2)1\nτ+|ξ|2+iε|ξ|2F˜F/ba∇dblL2\nτ,ξ\n/lessorsimilar/ba∇dbl˜F/ba∇dblX0,−1/2,∞/lessorsimilar/ba∇dblF/ba∇dblX0,−1/2,∞\n+.\nThus we complete the proof. /square\nLemma 3.3. Letn≥3,k∈Z,ε≥0. Assume u,Fsolves the equation\nut−i∆u−ε∆u=F(x,t), u(x,0) =u0.\nThen for any e∈Sn−1we have\n/ba∇dblPku/ba∇dblL2,∞\ne/lessorsimilar2k(n−1)/2/ba∇dblu0/ba∇dblL2+2k(n−2)/2sup\ne∈Sn−1/ba∇dblF/ba∇dblL1,2\ne, (3.7)\n/ba∇dblPk,eu/ba∇dblL∞,2\ne/lessorsimilar2−k/2/ba∇dblu0/ba∇dblL2+2−ksup\ne∈Sn−1/ba∇dblF/ba∇dblL1,2\ne, (3.8)\nwhere the implicit constant is independent of ε.\nProof.Ifε= 0, then the inequalities were proved in [14]. By the scaling and\nrotational invariance, we may assume k= 0 and e= (1,0,···,0). Then the second\ninequality follows from Proposition 2.5, 2.7 in [10]. We prove the first ineq uality by\nthe following two steps.\nStep 1: F= 0.\nFrom the fact\n|eεt∆f(·,t)(x)| ≤(εt)−n/2/integraldisplay\ne−|x−y|2\n2εt|f(y,t)|dy\n/lessorsimilar(εt)−n/2/integraldisplay\ne−|x−y|2\n2εt/ba∇dblf(y,t)/ba∇dblL∞\ntdy\nwe get\n/ba∇dbleit∆+εt∆P0u0/ba∇dblL2x1L∞\n¯x1,t/lessorsimilar/ba∇dbleit∆P0u0/ba∇dblL2x1L∞\n¯x1,t/lessorsimilar/ba∇dblu0/ba∇dbl2.\nStep 2: u0= 0.\nDecompose P0u=U1+···Unsuch that FxUiis supported in {|ξ| ∼1 :|ξi| ∼\n1}×R. Thus it suffices to show\n/ba∇dblUi/ba∇dblL2x1L∞\n¯x,t/lessorsimilar/ba∇dblF/ba∇dblL1,2\nei. (3.9)\nWe only show the estimate for U1. We still write u=U1. We assume FxFis\nsupported in {|ξ| ∼1 :ξ1∼1}×R. We have\nu(t,x) =/integraldisplay\nRn+1eitτeixξ\nτ+|ξ|2+iε|ξ|2/hatwideF(ξ,τ)dξdτ\n=/integraldisplay\nRn+1eitτeixξ\nτ+|ξ|2+iε|ξ|2/hatwideF(ξ,τ)(1{−τ−|¯ξ|2∼1}c+1−τ−|¯ξ|2∼1,|τ+|ξ|2|/lessorsimilarε\n+1−τ−|¯ξ|2∼1,|τ+|ξ|2|≫ε)dξdτ\n:=u1+u2+u3.\nForu1, we simply use the Plancherel equality and get\n/ba∇dbl∆u1/ba∇dblL2+/ba∇dbl∂tu1/ba∇dbl2≤ /ba∇dblF/ba∇dbl2,LANDAU-LIFSHITZ EQUATION 7\nand thus by Sobolev embedding and Bernstein’s inequality we obtain th e desired\nestimate. For u2, using the Lemma 2.1, Lemma 3.1 and Bernstein’s inequality we\nget\n/ba∇dblu2/ba∇dblL2x1L∞\n¯x1,t/lessorsimilar/ba∇dblu2/ba∇dbl˙X0,1/2,1/lessorsimilarε−1/2/ba∇dbl/hatwideF/ba∇dblL2/lessorsimilar/ba∇dblF/ba∇dblL1,2\ne1.\nNow we estimate u3. Since|τ+|ξ|2| ≫ε, we have\n1\nτ+|ξ|2+iε|ξ|2=1\nτ+|ξ|2+∞/summationdisplay\nk=1(−iε|ξ|2)k\n(τ+|ξ|2)k+1.\nMoreover, let s= (−τ−|¯ξ|2)1/2. Thenτ+|ξ|2=−(s−ξ1)(s+ξ1), and thus we get\n|s−ξ1| ≫ε,|s+ξ1| ∼1\n(τ+|ξ|2)−1=−1\n2s(1\ns−ξ1+1\ns+ξ1) =−1\n2s(s−ξ1)(1+s−ξ1\ns+ξ1).\nHence\n1\nτ+|ξ|2+iε|ξ|2=1\nτ+|ξ|2+∞/summationdisplay\nk=1(−iε|ξ|2)k\n(2s(ξ1−s))k+1\n+∞/summationdisplay\nk=1(−iε|ξ|2)k\n(2s(ξ1−s))k+1[(1+s−ξ1\ns+ξ1)k+1−1]\n:=a1(ξ,τ)+a2(ξ,τ)+a3(ξ,τ).\nInserting this identity into the expression of u3, then we have u3=u1\n3+u2\n3+u3\n3,\nwhere\nuj\n3=/integraldisplay\nRn+1eitτeixξaj(ξ,τ)/hatwideF(ξ,τ)1−τ−|¯ξ|2∼1,|τ+|ξ|2|≫εdξdτ, j = 1,2,3.\nForu1\n3, this corresponds to the case ε= 0 which is proved in [14]. For u3\n3, we can\ncontrol it similarly as u1, since\n|a3(ξ,τ)|/lessorsimilar∞/summationdisplay\nk=1εk|ξ|2k\n(2|s(s−ξ1)|)k+1(k+1)|s−ξ1|\n|s+ξ1|/lessorsimilar1.\nIt remains to control u2\n3. LetGk(x1,¯ξ,τ) =F−1\nx11−τ−|¯ξ|2∼11|ξ|∼1|ξ|2k/hatwideF. Note that\n/ba∇dblGk/ba∇dblL1,2\ne1/lessorsimilarck/ba∇dblF/ba∇dblL1,2\ne1.\nThen\nu2\n3=∞/summationdisplay\nk=1(−iε)k/integraldisplay\nR/integraldisplay\nRn+1eitτeixξ1|s−ξ1|≫ε\n(2s(ξ1−s))k+1[e−iy1ξ1Gk(y1,¯ξ,τ)]dξdτdy 1\n=∞/summationdisplay\nk=1(−iε)k/integraldisplay\nRTk\ny1(G(y1,·))(t,x)dy1\nwhere\nTk\ny1(f)(t,x) =/integraldisplay\nRn+1eitτeixξ1|s−ξ1|≫ε\n(2s(ξ1−s))k+1[e−iy1ξ1f(y1,¯ξ,τ)]dξdτ.8 Z. GUO AND C. HUANG\nWe have\nTk\ny1(f)(t,x) =/integraldisplay\nRn/integraldisplay\nRei(x1−y1)ξ11|s−ξ1|≫ε\n(ξ1−s)k+1dξ1·(2s)−k−1eitτei¯x·¯ξ[f(y1,¯ξ,τ)]d¯ξdτ\n=/integraldisplay\nRei(x1−y1)ξ11|ξ1|≫ε\n(ξ1)k+1dξ1·/integraldisplay\nRnei(x1−y1)s(2s)−k−1eitτei¯x·¯ξ[f(y1,¯ξ,τ)]d¯ξdτ.\nThen we get\n|Tk\ny1(f)(t,x)|/lessorsimilarM−kε−k/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRnei(x1−y1)s·(2s)−k−1eitτei¯x·¯ξ[f(y1,¯ξ,τ)]d¯ξdτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nMaking a change of variable η1=s=/radicalbig\n−τ−|¯ξ|2,dτ=−2η1dη1, we obtain\n/integraldisplay\nRnei(x1−y1)s·(2s)−k−1eitτei¯x·¯ξ[f(y1,¯ξ,τ)]d¯ξdτ\n=/integraldisplay\nRnei(x1−y1)η1·(2η1)−keit(η2\n1+|¯ξ|2)ei¯x·¯ξ[f(y1,¯ξ,η2\n1+|¯ξ|2)]d¯ξdτ.\nThus, by the linear estimate (see Lemma 3.1) we get\n/ba∇dblTk\ny1(f)/ba∇dbl\nL2x1L∞\n¯x,t/lessorsimilarM−kε−k/ba∇dblf/ba∇dbl2,\nwhich suffices to give the estimate for u2\n3. We complete the proof of the lemma. /square\nLemma 3.4. Letn≥3,k∈Z,ε≥0. Assume u,Fsolves the equation\nut−i∆u−ε∆u=F(x,t), u(x,0) =u0.\nThen for any e∈Sn−1we have\n/ba∇dblPku/ba∇dbl\nL∞\ntL2x∩L2\ntL2n\nn−2\nx/lessorsimilar/ba∇dblu0/ba∇dblL2+/ba∇dblF/ba∇dblNk, (3.10)\nwhere the implicit constant is independent of ε.\nProof.Since we have for t >0\n/ba∇dbleit∆+εt∆u0/ba∇dblL∞x/lessorsimilart−n/2/ba∇dblu0/ba∇dblL1x\n/ba∇dbleit∆+εt∆u0/ba∇dblL2x/lessorsimilar/ba∇dblu0/ba∇dblL2x\nwhere the implicit constant is independent of ε, then by the abstract framework of\nKeel-Tao [16] we get the Strichartz estimates\n/ba∇dblPku/ba∇dbl\nL∞\ntL2x∩L2\ntL2n\nn−2\nx/lessorsimilar/ba∇dblu0/ba∇dblL2+/ba∇dblF/ba∇dblL1\ntL2x,\nwith the implicit constant independent of ε.\nBy the same argument as in Step 2 of the proof of Lemma 3.3, we get\n/ba∇dblPku/ba∇dbl\nL∞\ntL2x∩L2\ntL2n\nn−2\nx/lessorsimilar/ba∇dblu0/ba∇dblL2+2−k/2sup\ne∈Sn−1/ba∇dblF/ba∇dblL1,2\ne.\nOn the other hand, by Lemma 2.1, Lemma 3.1 and Lemma 3.2, we get\n/ba∇dblPku/ba∇dbl\nL∞\ntL2x∩L2\ntL2n\nn−2\nx/lessorsimilar/ba∇dblPku/ba∇dblX0,1/2,1\n+/lessorsimilar/ba∇dblu0/ba∇dblL2+/ba∇dblF/ba∇dblX0,−1/2,1\n+.\nThus we complete the proof. /square\nGathering the above lemmas, we can get the following linear estimates :LANDAU-LIFSHITZ EQUATION 9\nLemma 3.5 (Linear estimates) .Assumen≥3,u,Fsolves the equation: for ε >0\nut−i∆u−ε∆u=F(x,t), u(x,0) =u0.\nThen for s∈R\n/ba∇dblu/ba∇dblFs∩Zs/lessorsimilar/ba∇dblu0/ba∇dbl˙Bs\n2,1+/ba∇dblF/ba∇dblNs, (3.11)\nwhere the implicit constant is independent of ε.\n4.Nonlinear estimates\nIn this section we prove some nonlinear estimates. The nonlinear ter m in the\nLandau-Lifshitz equation is\nG(u) =¯u\n1+|u|2n/summationdisplay\nj=1(∂xju)2.\nBy Taylor’s expansion, if /ba∇dblu/ba∇dbl∞<1 we have\nG(u) =∞/summationdisplay\nk=0¯u(−1)k|u|2kn/summationdisplay\nj=1(∂xju)2.\nSo we will need to do multilinear estimates.\nLemma 4.1. (1) Ifj≥2k−100andXis a space-time translation invariant Banach\nspace, then Q≤jPkis bounded on Xwith bound independent of j,k.\n(2) For any j,k,Q≤jPk,eis bounded on Lp,2\neandQ≤jis bounded on Lp\ntL2\nxfor\n1≤p≤ ∞, with bound independent of j,k.\nProof.See the proof of Lemma 5.4 in [8]. /square\nLemma 4.2. Assumen≥3,k1,k2,k3∈Z. Then\n/ba∇dblPk1uPk2v/ba∇dblL2\nt,x/lessorsimilar2(n−1)k1/22−k2/2/ba∇dblPk1u/ba∇dblYk1+Fk1/ba∇dblPk2v/ba∇dblFk2,(4.1)\n/ba∇dblPk3(Pk1uPk2v)/ba∇dblL2\nt,x/lessorsimilar2(n−2)min( k1,k2,k3)\n2/ba∇dblPk1u/ba∇dblYk1/ba∇dblPk2v/ba∇dblYk2. (4.2)\nProof.For the first inequality, we have\n/ba∇dblPk1uPk2v/ba∇dblL2\nt,x/lessorsimilarn/summationdisplay\nj=1/ba∇dblPk1uPk2,ejΘj\nk2v/ba∇dblL2\nt,x/lessorsimilarn/summationdisplay\nj=1/ba∇dblPk1u/ba∇dblL2,∞\nej/ba∇dblPk2,ejv/ba∇dblL∞,2\nej\n/lessorsimilar2(n−1)k1/22−k2/2/ba∇dblPk1u/ba∇dblYk1+Fk1/ba∇dblPk2v/ba∇dblFk2.\nFor the second inequality, if k3≤min(k1,k2), then\n/ba∇dblPk3(Pk1uPk2v)/ba∇dblL2\nt,x/lessorsimilar2k3(n−2)/2/ba∇dblPk1uPk2v/ba∇dbl\nL2\ntL2n\n2n−2\nx\n/lessorsimilar2k3(n−2)/2/ba∇dblPk1u/ba∇dbl\nL2\ntL2n\nn−2\nx/ba∇dblPk2v/ba∇dblL∞\ntL2x.\nIfk1≤min(k2,k3), then\n/ba∇dblPk3(Pk1uPk2v)/ba∇dblL2\nt,x/lessorsimilar/ba∇dblPk1u/ba∇dblL2\ntL∞x/ba∇dblPk2v/ba∇dblL∞\ntL2x\n/lessorsimilar2k1(n−2)/2/ba∇dblPk1u/ba∇dbl\nL2\ntL2n\nn−2\nx/ba∇dblPk2v/ba∇dblL∞\ntL2x.\nIfk2≤min(k1,k3), the proof is identical to the above case. /square10 Z. GUO AND C. HUANG\nLemma 4.3 (Algebra properties) .Ifs≥n/2, then we have\n/ba∇dbluv/ba∇dblYs/lessorsimilar/ba∇dblu/ba∇dblYs/ba∇dblv/ba∇dblYn/2+/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblYs.\nProof.We only show the case s=n/2. By the embedding ˙Bn/2\n2,1⊂L∞\nxwe get\n/ba∇dblu/ba∇dblL∞\nx,t≤ /ba∇dblu/ba∇dblL∞\nt˙Bn/2\n2,1/lessorsimilar/ba∇dblu/ba∇dblYn/2.\nThe Lebesgue component can be easily handled by para-product de composition and\nH¨ older’s inequality. Now we deal with Xs,b-type space. By (2.3) it suffices to show\n/summationdisplay\nk2nk/22−k/ba∇dblPk(fg)/ba∇dblX0,1+¯X0,1/lessorsimilar/ba∇dblf/ba∇dblYn/2/ba∇dblg/ba∇dblYn/2, (4.3)\nWe have\n/summationdisplay\nk2nk/22−k/ba∇dblPk(fg)/ba∇dblX0,1+¯X0,1\n/lessorsimilar/summationdisplay\nki2nk3/22−k3/ba∇dblPk3(Pk1fPk2g)/ba∇dblX0,1+¯X0,1\n/lessorsimilar(/summationdisplay\nki:k1≤k2+/summationdisplay\nki:k1>k2)2k3n/22−k3/ba∇dblPk3(Pk1fPk2g)/ba∇dblX0,1+¯X0,1\n:=I+II.\nBy symmetry, we only need to estimate the term I.\nAssumePk1f=Pk1f1+Pk1f2,Pk2g=Pk2g1+Pk2g2such that\n/ba∇dblPk1f1/ba∇dblX0,1+/ba∇dblPk1f2/ba∇dbl¯X0,1/lessorsimilar/ba∇dblPk1f/ba∇dblX0,1+¯X0,1,\n/ba∇dblPk2g1/ba∇dblX0,1+/ba∇dblPk2g2/ba∇dbl¯X0,1/lessorsimilar/ba∇dblPk2g/ba∇dblX0,1+¯X0,1.\nThen we have\nI/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k3/ba∇dblPk3(Pk1fjPk2g1)/ba∇dblX0,1\n+/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k3/ba∇dblPk3(Pk1fjPk2g2)/ba∇dbl¯X0,1\n:=I1+I2.\nWe only estimate the term I1since the term I2can be estimated in a similar way.\nWe have\nI1/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k3/ba∇dblPk3(Pk1fjPk2g1)/ba∇dblX0,1. (4.4)\nFirst we assume k3≤k1+5 in the summation of (4.4). We have\nI1/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k3(/ba∇dblPk3Q≤k1+k2+9(Pk1fjPk2g1)/ba∇dblX0,1\n+/ba∇dblPk3Q≥k1+k2+10(Pk1fjPk2g1)/ba∇dblX0,1)\n:=I11+I12.LANDAU-LIFSHITZ EQUATION 11\nFor the term I11we have\nI11/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12k3n/22−k32k1+k2/ba∇dblPk1fj/ba∇dblL∞\ntLnx/ba∇dblPk2g1/ba∇dbl\nL2\ntL2n\nn−2\nx\n/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12k3n/22−k32k22k1n/2/ba∇dblPk1fj/ba∇dblL∞\ntL2x/ba∇dblPk2g1/ba∇dbl\nL2\ntL2n\nn−2\nx\n/lessorsimilar/ba∇dblf/ba∇dblYn/2/ba∇dblg/ba∇dblYn/2.\nFor the term I12, we need to exploit the nonlinear interactions as in [8]. We have\nFPk3Q≥k1+k2+10(Pk1fjPk2g1)\n=χk3(ξ3)χ≥k1+k2+10(τ3+|ξ3|2)/integraldisplay\nξ3=ξ1+ξ2,τ3=τ1+τ2χk1(ξ1)/hatwidefj(τ1,ξ1)χk2(ξ2)/hatwideg1(τ2,ξ2).\nWe assume j= 1 since j= 2 is similar. On the plane {ξ3=ξ1+ξ2,τ3=τ1+τ2}we\nhave\nτ3+|ξ3|2=τ1+|ξ1|2+τ2+|ξ2|2−H(ξ1,ξ2) (4.5)\nwhereHis the resonance function in the product Pk3(Pk1fjPk2g1)\nH(ξ1,ξ2) =|ξ1|2+|ξ2|2−|ξ1+ξ2|2. (4.6)\nSince|H|/lessorsimilar2k1+k2, then one of Pk1fj,Pk2g1has modulation larger than the output\nmodulation, namely\nmax(|τ1+|ξ1|2|,|τ2+|ξ2|2|)/greaterorsimilar|τ3+|ξ3|2|.\nIfPk1fjhas larger modulation, then\nI12/lessorsimilar/summationdisplay\nki:k1≤k22nk3/22−k3/ba∇dbl2j3/ba∇dblPk3Qj3(Pk1fjPk2g1)/ba∇dblL2\nt,x/ba∇dbll2\nj3≥k1+k2+10\n/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22nk3/22−k3(/summationdisplay\nj3≥k1+k2+1022j3/ba∇dblQ≥j3Pk1fj/ba∇dbl2\nL2\nt,x)1/2/ba∇dblPk2g1/ba∇dblL∞\ntL2x\n/lessorsimilar/summationdisplay\nki:k1≤k22nk32−k3(/ba∇dblPk1f1/ba∇dblX0,1+/ba∇dblPk1f2/ba∇dbl¯X0,1)/ba∇dblPk2g1/ba∇dblYk2\n/lessorsimilar/ba∇dblf/ba∇dblYn/2/ba∇dblg/ba∇dblYn/2.\nIfPk2g1has larger modulation, then\nI12/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k3/ba∇dblPk1fj/ba∇dblL∞\nt,x(/summationdisplay\nj3≥k1+k222j3/ba∇dblPk2Q≥j3g1/ba∇dbl2\nL2\ntL2x)1/2\n/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k32nk1/2/ba∇dblPk1fj/ba∇dblYk1/ba∇dblPk2g1/ba∇dblX0,1\n/lessorsimilar/ba∇dblf/ba∇dblYn/2/ba∇dblg/ba∇dblYn/2.12 Z. GUO AND C. HUANG\nNow we assume k3≥k1+6 in the summation of (4.4) and thus |k2−k3| ≤4. We\nhave\nI1/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k3(/ba∇dblPk3Q≤k1+k2+9(Pk1fjPk2g1)/ba∇dblX0,1\n+/ba∇dblPk3Q≥k1+k2+10(Pk1fjPk2g1)/ba∇dblX0,1)\n:=˜I11+˜I12.\nBy Lemma 4.2 we get\n˜I11/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k3(/ba∇dblPk3Q≤k1+k2+9(Pk1fjPk2g1)/ba∇dblX0,1\n/lessorsimilar/summationdisplay\nki:k1≤k22nk3/22k12(n−2)k1/2/ba∇dblPk1fj/ba∇dblYk1/ba∇dblPk2g1/ba∇dblYk2/lessorsimilar/ba∇dblf/ba∇dblYn/2/ba∇dblg/ba∇dblYn/2.\nFor the term ˜I12, similarly as the term I12, one ofPk1fj,Pk2g1has modulation larger\nthan the output modulation. If Pk1fjhas larger modulation, then\n˜I12/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k3(/summationdisplay\nj322j3/ba∇dblPk1Q≥j3fj/ba∇dbl2\nL2\ntL∞x)1/2/ba∇dblPk2g1/ba∇dblL∞\ntL2x\n/lessorsimilar/ba∇dblf/ba∇dblYn/2/ba∇dblg/ba∇dblYn/2.\nIfPk2g1has larger modulation, then\n˜I12/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k3/ba∇dblPk1fj/ba∇dblL∞\ntL∞x(/summationdisplay\nj322j3/ba∇dblPk2Q≥j3g1/ba∇dbl2\nL2\ntL2x)1/2\n/lessorsimilar/ba∇dblf/ba∇dblYn/2/ba∇dblg/ba∇dblYn/2.\nThus, we complete the proof. /square\nLemma 4.4. We have\n/summationdisplay\nkj2k3(n−2)/2/ba∇dblPk3[un/summationdisplay\ni=1(∂xiPk1v∂xiPk2w)]/ba∇dblL2\nt,x/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.(4.7)\nProof.We have\nLHS of (4.7) /lessorsimilar/summationdisplay\nkj2k3(n−2)/2/ba∇dblPk3[P≥k3−10un/summationdisplay\ni=1(∂xiPk1v∂xiPk2w)]/ba∇dblL2\nt,x\n+/summationdisplay\nkj2k3(n−2)/2/ba∇dblPk3[P≤k3−10un/summationdisplay\ni=1(∂xiPk1v∂xiPk2w)]/ba∇dblL2\nt,x\n:=I+II.LANDAU-LIFSHITZ EQUATION 13\nBy symmetry, we may assume k1≤k2in the above summation. For the term II,\nsincen≥3, then we have\nII/lessorsimilar/ba∇dblu/ba∇dblYn/2/summationdisplay\nkj2k3(n−2)/2/ba∇dbl˜Pk3n/summationdisplay\ni=1(∂xiPk1v∂xiPk2w)]/ba∇dblL2\nt,x\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/summationdisplay\nk1,k3≤k2+52k3(n−2)/22k1+k22[(n−1)k1−k2]/2/ba∇dblPk1v/ba∇dblFk1/ba∇dblPk2w/ba∇dblFk2\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nFor the term I, ifk3≤k2+20, then we get from Lemma 4.2 that\nI/lessorsimilar/summationdisplay\nkj2nk3/22k3(n−2)/2/ba∇dblP≥k3−10u/ba∇dblL∞\ntL2x/ba∇dbln/summationdisplay\ni=1(∂xiPk1v∂xiPk2w)]/ba∇dblL2\nt,x\n/lessorsimilar/summationdisplay\nkj2nk3/22k3(n−2)/22(n+1)k1/22k2/2/ba∇dblP≥k3−10u/ba∇dblL∞\ntL2x/ba∇dblPk1v/ba∇dblFk1/ba∇dblPk2w/ba∇dblFk2\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nIfk3≥k2+20, then uhas frequency ∼2k3, and thus we get\nI/lessorsimilar/summationdisplay\nkj2k3(n−2)/2/ba∇dblPk3u/ba∇dblL∞\ntL2x/ba∇dbln/summationdisplay\ni=1(∂xiPk1v∂xiPk2w)]/ba∇dblL2\ntL∞x\n/lessorsimilar/summationdisplay\nkj2k3(n−2)/2/ba∇dblPk3u/ba∇dblL∞\ntL2x2nk2/2/ba∇dbln/summationdisplay\ni=1(∂xiPk1v∂xiPk2w)]/ba∇dblL2\ntL2x\n/lessorsimilar/summationdisplay\nkj2k3(n−2)/22k1+k22(n−1)k1/22(n−1)k2/2/ba∇dblPk3u/ba∇dblL∞\ntL2x/ba∇dblPk1v/ba∇dblFk1/ba∇dblPk2w/ba∇dblFk2\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nTherefore we complete the proof. /square\nLemma 4.5. We have\n/ba∇dblun/summationdisplay\ni=1(∂xiv∂xiw)/ba∇dblNn/2/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2∩Zn/2/ba∇dblw/ba∇dblFn/2∩Zn/2. (4.8)\nProof.By the definition of Nn/2, theL2component was handled by the previous\nlemma. We only need to control\n/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1un/summationdisplay\ni=1(Pk2∂xiv∂xiPk3w)]/ba∇dblNk4. (4.9)14 Z. GUO AND C. HUANG\nBy symmetry we may assume k2≤k3in the above summation. If in the above\nsummation we assume k4≤k1+40, then\n(4.9)/lessorsimilar/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1uPk2∂xiv∂xiPk3w]/ba∇dblL1\ntL2x\n/lessorsimilar/summationdisplay\nki2k1n/2/ba∇dblPk1u/ba∇dblL∞\ntL2x2k2/ba∇dblPk2v/ba∇dblL2\ntL∞x2k3/ba∇dblPk3w/ba∇dblL2\ntL∞x\n/lessorsimilar/summationdisplay\nki2k1n/2/ba∇dblPk1u/ba∇dblL∞\ntL2x2k2n/2/ba∇dblPk2v/ba∇dbl\nL2\ntL2n\nn−2\nx2k3n/2/ba∇dblPk3w/ba∇dbl\nL2\ntL2n\nn−2\nx\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nThus from now on we assume k4≥k1+ 40 in the summation of (4.9). We bound\nthe summation case by case.\nCase 1: k2≤k1+20\nIn this case we have k4≥k2+20 and hence |k4−k3| ≤5. By Lemma 4.2 we get\n(4.9)/lessorsimilar/summationdisplay\nki2k4n/22−k4/2/ba∇dblPk4[Pk1uPk2∂xiv∂xiPk3w]/ba∇dblL1,2\ne\n/lessorsimilar/summationdisplay\nki2k4n/22−k4/2/ba∇dblPk1uPk3∂xiw/ba∇dblL2\nx,t/ba∇dblPk2∂xiv/ba∇dblL2,∞\ne\n/lessorsimilar/summationdisplay\nki2k4n/22−k4/22(n−1)k1/22−k3/22(n−1)k2/2/ba∇dblPk1u/ba∇dblYk1/ba∇dblPk3∂xiw/ba∇dblFk3/ba∇dblPk2∂xiv/ba∇dblFk2\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nCase 2: k2≥k1+21\nIn this case we have k4≤k3+40. Let g=/summationtextn\ni=1(Pk2∂xiv·Pk3∂xiw). Then we have\n(4.9)/lessorsimilar/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1uQ≤k2+k3g]/ba∇dblNk4+/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1uQ≥k2+k3g]/ba∇dblNk4\n:=I+II.\nFirst we estimate the term II. We have\nII/lessorsimilar/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1Q≥k2+k3−10u·Q≥k2+k3g]/ba∇dblNk4\n+/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1Q≤k2+k3−10u·Q≥k2+k3g]/ba∇dblNk4\n:=II1+II2.LANDAU-LIFSHITZ EQUATION 15\nFor the term II1we have\nII1/lessorsimilar/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1Q≥k2+k3−10u·Q≥k2+k3g]/ba∇dblL1\ntL2x\n/lessorsimilar/summationdisplay\nki2k4n/2/ba∇dblPk1Q≥k2+k3−10u/ba∇dblL2\ntL∞x/ba∇dblQ≥k2+k3g]/ba∇dblL2\ntL2x\n/lessorsimilar/summationdisplay\nki2k4n/22k1n/2/ba∇dblPk1Q≥k2+k3−10u/ba∇dblL2\ntL2x/ba∇dblQ≥k2+k3g]/ba∇dblL2\ntL2x\n/lessorsimilar/summationdisplay\nki2k4n/22k1n/22−(k2+k3)/ba∇dblPk1Q≥2k1+10u/ba∇dblX0,1\n·2[(n−1)k2−k3]/22k2+k3/ba∇dblPk2v/ba∇dblFk2/ba∇dblPk3w/ba∇dblFk3\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nFor the term II2, sincek4≥k1+ 40, then we may assume ghas frequency of size\n2k4. The resonance function in the product Pk1u·Pk4gis of size /lessorsimilar2k1+k4. Thus the\noutput modulation is of size /greaterorsimilar2k2+k3. Then we get\nII2/lessorsimilar/summationdisplay\nki2k4n/22−(k2+k3)/2/ba∇dblPk4[Pk1Q≤k2+k3−10u·Q≥k2+k3g]/ba∇dblL2\nt,x\n/lessorsimilar/summationdisplay\nki2k4n/22−(k2+k3)/22k1n/2/ba∇dblPk1u/ba∇dblL∞\ntL2x·/ba∇dblg/ba∇dblL2\nt,x\n/lessorsimilar/summationdisplay\nki2k4n/22−(k2+k3)/22k1n/22[(n−1)k2−k3]/22k2+k3/ba∇dblPk1u/ba∇dblYk1/ba∇dblPk2v/ba∇dblFk2/ba∇dblPk3w/ba∇dblFk3\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nNow we estimate the term I. We have\nI/lessorsimilar/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1u·Q≤k2+k3n/summationdisplay\ni=1(Pk2∂xiQ≥k2+k3+40v·Pk3∂xiw)]/ba∇dblNk4\n+/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1u·Q≤k2+k3n/summationdisplay\ni=1(Pk2∂xiQ≤k2+k3+39v·Pk3∂xiw)]/ba∇dblNk4\n:=I1+I2.\nFor the term I1, since the resonance function in the product Pk2v·Pk3wis of size\n/lessorsimilar2k2+k3, then we may assume Pk3whas modulation of size /greaterorsimilar2k2+k3. Then we get\nI1/lessorsimilar/summationdisplay\nki2k4n/22−k4/2/ba∇dblPk4[Pk1u·Q≤k2+k3n/summationdisplay\ni=1(Pk2∂xiQ≥k2+k3+40v·Pk3∂xiQ≥k2+k3−5w)]/ba∇dblL1,2\ne\n/lessorsimilar/summationdisplay\nki2k4n/22−k4/2/ba∇dblPk1u/ba∇dblL∞\nt,x2k2+k3/ba∇dblPk2v/ba∇dblL2,∞\ne/ba∇dblPk3Q≥k2+k3−5w/ba∇dblL2\nt,x\n/lessorsimilar/summationdisplay\nki2k4n/22−k4/22(k2+k3)/22(n−1)k2/22k1n/2/ba∇dblPk1u/ba∇dblYk1/ba∇dblPk2v/ba∇dblFk2/ba∇dblPk3w/ba∇dblFk3\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.16 Z. GUO AND C. HUANG\nFinally, we estimate the term I2. For this term, we need to use the null structure\nobserved by Bejenaru [1]. We can rewrite\n−2∇u·∇v= (i∂t+∆)u·v+u·(i∂t+∆)v−(i∂t+∆)(u·v).(4.10)\nThen we have\nI2=/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1u·Q≤k2+k3(Pk2LQ≤k2+k3+39v·Pk3w)]/ba∇dblNk4\n+/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1u·Q≤k2+k3(Pk2Q≤k2+k3+39v·Pk3Lw)]/ba∇dblNk4\n+/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1u·Q≤k2+k3L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblNk4\n:=I21+I22+I23.\nFor the term I21, we have\nI21/lessorsimilar/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1u·Q≤k2+k3(Pk2LQ≤k2+k3+39v·Pk3w)]/ba∇dblL1\ntL2x\n/lessorsimilar/summationdisplay\nki2k4n/22k1n/2/ba∇dblPk1u/ba∇dblL∞\ntL2x2k2(n−2)/2/ba∇dblPk2Lv/ba∇dblL2\nt,x/ba∇dblPk3w/ba∇dbl\nL2\ntL2n\nn−2\nx\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblZn/2/ba∇dblw/ba∇dblFn/2.\nFor the term I22, we may assume whas modulation /lessorsimilar2k2+k3. Then we get\nI22/lessorsimilar/summationdisplay\nki2k4n/22−k4/2/ba∇dblPk4[Pk1u·Q≤k2+k3(Pk2Q≤k2+k3+39v·Pk3Q≤k2+k3+100Lw)]/ba∇dblL1,2\ne\n/lessorsimilar/summationdisplay\nki2k4n/22−k4/22k1n/2/ba∇dblPk1u/ba∇dblL∞\ntL2x/ba∇dblPk2v/ba∇dblL2,∞\ne/ba∇dblPk3Q≤k2+k3+100Lw)]/ba∇dblL2\nt,x\n/lessorsimilar/summationdisplay\nki2k4n/22−k4/22nk1/22(n−1)k2/22(k2+k3)/2/ba∇dblPk1u/ba∇dblYk1/ba∇dblPk2v/ba∇dblFk2/ba∇dblPk3w/ba∇dblX0,1/2,∞\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nNext we estimate the term I23. We have\nI23/lessorsimilar/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1u·Q[k1+k4+100,k2+k3]L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblNk4\n+/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1u·Q≤k1+k4+99L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblNk4\n:=I231+I232.\nFor the term I232we have\nI232/lessorsimilar/summationdisplay\nki2k4n/22−k4/2/ba∇dblPk4[Pk1u·Q≤k1+k4+99L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblL1,2\ne\n/lessorsimilar/summationdisplay\nki2k4n/22−k4/2/ba∇dblPk1u/ba∇dblL2,∞\ne2k1+k42[(n−1)k2−k3]/2/ba∇dblPk2v/ba∇dblFk2/ba∇dblPk3w/ba∇dblFk3\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.LANDAU-LIFSHITZ EQUATION 17\nFor the term I231we have\nI231/lessorsimilar/summationdisplay\nkik2+k3/summationdisplay\nj2=k1+k4+1002k4n/2/ba∇dblPk4Q≤j2−10[Pk1u·Qj2L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblNk4\n+/summationdisplay\nkik2+k3/summationdisplay\nj2=k1+k4+1002k4n/2/ba∇dblPk4Q≥j2−9[Pk1u·Qj2L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblNk4\n:=I2311+I2312.\nFor the term I2312we have\nI2312/lessorsimilar/summationdisplay\nkik2+k3/summationdisplay\nj2=k1+k4+100/summationdisplay\nj3≥k2−92k4n/22−j3/2\n·/ba∇dblPk4Qj3[Pk1u·Qj2L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblL2\nt,x\n/lessorsimilar/summationdisplay\nki2k4n/22k1n/22(k2+k3)/22[(n−1)k2−k3]/2/ba∇dblPk1u/ba∇dblYk1/ba∇dblPk2v/ba∇dblFk2/ba∇dblPk3w/ba∇dblFk3\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nFor the term I2311we have\nI2311/lessorsimilar/summationdisplay\nkik2+k3/summationdisplay\nj2=k1+k4+1002k4n/2/ba∇dblPk4Q≤j2−10[Pk1˜Qj2u·Qj2L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblL1\ntL2x\n/lessorsimilar/summationdisplay\nkik2+k3/summationdisplay\nj2=k1+k4+1002k4n/22k1n/2/ba∇dblPk1˜Qj2u/ba∇dblL2\nt,x2j22[(n−1)k2−k3]/2/ba∇dblPk2v/ba∇dblFk2/ba∇dblPk3w/ba∇dblFk3\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nTherefore, we complete the proof. /square\nCombining all the estimates above we get\nLemma 4.6 (Nonlinear estimates) .Assumeu∈Fn/2∩Zn/2with/ba∇dblu/ba∇dblYn/2≪1.\nThen/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble¯u\n1+|u|2n/summationdisplay\nj=1(∂xju)2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nNn/2/lessorsimilar/ba∇dblu/ba∇dblYn/2\n1−/ba∇dblu/ba∇dbl2\nYn/2/ba∇dblu/ba∇dblFn/2∩Zn/2/ba∇dblu/ba∇dblFn/2∩Zn/2.\nProof.SinceYn/2⊂L∞, then\n¯u\n1+|u|2n/summationdisplay\nj=1(∂xju)2=∞/summationdisplay\nk=0¯u(−1)k|u|2kn/summationdisplay\nj=1(∂xju)2.\nThe lemma follows from Lemma 4.5, Lemma 4.4 and Lemma 4.3. /square\n5.The limit behaviour\nIn this section we prove Theorem 1.1. It is equivalent to prove\nTheorem 5.1. Assumen≥3,ε∈[0,1]. There exists 0< δ≪1such for any\nφ∈˙Bn/2\n2,1with/ba∇dblφ/ba∇dbl˙Bn/2\n2,1≤δ, there exists a unique global solution uεto(1.5)such that\n/ba∇dbluε/ba∇dblFn/2∩Zn/2/lessorsimilarδ,18 Z. GUO AND C. HUANG\nwhere the implicit constant is independent of ε. The map φ→uεis Lipshitz from\nBδ(˙Bn/2\n2,1)toC(R;˙Bn/2\n2,1)and the Lipshitz constant is independent of ε. Moreover,\nfor anyT >0,\nlim\nε→0+/ba∇dbluε−u0/ba∇dblC([0,T];˙Bn/2\n2,1)= 0.\nFortheuniformglobalwell-posedness, wecanproveitbystandard Picarditeration\nargument by using the linear and nonlinear estimates proved in the pr evious section.\nIndeed, define\nΦu0(u) :=eit∆+εt∆u0\n−i/integraldisplayt\n0ei(t−s)∆+ε(t−s)∆/bracketleftbigg2¯u\n1+|u|2n/summationdisplay\nj=1(∂xju)2−2iε¯u\n1+|u|2n/summationdisplay\nj=1(∂xju)2/bracketrightbigg\nds.\nThenusingtheLemma3.5andLemma4.6wecanshowΦ u0isacontractionmapping\nin the set\n{u:/ba∇dblu/ba∇dblFn/2∩Zn/2≤Cδ}\nif/ba∇dblu0/ba∇dbl˙Bn/2\n2,1≤δwithδ >0sufficiently small. Thus wehaveexistence anduniqueness.\nMoreover, by standard arguments we immediately have the persist ence of regularity,\nnamely if u0∈˙Bs\n2,1for some s > n/2, thenu∈Fs∩Zsand\n/ba∇dblu/ba∇dblFs∩Zs/lessorsimilar/ba∇dblu0/ba∇dbl˙Bs\n2,1(5.1)\nuniformly with respect to ε∈(0,1].\nNow we prove the limit behaviour. Assume uεis a solution to the Landau-Lifshitz\nequation with small initial data φ1∈˙Bn/2\n2,1, anduis a solution to the Schr¨ odinger\nmap with small initial data φ2∈˙Bn/2\n2,1. Letw=uε−u,φ=φ1−φ2, thenwsolves\n(i∂t+∆)w=iε∆uε+/bracketleftbigg2¯uε\n1+|uε|2n/summationdisplay\nj=1(∂xjuε)2−2¯u\n1+|u|2n/summationdisplay\nj=1(∂xju)2/bracketrightbigg\n−2iε¯uε\n1+|uε|2n/summationdisplay\nj=1(∂xjuε)2, (5.2)\nw(0) =φ.\nFirst we assume in addition φ1∈˙B(n+4)/2\n2,1. By the linear and nonlinear estimates,\nfor anyT >0 we get\n/ba∇dblw/ba∇dblFn/2∩Zn/2/lessorsimilar/ba∇dblφ/ba∇dbl˙Bn/2\n2,1+εT/ba∇dbluε/ba∇dblL∞\nt˙B(n+4)/2\n2,1+δ2/ba∇dblw/ba∇dblFn/2∩Zn/2+ε/ba∇dbluε/ba∇dbl3\nFn/2∩Zn/2.\nThen we get by (5.1)\n/ba∇dblw/ba∇dblFn/2∩Zn/2/lessorsimilar/ba∇dblφ/ba∇dbl˙Bn/2\n2,1+εT/ba∇dblφ1/ba∇dbl˙B(n+4)/2\n2,1+εδ3. (5.3)\nNow we assume φ1=φ2=ϕ∈˙Bn/2\n2,1with small norm. For fixed T >0, we need to\nprove that ∀η >0, there exists σ >0 such that if 0 < ε < σthen\n/ba∇dblSε\nT(ϕ)−ST(ϕ)/ba∇dblC([0,T];˙Bn/2\n2,1)< η (5.4)LANDAU-LIFSHITZ EQUATION 19\nwhereSε\nTis the solution map corresponding to (5.2) and ST=S0\nT. We denote\nϕK=P≤Kϕ. Then we get\n/ba∇dblSε\nT(ϕ)−ST(ϕ)/ba∇dblC([0,T];˙Bn/2\n2,1)\n≤/ba∇dblSε\nT(ϕ)−Sε\nT(ϕK)/ba∇dblC([0,T];˙Bn/2\n2,1)\n+/ba∇dblSε\nT(ϕK)−ST(ϕK)/ba∇dblC([0,T];˙Bn/2\n2,1)+/ba∇dblST(ϕK)−ST(ϕ)/ba∇dblC([0,T];˙Bn/2\n2,1).\nFrom the uniform global well-posedness and (5.3), we get\n/ba∇dblSǫ\nT(ϕ)−ST(ϕ)/ba∇dblC([0,T];˙Bn/2\n2,1)/lessorsimilar/ba∇dblϕK−ϕ/ba∇dbl˙Bn/2\n2,1+εC(T,K,/ba∇dblϕ/ba∇dbl˙Bn/2\n2,1).(5.5)\nWe first fix Klarge enough, then let εgo to zero, therefore (5.4) holds.\nAcknowledgment. Z. Guo is supported in part by NNSF of China (No.11371037),\nand C. Huang is supported in part by NNSF of China (No. 11201498).\nReferences\n[1] I. Bejenaru, On Schr¨ odinger maps, Amer. J. Math. 130 (2008 ), 1033-1065.\n[2] I. Bejenaru, Global results for Schr¨ odinger maps in dimensions n≥3, Comm. Partial Differ-\nential Equations 33 (2008), 451-477.\n[3] I. Bejenaru, A. D. Ionescu, and C. E. Kenig, Global existence a nd uniqueness of Schr¨ odinger\nmaps in dimensions d≥4, Adv. Math. 215 (2007), 263-291.\n[4] I. Bejenaru, A. D. Ionescu, C. E. Kenigand D. Tataru, Global S chr¨ odingermapsin dimensions\nd≥2: small data in the critical Sobolevspaces, Annals ofMathematics1 73(2011), 1443-1506.\n[5] N.H. Chang,J.Shatah, K.Uhlenbeck, Schr¨ odingermaps, Com m.PureAppl. Math.53(2000),\n590-602.\n[6] W. Ding and Y. Wang, Local Schr¨ odinger flow into K¨ ahler manifold s, Sci. China Ser. A, 44\n(2001), 1446-1464.\n[7] B. Guo and S. Ding. Landau-Lifshitz equations, volume 1 of Front iers of Research with the\nChinese Academy of Sciences. World Scientific Publishing Co. Pte. Ltd ., Hackensack, NJ,\n2008.\n[8] Z. Guo, Spherically averaged maximal function and scattering fo r the 2D cubic derivative\nSchr¨ odinger equation, to appear Int. Math. Res. Notices.\n[9] Z. Guo and B. Wang, Global well posedness and inviscid limit for the K orteweg-de Vries-\nBurgers equation, J. Differential Equations 246 (2009) 3864-390 1.\n[10] L. Han, B. Wang and B. Guo, Inviscid limit for the derivative Ginzbu rg-Landau equation with\nsmall data in modulation and Sobolev spaces, Appl. Comput. Harmon. Anal. 32(2012),no.2,\n197-222.\n[11] T. Hmidi and S. Keraani, Inviscid limit for the two-dimensional N-S system in a critical Besov\nspace, Asymptot. Anal., 53 (3) (2007), 125-138.\n[12] C. Huang, B. Wang, Inviscid limit for the energy-critical complex Ginzburg-Landau equation,\nJ. Funct. Anal., 255 (2008), 681-725.\n[13] A. D. Ionescu and C. E. Kenig, Low-regularity Schr¨ odinger ma ps, Differential Integral Equa-\ntions 19 (2006), 1271-1300.\n[14] A. D. Ionescu and C. E. Kenig, Low-regularity Schr¨ odinger ma ps, II: global well-posedness in\ndimensions d≥3, Comm. Math. Phys., 271 (2007), 523-559.\n[15] C. E. Kenig, G. Ponce, and L. Vega, Smoothing effects and local existence theory for the\ngeneralized nonlinear Schr¨ odinger equations, Invent. Math., 134 (1998), 489-545.\n[16] M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. M ath., 120 (1998), 360–413.\n[17] M. Lakshmanan. The fascinating world of the Landau-Lifshitz- Gilbert equation: an overview.\nPhilos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 369 (193 9):1280-1300, 2011.\n[18] M. Lakshmanan and K. Nakamura, Landau-Lifshitz Equation of Ferromagnetism: Exact\nTreatment of the Gilbert Damping, Phy. Rev. Let. 53 (1984), NO. 2 6, 2497-2499.\n[19] L. D. Landau and E. M. Lifshitz, On the theory of the dispersion of magnetic permeability in\nferromagnetic bodies, Phys. Z. Sovietunion. 8 (1935) 153-169.20 Z. GUO AND C. HUANG\n[20] T. Tao, Global regularity of wave maps II. Small energy in two dim ensions, Commun. Math.\nPhys. 224 (2001), 443-544.\n[21] D. Tataru, Local andglobalresults forthe wavemaps I, Comm . PartialDifferential Equations,\n23 (1998), no. 9-10, 1781-1793.\n[22] B. Wang, Z. Huo, C. Hao and Z. Guo, Harmonic Analysis Method fo r Nonlinear Evolution\nEquations, I, World Scientific Press, 2011.\n[23] B. Wang, The limit behavior of solutions for the Cauchy problem of the Complex Ginzburg-\nLandau equation, Commu. Pure. Appl. Math., 55 (2002), 0481-050 8.\n[24] B. Wang and Y. Wang, The inviscid limit for the derivative Ginzburg- Landau equations, J.\nMath. Pures Appl., 83 (2004), 477-502.\nSchool of Mathematical Sciences, Monash University, VIC 38 00, Australia &\nLMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China\nE-mail address :zihua.guo@monash.edu\nSchool of Statistics and Mathematics, Central University o f Finance and Eco-\nnomics, Beijing 100081, China\nE-mail address :hcy@cufe.edu.cn"    },    {        "title": "1907.08454v2.A_cryogenic_memory_element_based_on_an_anomalous_Josephson_junction.pdf",        "content": "A cryogenic memory element based on an anomalous Josephson junction\nC. Guarcello1and F.S. Bergeret1, 2\n1Centro de Física de Materiales, Centro Mixto CSIC-UPV/EHU,\nPaseo Manuel de Lardizabal 5, 20018 San Sebastián, Spain\n2Donostia International Physics Center, Paseo Manuel de Lardizabal 4, 20018 San Sebastián, Spain\n(Dated: September 27, 2019)\nWe propose a non-volatile memory element based on a lateral ferromagnetic Josephson junction with spin-\norbit coupling and out-of-plane magnetization. The interplay between the latter and the intrinsic exchange\nfield of the ferromagnet leads to a magnetoelectric effect that couples the charge current through the junction\nand its magnetization, such that by applying a current pulse the direction of the magnetic moment in F can\nbe switched. The two memory states are encoded in the direction of the out-of-plane magnetization. With\nthe aim to determine the optimal working temperature for the memory element, we explore the noise-induced\neffects on the averaged stationary magnetization by taking into account thermal fluctuations affecting both the\nJosephson phase and the magnetic moment dynamics. We investigate the switching process as a function of\nintrinsic parameters of the ferromagnet, such as the Gilbert damping and strength of the spin-orbit coupling, and\nproposed a non-destructive readout scheme based on a dc-SQUID. Additionally, we analyze a way to protect the\nmemory state from external perturbations by voltage gating in systems with a both linear-in-momentum Rashba\nand Dresselhaus spin-orbit coupling.\nI. INTRODUCTION\nSuperconducting electronics is suggested as playing an im-\nportant role in the search of ultra-low-power computers [1–5].\nOne of the key challenges towards this objective is the fabrica-\ntion of a reliable and scalable cryogenic memory architecture.\nSuperconductor-ferromagnet-superconductor (SFS) junctions\nare promising structures suggested for such memories [6–15].\nIndeed, the interplay between the intrinsic exchange field and\nthe induced superconductivity in the ferromagnet leads to the\nso-called p-junction, i.e., a Josephson junction exhibiting an\nintrinsic p-phase shift in its ground state. Vertical ferromag-\nnetic multilayer structures are being used as Josephson mag-\nnetic memories. The two logic states of these memories usu-\nally correspond to states with different relative orientation of\nmagnetic layers, that in turn determines whether the junction\nis in the 0- or p-state. Readout schemes are commonly based\non distinguish between resistive and non-resistive states.\nHere, we suggest an alternative cryogenic memory element\nbased on a ferromagnetic anomalous Josephson junction, usu-\nally called j0-junction [16]. It consists of a SFS Josephson\njunction with a Rashba-like spin-orbit coupling (SOC). Its\nground state corresponds to a finite phase shift in its current-\nphase-relation 0 <j0<p. Such anomalous phase has been\nrecently detected experimentally in hybrid Josephson junc-\ntions fabricated with the topological insulator Bi 2Se3[17] and\nAl/InAs heterostructures [18]. Both materials have strong\nspin-orbit coupling and in those experiments, time reversal is\nbroken by an external magnetic field that acts as a Zeeman\nfield. In our proposal, the memory element is a Josephson\njunction with a ferromagnetic link, thus in principle time-\nreversal is broken intrinsically by the exchange field. As\ndemonstrated theoretically, in these junctions the magneti-\nzation of F can be controlled by passing an electric current\nthrough the device [19–24]. We propose to use such a junc-\ntion as a memory element with the information encoded in the\nmagnetization direction of the F layer. An important issue for\nthe memory element is the effects stemming from the unavoid-able thermal fluctuations on both phase and magnetization dy-\nnamics. Therefore we present an exhaustive analysis of the\nnoisy dynamics of a current-biased SFS Josephson junction,\nsee Fig. 1, considering the influence of stochastic thermal fluc-\ntuations. Indeed, we explore this current-induced magnetic\nbistability, in order to define two well-distinguishable logic\nstates, and investigate the robustness of such memory against\nnoise-induced effects. We also suggest a suitable non-invasive\nreadout scheme based on a current-controlled superconduct-\ning quantum interference device (SQUID), which does not\nrequire an additional magnetic flux to set the optimal oper-\nating point. Moreover, we discuss the intriguing possibility\nof effectively shielding the memory state by voltage gating,\nin a device formed by a ferromagnetic layer with a linear-in-\nmomentum Dresselhaus spin-orbit coupling term.\nThe work is organized as follows: In Sec. II, we present\nthe theoretical model used to describe the time evolution of\nboth the magnetic moment and the Josephson phase of a\ncurrent-driven SFS junction. In Sec. III we discuss the mag-\nSSFx yzIbiasMzImax\nMz\nFIG. 1. S/F/S Josephson junction driven by a rectangular bias current\npulses, Ibias, with amplitude Imax. The z-component of the magneti-\nzation, Mz, is the observable used to define the logic memory states\n0 and 1.arXiv:1907.08454v2  [cond-mat.supr-con]  26 Sep 20192\nnetic configuration of the junction after applying a current\npulse. Specifically we determine its stationary magnetiza-\ntion, as a function of the Gilbert damping parameter and the\nSOC strength. In Sec. IV, we focus on the effect of stochas-\ntic thermal fluctuations on both the phase and the magnetiza-\ntion dynamics. In Sec. V, we explore how a generic Rashba-\nDresselhaus type SOC can affect the stationary magnetization.\nIn Sec. VI, we discuss a feasible readout scheme based on a\ndc-SQUID magnetometer. In Sec. VII we present our conclu-\nsions.\nII. THE MODEL\nWe consider a similar setup as the one studied in Ref. 20. It\nconsists of a SFS junction in which the ferromagnet is a thin\nfilm with an out-of-plane magnetic anisotropy and a Rashba-\nlike SOC, see Fig. 1.\nBecause of the interplay between the exchange field and the\nSOC, the current-phase relation of the SFS Josephson junc-\ntion reads Ij=Icsin(j\u0000j0), where Icis the critical current\nof the junction, jis the Josephson phase difference, and j0\nis the so-called anomalous phase shift. The latter depends on\nseveral parameters of the system, such us the Rashba coeffi-\ncient a[25, 26], the transparency of S/F interfaces, the spin\nrelaxation, and the disorder degree of the junction. The ex-\nact dependence on these parameters is here not crucial, but\nthe geometric one. If we consider a two-dimensional SOC\nwith momenta in the plane of the F film, and the charge cur-\nrent flows in x-direction, then the phase shift j0is propor-\ntional to the y-component of the magnetic moment according\nto [16, 20, 27, 28]\nj0=rMy\nM; (1)\nwhere M=q\nM2x+M2y+M2zis the modulus of the magneti-\nzation vector, and the parameter rquantifies the SOC strength\nand contains the a-dependence. The specific dependence of\nron the SOC will be discussed in Sec. IV, when we consider\nthe coexistence of both Rashba and Dresselhaus SOC contri-\nbutions and their impact on the magnetization dynamics.\nAs discussed in Refs. 20, 22, 24, and 29, Eq. (1) estab-\nlishes a direct coupling between the magnetic moment and\nthe supercurrent. The time evolution of the magnetization can\nbe described in terms of the Landau–Lifshitz–Gilbert (LLG)\nequation [30, 31]\ndM\ndt=\u0000grM\u0002Heff+g\nM\u0012\nM\u0002dM\ndt\u0013\n; (2)\nwhere grdenotes the gyromagnetic ratio. The first term on\nthe right-hand side describes the precession motion around\nHeff, whereas the second term describes the dissipation, which\nis accounted by the phenomenological dimensionless Gilbert\ndamping parameter g. The i-th component of the effective\nfieldHeff, with i=x;y;z, can be calculated as [32]\nHeff;i=\u00001\nW¶F\n¶Mi; (3)where Wis the volume of the F layer, and Fis the free energy\nof the junction, which can be written as\nF=\u0000EJjIbias+Es(j;j0)+EM: (4)\nHere, EJ=F0Ic=(2p)withF0being the flux quantum, Ibiasis\nthe external current in units of Ic,Es(j;j0) =EJ[1\u0000cos(j\u0000\nj0)], and EM=\u0000KW\n2\u0010\nMz\nM\u00112\nis the magnetic energy depend-\ning on the anisotropy constant K. The ratio between the en-\nergy scales of the system will be indicated by the parameter\ne=EJ=(KW).\nThe effective magnetic field calculated from Eq. (3) reads\nHeff=K\nM[ersin(j\u0000rmy)ˆy+mzˆz]; (5)\nwith mx;y;z=Mx;y;z=M.\nSince the normalized components of the magnetization sat-\nisfy the condition m2\nx+m2\ny+m2\nz=1, it is convenient to\nwrite the LLG equations in spherical coordinates [33], so that\nthe normalized components of the magnetization can be ex-\npressed in terms of the polar and azimuthal angles qandf\nas\nmx(t) =sinq(t)cosf(t)\nmy(t) =sinq(t)sinf(t) (6)\nmz(t) =cosq(t):\nBy normalizing the time to the inverse of the ferromagnetic\nresonance frequency wF=grK=M, that is t=wFt, the LLG\nequations in spherical coordinates reduce to the following two\ncoupled equations [33]\ndq\ndt=1\n1+g2\u0010\neHeff;f+geHeff;q\u0011\n(7)\nsinqdf\ndt=1\n1+g2\u0010\ngeHeff;f\u0000eHeff;q\u0011\n; (8)\nwhere the qandfcomponents of the normalized effective\nfield are defined as\neHeff;q=ersin(j\u0000rmy)cosqsinf\u0000mzsinq (9)\neHeff;f=ersin(j\u0000rmy)cosf: (10)\nNext, we discuss how the Josephson phase j(t)of a mag-\nnetic junction responds to a driving bias current. The dy-\nnamics of an overdamped SFS Josephson junction can be\ndescribed through the modified resistively shunted junction\n(RSJ) model [34, 35] generalized to include the anomalous\nphase shift j0=rmy. The phase dynamics is described by the\nequation:\ndj\ndt=w[Ibias(t)\u0000sin(j\u0000rmy)]+rdmy\ndt; (11)\nwhere the time is still normalized to the inverse of the ferro-\nmagnetic resonance frequency and w=wJ=wF, with wJ=\n2pIcR=F0being the characteristic frequency of the junc-\ntion [34] ( Ris the normal-state resistance of the device).3\nmzst\n-1.0-0.500.51.0\nmzst\n-1.0-0.500.51.0(a)\n(d)(b)\n(e)(c)\n(f)\nFIG. 2. Stationary magnetization, mstz, as a function of randg, at\ndifferent values of the amplitude Imaxof the current pulse, in the\nabsence of noise fluctuations. The other parameters are: e=10,\nw=1,s=5, and mz(t=0) = + 1.\nLast term in the right-hand side of Eq. (11) stems from the\ntime derivative of the anomalous phase, ˙j0,cf.Eq. (1), and\nwas ignored in Refs. 20, 22, and 24, but has to be included in\norder to preserve gauge-invariance [35].\nWe assume that the SFS junction is driven by a rectangular\ncurrent pulse, Ibias, centered at tc:\nIbias(t) =\u001a\nImax;tc\u0000s\u0014t\u0014tc+s\n0; elsewhere :(12)\nHere, sis the width and Imaxis the intensity, in units of Ic,\nof the pulse, so that a value Imax>1 indicates a bias current\nlarger than the critical value.\nBecause of the magnetoelectric effect in a j0-junction, the\ncharge current induces an in-plane magnetic moment [27, 28,\n36–39], which in turn acts as a torque on the out-of plane mag-\nnetization of the F layer and eventually leads to its switch-\ning [20, 22].\nIn the next sections we search for an optimal combination\nof system parameters to induce the magnetization reversal.\nSpecifically, we explore the response of the magnetization by\nvarying gandrin suitable ranges, whereas the energy and\ntimescales ratios eandw, are fixed. The energy ratio eranges\nfrom e\u0018100 [20] in systems with weak magnetic anisotropy,\ntoe\u00181 for stronger anisotropy [40]. In our calculation we\nchoose an intermediate value e=10. The typical ferromagnet\nresonance frequency is wF'10 GHz, while the characteristic\nJosephson frequency, usually of the order of gigahertz, may\nbe tuned experimentally. Therefore we choose w=1. As\nlong as the injected bias current is below the critical value, the\nresults discussed in this work are only weakly affected by the\nvalue of w. In contrast, if Ibias>1 the magnetic switching\nwould become more unlikely as wincreases. In particular,\nforw\u001d1 the torque exerted by the Josephson current oscil-\nlates very fast, in comparison with the timescale of the mag-\nnetization [21]. This means that the magnetization would ex-\nperience an effective torque averaged over many oscillations,\n●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●-1.0-0.50.00.51.0mzstr=0.1 -γ= 0.25\nImax=0.8\n●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●-1.0-0.50.00.51.0mzstImax=0.9\n●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●-1.0-0.50.00.51.0mzstImax=1.\n●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●-1.0-0.50.00.51.0mzstImax=1.1\n●●●●\n●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●\n●●●●●●\n●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●\n●●●●●●●●●●●●●●●\n●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●\n●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●-1.0-0.50.00.51.0mzstImax=1.2\n●●●●\n●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●\n●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●\n●●●●●●●●●●●●\n●●●●●\n●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●\n●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●\n●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●\n●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●\n●●●●●●●●●●●●●●●●\n●●●●●●●●-1.0-0.50.00.51.0mzstImax=1.3\n●●●\n●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●-1.0-0.50.00.51.0mzstImax=1.4\n●●●\n●●●●●●●●●●●●●\n●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●\n●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●\n●●●●●●\n●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●\n●●●●●●\n●●●●●●●●●●●●●●●●●\n●●●●●●●●●●●●\n●●●●●●●●●●●●●●●●●●●●●●●●0 10 20 30 40 50-1.0-0.50.00.51.0\nσmzstImax=1.5FIG. 3. Stationary magnetization, mstz, as a function of the current\npulse width, s, at different values of the amplitude Imax2[0:8\u00001:5],\nin the absence of noise fluctuations. The other parameters are: r=\n0:1,g=0:25,e=10,w=1, and mz(t=0) = + 1.\nwhich results in a small contribution due to a partial cancella-\ntion of the net torque.\nAtt=0 we assume that the F magnetization points towards\nthez-direction, that is M= (0;0;1). With this initial values\nwe solve Eqs. (7)-(8) and (11) self-consistently for different\nvalues of the parameters. From the solution we determine the\nmagnetization direction after the current pulse.\nIII. THE DETERMINISTIC ANALYSIS\nWe first neglect the effect of thermal noise, as in Refs. 22\nand 24, and explore the magnetic switching of the junction.\nThe overall behavior of the stationary magnetization mst\nz,\nnamely, the value of mzat the time t=tmax=100, as a\nfunction of both the Gilbert damping parameter, g, and the\nSOC strength, r, at different current pulse intensities Imaxand\ns=5, is summarized in Fig. 2. We note that each contour plot\nis characterized by a dark fringes pattern, namely, we observe\nregions of the (g;r)parametric space in which the magneti-\nzation reversal systematically occurs, i.e., in which mst\nz=\u00001.\nThis means, for instance, that by increasing rat a fixed gwe\nobserve a sequence of mst\nz= +1 and mst\nz=\u00001 values.\nFor small enough rvalues, the magnetization reversal effect\nis absent. Interestingly, the pattern of dark fringes evidently4\nchanges for Imax>1, so that for Imax=1:6 the dark fringes\nmerge together in large areas of (g;r)values in which magne-\ntization reversal takes place.\nWith the aim of selecting a driving pulse suitable for a\nmemory application, we observe that the magnetization rever-\nsal effect should be sufficiently robust against small changes\nof both the current pulse intensity and the duration of the\npulse. In Fig. 3 we show the stationary magnetization as a\nfunction of the pulse width s, by changing the pulse am-\nplitude Imax. We have chosen the parameters r=0:1 and\ng=0:25. We observe that for a bias current below the crit-\nical value, the need for an accurate current-width regulation is\nsignificantly relaxed, since the magnetization reversal defini-\ntively occurs for any width above a specific value. Instead, for\ncurrent amplitudes higher than the critical value, the station-\nary magnetization versus sis highly scattered between the\ntwo possible values, mst\nz=\u00061, so that even a slight change\nin the pulse width may lead to a non-switching situation. To\nunderstand this behavior, we observe that, for a bias current\nhigher than Ic, a pulse sufficiently long can make the junction\nto switch to the resistive state, so that the Josephson phase\nrapidly evolves and a voltage drop across the device appears,\nsince Vµdj\ndt[34]. In this case, the steady magnetization\nstrongly depends on the dynamical state of the system when\nthe current pulse is switched off. Moreover, the higher the bias\ncurrent is, the faster the Josephson phase evolve and more pro-\nnounced will be the s-dependence of mst\nz. In view of a mem-\nory application, a current pulse smaller than the critical value\nis therefore recommended, in order to make the magnetization\nreversal unrelated on the pulse width s. For these reasons in\nthe subsequent analysis we set Imax=0:9 and s=5 and fo-\ncus on the thermal effect on the phase and the magnetization\ndynamics.\nIV . EFFECTS OF NOISE\nIn this section, we focus on the noisy dynamics of the junc-\ntion, specifically on how it affects the magnetization reversal.\nThe temperature can significantly influence the time evolution\nof the system, eventually inducing unwanted magnetization\nflip or preventing a stable magnetization reversal. Therefore,\nwe consider stochastic thermal fluctuations in both the phase\nand the magnetic moment dynamics. We start discussing the\neffect of a thermal noise source only on the phase within the\nRSJ model. In the second part of this section we include also\nthe thermal noise on the magnetization dynamics.\nA. Thermal current effects on the RSJ model\nThe phase dynamics can be directly perturbed by thermal\nfluctuations accounted by adding to the RSJ model, Eq. (11),\na Langevin Gaussianly distributed, delta correlated stochas-\ntic term, Ith(t). This “thermal current” has the usual white-\nnoise statistical properties that, in normalized units, can be\nFIG. 4. (a) Stationary magnetization, mstz, as a function of rand\ng. (b) Average stationary magnetization, mstz, as a function of rand\ng, atDI=0:01, calculated by averaging over Nexp=100 indepen-\ndent numerical repetitions. (c) Average stationary magnetization,\nmstz, as a function of the thermal current intensity, DI, atg=0:25\nandr=0:1 [namely, the (g;r)-values highlighted with a red circle\nin panel (b)], calculated by averaging over Nexp=1000 independent\nnumerical repetitions. The inset shows the normalized temperatures\ncorresponding to the noise intensities DI. For all panels Imax=0:9\nands=5, whereas the values of other parameters are the same used\nto obtain Fig. 2.\nexpressed as [34, 41, 42]\nhIth(t)i=0 (13)\nIth(t)Ith(t0)\u000b\n=2DId\u0000\nt\u0000t0\u0001\n: (14)\nHere, we introduced the dimensionless amplitude of thermal\ncurrent fluctuations defined as\nDI=kBT\nRwF\nI2c=1\nwkBT\nEJ: (15)\nFor example, if w=1 we obtain DI\u00180:04T\nIcmA\nK, so that, for\ninstance, a junction with Ic=1mA atT=250 mK is affected\nby a thermal fluctuation of intensity DI\u001810\u00002.\nBy taking into account the noise contribution, Eq. (11) be-\ncomes\ndj\ndt=w[Ibias(t)\u0000sin(j\u0000rmy)+Ith(t)]+rdmy\ndt:(16)\nIn Fig. 4 we compare the current-induced magnetization re-\nversal obtained without and with accounting of the noise ef-\nfects, see panel (a) and (b) respectively. We set the intensity\nof the current pulse Imax=0:9 and its width s=5.5\nIn Fig. 4(a) we show the behavior of mst\nzas a function of\nrandgin the deterministic case, namely, in the absence of\nnoise, DI=0. Here, we observe a contour plot composed by\nmany narrow dark fringes in which mst\nz=\u00001, see Fig. 4(a).\nThe situation drastically changes if we include the thermal\nnoise. In this case we focus on the average stationary magne-\ntization, mstz, which is computed by averaging the stationary\nmagnetization over Nexp=100 independent numerical runs.\nThe behavior of mstzas a function of randgforDI=0:01 is\nillustrated in Fig. 4(b). At small values of rthe magnetiza-\ntion reversal is still absent, whereas noise mostly affects the\nregions with large rwhere the averaged value of the magne-\ntization is mainly distributed around zero. Nevertheless, one\ncan still identify dark regions in which magnetization switch-\ning takes place. With a red circle we highlight in Fig. 4(b) the\nregion around the point (g;r) = ( 0:25;0:1)where the mag-\nnetization takes the largest negative average magnetization\nmstz'\u00001. In other words, the region with the most robust\nswitching.\nBy increasing the noise intensity the switching process is\nsuppressed, as shown in Fig. 4(c), at the optimal values r=0:1\nandg=0:25 in Fig. 4(b). In Fig. 4(c) the value of mstzis\nthe average over Nexp=1000 independent numerical repeti-\ntions. In the inset we show the normalized temperatures cor-\nresponding to the noise intensities DI, calculated by assuming\na junction with a temperature-dependent critical current and\nIc=100mA at low temperatures [43]. From this figure, one\nsees that mstz'\u00001 only for DI.0:01, that is for T.0:75Tc.\nFor higher noise intensities both the average magnetization\nand the error bar increase, approaching a zero magnetization\naverage only for DI&0:3.\nIn Fig. 5 we explore the time evolution of the different ob-\nservables with and without thermal noise. Specifically, we\nshow the response of the junction with g=0:25 and r=0:1\nto a current pulse with amplitude Imax=0:9 depicted in panel\n(a). In the absence of noise, DI=0, we plot in Fig. 5(b)\nthe time evolution of the phase and the supercurrent, and in\nFig. 5(c) the different components of the magnetic moment.\nDuring the current pulse, i.e., the yellow shaded region, the\nphase first increases, and then it goes to zero when the pulse\nis turned off, see Fig. 5(b). To understand the phase behav-\nior, we observe that, in the washboard-like picture [34], the\ntilting imposed by the bias current Imax=0:9 is not enough\nfor allowing the “particle” to overcome the nearest potential\nbarrier and switch the system to the finite voltage “running”\nstate. Instead, the phase-particle remains confined within a\npotential minimum, so that when the current is turned off, the\nslope of the washboard potential goes again to zero and the\nphase restores its initial position, i.e.,j!0.\nWe observe that the larger the bias current pulse is, the\nhigher is the washboard potential slope, and therefore for\nImax>1 the greater the speed of the phase particle, so that\nit can take a longer time to restore the initial position after\nthe current pulse is switched off. Moreover, a large bias cur-\nrent pulse may also longer switching times. Hence, a current\nImax<1 is, in general, more advantageous for a memory ap-\nplication.\nIn Fig. 5(c) we show how all components of the magne-\n0 10 20 30 400.00.20.40.60.81.0Ibias\nφ/π\nIφ\n0.00.20.40.60.8\nmx\nmy\nmz\n0 10 20 30 40-1.0-0.50.00.51.0mx,my,mz\n-0.20.00.20.40.60.81.0\nDI=0.05\n0 10 20 30 40-1.0-0.50.00.51.0\ntmx,my,mzDI=0.05(b)\n(c)\n(d)(a)\n(e)FIG. 5. Current pulse (a) and following time evolution of phase and\nJosephson current, see panel (b), and magnetization components, see\npanel (c), in the absence of noise and including a thermal current\ncontribution with amplitude DI=0:05, see panels (e) and (d). The\nvalues of other parameters are Imax=0:9,s=5,r=0:1, and g=\n0:25. The legends in panels (b) and (c) refer also to panels (d) and\n(e), respectively.\ntization are induced by the current pulse. Whereas mxand\nmyare generated during the current pulse, and they undergo a\ndamped oscillations around zero when the current is switched\noff, the z-component, after a transient regime, flips definitively\nto the value mz=\u00001. From this figure we can also estimate\nthe switching time tSW'10, as the time mzroughly takes to\napproach the value \u00001 after switching off the current.\nThe scenario described so far essentially persists also in the\nstochastic case, as shown in Figs. 5(d-e) for DI=0:05. There-\nfore, at the temperature that we are considering, the overall\nbehavior is still quite similar to the one obtained in the ab-\nsence of noise. In fact, the z-component of the magnetization\nflips again to the value \u00001, while the xandycomponents tend\nto oscillate around zero, without, however, vanishing defini-\ntively.\nThe magnetization switch can be achieved in a short time\nscale, by passing through the junction a sequence of current6\nFIG. 6. Time evolution of the magnetization mz, in response to a se-\nquence of three current pulses shown in the top panel, in the presence\nof a thermal current noise with amplitude DI=0:05. The values of\nthe other parameters are: Imax=0:9,s=5,r=0:1, and g=0:25.\npulses, as it is shown in Fig. 6. In the bottom panel we show\nthe time evolution of the magnetization mz, when the junction\nis excited by the three subsequent current pulses presented\nin the top panel, in the presence of a thermal current noise\nwith amplitude DI=0:05. In response to each current pulse,\nmzfollows first a transient regime, and then, as the current is\nswitched off, it approaches the steady value with an opposite\nsign.\nB. Effect of thermal noise on the magnetization dynamics\nThermal noise also affects directly the magnetization dy-\nnamics [44–49] via a stochastic field Hth, a sort of “thermal\nfield”, which is added to the effective magnetic field term in\nEq. (2), as done in Ref. [50]. Inclusion of the thermal noise in\nEq. (2) leads to [33]\ndM\ndt=\u0000grM\u0002(Heff+Hth)+g\nM\u0012\nM\u0002dM\ndt\u0013\n: (17)\nThis stochastic differential equation has to be solved numeri-\ncally by a stochastic integration prescription by keeping the\nmodulus of the magnetic moment constant during the time\nevolution (see Ref. 33 and references therein). For this pur-\npose it is again convenient to write the equations in spherical\ncoordinates, see Eq. (6), so that the stochastic LLG equation\nreads [33, 51]:\ndq\ndt=1\n1+g2h\neHeff;f+eHth;f+g\u0010\neHeff;q+eHth;q\u0011i\n(18)\nsinqdf\ndt=1\n1+g2h\ng\u0010\neHeff;f+eHth;f\u0011\n\u0000eHeff;q\u0000eHth;qi\n;(19)\nwhere\neHth;q=eHth;xcosqcosf+eHth;ycosqsinf\u0000eHth;zsinq(20)\neHth;f=\u0000eHth;xsinf+eHth;ycosf: (21)\nThe normalized field, eHth= (M=K)Hthis assumed to be a\n10-40.001 0.01 0.1 1-1.0-0.50.00.51.0\nDImzstr = 0.1  - γ = 0.25  - Nexp = 1000FIG. 7. Average stationary magnetization, mstz, as a function of the\nnoise intensity, DI, calculated by taking into account both the thermal\ncurrent and the thermal field noise contribution, and by averaging\nover Nexp=1000 independent numerical repetitions. The values of\nother parameters are Imax=0:9,s=5,r=0:1, and g=0:25.\nGaussianly distributed random field with the following statis-\ntical features\nD\neHth;i(t)E\n=0 (22)\nD\neHth;i(t)eHth;i(t0)E\n=2DHd\u0000\nt\u0000t0\u0001\n; (23)\nwhere i=x;y;zand\nDH=\u0012g\nMkBT\njgrjW\u0013\u0012M\nK\u00132\nwF=gkBT\nKW(24)\nis the dimensionless amplitude of thermal field fluctuations.\nIn all previous equations the time is still normalized to the\ninverse of wF.\nInterestingly, by recalling the definition of the parameter\ne=EJ=(KW), from Eqs. (15) and (24) we can easily obtain\nthe following relation between the normalized thermal noise\nintensities\nDH= (g ew)DI: (25)\nThus, by changing the magnetization energy, the Gilbert\ndamping parameter, or the magnetic resonance frequency we\ncan effectively modify the relative strength of the two noise\nmechanisms. This means that one could optimize the system\nparameters in such a way to make, for instance, the impact of\nthe thermal field negligible with respect to the thermal current.\nThis allows us to study the effects produced by these noise\nsources independently. In the following, even if we explicitly\nwrite only the value of DI, we are taking into account both\nthermal current and thermal field independent noise sources,\nwhich amplitudes are related by Eq. (25).\nThe overall effect of both the thermal current and field is\npresented in Fig. 7, where we show the behavior of mstz, calcu-\nlated by averaging over Nexp=1000 independent numerical\nruns, at different values of the noise intensity DI, and by set-\ntingImax=0:9,s=5,g=0:25, and r=0:1. We observe that\nthe average magnetization remains close to the value mstz'\u000017\nFIG. 8. Time evolution of phase and Josephson current (a) and mag-\nnetization components (b) as the system is excited by the current\npulse in Fig. 5. Here we are taking into account both a thermal cur-\nrent and a thermal field contribution, with noise intensity DI=0:005.\nThe values of other parameters are Imax=0:9,s=5,r=0:1, and\ng=0:25.\nonly for DI.0:003, that is for T.0:58Tc, see the inset\nof Fig. 4(c). For larger values of DI,mstzapproaches zero\nand hence the magnetization reversal probability is reduced,\nFig. 4(c). In view of the memory application, one should, in\nprinciple, carefully choose the F layer and its characteristics\n(such as its volume or the Gilbert damping parameter) in or-\nder to make the thermal field effect as small as possible. The\naim is to reduce the thermal field intensity in order to increase\nthe working temperature suitable for a memory application,\ne.g., through a lower Gilbert damping or a larger F volume,\naccording to Eq. (25).\nThe time evolution of j,Ij, and mi(with i=x;y;z), as\nthe junction dynamics if affected by both a thermal current\nand a thermal field, for r=0:1,g=0:25, and DI=0:005, is\nshown in Fig. 8. Here, we consider again the system excited\nby a current pulse with intensity Imax=0:9, as that one shown\nin Fig. 5(a). We observe that all noisy curves still resemble\nin shape the deterministic evolution presented in Figs. 5(b)-\n(c). The value tSW'10 is a quite good estimation for the\nswitching time of the device also in this noisy case.\nV . RASHBA-DRESSELHAUS SOC\nIn all previous analysis it was assumed a pure Rashba SOC.\nHowever, the theory of j0-junctions can be generalized for\nany linear-in-momentum SOC [27, 28], by using the SU(2)-\ncovariant formulation [28], where the SOC is described in\nterms of a SU(2) vector potential A. For a 2D SOC with\nboth Rashba and Dresselhaus contributions one obtains Ax=\n\u0000asy+bsxandAy=asx\u0000bsy(here, aandbare the\nRashba and Dresselhaus coefficients and sxandsyare the\nfirst two Pauli matrices).\nThe appearance of the anomalous phase is related to the\nexistence of a finite Liftshitz invariant term in the free en-\nergy [16, 52–54] which is proportional to Ti¶ij, where Tiis the i-th component of a polar vector which is odd under\ntime reversal, ¶iis the i-the derivative of the superconduct-\ning phase, and the sum over repeated indices is implied here\nand below. For the particular junction geometry sketched in\nFig. 1 the supercurrent, and hence the phase gradient, is fi-\nnite in x-direction. Thus, according to Eq.(5.17) of Ref. 27,\nthe anomalous phase can be written in the following compact\nform [27, 28]:\nj0=reb(ebmx+my): (26)\nHere, we defined the SOC coefficients ratio,eb=b=a, and\nthe parameter reb=r(1\u0000eb2), with rdepending this time on\nbothaandb. In the absence of the Dresselhaus SOC, that\nis wheneb=0 and reb!r, we recover Eq. (1). If both con-\ntributions are similar in magnitude, i.e., wheneb!1, since\nreb!0 the phase shift vanishes, i.e.,j0!0. This is a very\ninteresting situation that we explore in this section. In fact,\nwhereas the Dresselhaus contribution is due to the breaking\nof crystal inversion symmetries, the Rashba SOC stems from\nstructural broken symmetry and therefore can be controlled by\na gate voltage [55, 56]. In other words, a voltage gate can con-\ntrol the ratioebbetween Dresselhaus and Rashba coefficients,\nand hence the phase shift and the supercurrent flow, accord-\ning to Eq. (26). Specifically, by tuning asuch thateb'1 one\ncan fully decouple the phase and magnetic moment dynamics.\nSuch a process can be eventually used to protect the memory\nstate in one of the storage elements of a distributed architec-\nture.\nWe provide next a quantitative analysis of this situation, so\nthat by taking into account the generic j0, Eq. (26), into the\nexpression for the effective field, Eq. (5) becomes\nHeff=K\nM\u001a\neberebsinh\nj\u0000reb\u0010ebmx+my\u0011i\nˆx+\nerebsinh\nj\u0000reb\u0010ebmx+my\u0011i\nˆy+mzˆz\u001b\n:(27)\nTheqandfcomponents of the normalized effective field to\nbe included in LLG Eqs. (7)-(8), read\neHeff;q=eberebsinh\nj\u0000reb\u0010ebmx+my\u0011i\ncosqcosf (28)\n+erebsinh\nj\u0000reb\u0010ebmx+my\u0011i\ncosqsinf\u0000mzsinq\neHeff;f=\u0000eberebsinh\nj\u0000reb\u0010ebmx+my\u0011i\nsinf (29)\n+erebsinh\nj\u0000reb\u0010ebmx+my\u0011i\ncosf;\nwhereas the RSJ equation becomes\ndj\ndt=wn\nIbias(t)\u0000sinh\nj\u0000reb\u0010ebmx+my\u0011io\n(30)\n+reb\u0012\nebdmx\ndt+dmy\ndt\u0013\n:\nThe behavior of the stationary magnetization as a function of\nrandg, at different values ofeb2[0\u00001]is shown in Fig. 9.8\nmzst\n-1.0-0.500.51.0\nmzst\n-1.0-0.500.51.0(a)\n(d)(b)\n(e)(c)\n(f)\nFIG. 9. Stationary magnetization, mstz, as a function of randg, at\ndifferent values of the relative Dresselhaus coefficienteb=b=a, in\nthe absence of noise fluctuations, by imposing Imax=0:9 and s=5.\nThe current pulse intensity and width are chosen equal to\nImax=0:9 and s=5, respectively. As expected from the dis-\ncussion above, the region where no magnetization switching\noccurs, bright color in Fig. 9, increases by increasingebto-\nwards 1. Foreb=1, the j0behavior is fully suppressed, cf.\nEq. (26), and hence no magnetization switching takes place,\ndespite the current pulse flowing through the junction. Inter-\nestingly, we note that for intermediate values ofeb, the area of\nthe switching fringes, i.e., where mst\nz=\u00001, increases consid-\nerably.\nIn summary of this section, by tuning a=bone can decou-\nple the magnetic behavior from the Josephson dynamics and\nfreeze the memory state in order to protect it from external\ncurrent pulses and other perturbations. The major challenge\nin this regard is to find materials with a sizable magnetic mo-\nment and tunable by means of a gate voltage.\nVI. THE MEMORY READ-OUT\nAs discussed in previous section, the writing operation of\nthe proposed memory element can be performed by exciting\nthe junction with controlled current pulses. We could envis-\nage an array of j0-junction-based memory elements, each one\neventually having its own current line so that it can be written\nby sending individual current inputs. Alternatively, exploiting\nthe tuning of the SOC discussed in previous section, one could\ncontrol locally, via individual gates at each junction, several\nmemory elements connected in series to a common current\nline. In this way one could selectively write via a common\ncurrent pulse only a specific set of memory units.\nThe read-out of the memory state can be non-destructively\nperformed by direct measurement of the magnetization state\nthrough a dc-SQUID inductively coupled to the j0-junction.\nA SQUID is essentially a magnetic flux detector [57], which\ncan be employed to measure with a very high sensitivity any\nV V\nIbias MzIreadΦµ\nIbias MzIread\n (a) (b)ΦµSQ\nSQFIG. 10. SQUID-based memory readout and cartoon showing the\ncritical current interference pattern of the SQUID, in the cases of both\npositive and negative orientation along the z-axis of the magnetic\nmoment, see panel (a) and (b), respectively.\nphysical quantity that can be converted in a magnetic flux [58].\nWe suggest a SQUID sensor along the lines of the read-\nout scheme implemented in Ref. 13 for a p-junction mem-\nory. Our scheme is based on an asymmetric inductive dc-\nSQUID, which consists of a superconducting ring with a non-\nnegligible total inductance, L, with two Josephson junctions\nwith different critical currents, i.e.,Ic;16=Ic;2(here, we are ne-\nglecting for simplicity any asymmetry in the ring inductance).\nWith such asymmetric SQUID, one can avoid the use of an ad-\nditional magnetic flux to adjust the working point of the device\nin a high sensitivity position of the ISQ\nc\u0000Fcharacteristics,\nwhere ISQ\ncis the SQUID critical current and Fis the magnetic\nflux threading the loop. In fact, such asymmetric dc-SQUID\nshows non-negligible screening and asymmetry parameters,\nthat is bL=2p\nF0L\n2(Ic;1+Ic;2)6=0 and aI=Ic;1\u0000Ic;2\nIc;1+Ic;26=0, re-\nspectively. In this case, the ISQ\ncmaximum is not centered in\nF=0, but shifted from zero by DF, see Fig. 10, where [57]\n2pDF=F0bLaI. Accordingly, our readout SQUID demon-\nstrates a high sensitivity point of the ISQ\nc\u0000Fcharacteristics\nalso in F=0, that is in the absence of an external magnetic\nflux.\nWe assume that the unbiased critical current, ISQ\nc(F=0),\nlies in the positive branch of the critical current diffraction\npattern of the SQUID, that isdISQ\nc\ndF\f\f\f\nF=0>0, just like in the\ncase shown in Fig. 10. Thus, the magnetic moment mz= +1\ngenerates a positive magnetic flux F= +Fmthrough the loop,\nand gives a critical current higher than the unbiased value, i.e.,\nI+\nc=ISQ\nc(+Fm)>Ic(0), see the red dashed line in Fig. 10(a).\nConversely, if mz=\u00001 the magnetic flux through the loop\nis negative, i.e.,F=\u0000Fm, and the critical current is lower\nthan the unbiased value, I\u0000\nc=ISQ\nc(\u0000Fm)<Ic(0), see the blue\ndashed line in Fig. 10(b).\nThe SQUID readout loop is then sensed by passing a bit-\nread current with intensity Iread=ISQ\nc(0)through the device,\nsee the orange dashed line in Fig. 10, that is a current which\nlays in between I\u0000\ncandI+\nc. In this way, if the magnetic mo-\nment points in the negative z-direction, which encodes the9\n“1” logic state of the memory, the bit-read current makes the\nSQUID to switch to the voltage state, since Iread>I\u0000\nc. Con-\nsequently, a voltage drop appears across the readout SQUID\nin response to the bit-read current. For the opposite magnetic\nmoment orientation, which encodes the “0” logic state, the\nSQUID critical current is larger than the bit-read current, that\nisIread<I+\nc, so that the SQUID remains in the superconduct-\ning, zero-voltage state.\nTo provide an estimate of the dimensions of a SQUID loop\nsuitable to detect the magnetic moment reversal we consider\nthe simple case of a circular readout SQUID with radius a\nand a magnetic moment oriented along the z-axis and placed\non the symmetry axis of the SQUID at a height z0above the\nx\u0000yplane. In this case, the total magnetic flux through the\nloop can be estimated as [58, 59] Fm=m0Mz\n2a2\u000e\u0000\na2+z02\u00013=2.\nThis flux is maximum at z0=0 where it reads Fm=m0Mz\nd,\nwith d=2abeing the loop diameter. If we assume a satura-\ntion magnetization [60] M=8\u0002105A=m and a volume W=\n(10\u0002100\u0002100)nm3=10\u000022m3, we get a total magnetic\nmoment Mz=8\u000210\u000017Am2(that is Mz=8:6\u0002106mB,\nwithmBbeing the Bohr magneton), so that the total magnetic\nflux reads Fm=32p10\u000024d\u00001Wb m. Thus, for a SQUID with\ndiameter'0:5mm we obtain a measurable flux Fm'F0=10,\n(see vertical dot-dashed lines in Fig. 10).VII. CONCLUSIONS\nIn conclusion, we propose a non-volatile superconducting\nmemory based on a bistable magnetic behavior of a current-\nbiased j0-junction, that is a superconductor-ferromagnet-\nsuperconductor Josephson junction with a Rashba-like spin-\norbit coupling (SOC). The memory state is encoded in the\nmagnetization direction of the ferromagnetic layer, which can\nbe switched via controlled current pulses. 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Phys. 77, 1321–1373\n(2005)."    },    {        "title": "1507.06505v2.Nanomagnet_coupled_to_quantum_spin_Hall_edge__An_adiabatic_quantum_motor.pdf",        "content": "Nanomagnet coupled to quantum spin Hall edge: An adiabatic quantum\nmotor\nLiliana Arrachea\nDepartamento de F\u0013 \u0010sica, FCEyN, Universidad de Buenos Aires and IFIBA, Pabell\u0013 on I, Ciudad Universitaria, 1428 CABA\nand International Center for Advanced Studies, UNSAM, Campus Miguelete, 25 de Mayo y Francia, 1650 Buenos Aires,\nArgentina\nFelix von Oppen\nDahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universit at Berlin, 14195 Berlin, Germany\nAbstract\nThe precessing magnetization of a magnetic islands coupled to a quantum spin Hall edge pumps charge\nalong the edge. Conversely, a bias voltage applied to the edge makes the magnetization precess. We point\nout that this device realizes an adiabatic quantum motor and discuss the e\u000eciency of its operation based on\na scattering matrix approach akin to Landauer-B uttiker theory. Scattering theory provides a microscopic\nderivation of the Landau-Lifshitz-Gilbert equation for the magnetization dynamics of the device, including\nspin-transfer torque, Gilbert damping, and Langevin torque. We \fnd that the device can be viewed as\na Thouless motor, attaining unit e\u000eciency when the chemical potential of the edge states falls into the\nmagnetization-induced gap. For more general parameters, we characterize the device by means of a \fgure\nof merit analogous to the ZT value in thermoelectrics.\n1. Introduction\nFollowing Ref. [1], Meng et al. [2] recently showed\nthat a transport current \rowing along a quantum\nspin Hall edge causes a precession of the magneti-\nzation of a magnetic island which locally gaps out\nthe edge modes (see Fig. 1 for a sketch of the de-\nvice). The magnetization dynamics is driven by the\nspin transfer torque exerted on the magnetic island\nby electrons backscattering from the gapped region.\nIndeed, the helical nature of the edge state implies\nthat the backscattering electrons reverse their spin\npolarization, with the change in angular momen-\ntum transfered to the magnetic island. This e\u000bect\nis not only interesting in its own right, but may also\nhave applications in spintronics.\nCurrent-driven directed motion at the nanoscale\nhas also been studied for mechanical degrees of free-\ndom, as motivated by progress on nanoelectrome-\nchanical systems. Qi and Zhang [3] proposed that\na conducting helical molecule placed in a homo-\ngeneous electrical \feld could be made to rotate\naround its axis by a transport current and pointedout the intimate relations with the concept of a\nThouless pump [4]. Bustos-Marun et al. [5] devel-\noped a general theory of such adiabatic quantum\nmotors, used it to discuss their e\u000eciency, and em-\nphasized that the Thouless motor discussed by Qi\nand Zhang is optimally e\u000ecient.\nIt is the purpose of the present paper to em-\nphasize that the current-driven magnetization dy-\nnamics is another { perhaps more experimentally\nfeasible { variant of a Thouless motor and that\nthe theory previously developed for adiabatic quan-\ntum motors [5] is readily extended to this device.\nThis theory not only provides a microscopic deriva-\ntion of the Landau-Lifshitz-Gilbert equation for the\ncurrent-driven magnetization dynamics, but also al-\nlows one to discuss the e\u000eciency of the device and\nto make the relation with the magnetization-driven\nquantum pumping of charge more explicit.\nSpeci\fcally, we will employ an extension of the\nLandauer-B uttiker theory of quantum transport\nwhich includes the forces exerted by the electrons\non a slow classical degree of freedom [6, 7, 8, 9].\nPreprint submitted to Elsevier October 11, 2018arXiv:1507.06505v2  [cond-mat.mes-hall]  8 Sep 2015\u0001\u0001\u0001\u0001exeyezeθFigure 1: (Color online) Schematic setup. A nanomagnet\nwith magnetic moment Mcouples to a Kramers pair of edge\nstates of a quantum spin Hall insulator. The e\u000bective spin\ncurrent produces a spin-transfer toque and the magnetic mo-\nment precesses.\nMarkus B uttiker developed Landauer's vision of\nquantum coherent transport as a scattering prob-\nlem into a theoretical framework [10, 11] and ap-\nplied this scattering theory of quantum transport\nto an impressive variety of phenomena. These ap-\nplications include Aharonov-Bohm oscillations [12],\nshot noise and current correlations [11, 13, 14], as\nwell as edge-state transport in the integer Hall ef-\nfect [15] and topological insulators [16]. Frequently,\nB uttiker's predictions based on scattering theory\nprovided reference points with which other theo-\nries { such as the Keldysh Green-function formal-\nism [17, 18, 19, 20] or master equations [21] { sought\nto make contact.\nIn the present context, it is essential that scat-\ntering theory also provides a natural framework to\nstudy quantum coherent transport in systems un-\nder time-dependent driving. For adiabatic driving,\nB uttiker's work with Thomas and Pr^ etre [22] was\ninstrumental in developing a description of adia-\nbatic quantum pumping [4] in terms of scattering\ntheory [23, 24, 25, 26] which provided a useful back-\ndrop for later experiments [27, 28, 29, 30, 31]. Be-\nyond the adiabatic regime, Moskalets and B uttiker\ncombined the scattering approach with Floquet the-\nory to account for periodic driving [32]. These\nworks describe adiabatic quantum transport as a\nlimit of the more general problem of periodic driv-\ning and ultimately triggered numerous studies on\nsingle-particle emitters and quantum capacitors (as\nreviewed by Moskalets and Haack in this volume[33]).\nThe basic idea of the adiabatic quantum mo-\ntor [5] is easily introduced by analogy with the\nArchimedes screw, a device consisting of a screw\ninside a pipe. By turning the screw, water can be\npumped against gravity. This is a classical analog of\na quantum pump in which electrons are pumped be-\ntween reservoirs by applying periodic potentials to\na central scattering region. Just as the Archimedes\npump can pump water against gravity, charge can\nbe quantum pumped against a voltage. In addi-\ntion, the Archimedes screw has an inverse mode\nof operation as a motor : Water pushed through the\ndevice will cause the screw to rotate. The adiabatic\nquantum motor is a quantum analog of this mode\nof operation in which a transport current pushed\nthrough a quantum coherent conductor induces uni-\ndirectional motion of a classical degree of freedom\nsuch as the rotations of a helical molecule.\nThe theory of adiabatic quantum motors [5, 34]\nexploits the assumption that the motor degrees of\nfreedom { be they mechanical or magnetic { are\nslow compared to the electronic degrees of freedom.\nIn this adiabatic regime, the typical time scale of\nthe mechanical dynamics is large compared to the\ndwell time of the electrons in the interaction region\nbetween motor and electrical degrees of freedom. In\nthis limit, the dynamics of the two degrees of free-\ndom can be discussed in a mixed quantum-classical\ndescription. The motor dynamics is described in\nterms of a classical equation of motion, while a\nfully quantum-coherent description is required for\nthe fast electronic degrees of freedom.\nFrom the point of view of the electrons, the mo-\ntor degrees of freedom act as acpotentials which\npump charge through the conductor. Conversely,\nthe backaction of the electronic degrees of freedom\nenters through adiabatic reaction forces on the mo-\ntor degrees of freedom [6, 7, 8, 9]. When there is\njust a single (Cartesian) classical degree of freedom,\nthese reaction forces are necessarily conservative,\nakin to the Born-Oppenheimer force in molecular\nphysics [35]. Motor action driven by transport cur-\nrents can occur when there is more than one mo-\ntor degree of freedom (or a single angle degree of\nfreedom). In this case, the adiabatic reaction force\nneed no longer be conservative when the electronic\nconductor is subject to a bias voltage [6, 7, 8, 9].\nIn next order in the adiabatic approximation,\nthe electronic system also induces frictional and\nLorentz-like forces, both of which are linear in the\nslow velocity of the motor degree of freedom. In-\n2cluding the \ructuating Langevin force which ac-\ncompanies friction yields a classical Langevin equa-\ntion for the motor degree of freedom. This equation\ncan be derived systematically within the Keldysh\nformalism [35] and the adiabatic reaction forces ex-\npressed through the scattering matrix of the coher-\nent conductor [6, 7, 8].\nWhile these developments focused on mechani-\ncal degrees of freedom, it was also pointed out that\nthe scattering theory of adiabatic reaction forces\nextends to magnetic degrees of freedom [9]. In\nthis case, adiabaticity requires that the precessional\ntime scale of the magnetic moment is larger than\nthe electronic dwell time. The e\u000bective classical de-\nscription for the magnetic moment takes the form of\na Landau-Lifshitz-Gilbert (LLG) equation. Similar\nto nanoelectromechanical systems, the LLG equa-\ntion can be derived systematically in the adiabatic\nlimit for a given microscopic model and the coe\u000e-\ncients entering the LLG equation can be expressed\nalternatively in terms of electronic Green functions\nor scattering matrices [36, 37, 38, 39, 9]. In the fol-\nlowing, we will apply this general theory to a mag-\nnetic island coupled to a Kramers pair of helical\nedge states.\nThis work is organized as follows. Section 2\nreviews the scattering-matrix expressions for the\ntorques entering the LLG equation. Section 3 ap-\nplies this theory to helical edge states coupled to\na magnetic island and makes the relation to adia-\nbatic quantum motors explicit. Section 4 de\fnes\nand discusses the e\u000eciency of this device and de-\nrives a direct relation between charge pumping and\nspin transfer torque. Section 5 is devoted to con-\nclusions.\n2. S-matrix theory of spin transfer torques\nand Gilbert damping\n2.1. Landau-Lifshitz-Gilbert equation\nConsider a coherent (Landauer-B uttiker) con-\nductor coupled to a magnetic moment. The latter is\nassumed to be su\u000eciently large to justify a classical\ndescription of its dynamics but su\u000eciently small so\nthat we can treat it as a single macrospin. Then,\nits dynamics is ruled by a Landau-Lifshitz-Gilbert\nequation\n_M=M\u0002[\u0000@MU+Bel+\u000eB]: (1)\nNote that we use units in which Mis an angu-\nlar momentum and for simplicity of notation, Belas well as\u000eBdi\u000ber from a conventional magnetic\n\feld by a factor of gd, the gyromagnetic ratio of\nthe macrospin. The \frst term on the right-hand\nside describes the dynamics of the macrospin in the\nabsence of coupling to the electrons. It is derived\nfrom the quantum Hamiltonian\n^U=\u0000gd^M\u0001B+D\n2^M2\nz; (2)\nwhere M=h^Miis the uncoupled macrospin, Bthe\nmagnetic \feld, and D> 0 the easy-plane anisotropy\nof the macrospin. The coupling to the electrons\nleads to the additional e\u000bective magnetic \feld Bel.\nThis term can be derived microscopically from the\nHeisenberg equation of motion of the macrospin by\nevaluating the commutator of ^Mwith the interac-\ntion Hamiltonian between macrospin and electrons\nin the adiabatic approximation (see, e.g., Ref. [9]).\nKeeping terms up to linear order in the small mag-\nnetization \\velocity\" _M, we can write\nBel=B0(M)\u0000\r(M)_M: (3)\nHere, the \frst contribution B0can be viewed as the\nspin-transfer torque. The second term is a contribu-\ntion to Gilbert damping arising from the coupling\nbetween macrospin and electrons. In general, \rso\nderived is a tensor with symmetric and antisymmet-\nric components. However, it can be seen that only\nthe symmetric part plays a relevant role [9]. Finally,\nby \ructuation-dissipation arguments, the Gilbert\ndamping term is accompanied by a Langevin torque\n\u000eBwith correlator\nh\u000eBl(t)\u000eBk(t0)i=Dlk\u000e(t\u0000t0): (4)\nIts correlations are local in time as a consequence\nof the assumption of adiabaticity. As a result, we\n\fnd the LLG equation\n_M=M\u0002h\n\u0000@MU+B0\u0000\r_M+\u000eBi\n; (5)\nfor the macrospin M.\nThe spin-transfer torque, the Gilbert damping,\nand the correlator Dcan be expressed in terms\nof the scattering matrix of the coherent conductor,\nboth in and out of equilibrium [36, 37, 38, 39, 9].\nBefore presenting the S-matrix expressions, a few\ncomments are in order. First, the expression for the\nGilbert damping only contains the intrinsic damp-\ning originating from the coupling to the electronic\ndegrees of freedom. Coupling to other degrees of\nfreedom might give further contributions to Gilbert\n3damping which could be included phenomenologi-\ncally. Second, in the study of the nanomagnet cou-\npled to the helical modes we will consider the ex-\npressions to lowest order in the adiabatic approxi-\nmation presented in Sec. 2.2. The theory can actu-\nally be extended to include higher order corrections\n[9]. In Sec. 2.3 section, we brie\ry summarize the\nmain steps of the general procedure for complete-\nness.\n2.2. Coe\u000ecients of the LLG equation in the lowest\norder adiabatic approximation\nThis section summarizes the expressions for the\ncoe\u000ecients of the LLG equation that we will use\nto study the problem of the nanomagnet coupled\nto the helical edge states. These correspond to\nthe lowest order in the adiabatic approximation, in\nwhich we retain only terms linear in _MandeV.\nTo this order, we can write the coe\u000ecients of the\nLLG equation in terms of the electronic S-matrix\nfor a static macrospin M. The coupling between\nmacrospin and electronic degrees of freedom enters\nthrough the dependence of the electronic S-matrix\nS0=S0(M) on the (\fxed) macrospin. At this or-\nder, the spin-transfer torque and the Gilbert damp-\ning can be expressed as [36, 37, 38, 39, 9]\nB0(M) =X\n\u000bZd\"\n2\u0019if\u000bTr\u0014\n\u0005\u000b^Sy\n0@S0\n@M\u0015\n(6)\nand\n\rkl(M) =\u0000~X\n\u000bZd\"\n4\u0019f0\n\u000bTr\"\n\u0005\u000b@^Sy\n0\n@Mk@^S0\n@Ml#\ns;(7)\nrespectively. Finally, the \ructuation correlator Dis\nexpressed as [9]\nDkl(M) =~X\n\u000b;\u000b0Zd\"\n2\u0019f\u000b(1\u0000f\u000b0)\n\u0002Tr2\n4\u0005\u000b \n^Sy\n0@^S0\n@Mk!y\n\u0005\u000b0 \n^Sy\n0@^S0\n@Ml!3\n5\ns:(8)\nIn these expressions, \u000b=L;R denotes the reser-\nvoirs with electron distribution function f\u000b, \u0005\u000bis a\nprojector onto the channels of lead \u000b, and Tr traces\nover the lead channels.\n2.3. Corrections to the adiabatic approximation of\nthe S-matrix\nIn order to go beyond linear response in eVand\n_M, we must consider the electronic S-matrix in thepresence of the time-dependent magnetization M(t)\nand expand it to linear order in the magnetization\n\\velocity\" _M(t). This can be done, e.g., by starting\nfrom the full Floquet scattering matrix SF\n\u000b;\f(\"n;\")\nfor a periodic driving with period ![32]. The in-\ndices\u000band\flabel the scattering channels of the\ncoherent conductor and the arguments denote the\nenergies\"of the incoming electron in channel \u000band\n\"n=\"+n~!of the outgoing electron in channel \f.\nFor small driving frequency !, the Floquet scatter-\ning matrix can be expanded in powers of ~!,\n^SF(\"n;\") = ^S0\nn(\") +n~!\n2@^S0\nn(\")\n@\"\n+~!^An(\") +O(\"2): (9)\nHere ^S0\nn(\") is the Fourier transform of the frozen\nscattering matrix S0(M(t)) introduced above,\n^S0(M(t)) =1X\nn=\u00001e\u0000in!t^S0\nn(\"): (10)\nThe matrix ^An(\"), \frst introduced by Moskalets\nand B uttiker, is the \frst adiabatic correction to the\nadiabatic S-matrix and can be transformed in a sim-\nilar way to\n^A(t;\") =1X\nn=\u00001e\u0000in!t^An(\") = _M(t)\u0001^A(t;\"):(11)\nThe matrix ^An(\") can be straightforwardly calcu-\nlated from the retarded Green function of the device\n(see Refs. [20, 9]).\nWe are now in a position to give expressions for\nthe Gilbert damping to next order in the adiabatic\napproximation. (The spin-transfer torque and the\n\ructuation correlator remain unchanged.) To do\nso, we split the Gilbert matrix \rinto its symmetric\nand antisymmetric parts,\n\r=\rs+\ra: (12)\nStrictly speaking, it is only the symmetric part\nwhich corresponds to Gilbert damping. The anti-\nsymmetric part simply renormalizes the precession\nfrequency. One \fnds [9]\n\rkl\ns(M) =\u0000~X\n\u000bZd\"\n4\u0019f0\n\u000bTr\"\n\u0005\u000b@^Sy\n0\n@Mk@^S0\n@Ml#\ns\n+X\n\u000bZd\"\n2\u0019if\u000bTr\"\n\u0005\u000b \n@^Sy\n0\n@Mk^Al\u0000^Ay\nl@^S0\n@Mk!#\ns(13)\n4for the symmetric contribution. It can be seen, the\nsecond line is a pure nonequilibrium contribution\n(/eV~!). Similarly, the antisymmetric part of the\nGilbert damping can be written as [9]\n\rkl\na(M) =\u0000~X\n\u000bZd\"\n2\u0019if\u000b(\")\n\u0002Tr\"\n\u0005\u000b \n^Sy\n0@^Ak\n@Ml\u0000@^Ay\nk\n@Ml^S0!#\na;(14)\nwhich is really a renormalization of the precession\nfrequency as mentioned above.\n3. S-matrix theory of a nanomagnet coupled\nto a quantum spin Hall edge\nWe now apply the above theory to a magnetic\nisland coupled to a quantum spin Hall edge as\nsketched in Fig. 1. The quantum spin Hall edge sup-\nports a Kramers doublet of edge states. The mag-\nnetization M=M?cos\u0012ex+M?sin\u0012ey+Mzez\nof the magnetic island induces a Zeeman \feld JM\nacting on the electrons along the section of length\nLof the edge state which is covered by the magnet.\nThis Zeeman \feld causes backscattering between\nthe edge modes and induces a gap \u0001 = JM?~=2\n[1]. Linearizing the dispersion of the edge modes,\nthe electronic Hamiltonian takes the form [2]\n^H= (vp\u0000JMz)^\u001bz+\u0001(x) (cos\u0012^\u001bx+ sin\u0012^\u001by):(15)\nHere, the\u001bjdenote Pauli matrices in spin space\nand \u0001(x) is nonzero only over the region of length\nLcovered by the magnetic island. We have assumed\nfor simplicity that the spin Hall edge conserves \u001bz.\nThen, a static island magnetization induces a gap\nwhenever it has a component perpendicular to the\nz-direction. Indeed, ^His easily diagonalized for a\nspatially uniform coupling between edge modes and\nmagnet, and the spectrum\nEp=p\n(vp\u0000JMz)2+ \u00012 (16)\nhas a gap \u0001.\nIn the following, we assume that the easy-plane\nanisotropy D > 0 is su\u000eciently large so that the\nmagnetization entering the electronic Hamiltonian\ncan be taken in the xy-plane, i.e., Mz'0. (How-\never, we will have to keep Mzin the LLG equation\nwhen it is multiplied by the large anisotropy D.)\nThe electronic Hamiltonian (15) is equivalent to\nthe electronic Hamiltonian of the Thouless motor\nconsidered in Ref. [5]. Following this reference, wecan readily derive the frozen scattering matrix an-\nalytically [5],\n^S0=1\n\u0003\u0012\u0000iei\u0012\u0015 1\n1\u0000ie\u0000i\u0012\u0015\u0013\n; (17)\nwhere we have de\fned the shorthands\n\u0003 = cos \u001eL\u0000i\"p\n\"2\u0000\u00012sin\u001eL;\n\u0015=\u0001p\n\"2\u0000\u00012sin\u001eL (18)\nwith\n\u001eL(\") =L\n~vp\n\"2\u0000\u00012: (19)\nNote that these expressions are exact for any Land\nvalid for energies \"both inside and outside the gap.\nWe can now use this scattering matrix to eval-\nuate the various coe\u000ecients in the LLG equation,\nemploying the expressions given in Sec. 2.2. As-\nsuming zero temperature, we \fnd\nB0=eV\n2\u0019M\u0018(\u0016)e\u0012; (20)\nfor the spin transfer torque at arbitrary chemical\npotential\u0016. Here, we have de\fned the function\n\u0018(\u0016) =\u00012sin2\u001eL\nj\u00162\u0000\u00012jcos2\u001eL+\u00162sin2\u001eL(21)\nwith\u001eL=\u001eL(\u0016) (see Fig. 2). Below, we will iden-\ntify\u0018with the charge pumped between the reser-\nvoirs during one precessional period of the mag-\nnetization M. The vector B0points in the az-\nimuthal direction in the magnetization plane and\nindeed corresponds to a spin-transfer torque. Sim-\nilarly, we can substitute Eq. (17) into Eq. (13) for\nthe Gilbert damping and \fnd that the only nonzero\ncomponent of the tensor \ris\n\r\u0012\u0012=~\n2\u0019M2\u0018(\u0016): (22)\nSimilarly,\nD\u0012\u0012=~eV\n\u0019M2\u0018(\u0016) (23)\nis the only nonzero component of the \ructuation\ncorrelator. It is interesting to note that this yields\nan e\u000bective \ructuation-dissipation relation D\u0012\u0012=\n2Te\u000b\r\u0012\u0012with e\u000bective temperature Te\u000b=eV.\n5With these results, we can now write the LLG\nequation for the nanomagnet coupled to the helical\nedge state,\n_M =DM\u0002Mzez+\u0018(eV\u0000~_\u0012)\n2\u0019MM\u0002e\u0012\n+M\u0002\u000eB; (24)\nwhere\u0018=\u0018(\u0016), we have expressed _M'M_\u0012e\u0012, and\nassumed zero external magnetic \feld B. This com-\npletes our scattering-theory derivation of the LLG\nequation and generalizes the result obtained in Ref.\n[2] on phenomenological grounds in several respects.\nEquation (24) applies also for \fnite-length magnets\nand chemical potentials both inside and outside the\nmagnetization-induced gap of the edge-state spec-\ntrum. Moreover, the identi\fcation of the _\u0012-term\nas a damping term necessitates the inclusion of the\nLangevin torque \u000eB. Indeed, Ref. [2] refers to the\nentire term involving eV\u0000~_\u0012as the spin-transfer\ntorque. In contrast, our derivation produces the\nterm involving eValready in zeroth order in mag-\nnetization \\velocity\" _M, while the _\u0012term appears\nonly to linear order. Thus, the latter term is re-\nally a conrtribution to damping and related to the\nenergy dissipated in the electron system due to the\ntime dependence of the magnetization.\n4. E\u000eciency of the nanomagnet as a motor\nWhile the electronic Hamiltonian for the edge\nmodes is equivalent to that of the Thouless mo-\ntor discussed in Ref. [5], the LLG equation for the\nmacrospin di\u000ber from the equation of motion of the\nmechanical degrees of freedom discussed in Ref. [5].\nIn this section, we discuss the energetics and the\ne\u000eciency of the magnetic Thouless motor against\nthe backdrop of its mechanical cousin.\nThe dynamics of the macrospin is easily ob-\ntained from the LLG equation (24) [2]. For a large\nanisotropy and thus small Mz, we need to retain\nthez-component of Monly in combination with the\nlarge anisotropy D. Then, the steady-state value of\nMzis \fxed by the \u0012-component of the LLG equa-\ntion,\nMz=\u0000_\u0012\nD: (25)\nThe precessional motion of Mabout thez-axis is\ngoverned by the z-component of the LLG equation,\nwhich yields\n_\u0012=eV\n~(26)and hence Mz=\u0000eV=(~D). It is interesting to\nnote that the angular frequency _\u0012of the preces-\nsion is just given by the applied bias voltage, in-\ndependent of the damping strength. This should\nbe contrasted with the mechanical Thouless motor.\nHere, the motor degree of freedom satis\fes a New-\nton equation of motion which is second order in\ntime. Thus, the frequency of revolution is inversely\nproportional to the damping coe\u000ecient.\nIn steady state, the magnetic Thouless motor bal-\nances the energy provided by the voltage source\nthrough the spin-transfer torque B0against the dis-\nsipation through Gilbert damping due to the intrin-\nsic coupling between magnetic moment and elec-\ntronic degrees of freedom. It is instructive to look at\nthese contributions independently. The work per-\nformed by the spin-transfer torque per precessional\nperiod is given by\n\u0001Wspin\u0000transfer =Z2\u0019=_\u0012\n0dtB0\u0001_M: (27)\nWriting this as an integral over a closed loop of\nthe magnetization Mand inserting the S-matrix\nexpression (6), we \fnd\n\u0001Wspin\u0000transfer =X\n\u000bZd\"\n2\u0019if\u000b\n\u0002I\ndM\u0001Tr\"\n\u0005\u000b^Sy\n0@^S0\n@M#\n: (28)\nWithout applied bias, the integrand is just the gra-\ndient of a scalar function and the integral vanishes.\nThus, we expand to linear order in the applied bias\nand obtain\n\u0001Wspin\u0000transfer =ieV\n4\u0019\n\u0002X\n\u000bI\ndM\u0001Tr\"\n(\u0005L\u0000\u0005R)^Sy\n0@^S0\n@M#\n:(29)\nComparing Eq. (29) with the familiar S-matrix ex-\npression for the pumped charge [23], the right-hand\nside can now be identi\fed as the bias voltage multi-\nplied by the charge pumped between the reservoirs\nduring one revolution of the magnetization,\n\u0001Wspin\u0000transfer =QpV: (30)\nWith every revolution of the magnetization, a\nchargeQpis pumped between the reservoirs. The\ncorresponding gain QpVin electrical energy is driv-\ning the magnetic Thouless motor. This result can\nalso be written as\n_Wspin\u0000transfer =QpV\n2\u0019_\u0012 (31)\n6for the power provided per unit time by the voltage\nsource.\nThe relation between spin-transfer torque and\npumped charge also allows us to identify the func-\ntion\u0018(\u0016) appearing in the LLG equation as the\ncharge in units of epumped between the reservoirs\nduring one precessional period of the macrospin,\nQp=e\u0018: (32)\nThis can be obtained either by deriving the pumped\ncharge explicitly from the S-matrix expression or by\nevaluating Eq. (27) using the explicit expression Eq.\n(20).\nThe electrical energy gain is compensated by the\nenergy dissipated through Gilbert damping. The\ndissipated energy per period is given by\n\u0001WGilbert =Z2\u0019=_\u0012\n0dt_MT\r_MT\n= 2\u0019M2\r\u0012\u0012_\u0012: (33)\nUsing Eq. (22), this yields the dissipated energy\n\u0001WGilbert =\u0018~_\u0012 (34)\nper precessional period or\n_WGilbert =\u0018~\n2\u0019_\u00122(35)\nper unit time. These expressions have a simple\ninterpretation. Due to the \fnite frequency of the\nmagnetization precession, each pumped charge ab-\nsorbs on average an energy ~_\u0012which is then dissi-\npated in the reservoirs.\nArmed with these results, we can \fnally discuss\nthe e\u000eciency of a magnetic Thouless motor and fol-\nlow the framework introduced in Ref. [40] to de\fne\nan appropriate \fgure of merit (analogous to the ZT\nvalue of thermoelectrics). Imagine the same setup\nas in Fig. 1, but with an additional load coupled\nto the magnetization. We can now de\fne the e\u000e-\nciency of the magnetic Thouless motor as the ratio\nof the power delivered to the load and the electri-\ncal powerIVprovided by the voltage source. In\nsteady state, the power delivered to the load has\nto balance against the power provided by the elec-\ntrons, i.e., Bel\u0001_M. Thus, we can write the e\u000eciency\nas\n\u0011=_W\nIV; (36)\n00.20.40.60.81ξ,ηmax0\n0.5 1 1.5 2 2.5 3 µ/Δ\n00.20.40.60.81ξ,ηmaxFigure 2: (Color online) The parameter \u0018(dashed lines) en-\ntering the coe\u000ecients of the LLG equation and the maximal\ne\u000eciency\u0011max(solid lines) of the motor for a \fxed voltage\nV. Upper and lower panels correspond to nanomagnets of\nlengthL=~v=\u0001 andL= 10 ~v=\u0001, respectively.\nwhere\n_W= _Wspin\u0000torque\u0000_WGilbert\n=\u0018\n2\u0019eV_\u0012\u0000\u0018~\n2\u0019_\u00122: (37)\nThe total charge current \rowing along the topolog-\nical insulator edge averaged over the cycle is the\nsum of the dccurrentGVdriven by the voltage,\nwhereGis the dcconductance of the device, and\nthe pumping current Qp_\u0012=(2\u0019),\nI=GV+e\u0018\n2\u0019_\u0012: (38)\nWe can now optimize the e\u000eciency of the motor\nat a given bias Vas function of the frequency _\u0012of\nthe motor revolution. Note that due to the load,\nthe latter is no longer tied to the bias voltage eV.\nThis problem is analogous to the problem of the\noptimal e\u000eciency of a thermoelectric device which\nleads to the de\fnition of the important ZT value.\nThis analogy was discussed explicitly in Ref. [40].\nApplying the results of this paper to the present\ndevice yields the maximal e\u000eciency\n\u0011max=p1 +\u0010\u00001p1 +\u0010+ 1; (39)\nwith a \fgure of merit \u0010analogous to the ZT value\nde\fned by\n\u0010=e2\u0018(\u0016)\nhG(\u0016); (40)\n7where\u0018(\u0016) is de\fned in Eq. (21) and the conduc-\ntance reads\nG(\u0016) =e2\nhj\u00162\u0000\u00012j2\nj\u00162\u0000\u00012jcos2\u001eL+\u00162sin2\u001eL(41)\nas obtained from the Landauer-B uttiker equation.\nAs in thermoelectrics, the maximum e\u000eciency is\nrealized for \u0010!1 which requires a \fnite pumped\ncharge at zero conductance. Unlike thermoelectrics,\nthe motor e\u000eciency is bounded by \u0011= 1 instead of\nthe Carnot e\u000eciency. This re\rects the fact that\nelectrical energy can be fully converted into mag-\nnetic energy. Speci\fcally, unit e\u000eciency is reached\nin the limit of a true Thouless motor with zero\ntransmission when the Fermi energy falls into the\ngap and nonzero and quantized pumped charge per\nperiod. This can be realized to a good approxi-\nmation for a su\u000eciently long magnet, as seen from\nthe lower panel in Fig. 2. For chemical potentials\noutside the gap, the conductance and the pumped\ncharge exhibit Fabry-Perot resonances. This yields\na distinct sequence of maxima and minima in the\ne\u000eciency. For shorter magnets, the conductance\nremains nonzero within the gap, leading to lower\ne\u000eciencies. This is shown in the upper panel of\nFig. 2. Moreover, the Fabry-Perot resonances are\nwashed out, so that there is only a feature at the gap\nedge where the conductance vanishes while \u0018!1=2\nfor arbitrary L.\n5. Conclusions\nImplementing directional motion of a mechani-\ncal or magnetic degree of freedom is a fundamental\nproblem of nanoscale systems. An attractive gen-\neral mechanism relies on running quantum pumps\nin reverse. This is the underlying principle of adia-\nbatic quantum motors which drive periodic motion\nof a classical motor degree of freedom by applying a\ntransport current. In this paper, we emphasize that\na magnetic island coupled to a quantum spin Hall\nedge, recently discussed by Meng et al. [2], is just\nsuch an adiabatic quantum motor. We derive the\nLandau-Lifshitz-Gilbert equation for the magneti-\nzation dynamics from a general scattering-theory\napproach to adiabatic quantum motors, providing\na microscopic derivation of spin-transfer torque,\nGilbert damping, and Langevin torque. This ap-\nproach does not only provide a detailed microscopic\nunderstanding of the operation of the device but\nalso allows one to discuss its e\u000eciency. We \fndthat the device naturally approaches optimal e\u000e-\nciency when the chemical potential falls into the\nmagnetization-induced gap and the conductance is\nexponentially suppressed. This makes this system\na Thouless motor and possibly its most experimen-\ntally feasible variant to date.\nSeveral issues are left for future work. While we\nderived microscopic expressions for the Langevin\ntorque, we have not explored its consequences for\nthe motor dynamics. It should also be interesting\nto consider thermal analogs driven by a tempera-\nture gradient instead of a bias voltage. Inducing the\nmagnetization precession by a temperature gradient\nwould realize a quantum heat engine. Conversely,\nforcing a magnetic precession can be used to pump\nheat against a temperature gradient. Setups with\nseveral magnetic islands could be engineered to ef-\nfect exchange of charge and energy without employ-\ning a dcbattery. These devices have been explored\nin the literature on quantum pumps [41, 42, 43] and\ntheir e\u000eciencies could be analyzed in the thermo-\nelectric framework of Ref. [40].\nAcknowledgement\nWe thank Gil Refael and Ari Turner for dis-\ncussions. This work was supported by CON-\nICET, MINCyT and UBACyT (L.A.) as well\nas the Deutsche Forschungsgemeinschaft and the\nHelmholtz Virtual Institute New States of Matter\nand Their Excitations (F.v.O.). L.A. thanks the\nICTP Trieste for hospitality and the Simons Foun-\ndation for support. F.v.O. thanks the KITP Santa\nBarbara for hospitality during the \fnal preparation\nof this manuscript. 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B\n75, 245420.\n[42] Juergens, S., Haupt, F., Moskalets, M., and\nSplettstoesser, J., 2013, Thermoelectric performance of\na driven double quantum dot, Phys. Rev. B 87, 245423.\n[43] Moskalets, M. and B uttiker, M., 2009, Heat production\nand current noise for single- and double-cavity quantum\ncapacitors, Phys. Rev. B 80, 081302.\n9"    },    {        "title": "1507.08227v2.Spin_dynamics_and_relaxation_in_the_classical_spin_Kondo_impurity_model_beyond_the_Landau_Lifschitz_Gilbert_equation.pdf",        "content": "Spin dynamics and relaxation in the classical-spin Kondo-impurity model beyond the\nLandau-Lifschitz-Gilbert equation\nMohammad Sayad and Michael Pottho\u000b\nI. Institut f ur Theoretische Physik, Universit at Hamburg, Jungiusstra\u0019e 9, 20355 Hamburg, Germany\nThe real-time dynamics of a classical spin in an external magnetic \feld and locally exchange\ncoupled to an extended one-dimensional system of non-interacting conduction electrons is studied\nnumerically. Retardation e\u000bects in the coupled electron-spin dynamics are shown to be the source\nfor the relaxation of the spin in the magnetic \feld. Total energy and spin is conserved in the\nnon-adiabatic process. Approaching the new local ground state is therefore accompanied by the\nemission of dispersive wave packets of excitations carrying energy and spin and propagating through\nthe lattice with Fermi velocity. While the spin dynamics in the regime of strong exchange coupling\nJis rather complex and governed by an emergent new time scale, the motion of the spin for\nweakJis regular and qualitatively well described by the Landau-Lifschitz-Gilbert (LLG) equation.\nQuantitatively, however, the full quantum-classical hybrid dynamics di\u000bers from the LLG approach.\nThis is understood as a breakdown of weak-coupling perturbation theory in Jin the course of time.\nFurthermore, it is shown that the concept of the Gilbert damping parameter is ill-de\fned for the\ncase of a one-dimensional system.\nPACS numbers: 75.78.-n, 75.78.Jp, 75.60.Jk, 75.10.Hk, 75.10.Lp\nI. INTRODUCTION\nThe Landau-Lifshitz-Gilbert (LLG) equation1{3has\noriginally been considered to describe the dynamics of\nthe magnetization of a macroscopic sample. Nowadays it\nis frequently used to simulate the dynamics of many mag-\nnetic units coupled by exchange or magnetostatic interac-\ntions, i.e., in numerical micromagnetics.4The same LLG\nequation can be used on an atomistic level as well.5{9For\na suitable choice of units and for several spins Sm(t) at\nlattice sites m, it has the following structure:\ndSm(t)\ndt=Sm(t)\u0002B+X\nnJmnSm(t)\u0002Sn(t)\n+X\nn\u000bmnSm(t)\u0002dSn(t)\ndt: (1)\nIt consists of precession terms coupling the spin at site\nmto an external magnetic \feld Band, via exchange\ncouplingsJmn, to the spins at sites n. Those pre-\ncession terms typically have a clear atomistic origin,\nsuch as the Ruderman-Kittel-Kasuya-Yoshida (RKKY)\ninteraction10{12which is mediated by the magnetic po-\nlarization of conduction electrons. The non-local RKKY\ncouplingsJmn=J2\u001fmnare given in terms of the ele-\nments\u001fmnof the static conduction-electron spin suscep-\ntibility and the local exchange Jbetween the spins and\nthe local magnetic moments of the conduction electrons.\nOther possibilities comprise direct (Heisenberg) exchange\ninteractions, intra-atomic (Hund's) couplings as well as\nthe spin-orbit and other anisotropic interactions. The re-\nlaxation term, on the other hand, is often assumed as lo-\ncal,\u000bmn=\u000emn\u000b, and represented by purely phenomeno-\nlogical Gilbert damping constant \u000bonly. It describes the\nangular-momentum transfer between the spins and a usu-\nally unspeci\fed heat bath.On the atomistic level, the Gilbert damping must be\nseen as originating from microscopic couplings of the\nspins to the conduction-electron system (as well as to\nlattice degrees of freedom which, however, will not be\nconsidered here). There are numerous studies where the\ndamping constant, or tensor, \u000bhas been computed nu-\nmerically from a more fundamental model including elec-\ntron degrees of freedom explicitly13{15or even from \frst\nprinciples.16{21All these studies rely on two, partially\nrelated, assumptions: (i) The spin-electron coupling J\nis weak and can be treated perturbatively to lowest or-\nder, i.e., the Kubo formula or linear-response theory is\nemployed. (ii) The classical spin dynamics is slow as\ncompared to the electron dynamics. These assumptions\nappear as well justi\fed but they are also necessary to\nachieve a simple e\u000bective spin-only theory by eliminat-\ning the fast electron degrees of freedom.\nThe purpose of the present paper is to explore the\nphysics beyond the two assumptions (i) and (ii). Us-\ning a computationally e\u000ecient formulation in terms of\nthe electronic one-particle reduced density matrix, we\nhave set up a scheme by which the dynamics of classi-\ncal spins coupled to a system of conduction electrons can\nbe treated numerically exactly. The theory applies to ar-\nbitrary coupling strengths and does not assume a separa-\ntion of electron and spin time scales. Our approach is a\nquantum-classical hybrid theory22which may be charac-\nterized as Ehrenfest dynamics, similar to exact numerical\ntreatments of the dynamics of nuclei, treated as classical\nobjects, coupled to a quantum system of electrons (see,\ne.g., Ref. 23 for an overview). Some other instructive ex-\namples of quantum-classical hybrid dynamics have been\ndiscussed recently.24,25\nThe obvious numerical advantage of an e\u000bective spin-\nonly theory, as given by LLG equations of the form (1),\nis that in solving the equations of motion there is only\nthe time scale of the spins that must be taken care of. AsarXiv:1507.08227v2  [cond-mat.mes-hall]  28 Nov 20152\ncompared to our hybrid theory, much larger time steps\nand much longer propagation times can be achieved. Op-\nposed to ab-initio approaches16,17,26we therefore con-\nsider a simple one-dimensional non-interacting tight-\nbinding model for the conduction-electron degrees of free-\ndom, i.e., electrons are hopping between the nearest-\nneighboring sites of a lattice. Within this model ap-\nproach, systems consisting of about 1000 sites can be\ntreated easily, and we can access su\u000eciently long time\nscales to study the spin relaxation. An equilibrium state\nwith a half-\flled conduction band is assumed as the ini-\ntial state. The subsequent dynamics is initiated by a\nsudden switch of a magnetic \feld coupled to the classical\nspin. The present study is performed for a single spin,\ni.e., we consider a classical-spin Kondo-impurity model\nwith antiferromagnetic local exchange coupling J, while\nthe theory itself is general and can be applied to more\nthan a single or even to a large number of spins as well.\nAs compared to the conventional (quantum-spin)\nKondo model,27,28the model considered here does not\naccount for the Kondo e\u000bect and therefore applies to sit-\nuations where this is absent or less important, such as\nfor systems with large spin quantum numbers S, strongly\nanisotropic systems or, as considered here, systems in a\nstrong magnetic \feld. To estimate the quality of the\nclassical-spin approximation a priori is di\u000ecult.29{31For\none-dimensional systems, however, a quantitative study\nis possible by comparing with full quantum calculations\nand will be discussed elsewhere.32\nThere are di\u000berent questions to be addressed: For\ndimensional reasons, one should expect that linear-\nresponse theory, even for weak J, must break down at\nlong times. It will therefore be interesting to compare\nthe exact spin dynamics with the predictions of the LLG\nequation for di\u000berent J. Furthermore, the spin dynamics\nin the long-time limit can be expected to be sensitively\ndependent on the low-energy electronic structure. We\nwill show that this has important consequences for the\ncomputation of the damping constant \u000band that\u000bis\neven ill-de\fned in some cases. An advantage of a full\ntheory of spin and electron dynamics is that a precise\nmicroscopic picture of the electron dynamics is available\nand can be used to discuss the precession and relaxation\ndynamics of the spin from another, namely from the elec-\ntronic perspective. This information is in principle exper-\nimentally accessible to spin-resolved scanning-tunnelling\nmicroscope techniques33{36and important for an atom-\nistic understanding of nano-spintronics devices.37,38We\nare particularly interested in the physics of the system\nin the strong- Jregime or for a strong \feld Bwhere the\ntime scales of the spin and the electron dynamics become\ncomparable. This has not yet been explored but could\nbecome relevant to understand real-time dynamics in re-\nalizations of strong- JKondo-lattice models by means of\nultracold fermionic Yb quantum gases trapped in optical\nlattices.39,40\nThe paper is organized as follows: We \frst introduce\nthe model and the equations of motion for the exactquantum-classical hybrid dynamics in Sec. II and discuss\nsome computational details in Sec. III. Sec. IV provides\na comprehensive discussion of the relaxation of the clas-\nsical spin after a sudden switch of a magnetic \feld. The\nreversal time as a function of the interaction and the \feld\nstrength is analyzed in detail. We then set the focus on\nthe conduction-electron system which induces the relax-\nation of the classical spin by dissipation of energy. In Sec.\nV, the linear-response approach to integrate out the elec-\ntron degrees of freedom is carefully examined, including a\ndiscussion of the additional approximations that are nec-\nessary to re-derive the LLG equation and the damping\nterm in particular. Sec. VI summarizes the results and\nthe main conclusions.\nII. MODEL AND THEORY\nWe consider a classical spin SwithjSj= 1=2, which is\ncoupled via a local exchange interaction of strength Jto\nthe local quantum spin si0at the sitei0of a system of N\nitinerant and non-interacting conduction electrons. The\nconduction electrons hop with amplitude \u0000T(T > 0)\nbetween non-degenerate orbitals on nearest-neighboring\nsites of aD-dimensional lattice, see Fig. 1. Lis the\nnumber of lattice sites, and n=N=L is the average\nconduction-electron density.\nThe dynamics of this quantum-classical hybrid\nsystem22is determined by the Hamiltonian\nH=\u0000TX\nhiji;\u001bcy\ni\u001bcj\u001b+Jsi0S\u0000BS: (2)\nHere,ci\u001bannihilates an electron at site i= 1;:::;L with\nspin projection \u001b=\";#, andsi=1\n2P\n\u001b\u001b0cy\ni\u001b\u001b\u001b\u001b0ci\u001b0is\nthe local conduction-electron spin at i, where\u001bdenotes\nthe vector of Pauli matrices. The sum runs over the\ndi\u000berent ordered pairs hijiof nearest neighbors. Bis\nan external magnetic \feld which couples to the classical\nspin.\nTo be de\fnite, an antiferromagnetic exchange coupling\nJ > 0 is assumed. If Swas a quantum spin with\nS= 1=2, Eq. (2) would represent the single-impurity\nKondo model.27,28However, in the case of a classical spin\nconsidered here, there is no Kondo e\u000bect. The semiclas-\nsical single-impurity Kondo model thus applies to sys-\ntems where a local spin is coupled to electronic degrees\nof freedom but where the Kondo e\u000bect absent or sup-\npressed. This comprises the case of large spin quantum\nnumbersS, or the case of temperatures well above the\nKondo scale, or systems with a ferromagnetic Kondo cou-\nplingJ <0 where, for a classical spin, we expect a qual-\nitatively similar dynamics as for J >0.\nWe assume that initially, at time t= 0, the clas-\nsical spinS(t= 0) has a certain direction and that\nthe conduction-electron system is in the corresponding\nground state, i.e., the conduction electrons occupy the\nlowestNone-particle eigenstates of the non-interacting3\nHamiltonian Eq. (2) for the given S=S(t= 0) up to\nthe chemical potential \u0016. A non-trivial time evolution is\ninitiated if the initial direction of the classical spin and\nthe direction of the \feld Bare non-collinear.\nTo determine the real-time dynamics of the electronic\nsubsystem, it is convenient to introduce the reduced one-\nparticle density matrix. Its elements are de\fned as ex-\npectation values,\n\u001aii0;\u001b\u001b0(t)\u0011hcy\ni0\u001b0ci\u001bit; (3)\nin the system's state at time t. Att= 0 we have\u001a(0) =\n\u0002(\u0016\u0000T(0)). The elements of \u001a(0) are given by\n\u001ai\u001b;i0\u001b0(0) =X\nkUi\u001b;k\u0002(\u0016\u0000\"k)Uy\nk;i0\u001b0; (4)\nwhere \u0002 is the step function and where Uis the uni-\ntary matrix diagonalizing the hopping matrix T(0), i.e.,\nUyT(0)U=\"with the diagonal matrix \"given by the\neigenvalues of T(0). The hopping matrix at time tis\ncan be read o\u000b from Eq. (2). It comprises the physical\nhopping and the contribution resulting from the coupling\nterm. Its elements are given by\nTi\u001b;i0\u001b0(t) =\u0000T\u000ehii0i\u000e\u001b\u001b0+\u000eii0\u000ei0i0J\n2(S(t)\u001b)\u001b\u001b0:(5)\nHere\u000ehii0i= 1 ifi;i0are nearest neighbors and zero else.\nThere is a closed system of equations of motion for the\nclassical spin vector S(t) and for the one-particle density\nmatrix\u001a(t). The time evolution of the classical spin is\ndetermined via ( d=dt)S(t) =fS;Hclass:gby the classical\nHamilton function Hclass:=hHi. This equation of mo-\ntion is the only known way to consistently describe the\ndynamics of quantum-classical hybrids (see Refs. 22,41,42\nand references therein for a general discussion). The\nPoisson bracket between arbitrary functions AandBof\nthe spin components is given by,43,44\nfA;Bg=X\n\u000b;\f;\r\"\u000b\f\r@A\n@S\u000b@B\n@S\fS\r; (6)\nwhere the sums run over x;y;z and where \"\u000b\f\ris the\nfully antisymmetric \"-tensor. With this we \fnd\nd\ndtS(t) =Jhsi0it\u0002S(t)\u0000B\u0002S(t): (7)\nThis is the Landau-Lifschitz equation where the expec-\ntation value of the conduction-electron spin at i0is given\nby\nhsi0it=1\n2X\n\u001b\u001b0\u001ai0\u001b;i0\u001b0(t)\u001b\u001b0\u001b; (8)\nand where Jhsi0itacts as an e\u000bective time-dependent\ninternal \feld in addition to the external \feld B.\nS(t)JTi0BFIG. 1: (Color online) Classical spin S(t) coupled via an an-\ntiferromagnetic local exchange interaction of strength Jto a\nsystem of conduction electrons hopping with nearest-neighbor\nhopping amplitude Tover the sites of a one-dimensional lat-\ntice with open boundaries. The spin couples to the central\nsitei0of the system and is subjected to a local magnetic \feld\nof strength B.\nThe equation of motion for hsiitreads as\nd\ndthsiit=\u000eii0JS(t)\u0002hsiit\n+Tn:n:X\nj1\n2iX\n\u001b\u001b0(hcy\ni\u001b\u001b\u001b\u001b0cj\u001b0it\u0000c.c.);(9)\nwhere the sum runs over the nearest neighbors of i. The\nsecond term on the right-hand side describes the coupling\nof the local conduction-electron spin to its environment\nand the dissipation of spin and energy into the bulk of\nthe system (see below). Apparently, the system of equa-\ntions of motion can only be closed by considering the\ncomplete one-particle density matrix Eq. (3). It obeys a\nvon Neumann equation of motion,\nid\ndt\u001a(t) = [T(t);\u001a(t)] (10)\nas is easily derived, e.g., from the Heisenberg equation of\nmotion for the annihilators and creators.\nAs is obvious from the equations of motion, the real-\ntime dynamics of the quantum-classical Kondo-impurity\nmodel on a lattice with a \fnite but large number of sites L\ncan be treated numerically exactly (see also below). Nev-\nertheless, the model comprises highly non-trivial physics\nas the electron dynamics becomes e\u000bectively correlated\ndue to the interaction with the classical spin. In addi-\ntion, the e\u000bective electron-electron interaction mediated\nby the classical spin is retarded: electrons scattered from\nthe spin at time twill experience the e\u000bects of the spin\ntorque exerted by electrons that have been scattered from\nthe spin at earlier times t0<t.\nIII. COMPUTATIONAL DETAILS\nEqs. (5), (7), (8) and (10) represent a coupled non-\nlinear system of \frst-order ordinary di\u000berential equations\nwhich can be solved numerically. By blocking up the\nvon Neumann equation (10), the di\u000berential equations\nare written in a standard form _y=f(y(t);t), wherey(t)\nis a high-dimensional vector, such that an explicit Runge-\nKutta method can be applied. A high-order propagation4\ntechnique is used45which provides the numerically exact\nsolution up to 6-th order in the time step \u0001 t. For a typ-\nical system consisting of about L= 103sites this implies\nthat\u0018106coupled equations are solved.\nWe consider a one-dimensional system with open\nboundaries consisting of L= 1001 sites and a local per-\nturbation at the central site i0of the system, see Fig. 1.\nFor a half-\flled tight-binding conduction band the Fermi\nvelocityvF= 2Troughly determines the maximum speed\nof the excitations and de\fnes a \\light cone\".46,47This\nmeans that \fnite-size e\u000bects due to scattering at the sys-\ntem boundaries become relevant after a propagation time\ntmax\u0018500 (in units of 1 =T). A time step \u0001 t= 0:1 is\nusually su\u000ecient for reliable numerical results up to tmax,\ni.e., about 5000 time steps are performed. The compu-\ntational cost is moderate, and calculations can be per-\nformed in a few hours on a standard desktop computer.\nAssuming, for example, that B= (0;0;B), the Hamil-\ntonian is invariant under rotations around the zaxis. It\nis then easily veri\fed that not only the length of the spin\njSj= 1=2 is conserved but also the total number of con-\nduction electrons,\nNtot=X\ni\u001bhcy\ni\u001bci\u001bi; (11)\nthez-component of the total spin,\nStot;z=Sz+X\nihsizi; (12)\nas well as the total energy,\nEtot=hHi= tr (\u001a(t)T(t))\u0000BS(t): (13)\nDespite the fact that the model does not include a di-\nrect (e.g., Coulomb) interaction among the conduction\nelectrons, the average occupation numbers of the basis of\none-particle states in which the hopping matrix T(t) is\ndiagonal at time tarenotconserved. This is due to the\ne\u000bective retarded interaction mediated by the classical\nspin. Hence, the system is not integrable, unlike a free\nfermion gas. The conservation of the above-mentioned\nglobal observables serves as a sensitive check for the ac-\ncuracy of the numerical procedure.\nIV. COUPLED SPIN AND ELECTRON\nDYNAMICS\nA. Spin relaxation\nFig. 2 shows the real-time dynamics of the classical\nspin forJ= 1. Energy and time units are \fxed by the\nnearest-neighbor hopping T= 1 throughout the paper.\nInitially, for t= 0, the spin is oriented (almost) antipar-\nallel to the external local \feld B= (0;0;B) withB= 1,\ni.e., initially Sx(0) =1\n2sin#,Sy(0) = 0,Sz(0) =\u00001\n2cos#\n-0.5-0.3-0.10.10.30.5-0.5-0.3-0.10.10.30.5-0.5-0.3-0.10.10.30.5\nxyz\n10−1100101102\ntimet-0.50-0.250.000.250.50/vectorSSx\nSzFIG. 2: (Color online) Real-time dynamics of the classical\nspin. Upper panel: Bloch sphere representation. Lower panel:\nxandzcomponent of S(t) (jSj= 1=2). Calculations for\nexchange coupling J= 1 and \feld strength B= 1 and for\na system of L= 1001 sites. ( L= 1001 is kept \fxed for the\nrest of the paper). Energy and time units are \fxed by the\nnearest-neighbor hopping T= 1.\nwhere a non-zero but small polar angle #=\u0019=50 is nec-\nessary to slightly break the symmetry of the initial state\nand to start the dynamics.\nFor the same setup, the Landau-Lifschitz-Gilbert equa-\ntion would essentially predict two e\u000bects: \frst, a preces-\nsion of the classical spin around the \feld direction with\nLarmor frequency !L=Bz, and second, a relaxation\nof the spin to the equilibrium state with Sparallel to\nB=Bezfort!1 . Both e\u000bects are also found in\nthe full dynamics of the quantum-classical hybrid model.\nThe frequency of the oscillation of Sx(t) that is seen in\nFig. 2 is!L, andSzis reversed after a few hundred time\nunits. The precessional motion is easily explained by the\ntorque on the spin exerted by the \feld according to Eq.\n(7). The explanation of the damping e\u000bect is more in-\nvolved:\nEven for the high \feld strength considered here, the\nspin dynamics is slow as compared to the characteristic\nelectronic time scale such that it could be reasonable to\nassume the electronic system being in its instantaneous5\n10\u0000210\u00001100101102timet3.083.103.123.14~S(t)h~si0it\u0000(t)~B\u0000(t)J=46810J=20xy\nFIG. 3: (Color online) Time dependence of the angle \r(t)\nenclosed by S(t) andhsi0itforB= 0:1 and di\u000berent Jas\nindicated.\nground state at any instant of time and corresponding\nto the con\fguration of the classical spin. This, however,\nwould imply that the expectation value hsi0itof the local\nconduction-electron moment at i0is always strictly par-\nallel toS(t) and, hence, there would not be any torque\nmediated by the exchange coupling JonS(t).\nIn fact, the direction of hsi0itis always somewhat be-\nhind the \\adiabatic direction\", i.e., behind \u0000S(t): This\nis shown in Fig. 3 for a \feld of strength B= 0:1 where\nthe spin dynamics is by a factor 10 slower, compared to\nFig. 2, and for di\u000berent stronger exchange couplings J.\nEven in this case the process is by no means adiabatic,\nand the angle \r(t) betweenS(t) andhsi0itis close to but\nclearly smaller than \r=\u0019at any instant of time. This\nnon-adiabaticity results from the fact that the motion\nof the classical spin a\u000bects the conduction electrons in\na retarded way, i.e., it takes a \fnite time until the local\nconduction-electron spin hsi0itati0reacts to the motion\nof the classical spin.\nThis retardation e\u000bect results in a torque Jhsi0it\u0002\nS(t)6= 0 exerted on the classical S(t) in the +zdirection\nwhich adds to the torque due to Band which drives the\nspin to its new equilibrium direction. Hence, retardation\nis the physical origin of the Gilbert spin damping.\nWith increasing time, the deviation of \r(t) from the\ninstantaneous equilibrium value \r=\u0019increases in mag-\nnitude, i.e., the direction of hsi0itis more and more be-\nhind the adiabatic direction, and the torque increases. Its\nmagnitude is at a maximum at the same time when the\noscillatingx;ycomponents of S(t) are at a maximum\n(see Fig. 2). The z-component of the torque does not\nvanish before the spin has reached its new equilibrium\npositionS(t)/ez.\nFig. 3 shows results for di\u000berent J. Generally, non-\nadiabatic e\u000bects show up if the typical time scale of the\ndynamics is faster than the relaxation time, i.e., the time\nnecessary to transport the excitation away from the lo-\ncationi0where it is created initially. Roughly this time\nscale is set by the inverse hopping 1 =T. One therefore\nexpects that, for \fxed T, a stronger Jimplies a stronger\n0 20 40 60 80 100\nexchange coupling J050100150200250300350400reversal time τ100∗τ1/τ2\nτ1\nτ2FIG. 4: (Color online) Time for a spin reversal Sz=1\n2cos(\u0019\u0000\n#) =\u00001\n2cos(#)!Sz=1\n2cos(#) for#=#1=\u0019=50 (reversal\ntime\u001c1) and for#=#2=\u0019=25 (reversal time \u001c2). Calcula-\ntions as a function of Jfor \fxedB= 0:1.\nretardation of the conduction-electron dynamics. The re-\nsults for di\u000berent Jshown in Fig. 3 in fact show that the\nmaximum deviation of \r(t) from the adiabatic direction\n\r=\u0019increases with increasing J(for very strong Jthe\ndynamics becomes much more complicated, see below).\nThis results in a stronger torque on S(t) inzdirection\nand thus in a stronger damping. The picture is also qual-\nitatively consistent with the LLG equation as the Gilbert\ndamping constant \u000bincreases with J.\nThe spin (almost) reverses its direction after a \fnite\nreversal time \u001cwhich is shown in Fig. 4 as a function\nofJ. Calculations have been performed for an initial\ndirection of the classical spin with Sz(0) =\u0000(1=2) cos#\nwith two di\u000berent polar angles #1;2. If#is su\u000eciently\nsmall, the results for di\u000berent #are expected to di\u000ber by\naJandBindependent constant factor only. As Fig. 4\ndemonstrates, the ratio \u001c1=\u001c2is in fact nearly constant.\nFor weakJand up to coupling strengths of about J.30,\nwe \fnd that the reversal time decreases with increasing\nJ. With increasing J, the retardation e\u000bect increases, as\ndiscussed above, and the stronger damping results in a\nshorter reversal time.\n0 2 4 6 8 10\n1/B0100200300400reversal time τ\nJ= 2\nJ= 4\nFIG. 5: (Color online) Reversal time \u001c1as function of Bfor\n\fxedJ= 2 andJ= 4 as indicated.6\nThe prediction of the LLG equation for the reversal\ntime of a single spin \u001ccan be derived analytically48and\nis given by\n\u001c/1 +\u000b2\n\u000b1\nBln\f\f\f1=2\u0000Sz(0)\n1=2 +Sz(0)\f\f\f: (14)\nHowever, down to the smallest Jfor which\u001ccan be cal-\nculated reliably, our results for the full spin dynamics do\nnot scale as \u001c/1=J2as one would expect for weak J\nassuming that \u000b/J2(see discussion in Sec. V B).\nTheBdependence of the reversal time is shown in Fig.\n5. With increasing \feld strength, the classical spin S(t)\nprecesses with a higher Larmor frequency !L\u0019Baround\nthezaxis. Hence, non-adiabatic e\u000bects increase. The\nstronger the \feld, the more delayed is the precessional\nmotion of the local conduction-electrons spin hsi0it. This\nresults in a stronger torque in + zdirection exerted on\nthe classical spin. Therefore, the relaxation is faster and\nthe reversal time \u001csmaller. For weak and intermediate\n\feld strengths, \u001cis roughly proportional to 1 =B. This is\nconsistent with the prediction of the LLG equation, see\nEq. (14).\nIn the limit of very strong \felds one would expect an\nincrease of the reversal time with increasing Bsince the\n\feld term will eventually dominate the dynamics, i.e.,\nonly the precessional motion survives which implies a di-\nverging reversal time. In fact, for a \feld strength exceed-\ning a critical strength Bc, which depends on J, there is\nno full relaxation any longer, and \u001c=1. This strong-\nBregime cannot be captured by the LLG equation and\ndeserves further studies.\nThe strong- Jregime is interesting as well. For cou-\npling strengths exceeding J\u001930 the reversal time in-\ncreases withJ(see Fig. 4). Eventually, the reversal time\nmust even diverge. This is obvious as the dynamics is\ndescribed by a simple two-spin model in the limit J=1\nwhich cannot show spin relaxation. The corresponding\nequations of motion are obtained by from Eqs. (7) and\n(9) by setting t= 0 andB= 0:\nd\ndtS(t) =Jhsi0it\u0002S(t);\nd\ndthsi0it=JS(t)\u0002hsi0it (15)\nNote that we have jhsi0itj= 1=2 forJ!1 . The equa-\ntions are easily solved by exploiting the conservation of\nthe total spin Stot=S(t) +hsi0it. Both spins precess\nwith constant frequency !0=JStotaroundStot. Their\ncomponents parallel to Stotare equal, and their compo-\nnents perpendicular to Stotare anti-parallel and of equal\nlength.\nHowever, the two-spin dynamics of the J=1limit\nis not stable against small perturbations. Fig. 6 shows\nthe classical spin dynamics of the full model (with T=\n1 andB= 0:1) for a very strong but \fnite coupling\nJ= 100. Here, the motion of the classical spin gets\nvery complicated as compared with the highly regular\n-0.50.00.5\n/vectorSx\ny\nz\n-2.00.02.0\nJ/angbracketleft/vector si0/angbracketright×/vectorS\n-0.50.00.5\n−/vectorB×/vectorS\n10−1100101102\ntimet3.103.153.20\nγ=/negationslash(/vectorS/vector s0)FIG. 6: (Color online) Real-time dynamics in the strong- J\nregime:J= 100 and B= 0:1.First panel: Classical spin\nS(t).Second panel: Torque on S(t) due to the exchange\ninteraction. Third panel: Torque on S(t) due to the \feld\nterm. Fourth panel: Angle enclosed by S(t) andhsi0it\nbehavior in the weak- Jregime (cf. Fig. 2). In particular,\nthez-component of S(t) is oscillating on nearly the same\nscale as the xandycomponents. This characteristic\ntime scale \u0001 t\u00194 of the oscillation corresponds to a\nfrequency!\u00191:5 which di\u000bers by more than an order\nof magnitude from both, the Larmor frequency B= 0:1\nand from the exchange-coupling strength J= 100. Note\nthat the oscillation of Sz(t) is actually the reason for the\nambiguity in the determination of the precise reversal\ntime and gives rise to the error bars in Fig. 4 for strong\nJ.\nWe attribute the complexity of the dynamics to the\nfact that the torque due to the \feld term and the torque\ndue to the exchange coupling are of comparable magni-\ntudes, see second and third panel in Fig. 6. It is interest-\ning that even for strong J, where one would expect S(t)\nandhsi0itto form a tightly bound local spin-zero state,\nthere is actually a small deviation from perfect antiparal-\nlel alignment, i.e., \r(t)6=\u0019, as can be seen in the fourth\npanel of Fig. 6. This results in a \fnite z-component of the\ntorque onS(t) which leads to a very fast reversal with\nSz(t)\u0019+0:5 at timet\u00192:1. Contrary to the weak-\ncoupling limit, however, the z-component of the torque\nchanges sign at this point and drives the spin back to the7\n\u0000z-direction. At t\u00193:2, however, the z-component of\nS(t) once more reverses its direction. Here, the torque\ndue to the exchange coupling vanishes completely as S(t)\nandhsi0itare perfectly antiparallel (see the \frst zero of\n\r(t) in the fourth panel). The motion continues due to\nthe non-zero \feld-induced torque. This pattern repeats\nseveral times. S(t) mainly oscillates within a plane in-\ncluding and slowly rotating around the z-axis.\nEventually, there is a perfect relaxation of the classical\nspin for large tbut in a very di\u000berent way as compared to\nthe weak-coupling limit. While the deviation from \r=\u0019\nis small at any instant of time as for weak J, the most\napparent di\u000berence is perhaps that the new ground state\nis approached with an oscillating behavior of \r(t) around\n\r=\u0019, i.e.,hsi0itmay run behind or ahead ofS(t) as\nwell.\nLet us summarize at this point the main di\u000berences\nbetween the quantum-classical hybrid and the e\u000bective\nLLG dynamics of the classical spin: For weak JandB,\nthe qualitative behavior, precessional motion and relax-\nation, is the same in both approaches. Quantitatively,\nhowever, the LLG equation is inconsistent with the ob-\nservedJdependence of the reversal time when assuming\n\u000b/J2. TheBdependence of 1 =\u001cis linear as expected\nfrom the LLG approach. For strong J, the spin dynamics\nqualitatively di\u000bers from LLG dynamics and gets more\ncomplicated with a new time scale emerging. Absence of\ncomplete relaxation, as observed in the strong- Blimit, is\nalso not accessible to an e\u000bective spin-only theory.\nB. Energy dissipation\nTo complete the picture of the relaxation dynamics\nof the classical spin, the discussion should also comprise\nthe dynamics of the electronic degrees of freedom. The\nspin relaxation must be accompanied by a dissipation of\nenergy and spin into the bulk of the electronic system\nsince the total energy and the total spin are conserved\nquantities, see Eqs. (12) and (13), while conservation of\nthe total particle number, Eq. (11), is trivially ensured by\nthe particle-hole symmetric setup considered here where\nthe average conduction-electron number at every site is\ntime-independent:P\n\u001bhni\u001bit= 1.\nThe total energy is given by Etot=hHi, see Eq. (13),\nand is a sum over di\u000berent contributions, Etot=EB(t)+\nEhop(t)+Eint(t), namely the interaction energy with the\n\feld\nEB(t) =\u0000BS(t); (16)\nthe kinetic (hopping) energy of the conduction-electron\nsystem\nEhop(t) =\u0000TX\nhijiX\n\u001bhcy\ni\u001bcj\u001bit; (17)\nand the exchange-interaction energy\nEint(t) =Jhsi0itS(t): (18)\n-0.50.00.5\nEB\n-1.2724-1.2718-1.2712energyEhop/L\nEtot/L\n-0.68-0.67-0.66\nEint\n100101102\ntimet−1.30−1.25−1.20−1.15\nei0±1\nei0±3\nei0±2ei0±50ei0±100FIG. 7: (Color online) Di\u000berent contributions to the total\nenergy as functions of time for J= 5 andB= 1. Top panel :\nEB, energy of the classical spin in the external \feld. Second\npanel :Ehop=L, kinetic (hopping) energy of the conduction-\nelectron system per site, and Etot=L, total energy per site.\nThird panel :Eint, exchange-interaction energy. The fourth\npanel shows the time-dependence of the total-energy density\nat di\u000berent distances from the site i0.\nThe time dependence of those contributions is shown in\nFig. 7 forJ= 5 andB= 1.\nThe top panel of Fig. 7 shows that the system re-\nleases the interaction energy j2BSjof the classical spin\nin the external \feld by aligning the spin to the \feld di-\nrection. In the long-time limit, this energy is stored in\nthe conduction-electron system: The average kinetic en-\nergy per site ( L= 1001) increases by the same amount\nas shown in the second panel (note the di\u000berent scales).\nThe exchange-interaction energy changes with time but\nis the same for t= 0 andt!1 (see third panel). The\ntotal energy is constant (second panel).\nThe relaxation of the classical spin in the external \feld\nBimplies an energy \row away from the site i0into the\nbulk of the conduction-electron system such that locally ,\nin the vicinity of i0the system is in its new ground state.\nTo discuss this energy \row, it is convenient to consider\nthe total energy as a lattice sum Etot=P\niei(t) over the8\ni0−500 i0 i0+ 500\nsites0255075100125150175200225250time\n-1.3-1.2-1.1-1.0-0.9-0.8\ntotal energy density ei\nFIG. 8: (Color online) Spatiotemporal evolution of the total-\nenergy density ei(t), as de\fned in Eq. (19). Calculation for\nJ= 5,B= 1.\ntotal energy \\density\" de\fned as\nei(t) =\u0000Tn:n:(i)X\njX\n\u001bhcy\ni\u001bcj\u001bit+\u000eii0(Jhsi0it\u0000B)S(t);\n(19)\nwhere the sum over jruns over the nearest neighbors of\nsitei. The time dependence of the energy density in the\nvicinity ofi0and at distances 50 and 100 is shown in the\nfourth panel of Fig. 7.\nAt any site in the conduction-electron system, the\nenergy density increases from its ground-state value,\nreaches a maximum and eventually relaxes to the en-\nergy density of the new ground state. Since the latter\nis just the ground state with the reversed classical spin,\nthe new ground-state energy density is the same as in\nthe initial state at t= 0. As can be seen in the fourth\npanel of Fig. 7, there is also a slight spatial oscillation\nof the ground-state energy density which just re\rects the\nFriedel oscillations around the impurity at i0.\nComplete relaxation means that the excitation energy\nis completely removed from the vicinity of i0and trans-\nported into the bulk of the system. That this is in fact\nthe case can be seen by comparing the energy density at\ndi\u000berent distances from i0. It is also demonstrated by\nFig. 8 which visualizes the energy-current density which\nsymmetrically points away from i0: The total energy of\nthe excitation \rowing through each pair of sites i0\u0006\u0001iis\nconstant, i.e., the time-integrated energy \rux,R\ndtei(t)\nis the same for all i.\nAs is seen in Fig. 7 (fourth panel) and Fig. 8, there is\na considerable dispersion of the excitation wave packet\ncarrying the energy. For example, at i0+ 100 it takes\nmore than four times longer, as compared to i0+ 1, un-\ntil most of the excitation has passed through (note the\nlogarithmic time scale).\nThe broadening of the wave packet, due to dispersion,\nis asymmetric and bound by an upper limit for the speed\nof the excitation which is roughly set by the Fermi veloc-\nityvF= 2T. This Lieb-Robinson bound46,47determines\ni0\u0000500i0i0+5 0 0sites050100150200250300350400450500time \u0000 \u0000 \u0000 \u0000 \u0000-0.06-0.030.000.030.06hszii\ni0\u0000500i0i0+5 0 0sites050100150200250300350400450500time \u0000 \u0000 \u0000 \u0000 \u0000-0.04-0.020.000.020.04hsxiisx\nszFIG. 9: (Color online) Spatiotemporal evolution of the total-\nspin densityhsiit(upper panel: x-component, lower panel z-\ncomponent) for J= 5 andB= 0:1. See Fig. 10 for snapshots\nat times indicated by the arrows.\nthe \\light cone\" seen in Fig. 8.\nC. Spin dissipation\nThe same upper speed limit, given by the Fermi ve-\nlocity of the conduction-electron system, is also seen in\nthe spatiotemporal evolution of the conduction-electron\nspin densityhsiit. This is shown in Fig. 9 for a di\u000berent\nmagnetic \feld strength B= 0:1 where the classical spin\ndynamics is slower. Apparently, the wave packet of exci-\ntations emitted from the impurity not only carries energy\nbut also spin. It symmetrically propagates away from i0\nand, att\u0019300, reaches the system boundary where it\nis re\rected perfectly. Up to t= 500 there is hardly any\ne\u000bect visible in the local observables close to i0that is\na\u000bected by the \fnite system size.\nSnapshots of the conduction-electron spin dynamics\nare shown in Fig. 10 for the initial state at t= 0 and for\nstates at four later times t >0 which are also indicated\nby the arrows in Fig. 9. At t= 0 the conduction-electron\nsystem is in its ground state for the given initial direction\nof the classical spin. The latter basically points into the\n\u0000zdirection, apart from a small positive x-component\n(#=\u0019=50) which is necessary to break the symmetry9\n-0.0030.0000.003\n-0.0030.0000.003\n-0.0030.0000.003/angbracketleft/vector si/angbracketright\n-0.0030.0000.003sx sz\ni0−100i0−60i0−20-0.0030.0000.003\nt= 0 t= 80t= 60 t= 100sz sx\ni0i0+ 100 i0+ 500\nsites\nt= 250\nFIG. 10: (Color online) Snapshots the of conduction-electron magnetic moments hsiitat di\u000berent times tas indicated on the\nright and by the corresponding arrows in Fig. 9. Red lines: z-components ofhsiit. Blue lines: xcomponents. The pro\fles are\nperfectly symmetric to the impurity site i=i0but displayed up to distances ji\u0000i0j\u0014100 on the left-hand side and up to the\nsystem boundary, ji\u0000i0j\u0014500, on the right-hand side. Parameters J= 5;B= 0:1.\nof the problem and to initiate the dynamics. This tiny\ne\u000bect will be disregarded in the following.\nFrom the perspective of the conduction-electron sys-\ntem, the interaction term JSsi0acts as a local external\nmagnetic \feld JSwhich locally polarizes the conduction\nelectrons at i0. SinceJis antiferromagnetic, the local\nmomenthsi0ipoints into the + zdirection. At half-\flling,\nthe conduction-electron system exhibits pronounced an-\ntiferromagnetic spin-spin correlations which give rise to\nan antiferromagnetic spin-density wave structure aligned\nto thezaxis att= 0, see \frst panel of Fig. 10.\nThe total spin Stot= 0 att= 0, i.e., the classical\nspinSis exactly compensated by the total conduction-\nelectron spinhstoti=P\nihsii=\u0000Sin the ground state.\nThis can be traced back to the fact that for a D= 1-\ndimensional tight-binding system with an odd number of\nsitesL, withN=Land with a single static magnetic\nimpurity, there is exactly one localized state per spin pro-\njection\u001b, irrespective of the strength of the impurity po-\ntential (here given by JS= 0:5Jez). The number of \"\none-particle eigenstates therefore exceeds the number of\n#states by exactly one.\nSince the energy of the excitation induced by the ex-\nternal \feldBis completely dissipated into the bulk, the\nstate of the conduction-electron system at large t(but\nshorter than t\u0019500 where \fnite-size e\u000bects appear)\nmust locally, close to i0, resemble the conduction-electron\nground state for the reversed spin S= +0:5ez. This\nimplies that locally all magnetic moments hsiitmust re-\nverse their direction. In fact, the last panel in Fig. 10(left) shows that the new spin con\fguration is reached\nfort= 250 at sites with distance ji\u0000i0j.100, see\ndashed line, for example. For later times the spin con-\n\fguration stays constant (until the wave packet re\rected\nfrom the system boundaries reaches the vicinity of i0).\nThe reversal is almost perfect, e.g., hsi0it=0= 0:2649!\nhsi0it\u0015250=\u00000:2645. Deviations of the same order\nof magnitude are also found at larger distances, e.g.,\ni=i0\u0000100. We attribute those tiny e\u000bects to a weak de-\npendence of the local ground state on the non-equilibrium\nstate far from the impurity at t= 250, see right part of\nthe last panel in Fig. 10.\nThe other panels in Fig. 10 demonstrate the mecha-\nnism of the spin reversal. At short times (see t= 60,\nsecond panel) the perturbation of the initial equilibrium\ncon\fguration of the conduction-electron moments is still\nweak. For t= 80 andt= 100 one clearly notices the\nemission of the wave packet starting. Locally, the an-\ntiferromagnetic structure is preserved (see left part) but\nsuperimposed on this, there is an additional spatial struc-\nture of much longer size developing. This \fnally forms\nthe wave packet which is emitted from the central re-\ngion. Its spatial extension is about \u0001 \u0019300 as can be\nestimated for t= 250 (last panel on the right) where\nit covers the region 200 .i.500. The same can be\nread o\u000b from the upper part of Fig. 9. Assuming that\nthe reversal of each of the conduction-electron moments\ntakes about the same time as the reversal of the classi-\ncal spin, \u0001 is roughly given by the reversal time times\nthe Fermi velocity and therefore strongly depends on J10\nandB. For the present case, we have \u001c1\u0019150=Twhich\nimplies \u0001\u0019150\u00022 = 300 in rough agreement with the\ndata.\nIn the course of time, the long-wave length structure\nsuperimposed on the short-range antiferromagnetic tex-\nture develops a node. This can be seen for t= 100 and\ni\u001940 (fourth panel, see dashed line). The node marks\nthe spatial border between the new (right of the node,\ncloser toi0) and the original antiferromagnetic structure\nof the moments and moves away from i0with increasing\ntime.\nAt a \fxed position i, the reversal of the conduction-\nelectron moment hsiittakes place in a similar way as\nthe reversal of the classical spin (see both panels in Fig.\n9 for a \fxed i). During the reversal time, its xandy\ncomponents undergo a precessional motion while the z\ncomponent changes sign. Note, however, that during the\nreversaljhsiijgets much larger than its value in the initial\nand in the \fnal equilibrium state.\nV. EFFECTIVE CLASSICAL SPIN DYNAMICS\nA. Perturbation theory\nEqs. (7) and (9) do not form a closed set of equations of\nmotion but must be supplemented by the full equation of\nmotion (10) for the one-particle conduction-electron den-\nsity matrix. This implies that the fast electron dynamics\nmust be taken into account explicitly even if the spin\ndynamics is much slower. Hence, there is a strong mo-\ntivation to integrate out the conduction-electron degrees\nof freedom altogether and to take advantage from a much\nlarger time step within a corresponding spin-only time-\npropagation method. Unfortunately, a simple e\u000bective\nspin-only action can be obtained in the weak-coupling\n(small-J) limit only.13,14This weak-coupling approxima-\ntion is also implicit to all e\u000bective spin-only approaches\nthat consider the e\u000bect of conduction electrons on the\nspin dynamics.49\nIn the weak- Jlimit the electron degrees of freedom can\nbe eliminated in a straightforward way by using standard\nlinear-response theory:50We assume that the initial state\natt= 0 is given by the conduction-electron system in its\nground state or in thermal equilibrium and an arbitrary\nstate of the classical spin. This may be realized formally\nby suddenly switching on the interaction J(t) at time\nt= 0, i.e.,J(t) =J\u0002(t) and by switching the local \feld\nfrom some initial value Biniatt= 0 to a \fnal value B\nfort >0. The response of the conduction-electron spin\nati0and timet >0 (hsi0it= 0 fort= 0) due to the\ntime-dependent perturbation J(t)S(t) is\nhsi0it=JZt\n0dt0\u0005(ret)(t;t0)\u0001S(t0) (20)\nup to linear order in J. Here, the free ( J= 0) local\nretarded spin susceptibility of the conduction electrons\u0005(ret)(t;t0) is a tensor with elements\n\u0005(ret)\n\u000b\f(t;t0) =\u0000i\u0002(t\u0000t0)h[s\u000b\ni0(t);s\f\ni0(t0)]i; (21)\nwhere\u000b;\f =x;y;z . Using this in Eq. (7), we get an\nequation of motion for the classical spins only,\nd\ndtS(t) =S(t)\u0002B\n\u0000J2S(t)\u0002Zt\n0dt0\u0005(ret)(t\u0000t0)\u0001S(t0) (22)\nwhich is correct up to order J2.\nThis represents an equation of motion for the classical\nspin only. It has a temporally non-local structure and\nincludes an e\u000bective interaction of the classical spin at\ntimeS(t) with the same classical spin at earlier times\nt0< t. In the full quantum-classical theory where the\nelectronic degrees of freedom are taken into account ex-\nactly, this retarded interaction is mediated by a non-\nequilibrium electron dynamics starting at site i0and time\nt0and returning back to the same site i0at timet > t0.\nHere, for weak J, this is replaced by the equilibrium and\nhomogeneous-in-time conduction-electron spin suscepti-\nbility \u0005(ret)(t\u0000t0). Compared with the results of the\nfull quantum-classical theory, we expect that the pertur-\nbative spin-only theory breaks down after a propagation\ntimet\u00181=Jat the latest.\nUsing Wick's theorem,50the spin susceptibility is eas-\nily expressed in terms of the greater and the lesser equi-\nlibrium one-particle Green's functions, G>\nii;\u001b\u001b0(t;t0) =\n\u0000ihci\u001b(t)cy\ni\u001b0(t0)iandG<\nii;\u001b\u001b0(t;t0) =ihcy\ni\u001b0(t0)ci\u001b(t)i, re-\nspectively:\n\u0005(ret)\n\u000b\u000b0(t\u0000t0) = \u0002(t\u0000t0)1\n2\n\u0002Im tr 2\u00022h\n\u001b\u000bG>\ni0i0(t;t0)\u001b\u000b0G<\ni0i0(t0;t)i\n:(23)\nAssuming that the conduction-electron system is charac-\nterized by a real, symmetric and spin-independent hop-\nping matrix Tij(as given by the \frst term of Eq. (2)),\nG>andG<are unit matrices with respect to the spin\nindices. They are easily expressed as explicit functions\nofT(see Ref. 51, for example). With tr ( \u001b\u000b\u001b\u000b0) = 2\u000e\u000b\u000b0,\nwe \fnd\n\u0005(ret)\n\u000b\u000b0(t\u0000t0) = \u0002(t\u0000t0)\u000e\u000b\u000b0Im\n\u0002 \ne\u0000iT(t\u0000t0)\n1 +e\u0000\f(T\u0000\u0016)!\ni0i0 \ne\u0000iT(t0\u0000t)\ne\f(T\u0000\u0016)+ 1!\ni0i0(24)\nfor a conduction-electron system at inverse temperature\n\fand chemical potential \u0016.\nUsing Eq. (24) we have computed the spin suscepti-\nbility for the ground state ( \f=1) of the conduction-\nelectron system. This \fxes the kernel in the integro-\ndi\u000berential equation Eq. (22) which is solved numeri-\ncally by standard techniques.52We again consider the11\n10−1100101102\ntimet-0.50-0.250.000.250.50/vectorSJ= 1.0Sz\nSx\nFIG. 11: (Color online) Components Sx(t) andSz(t) of the\nclassical spin after a sudden switch of the \feld from xto\nzdirection at t= 0. Calculations for B= 1 andJ= 1.\nSolid lines: results of the linear-response dynamics, Eq. (22).\nDashed lines: results of the exact quantum-classical dynamics\nforL= 1001.\nsystem displayed in Fig. 1 with a single classical spin\ncoupled via Jto the central site i0of a chain consisting\nofL= 1001 sites. Finite-size artifacts do not show up\nbeforetmax= 500.\nFigs. 11 and 12 show the resulting linear-response dy-\nnamics of the classical spin after preparing the initial\nstate of the system with the classical spin pointing into\n+xdirection while B= (0;0;B). The external magnetic\n\feld induces a precessional motion of the classical spin:\nthere is a rapid oscillation of its xcomponent (and of its\nycomponent, not shown) with frequency !\u0019B(blue\nlines). Damping is induced by dissipation of energy and\nspin: for large times, the zcomponent aligns to the ex-\nternal \feld (red lines).\nFor weak coupling, up to J= 1 (Fig. 11), there is an al-\nmost perfect agreement between the results of the exact\nquantum-classical dynamics (full lines) and the linear-\nresponse theory (dashed lines) up to the maximum prop-\nagation time tmax= 500. We note that, compared to\nthe full theory, there is a tiny deviation of the linear-\nresponse result for the zcomponent of S(t) visible in\nFig. 11 for times t&10. Hence, on this level of accuracy,\nt1\u001910 sets the time scale up to which the linear-response\ntheory is valid. This may appear surprising as this im-\npliest1J= 10 for the \\small\" dimensionless parameter\nof the perturbation theory. One has to keep in mind,\nhowever, that even if the perturbation is \\strong\", its\ne\u000bects can be rather moderate since only non-adiabatic\nterms\u0018S(t)\u0002S(t0) contribute in Eq. (22).\nForJ= 3, see Fig. 12, damping of the classical spin\nsets in much earlier. Visible deviations of the linear-\nresponse theory from the full dynamics already appear\non a time scale that is almost two orders of magnitude\nsmaller as compared to the case J= 1. A simple rea-\nsoning based on the argument that the dimensionless\nexpansion parameter is t1Jfails as this disregards the\nstrong enhancement of retardation e\u000bects with increas-\n10−1100101\ntimet-0.50-0.250.000.250.50/vectorSJ= 3.0Sz\nSxFIG. 12: (Color online) The same as Fig. 11 but for a di\u000berent\nexchange coupling constant J= 3.\ningJ, which have been discussed in Sec. IV A. These\ne\u000bects make the perturbation much more e\u000bective, i.e.,\nlead to a torque, which is exerted by the conduction elec-\ntrons on the classical spin, growing stronger than linear\ninJ.\nWe conclude that linear-response theory is highly at-\ntractive formally as it provides a tractable spin-only e\u000bec-\ntive theory. On the other hand, substantial discrepancies\ncompared to the full (non-perturbative) theory show up\nas soon as damping e\u000bects become stronger. Note that,\nwith increasing time, these deviations must diminish and\ndisappear eventually since both, the full and the e\u000bec-\ntive theory, predict a fully relaxed spin state for t!1\n{ see lower panel of Fig. 12, for example. At least for\nsimple systems with a single classical spin, as considered\nhere, this implies that the e\u000bective theory provides qual-\nitatively reasonable results.\nB. Landau-Lifschitz-Gilbert equation\nTo derive the LLG equation, the linear-response the-\nory must be further simpli\fed:14,53As is obvious from\nEq. (22), the spin-susceptibility \u0005(ret)(t\u0000t0) can be inter-\npreted as an e\u000bective retarded self-interaction of the spin.\nWe assume that the electron dynamics is much faster\nthan the spin dynamics. On the time scale of the spin\ndynamics, the self-interaction then takes place almost in-\nstantaneously, i.e., the memory kernel \u0005(ret)\n\u000b\u000b0(t\u0000t0) =\n\u000e\u000b\u000b0\u0005(ret)(t\u0000t0) in Eq. (22) is peaked at t0\u0019t. We can\ntherefore approximate S(t0)\u0019S(t) + (t0\u0000t)_S(t) under\nthe integral in Eq. (22). This immediately yields:\ndS(t)\ndt=S(t)\u0002B+\u000b(t)S(t)\u0002dS(t)\ndt; (25)\nwhere\n\u000b(t) =J2Zt\n0d\u001c\u001c\u0005(ret)(\u001c) (26)12\nafter substituting t07!\u001c=t\u0000t0.\nEq. (25) takes the form of the standard LLG equation\nfor a single classical spin [cf. Eq. (1)] if \u000b(t) is replaced\nby\n\u000b\u0011lim\nt!1\u000b(t) =J2Z1\n0dtt\u0005(ret)(t): (27)\nNote that this is a necessary step to arrive at a constant\ndamping parameter which can again be justi\fed by not-\ning that \u0005(ret)(t) is peaked at t= 0.\nBefore proceeding, let us stress that Eq. (27) is, or\nis equivalent to, the standard expression used for com-\nputing the Gilbert damping constant in various studies:\nAfter Fourier transformation,\n\u0005(ret)(!) =Z\ndtei!t\u0005(ret)(t); (28)\none ends up with\n\u000b=\u0000iJ2@\n@!\u0005(ret)(!)\f\f\f\n!=0=J2@\n@!Im \u0005(ret)(!)\f\f\f\n!=0;\n(29)\nwhich has also been derived, e.g., in Refs. 14,54 in dif-\nferent contexts. The frequency-dependent spin correla-\ntion \u0005(ret)(!) in Eq. (29) can be obtained explicitly as\nthe Fourier transform of \u0005(ret)(t) given by Eq. (24). A\nstraightforward calculation yields:\n\u000b=\u0019\n2J2Z\nd!df(!)\nd!Aloc(!)Aloc(!); (30)\nwheref(!) = 1=(exp(\f!)+1) is the Fermi function, and\nAloc(!) =L\u00001X\nk\u000e(!+\u0016\u0000\"(k)) (31)\nforL!1 is the local one-particle spectral function.\nHere we have assumed periodic boundary conditions, i.e.,\nspatial homogeneity, such that the hopping matrix Tis\ndiagonalized by Fourier transformation:\nTij=X\nkUik\"(k)Uy\nkj(32)\nwithUik=L\u00001=2exp(ikRi). The eigenvalues of T\nare given by the tight-binding dispersion \"(k) of the\nconduction-electron Bloch band.\nWithin our simple tight-binding model for the\nconduction-electron system, Eqs. (29) and (30) are equiv-\nalent with Kambersky's breathing Fermi-surface the-\nory, related torque-correlation models and scattering\ntheory and have frequently been used for ab initio as\nwell as model computations of the Gilbert damping\nconstant.15,18,20,21,55{58\nLet us remark that Eq. (30) demonstrates that \u000b<0.\nThis results from the convention for the coupling of the\nmagnetic \feld to the spin, namely H=H(B= 0)\u0000BS\n[see Eq. 2)], which has been adopted here. As a conse-\nquence, the precession of S(t) aroundBis described by\naleft-hand helix, _S=S(t)\u0002B, and thus \u000bmust be\nnegative to describe damping.\n10−1100101102103\ntimet−0.10−0.050.000.050.10t·Πret(t)\nΠret(t)FIG. 13: (Color online) Local retarded spin susceptibility of\nthe conduction electrons \u0005(ret)(t) (red line) and t\u0005(ret)(t)\n(blue) as obtained from Eq. (33) for a one-dimensional\nconduction-electron system with L= 10000 sites and peri-\nodic boundary conditions.\nC. Ill-de\fned Gilbert damping\nThe above discussion shows that Eq. (27) represents\nthe fundamental de\fnition of the Gilbert damping con-\nstant\u000band that the limit t!1 is crucial to recover the\nLLG equation in its standard form. The existence of the\nlong-time limit, however, decisively depends on the long-\ntime behavior of the retarded spin-correlation function\n\u0005(ret)(t). Starting from Eq. (24), this is easily computed\nas\n\u0005(ret)(t) = \u0002(t)Im1\nL2X\nk;pe\u0000i\"(k)t\n1 +e\u0000\f(\"(k)\u0000\u0016)ei\"(p)t\ne\f(\"(p)\u0000\u0016)+ 1:\n(33)\nFig. 13 gives an example for the time-dependence of\n\u0005(ret)(t). The calculations have been done at half-\flling,\n\f=1forL= 104sites. We note that the susceptibility\nis in fact peaked at t= 0. For long times, it oscillates\nwith frequency !\u0005= 4 and decays as 1 =t. This is an im-\nportant observation as it implies that the limit in Eq. (27)\ndoes not exist and that, therefore, the damping constant\n\u000bis ill-de\fned.\nTo analyze the physical origin of the divergent integral,\nwe rewrite the spin susceptibility in Eq. (33) as\n\u0005(ret)(t) = \u0002(t)Im[A(unocc)\nloc(\u0000t)A(occ)\nloc(t)]: (34)\nIts long-time behavior is governed by the long-time be-\nhavior of the Fourier transform\nA(occ;unocc)\nloc(t) =Z\nd!ei!tA(occ;unocc)\nloc(!) (35)\nof the occupied, A(occ)\nloc(!)\u0011f(!)Aloc(!), and of the un-\noccupied,A(unocc)\nloc(!)\u0011f(!)Aloc(!), part of the spectral\ndensity, see Eq. (31).\nFor functions with smooth !dependence, the Fourier\ntransform generically drops to zero exponentially fast if13\nt!1 . A power-law decay, however, is obtained if there\nare singularities of A(occ;unocc)\nloc(!). We can distinguish\nbetween van Hove singularities, which are, e.g., of the\nform/\u0002(!\u0000!0)(!\u0000!0)k(withk>\u00001), and the step-\nlike singularity/\u0002(!\u0000!0) (i.e.k= 0), arising in the\nzero-temperature limit at !0= 0 due to the Fermi func-\ntion. Generally, a singularity of order kgives rise to the\nasymptotic behavior A(occ;unocc)\nloc(t)/t\u00001\u0000k, apart from\na purely oscillatory factor ei!0t. For the present case,\nthe van Hove singularities of A(occ;unocc)\nloc(!) at\u0006!0= 2\nexplain, via Eq. (34) the oscillation of \u0005(ret)(t) with fre-\nquency!\u0005= 2!0= 4.\nGenerally, the location of the van Hove singularity\non the frequency axis, i.e. !0, determines the oscilla-\ntion period while the decay of \u0005(ret)(t) is governed by\nthe strength of the singularity. Consider, as an exam-\nple, the zero-temperature case and assume that there are\nno van Hove singularities. The sharp Fermi edge im-\npliesA(occ;unocc)\nloc(t)/t\u00001, and thus \u0005(ret)(t)/t\u00002. The\nGilbert-damping constant is well de\fned in this case.\nThe strength of van Hove singularities depends on\nthe lattice dimension D.59For a one-dimensional lat-\ntice, we have van Hove singularities with k=\u00001=2, and\nthus \u0005(ret)(t)/t\u00001, consistent with Fig. (13). Here,\nthe strong van Hove singularity dominates the long-time\nasymptotic behavior as compared to the weaker Fermi-\nedge singularity. For D= 3, we have k= 1=2 and\n\u0005(ret)(t)/t\u00003if\f <1while for\f=1the Fermi-\nedge dominates and \u0005(ret)(t)/t\u00002. TheD= 2 case is\nmore complicated: The logarithmic van Hove singularity\n/lnj!jleads to \u0005(ret)(t)/t\u00002. This, however, applies\nto cases o\u000b half-\flling only. At half-\flling the van Hove\nand the Fermi-edge singularity combine to a singularity\n/\u0002(!) lnj!jwhich gives \u0005(ret)(t)/ln2(t)=t2. For \fnite\ntemperatures, we again have \u0005(ret)(t)/t\u00002.\nThe existence of the integral Eq. (27) depends on the\nt!1 behavior and either requires a decay as \u0005(ret)(t)/\nt\u00003or faster, or an asymptotic form \u0005(ret)(t)/ei!0t=t2\nwith an oscillating factor resulting from a non-zero po-\nsition!06= 0 of the van Hove singularity. For the one-\ndimensional case, we conclude that the LLG equation\n(with a time-independent damping constant) is based on\nan ill-de\fned concept. Also, the derivation of Eqs. (29)\nand (30) is invalid in this case as the !derivative and the\ntintegral do not commute. This conclusion might change\nfor the case of interacting conduction electrons. Here one\nwould expect a regularization of van Hove singularities\ndue to a \fnite imaginary part of the conduction-electron\nself-energy.\nVI. CONCLUSIONS\nHybrid systems consisting of classical spins coupled\nto a bath of non-interacting conduction electrons rep-\nresent a class of model systems with a non-trivial real-\ntime dynamics which is numerically accessible on longtime scales. Here we have considered the simplest vari-\nant of this class, the Kondo-impurity model with a clas-\nsical spin, and studied the relaxation dynamics of the\nspin in an external magnetic \feld. As a fundamental\nmodel this is interesting of its own but also makes con-\ntact with di\u000berent \felds, e.g., atomistic spin dynamics in\nmagnetic samples, spin relaxation in spintronics devices,\nfemto-second dynamics of highly excited electron systems\nwhere local magnetic moments are formed due to electron\ncorrelations, and arti\fcial Kondo systems simulated with\nultracold atoms in optical lattices.\nWe have compared the coupled spin and electron dy-\nnamics with the predictions of the widely used Landau-\nLifshitz-Gilbert equation which is supposed to cover the\nregime of weak local exchange Jand slow spin dynamics.\nFor the studied setup, the LLG equation predicts a rather\nregular time evolution characterized by spin precession,\nspin relaxation and eventually reversal of the spin on a\ntime scale\u001cdepending on J(and the \feld strength B).\nWe have demonstrated that this type of dynamics can be\nrecovered and understood on a microscopic level in the\nmore fundamental quantum-classical Kondo model. It is\ntraced back to a non-adiabatic dynamics of the electron\ndegrees of freedom and the feedback of the electronic sub-\nsystem on the spin. It turns out that the spin dynamics\nis essentially a consequence of the retarded e\u000bect of the\nlocal exchange. Namely, the classical spin can be seen as\na perturbation exciting the conduction-electron system\nlocally. This electronic excitation propagates and feeds\nback to the classical spin, but at a later time, and thereby\ninduces a spin torque.\nWe found that this mechanism drives the relaxation of\nthe system to its local ground state irrespective of the\nstrength of the local exchange J. As the microscopic dy-\nnamics is fully conserving, the energy and spin of the\ninitial excitation which is locally stored in the vicinity of\nthe classical spin, must be dissipated into the bulk of the\nsystem in the course of time. This dissipation could be\nuncovered by studying the relaxation process from the\nperspective of the electron degrees of freedom. Dissipa-\ntion of energy and spin takes place through the emission\nof a dispersive spin-polarized wave packet propagating\nthrough the lattice with the Fermi velocity. In this pro-\ncess the local conduction-electron magnetic moment at\nany given distance to the impurity undergoes a reversal,\ncharacterized by precession and relaxation, similar to the\nmotion of the classical spin.\nThe dynamics of the classical spin can be qualitatively\nvery di\u000berent from the predictions of the LLG equation\nfor strongJ. In this regime we found a complex mo-\ntion characterized by oscillations of the angle between\nthe classical spin S(t) and the local conduction-electron\nmagnetic moment at the impurity site hsi0iaround the\nadiabatic value \r=\u0019which takes place on an emergent\nnew time scale.\nIn the weak- Jlimit, the classical spin dynamics is qual-\nitatively predicted correctly by the LLG equation. At\nleast partially, however, this must be attributed to the14\nfact that the LLG approach, by construction, recovers\nthe correct \fnal state where the spin is parallel to the\n\feld. In fact, quantitative deviations are found during\nthe relaxation process. The LLG approach is based on\n\frst-order perturbation theory in Jand on the additional\nassumption that the classical spin is slow. To pinpoint\nthe source of the deviations, we have numerically solved\nthe integro-di\u000berential equation that is obtained in \frst-\norder-in-Jperturbation theory and compared with the\nfull hybrid dynamics. The deviations of the perturbative\napproach from the exact dynamics are found to gradually\nincrease with the propagation time (until the proximity\nto the \fnal state enforces the correct long-time asymp-\ntotics). This is the expected result as the dimensionless\nsmall parameter is Jt. However, with increasing Jthe\ntime scale on which perturbation theory is reliable de-\ncreases much stronger than 1 =Jdue to a strong enhance-\nment of retardation e\u000bects which make the perturbation\nmore e\u000bective and produce a stronger torque.\nGenerally, the perturbation can be rather ine\u000bective in\nthe sense that it produces a torque /S(t)\u0002S(t0) which\nis very weak if the process is nearly adiabatic. This ex-\nplains that \frst-order perturbation theory and the LLG\nequation is applicable at all for couplings of the order of\nhoppingJ\u0018T. For the present study this can also be\nseen as a fortunate circumstance since the regime of very\nweak couplings J\u001cTis not accessible numerically. In\nthis case the spin-reversal time scale gets so large that\nthe propagation of excitations in the conduction-electron\nsubsystem would by a\u000bected by backscattering from the\nedges of the system which necessarily must be assumed\nas \fnite for the numerical treatment.\nFor the one-dimensional lattice studied here, a di-\nrect comparison between LLG equation and the exact\nquantum-classical theory is not meaningful as the damp-\ning constant \u000bis ill-de\fned in this case. We could argue\nthat the problem results from the strength of the vanHove singularities in the conduction-electron density of\nstates which dictates the long-time behavior of the mem-\nory kernel of the integro-di\u000berential equation which is\ngiven by the equilibrium spin susceptibility. As the type\nof the van Hove singularity is characteristic for all sys-\ntems of a given dimension, we can generally conclude that\nthe LLG approach reduces to a purely phenomenological\nscheme in the one-dimensional case. However, it is an\nopen question, which will be interesting to tackle in the\nfuture, if this conclusion is still valid for systems where\nthe Coulomb interaction among the conduction electrons\nis taken into account additionally.\nThere are more interesting lines of research which are\nbased on the present work and could be pursued in the\nfuture. Those include systems with more than a single\nspin where, e.g., the e\u000bects of a time-dependent and\nretarded RKKY interaction can be studied additionally.\nWe are also working on a tractable extension of the the-\nory to account for longitudinal \ructuations of the spins\nto include time-dependent Kondo screening, and the\ncompetition with RKKY coupling, on a time-dependent\nmean-\feld level. 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On the other hand, in the case of only AC current, t he spin\ndynamics has a rather simple bifurcation diagram of attract ors. That is, for\nsmall Gilbert damping, when the AC current is below a critica l value, the\nattractor is a limit cycle; above the critical value, the att ractor is chaotic\n(turbulent). For normal Gilbert damping, the attractor is a lways a limit cycle\nin the physically interesting range of the AC current. We als o developed\na Melnikov integral theory for a theoretical prediction on t he occurrence of\nchaos. Our Melnikov prediction seems performing quite well in the DC case.\nIn the AC case, our Melnikov prediction seems predicting tra nsient chaos.\nThe sustained chaotic attractor seems to have extra support from parametric\nresonance leading to a turbulent state.\nContents\n1. Introduction 2\n2. Mathematical Formulation 3\n3. Isospectral Integrable Theory for the Heisenberg Equation 4\n4. A Melnikov Function 14\n5. Numerical Simulation 17\n6. Appendix: The Connection Between the Heisenberg Equation and the\nNLS Equation 23\nReferences 25\n1991Mathematics Subject Classification. Primary 35, 65, 37; Secondary 78.\nKey words and phrases. Magnetization reversal, spin-polarized current, chaos, D arboux\ntransformation, Melnikov function.\nc/circlecopyrt2008 (copyright holder)\n12 YUEHENG LAN AND Y. CHARLES LI\n1. Introduction\nThe greatest potential of the theory of chaos in partial different ial equations\nlies in its abundant applications in science and engineering. The variety of the spe-\ncific problems demands continuing innovation of the theory [ 17] [16] [18] [23] [24]\n[15] [20] [21]. In these representative publications, two theories were develop ed.\nThe theory developed in [ 17] [16] [18] involves transversal homoclinic orbits, and\nshadowing technique is used to prove the existence of chaos. This t heory is very\ncomplete. The theory in [ 23] [24] [15] [20] [21] deals with Silnikov homoclinic\norbits, and geometric construction of Smale horseshoes is employe d. This theory\nis not very complete. The main machineries for locating homoclinic orbit s are (1).\nDarbouxtransformations,(2). Isospectraltheory,(3). Per sistenceofinvariantman-\nifolds and Fenichel fibers, (4). Melnikov analysis and shooting techn ique. Overall,\nthe two theories on chaos in partial differential equations are resu lts of combining\nIntegrable Theory, Dynamical System Theory, and Partial Differe ntial Equations\n[19].\nIn this article, we are interested in the chaotic spin dynamics in a long n ano-\nmagnet diven by an electrical current. We hope that the abundant spin dynamics\nrevealed by this study can generate experimental studies on long n anomagets. To\nillustrate the general significance of the spin dynamics, in particular the magneti-\nzation reversal issue, we use a daily example: The memory of the har d drive of a\ncomputer. The magnetization is polarized along the direction of the e xternal mag-\nnetic field. By reversing the external magnetic field, magnetization reversal can\nbe accomplished; thereby, generating 0 and 1 binary sequence and accomplishing\nmemory purpose. Memory capacity and speed via such a technique h ave reached\ntheir limits. The “bit” writing scheme based on such Oersted-Maxwell magnetic\nfield(generatedbyanelectricalcurrent)encountersfundamen talproblemfromclas-\nsical electromagnetism: the long range magnetic field leads to unwan ted writing or\nerasing of closely packed neighboring magnetic elements in the extre mely high den-\nsity memory device and the induction laws place an upper limit on the mem ory\nspeed due to slow rise-and-decay-time imposed by the law of inductio n. Discovered\nby Slonczewski [ 35] and Berger [ 1], electrical current can directly apply a large\ntorque to a ferromagnet. If electrical current can be directly ap plied to achieve\nmagnetization reversal, such a technique will dramatically increase t he memory\ncapacity and speed of a hard drive. The magnetization can then be s witched\non the scale of nanoseconds and nanometers [ 39]. The industrial value will be\ntremendous. Nanomagnets driven by currents has been intensive ly studied recently\n[13, 10, 34, 7, 14, 12, 33, 8, 39, 11, 32, 37, 38, 4, 3, 26, 27, 29, 4 2, 31]. The\nresearches have gone beyond the original spin valve system [ 35] [1]. For instance,\ncurrent driven torques have been applied to magnetic tunnel junc tions [36] [6], di-\nlute magnetic semiconductors [ 40], multi-magnet couplings [ 10] [14]. AC currents\nwere also applied to generate spin torque [ 34] [7]. Such AC current can be used to\ngenerate the external magnetic field [ 34] or applied directly to generate spin torque\n[7].\nMathematically, the electrical current introduces a spin torque fo rcing term\nin the conventional Landau-Lifshitz-Gilbert (LLG) equation. The A C current can\ninduce novel dynamics of the LLG equation, like synchronization [ 34] [7] and chaos\n[25] [41]. Both synchronization and chaos are important phenomena to und erstand\nbefore implementing the memory technology. In [ 25] [41], we studied the dynamicsCHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 3\nof synchronization and chaos for the LLG equation by ignoring the e xchange field\n(i.e. LLG ordinary differential equations). When the nanomagnetic d evice has the\nsame order of length along every direction, exchange field is not impo rtant, and we\nhave a so-called single domain situation where the spin dynamics is gove rned by\nthe LLG ordinary differential equations. In this article, we will study what we call\n“long nanomagnet” which is much longer along one direction than othe r directions.\nIn such a situation, the exchange field will be important. This leads to a LLG\npartial differential equation. In fact, we will study the case where the exchange\nfield plays a dominant role.\nThe article is organized as follows: Section 2 presents the mathemat ical formu-\nlation of the problem. Section 3 is an integrable study on the Heisenbe rg equation.\nBased upon Section 3, Section 4 builds the Melnikov integral theory f or predicting\nchaos. Section 5 presents the numerical simulations. Section 6 is an appendix to\nSection 3.\n2. Mathematical Formulation\nTo simplify the study, we will investigate the case that the magnetiza tion de-\npends on only one spatial variable, and has periodic boundary condit ion in this\nspatial variable. The application of this situation will be a large ring sha pe nano-\nmagnet. Thus, we shall study the following forced Landau-Lifshitz -Gilbert (LLG)\nequation in the dimensionless form,\n(2.1)∂tm=−m×H−ǫαm×(m×H)+ǫ(β1+β2cosω0t)m×(m×ex),\nsubject to the periodic boundary condition\n(2.2) m(t,x+2π) =m(t,x),\nwheremis a unit magnetization vector m= (m1,m2,m3) in which the three\ncomponents are along( x,y,z) directions with unit vectors( ex,ey,ez),|m|(t,x) = 1,\nthe effective magnetic field Hhas several terms\nH=Hexch+Hext+Hdem+Hani\n=∂2\nxm+ǫaex−ǫm3ez+ǫbm1ex, (2.3)\nwhereHexch=∂2\nxmis the exchange field, Hext=ǫaexis the external field,\nHdem=−ǫm3ezisthe demagnetizationfield, and Hani=ǫbm1existhe anisotropy\nfield. For the materials of the experimental interest, the dimension less parameters\nare in the ranges\na≈0.05, b≈0.025, α≈0.02,\nβ1∈[0.01,0.3], β2∈[0.01,0.3] ; (2.4)\nandǫis a small parameter measuring the length scale of the exchange field . One\ncan also add an AC current effect in the external field Hext, but the results on the\ndynamics are similar.\nOur goal is to build a Melnikov function for the LLG equation around do main\nwalls. The roots of such a Melnikov function provide a good indication o f chaos.\nFor the rest of this section, we will introduce a few interesting nota tions. The\nPauli matrices are:\n(2.5)σ1=/parenleftbigg0 1\n1 0/parenrightbigg\n, σ2=/parenleftbigg0−i\ni0/parenrightbigg\n, σ3=/parenleftbigg1 0\n0−1/parenrightbigg\n.4 YUEHENG LAN AND Y. CHARLES LI\nLet\n(2.6) m+=m1+im2, m−=m1−im2,\ni.e.m+=m−. Let\n(2.7) Γ = mjσj=/parenleftbiggm3m−\nm+−m3/parenrightbigg\n.\nThus, Γ2=I(the identity matrix). Let\nˆH=−H−αm×H+βm×ex,Π =/parenleftbiggˆH3ˆH1−iˆH2\nˆH1+iˆH2−ˆH3/parenrightbigg\n.\nThen the LLG can be written in the form\n(2.8) i∂tΓ =1\n2[Γ,Π],\nwhere [Γ,Π] = ΓΠ −ΠΓ.\n3. Isospectral Integrable Theory for the Heisenberg Equati on\nSettingǫtozero,theLLG(2.1)reducestotheHeisenbergferromagneteq uation,\n(3.1) ∂tm=−m×mxx.\nUsing the matrix Γ introduced in (2.7), the Heisenberg equation (3.1) has the form\n(3.2) i∂tΓ =−1\n2[Γ,Γxx],\nwhere the bracket [ ,] is defined in (2.8). Obvious constants of motion of the\nHeisenberg equation (3.1) are the Hamiltonian,\n1\n2/integraldisplay2π\n0|mx|2dx ,\nthe momentum,/integraldisplay2π\n0m1m2x−m2m1x\n1+m3dx ,\nand the total spin,/integraldisplay2π\n0mdx .\nThe Heisenberg equation (3.1) is an integrable system with the followin g Lax pair,\n∂xψ=iλΓψ , (3.3)\n∂tψ=−λ\n2(4iλΓ+[Γ,Γx])ψ , (3.4)\nwhereψ= (ψ1,ψ2)Tis complex-valued, λis a complex parameter, Γ is the matrix\ndefined in (2.7), and [Γ ,Γx] = ΓΓ x−ΓxΓ. In fact, there is a connection be-\ntween the Heisenberg equation (3.1) and the 1D integrable focusing cubic nonlinear\nSchr¨ odinger (NLS) equation via a nontrivial gauge transformatio n. The details of\nthis connection are given in the Appendix.CHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 5\n3.1. A Simple Linear Stability Calculation. As shown in the Appendix,\nthe temporally periodic solutions of the NLS equation correspond to the domain\nwalls of the Heisenberg equation. Consider the domain wall\nΓ0=/parenleftbigg0e−iξx\neiξx0/parenrightbigg\n, ξ∈Z; i.e.m1= cosξx, m 2= sinξx, m 3= 0,\nwhich is a fixed point of the Heisenberg equation. Linearizing the Heise nberg equa-\ntion at this fixed point, one gets\ni∂tΓ =−1\n2[Γ0,Γxx]−1\n2[Γ,Γ0xx].\nLet\nΓ =/parenleftbigg\nm3 e−iξx(m1−im2)\neiξx(m1+im2) −m3/parenrightbigg\n,\nwe get\n∂tm1= 0, ∂tm2=m3xx+ξ2m3, ∂tm3=−m2xx−2ξm1x.\nLet\nmj=∞/summationdisplay\nk=0(m+\njk(t)coskx+m−\njk(t)sinkx),\nwherem±\njk(t) =c±\njkeΩt,c±\njkand Ω are constants. We obtain that\n(3.5) Ω =/radicalbig\nk2(ξ2−k2)\nwhich shows that only the modes 0 <|k|<|ξ|are unstable. Such instability is\ncalled a modulational instability, also called a side-band instability. Comp aring\nthe Heisenberg ferromagnet equation (3.1) and the Landau-Lifsh itz-Gilbert equa-\ntion (2.1), we see that if we drop the exchange field Hexch=∂2\nxmin the effective\nmagnetic field H(2.3), such a modulational instability will disappear, and the\nLandau-Lifshitz-Gilbert equation (2.1) reduces to a system of thr ee ordinary differ-\nential equations, which has no chaos as verified numerically. Thus th e modulational\ninstability is the source of the chaotic magnetization dynamics.\nIn terms of m±\njk, we have\nd\ndtm±\n1k= 0,d\ndtm±\n2k= (ξ2−k2)m±\n3k,d\ndtm±\n3k=k2m±\n2k∓2ξkm∓\n1k.\nChoosingξ= 2, we have for k= 0,\n\nm∓\n10\nm±\n20\nm±\n30\n=c1\n1\n0\n0\n+c2\n0\n1\n0\n+c3\n0\n4t\n1\n;\nfork= 1,\n\nm∓\n11\nm±\n21\nm±\n31\n=c1\n1\n±4\n0\n+c2\n0√\n3\n1\ne√\n3t+c3\n0\n−√\n3\n1\ne−√\n3t;\nfork= 2,\nm∓\n12\nm±\n22\nm±\n32\n=c1\n0\n0\n1\n+c2\n1\n0\n∓8t\n+c3\n0\n1\n4t\n;6 YUEHENG LAN AND Y. CHARLES LI\nfork>2,\n\nm∓\n1k\nm±\n2k\nm±\n3k\n=c1\n1\n±4/k\n0\n+c2\n0√\nk2−4cos(k√\nk2−4t)\nksin(k√\nk2−4t)\n\n+c3\n0\n−√\nk2−4sin(k√\nk2−4t)\nkcos(k√\nk2−4t)\n;\nwherec1,c2andc3are arbitrary constants.\nThe nonlinear foliation of the above linear modulational instability can b e es-\ntablished via a Darboux transformation.\n3.2. A Darboux Transformation. A Darboux transformation for (3.3)-\n(3.4) can be obtained.\nTheorem 3.1. Letφ= (φ1,φ2)Tbe a solution to the Lax pair (3.3)-(3.4) at ( Γ,ν).\nDefine the matrix\nG=N/parenleftbigg(ν−λ)/ν 0\n0 (¯ν−λ)/¯ν/parenrightbigg\nN−1,\nwhere\nN=/parenleftbigg\nφ1−φ2\nφ2φ1/parenrightbigg\n.\nThen ifψsolves the Lax pair (3.3)-(3.4) at ( Γ,λ),\n(3.6) ˆψ=Gψ\nsolves the Lax pair (3.3)-(3.4) at ( ˆΓ,λ), where ˆΓis given by\n(3.7) ˆΓ =N/parenleftbigg\ne−iθ0\n0eiθ/parenrightbigg\nN−1ΓN/parenleftbigg\neiθ0\n0e−iθ/parenrightbigg\nN−1,\nwhereeiθ=ν/|ν|.\nThe transformation (3.6)-(3.7) is called a Darboux transformation . This theo-\nrem can be proved either through the connection between the Heis enberg equation\nand the NLS equation (with a well-known Darboux transformation) [ 2], or through\na direct calculation.\nNotice also that ˆΓ2=I. Let/parenleftbigg\nΦ1−Φ2\nΦ2Φ1/parenrightbigg\n=N/parenleftbigg\ne−iθ0\n0eiθ/parenrightbigg\nN−1\n=1\n|φ1|2+|φ2|2/parenleftbigge−iθ|φ1|2+eiθ|φ2|2−2isinθ φ1φ2\n−2isinθφ1φ2eiθ|φ1|2+e−iθ|φ2|2/parenrightbigg\n. (3.8)\nThen\n(3.9) ˆΓ =/parenleftbigg\nˆm3 ˆm1−iˆm2\nˆm1+iˆm2−ˆm3/parenrightbigg\n,\nwhere\nˆm+= ˆm1+iˆm2=Φ12(m1+im2)−Φ2\n2(m1−im2)+2Φ1Φ2m3,\nˆm3=/parenleftbig\n|Φ1|2−|Φ2|2/parenrightbig\nm3−Φ1Φ2(m1+im2)−Φ1Φ2(m1−im2).CHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 7\nOne can generate the figure eight connecting to the domain wall, as t he nonlinear\nfoliation of the modulational instability, via the above Darboux trans fomation.\n3.3. Figure Eight Connecting to the Domain Wall . LetΓbethedomain\nwall\nΓ =/parenleftbigg\n0e−i2x\nei2x0/parenrightbigg\n,\ni.e.m1= cos2x,m2= sin2x, andm3= 0. Solving the Lax pair (3.3)-(3.4), one\ngets two Bloch eigenfunctions\n(3.10)ψ=eΩt/parenleftbigg\n2λexp{i\n2(k−2)x}\n(k−2)exp{i\n2(k+2)x}/parenrightbigg\n,Ω =−iλk , k =±2/radicalbig\n1+λ2.\nTo apply the Darboux transformation (3.7), we start with the two B loch functions\nwithk=±1,\nφ+=/parenleftbigg√\n3e−ix\nieix/parenrightbigg\nexp/braceleftBigg√\n3\n2t+i1\n2x/bracerightBigg\n,\n(3.11)\nφ−=/parenleftbigg−ie−ix√\n3eix/parenrightbigg\nexp/braceleftBigg\n−√\n3\n2t−i1\n2x/bracerightBigg\n.\nThe wise choice for φused in (3.7) is:\n(3.12) φ=/radicalbigg\nc+\nc−φ++/radicalbigg\nc−\nc+φ−=/parenleftbigg/parenleftbig√\n3eτ+iχ−ie−τ−iχ/parenrightbig\ne−ix\n/parenleftbig\nieτ+iχ+√\n3e−τ−iχ/parenrightbig\neix/parenrightbigg\n,\nwherec+/c−= exp{σ+iγ},τ=1\n2(√\n3t+σ), andχ=1\n2(x+γ). Then from the\nDarboux transformation (3.7), one gets\nˆm1+iˆm2=−ei2x/braceleftbigg\n1−2 sech2τcos2χ\n(2−√\n3 sech2τsin2χ)2/bracketleftbigg\nsech2τcos2χ\n+i/parenleftBig√\n3−2 sech2τsin2χ/parenrightBig/bracketrightbigg/bracerightbigg\n, (3.13)\nˆm3=2 sech2τtanh2τcos2χ\n(2−√\n3 sech2τsin2χ)2. (3.14)\nAst→ ±∞,\nˆm1→ −cos2x ,ˆm2→ −sin2x ,ˆm3→0.\nThe expressions (3.13)-(3.14) represent the two dimensional figu re eight separatrix\nconnecting to the domain wall ( m+=−ei2x,m3= 0), parametrized by σand\nγ. See Figure 1 for an illustration. Choosing γ= 0,π, one gets the figure eight\ncurve section of Figure 1. The spatial-temporal profiles correspo nding to the two\nlobes of the figure eight curve are shown in Figure 2. In fact, the tw o profiles\ncorresponding the two lobes are spatial translates of each other byπ. Inside one\nof the lobe, the spatial-temporal profile is shown in Figure 3(a). Out side the figure\neight curve, the spatial-temporal profile is shown in Figure 3(b). He re the inside\nand outside spatial-temporal profiles are calculated by using the int egrable finite\ndifference discretization [ 9] of the Heisenberg equation (3.1),\n(3.15)d\ndtm(j) =−2\nh2m(j)×/parenleftbiggm(j+1)\n1+m(j)·m(j+1)+m(j−1)\n1+m(j−1)·m(j)/parenrightbigg\n,8 YUEHENG LAN AND Y. CHARLES LI\nwherem(j) =m(t,jh),j= 1,···,N,Nh= 2π, andhis the spatial mesh size. For\nthe computation of Figure 3, we choose N= 128.\nγ\nσ\nFigure 1. The separatrix connecting to the domain wall m+=\n−ei2x,m3= 0.\n0246\n0102030−101\nxtm1\n(a)γ= 00246\n0102030−101\nxtm1\n(b)γ=π\nFigure 2. The spatial-temporal profiles corresponding to the two\nlobes of the figure eight curve.\nBy a translation x→x+θ, one can generate a circle of domain walls:\nm+=−ei2(x+θ), m3= 0,\nwhereθis the phase parameter. The three dimensional figure eight separa trix\nconnecting to the circle of domain walls, parametrized by σ,γandθ; is illustrated\nin Figure 4.\nIn general, the unimodal equilibrium manifold can be sought as follows: Let\nmj=cjcos2x+sjsin2x , j= 1,2,3,\nthen the uni-length condition |m|(x) = 1 leads to\n|c|= 1,|s|= 1, c·s= 0,\nwherecandsare the two vectors with components cjandsj. Thus the unimodal\nequilibrium manifold is three dimensional and can be represented as in F igure 5.CHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 9\n0246\n10203040−101\nxtm1\n(a) inside0123456\n10152025303540−101\nxtm1\n(b) outside\nFigure 3. The spatial-temporal profiles corresponding to the in-\nside and outside of the figure eight curve.\nγ\nσ\nθ\nFigure 4. The separatrix connecting to the circle of domain walls\nm+=ei2(x+θ),m3= 0.\nUsing the formulae (3.13)-(3.14), we want to build a Melnikov integral. The\nzeros of the Melnikov integral will give a prediction on the existence o f chaos. To\nbuild such a Melnikov integral, we need to first develop a Melnikov vecto r. This\nrequires Floquet theory of (3.3).\n3.4. Floquet Theory. Focusing on the spatial part (3.3) of the Lax pair\n(3.3)-(3.4), let Y(x) be the fundamental matrix solution of (3.3), Y(0) =I(2×2\nidentity matrix), then the Floquet discriminant is defined by\n∆ = trace Y(2π).\nThe Floquet spectrum is given by\nσ={λ∈C| −2≤∆(λ)≤2}.10 YUEHENG LAN AND Y. CHARLES LI\ns c\nFigure 5. A representation of the 3 dimensional unimodal equi-\nlibrium manifold.\nPeriodicandanti-periodicpoints λ±(whichcorrespondtoperiodicandanti-periodic\neigenfunctions respectively) are defined by\n∆(λ±) =±2.\nA critical point λ(c)is defined by\nd∆\ndλ(λ(c)) = 0.\nA multiple point λ(n)is a periodic or anti-periodic point which is also a critical\npoint. The algebraic multiplicity of λ(n)is defined as the order of the zero of\n∆(λ)±2 atλ(n). When the order is 2, we call the multiple point a double point,\nand denote it by λ(d). The order can exceed 2. The geometric multiplicity of λ(n)\nis defined as the dimension of the periodic or anti-periodic eigenspace atλ(n), and\nis either 1 or 2.\nCounting lemmas for λ±andλ(c)can be established as in [ 30] [23], which lead\nto the existence of the sequences {λ±\nj}and{λ(c)\nj}and their approximate locations.\nNevertheless, counting lemmas are not necessary here. For any λ∈C, ∆(λ) is a\nconstantofmotionofthe Heisenbergequation(3.1). Thisistheso- calledisospectral\ntheory.\nExample 3.2. For the domain wall m1= cos2x,m2= sin2x, andm3= 0; the\ntwo Bloch eigenfunctions are given in (3.10). The Floquet discriminant is given by\n∆ = 2cos/bracketleftBig\n2π/radicalbig\n1+λ2/bracketrightBig\n.\nThe periodic points are given by\nλ=±/radicalbigg\nn2\n4−1, n∈Z, nis even.\nThe anti-periodic points are given by\nλ=±/radicalbigg\nn2\n4−1, n∈Z, nis odd.\nThe choice of φ+andφ−correspond to n=±1 andλ=ν=i√\n3/2 withk=±1.\n∆′=−4πλ√\n1+λ2sin/bracketleftBig\n2π/radicalbig\n1+λ2/bracketrightBig\n.\n∆′′=−4π(1+λ2)−3/2sin/bracketleftBig\n2π/radicalbig\n1+λ2/bracketrightBig\n−8π2λ2\n1+λ2cos/bracketleftBig\n2π/radicalbig\n1+λ2/bracketrightBig\n.CHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 11\nλ\nFigure 6. The periodic and anti-periodic points corresponding to\nthe potential of domain wall m+=ei2x,m3= 0. The open circles\nare double points, the solid circle at the origin is a multiple point\nof order 4, and the two bars intersect the imaginary axis at two\nperiodic points which are not critical points.\nWhenn= 0, i.e.√\n1+λ2= 0, by L’Hospital’s rule\n∆′→ −8π2λ , λ=±i .\nThat is,λ=±iare periodic points, not critical points. When n=±1, we have two\nimaginary double points\nλ=±i√\n3/2.\nWhenn=±2,λ= 0 is a multiple point of order 4. The rest periodic and anti-\nperiodicpoints areallrealdouble points. Figure 6is anillustrationoft hese spectral\npoints.\n3.5. Melnikov Vectors. Starting from the Floquet theory, one can build\nMelnikov vectors.\nDefinition 3.3. An importantsequenceofinvariants Fjofthe Heisenbergequation\nis defined by\nFj(m) = ∆(λ(c)\nj(m),m).\nLemma 3.4. If{λ(c)\nj}is a simple critical point of ∆, then\n∂Fj\n∂m=∂∆\n∂m/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nλ=λ(c)\nj.\nProof. We know that\n∂Fj\n∂m=∂∆\n∂m/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nλ=λ(c)\nj+∂∆\n∂λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nλ=λ(c)\nj∂λ(c)\nj\n∂m.\nSince\n∂∆\n∂λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nλ=λ(c)\nj= 0,\nwe have\n∂2∆\n∂λ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nλ=λ(c)\nj∂λ(c)\nj\n∂m+∂2∆\n∂λ∂m/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nλ=λ(c)\nj= 0.12 YUEHENG LAN AND Y. CHARLES LI\nSinceλ(c)\njis a simple critical point of ∆,\n∂2∆\n∂λ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nλ=λ(c)\nj/ne}ationslash= 0.\nThus\n∂λ(c)\nj\n∂m=−/bracketleftBigg\n∂2∆\n∂λ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nλ=λ(c)\nj/bracketrightBigg−1\n∂2∆\n∂λ∂m/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nλ=λ(c)\nj.\nNotice that ∆ is an entire function of λandm[23], then we know that∂λ(c)\nj\n∂mis\nbounded, and\n∂Fj\n∂m=∂∆\n∂m/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nλ=λ(c)\nj.\n/square\nTheorem 3.5. As a function of two variables, ∆ = ∆(λ,m)has the partial deriva-\ntives given by Bloch functions ψ±(i.e.ψ±(x) =e±Λx˜ψ±(x), where˜ψ±are periodic\ninxof period 2π, andΛis a complex constant):\n∂∆\n∂m+=−iλ√\n∆2−4\nW(ψ+,ψ−)ψ+\n1ψ−\n1,\n∂∆\n∂m−=iλ√\n∆2−4\nW(ψ+,ψ−)ψ+\n2ψ−\n2,\n∂∆\n∂m3=iλ√\n∆2−4\nW(ψ+,ψ−)/parenleftbig\nψ+\n1ψ−\n2+ψ+\n2ψ−\n1/parenrightbig\n,\n∂∆\n∂λ=i√\n∆2−4\nW(ψ+,ψ−)/integraldisplay2π\n0/bracketleftbig\nm3/parenleftbig\nψ+\n1ψ−\n2+ψ+\n2ψ−\n1/parenrightbig\n−m+ψ+\n1ψ−\n1+m−ψ+\n2ψ−\n2/bracketrightbig\ndx ,\nwhereW(ψ+,ψ−) =ψ+\n1ψ−\n2−ψ+\n2ψ−\n1is the Wronskian.\nProof. Recall that Yis the fundamental matrix solution of (3.3), we have the\nequation for the differential of Y\n∂xdY=iλΓdY+i(dλΓ+λdΓ)Y , dY (0) = 0.\nUsing the method of variation of parameters, we let\ndY=YQ , Q (0) = 0.\nThus\nQ(x) =i/integraldisplayx\n0Y−1(dλΓ+λdΓ)Ydx ,\nand\ndY(x) =iY/integraldisplayx\n0Y−1(dλΓ+λdΓ)Ydx .\nFinally\nd∆ = trace dY(2π)\n=itrace/braceleftbigg\nY(2π)/integraldisplay2π\n0Y−1(dλΓ+λdΓ)Ydx/bracerightbigg\n. (3.16)\nLet\nZ= (ψ+ψ−)CHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 13\nwhereψ±are two linearly independent Bloch functions (For the case that the re is\nonly one linearly independent Bloch function, L’Hospital’s rule has to be used, for\ndetails, see [ 23]), such that\nψ±=e±Λx˜ψ±,\nwhere˜ψ±are periodic in xof period 2πand Λ is a complex constant (The existence\nof such functions is the result of the well known Floquet theorem). Then\nZ(x) =Y(x)Z(0), Y(x) =Z(x)[Z(0)]−1.\nNotice that\nZ(2π) =Z(0)E ,whereE=/parenleftbigg\neΛ2π0\n0e−Λ2π/parenrightbigg\n.\nThen\nY(2π) =Z(0)E[Z(0)]−1.\nThus\n∆ = trace Y(2π) = traceE=eΛ2π+e−Λ2π,\nand\ne±Λ2π=1\n2[∆±/radicalbig\n∆2−4].\nIn terms of Z,d∆ as given in (3.16) takes the form\nd∆ =itrace/braceleftbigg\nZ(0)E[Z(0)]−1/integraldisplay2π\n0Z(0)[Z(x)]−1(dλΓ+λdΓ)Z(x)[Z(0)]−1dx/bracerightbigg\n=itrace/braceleftbigg\nE/integraldisplay2π\n0[Z(x)]−1(dλΓ+λdΓ)Z(x)dx/bracerightbigg\n,\nfrom which one obtains the partial derivatives of ∆ as stated in the t heorem. /square\nIt turns out that the partial derivatives of Fjprovide the perfect Melnikov\nvectors rather than those of the Hamiltonian or other invariants [ 23], in the sense\nthatFjis the invariant whose level sets are the separatrices.\n3.6. An Explicit Expression of the Melnikov Vector Along the Figure\nEight Connecting to the Domain Wall. We continue the calculation in sub-\nsection 3.3 to obtain an explicit expression of the Melnikov vector alon g the figure\neight connecting to the domain wall. Apply the Darboux transformat ion (3.6) to\nφ±(3.11) atλ=ν, we obtain\nˆφ±=±¯ν−ν\n¯νexp{∓1\n2σ∓i1\n2γ}W(φ+,φ−)\n|φ1|2+|φ2|2\nφ2\n−φ1\n.\nIn the formula (3.6), for general λ,\ndetG=(ν−λ)(¯ν−λ)\n|ν|2, W(ˆψ+,ˆψ−) = detG W(ψ+,ψ−).\nIn a neighborhood of λ=ν,\n∆2−4 = ∆(ν)∆′′(ν)(λ−ν)2+ higher order terms in ( λ−ν).\nAsλ→ν, by L’Hospital’s rule\n√\n∆2−4\nW(ˆψ+,ˆψ−)→/radicalbig\n∆(ν)∆′′(ν)\nν−¯ν\n|ν|2W(φ+,φ−).14 YUEHENG LAN AND Y. CHARLES LI\nNotice, by the calculation in Example 3.2, that\nν=i√\n3\n2,∆(ν) =−2,∆′′(ν) =−24π2,\nthen by Theorem 3.5,\n∂∆\n∂m+/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nm=ˆm= 12√\n3πi\n(|φ1|2+|φ2|2)2φ22,\n∂∆\n∂m−/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nm=ˆm= 12√\n3π−i\n(|φ1|2+|φ2|2)2φ12,\n∂∆\n∂m3/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nm=ˆm= 12√\n3π2i\n(|φ1|2+|φ2|2)2φ1φ2,\nwhere ˆmis given in (3.13)-(3.14). With the explicit expression (3.12) of φ, we\nobtain the explicit expressions of the Melnikov vector,\n∂∆\n∂m+/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nm=ˆm=3√\n3π\n2isech2τ\n(2−√\n3 sech2τsin2χ)2/bracketleftbigg\n(1−2tanh2τ)cos2χ\n+i(2−tanh2τ)sin2χ−i√\n3 sech2τ/bracketrightbigg\ne−i2x, (3.17)\n∂∆\n∂m−/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nm=ˆm=3√\n3π\n2−isech2τ\n(2−√\n3 sech2τsin2χ)2/bracketleftbigg\n(1+2tanh2 τ)cos2χ\n−i(2+tanh2τ)sin2χ+i√\n3 sech2τ/bracketrightbigg\nei2x, (3.18)\n∂∆\n∂m3/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nm=ˆm=3√\n3π\n22isech2τ\n(2−√\n3 sech2τsin2χ)2/bracketleftbigg\n2 sech2τ−√\n3sin2χ\n−i√\n3tanh2τcos2χ/bracketrightbigg\n, (3.19)\nwhere again\nm±=m1±im2, τ=√\n3\n2t+σ\n2, χ=1\n2(x+γ),\nandσandγare two real parameters.\n4. A Melnikov Function\nThe forced Landau-Lifshitz-Gilbert (LLG) equation (2.1) can be re written in\nthe form,\n(4.1) ∂tm=−m×mxx+ǫf+ǫ2g\nwherefis the perturbation\nf=−am×ex+m3(m×ez)−bm1(m×ex)\n−αm×(m×mxx)+(β1+β2cosω0t)m×(m×ex),\ng=−αm×[m×(aex−m3ez+bm1ex)].\nThe Melnikov function for the forced LLG (2.1) is given as\nM=/integraldisplay∞\n−∞/integraldisplay2π\n0/bracketleftbigg∂∆\n∂m+(f1+if2)+∂∆\n∂m−(f1−if2)+∂∆\n∂m3f3/bracketrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nm=ˆmdxdt ,CHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 15\nwhere ˆmis given in (3.13)-(3.14), and∂∆\n∂w(w=m+,m−,m3) are given in (3.17)-\n(3.19). The Melnikov function depends on several external and int ernal parameters\nM=M(a,b,α,β 1,β2,ω0,σ,γ) whereσandγare internal parameters. We can split\nfas follows:\nf=af(a)+f(0)+bf(b)+αf(α)+β1f(β1)\n+β2/bracketleftbigg\ncos/parenleftbiggσω0√\n3/parenrightbigg\nf(c)+sin/parenleftbiggσω0√\n3/parenrightbigg\nf(s)/bracketrightbigg\n,\nwhere\nf(a)=−m×ex,\nf(0)=m3(m×ez),\nf(b)=−m1(m×ex),\nf(α)=−m×(m×mxx),\nf(β1)=m×(m×ex),\nf(c)= cos/parenleftbigg2√\n3ω0τ/parenrightbigg\nm×(m×ex),\nf(s)= sin/parenleftbigg2√\n3ω0τ/parenrightbigg\nm×(m×ex).\nThusMcan be splitted as\nM=aM(a)+M(0)+bM(b)+αM(α)+β1M(β1)\n+β2/bracketleftbigg\ncos/parenleftbiggσω0√\n3/parenrightbigg\nM(c)+sin/parenleftbiggσω0√\n3/parenrightbigg\nM(s)/bracketrightbigg\n, (4.2)\nwhereM(ζ)=M(ζ)(γ),ζ=a,0,b,α,β 1, andM(ζ)=M(ζ)(γ,ω0),ζ=c,s.\nIn general [ 19], the zeros of the Melnikov function indicate the intersection of\ncertain center-unstable and center-stable manifolds. In fact, t he Melnikov function\nis the leading order term of the distance between the center-unst able and center-\nstable manifolds. In some cases, such an intersection can lead to ho moclinic orbits\nand homoclinic chaos. Here in the current problem, we do not have an invariant\nmanifold result. Therefore, our calculation on the Melnikov function is purely from\na physics, rather than rigorous mathematics, point of view.\nIn terms ofthe variables m+andm3, the forced Landau-Lifshitz-Gilbert (LLG)\nequation (2.1) can be rewritten in the form that will be more convenie nt for the\ncalculation of the Melnikov function,\n∂tm+=i(m+m3xx−m3m+xx)+ǫf++ǫ2g+, (4.3)\n∂tm3=1\n2i(m+m+xx−m+m+xx)+ǫf3+ǫ2g3, (4.4)16 YUEHENG LAN AND Y. CHARLES LI\nwhere\nf+=f1+if2=af(a)\n++f(0)\n++bf(b)\n++αf(α)\n++β1f(β1)\n+\n+β2/bracketleftbigg\ncos/parenleftbiggσω0√\n3/parenrightbigg\nf(c)\n++sin/parenleftbiggσω0√\n3/parenrightbigg\nf(s)\n+/bracketrightbigg\n,\nf3=af(a)\n3+f(0)\n3+bf(b)\n3+αf(α)\n3+β1f(β1)\n3\n+β2/bracketleftbigg\ncos/parenleftbiggσω0√\n3/parenrightbigg\nf(c)\n3+sin/parenleftbiggσω0√\n3/parenrightbigg\nf(s)\n3/bracketrightbigg\n,\ng+=g1+ig2=αag(a)\n++αg(0)\n++αbg(b)\n+,\ng3=αag(a)\n3+αg(0)\n3+αbg(b)\n3,\nf(a)\n+=−im3,\nf(0)\n+=−im3m+,\nf(b)\n+=−i1\n2m3(m++m+),\nf(α)\n+=1\n2m+(m+m+xx−m+m+xx)+m3(m3m+xx−m+m3xx),\nf(β1)\n+=1\n2m+(m+−m+)−m2\n3,\nf(c)\n+= cos/parenleftbigg2√\n3ω0τ/parenrightbigg/bracketleftbigg1\n2m+(m+−m+)−m2\n3/bracketrightbigg\n,\nf(s)\n+= sin/parenleftbigg2√\n3ω0τ/parenrightbigg/bracketleftbigg1\n2m+(m+−m+)−m2\n3/bracketrightbigg\n,\nf(a)\n3=1\n2i(m+−m+),\nf(0)\n3= 0,\nf(b)\n3=1\n4i(m2\n+−m+2),\nf(α)\n3=m3xx|m+|2−1\n2m3(m+m+xx+m+m+xx),\nf(β1)\n3=1\n2m3(m++m+),\nf(c)\n3=1\n2cos/parenleftbigg2√\n3ω0τ/parenrightbigg\nm3(m++m+),\nf(s)\n3=1\n2sin/parenleftbigg2√\n3ω0τ/parenrightbigg\nm3(m++m+),CHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 17\ng(a)\n+=m2\n3−1\n2m+(m+−m+),\ng(0)\n+=m2\n3m+,\ng(b)\n+=1\n2m2\n3(m++m+)−1\n4m+(m2\n+−m+2),\ng(a)\n3=−1\n2m3(m++m+),\ng(0)\n3=−m3|m+|2,\ng(b)\n3=−1\n4m3(m++m+)2.\nDirect calculation gives that\nM(a)(γ) =M(0)(γ) =M(b)(γ) = 0, M(α)(γ) = 91.3343,\nandM(β1)andM(c)are real, while M(s)is imaginary. The graph of M(β1)is\nshown in Figure 7(a) (Notice that M(β1)is independent of ω0). The graph of M(c)\nis shown in Figure 7(b). The imaginary part of M(s)is shown in Figure 7(c). In\nthe case of only DC current ( β2= 0),M= 0 (4.2) leads to\n(4.5) α=−β1M(β1)/91.3343,\nwhereM(β1)(γ) is a function of the internal parameter γas shown in Figure 7(a).\nIn the general case ( β2/ne}ationslash= 0),M(s)(γ,ω0) = 0 determines curves\n(4.6) γ=γ(ω0) = 0, π/2, π,3π/2,\nandM= 0 (4.2) leads to\n(4.7) |β2|>/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftBig\n91.3343α+β1M(β1)/parenrightBig/slashbigg\nM(c)/vextendsingle/vextendsingle/vextendsingle/vextendsingle,\nwhereM(β1)andM(c)are evaluated along the curve (4.6), M(β1)=±43.858 (‘+’\nforγ=π/2,3π/2; ‘−’ forγ= 0, π),M(c)is plotted in Figure 8 (upper curve\ncorresponds to γ=π/2,3π/2; lower curve corresponds to γ= 0, π), and\ncos/parenleftbiggσω0√\n3/parenrightbigg\n=−91.3343α+β1M(β1)\nβ2M(c).\n5. Numerical Simulation\nIn the entire article, we use the finite difference method to numerica lly simulate\nthe LLG (2.1). Due to an integrable discretization [ 9] of the Heisenberg equation\n(3.1), the finite difference performs much better than Galerkin Fou rier mode trun-\ncations. As in (3.15), let m(j) =m(t,jh),j= 1,···,N,Nh= 2π, andhis the\nspatial mesh size. Without further notice, we always choose N= 128 (which pro-\nvides enough precision). The only tricky part in the finite difference d iscretization\nof (2.1) is the second derivative term in H, for the rest terms, just evaluate mat\nm(j):\n∂2\nxm(j) =2\nh2/parenleftbiggm(j+1)\n1+m(j)·m(j+1)+m(j−1)\n1+m(j−1)·m(j)/parenrightbigg\n.18 YUEHENG LAN AND Y. CHARLES LI\n0246\n024−50050\nγω0Mβ1\n(a)0246\n024−50050\nγω0M(c)\n(b)\n0246\n024−50050\nγω0M(s)\n(c)\nFigure 7. (a). The graph of M(β1)as a function of γ, andM(β1)\nis independent of ω0. (b). The graph of M(c)as a function of γ\nandω0. (c). The graph of the imaginarypart of M(s)as a function\nofγandω0.\n5.1. Only DC Current Case. In this case, β2= 0 in (2.1), and we choose\nβ1as the bifurcation parameter, and the rest parameters as:\n(5.1) a= 0.05,b= 0.025,α= 0.02,ǫ= 0.01.\nThe computation is first run for the time interval [0 ,8120π], then the figures are\nplotted starting from t= 8120π. The bifurcation diagram for the attractors, and\nthe typical spatial profiles on the attractors are shown in Figure 9 . This figure\nindiactes that interesting bifurcations happen over the interval β1∈[0,0.15] which\nis the physically important regime where β1is comparable with values of other\nparameters. There are six bifurcation thresholds c1···c6(Fig. 9). When β1<c1,\nthe attractor is the spatially uniform fixed point m1= 1 (m2=m3= 0). WhenCHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 19\n00.511.522.533.54−50−40−30−20−1001020304050\nω0M(c)\nFigure 8. The graphs of M(c)along the curves (4.6).\nβ1c2c3c1c c c4 5 6I II III IV V VI VII\nFigure 9. The bifurcation diagram for the attractors and typ-\nical spatial profiles on the attractors in the case of only DC\ncurrent where β1is the bifurcation parameter, c1···c6are the\nbifurcation thresholds, and c1∈[0,0.01],c2∈[0.0205,0.021],\nc3∈[0.0231,0.0232],c4∈[0.025,0.026],c5∈[0.08,0.1],c6∈\n[0.13,0.15].\nc1≤β1≤c2, the attractoris a spatially non-uniform fixed point as shown in Figur e\n10(a). When c2< β1< c3, the attractor is spatially non-uniform and temporally\nperiodic (a limit cycle) or quasiperiodic (a limit torus) as shown in Figure 1 0(b).\nHere as the value of β1is increased, first there is one basic temporal frequence,\nthen more frequencies enter and the temporal oscillation amplitude becomes bigger\nand bigger. When c3≤β1< c4, the attractor is chaotic, i.e. a strange attractor\nas shown in Figure 10(c). Even though the chaotic nature is not ver y apparent\nin Figure 10(c), due to the smallness of the perturbation paramete r together with\nsmallness of all other parameters, the temporal evolution is chaot ic, and we have\nused Liapunov exponent and power spectrum devices to verify this . Whenc4≤\nβ1< c5, the attractor is spatially non-uniform and temporally periodic (a limit\ncycle) as shown in Figure 11(a). The spatial modulation is small, and it is even not\napparentin Figure11(a). But itisapparentonthe individualtypical spatialprofiles\nas seen in region V in Figure 9. With the increase of β1, the spatial modulation\nbecomes smaller and smaller. When c5≤β1<c6, the attractor is spatially uniform\nand temporally periodic (a limit cycle, the so-called procession) as sho wn in Figure20 YUEHENG LAN AND Y. CHARLES LI\n0123456\n0510152025300.60.81\nxtm1\n(a)β1= 0.0205 spatially non-uniform fixed\npoint0123456\n05101520253000.51\nxtm1\n(b)β1= 0.023 spatially non-uniform and\ntemporally periodic or quasiperiodic attrac-\ntor\n0123456\n051015202530−101\nxtm1\n(c)β1= 0.0235 weak chaotic attractor\nFigure 10. The spatio-temporal profiles of solutions in the at-\ntractors in the case of only DC current.\n11(b). When β1≥c6, the attractor is the spatially uniform fixed point m1=−1\n(m2=m3= 0).\nWhenβ2= 0, the Melnikov function predicts that around β1= 0.041 (4.5),\nthere is probably chaos; while the numerical calculation shows that t here is an\ninterval [0.0231,0.026] forβ1where chaos is the attractor. Since the perturbation\nparameterǫ= 0.01 in the numerical calculation, the Melnikov function prediction\nseems in agreement with the numerical calculation.\nSome of the attractors in Figure 9 are attractors of the corresp onding ordinary\ndifferential equations by setting ∂x= 0 in (2.1), i.e. the single domain case. These\nattractors are the ones in regions I, VI and VII in Figure 9 [ 25] [41]. Because weCHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 21\n0123456\n0100200300400500600−101\nxtm1\n(a)β1= 0.026 spatially non-uniform and\ntemporally periodic attractor0123456\n0100200300400500600−0.500.5\nxtm1\n(b)β1= 0.1 spatially uniform and tempo-\nrally periodic attractor\nFigure 11. The spatio-temporal profiles of solutions in the at-\ntractors in the case of only DC current (continued).\nare studying the only DC current case, the ordinary differential eq uations do not\nhave any chaotic attractor [ 25] [41].\nOn the other hand, the partial differential equations (2.1) does ha ve a chaotic\nattractor (region IV in Figure 9).\nIn general, when β1<0, the Gilbert damping dominates the spin torque driven\nby DC current and m1= 1 is the attractor. When β1>0.15, the spin torque driven\nby DC current dominates the Gilbert damping, m1=−1 is the attractor, and we\nhave a magnetization reversal. In some technological applications, β1>0.15 may\ncorrespond too high DC current that can burn the device. On the o ther hand, in\nthe technologically advantageousinterval β1∈[0,0.15], magnetization reversalmay\nbe hard to achieve due to the sophisticated bifurcations in Figure 9.\n5.2. Only AC Current Case. In this case, β1= 0 in (2.1), and we choose\nβ2as the bifurcation parameter, and the rest parameters as:\n(5.2) a= 0.05,b= 0.025,α= 0.0015,ǫ= 0.01,ω0= 0.2.\nUnlike the DC case, here the figures are plotted starting from t= 0. It turns\nout that the types of attractors in the AC case are simpler than th ose of the\nDC case. When β2= 0, the attractor is a spatially non-uniform fixed point as\nshown in Figure 12. In this case, the only perturbation is the Gilbert d amping\nwhich damps the evolution to such a fixed point. When 0 < β2< β∗\n2where\nβ∗\n2∈[0.18,0.19], the attractor is a spatially non-uniform and temporally periodic\nsolution. When β2≥β∗\n2, the attractor is chaotic as shown in Figure 12. Our\nMelnikov prediction (4.7) predicts that when |β2|>0.003, certain center-unstable\nandcenter-stablemanifolds intersect. Ournumericsshowsthat s uch anintersection\nseems leading to transient chaos. Only when |β2|>β∗\n2, the chaos can be sustained\nas an attractor. It seems that such sustained chaotic attracto r gains extra support\nfrom parametric resonance due to the AC current driving [ 22], as can be seen from22 YUEHENG LAN AND Y. CHARLES LI\n0123456\n050001000015000−101\nxtm1\n(a)β2= 0 spatially non-uniform fixed point020004000600080001000012000−1−0.95−0.9−0.85−0.8−0.75−0.7−0.65−0.6−0.55−0.5\ntm1(x1)\n(b)β2= 0 temporal evolution at one spatial\nlocation\n0123456\n050001000015000−101\nxtm1\n(c)β2= 0.21 chaotic attractor020004000600080001000012000−1−0.8−0.6−0.4−0.200.20.40.60.81\ntm1(x1)\n(d)β2= 0.21 temporal evolution at one spa-\ntial location\nFigure 12. The attractors in the case of only AC current.\nthe turbulent spatial structure of the chaotic attractor (Fig. 1 2), which diverges\nquite far away from the initial condition. Another factor that may b e relevant\nis the fact that higher-frequency spatially oscillating domain walls hav e more and\nstronger linearly unstable modes (3.5). By properly choosing initial c onditions, one\ncan find the homotopy deformation from the ODE limit cycle (process ion) [25] [41]\nto the current PDE chaos as shown in Figure 13 at the same paramet er values.\nWe also simulated the case of normal Gilbert damping α= 0.02. For all values\nofβ2∈[0.01,0.3], the attractor is always non-chaotic. That is, the only attracto r\nwe can find is a spatially uniform limit cycle with small temporal oscillation a s\nshown in Figure 14.\nOf course, when neither β1norβ2is zero, the bifurcation diagram is a combi-\nnation of the DC only and AC only diagrams.CHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 23\nFigure 13. Homotopy deformation of the attractors under differ-\nent initial conditions. See http://www.math.missouri.edu/˜cli\nFigure 14. The attractor when α= 0.02,β2= 0.21 and\nall other parameters’ values are the same with Figure 12. See\nhttp://www.math.missouri.edu/˜cli\n6. Appendix: The Connection Between the Heisenberg Equatio n and\nthe NLS Equation\nIn this appendix, we will show the details on the connection between t he 1D\ncubic focusing nonlinear Schr¨ odinger (NLS) equation and the Heise nberg equation\n(3.1). The nonlinear Schr¨ odinger (NLS) equation\n(6.1) iqt+qxx+2|q|2q= 0,\nis a well-known integrable system with the Lax pair\n∂xφ= (iλσ3+U)φ , (6.2)\n∂tφ=−(2iλ2σ3+2λU+V)φ , (6.3)24 YUEHENG LAN AND Y. CHARLES LI\nwhereλis the complex spectral parameter, σ3is defined in (2.5), and\nU=/parenleftbigg0iq\ni¯q0/parenrightbigg\n, V=−i|q|2σ3+/parenleftbigg0qx\n−¯qx0/parenrightbigg\n.\nLemma 6.1. Ifφ= (φ1,φ2)Tsolves the Lax pair (6.2)-(6.3) at λ, then(−φ2,φ1)T\nsolves the Lax pair (6.2)-(6.3) at ¯λ. Whenqis even, i.e. q(−x) =q(x), then\n(φ2(−x),φ1(−x))Tsolves the Lax pair (6.2)-(6.3) at −¯λ. Whenλis real, and φis\na nonzero solution, then (−φ2,φ1)Tis another linearly independent solution. For\nany two solutions of the Lax pair, their Wronskian is indepen dent ofxandt.\nFor any real λ0, by the well-known Floquet theorem [ 28] and Lemma 6.1, there\nare always two linearly independent Floquet (or Bloch) eigenfunction sφ±to the\nLax pair (6.2)-(6.3) at λ=λ0, such that\nφ+=/parenleftbiggϕ1\nϕ2/parenrightbigg\n, φ−=/parenleftbigg−ϕ2\nϕ1/parenrightbigg\n,\nφ+(x+2π) =ρφ+(x), φ−(x+2π) = ¯ρφ−(x),|ρ|2= 1.\nSincetheWronskian W(φ+,φ−)isindependentof xandt, withoutlossofgenerality,\nwe chooseW(φ+,φ−) = 1. Then\nS=/parenleftbigg\nϕ1−ϕ2\nϕ2ϕ1/parenrightbigg\nis a unitary solution to the Lax pair at λ=λ0:\nS−1=SH=/parenleftbiggϕ1ϕ2\n−ϕ2ϕ1/parenrightbigg\n,|ϕ1|2+|ϕ2|2= 1.\nRecall the definition of Γ (2.7), let\nΓ =S−1σ3S=/parenleftbigg\n|ϕ1|2−|ϕ2|2−2ϕ1ϕ2\n−2ϕ1ϕ2|ϕ2|2−|ϕ1|2/parenrightbigg\n,\ni.e.\nm1+im2=−2ϕ1ϕ2, m3=|ϕ1|2−|ϕ2|2.\nNow for any φsolving the Lax pair (6.2)-(6.3) at λ, defineψas\nψ=S−1φ .\nThenψsolves the pair\nψx=i(λ−λ0)Γψ , (6.4)\nψt=−/braceleftbigg\n2i(λ2−λ2\n0)Γ+1\n2(λ−λ0)[Γ,Γx]/bracerightbigg\nψ . (6.5)\nThe compatibility condition of this pair leads to the equation\n(6.6) Γ t=−/braceleftbigg\n4λ0Γx+1\n2i[Γ,Γxx]/bracerightbigg\n.\nSettingλ0= 0 or performing the translation t=t,ˆx=x−4λ0t, equation (6.6)\nreduces to the Heisenberg equation (3.2). Therefore, the Gauge transformStrans-\nforms NLS equation into the Heisenberg equation. Periodicity in xmay not persist.CHAOTIC SPIN DYNAMICS OF A LONG NANOMAGNET 25\nExample 6.2. Consider the temporally periodic solution of the NLS equation\n(6.1),\nq=aeiθ(t), θ(t) = 2a2t+γ .\nThe corresponding Bloch eigenfunction of the Lax pair (6.2)-(6.3) a tλ= 0 is\nϕ=1√\n2/parenleftbigg\neiθ/2\ne−iθ/2/parenrightbigg\neiax.\nThen\nΓ =S−1σ3S=/parenleftbigg\n0 −e−i2ax\n−ei2ax0/parenrightbigg\n,\nwhich is called a domain wall.\nAcknowledgment : The second author Y. Charles Li is grateful to Professor\nShufeng Zhang, Drs. Zhanjie Li and ZhaoyangYang, and Mr. Jiexu an He for many\nhelpful discussions.\nReferences\n[1] L. Berger. Emission of Spin Waves by a Magnetic Multilaye r.Phys. Rev. B , 54:9353–9358,\n1996.\n[2] A. Calini. A Note on a B¨ acklund Transformation for the Co ntinuous Heisenberg Model. Phys.\nLett. A, 203: 333–344, 1995.\n[3] M. Covington et al. Current-Induced Magnetization Dyna mics in Current Perpendicular to\nthe Plane Spin Valves. Phy. Rev. B , 69:184406, 2004.\n[4] A. Fabian et al. 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Synchronization, Modification an d Chaos Induced by Spin-polarized\nCurrents. Phys. Rev. B. , 74:054417, 2006.\n[26] Z. Li and S. Zhang. Magnetization Dynamics with a Spin-T ransfer Torque. Phys. Rev. B ,\n68:024404, 2003.\n[27] Z. Li and S. Zhang. Thermally Assisted Magnetization Re versal in the Presence of a Spin-\nTransfer Torque. Phys. Rev. B , 69:134416, 2004.\n[28] W. Magnus, S. Winkler. Hill’s Equation . Dover, NY, 1979.\n[29] J. Miltat et al. Spin Transfer into an Inhomogeneous Mag netization Distribution. J. Appl.\nPhys., 89:6982, 2001.\n[30] B. Mityagin. Spectral Expansions of One-Dimensional P eriodic Dirac Operators. Dynamics\nof PDE, 1, no.2:125, 2004.\n[31] E. B. Myers et al. Current-Driven Switching of Domains i n Magnetic Multilayer Devices.\nScience, 285:867–870, 1999.\n[32] B. Ozyilmaz et al. Current-Induced Magnetization Reve rsal in High Magnetic Fields in\nCo/Cu/Co Nanopillars. Phys. Rev. Lett. , 93:067203, 2003.\n[33] M. R. Pufall et al. Materials Dependence of the Spin-Mom entum Transfer Efficiency and\nCritical Current in Ferromagnetic Metal/Cu Multilayers. Appl. Phys. Lett. , 83:323, 2003.\n[34] A. Slavin, et al. Nonlinear Self-Phase-Locking Effect i n an Array of Current-Driven Magnetic\nNanocontact. Phys. Rev. B , 72: 092407, 2005.\n[35] J.C.Slonczewski.Current-DrivenExcitation ofMagne tic Multilayers. J. Magn. Magn. Mater ,\n159:L1–L7, 1996.\n[36] J. Z. Sun. Current-Driven Magnetic Switching in Mangan ite TrilayerJunctions. J. Mag. Mag.\nMater., 202:157, 1999.\n[37] S. Urazhdin et al. Current-Driven Magnetic Excitation s in Permalloy-Based Multilayer\nNanopillars. Phys. Rev. Lett. , 91:146803, 2003.\n[38] J. E. Wegrowe. Magnetization Reversal and Two-Level Fl uctuations by Spin Injection in a\nFerromagnetic Metallic Layer. Phy. Rev. B , 68:2144xx, 2003.\n[39] S. A. Wolf et al. Spintronics: a Spin-Based Electronics Vision for the Future. Science,\n294:1488, 2001.\n[40] M. Yamanouchi et al. Current-Induced Domain-Wall Swit ching in a Ferromagnetic Semicon-\nductor Structure. Nature, 428:539, 2004.\n[41] Z. Yang, S. Zhang, Y. Li. Chaotic Dynamics of Spin Valve O scillators. Phys. Rev. Lett. , 99:\n134101, 2007.\n[42] J. G. Zhu et al. Spin Transfer Induced Noise in CPP Read He ads.IEEE Trans. on Magn. ,\n40:182, 2004.\nDepartment of Mechanical Engineering, University of Califo rnia, Santa Barbara,\nCA 93106\nE-mail address :yueheng lan@yahoo.com\nDepartment of Mathematics, University of Missouri, Columbi a, MO 65211\nE-mail address :cli@math.missouri.edu"    },    {        "title": "2312.10451v2.Spin_torque_nano_oscillator_based_on_two_in_plane_magnetized_synthetic_ferrimagnets.pdf",        "content": "1 \n The following article has been accepted by Journal of Applied Physics . After it is published, it \nwill be found at Link . \n \nSpin -torque nano -oscillator based on two in-plane magnetized \nsynthetic ferrimagnets  \n \nE. Monteblanco1,a), F. Garcia -Sanchez1, M. Romera1,2, D. Gusakova1, L. D. Buda -Prejbeanu1, \nU. Ebels1 \n1Univ. Grenoble Alpes, CEA, CNRS, Grenoble INP *, INAC, SPINTEC, F -38000 Grenoble, France  \n2GFMC, Departamento de Física de Materiales, Universidad Complutense, Madrid, Spain.  \n* Institute of Engineering Univ. Grenoble Alpes  \na) Electronic email: nmonteblanco@gmail.com  \n \n  \nWe report the dynamic characterization of the spin -torque -driven in- plane precession modes of a spin -torque nano-\noscillator based on two different synthetic ferrimagnets: a pinned one  characterized by a strong RKKY interaction \nwhich is exchange coupled to an  antiferromagnetic layer; and a second one, non -pinned characterized by weak \nRKKY coupling. The microwave properties associated with  the steady -state precession of both SyFs are \ncharacterized by high spectral purity and power spectral density. However, f requency dispersion diagrams of the \ndamped and spin transfer torque modes reveal drastically different dynamical behavior  and microwave emission \nproperties in both SyFs. In particular, the weak coupling between the magnetic layers of the non- pinned SyF raises \ndiscontinuous dispersion diagrams suggesting a strong influence of mode crossing. An interpretation of the \ndifferent dynamical features observed in the damped and spin torque modes of both SyF systems was obtained by \nsolving simultaneously, in a macros pin approach, a linearized version of the Landau -Lifshitz -Gilbert equation \nincluding the spin transfer torque term.  \n \n \nI. INTRODUCTION  \n \nThe exceptional and multi -functional properties of spin- torque1-2 nano- oscillators (STOs) made them \npromising candidates for a wide range of emerging technologies  which span from microwave emitters3 \nand detectors4-5 to neuromorphic  computing  systems6-10. These devices use the  transfer of angular \nmomentum from a  spin-polarized  current  to the local magnetization of a ferromagnetic layer11,12 to \ninduce self -sustained oscillations of the magnetization, which translate into a microwave signal whose  \nfrequency can be finely tuned with the applied direct current13-17. A widely studied structure of the STO \nis of the type (AF/SyF/MgO/SL)  with in -plane magnetization , including a synthetic ferrimagnet \nstructure (called  SyF- Polarizer) pinned via  exchange bias to an antiferromagnetic layer (AF) and a sin gle \nferromagnetic layer (SL)18-20. A standard SyF layer is composed of two ferromagnetic layers, a top (TL) \nand bottom (BL) layer coupled antiferromagnetically through a thin metallic layer via the Ruderman-\nKittel- Kasuya- Yosida (RKKY) interaction.21,22 In STO s, spin-polarized electrons affect damped \noscillations by modifying  damping and thus linewidth and amplitudes  and overcoming a critical current \ndensity (j c) polarized electrons can induce the steady -state oscillations (STT excitations) . Steady -state \noscillations can be obtained  in both parts,  SyF or SL, by changing the polarity of the current. The STT \nexcitations of a SyF structure  present some advantages in comparison with the SL excitations, as the \nhigher spectral purity (smaller linewidth), zero field excitations23,24, frequency tuning as a function of \nthe current with the possibility to change from redshift (df/dj app<0) to blueshift (df/dj app>0) applying an \nin-plane field and achieving large thermal stability25-29. However, since the SyF is pinned by an \nantiferromagnet, achieving  steady -state excitations require s a relatively large critical current. The use of \nexchange -coupled layers with perpendicular anisotropy30,31 or in- plane magnetized magnetic layers32-34 \nhave been shown to increase the magnetic stiffness and can potentially improve the performance of the \noscillator . Replacing  the standard free layer SL  by an unpinned SyF layer is thus of potential interest \ntowards  improving the microwave properties of  spin torque oscillators .  \n \nIn this article, we report the main static and dynamic features of a spin torque oscillator  based on two \nSyFs  with the following structure : IrMn(6.1)/SyF -Polarizer/MgO(0.9)/SyF -FL, w here numbers 2 \n represent thickness in nanometers. The composition of the SyF- Polarizer and SyF -FL are  \nCoFe(1.8)/Ru(0.4)/CoFe B(2) and CoFe(0.5)/CoFeB(3.4)/Ru/CoFe(3.6)  respectively. The structure of \nthis manuscript is the following: section II introduces the numerical techniques used to predict some \nfeatures of the STO dispersion diagrams and it is devoted to the macrospin analysis of the STO structure, \nsection III presents the experi mental characterization of the STO dynamics. Section IV presents the \ndiscussion and conclusions. In the following, the nano- oscillator based on a double  SyF will be called  \nD-SyF for simplicity.  \n \n \nII. NUMERICAL ANALYSIS  \n \nTwo types of numerical studies were performed in the framework of the macrospin approximation: (1) \ncomputation of the hysteresis loops (MH) and magnetoresistance loops (MR) using the minimization of \nthe energy and (2) stability analysis based on the linearization of the Landau- Lifshitz -Gilbert (LLG) \nequation enhanced with the spin transfer torque term (Slonczewski term) around the equilibrium \npositions, in order to find instabilities due to the applied current and fiel d. The D -SyF under  study \npresent s the following structure: AF/SyF -Polarizer/insulator barrier/SyF -Free layer, wh ere the bottom \nlayer of the SyF -Polarizer  is pinned into the positive x direction by an exchange bias field (H eb) induced \nby an antiferromagnetic layer, see Figure 1.  \n \nFig. 1. (a) Schematics of the double SyF oscillator structure with the labels used in this article. Negative applied \ncurrent corresponds to electrons flowing from SyF -FL to SyF- Polarizer.  (b) Schematic  of the standard RF setup.   \n  \nII.1. Simulation of static hysteresis loops  \n In order to understand the complex frequency dispersion diagrams of the D -SyF oscillator, its hysteresis \nloop has been simulated as the first step considering the different coupling between the ferromagnetic \nlayers. The total free energy density of the coupled system is written as 𝜎𝜎\n𝑡𝑡𝑡𝑡𝑡𝑡=𝛴𝛴\n𝑖𝑖𝜎𝜎𝑡𝑡𝑡𝑡𝑡𝑡,𝑖𝑖=𝛴𝛴\n𝑖𝑖(𝜎𝜎𝑖𝑖𝑖𝑖𝑡𝑡,𝑖𝑖+\n𝜎𝜎𝑒𝑒𝑒𝑒𝑡𝑡,𝑖𝑖) where the internal and external component for each layer are defined as follow,  \n \n 𝜎𝜎𝑖𝑖𝑖𝑖𝑡𝑡,𝑖𝑖=𝜎𝜎𝑧𝑧𝑒𝑒𝑒𝑒𝑧𝑧 ,𝑖𝑖+𝜎𝜎𝑎𝑎𝑖𝑖,𝑖𝑖+𝜎𝜎𝑑𝑑,𝑖𝑖+𝜎𝜎𝑒𝑒𝑒𝑒,𝑖𝑖        (1) \n 𝜎𝜎𝑒𝑒𝑒𝑒𝑡𝑡,𝑖𝑖=𝜎𝜎𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 ,𝑖𝑖+𝜎𝜎𝑑𝑑𝑖𝑖𝑑𝑑,𝑖𝑖𝑖𝑖        (2) \n \nThe magnetic layers are labelled as i, j =1 to 4 and i≠ j, corresponding to the structure on Figure 1. The \nmodel includes the different internal energies contributions such as the demagnetizing σ d and \nmagnetocrystalline anisotropies  σan, exchange bias σeb (only for the BL of the SyF pinned) and the \nZeeman terms σzeem. Also, we include the Ruderman- Kittel- Kasuya -Yosida (RKKY) interlayer \nexchange interaction  σRKKY (internal to each SyF) and the dipolar stray field35-36 σdip, see Eq. (2). The \nRKKY coupling is taken into account only between the layers 1- 2 and 3- 4 (σRKKY,1(3) = σRKKY,2(4) ) and the \n3 \n dipolar field between the four layers ( σdip,ij= σdip,ji). More details about each term of the equations are \nfound in Ref. 34 and 38. Two different RKKY coupling constants were considered: J RKKY =-0.1mJ/m2 \nand -1.5mJ/m2 for the SyF -FL (weak coupling)  and SyF -Polarizer  (strong coupling)  layer respectively . \nThe influence of a high or weak RKKY coupling in the hysteresis loops (MH) and magnetoresistance \ncurves (MR) has been shown in previous studies34. The structure considered is not compensated, i.e., \nthe product (M S*t*S) for the TL and BL of each SyF are not close ,  (M S*t*S)TL,SyF- FL= 35.12 μA nm2, \n(M S*t*S)BL,SyF- FL=28.79 μA nm2, (M S*t*S)TL,SyF -Polarizer =14.06 μA nm2, and (M S*t*S)BL,SyF -Polarizer =17.56 \nμA nm2. The net magnetic moment of the SyF -FL is 6.33  μA nm2 and for the SyF -Polarizer is 3.5  μA \nnm2 so we should consider  the stray magnetic field (dipolar field) between both SyFs.  This is \nfundamental to understanding the magnetization dynamics of this structure.  \n \n \n \n SyF Pinned Layer  SyF Free Layer  \nParameters  1 2 3 4 \nMS(kA/m)  1470  1060  1112.5  1470  \nK (J/m3) 7957.75  \nt (nm)  1.8 2.0 3.9 3.6 \nMS*t *S \n(µA.nm2) 17.56  14.06  28.79  35.12  \nNx \nNy \nNz 0.024952  \n0.046343  \n0.928705  0.027061  \n0.050280  \n0.922658  0.044620  \n0.083119  \n0.872260  0.042085  \n0.078372  \n0.879543  \np - -m3 m2 - \nη 0.3 \nα 0.02 \nS (nm2) 6636.63  \nHeb (kA/m)  79.5 0 0 0 \nJRKKY (mJ/m2) -1.5 -0.1 \n \nTable I. Parameters used in the numerical simulations . Here M S is the saturation magnetization, K u is a uniaxial \nmagneto crystalline anisotropy constant (//ox axis in the plane), t is the film thickness,  S is the surface,  Nx, Ny, and \nNz are demagnetization factors, α is the damping constant. H eb is the exchange bias field that acts on the BL of the \nSyF- Polarizer and  η is the spin efficiency.   \n \nNumerical simulation of the hysteresis loop and the magnetoresistance of the D -SyF oscillator are shown \nin Figure 2(a). These curves provide information about the relative orientation of the magnetizations of \nthe different layers, as a function of the applied field. The MR was calculated with the scalar product of \nthe magnetizations adjacent to the insulator barrier  i.e. layers 2 and 3. The parameters  are listed in Table \nI. In our samples, the anisotropy axis corresponds to the longer axis of the ellipse,  and it is parallel to \nthe X -axis. Arrows on Figure 2 represent the SyF -FL and SyF- Polarizer magnetization s respectively \nfollowing the color convention of the layers in Figure 1. For relatively large positive values of the \napplied field [100mT, 400mT] the magnetization of both layers of the SyF -FL are parallel to the external \nfield while the magnetization of the TL of the SyF -pinned is pointing in opposite direction, \ncorresponding to the antiparallel state in the magnetoresistance curve (Figure 2b). Upon sweeping the \nfield from positive to negative values,  a first magnetization switching is observed at μ oHsw ≈ +8.4 mT \n(Hsat+SyF- FLin Figure 2(a)). It corresponds to the  switching of the  magnetization of the BL of the SyF -FL, \nwhich presents a lower net magnetic moment in comparison with the TL.  The magnetization switching \nof the BL of the SyF -FL is accompanied by a change from the antiparallel (high resistance) to the parallel \n(low resistance) state in the magnetoresistance curve,  see Figure 2(b).  Upon sweeping further the applied \nfield towards negative values, the magnetization of the TL layer of the SyF -FL is inverted at Hsat-SyF- FL \n(Figure 2(a)), which does not affect  the magnetoresistance  (Figure 2(b)).  From now on, we  refer to th e \nregion of around 100mT between the two character istic saturati on field values ( Hsat-SyF-FL and Hsat+SyF-FL) \nas “plateau”. It corresponds to the P state in the RH curve. Sweeping the field in the opposite direction  \n(from negative to positive  values ) the bottom layer of the SyF -FL is reversed first again leading to a 4 \n similar scenario except because  the plateau region is shifted towards positive field  value s and now \ncorresponds to the antiparallel state (Figure 2(b)) . \n \n \nII.2. Dynamics features, the LLG equation  \n \nThe magnetization dynamics of the D -SyF structure is described in a macrospin approach solving the \ngeneralized Landau- Lifshitz -Gilbert (LLG) equations enhanced by the spin torque term34,38. The \nequation for each ferromagnetic layer is written as follow s, \n \n𝑑𝑑𝒎𝒎𝑖𝑖\n𝑑𝑑𝑡𝑡=−𝛾𝛾0(𝒎𝒎𝑖𝑖×𝑯𝑯𝑖𝑖𝑒𝑒𝑒𝑒𝑒𝑒)+𝛼𝛼𝑖𝑖�𝒎𝒎𝑖𝑖×𝑑𝑑𝒎𝒎𝑖𝑖\n𝑑𝑑𝑡𝑡�+�𝑑𝑑𝒎𝒎𝑖𝑖\n𝑑𝑑𝑡𝑡�\n𝑆𝑆𝑆𝑆𝑆𝑆  (3) \n�𝑑𝑑𝒎𝒎𝑖𝑖\n𝑑𝑑𝑡𝑡�\n𝑆𝑆𝑆𝑆𝑆𝑆=𝛾𝛾0𝑗𝑗𝑎𝑎𝑑𝑑𝑑𝑑𝐺𝐺(𝜂𝜂)𝒎𝒎𝑖𝑖×(𝒎𝒎𝑖𝑖×𝐩𝐩𝑖𝑖)  (4) \n \nwhere the layers are identified by the number i =1 to 4, see structure in Figure 1(a). The vector m i=Mi/MSi \nis the normalized magnetization vector, M Si is the corresponding saturation magnetization, γ 0 is the \ngyromagnetic ratio, α i is the Gilbert damping constant. All the parameters are listed in Table I. \nHeffi=Hinti+Hcouplingi  is the effective field, composed of the internal field Hinti and the coupling field \nHcouplingi of the ith layer. The effective field is derived from the energy term of equations (1) and (2) for \neach layer. The last term is the spin torque term that affects the damping (second term in (3)).  For this \nmodel we have not taken into account the effects of a mutual spin through the Ru spacer  \n \nDuring simulations, three types of dynamic couplings are considered : dynamic RKKY interaction \nbetween  the two magnetic layers of each SyF, dynamic dipolar interaction between the four magnetic \nlayers of the oscillator , and the mutual spin transfer torque (MSTT) only between the BL of the SyF -FL \nand the TL of the SyF -Polarizer , see Eq (4) . Dynamic RKKY interaction and dynamic dipolar interaction \nare conservative couplings included in the precession term of the LLG equation (first term in Eq. (3)) , \nwithout an energy loss . The  Gilbert term and the MSTT , are dissipative couplings  considered by the \ndamping and the spin-transfer torque term  (second and last term in Eq. (3)) . Here j app is the applied \ncurrent density, the pre -factor G (𝜂𝜂) is given by Eq. 2 in Ref. 38, with the spin polarization efficiency 𝜂𝜂 \nand the unit vector p i represents the direction of the spin polarization vector of electrons. We are only \nconsidering the STT between layers 2 and 3 with the polarizer of layer 2 is being layer 3 and the opposite \nas shown in Table I.  The equation (3) should be read as follow s: the spin-polarized electrons from the \nmj layer reach the layer mi where the intrinsic damping starts to be counteracted  (japp<jc) (only for the \ncorrect sign of current ) due to the spin transfer torque. These damped excitations produced by the applied \ncurrent are called the STT damped modes in this manuscript . There are four STT  damped modes such \nas the result of the hybridization of the isolated SyFs modes, due to the dipolar coupling present between the layers in the structure. Overcoming a critical current value (j\nc) the damping is completely \ncounteracted, and the magnetization state is destabilized, leading to switching to another stable state or into steady st ate oscillations (STT excitations). All excitations involve all four layers of the oscillator.  \n   \nIn the case of positive applied current (j\napp>jc>0) the BL of the SyF -FL magnetization is driven by the \nspin transfer torque of spin -polarized  current and this leads to excitations dominated by the SyF -FL, in \nwhich the SyF -Polarizer participate following the dynamics due to the dipolar coupling. For the case of \nthe SyF -Polarizer dominant precession (j app<jc<0), it is the TL magnetization of the SyF -Polarizer which \nis controlled by the spin transfer torque. As this structure is coupled by the dipolar field, the four layers \nmove together at the same frequency. In the absence of current (j app=0) the spin torque term disappears, \nand we can simulate the excitation modes in the linear regime.  5 \n  \nFig. 2. (a) -(b) Hysteresis and magnetoresistance curves of the D -SyF oscillator. (c) Dispersion diagram  of \nhybridized damped modes  f1,f2, f3, and f4 for j app=0. The magnetic field has been swept from positive to negative \nvalues. The regions (i) -(iv) are defined by the crossing of modes.  \n \nTo calculate the STT damped modes (j app≠0), the  set of equations (3) are rewritten in the form of a \nmatrix, giving a periodic solution for mi and linearizing around an equilibrium position, \nmi=(M eq,0,0)+(0,n y,i,nz,i)ept. The  corresponding equilibrium positions were extracted from the hysteresis \nloop. The eigenvalue of the characteristic matrix will be the complex number p=λ+iω, with a real part λ \ncorresponding to the decay rate of the excitation modes and the imaginary part ω corresponding to the \n2π frequency of the excitations. The eigenvectors (0,n y,i,nz,i) define the character of the oscillation mode \nand the amplitude of oscillation for each layer. Two types of instabilities can be described using the \nlinearization. The first one corresp onds to spin wave mode softening (f>0) and will coincide with the \nmagnetization switching32. In the second type the real part of the characteristic eigenvalue can cross the \nzero axis.  To monitor such instability, we have calculated the decay rates  of the system as a function of \nthe current density and the applied field λ(j app,Happ). This provides us with the main criteria to find the \ncritical current density of the STT regime. A negative value means that the system relaxes  into a stable \nstate (damped regime), while a positive value indicates that the system becomes unstable, where one of \nthe possibilities is the  STT regime  (auto oscillations). The passage from negative to positive defi nes j c, \ni.e. the current where the decay rate  is zero: λ(j app,Happ)=0→ japp=jc. This method does not consider \ntemperature. Parameters used for the simulations were extracted from the experimental devices and correspond to SyF structure s, which are not magnetically compensated.  \n The dipolar field was calculated numerically for both SyFs in the AP magnetic configuration (H\napp= 0), \nfinding for the SyF -FL values around ±0.2 mT, for the BL and TL respectively. In the case of the SyF -\nPolarizer, the dipolar field calculated in the same X direction was - 2.1 mT and - 1.2 mT, for TL and BL \nrespectively. Moreover, when the SyF -FL is already saturated (positive X direction) the dipolar field in \nthe SyF -Polarizer increases up  to -21.9 mT and -18.9mT. Therefore, due to the non- compensated SyF \nstructures, we always expect  the influence of the dynamical dipolar coupling in the magnetization \ndynamics.  \n Figure 2(c) shows the calculated STT damped  mode  frequencies at j\napp=0 obtained by sweeping the \napplied field from positive to negative values. These modes are labeled  f1, f2, f3, and f4 from low to \nhigh frequency. The dispersion diagram  show s a well -defined mode splitting between modes 1 and 2, \nas well as two  mode s anti-crossings between modes 1 and 2, and 2 and 3 respectively. These effects are \nreminiscent of the splitting between the acoustic and optical -like modes provoked by the conservative \nRKKY inter action on single SyFs structures37. Here, the dipolar field between the four layers is \n6 \n responsible for the splitting and anti -crossings of the frequency dispersion diagram ,35 see Figure 2(c). \nDue to the crossing of modes, we define four regions, indicated in Figure 2(c) as (i) -(iv). The region (i)  \nis located for negative fields in the P state (low resistance state), and the three regions (ii) -(iv) on the AP \nstate (high resistance state of the structure and interesting region to study the STT modes), see MR curve \nin Figure 2(b). In the following section, we study the evolution of the damped  modes upon increasing \nthe applied current and their stability . \n \nII.3. Magnetization stability analysis  \n \nWe start the study of the stability of the magnetization precession around the equilibrium positions. \nUsing the decay rate  λ criteria, λ(j app,Happ)<0→STT damped modes, λ(j app,Happ)>0→STT modes and \nλ(japp,Happ)=0→ japp=jc introduced before, we can distinguish between stable and unstable regimes, by \ncomputing the critical current for each value of the applied field . Figure 3(a) shows the corresponding \ndecay rate s λ(0,H app) of the damped  modes f1,f2,f3,  and f4 respectively. We will respect the color of \nmodes defined in Figure 2(c). As it was expected for j app=0, the decay rate of the four modes remain \nnegative s, corresponding to the stability for the damped  modes. Since the frequency dispersion vs. field \nwas divided into  several regions (i)-(iv), due to the crossing of the damped modes, we notice that the \ndecay rates suffer inflections  or in some cases abrupt jumps from one of these regions to another one. It \nis important to remark that the mode f4 is less affected by the crossing of modes, showing the larger \ndecay rate and without distortions in the whole range of applied field.   \n \n \n \n \nFig. 3. Decay rate  versus applied field dispersion diagrams, for the corresponding hybridized damped modes f1, \nf2, f3, and f4. Colors correspond to the modes defined in Figure 2(c). In (a) below the critical current j app=0, in (b) \nfor the SyF -FL dominant precession, j app=1x1012 A/m2 and in (c) for the SyF -Polarizer  dominant precession, j app=-\n1x1012 A/m2. (d) State diagram H app vs j app. Positive current corresponds to an electron flow from the TL of the \nSyF- Polarizer (red arrow) to the SyF -FL (green arrows). The light grey  and black  regions correspond to the \n7 \n unstable region of excitations, and the dark grey to the stable region. The black region corresponds  to the switching \nof the SyF -FL. \n \nWhen the positive current density is increased up to japp=1x1012 A/m2, the decay rate tendencies of the \nsystem change, see Figure 3( b). It is noticed that the decay rate for the modes f1 and f2 (dominated by \nthe SyF -FL) are already positive λ>0, which is an indication that these modes evolve into the STT regime  \ndue to the damping compensation by the increase of the applied current. As this method is a linearization \nof the LLG equation, small magnetization precession, it is not possible to study the tendency of the \ndecay rate when the system is already in the steady state regime, large magnetization precession. \nApplying negative current density, j app=-1x1012 A/m2, the system reaches the SyF -Polarizer dominant \nprecession; see Figure 3( c). The decay rate is already positive for the modes f1, f2 and f3 (dominated by \nthe SyF -Polarizer), generating STT modes in the frequency field diagram. For both senses of current \ndensity, it is observed a crossing of the decay rate in the (i) region. In conclusion, it is possible to obtain \nthe critical current of the STT modes using the criteria λ(j app,Happ)=0 ( for positive and negative current \ndensity) and we can predict which of the modes will be stable or unstable. Sweeping the current density \nfor each value of magnetic field we obtain the critical lines to build the state diagram H app vs j app, shown \nin Fig ure 3(d).  \n \nThe magnetic field was swept from positive to negative values. i.e . from the positive saturation \nmagnetization of the SyF -FL until its plateau region in the P state (low resistance). The arrows \ncorrespond to the orientation of the magnetization of the four layers of the structure. The stable and \nunstable dynamical states are indicated by the dark and light grey regions, respectively. Excitations \ndominated by the SyF -FL (SyF- Polarizer) are observed in the region of positive (negative) current. \nYellow lines indicate the critical current densities, j c,SyF -FL and j c,SyF- Polari zer. The critical current values \nobtained and shown in Figure 3(d)  are referential due to the simplification of the model and to the \nparameters used in performing simulations (exchange bias, RKKY coupling, thicknesses, M S etc). The \ncritical currents should be taken as an approximation when compared to experimental results . The \ndashed yellow lines represent the border with the switching of the SyF -FL magnetizations, represented \nby the dark black  region. In the SyF -FL dominant precession region, two small def lections can be \nobserved, which corresponds to the splitting shown in Figure 1(c). The critical current for the SyF -FL \nin the saturated state  is around 0.9x1012 A/m2. \n \nThis numerical simulation framework  allows to  predict the magnetization dynamic behavior of the \ncoupled SyFs of the STO structure  and the critical current as a function of the applied field. As w e will \nverify  in the next section, this framework  provides useful information to understand the complex \nfrequency dispersion diagrams studied experimentally, a fundamental issue in designing STO devices.  \n \nIII. EXPERIMENTAL SECTION  \n In this section, we present  the static and dynamic  features of the D-SyF nano- oscillator . Measurements  \nwere carried out  using a standard microwave measurement setup.  \n  \nIII.1. Static measurements \n \nThe experimental results are obtained for magnetic tunnel junctions with the following structure: \nIrMn(6.1)/SyF -Polarizer/MgO(0.9)/SyF -FL, where numbers represent thickness  in nanometers. The \ncomposition of the SyF- Polarizer and SyF -FL are  CoFe(1.8)/Ru(0.4)/CoFe B(2) and \nCoFe(0.5)/CoFeB(3.4)/Ru/CoFe(3.6)  respectively. The thickness of Ru in SyF -FL was selected to \nachieve a weak  and negative RKKY coupling.  Samples were grown by sputter -deposition and patterned \ninto elliptical nanopillars  (130nm x 65nm ) with and area of 6636.63nm\n2. The  uniaxial shape anisotropy \nstabilizes the magnetization in the direction of the longest axis.  We have characterized tens of devices \nwhich we classified in two categories depending on whether the TMR is above or below 50% (High-TMR and Low -TMR devices respectively)\n16. The MgO barrier of the latter is considered to have \ninhomogeneities and/or pin holes either caused during deposition or electrical characterization. In this 8 \n work, we mainly focus on HTMR devices. However, due to the slightly  high critical current of STT \nexcitations dominated by SyF -Polarizer, those cannot be achieved  easily  in HTMR devices applying \nvoltages below 400 mV  (MgO barrier degradation) . Thus, LTMR devices characterized by a smaller \nresistance are used to characterize STT excitations dominated by the SyF -Polarizer . Figure 4(a) shows  \nthe MR curve s of a High -TMR (HTMR, red curve) device and a Low -TMR (LTMR, black curve) device \nrespectively. The MR curves are in good agreement with the numerical simulations (Figure 2(b) ), which \nprovide information about the magnetic orientations of the layers as a function of the applied field.  \n \nIt has been found during the optimization of D -SyF structures, (not shown in this article) that the \nroughness in a multilayer structure gives less control of the thicknesses of the layers on top, thus the \nRKKY coupling of the SyF -FL becomes weak, reducing the size of the plateau. As it was shown in \nprevious studies34, the size of the plateau region of a SyF is directly related to  the strength of its RKKY \ncoupling and increased by  the exchange bias field. The SyF -FL plateau is located between two \ncharacteris tic fields, the Hsat-SyF- FL and Hsat+SyF-FL=H sw, however , it is not evident in the MR curve due to \nthe weak RKKY coupling. The plateau size will be  estimated in the next section measuring the STT \ndamped modes. Both  magnetizations of the SyF -Polarizer remain in its AP configuration ( plateau  \nregion)  for a quite large range of applied field (>500mT) thus  we can consider this static configuration \nfor all our measurements, until the spin flop field (Hsf-+SyF- Polarizer ).  \n \n \nIII.2. Dynamic Measurements  \n \nIn this section, the dynamical features  of the D -SyF oscillator device are presented . Section ( A) shows \nthe study of the STT  damped modes  on the plateau region of the SyF -FL. Section (B) corresponds  to the \nstudy of the STT  damped modes  on the AP region (high resistance state) . Section  (C) is focused on the \nanalysis of the STT modes for SyF -FL and SyF- Polarizer dominant precession. The same sign \nconvention of the numerical simulations  is considered:  electrons flow from the SyF -Polarizer towards \nthe SyF -FL for positive applied current, promoting STT excitations in the SyF -FL. It is worth noting \nthat during the STT measurements, the STT damped modes  are also excited.  \n \nA. STT damped modes, positive current  \n \nFirst, the STT damped  modes were measured  for a HTMR device (TMR= 60%) around zero applied \nfield. The corresponding excitation frequency dispersion as a function of the applied field for positive \napplied current I app=1 mA is shown in Figure 4(b). The power spectral density (PSD) of the STT  damped \nmodes is  shown on a logarithmic scale. The magnetic field was swept from positive to negative values.  \nWe identify three different regimes in Figure 4(b) corresponding to the three different magnetic \nconfigurations of the SyF -FL (plateau  (i) and both saturation regions ) described by the numerical \nsimulations (Figure 2). Regions  (i) and (ii) in Figure 4(b) correspond to regions (i) and (ii) in Figure \n2(c). Experimentally, we identified region (i) between -27 mT<μ 0Happ<10 mT. Out of this range, the \nSyF- FL is saturated.  \n \n 9 \n  \nFig. 4. (a) MR curves of an elliptical device (130x65nm2) for HTMR device in  red (TMR  60%) and LTMR in blue \n(TMR  28%) respectively. (b) Frequency vs. applied field of the dB ( 10 log of power spectral density (nV2/Hz)) \nfor positive applied current (I app=1 mA). The regions  (i) and (ii) are identified in agreement with numerical \nsimulations and the arrows correspond to the magnetic direction of magnetizations. The STT  damped  fundamental \nmodes f1, f2 and f3 are identified,  and the higher order damped modes f 11, f21, and the harmonics 2f1, 2f2. The \ndashed lines are included to identify the modes only as a visual reference.  Linewidth as a function of the applied \nfield of the damped f1 and f 2 STT  damped  modes for ( c) positive and ( d) negative applied currents (I app =±1 mA).  \n \nBy comparing with the numerical simulations (Figure 2(c)), we can identify the STT  damped  modes f1, \nf2, f3 in regions  (i) and (ii) in Figure 4(b), as well as other harmonics  (2f1 is the 2nd harmonic of the f1 \nmode) . In region (i)  we observe the splitting and curvature of modes, which indicates  weak conservative \nRKKY coupling  of the SyF -FL, according to simulations. In region (ii)  we observe other  higher damped \nmodes  such as f11, f21, and the harmonics (2f1, 2f2) , in agreement  with results  given in Ref. 39, where \nthe finite size of the device generates quantized spin waves.  As we will see in the next section, the \nappearance of higher -order  modes has negative consequences on the microwave properties of  the D -\nSyF oscillator.  \n \nAs can be seen in Figure 4(b), the intensi ty of the different modes change s with the applied field (and \napplied current). As previously discussed , the amplitude of the magnetization precession and the \nlinewidth should be proportional to the absolute value of the decay rate  λ(japp,Happ) of the STT  damped \nmodes in the linear regime. The  tendency of the linewidth observed experimentally ( Figure s 4(c) and \n4(d)) agrees well  with the evolution of  modes 1 and 2 obtained by numerical simulations in the region \n(i) (Figure s 3(b) and 3(c) ), indicating a good correspondence between our model and experiments . Large \nvalues of the linewidth (≈400- 800 MHz ), for a quality factor Q=∆f/f ≈50-100, corresponding to STT \ndamped  modes  (linear regime)  were measured.  \n \n10 \n  \nFig. 5. Frequency vs. applied field diagram for regions (i) -(iv) for the device with TMR=60%, for I app=-1 mA. \nDashed lines correspond to the damped f1, f2, and f3 modes only as a visual reference. Arrows indicate the \nmagnetization directions on the magnetic layers for the different regions.  \n \n \nB. STT damped modes negative current  \n STT damped modes in both SyFs structures were measured by applying -1mA  and they are introduced \nin the frequency dispersion diagram in Figure 5. For applied magnetic fields >10mT the SyF -FL is \ncompletely saturated (green arrows) while the SyF pinned remains in the AP magnetic configuration. \nBy comparing with numerical simulations in Figure 2(c), we can also identify the regions (ii)-(iv) and \nmodes f1, f2, and f3. The experimental  STT damped  modes follow the frequency- field dependence  of \nthe simulated modes, and they are accompanied by additional  higher -order  modes\n27 (f11, f31, f22). At low \npositive fields, modes  f1 and f2 are separated 2.4GHz, as a result of RKKY and dipolar interactions  \nwithin the SyF -FL (conservative couplings). Two additional mode splits are observed  around 100mT, \nbetween modes  f1 and f2 and between modes  f2 and f3 respectively , in agreement  with the  numerical \nanalysis, see region (iii) in  Figure 2( c). In the next section, we discuss the evolution of the different STT  \ndamped  modes into STT excitations in the SyF -FL or the SyF- Polarizer upon increasing the applied \ncurrent for positive and negative values  respectively .  \n \nC. STT excitation modes  \n \nC.1 HTMR device  : SyF -FL dominantes  STT mode  \n \nSteady -state oscillations (STT modes) were first analyzed by applying positive current, which favors \nSTT modes dominated by the SyF -FL. Figure 6(a) shows the frequency -field dispersion curve with an \napplied current of I app=0.92 mA. The observed SyF -FL dominated STT modes correspond to the \nevolution of the STT  damped modes f1 and f2 on region (ii)  and (iii), as predicted by the stability \nanalysis (Figure 3) . The fundamental STT mode f1 shows several discontinuities around 40, 58 and 82 \nmT, in agreement with the macro -spin simulations  (Figure 2(c)).  These discontinuities are interpreted \nto be due to interactions between the steady state mode and other damped modes of the system through \nhigher harmonics29-35. Indeed, modes f 3, f11 and f 31 cross the second harmonic of the STT mode f1 (lines \n11 \n in Figure 5 and 6(a)). Non -linear mode interactions through higher harmonics can produce \ndiscontinuities and kinks in the fundamental STT mode f140-41.  \n  \nFigure 6(b) displays t he linewidth of the STT mode  f1. The linewidth of STT mode 1 increases each \ntime there is a discontinuity in the f -H dispersion, as expected from the interaction with other damped \nmodes via higher harmonics26. The STT mode f2 is characterized by a much lower linewidth, reaching \na minimum of 42 MHz , Q≈4.94, around 100 mT . At low fields (region (i) ), the STT damped  mode in  \nthe SyF -FL plateau  shows a continuous frequency- field dispersion, without jumps or kinks, since t here \nare no mode crossings  in this range of field. Exciting STT modes in this region would potentially offer \nexcitations at zero fields without linewidth enhancements due to undesired mode interactions. Due to \nthe switching of the SyF -FL for an  applied current below the critical current, it was not possible to obtain \nSTT in the region (i) . \n \n \nFig. 6. (a) Frequency dispersion as a function of the field for the SyF -FL dominant precession, I app=0.92  mA. The \ncircles indicate the kink and jumps in the STT f1 mode. (b) Frequency (black) and linewidth (blue) versus applied \nfield. Deviations in frequency correspond to an abrupt increase of ∆f. (c) Frequency dispersion as a function of the \napplied field for the SyF -FL dominant precession at 0. 6, 0.92 and 0.94 mA . Inset: peaks of modes interactions \naround the bi -stable region ( around 82 mT) . (d) Frequency dispersion as a function of the  applied current for three \nvalues of applie d field. Device HTMR ( TMR= 60%).   \n \nTo study the discontinuities of the STT modes , the frequency field dispersion is plotted for three \ndifferent values of the applied current (Figure 6(c)). The evolution from a continuous STT mode at \n0.6mA ( black curve ), into a discontinuous STT mode with jumps and kinks at 0.94mA ( blue curve). The \nPSD spectra at different fields around 82 mT are shown in the inset. The PSD spectra shows a region of \nbi-stability  where two modes co -exist . This behavior has been reported to  come from  the interaction \nbetween a STT mode with other damped modes of the system through higher harmonics37,39,41,42, where \nthe apparent mode co -existence is indeed thermally activated mode hopping. This kind of discontinuities  \n12 \n and jumps are observed only for large values of current, which implies  a large amplitude of the \nmagnetization precession  and thus  large dipolar interaction .  \n \nAn interesting feature of the magnetization dynamic on SyF pinned structures is the possibility to tune \nthe frequency -current  dependence  from redshift (df/dj app<0) to blueshift (df/dj app>0) by applying an in-\nplane field . Figure 6 (d) shows th e frequency as a function of the applied current for μ0Happ=20, 70 and \n85 mT  respectively. A transition from redshift to blueshift is observed upon increasing field from 20 to \n85 mT. Interestingly both curves are continuous with n o abrupt discontinuities . However, at μ 0Happ= 70 \nmT, a bi -stable region characterized by mode  is observed around I app=0.82 mA (see dashed line).   \n \n \nC.2. LTMR device   \n \nAchieving SyF- Polarizer dominant precession requires increasing the applied current  above the \nbreakdown voltage in HTMR devices. Thus, an LTMR device (TMR= 28%) is used to pursue the study \nof the STT excitations dominated by the SyF -Polarizer. Figure 7 displays the STT modes of a LTMR \ndevice for positive (Figure 7(a)) and negative (Figure 7(b)) currents ( ±1.4 mA). The SyF- FL dominant \nexcitations  (Figure 7(a)) exhibit  several discontinuities and kinks  in the STT modes f1 and f2 mode s and \nhigher  harmonic s (2f1). Indeed, more discontinuities than the equivalent  frequency- field dispersion of \na HTMR device ( Figure 6(a)) are observed . This is expected since the existence of pinholes in the tunnel \nbarrier increases interlayer interaction. The frequency -field dependence becomes flat in the field region \naround 100mT  because of  the interaction between higher harmonics f31 and 2f1. For  negative current , \nthe SyF -PL dominant precession shows a gap between the STT f3 and STT f1 mode s (Figure 7 (b)).  \n \n \nFig. 7. (a) SyF -FL dominant precession, I app=+1.4  mA and (b) SyF -PL dominant precession, for I app=-1.4 mA. \nInteractions between the harmonics of the STT f1 and f2 mode s with the damped or higher order modes will be \n13 \n transmitted as a form of kinks, deviations or jumps of the normal tendency of the frequency and linewidth versus \napplied field dispersion, generating discontinuities in the linewidth. Device with LTMR ( TMR= 28%).  \n \nThe general tendency of the linewidth  not shown  is to decrease upon increasing the  applied field , \nalthough several discontinuities (regions of linewidth enhancement) are observed corresponding to the interaction  of the S TT f1, f2 , and f3 modes with damped and higher order modes, as in the HTMR device \n(Figure 6(b)).  The local minimum linewidths measured for the SyF -FL dominant precession  at 1.4  mA, \nwere 20.5 MHz (Q≈1.73)  at 112 mT, 13.6 MHz  (Q≈1.1) at 194 mT and 12.9 MHz (Q≈1.17)  at 214  mT. \nIn the case of the  SyF- Polarizer dominant precession  at -1.4 mA , the minimum linewidths were 88MHz  \n(Q≈8) at 46 mT, 53 MHz (Q≈5.57) at 150 mT and 46 MHz  (Q≈5.41)  at 214 mT. Interestingly, t he \nminimum values of the linewidth  associated to the SyF-FL dominant  precession are smaller  than the \nminimum linewidth associated to the precession of a single free layer o n standard  in-plane magnetized \nspin torque oscillators\n22. For the case of the SyF -Polarizer, the minimal linewidth was around ≈46  MHz. \nWe note that continuous  excitation  branches can be found for specific conditions in the case of the SyF -\nPolarizer.  \n \nFinally, we analyze the redshift and blueshift regimes associated to  SyF-FL dominant excitations. We \nnote that the redshift and blueshift regimes have  been studied only on SyF pinned structures23,25 so far. \nFigure 8 displays the frequency- current  dependence at different magnetic fields for the SyF -FL \ndominant regime. In (a) -(b) (20 mT -plateau region and 54 mT -P state respectively) , the applied current \nremains below the critical  current, therefore STT damped  mode shown in (a)  the standard redshift \nbehavior  with large linewidth Δf≈395- 550 MHz  (Q≈79-110). In (b) t he STT damped mode shows the \nblueshift regime with a reduction of the linewidth by the current from Δf≈480 MHz  (Q≈80) for low \ncurrent for Iapp <1 mA to 125 MHz  (Q≈17.8)  for Iapp>1.2 mA. Upon increasing the field, the critical \ncurrent gets smaller. Thus, at μ 0Happ=70.7  mT in (c) the system evolves into the STT excitations. The \nfrequency dispersion shows discontinuities and a transition from a blueshift regime to a redshift regime upon increasing current , around 1.1 mA . Looking at Figures  7 and 5, we observe that this discontinuity \nis a product of the crossing of the STT mode f1 with the damped mode f3, however , this transition is \nopposite to the previous observed (redshift into blueshift) , and due to the complexity of these coupled \nsystems. The mode t ransition into the redshift regime is accompanied by a reduction of the linewidth \ninto Δf≈75 MHz (Q≈10) around 1.25 mA. At large fields (Figure 8(d) ) a redshift to blueshift transition \nis observed upon increasing current, and a very low linewidth of Δf≈28  MHz  (Q≈2.54) is obtained in \nthe blueshift regime . This region of  STT excitations at  large  fields and applied current in  LTMR devices \nis of potential interest for applications as it offers the possibility of selecting excitations in the redshift \nor in the  blueshift regime.  \n 14 \n  \nFig. 8. Frequency as a function of the current density for I app>0 for different field values (SyF -FL dominant \nprecession). (a) STT damped mode f1 on the SyF -FL plateau  region . (b) In the P state close to the switching field. \nThe system shows a blueshift regime and large linewidth . In (c) the system shows a blueshift regime until \nIapp=1.1mA. Overcoming this value of current , the regime changes towards a redshift regime. In (d) the STT f1 \nmode shows first a redshift regime and after the splitting a blueshift regime with a minimum local linewidth of 28  \nMHz at 1.25  mA. \n \n \nIV. C ONCLUSIONS  \n In this manuscript, we conducted a comprehensive investigation into the spin- transfer torque damped \nmodes  and steady- state oscillations of a spintronic nano -oscillator employing two SyF structures. The \nstudy involved both numerical simulations and experimental analyses.  Numerical simulation s were \ncarried out using two different values of R KKY coupling, J\nRKKY ≈-0.1mJ/m2 and J RKKY ≈-1.5 mJ/m2 for \nthe SyF -FL and SyF- Polarizer respectively. The small RKKY coupling of a SyF -FL eliminates the spin \nflop region, producing an abrupt switching of the magnetization layers  and introducing a small plateau \nregion (≈40 mT), in comparison with the corresponding of the pinned SyF- Polarizer (>600 mT). Static  \nand dynamical experimental measurement s confirmed the weak RKKY coupling of the SyF -FL on top \nof the structure. The experimental STT  damped  hybridized modes  have been identified using numerical \nmodeling. W e found that it is possible to find the tendency of the linewidth and the PSD of the STT  \ndamped  hybridized modes following the tendency of the decay rate (λ), also we can obtain the critical \ncurrent of the STT modes. This study shows that the analysis of the stability predicts the state diagram \nof different structures, with an arbitrary number of layers. The frequency versus field dispersion \ndiagrams for b oth devices (TMR 60% and 28%) show several discontinuities, attributed to the crossing \nof the STT mode or its harmonics with other damped hybridized or higher order modes, as was reported \n15 \n in Ref. 2 4, 36, 37 and 39. The introduction of a SyF -FL instead of a single layer in the STO based will \ngenerate more damped modes which produce more discontinuities  in the STT mode s. This last fact is  \nalso attributed to the dipolar coupling between the  magnetic layers. The frequency current tuning shows \ntwo regimes  for the SyF FL, the redshift (df/dI app<0) and a blueshift (df/dI app>0) which can be selected \nfor different values of applied field. In the case of the SyF -Polarizer  STT modes,  a flat regi on (df/dI app≈0) \nwith a small redshift and blueshift regimes was found. We found the minimum linewidth on a LTMR \ndevice, around ≈12.9 MHz  (Q≈1.17)  for SyF- FL and around ≈46 MHz (Q≈5.41)  for the SyF -Polarizer \ndominant precession, evidence of the stability of the system.  \n \nV. A CKNOWLEDGE  \n \nM.R. acknowledges financial support from Spanish MIC, AEI and FEDER through Grant No. PID2020-\n116181RB -C33 (MCI/AEI/FEDER, UE) and from Comunidad de Madrid (Atracción de Talento Grant \nNo. 2018- T1/IND -11935).  \n \n \nVI. REFERENCES  \n \n1 N. Locatelli, V. Cros, and J. Grollier, Nat Mater 13, 11 (2014).  \n2 Kent, A., Worledge, D. A new spin on magnetic memories. Nature Nanotech  10, 187–191 (2015)  \n3 A. Dussaux et al., Nat. 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Langner,1,2C.L.S. Kantner,1,2Y.H. Chu,3L.M.\nMartin,2P. Yu,1R. Ramesh,1,4and J. Orenstein1,2\n1Department of Physics, University of California, Berkeley , CA 94720\n2Materials Science Division, Lawrence Berkeley National La boratory, Berkeley, CA 94720\n3Department of Materials Science and Engineering,\nNational Chiao Tung University, HsinChu, Taiwan, 30010\n4Department of Materials Science and Engineering,\nUniversity of California, Berkeley, CA 94720\n(Dated: October 26, 2018)\nAbstract\nWe report the observation of ferromagnetic resonance (FMR) in SrRuO 3using the time-resolved\nmagneto-optical Kerr effect. The FMR oscillations in the ti me-domain appear in response to a sudden,\noptically induced change in the direction of easy-axis anis tropy. The high FMR frequency, 250 GHz, and\nlarge Gilbert damping parameter, α≈1, are consistent with strong spin-orbit coupling. We find th at the\nparameters associated with the magnetization dynamics, in cludingα, have a non-monotonic temperature\ndependence, suggestive of alink to theanomalous Hall effec t.\nPACS numbers: 76.50.+g,78.47.-p,75.30.-m\n1Understanding and eventually manipulating the electron’s spin degree of freedom is a major\ngoal of contemporary condensed matter physics. As a means to this end, considerable attention\nis focused on the spin-orbit (SO) interaction, which provid esa mechanism for control of spin po-\nlarization by applied currents or electric fields [1]. Despi te this attention, many aspects of SO\ncoupling are not fully understood, particularly in itinera nt ferromagnets where the same elec-\ntrons are linked to both rapid current fluctuations and slow s pin dynamics. In these materials,\nSO coupling is responsible for spin-wave damping [2, 3], spi n-current torque [4, 5], the anoma-\nlous Hall effect (AHE) [6], and magnetocrystalline anisotr opy (MCA) [7]. Ongoing research is\naimed toward a quantitative understanding of how bandstruc ture, disorder, and electron-electron\ninteractionsinteracttodeterminethesizeandtemperatur edependenceoftheseSO-driveneffects.\nSrRuO 3(SRO) is a material well known for its dual role as a highly cor related metal and\nan itinerant ferromagnet with properties that reflect stron g SO interaction [8, 9, 10]. Despite\nits importance as a model SO-coupled system, there are no pre vious reports of ferromagnetic\nresonance (FMR) in SRO. FMR is a powerful probe of SO coupling in ferromagnets, providing\na means to measure both MCA and the damping of spin waves in the small wavevector regime\n[11]. HerewedescribedetectionofFMRbytime-resolvedmag netoopticmeasurementsperformed\non high-quality SRO thin films. We observe a well-defined reso nance at a frequency ΩFMR=\n250 GHz. This resonant frequency is an order of magnitude hig her than in the transition metal\nferromagnets,which accountsforthenonobservationbycon ventionalmicrowavetechniques.\n10-200nmthickSROthinfilmsweregrownviapulsedlaserdepo sitionbetween680-700◦Cin\n100 mTorr oxygen. High-pressure reflection high-energy ele ctron diffraction (RHEED) was used\nto monitor the growth of the SRO film in-situ. By monitoring RH EED oscillations, SRO growth\nwas determined to proceed initially in a layer-by-layer mod e before transitioning to a step-flow\nmode. RHEED patterns and atomic force microscopy imaging co nfirmed the presence of pristine\nsurfaces consisting of atomically flat terraces separated b y a single unit cell step ( 3.93 ˚A). X-ray\ndiffractionindicatedfullyepitaxialfilmsandx-rayreflec tometrywasusedtoverifyfilmthickness.\nBulk magnetization measurements using a SQUID magnetomete r indicated a Curie temperature,\nTC, of∼150K.\nSensitive detection of FMR by the time-resolved magnetoopt ic Kerr effect (TRMOKE) has\nbeen demonstrated previously [12, 13, 14]. TRMOKE is an all o ptical pump-probe technique in\nwhichtheabsorptionofan ultrashortlaserpulseperturbst hemagnetization, M, ofaferromagnet.\nThe subsequent time-evolutionof Mis determined from the polarization state of a normally inci -\n2dent, time-delayed probe beam that is reflected from the phot oexcited region. The rotation angle\nof the probe polarization caused by absorption of the pump, ∆ΘK(t), is proportional to ∆Mz(t),\nwherezisthedirectionperpendiculartotheplaneofthefilm[15].\nFigs. 1a and 1b show ∆ΘK(t)obtained on an SRO film of thickness 200 nm. Very similar\nresults are obtained in films with thickness down to 10 nm. Two distinct types of dynamics are\nobserved,dependingonthetemperatureregime. Thecurvesi nFig. 1aweremeasuredattempera-\nturesnearT C. Therelativelyslowdynamicsagreewithpreviousreportsf orthisTregime[16]and\nare consistent with critical slowing down in the neighborho od of the transition [17]. The ampli-\ntudeofthephotoinducedchangeinmagnetizationhasalocal maximumnearT=115K.Distinctly\ndifferentmagnetizationdynamicsareobservedasTisreduc edbelowabout80K,asshowninFig.\n1b. The TRMOKE signal increases again, and damped oscillati ons with a period of about 4 ps\nbecomeclearly resolved.\nFIG. 1: Change in Kerr rotation as a function of time delay fol lowing pulsed photoexcitation, for several\ntemperatures below the Curie temperature of 150 K. Top Panel : Temperature range 100 K <T<150 K.\nBottom panel: Temperature range 5K <T<80 K.Signal amplitude and oscillations grow with decreasin g\nT.\nIn order to test if these oscillations are in fact the signatu re of FMR, as opposed to another\n3photoinduced periodic phenomenon such as strain waves, we m easured the effect of magnetic\nfield on theTRMOKE signals. Fig. 2a shows ∆ΘK(t)for several fields up to 6 T applied normal\nto the film plane. The frequency clearly increases with incre asing magnetic field, confirming that\ntheoscillationsare associatedwithFMR.\nThemechanismfortheappearanceofFMRinTRMOKEexperiment siswellunderstood[14].\nBeforephotoexcitation, Misorientedparallelto hA. Perturbationofthesystembythepumppulse\n(by local heating for example) generates a sudden change in t he direction of the easy axis. In the\nresulting nonequilibrium state, MandhAare no longer parallel, generating a torque that induces\nMto precess at the FMR frequency. In the presence of Gilbert da mping,Mspirals towards the\nnewhA, resultingin thedamped oscillationsof Mzthat appearin theTRMOKE signal.\nTo analyze the FMR further we Fourier transform (FT) the time -domain data and attempt to\nextract thereal and imaginary parts of the transversesusce ptibility,χij(ω). Themagnetization in\nthetime-domainisgivenby therelation,\n∆Mi(t) =/integraldisplay∞\n0χij(τ)∆hj\nA(t−τ)dτ, (1)\nwhereχij(τ)is the impulse response function and ∆hA(t)is the change in anisotropy field.\nIf∆hA(t)is proportional to the δ-function, ∆Mi(t)is proportional χij(τ)and the FT of the\nTRMOKE signal yields χij(ω)directly. However, for laser-induced precession one expec ts that\n∆hA(t)will be more like the step function than the impulse function , as photoinduced local\nheating can be quite rapid compared with cooling via thermal conduction from the laser-excited\nregion. When ∆hA(t)is proportional to the step function, χij(ω)is proportionalto the FT of the\ntimederivativeof ∆Mi(t), ratherthan ∆Mi(t)itself. Inthiscase, theobservable ωIm{∆ΘK(ω)}\nshouldbecloselyrelated tothereal, ordissipativepart of χij(ω).\nInFig. 2bweplot ωIm{∆ΘK(ω)}foreachofthecurvesshowninFig. 2a. Thespectrashown\nin Fig. 2b do indeed exhibit features that are expected for Re χij(ω)near the FMR frequency. A\nwell-definedresonancepeakisevident,whosefrequencyinc reaseswithmagneticfieldasexpected\nforFMR.TheinsettoFig. 2bshows ΩFMRasafunctionofappliedmagneticfield. Thesolidline\nthroughthedatapointsisafitobtainedwithparameters |hA|=7.2Tandeasyaxisdirectionequal\nto 22 degrees from the film normal. These parameter values agr ee well with previous estimates\nbased onequilibriummagnetizationmeasurements[8, 10].\nAlthough the spectra in Fig. 2b are clearly associated with F MR, the sign change at low fre-\nquency is not consistent with Re χij(ω), which is positive definite. We have verified that the\n4FIG. 2: Top panel: Change in Kerr rotation as a function of tim e delay following pulsed photoexcitation at\nT=5K,forseveral valuesofappliedmagneticfieldranging up to6Tesla. Bottompanel: Fourier transforms\nof signals shown intop panel. Inset: FMRfrequency vs. appli ed field.\nnegativecomponentis always present in the spectra and is no t associated with errors in assigning\nthet=0pointinthetime-domaindata. Theoriginofnegativecomp onentoftheFTismadeclearer\nby referring back to the time domain. In Fig. 3a we show typica l time-series data measured in\nzero field at 5 K. Forcomparison we show theresponseto a step f unction changein theeasy axis\ndirection predicted by the Landau-Lifschitz-Gilbert (LLG ) equations [18]. It is clear that, if the\nmeasured and simulated responses are constrained to be equa l at large delay times, the observed\n∆ΘK(t)ismuchlarger thantheLLGpredictionat smalldelay.\nWe have found that ∆ΘK(t)can be readily fit by LLG dynamics if we relax the assumption\nthat∆hA(t)is a step-function, in particular by allowing the change in e asy axis direction to\n”overshoot” at short times. The overshoot suggests that the easy-axis direction changes rapidly\nas the photoexcited electrons approach quasiequilibriumw ith the phonon and magnon degrees of\nfreedom. The red line in Fig. 3a shows the best fit obtained by m odeling∆hA(t)byH(t)(φ0+\nφ1e−t/τ), whereH(t)is the step function, φ0+φ1is the change in easy-axis direction at t= 0,\nandτis the time constant determining the rate of approach to the a symptotic value φ0. The fit is\n5FIG. 3: Components of TRMOKE response in time (top panel) and frequency (bottom panel) domain.\nBlacklinesaretheobserved signals. Greenlineinthetoppa nel isthesimulated response toastepfunction\nchange in easy-axis direction. Best fits to the ”overshoot” m odel described in the text are shown in red.\nBluelines are thedifference between the measured and best- fit response.\nclearly much better when the possibility of overshoot dynam ics in∆hA(t)is included. The blue\nline shows the difference between measured and simulated re sponse. With the exception of this\nveryshortpulsecenterednear t=0,theobservedresponseisnowwelldescribedbyLLGdynami cs.\nInprinciple,analternateexplanationforthediscrepancy withthestep-functionassumptionwould\nbe to consider possible changes in the magnitude as well as di rection of M. However, we have\nfound that fitting the data then requires |M(t)|to be larger at t >20ps than|M(t <0)|, a\nphotoinducedincrease thatisunphysicalfora systemin ast ableFM phase.\nIn Fig. 3b we compare data and simulated response in the frequ ency domain. With the al-\nlowance for an overshoot in ∆hA(t)the spectrum clearly resolves into two components. The\npeak at 250 GHz and the sign change at low frequency are the bot h part of the LLG response to\n∆hA(t). The broad peak or shoulder centered near 600 GHz is the FT of t he short pulse compo-\nnentshowninFig. 3a. Wehavefoundthiscomponentisessenti allylinearinpumppulseintensity,\n6and independent of magnetic field and temperature - observat ions that clearly distinguish it from\nthe FMR response. Its properties are consistent with a photo induced change in reflectivity due to\nband-filling,whichiswell-knowntocross-coupleintotheT RMOKEsignalofferromagnets [19].\nByincludingovershootdynamicsin ∆hA(t),weareabletodistinguishstimulusfromresponse\nin the observed TRMOKE signals. Assuming LLG dynamics, we ca n extract the two parameters\nthatdescribetheresponse: ΩFMRandα;andthetwoparametersthatdescribethestimulus: φ1/φ0\nandτ. In Fig. 4 we plot all four parameters as a function of tempera ture from 5 to 80 K. The\nT-dependence of the FMR frequency is very weak, with ΩFMRdeviating from 250 GHz by only\nabout 5%overthe range ofthe measurement. TheGilbert damping param eterαis of order unity\nat all temperatures, avaluethatis approximatelyafactor 102largerthan found intransitionmetal\nferromagnets. Over the same T range the decay of the easy axis overshoot varies from about 2\nto 4 ps. We note that the dynamical processes that characteri ze the response all occur in strongly\noverlapping time scales, that is the period and damping time of the FMR, and the decay time of\nthehAovershoot,areeach inthe2-5ps range.\nWhileΩFMRisessentiallyindependentofT,theparameters α,φ1/φ0andτexhibitstructurein\ntheirT-dependencenear40K.Thisstructureisreminiscent oftheT-dependenceoftheanomalous\nHallcoefficient σxythathasbeenobservedinthinfilmsofSRO[20,21,22]. Forcom parison,Fig.\n4dreproduces σxy(T)reportedinRef. [20]Thesimilaritybetween theT-dependen ceofAHEand\nparameters related to FMR suggests a correlation between th e two types of response functions.\nRecently Nagaosa and Ono [23] have discussed the possibilit y of a close connection between\ncollective spin dynamics at zero wavevector (FMR) and the of f-diagonal conductivity (AHE). At\na basic level,both effects are nonzero only in the presence o f both SO couplingand time-reversal\nbreaking. However, the possibilityof a more quantitativec onnection is suggested by comparison\nof the Kubo formulas for the two corresponding functions. Th e off-diagonal conductivity can be\nwrittenin theform [24],\nσxy(ω) =i/summationdisplay\nm,n,kJx\nmn(k)Jy\nnm(k)fmn(k)\nǫmn(k)[ǫmn(k)−ω−iγ], (2)\nwhereJi\nmn(k)is current matrix element between quasiparticle states wit h band indices n,mand\nwavevector k. The functions ǫmn(k)andfmn(k)are the energy and occupation difference, re-\nspectively,between such states, and γis a phenomenologicalquasiparticledamping rate. FMR is\nrelated to theimaginary part of theuniformtranverse susce ptibility,with thecorresponding Kubo\n7FIG. 4: Temperature dependence of (a) FMR frequency (triang les) and damping parameter (circles), (b)\novershoot decay time, (c) ratio of overshoot amplitude to st ep-response amplitude ( φ1/φ0), and (d) σxy\n(adapted from [20]).\nform,\nImχxy(ω) =γ/summationdisplay\nm,n,kSx\nmn(k)Sy\nnm(k)fmn(k)\n[ǫmn(k)−ω]2+γ2, (3)\nwhereSi\nmnisthematrixelementofthespinoperator. Ingeneral, σxy(ω)andχxy(ω)areunrelated,\nas they involvecurrent and spin matrix elements respective ly. However, it has been proposed that\nin several ferromagnets, including SRO, the k-space sums in Eqs. 2 and 3 are dominated by a\nsmall number of band-crossings near the Fermi surface [22, 2 5]. If the matrix elements Si\nmnand\nJi\nmnvary sufficiently smoothly with k, thenσxy(ω)andχxy(ω)may be closely related, with both\nproperties determined by thepositionofthechemical poten tialrelativeto theenergy at which the\n8bandscross. Furthermore,asGilbertdampingisrelatedtot hezero-frequencylimitof χxy(ω),i.e.,\nα=ΩFMR\nχxy(0)∂\n∂ωlim\nω→∞Imχxy(ω), (4)\nand AHE is the zero-frequency limit of σxy(ω), the band-crossing picture implies a strong corre-\nlationbetween α(T)andσxy(T).\nIn conclusion,we havereported the observationof FMR in the metallictransition-metaloxide\nSrRuO 3. Both the frequency and damping coefficient are significantl y larger than observed in\ntransition metal ferromagnets. Correlations between FMR d ynamics and the AHE coefficient\nsuggest that both may be linked to near Fermi surface band-cr ossings. Further study of these\ncorrelations, as Sr is replaced by Ca, or with systematic var iation in residual resistance, could be\na fruitful approach to understanding the dynamics of magnet ization in the presence of strong SO\ninteraction.\nAcknowledgments\nThis research is supported by the US Department of Energy, Of fice of Science, under contract\nNo. DE-AC02-05CH1123. Y.H.C. would also like to acknowledg e the support of the National\nScience Council,R.O.C., underContract No. NSC97-3114-M- 009-001.\n[1] I.ˆZuti´ c, J. Fabian, and S.DasSarma, Rev.Mod. Phys. 76, 323 (2004).\n[2] V. Korenman and R.E.Prange, Phys. Rev.B 6, 2769 (1972).\n[3] V. Kambersk´ y, Can. J. Phys. 48, 2906 (1970).\n[4] J. C.Slonczewski, J. Magn. Magn. Mater. 159, L1(1996).\n[5] L.Berger, Phys. Rev.B 54, 9353 (1996).\n[6] J. M.Luttinger and R. Karplus, Phys. Rev. 94, 782 (1954).\n[7] H.Brooks, Phys. Rev. 58, 909 (1940).\n[8] L. Klein, J. S. Dodge, C. H. Ahn, J. W. Reiner, L. Mieville, T. H. Geballe, M. 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Tokura,\nPhys. Rev. Lett. 94, 087202 (2005).\n[17] T. Kise, T. Ogasawara, M. Ashida, Y. Tomioka, Y. Tokura, and M. Kuwata-Gonokami, Phys. Rev.\nLett.85, 1986 (2000).\n[18] W.F.Brown, Micromagnetics (Krieger, 1963).\n[19] B.Koopmans,M.vanKampen,J.T.Kohlhepp,andW.J.M.de Jonge,Phys.Rev.Lett. 85,844(2000).\n[20] R. Mathieu, A. Asamitsu, H. Yamada, K. S. Takahashi, M. K awasaki, Z. Fang, N. Nagaosa, and\nY. Tokura, Phys. Rev. Lett. 93, 016602 (2004).\n[21] L. Klein, J. R. Reiner, T. H. Geballe, M. R. Beasley, and A . Kapitulnik, Phys. Rev. B 61, R7842\n(2000).\n[22] Z. Fang, N. Nagaosa, K. Takahashi, A. Asamitsu, R. Mathi eu, T. Ogasawara, H. Yamada,\nM. Kawasaki, Y. Tokura, and K.Terakura, Science 302, 92(2003).\n[23] M. Onoda, A.S.Mishchenko, and N. Nagaosa, J.Phys. Soc. Jap.77, 013702 (2008).\n[24] M. Onoda and N.Nagaosa, J. Phys.Soc. Jap. 71, 19 (2002).\n[25] X.Wang, J.R. Yates, I. Souza, and D.Vanderbilt, Phys.R ev. B.74, 195118 (2006).\n10"    },    {        "title": "1109.4964v1.Hole_spin_relaxation_and_coefficients_in_Landau_Lifshitz_Gilbert_equation_in_ferromagnetic_GaMnAs.pdf",        "content": "arXiv:1109.4964v1  [cond-mat.mtrl-sci]  22 Sep 2011Hole spin relaxation and coefficients in Landau-Lifshitz-Gi lbert equation in\nferromagnetic GaMnAs\nK. Shen and M. W. Wu∗\nHefei National Laboratory for Physical Sciences at Microsc ale and Department of Physics,\nUniversity of Science and Technology of China, Hefei, Anhui , 230026, China\n(Dated: April 19, 2022)\nWe investigate the temperature dependence of the coefficient s in the Landau-Lifshitz-Gilbert\nequation in ferromagnetic GaMnAs by employing the Zener mod el. We first calculate the hole spin\nrelaxation time based on the microscopic kinetic equation. We find that the hole spin relaxation\ntime is typically several tens femtoseconds and can present a nonmonotonic temperature dependence\ndue to the variation of the interband spin mixing, influenced by the temperature related Zeeman\nsplitting. With the hole spin relaxation time, we are able to calculate the coefficients in the Landau-\nLifshitz-Gilbert equation, such as the Gilbert damping, no nadiabatic spin torque, spin stiffness and\nvertical spin stiffness coefficients. We find that the nonadiab atic spin torque coefficient βis around\n0.1∼0.3 at low temperature, which is consistent with the experimen t [Adam et al., Phys. Rev. B\n80, 193204 (2009)]. As the temperature increases, βmonotonically increases and can exceed one\nin the vicinity of the Curie temperature. In the low temperat ure regime with β <1, the Gilbert\ndamping coefficient αincreases with temperature, showing good agreement with th e experiments\n[Sinovaet al., Phys. Rev. B 69, 085209 (2004); Khazen et al.,ibid.78, 195210 (2008)]. Furthermore,\nwe predict that αdecreases with increasing temperature once β >1 near the Curie temperature.\nWe also find that the spin stiffness decreases with increasing temperature, especially near the Curie\ntemperature due to the modification of the finite β. Similar to the Gilbert damping, the vertical\nspin stiffness coefficient is also found to be nonmonotonicall y dependent on the temperature.\nPACS numbers: 72.25.Rb, 75.50.Pp, 72.25.Dc, 75.30.Gw\nI. INTRODUCTION\nThe ferromagnetic semiconductor, GaMnAs, has been\nproposed to be a promising candidate to realize all-\nsemiconductor spintronic devices,1,2where the existence\nof the ferromagnetic phase in the heavily doped sample\nsustains seamless spin injection and detection in normal\nnon-magneticsemiconductors.3,4Oneimportant issuefor\nsuch applications lies in the efficiency of the manipula-\ntion of the macroscopic magnetization, which relies on\nproperties of the magnetization dynamics. Theoretically,\nthe magnetization dynamics can be described by the ex-\ntended Landau-Lifshitz-Gilbert (LLG) equation,5–10\n˙n=−γn×Heff+αn×˙n−(1−βn×)(vs·∇)n\n−γ\nMdn×(Ass−Av\nssn×)∇2n, (1)\nwithnandMdstanding for the direction and magni-\ntude of the magnetization, respectively. Heffis the ef-\nfective magnetic field and/or the external field. The sec-\nond term on the right hand side of the equation is the\nGilbert damping torque with αdenoting the damping\ncoefficient.5,6The third one describes the spin-transfer\ntorque induced by the spin current vs.7,8As reported,\nthe out-of-plane contribution of the spin-transfer torque,\nmeasured by the nonadiabatic torque coefficient β, can\nsignificantly ease the domain wall motion.7,8In Eq.(1),\nthespinstiffnessandverticalspinstiffnesscoefficientsare\nevaluated by AssandAv\nssrespectively, which are essen-\ntially important for the static structure of the magnetic\ndomain wall.10Therefore, for a thorough understandingof properties of the magnetization dynamics, the exact\nvalues of the above coefficients are required.\nIn the past decade, the Gilbert damping and nonadia-\nbatic torque coefficients have been derived via many mi-\ncroscopic approaches, such as the Blotzmann equation,11\ndiagrammatic calculation,12,13Fermi-surface breathing\nmodel14–16andkineticspinBlochequations.10,17Accord-\ning to these works, the spin lifetime of the carriers was\nfound to be critical to both αandβ. However, to the\nbest of our knowledge, the microscopic calculation of the\nhole spin lifetime in ferromagnetic GaMnAs is still ab-\nsent in the literature, which prevents the determination\nofthe values of αandβfrom the analyticalformulas. Al-\nternatively, Sinova et al.18identified the Gilbert damp-\ning from the susceptibility diagram of the linear-response\ntheory and calculated αas function of the quasiparticle\nlifetime and the hole density. Similar microscopic calcu-\nlation on βwas later given by Garate et al..19In those\nworks, the quasiparticle lifetime was also treated as a\nparameter instead of explicit calculation. Actually, the\naccurate calculation of the hole spin and/or quasiparti-\ncle lifetime in ferromagnetic GaMnAs is difficult due to\nthe complex band structure of the valence bands. In the\npresentwork,weemploythe microscopickineticequation\ntocalculatethespinlifetimeoftheholegasandtheneval-\nuateαandβin ferromagnetic GaMnAs. For the velocity\nof the domain-wall motion due to the spin current, the\nratioβ/αisanimportantparameter,whichhasattracted\nmuch attention.12,19,20Recently, a huge ratio ( ∼100) in\nnanowire was predicted from the calculation of the scat-\ntering matrix by Hals et al..20By calculating αandβ, we2\nare able to supply detailed information of this interesting\nratio in bulk material. Moreover, the peak-to-peak fer-\nromagnetic resonance measurement revealed pronounced\ntemperature and sample preparation dependences of the\nGilbert damping coefficient.18,21,22For example, in an-\nnealed samples, αcan present an increase in the vicinity\nof the Curie temperature,18,21which has not been stud-\nied theoretically in the literature. Here, we expect to\nuncover the underlying physics of these features. In ad-\ndition, the nonadiabatic torque coefficient βin GaMnAs\nhas been experimentally determined from the domain-\nwall motion and quite different values were reported by\ndifferent groups, from 0.01 to 0.36,23,24which need to be\nverified by the microscopic calculation also. Moreover,\nto the best of our knowledge, the temperature depen-\ndence of βhas not been studied theoretically. We will\nalso address this issue in the present work.\nIn the literature, the spin stiffness in GaMnAs was\nstudied by K¨ onig et al.,25,26who found that Assincreases\nwith hole density due to the stronger carrier-mediated\ninteraction between magnetic ions, i.e., Ass=Nh/(4m∗)\nwithNhandm∗being the density and effective mass\nof hole gas, separately. However, as shown in our pre-\nvious work, the stiffness should be modified as Ass∼\nNh/[4m∗(1+β2)] in ferromagnetic GaMnAs with a finite\nβ.10As a result, Assas well as the vertical spin stiffness\nAv\nss=βAssmay show a temperature dependence intro-\nduced by β. This is also a goal of the present work.\nFor a microscopic investigation of the hole dynamics,\nthe valence band structure is required for the descrip-\ntion of the occupied carrier states. In the literature,\nthe Zener model27based on the mean-field theory has\nbeen widely used for itinerant holes in GaMnAs,28–31\nwhere the valence bands split due to the mean-field p-\ndexchange interaction. In the present work, we utilize\nthis model to calculate the band structure with the ef-\nfective Mn concentration from the experimental value of\nthe low-temperaturesaturatemagnetization in GaMnAs.\nThe thermal effect on the band structure is introduced\nviathe temperaturedependence ofthe magnetizationfol-\nlowing the Brillouin function. Then we obtain the hole\nspin relaxation time by numerically solving the micro-\nscopic kinetic equations with the relevant hole-impurity\nand hole-phonon scatterings. The carrier-carrier scatter-\ning is neglected here by considering the strongly degen-\nerate distribution of the hole gas below the Curie tem-\nperature. We find that the hole spin relaxation time\ndecreases/increases with increasing temperature in the\nsmall/large Zeeman splitting regime, which mainly re-\nsults from the variation of the interband spin mixing.\nThen we study the temperature dependence of the co-\nefficients in the LLG equation, i.e., α,β,AssandAv\nss,\nby using the analytical formulas derived in our previous\nworks.10,17Specifically, we find that βincreases with in-\ncreasing temperature and can exceed one in the vicinity\nof the critical point, resulting in very interesting behav-\niors of other coefficients. For example, αcan present an\ninteresting nonmonotonic temperature dependence withthe crossoveroccurringat β∼1. Specifically, αincreases\nwithtemperatureinthelowtemperatureregime,whichis\nconsistent with the experiments. Near the Curie temper-\nature, an opposite temperature dependence of αis pre-\ndicted. Similar nonmonotonic behavior is also predicted\nin the temperature dependence of Av\nss. Our results of β\nandAssalso show good agreement with the experiments.\nThis work is organized as follows. In Sec.II, we setup\nour model and lay out the formulism. Then we show\nthe band structure from the Zener model and the hole\nspin relaxation time from microscopic kinetic equations\nin Sec.III. The temperature dependence of the Gilbert\ndamping, nonadiabatic spin torque, spin stiffness and\nvertical spin stiffness coefficients are also shown in this\nsection. Finally, we summarize in Sec.IV.\nII. MODEL AND FORMULISM\nInthesp-dmodel, theHamiltonianofholegasinGaM-\nnAs is given by31\nH=Hp+Hpd, (2)\nwithHpdescribing the itinerant holes. Hpdis thesp-d\nexchange coupling. By assuming that the momentum k\nis still a good quantum number for itinerant hole states,\none employsthe Zenermodel and utilizes the k·ppertur-\nbation Hamiltonian to describe the valence band states.\nSpecifically, we take the eight-band Kane Hamiltonian\nHK(k) (Ref.32) in the present work. The sp-dexchange\ninteraction reads\nHpd=−1\nN0V/summationdisplay\nl/summationdisplay\nmm′kJmm′\nexSl·∝an}b∇acketle{tmk|ˆJ|m′k∝an}b∇acket∇i}htc†\nmkcm′k,\n(3)\nwithN0andVstanding for the density of cation sites\nand the volume, respectively. The cation density N0=\n2.22×1022cm−3. The eight-band spin operator can be\nwritten as ˆJ= (1\n2σ)⊕J3/2⊕J1/2, where1\n2σ,J3/2and\nJ1/2represent the total angular momentum operators of\nthe conduction band, Γ 8valence band and Γ 7valence\nband, respectively. Jmm′\nexstands for the matrix element\nof the exchange coupling, with {m}and{m′}being the\nbasis defined as the eigenstates of the angular momen-\ntum operators ˆJ. The summation of “ l” is through all\nlocalized Mn spins Sl(atrl).\nThen we treat the localized Mn spin in a mean-field\napproximation and obtain\n¯Hpd=−xeff∝an}b∇acketle{tS∝an}b∇acket∇i}ht·/parenleftBigg/summationdisplay\nmm′kJmm′\nex∝an}b∇acketle{tmk|ˆJ|m′k∝an}b∇acket∇i}htc†\nmkcm′k/parenrightBigg\n,\n(4)\nwhere∝an}b∇acketle{tS∝an}b∇acket∇i}htrepresents the average spin polarization of\nMn atoms with uncompensated doping density NMn=\nxeffN0. Obviously, ¯Hpdcan be reduced into three blocks\nasˆJ, i.e.,¯Hmm′\npd(k) = ∆mmn·∝an}b∇acketle{tmk|ˆJ|m′k∝an}b∇acket∇i}htwith the Zee-\nman splitting of the m-band ∆mm=−xeffSdJmm\nexM(T)\nM(0).3\nHere,nis the direction of ∝an}b∇acketle{tS∝an}b∇acket∇i}ht. For a manganese ion, the\ntotal spin Sd= 5/2. The temperature-dependent spon-\ntaneous magnetization M(T) can be obtained from the\nfollowing equation of the Brillouin function33\nBSd(y) =Sd+1\n3SdT\nTcy, (5)\nwherey=3Sd\nSd+1M(T)\nM(0)Tc\nTwithTcbeing the Curie\ntemperature. Here, BSd(y) =2Sd+1\n2Sdcoth(2Sd+1\n2Sdy)−\n1\n2Sdcoth(1\n2Sdy).\nThe Schr¨ odinger equation of the single particle Hamil-\ntonian is then written as\n/bracketleftbig\nHK(k)+¯Hpd(k)/bracketrightbig\n|µ,k∝an}b∇acket∇i}ht=Eµk|µ,k∝an}b∇acket∇i}ht.(6)\nOne obtains the band structure and wave functions from\nthe diagonalization scheme. In the presence of a finite\nZeemansplitting, thestructureofthe valencebandsdevi-\nates from the parabolic dispersion and becomes strongly\nanisotropicaswewillshowinthenextsection. Moreover,\nthe valence bands at Fermi surface are well separated in\nferromagnetic GaMnAs because of the high hole density\n(>1020cm−3) and Zeeman splitting, suggesting that the\nFermi golden rule can be used to calculate the lifetime of\nthe quasiparticlestates. For example, the contribution of\nthe hole-impurity scattering on the µth-band state with\nenergyǫcan be expressed by\n[τhi\nµ,p(ǫ)]−1= 2π/summationdisplay\nνni\nDµ(ǫ)/integraldisplayd3k\n(2π)3/integraldisplayd3q\n(2π)2δ(ǫ−ǫµk)\n×δ(ǫµk−ǫνq)U2\nk−q|∝an}b∇acketle{tµk|νq∝an}b∇acket∇i}ht|2f(ǫµk)[1−f(ǫνq)],(7)\nwhereDµ(ǫ) stands for the density of states of the µth\nband.f(ǫµk) satisfies the Fermi distribution in the equi-\nlibrium state. The hole-impurity scattering matrix ele-\nmentU2\nq=Z2e4/[κ0(q2+κ2)]2withZ= 1.κ0and\nκdenote the static dielectric constant and the screening\nconstant under the random-phase approximation,34re-\nspectively. Similar expression can also be obtained for\nthe hole-phonon scattering.\nHowever, it is very complicated to carry out the multi-\nfold integrals in Eq.(7) numerically for an anisotropic\ndispersion. Also the lifetime of the quasiparticle is not\nequivalent to the spin lifetime of the whole system, which\nis required to calculate the LLG coefficients according\nto our previous work.10,17Therefore, we extend our ki-\nnetic spin Bloch equation approach35to the current sys-\ntem to study the relaxation of the total spin polarization\nas follows. By taking into account the finite separation\nbetween different bands, one neglects the interband co-\nherence and focuses on the carrier dynamics of the non-\nequilibrium population. The microscopic kinetic equa-\ntion is then given by\n∂tnµ,k=∂tnµ,k/vextendsingle/vextendsinglehi+∂tnµ,k/vextendsingle/vextendsinglehp, (8)\nwithnµ,kbeing the carrier occupation factor at the µth\nband with momentum k. The first and second terms onthe right hand side stand for the hole-impurity and hole-\nphonon scatterings, respectively. Their expressions can\nbe written as\n∂tnµ,k/vextendsingle/vextendsinglehi=−2πni/summationdisplay\nν,k′U2\nk−k′(nµk−nνk′)|∝an}b∇acketle{tµk|νk′∝an}b∇acket∇i}ht|2\n×δ(Eµk−Eνk′), (9)\nand\n∂tnµ,k/vextendsingle/vextendsinglehp=−2π/summationdisplay\nλ,±,ν,k′|Mλ\nk−k′|2δ(Eνk′−Eµk±ωλ,q)\n×[N±\nλ,q(1−nνk′)nµk−N∓\nλ,qnνk′(1−nµk)]|∝an}b∇acketle{tµk|νk′∝an}b∇acket∇i}ht|2,(10)\nwithN±\nλ,q= [exp(ωλ,q/kBT)−1]−1+1\n2±1\n2. Thedetailsof\nthe hole-phonon scattering elements |Mλ\nq|2can be found\nin Refs.36–38. From an initial condition with a small\nnon-equilibrium distribution, the temporal evolution of\nthe hole spin polarization is carried out by\nJ(t) =1\nNh/summationdisplay\nµ,k∝an}b∇acketle{tµk|ˆJ|µk∝an}b∇acket∇i}htnµ,k(t),(11)\nfrom the numerical solution of Eq.(8). The hole spin\nrelaxation time can be extracted from the exponential\nfitting of Jwith respect to time. One further calculates\nthe concerned coefficients such as α,β,AssandAv\nss.\nIII. NUMERICAL RESULTS\nIn the Zener model, the sp-dexchange interaction con-\nstantsJmm\nexareimportant parametersfor the band struc-\nture. In the experimental works, the p-dexchange cou-\npling constant Jpp\nexwas reported to vary from −1 eV\nto 2.5 eV, depending on the doping density.39–41In\nferromagnetic samples, Jpp\nexis believed to be negative,\nwhich was demonstrated by theoretical estimation Jpp\nex≈\n−0.3 eV (Ref.42). In our calculation, the antiferromag-\nneticp-dinteraction Jpp\nexis chosen to be −0.5 eV or\n−1.0 eV. The ferromagnetic s-dexchange coupling con-\nstant is taken to be Jss\nex= 0.2 eV (Ref.31).\nAnother important quantity for determining the Zee-\nman splitting is the macroscopic magnetization or the\neffective concentration of the Mn atoms. As deduced\nfrom the low-temperature saturate magnetization, only\naround 50% Mn atoms can contribute to the ferromag-\nnetic magnetization, which hasbeen recognizedasthe in-\nfluence of the compensation effect due to the deep donors\n(e.g., Asantisites)ortheformationofsixfold-coordinated\ncenters defect Mn6As(Ref.43). As only the uncompen-\nsated Mn atoms can supply holes and contribute to the\nferromagneticmagnetization,44one can also estimate the\ntotal hole density from the saturate magnetization.45\nHowever, the density of the itinerant hole can be smaller\nthan the effective Mn concentration because of the local-\nized effect in such disordered material. It was reported4\nTc Ms NMn\n(K) (emu ·cm−3) (1020cm−3)\nAa130 38 8\nBa157 47 10\nCb114 33 6.9\nDc110 – –\nEd139 53.5 11.5\naRef. 21,bRef. 23,cRef. 18,dRef. 45\nTABLE I: The parameters obtained from the experiments\nfordifferentsamples: A:Ga 0.93Mn0.07As/Ga 0.902In0.098As; B:\nGa0.93Mn0.07As/GaAs; C: Ga 0.93Mn0.07As/Ga 1−yInyAs; D:\nGa0.92Mn0.08As; E: Ga 0.896Mn0.104As0.93P0.07.Msstands for\nthe saturate magnetization at zero temperature M(0).\nthat the hole density is only 15-30% of the total concen-\ntration of the Mn atoms.43\nIn our calculation, the magnetization lies along the\nprinciple axis chosen as [001]-direction.31The conven-\ntional parameters are mainly taken from those of GaAs\nin Refs.46 and 47. Other sample-dependent parame-\nters such as the Curie temperature and effective Mn\nconcentration are picked up from the experimental\nworks.18,21,23,45For sample A, B and E (C), only the\nsaturate magnetization at 4 (104) K was given in the\nreferences. Nevertheless, one can extrapolate the zero\ntemperature magnetization Msfrom Eq.(5). The effec-\ntive Mn concentrations listed in TableI are derived from\nNMn=Ms/(gµBSd). It is clearto see that all of these ef-\nfectiveMnconcentrationsaremuchsmallerthanthedop-\ning density ( ≥1.5×1021cm−3) due to the compensation\neffect as discussed above. Since the saturate magneti-\nzation of sample D is unavailable, we treat the effective\nMn concentration as a parameter in this case. More-\nover, since the exact values of the itinerant hole densities\nare unclear in such strongly disordered samples, we treat\nthem as parameters. Two typical values are chosen in\nour numerical calculation, i.e., Nh= 3×1020cm−3and\n5×1020cm−3. The effective impurity density is taken to\nbe equal to the itinerant hole density.\nFor numerical calculation of the hole spin dynamics,\nthe momentum space is partitioned into blocks. Com-\npared to the isotropic parabolic dispersion, the band\nstructure in ferromagnetic GaMnAs is much more com-\nplex as we mentioned above [referred to Figs.1(b) and\n4]. Therefore, we need to extend the partition scheme\nused in isotropic parabolic dispersion48into anisotropic\ncase. In our scheme, the radial partition is still carried\nout with respect to the equal-energy shells, while the an-\ngular partition is done by following Ref.48. In contrast\nto the isotropic case, the number of states in one block is\ngenerally different from that in another block even both\nof them are on the same equal-energy shell. We calculate\nthe number of states of each block from its volume inmomentum space.\n 0 10 20 30 40 50 60 0.2  0.4  0.6  0.8  1∆pp (meV)T/Tc\n(a)\n6.9×1020 cm-3\n8×1020 cm-3\n1×1021 cm-3\n-0.4-0.3-0.2-0.1 0 0.1-0.1  0  0.1  0.2\nE (eV)k (2π/a ) [111] [001]\n(b)\n 0 51015\n-0.1 -0.05  0 0.05  0.1  0.15  0.2DOS (1020/eVcm3)\nE (eV)T = 0.1 TcNMn = 8×1020 cm-3\n(c)\n-0.05  0 0.05  0.1  0.15  0.2 0 5 10 15\nDOS (1020/eVcm3)\nE (eV)T = 0.99 Tc\n(d)\nFIG. 1: (Color online) (a) Zeeman energy as function of tem-\nperature. (b)Thevalencebandstructurewith∆pp= 45meV.\nThe blue dashed curve illustrates the Fermi level for the hol e\ndensityNh= 3×1020cm−3, while the green dotted one gives\nNh= 5×1020cm−3. The density of states as function of\nenergy at (c) T/Tc= 0.1 and (d) T/Tc= 0.99 for the uncom-\npensated Mn density NMn= 8×1021cm−3. In (d), the blue\ndashed curve stands for the upper heavy hole band from the\nspherical approximation and the corresponding DOS from the\nanalytical formula (√\n2E[√\nm∗/(2π/planckover2pi1)]3) is given as the green\ndotted curve. Here, Jpp\nex=−0.5 eV.\nA. Density of states\nBy solving Eq.(5), one obtains the magnetization at\nfinite temperature M(T) and the corresponding Zeeman\nenergy ∆pp. In Fig.1(a), the Zeeman splitting from\nJpp\nex=−0.5 eV is plotted as function of the temperature.\nIt is seen that the Zeeman energy is tens of milli-electron\nvolts at low temperature and decreases sharply near the\nCurie temperature due to the decrease of the magnetiza-\ntion. To show the anisotropicnonparabolicfeature of the\nband structure in the presence of the Zeeman splitting,\nwe illustrate the valence bands along [001]- and [111]-\ndirections in Fig.1(b), which are obtained from Eq.(6)\natT/Tc= 0.1 forNMn= 8×1020cm−3. In this case, the\nZeeman splitting ∆pp= 45 meV. The Fermi levels for the\nhole densities Nh= 3×1020cm−3and 5×1020cm−3are\nshown as blue dashed and green dotted curves, respec-\ntively. As one can see that all of the four upper bands\ncan be occupied and the effective mass approximation\nobviously breaks down.\nBy integrating over the volume of each equal-energy\nshell, one obtains the density of states (DOS) of each\nband as function of energy in Fig.1(c) and (d). Here the\nenergy is defined in the hole picture so that the sign of5\nthe energy is opposite to that in Fig.1(b). It is seen that\nthe DOS of the upper two bands are much larger than\nthose of the other bands, regardless of the magnitude of\nthe Zeeman splitting. For T/Tc= 0.99, the systems ap-\nproaches the paramagnetic phase and the nonparabolic\neffect is still clearly seen from the DOS in Fig.1(d), es-\npecially in the high energy regime. Moreover, the pro-\nnounced discrepancy of the DOS for the two heavy hole\nbands suggests the finite splitting between them. We\nfind that these features are closely connected with the\nanisotropy of the valence bands, corresponding to the\nLuttinger parameters γ2∝ne}ationslash=γ3in GaAs.49In our calcu-\nlation, we take γ1= 6.85,γ2= 2.1 andγ3= 2.9 from\nRef.47. As a comparison, we apply a spherical approx-\nimation ( γ1= 6.85 andγ2=γ3= ¯γ= 2.5) and find\nthat the two heavy hole bands become approximately\ndegenerate.38The DOS of the upper heavy hole band\nis shown as the blue dashed curve in Fig.1(d), where\nwe also plot the corresponding DOS from the analyti-\ncal expression, i.e.,√\n2E[√\nm∗/(2π/planckover2pi1)]3, as the green dot-\nted curve. Here, we use the heavy-hole effective mass\nm∗=m0/(γ1−2¯γ) withm0denoting the free electron\nmass. The perfect agreement between the analytical and\nour numerical results under the spherical approximation\nsuggests the good precision of our numerical scheme.\n 0 0.1 0.2 0.3 0.4 0.5 0.6\n 0  10  20  30  40  50  60Equilibrium Hole Polarization\n∆pp (meV)A: Nh=3×1020 cm-3\n5×1020 cm-3\nB: Nh=3×1020 cm-3\n5×1020 cm-3\nFIG. 2: (Color online) The equilibrium hole spin polarizati on\nas function of Zeeman splitting for sample A and B. Here,\nJpp\nex=−0.5 eV.\nB. Hole spin relaxation\nIn this part, we investigate the hole spin dynamics by\nnumericallysolvingthe microscopickinetic equation, i.e.,\nEq.(8). By taking into account the equilibrium hole spin\npolarization, we fit the temporal evolution of the total\nspin polarization along [001]-direction by\nJz(t) =J0\nz+J′\nze−t/τs, (12)\nwhereJ0\nzandJ′\nzcorrespondto the equilibrium and non-\nequilibrium spin polarizations, respectively. τsisthe hole\nspin relaxation time.In all the cases of the present work, the equilibrium\nhole spin polarization for a fixed hole density is found\nto be approximately linearly dependent on the Zeeman\nsplitting. In Fig.2, J0\nzin samples A and B (similar be-\nhaviorforothers) areplotted asfunction ofZeemansplit-\nting, where the exchange coupling constant Jpp\nexis taken\nto be−0.5 eV. One notices that the average spin polar-\nizationbecomessmallerwith the increaseofthe holeden-\nsity, reflecting the large interband mixing for the states\nin the high energy regime.\n 30 40 50 60 70 80\n 0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1τs (fs)\nT/Tc(a)Jexpp = -0.5 eV\nA: Nh=3×1020 cm-3\n5×1020 cm-3\nB: Nh=3×1020 cm-3\n5×1020 cm-3\n 40 50 60 70 80\n 0  0.2  0.4  0.6  0.8  1  1.2 0  20  40  60  80  100  120τs (fs)\nT/Tc∆pp (meV)\n(b)\nJexpp = -1 eV0.4 Tc0.99 Tc\nB: Nh=3×1020 cm-3\n5×1020 cm-3\nFIG. 3: (Color online) (a) Spin relaxation time as function o f\ntemperaturewith Jpp\nex=−0.5eVfor sampleAandB.(b)Spin\nrelaxation time as function of temperature and Zeeman split -\nting obtained from the calculation with Jpp\nex=−1 eV for sam-\nple B. The inset at the left (right) upper corner illustrates the\nband structure from [001]-direction to [111]-direction [r efer to\nFig.1(b)] for T/Tc= 0.4 (0.99) and ∆pp= 105 (16.7) meV.\nThe Fermi levels of Nh= 3×1020cm−3and 5×1020cm−3\nare shown as the blue dashed and green dotted curves in the\ninsets, separately.\nThe temperature dependence of the hole spin relax-\nation time in samples A and B with Jpp\nex=−0.5 eV is\nshown in Fig.3(a), where the spin relaxation time mono-\ntonically decreases with increasing temperature. This\nfeature can be understood from the enhancement of\nthe interband mixing as the Zeeman splitting decreases\n(shownbelow).50Togainacompletepicture ofthe roleof\nthe Zeeman splitting on the hole spin relaxation in fer-6\nromagnetic GaMnAs, we also carry out the calculation\nwith the exchange constant Jpp\nex=−1 eV.31,39Very in-\nterestingly, onefinds that the holespin relaxationtime at\nlow temperature increases with increasing temperature,\nresulting in a nonmonotonic temperature dependence of\nthe hole spin relaxation time in sample B. The results\nin this case are shown as solid curves in Fig.3(b), where\nwe also plot the Zeeman splitting dependence of the hole\nspin relaxation time as dashed curves. It is seen that\nthe hole spin relaxation time for the hole density Nh=\n3×1020cm−3first increases with increasing temperature\n(alternativelyspeaking,decreasingZeemansplitting)and\nstarts to decrease at around 0 .8Tcwhere the Zeeman\nsplitting ∆pp= 70 meV. To understand this feature, we\nshowthetypicalbandstructureinthe increase(decrease)\nregime of the hole relaxation time at T/Tc= 0.4 (0.99),\ncorresponding to ∆pp= 105 (16.7) meV, in the inset at\nthe left (right) upper corner. The Fermi levels of the\nhole density 3 ×1020cm−3are labeled by blue dashed\ncurves. One finds that the carrier occupations in the\nincrease and decrease regimes are quite different. Specif-\nically, all of the four upper bands are occupied in the\ndecrease regime while only three valence bands are rele-\nvant in the increase regime.\nOne may naturally expect that the increase regime\noriginates from the contribution of the fourth band via\nthe inclusion of the additional scattering channels or the\nmodification of the screening. However, we rule out this\npossibilitythroughthecomputationwiththefourthband\nartificially excluded, where the results are qualitatively\nthe same as those in Fig.3(b). Moreover, the variations\nof the screening and the equilibrium distribution at fi-\nnite temperature are also demonstrated to be irrelevant\nto the present nonmonotonic dependence by our calcula-\ntion (not shown here). Therefore, the interesting feature\nhas to be attributed to the variations of the band dis-\ntortion and spin mixing due to the exchange interaction.\nThis is supported by our numerical calculation, where\nthe nonmonotonic behavior disappears once the effect of\nthe interband mixing is excluded by removing the wave-\nfunction overlaps |∝an}b∇acketle{tµk|νk′∝an}b∇acket∇i}ht|2in Eqs.(9) and (10) (not\nshown here).\nFor a qualitative understanding of the nonmonotonic\ntemperature dependence of the hole spin relaxation time,\nwe plot the Fermi surface in the kx-kz(ky= 0) and kx-\nky(kz= 0) planes at Nh= 3×1020cm−3in Fig.4. We\nchoose typical Zeeman splittings in the increase regime\n(∆pp= 105meV), the decreaseregime(∆pp= 16.7meV)\nand also the crossover regime (∆pp= 70 meV). One no-\ntices that the Fermi surfacesin Fig.4(a) and (d) arecom-\nposed of three closed curves, meaning that only three\nbands are occupied for ∆pp= 105 meV [also see the in-\nset of Fig.3(b)]. For the others with smaller Zeeman\nsplittings, all of the four upper bands are occupied. The\nspin expectation of each state at Fermi surface is repre-\nsented by the color coding. Note that the spin expecta-\ntion of the innermost band for ∆pp= 70 meV is close to\n−1.5 [see Fig.4(b) and (e)], suggesting that this band is-1.5-1-0.5 0 0.5 1 1.5ξ∆pp = 105 meV  70 meV  16.7 meV\n(a)-0.2-0.10.00.10.2kz (2π/a)\n-1.5-1-0.5 0 0.5 1 1.5\n(b)\n-1.5-1-0.5 0 0.5 1 1.5\n(c)\n-1.5-1-0.5 0 0.5 1 1.5\n(d)\n-0.2 -0.1 0.0 0.1 0.2-0.2-0.10.00.10.2ky (2π/a)-1.5-1-0.5 0 0.5 1 1.5\n(e)\n-0.2 -0.1 0.0 0.1 0.2\nkx (2π/a)-1.5-1-0.5 0 0.5 1 1.5\n(f)\n-0.2 -0.1 0.0 0.1 0.2\nFIG.4: (Color online)TheFermisurface inthe kx-kz(ky= 0)\nandkx-ky(kz= 0) planes with ∆pp=105 meV (a,d), 70 meV\n(b,e) and 16.7 meV (c,f). The color coding represents the\nspin expectation of each state, ξ=/angbracketleftµ|Jz|µ/angbracketright. Here, Nh=\n3×1020cm−3.\nthe spin-down heavy hole band and the mixing of other\nspin components in this band is marginal. Therefore,\nthe spin-flip scattering related to this band is weak and\ncan not result in the increase of the hole spin relaxation\ntime mentioned above. By comparing the results with\n∆pp= 105 meV and 70 meV, one notices that the spin\nexpectation of the Fermi surface of the outermost band\nis insensitive to the Zeeman splitting. Therefore, this\nband can not be the reason of the increase regime also.\nMoreover, for the second and third bands in Fig.4(a)\nand (b), from the comparable color coding between the\ntwo figures in this regime [also see Fig.4(d) and (e) with\nkz= 0], one finds that the spin expectation for the\nstates with small kzis also insensitive to the Zeeman\nsplitting. However, for the states with large kz, the spin\nexpectation of the spin-down states ( ξ <0) approaches\na large magnitude ( −1.5) with decreasing Zeeman split-\nting, suggestingthe decrease ofthe mixing from the spin-\nup states. As a result, the interband spin-flip scattering\nfrom/to these states becomes weak and the hole spin re-\nlaxation time increases. In the decrease regime of the\nhole spin relaxation time, Fig.4(c) and (f) show that the\ntwo outer/inner bands approach each other, leading to a\nstrong and anisotropic spin mixing. Therefore, the spin-\nflip scattering becomes more efficient in this regime and\nthe spin relaxation time decreases. One may suppose\nthat the nonmonotonic temperature dependence of the\nhole spin relaxation time can also arise from the varia-\ntionofthe shapeofthe Fermisurface, accordingtoFig.4.\nHowever, this variation itself is not the key of the non-\nmonotonic behavior, because the calculation with this\neffect but without band mixing can not recover the non-\nmonotonic feature as mentioned in the previous para-\ngraph. For the hole density Nh= 5×1020cm−3, the\nstructures of the Fermi surface at ∆pp= 105 meV are\nsimilartothoseinFig.4(b)and(e). Thisexplainstheab-\nsence of the increase regime for this density in Fig.3(b).\nMoreover,weshouldpoint outthat the increaseregime7\nof the hole spin relaxation time in sample A for Jpp\nex=\n−1 eV is much narrower than that in sample B. The\nreason lies in the fact of lower effective Mn density in\nsample A, leading to the smaller maximal Zeeman split-\nting∼90 meV, only slightly larger than the crossover\nvalue 70 meV at Nh= 3×1020cm−3.\nAs a summary of this part, we find different temper-\nature dependences of the hole spin relaxation time due\nto the different values of effective Mn concentration, hole\ndensity and exchange coupling constant Jpp\nex. In the case\nwith large coupling constant and high effective Mn con-\ncentration, the interband spin mixing can resultin a non-\nmonotonic temperature dependence of the hole spin re-\nlaxation time. Our results suggest a possible way to esti-\nmate the exchange coupling constant with the knowledge\nof itinerant hole density, i.e., by measuring the temper-\nature dependence of the hole spin relaxation time. Al-\nternatively, the discrepancy between the hole relaxation\ntime from different hole densities in Fig.3(b) suggests\nthat one can also estimate the itinerant hole density if\nthe exchange coupling constant has been measured from\nother methods.\nC. Gilbert damping and non-adiabatic torque\ncoefficients\nFacilitated with the knowledge of the hole spin re-\nlaxation time, we can calculate the coefficients in the\nLLG equation. According to our previous works,10,17\nthe Gilbert damping and nonadiabatic spin torque co-\nefficients can be expressed as\nα=Jh/[NMn|∝an}b∇acketle{tS∝an}b∇acket∇i}ht|(β+1/β)], (13)\nand\nβ= 1/(2τs∆pp), (14)\nrespectively. In Eq.(13), Jhrepresents the total equi-\nlibrium spin polarization of the itinerant hole gas, i.e.,\nJh=NhJ0\nzwithJ0\nzbeing the one defined in Eq.(12) in\nour study. The average spin polarization of a single Mn\nion is given by |∝an}b∇acketle{tS∝an}b∇acket∇i}ht|=SdM(T)/M(0).\nIn Fig.5(a), (c) and (e), the nonadiabatic spin torque\ncoefficients βin sample A-C are plotted as function of\ntemperature with Jpp\nex=−0.5 eV and −1.0 eV. Our re-\nsults in sample C show good agreement with the experi-\nmental data (plotted as the brown square) in Fig.5(e).23\nAt low temperature, the value of βis around 0.1 ∼0.3,\nwhich is also comparable with the previous theoretical\ncalculation.19Very interestingly, one finds that βsharply\nincreases when the temperature approaches the Curie\ntemperature. This can be easily understood from the\npronounced decreases of the spin relaxation time and\nthe Zeeman splitting in this regime [see Figs.1(a) and\n3]. By comparing the results with different values of\nthe exchange coupling constant, one finds that βfrom\nJpp\nex=−1 eV is generally about one half of that ob-\ntained from Jpp\nex=−0.5 eV because of the larger Zeeman 0 0.5 1 1.5 2 2.5 3\n 20  40  60  80  100  120β\nT (K)Sample A(a)-0.5 eV, Nh=3×1020 cm-3\n5×1020 cm-3\n-1.0 eV, Nh=3×1020 cm-3\n5×1020 cm-3\n 0 0.01 0.02 0.03 0.04\n 20  40  60  80  100  120α\nT (K)(b)\n 0 0.5 1 1.5 2 2.5 3\n 20  40  60  80 100  120  140  160β\nT (K)Sample B(c)\n 0 0.01 0.02 0.03 0.04\n 20  40  60  80 100  120  140  160α\nT (K)(d)\n 0 0.5 1 1.5 2 2.5 3\n 20  40  60  80  100  120β\nT (K)Sample C(e)\n 0 0.01 0.02 0.03 0.04 0.05\n 20  40  60  80  100  120α\nT (K)(f)\nFIG. 5: (Color online) βandαas function of temperature\nwithJpp\nex=−0.5 eV and −1.0 eV in sample A-C. In (b) and\n(d), the dots represent the experimental data from ferromag -\nnetic resonance measurement for [001] (brown solid upper tr i-\nangles), [110] (orange solid circles), [100] (green open sq uares)\nand [1-10] (black open lower triangles) dc magnetic-field or i-\nentations (Ref.21). The brown solid square in (e) stands for\nthe experimental result from domain-wall motion measure-\nment (Ref.23).\nsplitting. Moreover, one notices that the nonmonotonic\ntemperature dependence of the hole spin relaxation time\nin Fig.3(b) is not reflected in βdue to the influence of\nthe Zeeman splitting. In all cases, the values of βcan\nexceed one very near the Curie temperature.\nThe results of the Gilbert damping coefficient from\nEq.(13) are shown as curves in Fig.5(b), (d) and (f).\nThe dots in these figures are the reported experimental\ndata from the ferromagnetic resonance along different\nmagnetic-field orientations.21Both the magnitude and\nthetemperaturedependenceofourresultsagreewellwith\nthe experimental data. From Fig.2, one can conclude\nthat the prefactor in Eq.(13), Jh/(NMn|∝an}b∇acketle{tS∝an}b∇acket∇i}ht|), is almost\nindependent of temperature. Therefore, the temperature\ndependence of αmainly results from the nonadiabatic\nspin torque coefficient β. Specifically, αis insensitive\nto the temperature in the low temperature regime and\nit gradually increases with increasing temperature due8\nto the increase of β. Moreover, we predict that αbe-\ngins to decrease with increasing temperature once βex-\nceeds one. This crossover lying at β≈1 can be expected\nfrom Eq.(13). By comparing the results with different\nvalues of Jpp\nex, one finds that the value of αis robust\nagainst the exchange coupling constant in the low tem-\nperature regime. In this regime, β≪1 and one can sim-\nplify the expression of the Gilbert damping coefficient as\nα≈Nh\nNMnSdJ0\nz\n(τs∆pp). Since the total hole spin polariza-\ntion is proportional to the Zeeman splitting (see Fig.2)\nandτsis only weakly dependent on the Zeeman split-\nting (see Fig.3) in this regime, the increase of Jpp\nexdoes\nnot show significant effect on α. However, at high tem-\nperature, the scenario is quite different. For example,\none has the maximum of the Gilbert damping coefficient\nαm≈Nh\n2NMn|/angbracketleftS/angbracketright|J0\nz∝Jpp\nexatβ= 1.\n 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5\n 20  40  60  80  100β\nT (K)Sample D(a)-0.5 eV, Nh=3×1020 cm-3\n5×1020 cm-3\n-1.0 eV, Nh=3×1020 cm-3\n5×1020 cm-3\n 0 0.02 0.04 0.06 0.08 0.1\n 20  40  60  80  100α\nT (K)(b)\nFIG. 6: (Color online) βandαas function of temperature by\ntakingNMn= 5×1020cm−3withJpp\nex=−0.5 eV and −1.0 eV\nin sample D. The dots are from ferromagnetic resonance mea-\nsurement (Ref.18) for [001] (brown solid upper triangles) a nd\n[110] (orange solid circles) dc magnetic-field orientation s.\nSince the effective Mn concentration of sample D is\nunavailable as mentioned above, we here take NMn=\n5×1020cm−3. The results are plotted in Fig.6. It is\nseen that the Gilbert damping coefficients from our cal-\nculation with Jpp\nex=−1 eV agree with the experiment\nvery well. As reported, the damping coefficient in this\nsample is much larger ( ∼0.1) before annealing.18The\nlarge Gilbert damping coefficient in the as-grown sample\nmay result from the direct spin-flip scattering between\nthe holes and the random Mn spins, existing in low qual-\nity samples. In the presence of this additional spin-flip\nchannel, the hole spin relaxation time becomes shorter\nand results in an enhancement of αandβ(forβ <1).\nMoreover, in the low temperature regime, a decrease of\nthe Gilbert damping coefficient was observed by increas-\ning temperature,18which is absent in our results. This\nmay originate from the complicated localization or cor-\nrelation effects in such a disordered situation. The quan-\ntitatively microscopic study in this case is beyond the\nscope of the present work.\nIn addition, one notices that βin Ref.24 was deter-\nmined to be around 0.01, which is one order of magni-\ntude smaller than our result. The reason is because of\nthe incorrectparameterused in that work, aspointed outby Adam et al..23\n 0 0.2 0.4 0.6 0.8 1\n 0  20  40  60  80  100  120  140Ass(v) (pJ/m)\nT (K)Nh=3×1020 cm-3, 1.0m0\n1×1020 cm-3, 1.0m0\n1×1020 cm-3, 0.5m0\nFIG. 7: (Color online) Spin stiffness (vertical spin stiffnes s)\ncoefficient as function of temperature is plotted as curves\nwith (without) symbols. The calculation is carried out with\nJpp\nex=−0.5 eV in sample E. The effective mass is taken to be\n1.0 (0.5) m0as labeled in the figure. The brown solid (from\nthe period of the domains) and open (from the hysteresis cy-\ncle) squares are the experimental data of spin stiffness from\nRef.45.\nD. Spin stiffness and vertical spin stiffness\nIn this subsection, we calculate the spin stiffness and\nvertical spin stiffness coefficients according to our previ-\nous derivation10\nAss=Nh/[4m∗(1+β2)] (15)\nand\nAv\nss=Nhβ/[4m∗(1+β2)]. (16)\nSince the effective mass m∗is a rough description for the\nanisotropic valence bands in the presence of a large Zee-\nman splitting, it is difficult to obtain the accurate value\nof the stiffness coefficients from these formulas. Nev-\nertheless, one can still estimate these coefficients with\nthe effective mass taken as a parameter. The results are\nplotted in Fig.7. By fitting the DOS of the occupied hole\nstates,wefind m∗≈m0, whichisconsistentwiththepre-\nvious work.31The spin stiffness and vertical spin stiffness\ncoefficients with Nh= 3×1020cm−3(1×1020cm−3) are\nplotted as the red solid (blue dashed) curves with and\nwithout symbols, respectively. The sudden decrease of\nAssoriginates from the increase of βin the vicinity of\nthe Curie temperature (see Fig.5). Our results are com-\nparable with the previous theoretical work from 6-band\nmodel.26As a comparison, we take m∗= 0.5m0, which\nis widely used to describe the heavy hole in the low en-\nergy regime in the absence of the Zeeman splitting.51\nThe spin stiffness becomes two times larger. Moreover,9\nAv\nssis found to present a nonmonotonic behavior in the\ntemperature dependence as predicted by Eq.(16).\nIn Fig.7, we also plot the experimental data of the\nspin stiffness coefficient from Ref.45. It is seen that these\nvalues of Assare comparable with our results and show\na decrease as the temperature increases. However, one\nnotices that the experimental data is more sensitive to\nthe temperature especially for those determined from the\ndomain period in the low temperature regime. This may\noriginate from the strong anisotropic interband mixing\nand inhomogeneity in the real material.\nIn Ref.10, we have shown that the vertical spin stiff-\nness can lead to the magnetization rotated around the\neasy axis within the domain wall structure by ∆ ϕ=\n(/radicalbig\n1+β2−1)/βin the absence of the demagnetization\nfield. For β= 1, ∆ϕ≈0.13π, while ∆ ϕ=β/2→0 for\nβ≪1. As illustrated above, βis always larger than 0.1.\nTherefore, the vertical spin stiffness can present observ-\nable modification of the domain wall structure in GaM-\nnAs system.10\nIV. SUMMARY\nIn summary, we theoretically investigate the tempera-\nture dependence of the LLG coefficients in ferromagnetic\nGaMnAs, based on the microscopic calculation of the\nhole spin relaxation time. In our calculation, we employ\nthe Zener model with the band structure carried out by\ndiagonalizing the 8 ×8 Kane Hamiltonian together with\nthe Zeeman energy due to the sp-dexchange interaction.\nWe find that the hole spin relaxation time can present\ndifferent temperature dependences, depending on the ef-fective Mn concentration, hole density and exchangecou-\npling constant. In the case with high Mn concentra-\ntion and large exchange coupling constant, the hole spin\nrelaxation time can be nonmonotonically dependent on\ntemperature, resulting from the different interband spin\nmixings in the large and small Zeeman splitting regimes.\nThese features are proposed to be for the estimation of\nthe exchange coupling constant or itinerant hole density.\nBysubstituting the hole relaxationtime, we calculatethe\ntemperature dependence of the Gilbert damping, nona-\ndiabatic spin torque, spin stiffness, and vertical spin stiff-\nness coefficients. We obtain the nonadiabatic spin torque\ncoefficient around 0 .1∼0.3 at low temperature, which\nis consistent with the experiment. As the temperature\nincreases, this coefficient shows a monotonic increase. In\nthe low temperature regime, the Gilbert damping co-\nefficient increases with temperature, which shows good\nagreement with the experiments. We predict that the\nGilbert damping coefficient can decrease with increasing\ntemperatureoncethenonadiabaticspintorquecoefficient\nexceed one in the vicinity of the Curie temperature. 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R¨ uhle, and K. Ploog, Phys. Rev. B 38,\n1947 (1988)."    },    {        "title": "0807.2901v1.Current_induced_dynamics_of_spiral_magnet.pdf",        "content": "arXiv:0807.2901v1  [cond-mat.str-el]  18 Jul 2008Current-induced dynamics of spiral magnet\nKohei Goto,1,∗Hosho Katsura,1,†and Naoto Nagaosa1,2,‡\n1Department of Applied Physics, The University of Tokyo,\n7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan\n2Cross-Correlated Materials Research Group (CMRG), ASI, RI KEN, Wako 351-0198, Japan\nWe study the dynamics of the spiral magnet under the charge cu rrent by solving the Landau-\nLifshitz-Gilbert equation numerically. In the steady stat e, the current /vectorjinduces (i) the parallel shift\nof the spiral pattern with velocity v= (β/α)j(α,β: the Gilbert damping coefficients), (ii) the\nuniform magnetization Mparallel or anti-parallel to the current depending on the ch irality of the\nspiral and the ratio β/α, and (iii) the change in the wavenumber kof the spiral. These are ana-\nlyzed by the continuum effective theory using the scaling arg ument, and the various nonequilibrium\nphenomena such as the chaotic behavior and current-induced annealing are also discussed.\nPACS numbers: 72.25.Ba, 71.70.Ej, 71.20.Be, 72.15.Gd\nThe current-induced dynamics of the magnetic struc-\ntureisattractingintensiveinterestsfromtheviewpointof\nthe spintronics. A representative example is the current-\ndriven motion of the magnetic domain wall (DW) in fer-\nromagnets [1, 2]. This phenomenon can be understood\nfrom the conservation of the spin angular momentum,\ni.e., spin torque transfer mechanism [3, 4, 5, 6, 7]. The\nmemorydevicesusing this current-inducedmagneticDW\nmotion is now seriously considered [8]. Another example\nis the motion of the vortex structure on the disk of a\nferromagnet, where the circulating motion of the vortex\ncore is sometimes accompanied with the inversion of the\nmagnetization at the core perpendicular to the disk [9].\nTherefore, the dynamics of the magnetic structure in-\nduced by the current is an important and fundamental\nissue universal in the metallic magnetic systems. On the\notherhand, thereareseveralmetallicspiralmagnetswith\nthe frustrated exchange interactions such as Ho metal\n[10, 11], and with the Dzyaloshinskii-Moriya(DM) inter-\naction suchas MnSi [12, 13, 14], (Fe,Co)Si [15], and FeGe\n[16]. Thequantumdisorderingunderpressureorthenon-\ntrivial magnetic textures have been discussed for the lat-\nter class of materials. An important feature is that the\ndirection of the wavevector is one of the degrees of free-\ndom in addition to the phase of the screw spins. Also the\nnon-collinear nature of the spin configuration make it an\ninteresting arena for the study of Berry phase effect [17],\nwhichappearsmostclearlyinthecouplingtothecurrent.\nHowever, the studies on the current-induced dynamics of\nthe magnetic structures with finite wavenumber, e.g., an-\ntiferromagnet and spiral magnet, are rather limited com-\npared with those on the ferromagnetic materials. One\nreason is that the observation of the magnetic DW has\nbeen difficult in the case of antiferromagnets or spiral\nmagnets. Recently, the direct space-time observation of\nthe spiral structure by Lorentz microscope becomes pos-\nsible for the DM induced spiralmagnets [15, 16] since the\nwavelength of the spiral is long ( ∼100nm). Therefore,\nthe current-induced dynamics of spiral magnets is now\nan interesting problem of experimental relevance.In this paper, we study the current-induced dynamics\nof the spiral magnet with the DM interaction as an ex-\nplicit example. One may consider that the spiral magnet\ncan be regarded as the periodic array of the DW’s in fer-\nromagnet,butithasmanynontrivialfeaturesunexpected\nfrom this naive picture as shown below.\nThe Hamiltonian we consider is given by [18]\nH=/integraldisplay\nd/vector r/bracketleftBigJ\n2(/vector∇/vectorS)2+γ/vectorS·(/vector∇×/vectorS)/bracketrightBig\n,(1)\nwhereJ >0 is the exchange coupling constant and γis\nthe strength of the DM interaction. The ground state of\nHis realized when /vectorS(/vector r) is a proper screw state such that\n/vectorS(/vector r) =S(/vector n1cos/vectork·/vector r+/vector n2sin/vectork·/vector r), (2)\nwhere the wavenumber /vectork=/vector n3|γ|/J, and/vector ni(i= 1,2,3)\nform the orthonormal vector sets. The ground state en-\nergy is given by −VS2γ2/2JwhereVis the volume of\nthe system. The sign of γis equal to that of( /vector n1×/vector n2)·/vector n3,\ndetermining the chirality of the spiral.\nThe equation of motion of the spin under the current\nis written as\n˙/vectorS=gµB\n¯h/vectorBeff×/vectorS−a3\n2eS(/vectorj·/vector∇)/vectorS+a3\n2eSβ/vectorS×(/vectorj·/vector∇)/vectorS+α\nS/vectorS×˙/vectorS\n(3)\nwhere/vectorBeff=−δH/δ/vectorSis the effective magnetic field and\nα,βare the Gilbert damping constants introduced phe-\nnomenologically [19, 20].\nWe discretize the Hamiltonian Eq.(1) and the equation\nof motion Eq.(3) by putting spins on the chain or the\nsquare lattice with the lattice constant a, and replacing\nthe derivative by the difference. The length of the spin\n|/vectorSi|is a constant of motion at each site i, and we can\neasily derive ˙H=δH\nδ/vectorS·˙/vectorS=−α|˙/vectorS|2from Eq.(3), i.e., the\nenergy continues to decrease as the time evolution.\nWe start with the one-dimensional case along x-axis\nas shown in Fig.1. The discretization means replacing\n∂x/vectorS(x) by (/vectorSi+1−/vectorSi−1)/2a, and∂2\nx/vectorS(x) by (/vectorSi+1−2/vectorSi+2\n/vectorSi−1)/a2. We note that the wavenumber which mini-\nmizes Eq.(1) is k=k0= arcsin( γ/J) on the discretized\none-dimensional lattice. Numerical study of Eq.(3) have\nbeen done with gµB/¯h= 1, 2e= 1,S= 1,a= 1\nJ= 2, and γ= 1.2. In this condition, the wavelength\nof the spiral λ= 2π/k0≈11.6 is long compared with\nthe lattice constant a= 1, and we choose the time scale\n∆t/(1 +α2) = 10−2. We have confirmed that the re-\nsults do not depend on ∆ teven if it is reduced by the\nfactor 10−1or 10−2. The sample size Lis 104with the\nopen boundary condition. As we will show later, the\ntypical value of the current is j∼2γand in the real\nsituation with the wavelength λ[nm], the exchange cou-\npling constant J[eV] and the lattice constant a[nm], it is\nj≈3.2×1015J/(λa)[A/m2]. Substituting J= 0.02,\nλ= 100,a= 0.5 into above estimatation, the typi-\ncal current is 1012[A/m2], and the unit of the time is\n∆t=J/¯h≈30[ps].\nThe Gilbert damping coefficients α,βare typically\n10−3∼10−1in the realistic systems. In most of the cal-\nculations, however,wetake α= 5.0toaccelaratethecon-\nvergence to the steady state. The obtained steady state\ndepends only on the ratio β/αexcept the spin configura-\ntions near the boundaries as confirmed by the simlations\nwithα= 0.1. We employ the two types of initial condi-\ntion, i.e., the ideal proper screw state with the wavenum-\nberk0, and the random spin configurations. The differ-\nence of the dynamics in these two cases are limited only\nin the early stage ( t <5000∆t).\nNow we consider the steady state with the constant\nvelocity for the shift of the spiral pattern obtained after\nthe time of the order of 105∆t. One important issue here\nis the current-dependence of the velocity, which has been\ndiscussed intensivelyfor the DW motion in ferromagnets.\nIn the latter case, there appears the intrinsic pinning in\nthe case of β= 0 [4], while the highly nonlinear behavior\nforβ/α/negationslash= 0 [20]. In the special case of β=α, the trivial\nsolution corresponding to the parallel shift of the ground\nstate configuration of Eq.(1) with the velocity v=jis\nconsidered to be realized [5]. Figure 2 shows the results\nfor the velocity, the induced uniform magnetization Sx\nalongx-axis, and the wavevector kof the spiral in the\nsteady state. The current-dependence of the velocity for\nthe cases of β= 0.1,0.5α,αand 2αis shown in Fig.\n2(a). Figure 2(b) shows the β/α-dependence of the ve-\nlocity for the fixed current j= 1.2. It is seen that the\nvelocity is almost proportional to both the current jand\nthe ratio β/α. Therefore, we conclude that the velocity\nv= (β/α)jwithout nonlinear behavior up to the current\nj∼2γ, which is in sharp contrast to the case of the DW\nmotion in ferromagnets. The unit of the velocity is given\nbya/∆t, which is of the order of 20[m /s] fora≈5[˚A]\nand ∆t≈30[ps]. In Fig. 2(c) shown the wavevector k\nof the spiral under the current j= 1.2 for various values\nofβ/α. It shows a non-monotonous behavior with the\nmaximum at β/α≈0.2, and is always smaller than the\nFIG. 1: Spin configurations in the spiral magnet (a) in\nthe equilibrium state, and (b) under the current. Under\nthe current /vectorj, the uniform magnetization Sxalong the spi-\nral axis/current direction is induced together with the ro-\ntation of the spin, i.e., the parallel shift of the spiral pat -\ntern with the velocity v. Note that the magnetization\nis anti-parallel/parallel to the current direction with po si-\ntive/negative γforβ < α, while it is reversed for β > α,\nand the wavenumber kchanges from the equilibrium value.\n 0  0.5  1  1.5  2  2.5  3  3.5  4 \n 0  0.5  1  1.5  2  2.5v\njβ/α=2.0 \nβ/α=1.0 \nβ/α=0.5 \nβ/α=0.1 (a) \n 0  0.5  1  1.5  2 \n 0  0.5  1  1.5  2 v / j \nβ / α(b) \n 0  0.2  0.4  0.6  0.8  1 \n 0  0.5  1  1.5  2 k\nβ / α(c) \n-1 -0.5  0  0.5  1 \n 0  0.5  1  1.5  2 Sx\nβ / α(d) \nFIG. 2: For the case of γ= 1.2, the numerical result is shown.\n(a) The steady state velocity vas a function of the current\njwith the fixed values of β= 0.1,0.5α,α, and 2α. (b) The\nvelocityvas a function of β/αfor a fixed value of the current\nj= 1.2.visalmost proportional to β/α. (c)Thewavenumber\nkas a function of β/α. The dotted line shows k0in the\nequilibrium. (d) The uniform magnetization Sxalong the\ncurrent direction as a function of β/αfor a fixed value of\nj= 1.2.\nwavenumber k0in the equilibrium shown in the dotted\nline. Namely, the period of the spiral is elongated by the\ncurrent. As shown in Fig. 2(d), there appears the uni-\nform magnetization Sxalong the x-direction. Sxis zero\nand changes the sign at β/α= 1. With the positive γ\n(as in the case of Fig. 2(d)), Sxis anti-parallel to the\ncurrentj//xwithβ < αand changes its direction for\nβ > α. For the negative γ, the sign of Sxis reversed. As\nfor the velocity /vector v, on the other hand, it is always parallel\nto the current /vectorj.3\nNow we present the analysis of the above results in\nterms of the continuum theory and a scaling argument.\nFor one-dimensional case, the modified LLG Eq.(3) can\nbe recast in the following form:\n˙/vectorS=−J/vectorS×∂2\nx/vectorS−(2γSx+j)∂x/vectorS+/vectorS×(α˙/vectorS+βj∂x/vectorS).(4)\nIt is convenientto introduceamovingcoordinates ˆξ(x,t),\nˆη, andˆζ(x,t) (see Fig.1) [21]. They are explicitly defined\nthrough ˆ x, ˆyand ˆzas\nˆζ(x,t) = cos( k(x−vt)+φ)ˆy+sin(k(x−vt)+φ)ˆz,\nˆξ(x,t) =−sin(k(x−vt)+φ)ˆy+cos(k(x−vt)+φ)ˆz,\nand ˆη= ˆx. We restrict ourselves to the following ansatz:\n/vectorS(x,t) =Sxˆη+/radicalbig\n1−S2xˆζ(x,t), (5)\nwhereSxis assumed to be constant.\nBy substituting Eq.(5) into Eq.(4), we obtain vas\nv=β\nαj, (6)\nby requiring that there is no force along ˆ η- andˆζ-\ndirections acting on each spin. In contrast to the DW\nmotioninferromagnets,thevelocity vbecomeszerowhen\nβ→0 even for large value of the current. The numerical\nresults in Fig. 2(a), (b) show good agreement with this\nprediction Eq.(6).\nOn the other hand, the magnetization Sxalongx-axis\nis given by\nSx=β/α−1\n2γ−Jkj, (7)\nonce the wavevector kis known. Here we note that the\nabove solution is degenerate with respect to k, which\nneeds to be determined by the numerical solution. From\nthe dimensional analysis, the spiral wavenumber kis\ngiven by the scaling form, k=k0g(j/(2γ),β/α) with\nthe dimensionless function g(x,y) and also is Sxthrough\nEq.(7).\nMotivated by the analysis above, we study the γ-\ndependence of the steady state properties. In Fig.3,\nwe show the numerical results for k/k0andSxas the\nfunctions of j/2γin the cases of β/α= 0.1,0.5 and\n2. Roughly speaking, the degeneracies of the data are\nobtained approximately for each color points (the same\nβ/αvalue) with different γvalues. The deviation from\nthe scaling behavior is due to the discrete nature of the\nlattice model, which is relevant to the realistic situation.\nForβ/α= 0.1(black points in Fig.3), kremainsconstant\nandSxis induced almost proportional to the current up\nj/2γ≈0.4, where the abrupt change of koccurs. For\nβ/α= 0.5 (blue points) and β/α= 2.0 (red points), the\nchanges of kandSxare more smooth. A remarkable\nresult is that the spin Sion the lattice point iis well 0  0.2  0.4  0.6  0.8  1  1.2  1.4  1.6 \n 0  0.2  0.4  0.6  0.8  1  1.2  1.4  1.6 k / k 0\nj / 2γβ/α=2.0, γ=1.2 \nβ/α=2.0, γ=1.0 \nβ/α=2.0, γ=0.8 \nβ/α=2.0, γ=0.6 \nβ/α=0.5, γ=1.2 \nβ/α=0.5, γ=1.0 \nβ/α=0.5, γ=0.8 \nβ/α=0.5, γ=0.6 \nβ/α=0.1, γ=1.2 \nβ/α=0.1, γ=1.0 \nβ/α=0.1, γ=0.8 \nβ/α=0.1, γ=0.6 (a) \n-1 -0.8 -0.6 -0.4 -0.2  0  0.2  0.4  0.6  0.8  1 \n 0  0.2  0.4  0.6  0.8  1  1.2  1.4  1.6 Sx \nj / 2γβ/α=2.0, γ=1.2 \nβ/α=2.0, γ=1.0 \nβ/α=2.0, γ=0.8 \nβ/α=2.0, γ=0.6 \nβ/α=0.5, γ=1.2 \nβ/α=0.5, γ=1.0 \nβ/α=0.5, γ=0.8 \nβ/α=0.5, γ=0.6 \nβ/α=0.1, γ=1.2 \nβ/α=0.1, γ=1.0 \nβ/α=0.1, γ=0.8 \nβ/α=0.1, γ=0.6 (b) \nFIG.3: Thescalingplotfor (a) k/k0wherek0isthewavenum-\nber in the equilibrium without the current, and (b) Sxin the\nsteady state as the function of j/2γ. The black, blue, and\nred color points correspond to β/α= 0.1, 0.5 and 2.0, respec-\ntively. The curves in (b) indicate Eq.(7) calculated from th e\nkvalues in (a), showing the good agreement with the data\npoints.\ndescribed by Eq.(5) at x=xi, and hence the relation\nEq.(7) is well satisfied as shown by the curves in Fig.\n3(b), even though the scaling relation is violated to some\nextent. For larger values of jbeyond the data points,\ni.e.,j/2γ >0.75 forβ/α= 0.1,j/2γ >1.5 forβ/α= 0.5\nandj/2γ >0.9 forβ/α= 2.0, the spin configuration is\ndisordered from harmonic spiral characterized by a sin-\ngle wavenumber k. The spins are the chaotic funtion of\nboth space and time in this state analogousto the turbu-\nlance. This instability is triggered by the saturated spin\nSx=±1, occuring near the edge of the sample.\nNext, we turn to the simulations on the two-\ndimensional square lattice in the xy-plane. In this case,\nthe direction of the spiral wavevector becomes another\nimportant variable because the degeneracy of the ground\nstate energy occurs.\nStarting with the random spin configuration, we sim-\nulate the time evolution of the system without and with\nthe current as shown in Fig.4. Calculation has been done\nwith the same parameters as in the one-dimensional case4\n(a)/vectorj= 0\n(b)/vectorj= (0.3,0)\n(c)/vectorj= (0.3/√\n2,0.3/√\n2)\ncolor box of Sz(r)\ncolor box of |Sz(k)|2\nFIG. 4: The time evolution of the zcomponent Szof the\nspin from the random initial configuration of the 102×102\nsection in the middle of the sample is shown in the case of\n(a)j= 0, (b) j= 0.3 along the x-axis, (c) j= 0.3 along\nthe (1,1)-direction. From the left, t= 102∆t, 1.7×103∆t,\n5×103∆t. The rightmost panels show the spectral intensity\n|Sz(/vectork,5×103∆t)|2from the whole sample of the size 210×210\nin the momentum space /vectork= (kx,ky).\nwhereγ= 1.2,β= 0, and the system size is 210×210.\nIn the absence of the current, the relaxation of the\nspins into the spiral state is very slow, and many dislo-\ncations remain even after a long time. Correspondingly,\nthe energy does not decrease to the ground state value\nbut approaches to the higher value with the power-law\nlike long-time tail. The momentum-resolved intensity is\ncircularly distributed with the broad width as shown in\nFig. 4(a) corresponding to the disordered direction of\n/vectork. This glassy behavior is distinct from the relaxation\ndynamics of the ferromagnet where the large domain for-\nmation occurs even though the DW’s remain. Now we\nput the current along the ˆ x(Fig. 4(b)) and (ˆ x+ˆy) (Fig.\n4(c)) directions. It is seen that the direction of /vectorkis con-\ntrolled by the current also with the radial distribution\nin the momentum space being narrower than that in the\nabsence of j(Fig. 4(a)). This result suggests that the\ncurrentjwith the density ∼1012[A/m2] of the time du-\nration∼0.1[µsec] can anneal the directional disorder of\nthe spiral magnet. After the alignment of /vectorkis achieved,\nthe simulations on the one-dimensional model described\nabove are relevant to the long-time behavior.To summarize, we have studied the dynamics of the\nspiral magnet with DM interaction under the current j\nby solving the Landau-Lifshitz-Gilbert equation numeri-\ncally. In the steady state under the charge current j, the\nvelocityvis given by ( β/α)j(α,β: the Gilbert-damping\ncoefficients), the uniform magnetization is induced par-\nallel or anti-parallel to the current direction, and period\nof the spiral is elongated. The annealing effect especially\non the direction of the spiral wavevector /vectorkis also demon-\nstrated.\nTheauthorsaregratefultoN.FurukawaandY.Tokura\nfor fruitful discussions. This work was supported in part\nby Grant-in-Aids (Grant No. 15104006, No. 16076205,\nand No. 17105002) and NAREGI Nanoscience Project\nfromtheMinistryofEducation, Culture, Sports, Science,\nandTechnology. HK wassupported bythe JapanSociety\nfor the Promotion of Science.\n∗Electronic address: goto@appi.t.u-tokyo.ac.jp\n†Electronic address: katsura@appi.t.u-tokyo.ac.jp\n‡Electronic address: nagaosa@appi.t.u-tokyo.ac.jp\n[1] L. Berger, J. Appl. Phys. 49, 2156 (1978).\n[2] L. Berger, Phys. Rev. B 33, 1572 (1986).\n[3] J. C. Slonczewski, Int. J. Magn. 2, 85 (1972).\n[4] G. Tatara and H. Kohno, Phys. Rev. Lett. 92,\n086601(2004).\n[5] E. Barnes and S. Maekawa, Phys. Rev. Lett. 95, 107204\n(2005).\n[6] A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu,\nand T. Shinjo, Phys. Rev. Lett. 92, 077205 (2004).\n[7] M. Yamanouchi, D. Chiba, F. Matsukura, and H. Ohno,\nNature428,539 (2004).\n[8] M. Hayashi, L. Thomas, C. Rettner, R. Moriya, and S.\nS. P. Parkin, Nat. Phys. 3, 21 (2007).\n[9] K. Yamada et al., Nat. Mat. 6, 269 (2007).\n[10] W. C. Koehler, J. Appl. Phys 36, 1078 (1965).\n[11] R. A. Cowley et al., Phys. Rev. B 57, 8394 (1998).\n[12] M. Bode, M. Heide, K. von Bergmann, P. Ferriani, S.\nHeinze, G. Bihlmayer, A. Kubetzka, O. Pietzsch, S.\nBlugel, and R. Wiesendanger, Nature (London) 447, 190\n(2007).\n[13] C. Pfleiderer, S. R. Julian, and G. G. Lonzarich, Nature\n(London) 414, 427 (2001).\n[14] N. Doiron-Leyraud, I. R. Waker, L. Taillefer, M. J.\nSteiner, S. R. Julian, and G. G. Lonzarich, Nature (Lon-\ndon) 425, 595 (2003).\n[15] M. Uchida, Y. Onose, Y. Matsui, and Y. Tokura, Science\n311, 359 (2006).\n[16] M. Uchida et al., Phys. Rev. B 77, 184402 (2008).\n[17] M. V. Berry, Proc. R. Soc. London A 392, 45 (1984).\n[18] L. D. Landau, in Electrodynamics of Continuous Media\n(Pergamon Press, 1984), p178.\n[19] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n[20] A. Thiaville et al., Europhys. Lett. 69, 990 (2005).\n[21] T. Nagamiya, in Solid State Physics , edited by F. Seitz,\nD. Turnbull, and H. Ehrenreich (Academic Press, New\nYotk, 1967), Vol. 20, p. 305."    },    {        "title": "1110.3387v2.Atomistic_spin_dynamic_method_with_both_damping_and_moment_of_inertia_effects_included_from_first_principles.pdf",        "content": "Atomistic spin dynamic method with both damping and moment of inertia e\u000bects\nincluded from \frst principles\nS. Bhattacharjee, L. Nordstr om, and J. Fransson\u0003\nDepartment of Physics and Astronomy, Box 516, 75120, Uppsala University, Uppsala, Sweden\n(Dated: October 28, 2018)\nWe consider spin dynamics for implementation in an atomistic framework and we address the\nfeasibility of capturing processes in the femtosecond regime by inclusion of moment of inertia. In the\nspirit of an s-d-like interaction between the magnetization and electron spin, we derive a generalized\nequation of motion for the magnetization dynamics in the semi-classical limit, which is non-local in\nboth space and time. Using this result we retain a generalized Landau-Lifshitz-Gilbert equation,\nalso including the moment of inertia, and demonstrate how the exchange interaction, damping, and\nmoment of inertia, all can be calculated from \frst principles.\nPACS numbers: 72.25.Rb, 71.70.Ej, 75.78.-n\nIn recent years there has been a huge increase in the in-\nterest in fast magnetization processes on a femto-second\nscale, which has been initialized by important develop-\nments in experimental techniques [1{5], as well as po-\ntential technological applications [6]. From a theoretical\nside, the otherwise trustworthy spin dynamical (SD) sim-\nulation method fails to treat this fast dynamics due to\nthe short time and length scales involved. Attempts have\nbeen made to generalize the mesoscopic SD method to an\natomistic SD, in which the dynamics of each individual\natomic magnetic moment is treated [7, 8]. While this\napproach should in principle be well suited to simulate\nthe fast dynamics observed in experiments, it has not\nyet reached full predictive power as it has inherited phe-\nnomenological parameters, e.g. Gilbert damping, from\nthe mesoscopic SD. The Gilbert damping parameter is\nwell established in the latter regime but it is not totally\nclear how it should be transferred to the atomic regime.\nIn addition, very recently it was pointed out that the mo-\nment of inertia, which typically is neglected, plays an im-\nportant role for fast processes [9]. In this Letter we derive\nthe foundations for an atomistic SD where all the rele-\nvant parameters, such as the exchange coupling, Gilbert\ndamping, and moment of inertia, can be calculated from\n\frst principles electronic structure methods.\nUsually the spin dynamics is described by the phe-\nnomenological Landau-Lifshitz-Gilbert (LLG) equation\n[10, 11] which is composed of precessional and damping\nterms driving the dynamics to an equilibrium. By in-\ncluding the moment of inertia, we arrive at a generalized\nLLG equation\n_M=M\u0002(\u0000\rB+^G_M+^IM) (1)\nwhere ^Gand^Iare the Gilbert damping and the moment\nof inertia tensors, respectively. In this equation the e\u000bec-\ntive \feld Bincludes both the external and internal \felds,\nof which the latter includes the exchange coupling and\nanisotropy e\u000bects. Here, we will for convenience include\nthe anisotropy arising from the classical dipole-dipole in-\nteraction responsible for the shape anisotropy as a partof the external \feld. The damping term in the LLG\nequation usually consists of a single damping parame-\nter, which essentially means that the time scales of the\nmagnetization variables and the environmental variables\narewell separated . This separation naturally brings a\nlimitation to the LLG equation concerning its time scale\nwhich is restricting it to the mesoscopic regime.\nThe addition of a moment of inertia term to the LLG\nequation can be justi\fes as follows. A general process\nof a moment Munder the in\ruence of a \feld Fis al-\nways endowed with inertial e\u000bects at higher frequencies\n[12]. The \feld Fand moment Mcan, for example, be\nstress and strain for mechanical relaxation, electric \feld\nand electric dipole moment in the case of dielectric re-\nlaxation, or magnetic \feld and magnetic moment in the\ncase of magnetic relaxation. In this Letter we focus on\nthe latter case | the origin of the moment of inertia in\nSD. The moment of inertia leads to nutations of the mag-\nnetic moments, see Fig. 1. Its wobbling variation of the\nazimuthal angle has a crucial role in fast SD, such as fast\nmagnetization reversal processes.\nIn the case of dielectric relaxation the inertial e\u000bects\nare quite thoroughly mentioned in the literature [13, 14],\nespecially in the case of ferroelectric relaxors. Co\u000bey et\nal.[14] have proposed inertia corrected Debye's theory of\ndielectric relaxation and showed that by including inertial\nB\nprecessionĜ\ndampingÎ\nnutation\nFIG. 1: The three contributions in Eq. (1), the bare preces-\nsion arising from the e\u000bective magnetic \feld, and the super-\nimposed e\u000bects from the Gilbert damping and the moment of\ninertia, respectively.arXiv:1110.3387v2  [cond-mat.stat-mech]  1 Feb 20122\ne\u000bects, the unphysical high frequency divergence of the\nabsorption co-e\u000ecient is removed.\nVery recently Ciornei et al [9] have extended the LLG\nequation to include the inertial e\u000bects through a mag-\nnetic retardation term in addition to precessional and\ndamping terms. They considered a collection of uni-\nformly magnetized particles and treated the total angular\nmomentum Las faster variable. They obtained Eq. (1)\nfrom a Fokker-Plank equation where the number den-\nsity of magnetized particles were calculated by integrat-\ning a non-equilibrium distribution function over faster\nvariables such that faster degrees of freedom appear as\nparameter in the calculation.\nThe authors showed that at very short time scales the\ninertial e\u000bects become important as the precessional mo-\ntion of magnetic moment gets superimposed with nuta-\ntion loops due to inertial e\u000bects. It is pointed out that\nthe existence of inertia driven magnetization dynamics\nopen up a pathway for ultrafast magnetic switching [15]\nbeyond the limitation [16] of the precessional switching.\nIn practice, to perform atomistic spin dynamics simu-\nlations the knowledge of ^Gand^Iis necessary. There are\nrecent proposals [17, 18] of how to calculate the Gilbert\ndamping factor from \frst principles in terms of Kubo-\nGreenwood like formulas. Here, we show that similar\ntechniques may by employed to calculate the moment of\ninertia tensor ^I. Finally, we present a microscopical jus-\nti\fcation of Eq. (1), considering a collective magnetiza-\ntion density interacting locally with electrons constitut-\ning spin moments. Such a description would in principle\nbe consistent with the study of magnetization dynam-\nics where the exchange parameters are extracted from\n\frst-principles electronic structure calculations, e.g den-\nsity functional theory (DFT) methods. We \fnd that in\nan atomistic limit Eq. (1) actually has to be general-\nized slightly as both the damping and inertia tensors are\nnaturally non-local in the same way as the exchange cou-\npling included in the e\u000bective magnetic \feld B. From\nour study it is clear that both the damping and the mo-\nment of inertia e\u000bects naturally arise from the retarded\nexchange interaction.\nWe begin by considering the magnetic energy E=M\u0001\nB. Using that its time derivative is _E=M\u0001_B+_M\u0001B\nalong with Eq. (1), we write\n_E=M\u0001_B+1\n\r_M\u0001\u0010\n^G_M+^IM\u0011\n: (2)\nRelating the rate of change of the total energy to the\nHamiltonianH, through _E=hdH=dti, and expanding\ntheHlinearly around its static magnetization M0, with\nM(t) =M0+\u0016(t), we can writeH\u0019H 0+\u0016(t)\u0001r\u0016H0,\nwhereH0=H(M0). Then the rate of change of the total\nenergy equals _E= _\u0016\u0001hr\u0016Hito the \frst order. Following\nRef. [19] and assuming su\u000eciently slow dynamics such\nthat\u0016(t0) =\u0016(t)\u0000\u001c_\u0016(t) +\u001c2\u0016(t)=2,\u001c=t\u0000t0, we canwrite the rate of change of the magnetic energy as\n_E= lim\n!!0_\u0016i[\u001fij(!)\u0016j+i@!\u001fij(!) _\u0016j\u0000@2\n!\u001fij(!)\u0016j=2]:\n(3)\nHere,\u001fij(!) =R\n(\u0000i)\u0012(\u001c)h[@iH0(t);@jH0(t0)]iei!\u001cdt0,\n\u001c=t\u0000t0, is the (generalized) exchange interaction\ntensor out of which the damping and moments of in-\nertia can be extracted. Summation over repeated in-\ndices (i;j=x;y;z ) is assumed in Eq. (3). Equat-\ning Eqs. (2) and (3) results in an internal contribu-\ntion to the e\u000bective \feld about which the magnetiza-\ntion precesses Bint=\u0016lim!!0\u001f(!), the damping term\n^G=\rlim!!0i@!\u001f(!) as well as the moment of inertia\n^I=\u0000\rlim!!0@2\n!\u001f(!)=2.\nFor a simple order of magnitude estimate of the damp-\ning and inertial coe\u000ecients, ^Gand^I, respectively, we\nmay assume for a state close to a ferromagnetic state\nthat the spin resolved density of electron states \u001a\u001b(\")\ncorresponding to the static magnetization con\fguration\nH0is slowly varying with energy. At low temperatures\nwe, then, \fnd\n^G\u00182\r\u0019sp [h@iH0i\u001ah@jH0i\u001a]\"=\"F; (4)\nin agreement with previous results [19]. Here, sp denotes\nthe trace over spin 1/2 space. By the same token, the\nmoment of inertia is estimated as\n^I\u0018\u0000(\r=D) sp [h@iH0i\u001ah@jH0i\u001a]\"=\"F; (5)\nwhere 2Dis the band width of density of electron states\nof the host material. Typically, for metallic systems the\nband width 2 D\u00181|10 eV, which sets the time-scale\nof the inertial contribution to the femto second (10\u000015\ns) regime. It, therefore, de\fnes magnetization dynamics\non a time-scale that is one or more orders of magnitude\nshorter compared to e.g. the precessional dynamics of the\nmagnetic moment.\nNext, we consider the physics leading to the LLG equa-\ntion given in Eq. (1). As there is hardly any microscopical\nderivation of the LLG equation in the literature, we in-\nclude here, for completeness the arguments that leads to\nthe equation for the spin-dynamics from a quantum \feld\ntheory perspective.\nIn the atomic limit the spin degrees of freedom are\ndeeply intertwined with the electronic degrees of free-\ndom, and hence the main environmental coupling is the\none to the electrons. In this study we are mainly con-\ncerned with a mean \feld description of the electron\nstructure, as in the spirit of the DFT. Then a natural\nand quite general description of the magnetic interac-\ntion due to electron-electron interactions on the atomic\nsite around rwithin the material is captured by the s-d-\nlike modelHint=\u0000R\nJ(r;r0)M(r;t)\u0001s(r0;t)drdr0, where\nJ(r;r0) represents the interaction between the magneti-\nzation density Mand the electron spin s. From a DFT3\nperspective the interaction parameter J(r;r0) is related\nto the e\u000bective spin dependent exchange-correlation func-\ntionalBxc[M(r0)](r). For generality we assume a fully\nrelativistic treatment of the electrons, i.e. including the\nspin-orbit coupling. In this interaction the dichotomy of\nthe electrons is displayed, they both form the magnetic\nmoments and provide the interaction among them.\nOwing to the general non-equilibrium conditions in the\nsystem, we de\fne the action variable\nS=I\nCHintdt+SZ+SWZWN (6)\non the Keldysh contour [20{22]. Here, the ac-\ntionSZ=\u0000\rH\nCR\nBext(r;t)\u0001M(r;t)dtdrrepresents\nthe Zeeman coupling to the external \feld Bext(r;t),\nwhereas the Wess-Zumino-Witten-Novikov (WZWN)\ntermSWZWN =RH\nCR1\n0M(r;t;\u001c)\u0001[@\u001cM(r;t;\u001c)\u0002\n@tM(r;t;\u001c)]d\u001cdtjM(r)j\u00002drdescribes the Berry phase\naccumulated by the magnetization.\nIn order to acquire an e\u000bective model for the magne-\ntization density M(r;t), we make a second order [23] ex-\npansion of the partition function Z[M(r;t)]\u0011trTCeiS,\nand take the partial trace over the electronic degrees of\nfreedom in the action variable. The e\u000bective action \u000eSM\nfor the magnetization dynamics arising from the mag-\nnetic interactions described in terms of Hint, can, thus,\nbe written\n\u000eSM=\u0000I Z\nM(r;t)\u0001D(r;r0;t;t0)\u0001M(r0;t0)drdr0dtdt0;\n(7)\nwhereD(r;r0;t;t0) =R\nJ(r;r1)(\u0000i)hTs(r1;t)s(r2;t0)i\u0002\nJ(r2;r0)dr1dr2is a dyadic which describes the electron\nmediated exchange interaction.\nConversion of the Keldysh contour integrations into\nreal time integrals on the interval ( \u00001;1) results in\nS=Z\nM(fast)(r;t)\u0001[M(r;t)\u0002_M(r;t)]dtjM(r)j\u00002dr\n+Z\nM(fast)(r;t)\u0001Dr(r;r0;t;t0)\u0001M(r0;t0)drdr0dtdt0\n\u0000\rZ\nBext(r;t)\u0001M(fast)(r;t)dtdr; (8)\nwithM(fast)(r;t) =Mu(r;t)\u0000Ml(r;t) and M(r;t) =\n[Mu(r;t) +Ml(r;t)]=2 which de\fne fast and slow vari-\nables, respectively. Here, Mu(l)is the magnetization den-\nsity de\fned on the upper (lower) branch of the Keldysh\ncontour. Notice that upon conversion into the real time\ndomain, the contour ordered propagator Dis replaced by\nits retarded counterpart Dr.\nWe obtain the equation of motion for the (slow) mag-\nnetization variable M(r;t) in the classical limit by mini-\nmizing the action with respect to M(fast)(r;t), cross mul-\ntiplying by M(r;t) under the assumption that the totalmoment is kept constant. We, thus, \fnd\n_M(r;t) =M(r;t)\u0002\u0012\n\u0000\rBext(r;t)\n+Z\nDr(r;r0;t;t0)\u0001M(r0;t0)dt0dr0\u0013\n:(9)\nEq. (9) provides a generalized description of the semi-\nclassical magnetization dynamics compared to the LLG\nEq. (1) in the sense that it is non-local in both time and\nspace. The dynamics of the magnetization at some point\nrdepends not only on the magnetization locally at r,\nbut also in a non-trivial way on the surrounding magne-\ntization. The coupling of the magnetization at di\u000berent\npositions in space is mediated via the electrons in the\nhost material. Moreover, the magnetization dynamics is,\nin general, a truly non-adiabatic process in which the\ninformation about the past is crucial.\nHowever, in order to make connection to the magne-\ntization dynamics as described by e.g. the LLG equa-\ntion as well as Eq. (1) above, we make the following\nconsideration. Assuming that the magnetization dy-\nnamics is slow compared to the electronic processes in-\nvolved in the time-non-local \feld D(r;r0;t;t0), we ex-\npand the magnetization in time according to M(r0;t0)\u0019\nM(r0;t)\u0000\u001c_M(r0;t) +\u001c2M(r0;t)=2. Then for the inte-\ngrand in Eq. (9), we get\nDr(r;r0;t;t0)\u0001M(r0;t0) =\nDr(r;r0;t;t0)\u0001[M(r0;t)\u0000\u001c_M(r0;t) +\u001c2\n2M(r0;t)]:(10)\nHere, we observe that as the exchange coupling for the\nmagnetization is non-local and mediated through D, this\nis also true for the damping (second term) and the inertia\n(third term).\nIn order to obtain an equation of the exact same\nform as LLG in Eq. (1) we further have to assume\nthat the magnetization is close to a uniform ferromag-\nnetic state, then we can justify the approximations\n_M(r0;t)\u0019_M(r;t) and M(r0;t)\u0019M(r;t). When\nBint=\u0000R\nD(r;r0;t;t0)\u0001M(r0;t)dr0dt0=\ris included in\nthe total e\u000bective magnetic \feld B, the tensors of Eq. (1)\n^Gand^Ican be identi\fed with \u0000R\n\u001cD(r;r0;t;t0)dr0dt0\nandR\n\u001c2D(r;r0;t;t0)dr0dt0=2, respectively. From a \frst\nprinciples model of the host materials we have, thus, de-\nrived the equation for the magnetization dynamics dis-\ncussed in Ref. 9, where it was considered from purely\nclassical grounds. However it is clear that for a treatment\nof atomistic SD that allows for all kinds of magnetic or-\nders, not only ferromagnetic, Eq. (1) is not su\u000ecient and\nthe more general LLG equation of Eq. (9) together with\nEq. (10) has to be used.\nWe \fnally describe how the parameters of Eq. (1)\ncan be calculated from a \frst principles point of view.\nWithin the conditions de\fned by the DFT system, the\ninteraction tensor Dris time local which allows us to4\nwrite lim \"!0i@\"Dr(r;r0;\") =R\n\u001cDr(r;r0;t;t0)dt0and\nlim\"!0@2\n\"Dr(r;r0;\") =\u0000R\n\u001c2Dr(r;r0;t;t0)dt0, where\nDr(r;r0;\") = 4 spZ\nJr\u001aJ\u001a0r0f(!)\u0000f(!0)\n\"\u0000!+!0+i\u000e\n\u0002\u001bImGr\n\u001a0\u001a(!)\u001bImGr\n\u001a\u001a0(!0)d!\n2\u0019d!0\n2\u0019d\u001ad\u001a0:(11)\nHere,Jrr0\u0011J(r;r0) whereas Gr\nrr0(!)\u0011Gr(r;r0;!) is\nthe retarded GF, represented as a 2 \u00022-matrix in spin-\nspaces. We notice that the above result presents a general\nexpression for frequency dependent exchange interaction.\nUsing Kramers-Kr onig's relations in the limit \"!0, it\nis easy to see that Eq. (11) leads to\nDr(r;r0; 0) =\u00001\n\u0019sp ImZ\nJr\u001aJ\u001a0r0f(!)\n\u0002\u001bGr\n\u001a0\u001a(!)\u001bGr\n\u001a\u001a0(!)d!d\u001ad\u001a0;(12)\nin agreement with e.g. Ref. [24]. We can make connection\nwith previous results, e.g. Refs. 25, 26, and observe that\nEq. (11) contains the isotropic Heisenberg, anisotropic\nIsing, and Dzyaloshinsky-Moriya exchange interactions\nbetween the magnetization densities at di\u000berent points\nin space [22], as well as the onsite contribution to the\nmagnetic anisotropy.\nUsing the result in Eq. (11), we \fnd that the damping\ntensor is naturally non-local and can be reduced to\n^G(r;r0) =1\n\u0019spZ\nJr\u001aJ\u001a0r0f0(!)\n\u0002\u001bImGr\n\u001a0\u001a(!)\u001bImGr\n\u001a\u001a0(!)d!d\u001ad\u001a0;(13)\nwhich besides the non-locality is in good accordance with\nthe results in Refs. [17, 25], and is closely connected to\nthe so-called torque-torque correlation model [27]. With\ninclusion of the the spin-orbit coupling in Gr, it has been\ndemonstrated that Eq. (13) leads to a local Gilbert damp-\ning of the correct order of magnitude for the case of fer-\nromagnetic permalloys [17].\nAnother application of Kramers-Kr onig's relations\nleads, after some algebra, to the moment of inertia tensor\n^I(r;r0) = spZ\nJr\u001aJ\u001a0r0f(!)\u001b[ImGr\n\u001a0\u001a(!)\u001b@2\n!ReGr\n\u001a\u001a0(!)\n+ ImGr\n\u001a\u001a0(!)\u001b@2\n!ReGr\n\u001a0\u001a(!)]d!\n2\u0019d\u001ad\u001a0;(14)\nwhere we notice that the moment of inertia is not sim-\nply a Fermi surface e\u000bect but depends on the electronic\nstructure as a whole of the host material. Although the\nstructure of this expression is in line with the exchange\ncoupling in Eq. (12) and the damping of Eq. (13), it is\na little more cumbersome to compute due the presence\nof the derivatives of the Green's functions. Note that it\nis not possible to get completely rid of the derivatives\nthrough partial integration. These derivatives also makethe moment of inertia very sensitive to details of the elec-\ntronic structure, which has a few implications. Firstly the\nmoment of inertia can take large values for narrow band\nmagnetic materials, such as strongly correlated electron\nsystems, where these derivatives are substantial. For\nsuch systems the action of moment of inertia can be im-\nportant for longer time scales too, as indicated by Eq. (5).\nSecondly, the moment of inertia may be strongly depen-\ndent on the reference magnetic ordering for which it is\ncalculated. It is well known that already the exchange\ntensor parameters may depend on the magnetic order.\nIt is the task of future studies to determine how trans-\nferable the moment of inertia tensor as well as damping\ntensor are in-between di\u000berent magnetic ordering.\nIn conclusion, we have derived a method for atomistic\nspin dynamics which would be applicable for ultrafast\n(femtosecond) processes. Using a general s-d-like interac-\ntion between the magnetization density and electron spin,\nwe show that magnetization couples to the surrounding\nin a non-adiabatic fashion, something which will allow for\nstudies of general magnetic orders on an atomistic level,\nnot only ferromagnetic. By showing that our method\ncapture previous formulas for the exchange interaction\nand damping tensor parameter, we also derive a formula\nfor calculating the moment of inertia from \frst principles.\nIn addition our results point out that all parameters are\nnon-local as they enter naturally as bilinear sums in the\nsame fashion as the well established exchange coupling.\nOur results are straight-forward to implement in existing\natomistic SD codes, so we look on with anticipation to\nthe \frst applications of the presented theory which would\nbe fully parameter-free and hence can take a large step\ntowards simulations with predictive capacity.\nSupport from the Swedish Research Council is ac-\nknowledged. We are grateful for fruitful and encouraging\ndiscussions with A. Bergman, L. Bergqvist, O. Eriksson,\nC. Etz, B. Sanyal, and A. Taroni. J.F. also acknowledges\ndiscussions with J. -X. Zhu.\n\u0003Electronic address: Jonas.Fransson@physics.uu.se\n[1] E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot,\nPhys. Rev. Lett. 76, 4250 (1996).\n[2] G. P. Zhang and W. H ubner, Phys. Rev. Lett. 85, 3025\n(2000).\n[3] M. G. M unzenberg, Nat. Mat. 9, 184 (2010);\n[4] S. L. Johnson et al. unpublished , arXiv:1106.6128v2.\n[5] A. Kirilyuk, A. V. Kimel, and T. Rasing, Rev. Mod.\nPhys. 82, 2731 (2010).\n[6] J. \u0017Akerman, Science, 308, 508 (2005);\n[7] U. Nowak, O. N. Mryasov, R. Wieser, K. Guslienko, and\nR. W. Chantrell, Phys. Rev. B, 72, 172410 (2005).\n[8] B. Skubic, J. Hellsvik, L. Nordstr om, and O. Eriksson,\nJourn. of Phys.: Cond. Matter, 20, 315203 (2008).\n[9] M. -C. Ciornei, J. M. Rub\u0013 \u0010, and J. -E. Wegrowe, Phys.\nRev. B, 83, 020410(R) (2011).5\n[10] L. Landau and E. Lifshitz, Phys. Z. Sowjetunion, 8, 153\n(1935).\n[11] T. L. Gilbert, IEEE Trans. Mag. 40, 3443 (2004).\n[12] R. A. Sack, Proc. Phys. Soc. B, 70402 (1957).\n[13] W. T. Co\u000bey, Journal of Molecular Liquids, 114, 5\n(2004).\n[14] W.T. Co\u000bey, Yu. P. Kalmykov and S.V. Titov, Phys.\nRev. E, 65, 032102 (2002)\n[15] A. V. Kimel, B. A. Ivanov, R. V. Pisarev, P. A. Usachev,\nA. Kirilyuk, and Th. Rasing, Nature Phys. 5, 727 (2009).\n[16] W. F. Brown, Phys. Rev. 130, 1677 (1963).\n[17] H. Ebert, S. Mankovsky, D. Kodderitzsch, P. J. Kelly,\nPhys. Rev. Lett. 107, 066603 (2011).\n[18] K. Gilmore, Y. U. Idzerda, and M. D. Stiles, Phys. Rev.\nLett. 99, 027204 (2007).\n[19] A. Brataas, Y. Tserkovnyak, and G. E.W. Bauer, Phys.\nRev. Lett. 101, 037207 (2008).[20] J. -X. Zhu, Z. Nussinov, A. Shnirman, and A. V. Bal-\natsky, Phys. Rev. Lett. 92, 107001 (2004).\n[21] J. Fransson and J. -X. Zhu, New J. Phys. 10, 013017\n(2008).\n[22] J. Fransson, Phys. Rev. B, 82, 180411(R) (2010).\n[23] If higher order spin interaction terms than bilinear are\nneeded it is straight forward to include higher order in\nthis expansion.\n[24] V. P. Antropov, M. I. Katsnelson, M. van Schilfgaarde,\nand B. N. Harmon, Phys. Rev. Lett. 75, 729 (1995).\n[25] V. P. Antropov, M. I. Katsnelson, and A. I. Liechtenstein,\nPhysica B, 237-238 , 336 (1997).\n[26] M. I. Katsnelson and A. I. Lichtenstein, J. Phys.: Con-\ndens. Matter, 16, 7439 (2004).\n[27] V. Kambersk\u0013 y, Phys. Rev. B, 76, 134416 (2007)."    },    {        "title": "1703.07310v2.Using_rf_voltage_induced_ferromagnetic_resonance_to_study_the_spin_wave_density_of_states_and_the_Gilbert_damping_in_perpendicularly_magnetized_disks.pdf",        "content": "Using rf voltage induced ferromagnetic resonance to study the spin-wave density of states and the\nGilbert damping in perpendicularly magnetized disks\nThibaut Devolder\u0003\nCentre de Nanosciences et de Nanotechnologies, CNRS, Univ. Paris-Sud,\nUniversit ´e Paris-Saclay, C2N-Orsay, 91405 Orsay cedex, France\n(Dated: September 18, 2018)\nWe study how the shape of the spinwave resonance lines in rf-voltage induced FMR can be used to extract the\nspin-wave density of states and the Gilbert damping within the precessing layer in nanoscale magnetic tunnel\njunctions that possess perpendicular magnetic anisotropy. We work with a field applied along the easy axis to\npreserve the cylindrical symmetry of the uniaxial perpendicularly magnetized systems. We first describe the\nexperimental set-up to study the susceptibility contributions of the spin waves in the field-frequency space. We\nthen identify experimentally the maximum device size above which the spinwaves confined in the free layer can\nno longer be studied in isolation as the linewidths of their discrete responses make them overlap into a continuous\ndensity of states. The rf-voltage induced signal is the sum of two voltages that have comparable magnitudes: a\nfirst voltage that originates from the linear transverse susceptibility and rectification by magneto-resistance and a\nsecond voltage that arises from the non-linear longitudinal susceptibility and the resultant time-averaged change\nof the exact micromagnetic configuration of the precessing layer. The transverse and longitudinal susceptibility\nsignals have different dc bias dependences such that they can be separated by measuring how the device rectifies\nthe rf voltage at different dc bias voltages. The transverse and longitudinal susceptibility signals have different\nlineshapes; their joint studies in both fixed field-variable frequency, or fixed frequency-variable field configura-\ntions can yield the Gilbert damping of the free layer of the device with a degree of confidence that compares\nwell with standard ferromagnetic resonance. Our method is illustrated on FeCoB-based free layers in which\nthe individual spin-waves can be sufficiently resolved only for disk diameters below 200 nm. The resonance\nline shapes on devices with 90 nm diameters are consistent with a Gilbert damping of 0:011. A single value\nof the damping factor accounts for the line shape of all the spin-waves that can be characterized. This damp-\ning of 0.011 exceeds the value of 0.008 measured on the unpatterned films, which indicates that device-level\nmeasurements are needed for a correct evaluation of dissipation.\nThe frequencies of the magnetization eigenmodes of\nmagnetic body reflect the energetics of the magnetization.\nAs a result the frequency-based methods – the ferromag-\nnetic resonances (FMR)1and more generally the spin-wave\nspectroscopies– are particularly well designed for the metrol-\nogy of the various magnetic interactions. In particular, mea-\nsuring the Gilbert damping parameter \u000bthat describes the\ncoupling of the magnetization dynamics to the thermal bath,\nspecifically requires high frequency measurements. There are\ntwo main variants of these resonance techniques. The so-\ncalled conventional FMR and its modern version the vector\nnetwork analyzer2(VNA)-FMR are established technique to\nharness the coupling of microwave photons to the magneti-\nzation eigenmodes to measure to anisotropy fields1, demag-\nnetizing fields, exchange stiffness3, interlayer exchange4and\nspin-pumping5, most often at film level. More recent meth-\nods, like the increasingly popular spin-transfer-torque-(STT)-\nFMR, are developed6to characterize the magnetization dy-\nnamics of magnetic bodies embodied in electrical devices pos-\nsessing a magneto-resistance of some kind.\nIn conventional FMR or VNA-FMR, the community is well\naware that the line shape of a resonance is more complicated\nthan simple arguments based on the Landau-Lifshitz-Gilbert\nequation would tell. There are for instance substantial contri-\nbutions from microwave shielding effects7(”Eddy currents”)\nfor conductive ferromagnetic films8or ferromagnetic films in\ncontact with (or capacitively coupled to) a conductive layers.\nA hint to these effect is for instance to compare the lineshapes8\nfor the quasi-uniform precession mode and the first perpen-\ndicular standing spin wave modes that occur in different res-onance conditions. Note that the experimental lineshapes are\nalready complex in VNA-FMR despite the fact that the dy-\nnamics is induced by simple magnetic fields supposedly well\ncontrolled.\nIn contrast, STT-FMR methods rely on torques [spin-orbit\ntorques (SOT)9or STT] that have less hindsight that magnetic\nfields or that are the targeted measurements. These torques are\nrelated to the current across the device and the experimental\nanalysis generally assumes that this current is in phase with\nthe applied voltage. This implicitly assumes that the sam-\nple is free of capacitive and inductive responses, even at the\nmicrowave frequencies used for the measurement. A careful\nanalysis is thus needed when the STT-FMR methods analyze\nthe phase of the device response to separate the contribution of\nthe different torques6,10,11. Besides, the quasi-uniform mode\nis often the sole to be analyzed despite that fact that the line\nshapes of the higher frequency modes can be very different10.\nFinally, an external field is generally applied in a direction\nthat is not a principal direction of the magnetization energy\nfunctional12. While this maximizes the signal, this unfortu-\nnately makes numerical simulation unavoidable to model the\nexperimental responses.\nWith the progress in MTJ technologies, much larger\nmagneto-resistance are now available13, such that signals can\nbe measured while maintaining sample symmetries, for in-\nstance with a static field applied collinearly to the magne-\ntization. In addition, high anisotropy materials can now be\nincorporated in these MTJs. This leads to a priori much\nmore uniform magnetic configurations in which analytical de-\nscriptions are more likely to apply. In this paper, we revisitarXiv:1703.07310v2  [cond-mat.mtrl-sci]  4 Sep 20172\nrf-voltage induced FMR in a situation where the symmetry\nis chosen so that all torques should yield a priori the same\ncanonical lineshape for all spinwaves excited in the system.\nWe use PMA MTJ disks of sizes 500 nm, on which a quasi-\ncontinuum of more that 20 different spin-wave modes can be\ndetected, down to sizes of 60 nm where only a few discrete\nspinwave modes can be detected. We discuss the lineshapes\nof the spin-wave signals with the modest objective of deter-\nmining if at least the Gilbert damping of the dynamically ac-\ntive magnetic layer can be reliably extracted. We show that\nthe linear transverse susceptibility and the non-linear longitu-\ndinal susceptibilities must both be considered when a finite dc\nvoltage is applied through the device. We propose a method-\nology and implement it on a nanopillars made with a stan-\ndard MgO/FeCoB/MgO free layer system in which we obtain\na Gilbert damping of 0:011\u00060:0003 . This exceeds the value\nof 0.008 measured on the unpatterned film, which indicates\nthat device-level measurements are needed for a correct eval-\nuation of dissipation.\nThe paper is organized as follows:\nThe first section lists the experimental considerations, includ-\ning the main properties of the sample, the measurement set-\nup and the mathematical post-processing required for an in-\ncreased sensitivity. The second section discusses the origins\nof the measured resonance signals and their main properties.\nThe third section describes how the device diameter affects the\nspin-wave signals in rf-voltage-induced ferromagnetic reso-\nnance. The last section describes how the voltage bias depen-\ndence of the spinwave resonance signals can be manipulated\nto extract the Gilbert damping of the dynamically active mag-\nnetic layer. After the conclusion, an appendix details the main\nfeatures of the spectral shapes expected in ideal perpendicu-\nlarly magnetized systems.\nI. EXPERIMENTAL CONSIDERATIONS\nA. Magnetic tunnel junctions samples\nWe implement our characterization technique on the sam-\nples described in detail in ref. 14. They are tunnel junctions\nwith an FeCoB-based free layer and a hard reference system\nbased on a well compensated [Co/Pt]-based synthetic antifer-\nromagnet. All layers have perpendicular magnetic anisotropy\n(PMA). The perpendicular anisotropy of the thick ( t= 2nm)\nfree layer is ensured by a dual MgO encapsulation and an\niron-rich composition. After annealing, the free layer has an\nareal moment of Mst\u00191:8mA and an effective perpendic-\nular anisotropy field \u00160(Hk\u0000Ms)= 330 mT. Before pat-\ntering, standard ferromagnetic resonance measurements in-\ndicated a Gilbert damping parameter of the free layer being\n\u000b= 0:008. Depending on the size of the patterned device,\nthe tunnel magnetoresistance (TMR) is 220 to 250%, for a\nstack resistance-area product is RA = 12 \n:\u0016m2. The de-\nvices are circular pillars with diameters varied from 60 to 500\nnm. The materials, processing and device rfcircuitry were\noptimized for fast switching14spin-transfer-torque magnetic\nrandom access memories (STT-MRAM15) ; the quasi-static\ndcRF50 ΩPulse modulation ac, 50 kHz\n50 ΩVac ~ 40 mVLI75AamplifierMSG3697synthetiser\nK2400sourcemeter\nSR830lock-inamplifier×100-10 dB attenuation~ 0 dBm~ 14 µAdc10 mVdcMTJ device300 nm~700 ΩFIG. 1. (Color online). Sketch of the experimental set-up with an\n300\u0002300\u0016m2optical micrograph of the device circuitry. The given\nnumbers are the typical experimental parameters for a 300 nm diam-\neter junction. Inset: resistance versus out-of-plane field hysteresis\nloop for a device with 300 nm diameter.\ndcswitching voltage is \u0019600mV . In the present report, the\napplied voltages shall never exceed 100 mV to minimize spin-\ntransfer-torque effects. The fields will always be applied along\n(z) which is the easy magnetization axis. The sample will be\nmaintained in the antiparallel (AP) state.\nB. Measurement set-up\nThe pillars are characterized in a set-up (Fig. 1) inspired\nfrom spin-torque diode experiments6but an electrical band-\nwidth increased to 70 GHz. The objective is to identify the\nregions in theffrequency, fieldgspace in which the magneti-\nzation is responding in a resonant manner. The device is at-\ntacked with an rfvoltageVrf. A 10 dB attenuator is inserted\nat the output port of the synthesizer to improve its impedance\nmatching so as to avoid standing waves in the circuit. This im-\nproves the frequency flatness of the amplitude of the stimulus\narriving at the device. To ease the detection of the sample’s re-\nsponse, the rfvoltage is pulse-modulated at an acfrequency\n!ac=(2\u0019) = 50 kHz (Fig. 1). The current passing through\nthe MTJ has thus frequency components at the two sidebands\n!rf\u0006!ac. The acvoltage which appears across the device\nis amplified and analyzed by a lock-in amplifier. We shall\ndiscuss the origin of this acvoltage in section II. Optionally,\nthe device is biased using a dcsourcemeter supplying Vdcand\nmeasuringIdc.\nFigure 2 shows a representative map of thedVac\ndHzresponse\nobtained on a pillar of diameter 300 nm with Vdc= 10 mV.\nAs positive fields are parallel to the free layer magnetization,\nthe spin waves of the free layer appear with a positive fre-\nquency versus field slope, expected to be the gyromagnetic3\n⦰ 300 nm\nFIG. 2. Field derivative of the rectified voltagedVac\ndHzin the\nffrequency-fieldgparameter space for a 300 nm diameter device in\nthe AP state when the field is parallel to the free layer magnetization.\nThe linear features with positive (resp. negative) slopes correspond\nto free layer (resp. reference layers) confined spin-wave modes.\nBlack and white colors correspond to signals exceeding \u00060:01V/T.\nThe one-pixel high horizontal segments are experimental artefacts\ndue to transient changes of contact resistances.\nratio\r0of the free layer material (see appendix). Conversely,\nthe reference layer eigenmodes appear with a negative slope,\nexpectedly\u0000\r0, where this time \r0is gyromagnetic ratio of\nthe reference layer material combination. Working in the AP\nstate is thus a convenient way to easily distinguish between\nthe spinwaves of the free layer and of the reference layers.\nNote that the gyromagnetic ratios \r0of the free layer mode\nand the reference layer modes differ slightly owing to their\ndifference chemical nature. The free layer has a Land ´e factor\ng= 2:085\u00060:015where the error bar is given by the precision\nof the field calibration; the reference layer modes are consis-\ntent with a 1.2% larger gyromagnetic ratio. The accuracy of\nthis latter number is limited only by the signal-to-noise ratio\nin the measurement of the reference layer properties. Looking\nat Fig. 2, one immediately notices that the linewidths of the\nreference layer modes are much broader than that of the free\nlayer. While the linewidh of the reference layer modes will not\nbe analyzed here, we mention that this increased linewidth is\nto be expected for reference layers that contain heavy metals\n(Pt, Ru) with large spin-orbit couplings, hence larger damping\nfactors16.C. Experimental settings\nIn practice, we choose an applied field interval of\n[\u0000110;110mT]that is narrow enough to stay in a state whose\nresistance is very close to that of the remanent AP state. The\nfrequency!rf=(2\u0019)is varied from 1 to 70 GHz; we gener-\nally could not detect signals above 50 GHz. The practical\nfrequency range 2\u0019\u000250GHz=\r0\u00191:6T is much wider that\nour accessible field range. For wider views of the experimen-\ntal signals (for instance when the spin-wave density of states\nis the studied thing), we shall thus prefer to plot them versus\nfrequency than versus field. The response is recorded pixel by\npixel in in theffrequency, fieldgspace. The typical pixel size\nisf\u000eHz\u0002\u000efg=f1 mT\u000250 MHzg. The field and frequency\nresolutions are thus comparable (indeed 2\u0019\u0002\u000ef=\r 0= 1:7\nmT).\nD. Signal conditioning\n1. Mathematical post-treatments\nFinally, despite all our precautions to suppress the rectify-\ning phenomena that do not originate from magnetization dy-\nnamics, we have to artificially suppress the remaining ones.\nThis was done by mathematical differentiation, and we gener-\nally plotdVac\ndfordVac\ndHzin the experimental figures (Figs. 2-5).\n2. Dynamic range improvement by self-conformal averaging\nA special procedure (Fig. 3) is applied when a better signal\nto noise ratio is desired while the exact signal lineshape and\namplitudes are not to meant to be looked at. This procedure\nharnesses the fact that the normalized shape of the sample’s\nresponse is essentially self-conformal when moving across a\nline withd!\ndHz=\r0in theffrequency, fieldgparameter space\n(see appendix). The procedure consists in calculating the fol-\nlowing primitive:\ns(f0) =1\n2\r0HmaxzZ\ncontourdVac\ndHzdf ; (1)\nin which the integration contour is the segment linking the\npoints (\u0000Hmax\nz;f0\u0000\r0Hmax\nz) and (Hmax\nz;f0+\r0Hmax\nz)\nin theffield, frequencygparameter space. Such contours ap-\npear as pixel columns in Fig. 3(b). This primitive (eq. 1) is\nefficient to reveal the free layer spin-wave modes that yield an\notherwise too small signal. For instance when only 7 modes\ncan be detected in single field spectra [Fig. 3(a)], the aver-\naging procedure can increase this number to typically above\n25. The averaging procedure is also effective in suppressing\nthe signals of the reference layer as these laters average out\nover a contour designed for the free layer mode when in the\nAP state. However as the linewidth of the free layer modes\nis proportional to the frequency, it is not constant across the\ncontour; the higher signal to noise ratio is thus unfortunately4\nf\u0000\u00000Hz2⇡Hz(b)(a)\nFIG. 3. (Color online). Illustration of the dynamic range improve-\nment by self-conformal averaging (section I D 2). The procedure is\nimplemented on a 300 nm diameter device to evidence the free layer\nmodes. Bottom panel: field derivative of the acsignal in the rotated\nframe in which the modes withdf\ndHz=\r0\n2\u0019should appear as vertical\nlines. Top panel: comparison of a single field frequency scan (red)\nwith the average over all scans as performed in the !=\r0Hzdi-\nrection. Note that the signal of the lowest frequency mode (which\ncorresponds to the quasi-uniform precession) disappears near zero\nfield, at 5 mT (see the apparent break in the middle of the most left\nline in the bottom panel).\nobtained at the expense of a distorted (and unphysical) line-\nshape. Note also that this procedure can not be applied to the\nquasi-uniform precession mode as will be explained in section\nII D 2).\nII. ORIGIN AND NATURE OF THE RECTIFIED SIGNAL\nLet us now discuss the origin of the demodulated acvolt-\nage. In this section, we assume that the reference layer mag-\nnetization is static but not necessarily uniformly magnetized.\nWe can thus express any change of the resistance by writing\n\u000eR=\u000eR\n\u000eM\u000eM where\u000ehas to be understood as a functional\nderivative with respect to the free layer magnetization distri-\nbution.\nA. The two origins of the rectified signals\nTheacsignal can contain two components V1;acandV2;ac\nof different physical origins17. The first component is the\n’standard’ STT-FMR signal: the pulse-modulated rfcurrent is\nat the frequency sidebands !rf\u0006!acand it rectifies to acany\noscillation of the resistance \u000eRrfoccurring at the frequency\n!rf. We simply have V1;ac=\u000eRrf\u0002i!rf\u0006!ac.\nThe second acsignal (V2;ac) is related to the change of the\ntime-averaged resistance due to the population of spinwavescreated when the rfcurrent is applied12. Indeed the time-\naveraged magnetization distribution is not the same when the\nrfisonoroff. This change of resistance \u000eRaccan revealed by\nthe (optional) dccurrentIdcpassing through the sample, i.e.\nV2;ac=\u000eRac\u0002Idc.\nNote that a third rectification channel18can be obtained by\na combination of spin pumping and inverse spin Hall effect in\nin-plane magnetized systems19. This third rectification chan-\nnel yields symmetric lorentzian lines when applied to PMA\nsystems in out-of-plane applied fields (see eq. 23 in ref. 18).\nBesides, the spin-pumping is known to be largely suppressed\nby the MgO tunnel barrier20, such that we will consider that\nwe can neglect this third rectification channel from now on. In\nsummary, we have:\nV1;ac=Vrf\nR+ 50\u000eR\n\u000eM\u000eMrf and (2)\nV2;ac=Vdc\nR+ 50\u000eR\n\u000eM\u000eMac (3)\nThis has important consequences.\nB. Compared signal amplitudes in the P and AP states\nThe first important consequence of Eq. 2 and 3 is that the\nsignal amplitude depends on the nature of the micromagnetic\nconfiguration. As intuitive, both V1;acandV2;acscale with\nhow much the instantaneous device resistance depends on its\ninstantaneous micromagnetic configuration. This is expressed\nby the sensitivity factor\u000eR\n\u000eMwhich is essentially a magneto-\nresistance. We expect no signal when the resistance is insen-\nsitive to the magnetization distribution at first order (i.e. when\n\u000eR\n\u000eM\u00110).\nIn our samples, the shape of the hysteresis loop (Fig. 1)\nseems to indicate that the free layer magnetization is very uni-\nform when in the Parallel state. Consistently, the experimental\nrectified signal were found to be weak signals when in the P\nstate. Conversely, there is a pronounced curvature in the AP\nbranch of the R(Hz)hysteresis loop (see one example in the\ninset of Fig. 1). This indicates that the resistance is much de-\npendent on the exact magnetization configuration when in the\nAP state. Consistently, this larger\u000eR\n\u000eMin the AP state is proba-\nbly the reason why the rectified signal is much easier to detect\nin the AP state for our samples.\nC. Bias dependence of the rectified signals\nThe second important consequence of Eqs. 2-3 concerns the\ndependence of the rectified acsignalsV1;acandV2;acon the\ndcandrfstimuli. As\u000eMrfscales with the applied rftorque\naccording to a linear transverse susceptibility ( <e(\u001fxx), see\nappendix),V1;acis expected to scale with the rfpowerVrf2\n(see Eq. 2) independently from the dcbias, i.e. we have\nV1;ac/Vrf2:5\nIn contrast,\u000eMacis related to a longitudinal susceptibility and\nis thus quadratic with the rftorques (see appendix). Using\nEq. 3, we thus expect the following bias dependence\nV2;ac/Vrf2Vdc:\nD. Peculiarities of the quasi-uniform precession (QUP) mode\nThe last important consequence of Eqs. 2-3 concerns\nspecifically the quasi-uniform precession (QUP) mode that\nshows a peculiar acsignal.\n1. Quasi-absence of STT-FMR like signal for the quasi-uniform\nprecession mode\nIn the idealized macrospin case (see appendix) the uniform\nprecession is perfectly circular with no rfvariation of Mzat\nany order. If the fixed layer was uniformly magnetized along\nexactly (z), this would lead V1;ac= 0 such that the signal of\nthe QUP mode would be given by purely V2;ac. This qual-\nitatively ’pure V2;accharacter’ is confirmed experimentally\nby the fact that the signal of the QUP mode systematically\nchanges sign with Vdcin our sample series (not shown).\n2. Strong dependence of the QUP signal amplitude with the\napplied field\nIn addition, the experimental signal of the quasi-uniform\nmode is found to disappear at low fields [see Figs. 2, 3(b) and\n4(a, b)] exactly at the apex of the AP branch of the hysteresis\nloop (Fig. 1), i.e. for the field leading specifically todR\ndHz= 0.\nMoreover, the amplitude of the acsignal of the QUP mode ap-\npears to be essentially linearly correlated with the loop slope\ndR\ndHz(not shown). For instance, the V2;acof the QUP mode\nchanges sign when the applied field crosses the apex of the\nR(Hz)loops (Fig. 1).\nThe reason stems probably from the sensitivity factor\u000eR\n\u000eM\nand its correlation with the loop slopedR\ndHz; in some sense, a\nlarge loop slope should translate in a large sensitivity factor.\nWhile a numerical evaluation of this correlation goes beyond\nthe scope of this paper, we stress that if the magnetization was\nperfectly uniform there would be a one-to-one correlation be-\ntween loop slopedR\ndHzand magnetoresistance sensitivity fac-\ntor\u000eR\n\u000eM. This trend remains qualitatively true for the QUP\nmode. Indeed as the hysteresis loop is monitoring the spatial\naverage of the magnetization, it is more insightful for the uni-\nform mode than for any other (higher order) modes whose dy-\nnamic profiles spatially average to essentially zero21; the cor-\nrelation betweendR\ndHzand\u000eR\n\u000eMis thus expected to be maximal\nfor quasi-uniform changes of the magnetization configuration.\nWhile this property – the disappearance of the QUP mode\nsignal whendR\ndHz= 0 – can be used to distinguish the QUP\nmode from the higher order spin waves, the pronounced field\ndependence of the QUP signal complicates the analysis, as it\nprevents to conveniently analyze the field derivative of the acsignal (I D 1). In the remainder of this paper we shall focus on\nonly higher order modes to avoid such difficulties.\nE. Signals for non-uniform spin-waves\nBefore analyzing the spin-wave density of states (section\nIII), let us comment on the amplitude of the STT-FMR-like\nsignalV1;acfor the non-uniform spin-waves. In the perpen-\ndicular magnetization state, these spin-waves have a circular\nprecession22. By symmetry, the resistance is not expected to\nchange during a period of circular precession when in the per-\nfect collinear cases and for radial spin waves maintaining the\ncylindrical symmetry of the system. In other words, when\nthe dynamical magnetization of the eigenmode maintains the\ncylindrical symmetry and when the free and reference layers\nequilibrium magnetizations follow ~Mfree\u0002~Mref=~0every-\nwhere in the (xy) plane, with \u0002being the conventional vec-\ntor product) the device resistance is not expected to oscillate.\nWhile we can not identify to what extent we depart from this\nideal situation, we speculate that this perfect collinearity does\nnot happen in practice at least because of finite thermal fluctu-\nations. The effect of thermal fluctuations on the device resis-\ntance is not averaged out for non-uniform spin-waves, while\nit could be essentially averaged out for the QUP mode ana-\nlyzed earlier. In practice a finite variation of the resistance\n\u000eRrf6= 0 is always present during a precession period for a\nnon-uniform spin-wave. This provides a finite sensitivity to\nany spin-wave mode. This resistance variation at !rfhas the\nspectral shape of a transverse susceptibility term <e(\u001fxx)(see\nappendix).\nIII. SPIN WA VE DENSITY OF STATES AGAINST\nLATERAL CONFINEMENT\nAny reliable analysis of a spectral lineshape or linewidth re-\nquires to determine priorly how many spin-waves contribute\nto the lineshape under study. Therefore, before discussing the\nlineshapes of the individual spin-wave modes, let us determine\nhow the lateral confinement influences the measured rectified\nsignal. The impact of the device diameter on the spectral sig-\nnals is reported in Fig. 4.\nA. Spin-waves within the references layers\nFig. 4 indicates that the modes of the reference layers have\nfrequencies that are almost not affected by the device diam-\neter. This fact is related to the well compensated character\nof the synthetic antiferromagnet that composes the reference\nlayers. Indeed the internal demagnetizing fields compensate\nto some extent, such that they do not influence the frequency\nof the acoustical mode of a SAF as much as the anisotropies\nand the interlayer exchange couplings do.6\nSignal Spectral Peak-to-peak Full Width Zero crossings\nshape separation at Half Maximum separation\nExpected signals and their stimulus dependence:\nV1;ac/V2\nrf <e(\u001fxx) 2\u000b! - -\nV2;ac/V2\nrfVdc \u0001Mz - 2\u000b! -\nSignal extraction procedure from experiments:\nd\ndHzVexp\n1;acestimated fromdVexp\nac\ndHz\f\f\f\f\nVdc=0d<e(\u001fxx)\nd!- - 2\u000b!\nd\ndHzVexp\n2;acestimated from\"\ndVexp\nac\ndHz\f\f\f\f\nVdc6=0\u0000dVexp\nac\ndHz\f\f\f\f\nVdc=0#\nd=m(\u001fxx)\nd!2p\n3\u000b! - -\nTABLE I. Summary of the expected lineshapes and linewidths for the different signals that can be encountered in rf-voltage-induced FMR\nexperiments. An hyphen is inserted when the concept is not applicable.\nB. Spin-waves within the free layer\nConversely, the frequencies of the modes of the free layer\nare strongly affected by the device diameter (Fig. 4). First,\nthe modes are pushed to higher frequencies as the device\nis shrunk. At remanence, the lowest frequency mode is at\nfQUP= 12:3GHz for a diameter of 500 nm; it reaches 19.5\nGHz for 60 nm devices (not shown). Second, the frequency\nspacing between the free layer modes increases substantially\nwhen downsizing the device.\nThe first effect – increase of fQUPat downscaling – is in-\ndicative of a dependence of some effective fields with the\ndevice diameter. Among the effective fields, the only ones\nthat vary with the diameter are the exchange fields (positive\ncontribution to the frequency fQUPif magnetization is non-\nuniform), the demagnetizing fields (positive contribution to\nthe frequency fQUPat downscaling) and the local effective\nanisotropy fields in case some process damages alter locally\nthe interface anisotropy at the perimeter of the free layer (neg-\native contribution to the frequency fQUPat downscaling) or\nalter the local magnetization of the rim of the free layer (pos-\nitive contribution to the frequency fQUPat downscaling). The\nexchange fields are related to the non uniformities of either the\nstatic configuration –the fact that the AP state is not perfectly\nuniform as inferred previously from the loop in Fig. 1– or non\nuniformities of the dynamic magnetization –i.e. the fact that\nthe quasi-uniform mode is not a strictly uniform mode–. If\nthe frequency increase was due to the sole demagnetizing ef-\nfects, it could be estimated from the demagnetizing factors of\ndisks23which areNz\u00191\u0000(3\u0019=8)t=a, wheretandaare\nthe thickness and radius of the free layer. However a fQUP\nagainst 1=aplot (not shown) has a perceivable curvature near\nall sizes; an unwise linear fit through fQUPagainst the ex-\npected\rNzwould give a slope of \u00192:5T, which is obvi-\nously too large for the magnetization of the free layer. This\nindicates that the sole change of the global shape anisotropy\nwith the device diameter is insufficient to account for the in-crease offQUPat downscaling: exchange contributions or non\nuniformities induced by process damages also contribute to\nthe frequencies. Exchange contributions should not contribute\nfor the largest devices, however even for those devices the ex-\nperimental frequencies are larger than the ones expected from\nglobal shape anisotropy only, which argues for some process\ndamages. Since the TMR is almost independent of the de-\nvice size14, we can reasonably assume that the MgO/FeCoB\ninterface is not substantially affected by the patterning and\nthat consequently the interface anisotropy is essentially pre-\nserved at the rim of the free layer. We conclude that part of\nthe increase of fQUPat downscaling is due to a reduced mag-\nnetization (magnetically ”dead” or weak zone) near the edges\nof the free layer. This interpretation is probably very much\nstack and process technology dependent, hence it should not\nbe considered as general.\nThe second effect – the increased frequency spacing be-\ntween the modes at small diameters – is the expected effect\nof the confinement of the spin waves and the resulting in-\ncrease of the exchange contribution to the mode frequencies17.\nThe eigenmodes of perpendicularly magnetized circular disks\nare well understood and can be described analytically in a\nsemi-quantitative manner21,24–28. The frequency spacing be-\ntween the lowest frequency modes scales with \r0HJwhere\nHJ=2Ak2\n\u00160MSis a generalized exchange field with Athe ex-\nchange stiffness. The effective wavevector kis reminiscent of\nthe lateral confinement and reads k2= (u2\n2\u0000u2\n1)=a2\u00199=a2\nwhereu1andu2are the first zeros of the first and second\nBessel functions21. The lowest frequency spinwave modes\ncan be resolved only if their frequency spacing is comparable\nor greater than their linewidth 2\u000b\r0(Hz+Hk\u0000Ms+HJ)\n(see appendix).\nThis condition can be used to define a critical device diam-\neter:\na2\ncrit=9A\n\u000b\u00160MS(Hz+Hk\u0000Ms)(4)\nFor large devices with a\u001dacritwe expect to observe a quasi-7\ncontinuum of overlapping modes above fQUP, while discrete\nnon-overlapping modes are anticipated in the opposite limit.\nTypical parameters of an FeCoB-based free layer include a\nmagnetization of \u00160Ms= 1:2T and a Gilbert damping of29\n\u000b= 0:01. From the quasi-uniform mode frequency, we can\nget our effective anisotropy which is Hk\u0000Ms= 330 kA/m.\nIf the exchange stiffness of the free layer was bulk-like (i.e.\nA= 22 pJ/m) like in ref. 17, the critical diameter would\nbe2acrit= 444 nm. In practice the small frequency spac-\nings between the modes of our samples indicates that the ex-\nchange stiffness of our free layer is in the range of 6-7 pJ/m\ni.e. well below the bulk value. This estimate of the exchange\nstiffness was deduced assuming perfectly pinned boundary\nconditions for the spin-waves at the device edge, which is a\nquestionable30assumption. However the exchange stiffness is\nanyway weak in the free layer and this can be also qualita-\ntively seen directly from the spin wave spectroscopy: indeed\nthe frequency spacing of the lowest modes of the reference\nlayer system is typically twice larger that the frequency spac-\ning of the lowest modes of the free layer [see for instance\nFig. 3(b)]. While the reason for this small value of the free\nlayer exchange stiffness is not entirely clear, we emphasize\nthat having such a small exchange stiffness is not uncommon\nin magnetic systems that comprise only a small number of\natomic layers, starting for instance from 2 pJ/m for a single\nlayer of iron31. Anyway with these parameters, we expect\na clear separation of the lowest frequency modes at rema-\nnence provided that the device diameter is much smaller than\n2acrit= 250 nm.\nIn practice for 300 and 500 nm devices a fine structure\ncan still be detected in the spin-wave density of states [see\nFig. 4(c)] but it is hard to count the modes and guess their\nfrequencies out of this fine structure. In the remainder of this\npaper, we shall thus only consider devices of diameter less\nthan 200 nm, in which the different spin-wave modes can be\nunambiguously resolved [see Fig. 4(e-f)].\nIV . LINESHAPE EVOLUTIONS WITH BIAS AND\nEXTRACTION OF THE GILBERT DAMPING\nLet us now compare the shapes of the experimental rectified\nsignal with those expected (see appendix). For that purpose\nwe harness the different bias dependences (Table I) of the rec-\ntified signals V1;acandV2;acto isolate each of them in the\nexperimental signal Vac. We identify V1;acto the experimen-\ntal curveVacmeasured at Vdc= 0, and we construct an esti-\nmate ofV2;acby subtracting Vacmeasured at Vdc= 0 from\nthat measured at Vdc6= 0 (see Table I). Note because of this\nsubtraction, any dependence of the spin-wave frequency with\nthedcvoltage will prevent the measurement of the voltage de-\npendence of the damping factor or of the voltage dependence\nof the exciting torques.\nWe illustrate this procedure in Fig. 5 in which we plot the\nfield derivatives of the so-calculated V1;acandV2;acrectified\nsignals in both fixed field or fixed frequency experimental con-\nditions for a device of diameter 90 nm. We center the curves\non the second lowest frequency eigenmode since it provides\n(a)(b)\n(c)(e)(f)(d)FIG. 4. (Color online). Dependence of the field derivative of the ac\nvoltage over the device diameter. Top panels: full spectral depen-\ndence in the window [6;40GHz]\u0002[\u000090;90mT]for 90 nm (left)\nand 500 nm (right) diameter devices. Bottom panels: frequency de-\npendence of the field derivative of the acvoltage after self-conformal\naveraging for various device sizes. The signal amplitudes have been\nnormalized to ease their comparison.\nthe largest signals and it is reasonably separated from both the\nquasi-uniform precession mode and from the other high order\nmodes. The obtained experimental V1;accurves [see Fig. 5\n(a) and (d)] have the expected line shapes (see appendix) with\na negative peak surrounded by two tiny positive halos (areas\nshaded in red). The separation between the two zero crossings\nis9:5\u00060:5mT or 285\u000610MHz. The obtained experimental\nV2;accurves also have the expected line shape of the deriva-\ntive of a Lorentzian distribution (see appendix). As expected,\nthe sign of the response changes with the dcbias voltage. The\nseparation between the positive and negative maxima of the\ndistribution are 6:1\u00060:5mT and 170\u000610MHz.\nThese four different ways of measuring the linewidths\n[Fig. 5 (a, d, b, e)] are consistent with a free layer damping\nof\u000b= 0:011\u00060:0003 . Indeed this value of damping would\npredict linewidths of 2\u000bf= 295 MHz and 2\u00160\u000b!=\r 0=\n9:54mT (materialized as black bars in [Fig. 5 (a, d) and (d)])\nand1:15\u000bf= 171 MHz or 1:15\u000b!=\r 0= 6:21mT (material-\nized as blue bars in [Fig. 5 (b, c, e, f)]).\nThis proposed value of damping is also consistent with the\nlinewidths of higher order spin waves that appear at larger fre-\nquencies but with a lower signal. This is illustrated in fig. 6\nwhere a comparison is drawn between the values of the damp-\ning estimated for each applied field from the second (and most\nintense) mode and from the third mode for a device of diame-\nter 80 nm. The used procedure is a direct fit of the experimen-8\ntal lineshapes to the derivative of Eq. 7 with the damping, the\nresonance frequency and the signal amplitude as free parame-\nters. For this specific device, the estimates of the damping pa-\nrameter are subjected to a random error of standard deviation\n0.0018 around a mean value of \u000b= 0:0119 . It is interesting\nto note that the non-local contributions32,33to the damping ex-\npected for the relatively large wavevectors of the second and\nthird spin-wave modes seem to be too small to be observed\nin our samples. Note that as mentioned earlier, the same pro-\ncedure can in principle not be applied to the quasi-uniform\nmode since it exhibits a strong dependence of the mode am-\nplitude with the field which invalidates the procedure to some\nextent. For the sake of completeness of this paper, we have\nanyway fitted the experimental QUP lineshapes with the field\nderivative of Eq. 8; this is not possible near zero field, as the\ncorresponding signal vanishes. The value of the Gilbert damp-\ning that would be illegitimately deduced would be 0:01, i.e.\n20% lower than the correct value. Besides, the estimates from\nthe QUP mode would exhibit a substantially larger spread in\nthe fit results [compare the histograms in in fig. 6(b)]. For\nthese two reasons, we consider that the reliable estimate of the\ndamping is the one extracted from the non-uniform modes.\nAbove 100 mV of dcbias, the amplitude of the constructed\nexperimental V2;acstart to depart from proportionality with\nVdcand a frequency shift is observed, as expected when dc\nfield-like spin torques are applied. This comes with by a dis-\ntortion of the line shape, probably linked to the modification of\nthe spin waves lifetimes by spin-transfer torque as commonly\nobserved in in-plane magnetized MTJs34.\nV . SUMMARY AND CONCLUSIONS\nIn this paper, we have studied how to use rf-voltage-\ninduced ferromagnetic resonance to study the spin-wave den-\nsity of states and the Gilbert damping in perpendicularly mag-\nnetized disks embodied in magnetic tunnel junctions. We have\napplied the field along the easy axis to preserve the cylindrical\nsymmetry of the magnetization energy functional. The inter-\nest of this configuration is that all the current-induced torques\nthat potentially excite the dynamics yield the same type of\nsusceptibility spectral shape. Additionally, this configuration\nis the sole in which the applied field and the frequency play\nsimilar roles near FMR so that consistency crosschecks be-\ntween variable-field and variable-frequency experiments can\nbe performed to reveal and suppress potential experimental\nartefacts.\nWorking in a situation in which the fixed layer and the free\nlayer are oppositely magnetized in a convenient way to clas-\nsify the spin-waves according to their hosting layer, as the two\nsub-systems have opposite eigenmode frequency-versus-field\nslopes. The dcbias dependence of the signal of the quasi-\nuniform mode is peculiar and can be used to ambiguously\nidentify the free layer quasi-uniform mode in the manifold of\nspin-waves. The non-uniform (higher order) spin-waves are\neasier to analyze, as their amplitudes weakly depend on the\napplied field so that field differentiation can be used safely for\nbackground subtraction. Optionally, the dynamic range of theexperiment can be improved by self-conformal averaging of\nthe resonance spectra.\nThe unambiguous identification of the spin-wave frequen-\ncies requires devices that are sufficiently small to avoid that\nthe spinwave modes overlap into a quasi-continuous density\nof states. The critical device size is set by the exchange\nstiffness, the damping, the magnetization and the effective\nanisotropy field. In practice, device diameters below 200 nm\nare needed in our low-damped FeCoB-based PMA system.\nFor each spin-wave mode, the rf-voltage-induced spin-\nwave spectra contain contributions from two different phys-\nical mechanisms. The first one is the standard STT-FMR-like\nsignal, whose spectral shape is a linear transverse susceptibil-\nity term. It is independent from the dcvoltage applied across\nthe MTJ. The second one is a variation of the time-averaged\nmagnetic configuration when the rfvoltage is applied. It is\nproportional to the dcvoltage applied across the MTJ and it\nhas the spectral shape of a non-linear longitudinal susceptibil-\nity. The bias dependence can be used to separate these two\nsignals. The analysis of their spectral shape yields the Gilbert\ndamping within the precessing layer. A single value of the\ndamping factor is found to account for the lineshapes of all\nstudied spin-waves.\nThe spectra of rf-voltage-induced rectified voltages for a\nvanishing dcvoltage bias are in principle sufficient to get the\nGilbert damping of the dynamically active layer. However\nas microwave methods are prone to artefacts, a consistency\ncheck exploiting the bias dependence of the resonance spectra\nis useful for a consolidation of the numerical estimation of the\ndamping.\nThis work was supported in part by the Samsung Global\nMRAM Innovation Program, who provided also the samples.\nCritical discussions with Vladimir Nikitin, Jean-Paul Adam,\nJoo-V on Kim and Paul Bouquin are acknowledged.\nVI. APPENDIX: SUSCEPTIBILITIES IN AN IDEALIZED\nPMA FILM\nIn this appendix, our aim is to determine the transverse and\nlongitudinal microwave susceptibility versus frequency fand\nstatic fieldHzfor a PMA film as a response to an harmonic\ntransverse field hxcos(!t)~ ex. We shall write the equations\nwith this transverse field hxbut any other effect that yields a\ntorque possessing a component transverse to the static mag-\nnetization will yield similar lineshapes. This includes current-\ninduced Oersted-Ampere fields but also Slonczewski STT and\nfield-like STT as soon as the reference layer magnetization\n~Mrefis not strictly collinear with that of the free layer ~Mfree.\nThe susceptibility tensor will be used to deduce the shape of\nthe line expected in rf-voltage-induced FMR, as summarized\nin Table I. Throughout this appendix, we assume a dcfield\nHzperfectly perpendicular to the plane and a free layer mag-\nnetization~M=Mz~ ez+mx~ ex+my~ ey, where the transverse\nterms are assumed small and written as complex numbers in\nthe frequency space. For conveniency, we will use the nota-\ntionH0=Hz+Hk\u0000Ms. We shall also write the frequencies\nin field units and define !0=!=\r 0. This is meant to empha-9\nddHzVexp2,a cforVdc= 100 mVddHzVexp1,a cforVdc=0\nddHzVexp2,a cforVdc=\u0000100 mV\nFIG. 5. (Color online). Rectified acsignals versus frequency at fixed\napplied field (left panels) or versus field at fixed frequency (right\npanels). The plotted data ared\ndHzV1;ac(black, panels a and d) and\nd\ndHzV2;ac(blue, panels b, c, e and f) as estimated according to the\nformulas of Table I. The arbitrary vertical scale is the same for all\npanels. The dotted black lines are separated by 9.5 mT and 270 MHz.\nThe dotted blue lines are separated by 6.1 mT or 170 MHz. These\ndotted lines correspond to the expected linewidth for a damping of\n0.011. The panel (b) is measured for a device different from that of\nthe other panels.\nsize the fact that the generalized field H0and the generalized\nfrequency!0play very similar roles in the FMR of perpendic-\nularly magnetized macrospin when near resonnance. We shall\nsystematically assume that \u000b\u001c1and only keep the lowest\norder of the damping terms in the equations. In sections VI A\n-VI C we make the macrospin approximation. This approxi-\nmation is discussed in section VI D.\nA. Transverse linear susceptibility\nFollowing the usual procedure, we project the linearized\nLandau-Lifshitz-Gilbert equation along ~ exet~ ey:\n\u001a\nH0(my+\u000bmx) +imx!0=hxMs\u000b\nH0(mx\u0000\u000bmy)\u0000imy!0=hxMs\nWe then invert this system of equations to get the susceptibil-\nitiesmx=\u001fxxhxandmy=\u001fyxhx. They are:\n\u001fxx=MsH0\nH02\u0000!02+ 2i\u000bH0!0(5)\nand\n\u001fyx=\u0000iMs!0\nH02\u0000!02+ 2i\u000bH0!0(6)\n-100- 500 5 01 000.0000.0050.0100.0150.0200\n.0080 .0100 .0120 .014Third modeSecond modeOccurrence (a. u.)D\namping parameterQuasi-uniform mode(\nb)DampingF\nield (mT)(a)FIG. 6. (Color online). Statistical view of the Gilbert damping\nparameters obtained from the fitting of the three lowest spin-wave\nresonances of a device of 90 nm diameter. For each applied field,\nthe spectral line shapes of the quasi-uniform mode is fitted using\nd\ndHzV2;acfunction (red symbols) while the lineshapes of the two\nnext modes are fitted withd\ndHzV1;acfunction (blue and black). The\npanel (a) gathers the values of the damping parameters that best fit\neach spectrum recorded at a given applied field. Panel (b) displays\nthe histogram of the distribution of these estimates of the Gilbert\ndamping for the non-uniform modes (dashed-dotted line histogram\nand its blue Gaussian guide to the eye, of half width 0.0009) and\na Gaussian fit (red curve) of the distribution for the quasi-uniform\nmode, of half width 0.0015.\nSeveral points are worth to remind:\n(i) The dctransverse susceptibilities are \u001fdc\nxx=Ms=H0and\n\u001fdc\nyx= 0.\n(ii) The in-phase transverse susceptibility \u001fxxis peaked at\nthe FMR condition !0=H0. It reaches\u001fFMR\nxx =\u00001\n2i\u000b\u001fdc\nxx.\n(iii) When near the FMR condition, we have \u001fyx\u0019\u0000i\u001fxx.\nHence the two transverse components of the magnetization\nare in quadrature and the forced precession is essentially\ncircular. Note that this holds true despite the fact that\nthe pumping field is linearly polarized (i.e. along (x)only).\nIt would also remain true for other (e.g STT) pumping torques.\nFrom Eq. 5 we deduce the classical expressions for the real\nand imaginary parts of the transverse susceptibility:\n<e(\u001fxx) =MsH0(H02\u0000!02)\n4\u000b2H02!02+ (H02\u0000!02)2(7)\n=m(\u001fxx) =\u00002\u000bMs!0H02\n4\u000b2H02!02+ (H02\u0000!02)2(8)\nThe lineshapes given by the above expressions are shown\nin Fig. 7. Their main properties are summarized in Table I.10\nB. Longitudinal non-linear susceptibility\nLet us now express the non-linear change of the longitu-\ndinal magnetization \u0001Mzthat occurs due to the precession.\nThis can be viewed as an rf-induced reduction of the rema-\nnence. Using the circularity of the precession near the FMR\nresonance and the conservation of the magnetization norm to\nsecond order in mx;y, one gets:\n\u0001Mz\u0019jjhxjj2\n2MSM2\nsH02\n[H02\u0000!02]2+ 4\u000b2H02!02(9)\nwherejjhxjjis the (constant) amplitude of the applied rffield.\nIt is worth noticing that the longitudinal loss of the magneti-\nzation is stationary (constant in time) despite the fact that the\nmagnetization precesses continuously.\nC. Lineshapes and linewidths of the susceptibiliies and their\nderivatives\nThe different susceptibility expressions are plotted in Fig. 7.\nThe lineshape of the functions \u0001Mzand\u001fxxare essentially\ndetermined by their denominator which are the fast varying\nfunctions of eqs. 8 and 9. As \u001fxxand\u0001Mzare two signatures\nof the same resonance process, their denominators are equal\n(see eqs. 8 and 9) and a simple algebra confirms that =m(\u001fxx)\nand\u0001Mzlead to the same frequency or field linewidths (Half\nWidth at Half Maximum) which are:\n\u0001!= 2\u000b!or equivalently, \u0001Hz= 2\u000bH0(10)\nNote that \u0001!(resp. \u0001Hz) is also the frequency (resp. field)\nspacing between the positive maximum and negative maxi-\nmum of<e(\u001fxx)about the FMR condition.\nWe stress that this linewidth is different from that obtained\nby conventional FMR in which people examine the derivative\nof the absorption signald=m(\u001fxx)\ndHzversusHz. The peak-to-\npeak separation of the conventional FMR signal is \u0001Hz=\n2p\n3\u000bH0=2p\n3\u000b!0. The factor 2=p\n3is 1.1547. The spectral\nshapes of<e(\u001fxx)and\u0001Mzas deduced above are to be used\nin the main part of the paper to describe respectively V1;ac\nandV2;ac(Table I).\nD. Linewidth beyond the macrospin approximation\nSome words of caution are needed as the calculations done\nso far the appendix are for the uniform precession mode of a\nmacrospin, while most of our experimental results were ob-\ntained on the higher order (non-uniform) spin-waves. When\nmodeling non-uniform spin-waves, one needs to take into ac-\ncount additional exchange and dipole-dipole terms.\nFor non-uniform spin-waves in perpendicularly magnetized\nsystem, the exchange fields related to the non uniformity of\nthe dynamical magnetizations mxandmycan be added to H0\nto form a new generalized field ~H=H+Hk\u0000Ms+2Ak2\n\u00160MS,\n/s48/s46/s57/s48 /s48/s46/s57/s53 /s49/s46/s48/s48 /s49/s46/s48/s53 /s49/s46/s49/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s45/s49/s48/s48/s46/s57/s48 /s48/s46/s57/s53 /s49/s46/s48/s48 /s49/s46/s48/s53 /s49/s46/s49/s48\n/s77\n/s122\n/s82/s101 /s40 /s41\n/s112/s112/s32/s82/s101/s115/s111/s110/s97/s110/s99/s101/s45/s108/s105/s107/s101\n/s71/s101/s110/s101/s114/s97/s108/s105/s122/s101/s100/s32/s102/s105/s101/s108/s100/s32/s111/s114/s32/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40 /s39/s32/s111/s114/s32/s72/s39/s41/s70/s87 /s72/s77/s100/s100/s73/s109 /s40 /s41\n/s100/s100 /s77\n/s122\n/s111/s114/s100/s100/s82/s101 /s40 /s41\n/s73/s109 /s40 /s41/s32\n/s32/s76/s111/s115/s115/s45/s108/s105/s107/s101/s111/s114FIG. 7. (Color online). Transverse susceptibilities, longitudinal sus-\nceptibility and their derivatives in the PMA macrospin model. The\ncurves are plotted for \u000b= 0:02and a resonance condition of unity.\nThe responses amplitudes have been normalized to ease the compar-\nison between the different lineshapes. The black horizontal segment\nin the bottom panel is the FWHM of =m(\u001fxx)and\u0001Mzor equiva-\nlently the peak-to-peak separation of <e(\u001fxx)(also sketched as the\nblack square dots). The blue segment sketches the peak-to-peak sep-\naration ofd=m(\u001fxx)\nd!0 (also sketched as the empty blue square dots).\nThe area shaped in red is the positive halo that surrounds the main\n(negative) peak ind<e(\u001fxx)\nd!0.\nwherekis a generalized wavevector21,26,28. The additional\nexchange contributions act on mxandmyon equal footing,\nhence they maintain the circularity of the precession. The\nGilbert linewidth for a non-uniform spin-wave is simply35,36:\n\u0001!=\u000b!k@!k\n\r0@~H(11)\nWhere this equation holds as ~His a circular term. As a result\nof Eq. 11 the proportionality of an eigemode linewidth to its\nfrequency (Eq. 10) is not broken by the exchange contribu-\ntions in non-uniform spin-waves.\nConversely, the directional nature of the dipole-dipole in-\nteraction is such that the related effective fields act differently\non the two dynamical magnetizations mxandmyfor spin-\nwaves having a non-radial character. As a result the dynamic\ndemagnetizing fields can not be simply added to the circu-\nlar precession H0term and they induce some ellipticity of\nthe precession of the non-uniform spin-waves. Dipole-dipole\ninteractions make the eigenmode frequency non-linear with\nthe field (see Eq. 52 in ref. 22). For the lowest lying non-\nuniform spin-wave the dipole-dipole stiffness field is MSkt=2\nwithk\u0019\u0019=a. It is negligible against the generalized field\n~Hfor the thickness used in practice in PMA systems meant\nfor spin-torque applications. The precession stays therefore\nessentially circular for all the modes observed experimentally11\nhere; we will thus consider that it is legitimate to use Eq. 10\nand deduce the damping from the ratio of the half frequency\nlinewidth to the eigenmode frequency.\nFinally, we would like to mention that our method is notrestricted to the PMA materials only: it should hold when the\nconsidered spin-waves are quasi-circular. In particular, this is\nthe case of exchange-dominated spin waves in in-plane mag-\nnetized systems37.\n\u0003thibaut.devolder@u-psud.fr\n1C. Kittel, Physical Review 73, 155 (1948).\n2C. Bilzer, T. Devolder, P. Crozat, C. Chappert, S. Cardoso, and\nP. P. Freitas, Journal of Applied Physics 101, 074505 (2007).\n3T. Kubota, J. Hamrle, Y . Sakuraba, O. Gaier, M. Oogane,\nA. Sakuma, B. Hillebrands, K. Takanashi, and Y . 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Hillebrands, and\nE. T. Papaioannou, arXiv:1702.05298 [cond-mat] (2017), arXiv:\n1702.05298.20A. A. Baker, A. I. Figueroa, D. Pingstone, V . K. Lazarov, G. v. d.\nLaan, and T. Hesjedal, Scientific Reports 6, srep35582 (2016).\n21O. Klein, G. de Loubens, V . V . Naletov, F. Boust, T. Guillet,\nH. Hurdequint, A. Leksikov, A. N. Slavin, V . S. Tiberkevich, and\nN. Vukadinovic, Physical Review B 78, 144410 (2008).\n22B. A. Kalinikos and A. N. Slavin, Journal of Physics C: Solid State\nPhysics 19, 7013 (1986).\n23K. Mizunuma, M. Yamanouchi, H. Sato, S. Ikeda, S. Kanai,\nF. Matsukura, and H. Ohno, Applied Physics Express 6, 063002\n(2013).\n24G. N. Kakazei, P. E. Wigen, K. Y . Guslienko, V . Novosad, A. N.\nSlavin, V . O. Golub, N. A. Lesnik, and Y . Otani, Applied Physics\nLetters 85, 443 (2004).\n25R. E. Arias and D. L. Mills, Physical Review B 79, 144404 (2009).\n26V . V . Naletov, G. de Loubens, G. Albuquerque, S. Borlenghi,\nV . Cros, G. Faini, J. Grollier, H. Hurdequint, N. 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Vansteenkiste, B. V . Waeyenberge, A. N. Kuchko,\nV . V . Kruglyak, and H. Fangohr, Physical Review B 92, 054430\n(2015).\n34S. Petit, C. Baraduc, C. Thirion, U. Ebels, Y . Liu, M. Li, P. Wang,\nand B. Dieny, Physical Review Letters 98, 077203 (2007).\n35A. N. Slavin and P. Kabos, IEEE Transactions on Magnetics 41,\n1264 (2005).\n36J.-V . Kim, Physical Review B 73, 174412 (2006).\n37M. Dvornik and V . V . Kruglyak, Physical Review B 84, 140405\n(2011)."    },    {        "title": "1704.01559v1.Relativistic_theory_of_magnetic_inertia_in_ultrafast_spin_dynamics.pdf",        "content": "Relativistic theory of magnetic inertia in ultrafast spin dynamics\nRitwik Mondal,\u0003Marco Berritta, Ashis K. Nandy, and Peter M. Oppeneer\nDepartment of Physics and Astronomy, Uppsala University, P. O. Box 516, SE-75120 Uppsala, Sweden\n(Dated: November 12, 2018)\nThe influence of possible magnetic inertia effects has recently drawn attention in ultrafast mag-\nnetization dynamics and switching. Here we derive rigorously a description of inertia in the\nLandau-Lifshitz-Gilbert equation on the basis of the Dirac-Kohn-Sham framework. Using the Foldy-\nWouthuysen transformation up to the order of 1=c4gives the intrinsic inertia of a pure system\nthrough the 2ndorder time-derivative of magnetization in the dynamical equation of motion. Thus,\nthe inertial damping Iis a higher order spin-orbit coupling effect, \u00181=c4, as compared to the\nGilbert damping \u0000that is of order 1=c2. Inertia is therefore expected to play a role only on ultra-\nshort timescales (sub-picoseconds). We also show that the Gilbert damping and inertial damping\nare related to one another through the imaginary and real parts of the magnetic susceptibility tensor\nrespectively.\nPACS numbers: 71.15.Rf, 75.78.-n, 75.40.Gb\nI. INTRODUCTION\nThe foundation of contemporary magnetization dy-\nnamics is the Landau-Lifshitz-Gilbert (LLG) equation\nwhich describes the precession of spin moment and a\ntransverse damping of it, while keeping the modulus of\nmagnetization vector fixed [1–3]. The LLG equation of\nmotion was originally derived phenomenologically and\nthe damping of spin motion has been attributed to rela-\ntivistic effects such as the spin-orbit interaction [1, 4–6].\nIn recent years there has been a flood of proposals for the\nfundamental microscopic mechanism behind the Gilbert\ndamping: the breathing Fermi surface model of Kamber-\nský, where the damping is due to magnetization preces-\nsion and the effect of spin-orbit interaction at the Fermi\nsurface [4], the extension of the breathing Fermi surface\nmodel to the torque-torque correlation model [5, 7], scat-\ntering theory description [8], effective field theories [9],\nlinear response formalism within relativistic electronic\nstructure theory [10], and the Dirac Hamiltonian theory\nformulation [11].\nFor practical reasons it was needed to extend the orig-\ninal LLG equation to include several other mechanisms\n[12, 13]. To describe e.g. current induced spin-transfer\ntorques, the effects of spin currents have been taken\ninto account [14–16], as well as spin-orbit torques [17],\nand the effect of spin diffusion [18]. A different kind of\nspin relaxation due to the exchange field has been intro-\nduced by Bar’yakhtar et al.[19]. In the Landau-Lifshitz-\nBar’yakhtar equation spin dissipations originate from the\nspatial dispersion of exchange effects through the second\norder space derivative of the effective field [20, 21]. A\nfurther recent work predicts the existence of extension\nterms that contain spatial as well temporal derivatives of\nthe local magnetization [22].\nAnother term, not discussed in the above investiga-\ntions, is the magnetic inertial damping that has recently\n\u0003ritwik.mondal@physics.uu.sedrawnattention[23–25]. Originally, magneticinertiawas\ndiscussed following the discovery of earth’s magnetism\n[26]. Within the LLG framework, inertia is introduced\nas an additional term [24, 27–29] leading to a modified\nLLG equation,\n@M\n@t=\u0000\rM\u0002He\u000b+M\u0002\u0012\n\u0000@M\n@t+I@2M\n@2t\u0013\n;(1)\nwhere \u0000is the Gilbert damping constant [1–3], \rthe gy-\nromagnetic ratio, He\u000bthe effective magnetic field, and I\nis the inertia of the magnetization dynamics, similar to\nthe mass in Newton’s equation. This type of motion has\nthe same classical analogue as the nutation of a spinning\nsymmetric top. The potential importance of inertia is il-\nlustrated in Fig. 1. While Gilbert damping slowly aligns\nthe precessing magnetization to the effective magnetic\nfield, inertial dynamics causes a trembling or nutation of\nthe magnetization vector [24, 30, 31]. Nutation could\nconsequently pull the magnetization toward the equa-\ntor and cause its switching to the antiparallel direction\n[32, 33], whilst depending crucially on the strength of\nthe magnetic inertia. The parameter Ithat character-\nizes the nutation motion is in the most general case a\ntensor and has been associated with the magnetic suscep-\ntibility [29, 31, 33]. Along a different line of reasoning,\nFähnle et al.extended the breathing Fermi surface model\nto include the effect of magnetic inertia [27, 34]. The\ntechnological importance of nutation dynamics is thus\nits potential to steer magnetization switching in memory\ndevices [23–25, 32] and also in skyrmionic spin textures\n[35]. Magnetization dynamics involving inertial dynam-\nics has been investigated recently and it was suggested\nthat its dynamics belongs to smaller time-scales i.e., the\nfemtosecond regime [24]. However, the origin of inertial\ndamping from a fundamental framework is still missing,\nand, moreover, although it is possible to vary the size of\nthe inertia in spin-dynamics simulations, it is unknown\nwhat the typical size of the inertial damping is.\nNaturally the question arises whether it is possible\nto derive the extended LLG equation including iner-arXiv:1704.01559v1  [cond-mat.other]  20 Mar 20172\nMHeff\nPrecession\nNutation\nFigure 1. (Color online) Schematic illustration of magnetiza-\ntion dynamics. The precessional motion of Maround He\u000bis\ndepicted by the blue solid-dashed curve and the nutation is\nshown by the red curve.\ntia while starting from the fully relativistic Dirac equa-\ntion. Hickey and Moodera [36] started from a Dirac\nHamiltonian and obtained an intrinsic Gilbert damping\nterm which originated from spin-orbit coupling. How-\never they started from only a part of the spin-orbit cou-\npling Hamiltonian which was anti-hermitian [37, 38]. A\nrecent derivation based on Dirac Hamiltonian theory for-\nmulation [11] showed that the Gilbert damping depends\nstrongly on both interband and intraband transitions\n(consistent with Ref. [39]) as well as the magnetic sus-\nceptibility response function, \u001fm. This derivation used\nthe relativistic expansion to the lowest order 1=c2of the\nhermitian Dirac-Kohn-Sham (DKS) Hamiltonian includ-\ning the effect of exchange field [40].\nIn this article we follow an approach similar to that of\nRef. [11] but we consider higher order expansion terms\nof the DKS Hamiltonian up to the order of 1=c4. This is\nshown to lead to the intrinsic inertia term in the modi-\nfied LLG equation and demonstrates that it stems from\na higher-order spin-orbit coupling term. A relativistic\norigin of the spin nutation angle, caused by Rashba-like\nspin-orbit coupling, was previously concluded, too, in the\ncontext of semiconductor nanostructures [41, 42].\nIn the following, we derive in Sec. II the relativistic\ncorrection terms to the extended Pauli Hamiltonian up\nto the order of 1=c4, which includes the spin-orbit inter-\naction and an additional term. Then the corresponding\nmagnetization dynamics is computed from the obtained\nspin Hamiltonian in Sec. III, which is shown to contain\nthe Gilbert damping and the magnetic inertial damping.\nFinally, we discuss the size of the magnetic inertia in re-\nlation to other earlier studies.II. RELATIVISTIC HAMILTONIAN\nFORMULATION\nWe start our derivation with a fully relativistic par-\nticle, a Dirac particle [43] inside a material and in the\npresence of an external field, for which we write the DKS\nHamiltonian:\nH=c\u000b\u0001(p\u0000eA) + (\f\u0000 1)mc2+V 1\n=O+ (\f\u0000 1)mc2+E; (2)\nwhereVis the effective crystal potential created by the\nion-ion, ion-electron and electron-electron interactions,\nA(r;t)is the magnetic vector potential from the external\nfield,cis the speed of light, mis particle’s mass and 1\nis the 4\u00024unit matrix. \u000band\fare the Dirac matrices\nthat have the form\n\u000b=\u0012\n0\u001b\n\u001b0\u0013\n; \f =\u0012\n10\n0\u00001\u0013\n;\nwhere\u001bis the Pauli spin matrix vector and 1is2\u00022unit\nmatrix. TheDiracequationisthenwrittenas i~@ (r;t)\n@t=\nH (r;t)for a Dirac bi-spinor  . The quantityO=c\u000b\u0001\n(p\u0000eA)defines the off-diagonal, or odd terms in the\nmatrix formalism and E=V 1are the diagonal, i.e., even\nterms. The latter have to be multiplied by a 2\u00022block\ndiagonal unit matrix in order to bring them in a matrix\nform. To obtain the nonrelativistic Hamiltonian and the\nrelativistic corrections one can write down the Dirac bi-\nspinor in double two component form as\n (r;t) =\u0012\n\u001e(r;t)\n\u0011(r;t)\u0013\n;\nand substitute those into the Dirac equation. The up-\nper two components represent the particle and the lower\ntwo components represent the anti-particle. However the\nquestion of separating the particle’s and anti-particle’s\nwave functions is not clear for any given momentum. As\nthe part\u000b\u0001pis off-diagonal in the matrix formalism, it\nretains the odd components and thus links the particle-\nantiparticle wave function. One way to eliminate the an-\ntiparticle’s wave function is by an exact transformation\n[44] which gives terms that require a further expansion in\npowers of 1=c2. Another way is to search for a represen-\ntation where the odd terms become smaller and smaller\nand one can ignore those with respect to the even terms\nand retain only the latter [45]. The Foldy-Wouthuysen\n(FW) transformation [46, 47] was the very successful at-\ntempt to find such a representation.\nIt is an unitary transformation obtained by suitably\nchoosing the FW operator,\nUFW=\u0000i\n2mc2\fO: (3)\nThe minus sign in front of the operator is because \fand\nOanti-commute with each other. The transformation of\nthewavefunctionadoptstheform  0(r;t) =eiUFW (r;t)3\nsuch that the probability density remains the same,\nj j2=j 0j2. The time-dependent FW transformation\ncan be expressed as [45, 48]\nHFW=eiUFW\u0012\nH\u0000i~@\n@t\u0013\ne\u0000iUFW+i~@\n@t:(4)\nThe first term can be expanded in a series as\neiUFWHe\u0000iUFW=H+i[UFW;H] +i2\n2![UFW;[UFW;H]]\n+::::+in\nn![UFW;[UFW;:::[UFW;H]:::]] +::: :(5)\nThe time dependency enters through the second term of\nEq. (4) and for a time-independent transformation one\nworks with@UFW\n@t= 0. It is instructive to note that the\naim of the whole procedure is to make the odd termssmaller and one can notice that as it goes higher and\nhigher in the expansion, the corresponding coefficients\ndecrease of the order 1=c2due to the choice of the unitary\noperator. After a first transformation, the new Hamilto-\nnian will contain new even terms, E0, as well as new odd\nterms,O0of1=c2or higher. The latter terms can be used\nto perform a next transformation having the new unitary\noperator as U0\nFW=\u0000i\n2mc2\fO0. After a second transfor-\nmation the new Hamiltonian, H0\nFWis achieved that has\nthe odd terms of the order 1=c4or higher. The trans-\nformation is a repetitive process and it continues until\nthe separation of positive and negative energy states are\nguaranteed.\nAfter a fourth transformation we derive the new trans-\nformed Hamiltonian with all the even terms that are cor-\nrect up to the order of1\nm3c6as [48–50]\nH000\nFW= (\f\u0000 1)mc2+\f\u0012O2\n2mc2\u0000O4\n8m3c6\u0013\n+E\u00001\n8m2c4h\nO;[O;E] +i~_Oi\n+\f\n16m3c6fO;[[O;E];E]g+\f\n8m3c6n\nO;h\ni~_O;Eio\n+\f\n16m3c6n\nO;(i~)2Oo\n: (6)\nNote that [A;B]defines the commutator, while fA;Bgrepresents the anti-commutator for any two operators Aand\nB. A similar Foldy-Wouthuysen transformation Hamiltonian up to an order of 1=m3c6was derived by Hinschberger\nand Hervieux in their recent work [51], however there are some differences, for example, the first and second terms in\nthe second line of Eq. (6) were not given. Once we have the transformed Hamiltonian as a function of odd and even\nterms, the final form is achieved by substituting the correct form of odd terms Oand calculating term by term.\nEvaluating all the terms separately, we derive the Hamiltonian for only the positive energy solutions i.e. the upper\ncomponents of the Dirac bi-spinor as a 2\u00022matrix formalism [40, 51, 52]:\nH000\nFW=(p\u0000eA)2\n2m+V\u0000e~\n2m\u001b\u0001B\u0000(p\u0000eA)4\n8m3c2\u0000e~2\n8m2c2r\u0001Etot+e~\n8m3c2n\n(p\u0000eA)2;\u001b\u0001Bo\n\u0000e~\n8m2c2\u001b\u0001[Etot\u0002(p\u0000eA)\u0000(p\u0000eA)\u0002Etot]\n\u0000e~2\n16m3c4f(p\u0000eA);@tEtotg\u0000ie~2\n16m3c4\u001b\u0001[@tEtot\u0002(p\u0000eA) + (p\u0000eA)\u0002@tEtot]; (7)\nwhere@t\u0011@=@tdefines the first-order time derivative.\nThehigherorderterms( 1=c6ormore)willinvolvesimilar\nformulations and more and more time derivatives of the\nmagnetic and electric fields will appear that stem from\nthe time derivative of the odd operator O[48, 51].\nThe fields in the last Hamiltonian (7) are defined as\nB=r\u0002A, the external magnetic field, Etot=Eint+\nEextare the electric fields where Eint=\u00001\nerVis the\ninternal field that exists even without any perturbation\nandEext=\u0000@A\n@tis the external field (only the temporal\npart is retained here because of the Coulomb gauge).\nThe spin Hamiltonian\nThe aim of this work is to formulate the magnetiza-\ntion dynamics on the basis of this Hamiltonian. Thus,we split the Hamiltonian into spin-independent and spin-\ndependentpartsandconsiderfromnowonelectrons. The\nspin Hamiltonian is straightforwardly given as\nHS(t) =\u0000e\nmS\u0001B+e\n4m3c2n\n(p\u0000eA)2;S\u0001Bo\n\u0000e\n4m2c2S\u0001[Etot\u0002(p\u0000eA)\u0000(p\u0000eA)\u0002Etot]\n\u0000ie~\n8m3c4S\u0001[@tEtot\u0002(p\u0000eA) + (p\u0000eA)\u0002@tEtot];\n(8)\nwhere the spin operator S= (~=2)\u001bhas been used. Let\nus briefly explain the physical meaning behind each term\nthat appears inHS(t). The first term defines the Zee-\nman coupling of the electron’s spin with the externally\napplied magnetic field. The second term defines an indi-\nrect coupling of light to the Zeeman interaction of spin\nand the optical B-field, which can be be shown to have4\nthe form of a relativistic Zeeman-like term. The third\nterm implies a general form of the spin-orbit coupling\nthat is gauge invariant [53], and it includes the effect of\nthe electric field from an internal as well as an external\nfield. Thelasttermisthenewtermofrelevanceherethat\nhas only been considered once in the literature by Hin-\nschberger et al.[51]. Note that, although the last term\nin Eq. (8) contains the total electric field, only the time-\nderivative of the external field plays a role here, because\nthe time derivative of internal field is zero as the ionic\npotential is time independent. In general if one assumes\na plane-wave solution of the electric field in Maxwell’s\nequation asE=E0ei!t, the last term can be written as\ne~!\n8m3c4S\u0001(E\u0002p)and thus adopts the form of a higher-\norder spin-orbit coupling for a general E-field.\nThe spin-dependent part can be easily rewritten in a\nshorter format using the identities:\nA\u0002(p\u0000eA)\u0000(p\u0000eA)\u0002A= 2A\u0002(p\u0000eA)\n+i~r\u0002A (9)\nA\u0002(p\u0000eA) + (p\u0000eA)\u0002A=\u0000i~r\u0002A(10)\nfor any operator A. This allows us to write the spin\nHamiltonian as\nHS=\u0000e\nmS\u0001B+e\n2m3c2S\u0001B\u0014\np2\u00002eA\u0001p+3e2\n2A2\u0015\n\u0000e\n2m2c2S\u0001\u0002\nEtot\u0002(p\u0000eA)\u0003\n+ie~\n4m2c2S\u0001@tB\n+e~2\n8m3c4S\u0001@ttB: (11)\nHere, the Maxwell’s equations have been used to derive\nthe final form that the spatial derivative of the electric\nfield will generate a time derivative of a magnetic field\nsuch that r\u0002Eext=\u0000@B\n@t, whilst the curl of a internal\nfield results in zero as the curl of a gradient function is\nalways zero. The final spin Hamiltonian (11) bears much\nimportance for the strong laser field-matter interaction\nas it takes into account all the field-spin coupling terms.\nIt is thus the appropriate fundamental Hamiltonian to\nunderstand the effects of those interactions on the mag-\nnetization dynamics described in the next section.\nIII. MAGNETIZATION DYNAMICS\nIn general, magnetization is given by the magnetic mo-\nment per unit volume in a magnetic solid. The magnetic\nmomentisgivenby g\u0016BhSi,wheregistheLandég-factor\nand\u0016Bis the unit of Bohr magneton. The magnetization\nis then written\nM(r;t) =X\njg\u0016B\n\nhSji; (12)\nwhere \nis the suitably chosen volume element, the sum\njgoes over all electrons in the volume element, and h::i\nis the expectation value. To derive the dynamics, wetake the time derivative in both the sides of Eq. (12)\nand, withintheadiabaticapproximation, wearriveatthe\nequation of motion for the magnetization as [36, 54, 55]\n@M\n@t=X\njg\u0016B\n\n1\ni~h\u0002\nSj;HS(t)\u0003\ni:(13)\nNow the task looks simple, one needs to substitute the\nspin Hamiltonian (11) and calculate the commutators in\norder to find the equation of motion. Note that the dy-\nnamics only considers the local dynamics as we have not\ntaken into account the time derivative of particle density\noperator (for details, see [11]). Incorporating the latter\nwould give rise the local as well as non-local processes\n(i.e., spin currents) within the same footing.\nThe first term in the spin Hamiltonian produces the\ndynamics as\n@M(1)\n@t=\u0000\rM\u0002B; (14)\nwith\r=gjej=2mdefines the gyromagnetic ratio and\nthe Landé g-factor g\u00192for spins, the electronic charge\ne < 0. Using the linear relationship of magnetization\nwith the magnetic field B=\u00160(H+M), the latter is\nreplaced in Eq. (14) to get the usual form in the Landau-\nLifshitz equations, \u0000\r0M\u0002H, where\r0=\u00160\ris the\neffective gyromagnetic ratio. This gives the Larmor pre-\ncessionofmagnetizationaroundaneffectivefield H. The\neffective field will always have a contribution from a ex-\nchange field and the relativistic corrections to it, which\nhas not been explicitly taken into account in this article,\nas they are not in the focus here. For detailed calcula-\ntions yet including the exchange field see Ref. [11].\nThe second term in the spin Hamiltonian Eq. (11) will\nresult in a relativistic correction to the magnetization\nprecession. Within an uniform field approximation (A=\nB\u0002r=2), the corresponding dynamics will take the form\n@M(2)\n@t=\r\n2m2c2M\u0002BD\np2\u0000eB\u0001L+3e2\n8(B\u0002r)2E\n;\n(15)\nwithL=r\u0002pthe orbital angular momentum. The\npresence of \r=2m2c2implies that the contribution of this\ndynamics to the precession is relatively small, while the\nleading precession dynamics is given by Eq. (14). For\nsake of completeness we note that a relativistic correc-\ntion to the precession term of similar order 1=m2c2was\nobtained previously for the exchange field [11].\nThe next term in the Hamiltonian is a bit tricky to\nhandle as the third term in Eq. (11) is not hermitian, not\neven the fourth term which is anti-hermitian. However\ntogether they form a hermitian Hamiltonian [11, 37, 38].\nTherefore one has to work together with those terms and\ncannot only perform the dynamics with an individual\nterm. In an earlier work [11] we have shown that taking\nanuniformmagneticfieldalongwiththegauge A=B\u0002r\n2\nwill preserve the hermiticity. The essence of the uniform\nfield lies in the assumption that the skin depth of the5\nelectromagnetic field is longer than the thickness of the\nthin-film samples used in experiments. The dynamical\nequation of spin motion with the second and third terms\nthus thus be written in a compact form for harmonic ap-\nplied fields as [11]\n@M(3;4)\n@t=M\u0002\u0012\nA\u0001@M\n@t\u0013\n; (16)\nwith the intrinsic Gilbert damping parameter Athat is\na tensor defined by\nAij=\r\u00160\n4mc2X\nn;kh\nhripk+pkrii\u0000hrnpn+pnrni\u000eiki\n\u0002\u0010\n1+\u001f\u00001\nm\u0011\nkj:(17)\nHere\u001fmis the magnetic susceptibility tensor of rank 2\n(a3\u00023matrix) and 1is the 3\u00023unit matrix. Note that\nfor diagonal terms i.e., i=kthe contributions from the\nexpectation values of rkpicancel each other. The damp-\ning tensor can be decomposed to have contributions from\nan isotropic Heisenberg-like, anisotropic Ising-like and\nDzyaloshinskii-Moriya-like tensors. The anti-symmetric\nDzyaloshinskii-Moriya contribution has been shown to\nlead to a chiral damping of the form M\u0002(D\u0002@M=@t)\n[11]. Experimental observations of chiral damping have\nbeen reported recently [56]. The other cross term having\nthe formE\u0002Ain Eq. (11) is related to the angular mo-\nmentum of the electromagnetic field and thus provides\na torque on the spin that has been at the heart of an-\ngular magneto-electric coupling [53]. A possible effect in\nspin dynamics including the light’s angular momentum\nhas been investigated in the strong field regime and it\nhas been shown that one has to include this cross term\nin the dynamics in order to explain the qualitative and\nquantitative strong field dynamics [57].\nFor the last term in the spin Hamiltonian (11) it is\nrather easy to formulate the spin dynamics because it is\nevidently hermitian. Working out the commutator with\nthe spins gives a contribution to the dynamics as\n@M(5)\n@t=\u000eM\u0002@2B\n@t2; (18)\nwith the constant \u000e=\r~2\n8m2c4.\nLet us work explicitly with the second-order time\nderivative of the magnetic induction by the relation B=\n\u00160(H+M), using a chain rule for the derivative:\n@2B\n@t2=@\n@t\u0010@B\n@t\u0011\n=\u00160@\n@t\u0010@H\n@t+@M\n@t\u0011\n=\u00160\u0010@2H\n@t2+@2M\n@t2\u0011\n: (19)\nThis is a generalized equation for the time-derivative of\nthe magnetic induction which can be used even for non-\nharmonic fields. The magnetization dynamics is then\ngiven by\n@M(5)\n@t=\u00160\u000eM\u0002\u0010@2H\n@t2+@2M\n@t2\u0011\n:(20)Thus the extended LLG equation of motion will have\nthese two additional terms: (1) a field-derivative torque\nand (2) magnetization-derivative torque, and they ap-\npear with their 2ndorder time derivative. It deserves to\nbe noted that, in a previous theory we also obtained a\nsimilar term–a field-derivative torque in 1storder-time\nderivative appearing in the generalized Gilbert damping.\nSpecifically, the extended LLG equation for a general\ntime-dependent field H(t)becomes\n@M\n@t=\u0000\r0M\u0002H+M\u0002h\n\u0016A\u0001\u0010@H\n@t+@M\n@t\u0011i\n+\u00160\u000eM\u0002\u0010@2H\n@t2+@2M\n@t2\u0011\n; (21)\nwhere \u0016AisamodifiedGilbertdampingtensor(fordetails,\nsee [11]).\nHowever for harmonic fields, the response of the ferro-\nmagnetic materials is measured through the differential\nsusceptibility, \u001fm=@M=@H, because there exists a net\nmagnetization even in the absence of any applied field.\nWith this, the time derivative of the harmonic magnetic\nfield can be further written as:\n@2H\n@t2=@\n@t\u0010@H\n@M@M\n@t\u0011\n=@\n@t\u0010\n\u001f\u00001\nm\u0001@M\n@t\u0011\n=@\u001f\u00001\nm\n@t\u0001@M\n@t+\u001f\u00001\nm\u0001@2M\n@t2: (22)\nIn general the magnetic susceptibility is a spin-spin re-\nsponse function that is wave-vector and frequency depen-\ndent. Thus, Eq. (18) assumes the form with the first and\nsecond order time derivatives as\n@M(5)\n@t=M\u0002\u0012\nK\u0001@M\n@t+I\u0001@2M\n@t2\u0013\n;(23)\nwhere the parameters Iij=\u00160\u000e\u0000\n1+\u001f\u00001\nm\u0001\nijandKij=\n\u00160\u000e@t(\u001f\u00001\nm)ijare tensors. The dynamics of the second\nterm is that of the magnetic inertia that operates on\nshorter time scales [25].\nHaving all the required dynamical terms, finally the\nfull magnetization dynamics can be written by joining\ntogether all the individual parts. Thus the full magneti-\nzation dynamics becomes, for harmonic fields,\n@M\n@t=M\u0002\u0012\n\u0000\r0H+ \u0000\u0001@M\n@t+I\u0001@2M\n@t2\u0013\n:(24)\nNote that the Gilbert damping parameter \u0000has two con-\ntributions, one from the susceptibility itself, Aij, which\nis of order 1=c2and an other from the time derivative of\nit,Kijof order 1=c4. Thus, \u0000ij=Aij+Kij. However we\nwill focus on the first one only as it will obviously be the\ndominant contribution, i.e., \u0000ij\u0019Aij. Even though we\nconsider only the Gilbert damping term of order 1=c2in\nthe discussions, we shall explicitly analyze the other term\nof the order 1=c4. For an ac susceptibility i.e., \u001f\u00001\nm/ei!t\nwe find thatKij/\u00160\u000e@t(\u001f\u00001\nm)ij/i\u00160!\u000e\u001f\u00001\nm, which\nsuggests again that the Gilbert damping parameter of6\nthe order 1=c4will be given by the imaginary part of the\nsusceptibility,Kij/\u0000\u00160!\u000eIm\u0000\n\u001f\u00001\nm\u0001\n.\nThe last equation (24) is the central result of this\nwork, as it establishes a rigorous expression for the in-\ntrinsic magnetic inertia. Magnetization dynamics in-\ncluding inertia has been discussed in few earlier articles\n[24, 30, 31, 58]. The very last term in Eq. (24) has been\nassociated previously with the inertia magnetization dy-\nnamics [32, 59, 60]. As mentioned, it implies a magne-\ntization nutation i.e., a changing of the precession angle\nas time progresses. Without the inertia term we obtain\nthe well-known LLG equation of motion that has already\nbeen used extensively in magnetization dynamics simu-\nlations (see, e.g., [61–65]).\nIV. DISCUSSIONS\nMagnetic inertia was discussed first in relation to the\nearth’s magnetism [26]. From a dimensional analysis,\nthe magnetic inertia of a uniformly magnetized sphere\nundergoing uniform acceleration was estimated to be of\nthe order of 1=c2[26], which is consistent with the here-\nobtained relativistic nature of magnetic inertia.\nOur derivation based on the fundamental Dirac-Kohn-\nSham Hamiltonian provides explicit expressions for both\nthe Gilbert and inertial dampings. Thus, a comparison\ncan be made between the Gilbert damping parameter\nand the magnetic inertia parameter of a pure system.\nAs noticed above, both the parameters are given by the\nmagnetization susceptibility tensor, however it should be\nnoted that the quantiy hr\u000bp\fiis imaginary itself, because\n[11],\nhr\u000bp\fi=\u0000i~\n2mX\nn;n0;kf(Enk)\u0000f(En0k)\nEnk\u0000En0kp\u000b\nnn0p\f\nn0n:(25)\nThus the Gilbert damping parameter should be given by\nthe imaginary part of the susceptibility tensor [36, 66].\nOn the other hand the magnetic inertia tensor must be\ngiven by the real part of the susceptibility [31]. This is\nin agreement with a recent article where the authors also\nfound the same dependence of real and imaginary parts\nof susceptibility to the nutation and Gilbert damping re-\nspectively [33]. In our calculation, the Gilbert damping\nand inertia parameters adopt the following forms respec-tively,\n\u0000ij=i\r\u00160\n4mc2X\nn;k[hripk+pkrii\u0000hrnpn+pnrni\u000eik]\n\u0002Im\u0000\n\u001f\u00001\nm\u0001\nkj\n=\u0000\u00160\r~\n4mc2X\nn;k\u0014hripk+pkrii\u0000hrnpn+pnrni\u000eik\ni~\u0015\n\u0002Im\u0000\n\u001f\u00001\nm\u0001\nkj\n=\u0000\u0010X\nn;k\u0014hripk+pkrii\u0000hrnpn+pnrni\u000eik\ni~\u0015\n\u0002Im\u0000\n\u001f\u00001\nm\u0001\nkj;(26)\nIij=\u00160\r~2\n8m2c4h\n1+Re\u0000\n\u001f\u00001\nm\u0001\nkji\n=\u0010~\n2mc2h\n1+Re\u0000\n\u001f\u00001\nm\u0001\nkji\n; (27)\nwith\u0010\u0011\u00160\r~\n4mc2. Note that the change of sign from damp-\ningtensortotheinertiatensorthatisalsoconsistentwith\nRef. [33], and also a factor of 2 present in inertia. How-\never, most importantly, the inertia tensor is ~=mc2times\nsmallerthan the damping tensor as is revealed in our\ncalculations. Considering atomic units we can evaluate\n\u0010\u0018\u00160\n4c2\u00180:00066\n4\u00021372\u00188:8\u000210\u00009;\n\u0010~\n2mc2\u0018\u0010\n2c2\u00188:8\u000210\u00009\n2\u00021372\u00182:34\u000210\u000013:\nThis implies that the intrinsic inertial damping is typi-\ncally 4\u0002104times smaller than the Gilbert damping and\nit is not an independently variable parameter. Also, be-\ncause of its smallness magnetic inertial dynamics will be\nmore significant on shorter timescales [24].\nA further analysis of the two parameters can be made.\nOnecanusetheKramers-Kronigtransformationtorelate\nthe real and imaginary parts of a susceptibility tensor\nwith one another. This suggests a relation between the\ntwo parameters that has been found by Fähnle et al.[34],\nnamelyI=\u0000\u0000\u001c,where\u001cisarelaxationtime. Weobtain\nhere a similar relation, I/\u0000 \u0000\u0016\u001c, where \u0016\u001c=~=mc2has\ntime dimension.\nEven though the Gilbert damping is c2times larger\nthan the inertial damping, the relative strength of the\ntwo parameters also depends on the real and imaginary\nparts of the susceptibility tensor. In special cases, when\nthe real part of the susceptibility is much higher than\nthe imaginary part, their strength could be comparable\nto each other. We note in this context that there exist\nmaterials where the real part of the susceptibility is 102\u0000\n103times larger than the imaginary part.\nFinally, we emphasize that our derivation provides the\nintrinsic inertial damping of a pure, isolated system. For\nthe Gilbert damping it is already well known that en-\nvironmental effects, such as interfaces or grain bound-\naries, impurities, film thickness, and even interactions of7\nthe spins with quasi-particles, for example, phonons, can\nmodify the extrinsic damping (see, e.g., [67–69]). Simi-\nlarly, it can be expected that the inertial damping will\nbecome modified through environmental influences. An\nexample of environmental effects that can lead to mag-\nnetic inertia have been considered previously, for the case\nof a local spin moment surrounded by conduction elec-\ntrons, whose spins couple to the local spin moment and\naffect its dynamics [31, 32].\nV. CONCLUSIONS\nIn conclusion, we have rigorously derived the magne-\ntization dynamics from the fundamental Dirac Hamilto-\nnian and have provided a solid theoretical framework for,\nand established the origin of, magnetic inertia in pure\nsystems. We have derived expressions for the Gilbert\ndamping and the magnetic inertial damping on the same\nfooting and have shown that both of them have a rela-\ntivistic origin. The Gilbert damping stems from a gen-\neralized spin-orbit interaction involving external fields,\nwhiletheinertialdampingisduetohigher-order(in 1=c2)\nspin-orbit contributions in the external fields. Both have\nbeen shown to be tensorial quantities. For general time\ndependent external fields, a field-derivative torque with\na 1storder time derivative appears in the Gilbert-type\ndamping, and a 2ndorder time-derivative field torque ap-\npears in the inertial damping.\nIn the case of harmonic external fields, the expressions\nof the magnetic inertia and the Gilbert damping scalewith the real part and the imaginary part, respectively,\nof the magnetic susceptibility tensor, and they are op-\nposite in sign. Alike the Gilbert damping, the magnetic\ninertia tensor is also temperature dependent through the\nmagnetic response function and also magnetic moment\ndependent. Importantly, we find that the intrinsic iner-\ntial damping is much smaller than the Gilbert damping,\nwhich corroborates the fact that magnetic inertia was\nneglected in the early work on magnetization dynamics\n[1–3, 19]. This suggests, too, that the influence of mag-\nnetic inertia will be quite restricted, unless the real part\nof the susceptibility is much larger than the imaginary\npart. Another possibility to enhance the magnetic iner-\ntia would be to use environmental influences to increase\nits extrinsic contribution. 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Lett.\n102, 102401 (2013)."    },    {        "title": "1609.07901v1.Relativistic_theory_of_spin_relaxation_mechanisms_in_the_Landau_Lifshitz_Gilbert_equation_of_spin_dynamics.pdf",        "content": "arXiv:1609.07901v1  [cond-mat.mtrl-sci]  26 Sep 2016Relativistic theory of spin relaxation mechanisms in the La ndau-Lifshitz-Gilbert equation\nof spin dynamics\nRitwik Mondal,∗Marco Berritta, and Peter M. Oppeneer\nDepartment of Physics and Astronomy, Uppsala University, P .O. Box 516, Uppsala, SE-75120, Sweden\n(Dated: October 14, 2018)\nStarting from the Dirac-Kohn-Sham equation we derive the re lativistic equation of motion of\nspin angular momentum in a magnetic solid under an external e lectromagnetic field. This equation\nof motion can be rewritten in the form of the well-known Landa u-Lifshitz-Gilbert equation for a\nharmonic external magnetic field, and leads to a more general magnetization dynamics equation for\na general time-dependent magnetic field. In both cases with a n electronic spin-relaxation term which\nstems from the spin-orbit interaction. We thus rigorously d erive, from fundamental principles, a\ngeneral expression for the anisotropic damping tensor whic h is shown to contain an isotropic Gilbert\ncontribution as well as an anisotropic Ising-like and a chir al, Dzyaloshinskii-Moriya-like contribution.\nThe expression for the spin relaxation tensor comprises fur thermore both electronic interband and\nintraband transitions. We also show that when the externall y applied electromagnetic field possesses\nspin angular momentum, this will lead to an optical spin torq ue exerted on the spin moment.\nPACS numbers: 75.78.-n, 76.20.+q, 71.15.Rf\nI. INTRODUCTION\nIn their seminal 1935-paper, L. D. Landau and E. M.\nLifshitz proposed the equation of motion governing the dy-\nnamics of a continuum magnetization [1]. Eighty years after\nits original formulation, the Landau-Lifshitz (LL) equati on\ncontinues to play a fundamental role in the understanding\nof magnetization dynamics [2] and forms the cornerstone of\ncontemporary micromagnetic simulations (see, e.g., Refs.\n[3, 4]).\nOriginally, the Landau-Lifshitz equation was derived on\nthe basis of phenomenological considerations [1]. It define s\nthe time-evolution of a volume magnetization M(r,t)as\n∂M\n∂t=−γM×Heff−λM×[M×Heff],(1)\nwhereγis the gyromagnetic ratio, Heffis the effective mag-\nnetic field, and λis an isotropic damping parameter. The\nfirst term describes the precession of the local magnetiza-\ntionM(r,t)around the effective field Heff. The second\nterm describes the magnetization relaxation such that the\nmagnetization vector relaxes to the direction of the effecti ve\nfield. The damping term in the LL equation was reformu-\nlated by Gilbert [5, 6] to give the Landau-Lifshitz-Gilbert\n(LLG) equation,\n∂M\n∂t=−γM×Heff+αM×∂M\n∂t, (2)\nwhereαis the Gilbert damping constant. Note that both\ndamping parameters αandλare here scalars, which cor-\nresponds to the assumption of an isotropic medium. Both\nLL and LLG equations preserve the length of the magneti-\nzation during the dynamics and are mathematically equiv-\nalent (see, e.g. [7]).\nA number of explanations have been proposed for the\nmicroscopic origin of the spin relaxation in magnetic met-\nals [8–18]. Already in their original work Landau and Lif-\nshitz attributed the damping constant to relativistic effects\n∗Ritwik.Mondal@physics.uu.se[1]. More specific microscopic theories of spin relaxation\nin ferromagnetic metals have been developed in the last\ndecennia. Kamberský proposed the breathing Fermi sur-\nface model [8] and the related torque-correlation model\n[14, 19]. Brataas et al. proposed a scattering theory for-\nmulation [15] of the Gilbert damping which is equiva-\nlent to a Kubo linear-response formulation. A different\nform of the relaxation term caused by spatial dispersion of\nthe exchange interaction—this in contrast to the isotropic\nmedium assumption made in the LL equation—was pro-\nposed by Bar’yakhtar and co-workers [10, 20, 21].\nMore recently the debate on what the appropriate the-\nory to describe damping would be has focused on first-\nprinciples electronic structure calculations and, in how\nfar these could provide quantitative values of the Gilbert\ndamping [22–30]. Recent ab initio calculations of the\nGilbert damping constant for transition-metal alloys pre-\ndicted values that correspond to the experimental values\nwithin a range of a factor of two to three [22–24, 26–28],\nwith significant deviations however for the pure elemen-\ntal ferromagnets. This indicates that there is still a need\nto improve the fundamental understanding of the origin\nof spin-moment relaxation. Also, very recent publications\nhave questioned the existing understanding of the Gilbert\ndamping [31, 32].\nHere we develop a theoretical description of spin relax-\nation on the basis of the relativistic Density Functional\nTheory (DFT). To this end, we start from the relativis-\ntic Dirac-Kohn-Sham (DKS) equation that adequately de-\nscribes the electronic states in a magnetic solid. From thes e\nwe derive the general equation of motion for spin angular\nmomentum, which adopts the form of the LLG equation.\nWithin this framework we obtain explicit expressions for\nthe tensorial form of the Gilbert damping term, which we\nfind to contain an isotropic Gilbert-like contribution and\nanisotropic Ising-like and chiral Dzyaloshinskii-Moryia -like\ncontributions. Our derivation follows similar steps as a pr e-\nvious derivation by Hickey and Moodera [17], however, as\ndiscussed below, it includes previously missing terms and\nthus leads to different expressions for the spin relaxation.2\nII. THE RELATIVISTIC DIRAC HAMILTONIAN\nAs mentioned before, relativistic effects such as the spin-\norbit interaction are at the heart of spin angular momen-\ntum dissipation in solids. To examine how these fundamen-\ntal physical interactions lead to magnetization damping we\nchoose therefore to start from the most general relativisti c\nHamiltonian, the DKS Hamiltonian. This Hamiltonian de-\nscribes the one-electron quantum state in an effective spin-\npolarized field due to other electrons and nuclei in the solid ,\nin addition to externally applied fields. For spin-polarize d\nelectrons in a magnetic material the DKS Hamiltonian is\ngiven as [33–35]\nHD=cα·(p−eA)+/parenleftbig\nβ−1/parenrightbig\nmc2+V1+eΦ1\n−µBβΣ·Bxc. (3)\nHereVis the unpolarized Kohn-Sham selfconsistent po-\ntential,Bxcis the spin-polarized part of the exchange-\ncorrelation potential in the material, A=A(r,t)is the vec-\ntor potential of an externally applied electromagnetic fiel d,\neΦ(r,t)is the scalar potential of this field, p=−i/planckover2pi1∇, and\nµBise/planckover2pi1\n2m, the Bohr magneton. 1is the4×4identity matrix\nandα,β, andΣare the well-known Dirac matrices in Dirac\nbi-spinor space, which contain the Pauli spin matrices σ\nand the2×2identity matrix. At this point, it is important\nto observe that there are two fundamentally different fields\npresent in the DKS Hamiltonian. There are the Maxwellfields, that is, (implicitly) the external magnetic inducti on\nB(r,t) =∇×A(r,t)as well as the external electric field,\nE(r,t) =−∂A(r,t)\n∂t−∇Φ. The strongest field in a magnetic\nmaterial is however the exchange field, which stems from\nthe Pauli exclusion principle. The exchange field Bxcis\nfundamentally different from the standard magnetic induc-\ntion, as it obviously acts only on the spin degree of freedom\n(see, e.g., [34]) and does not couple to the orbital angular\nmomentum. Also, it doesn’t fulfill the Maxwell equations\nas the auxiliary electromagnetic field (e.g., ∇·B= 0) and\nit cannot be included as a vector potential Axcin the linear\nmomentum, i.e. p−eAxc, but instead needs to be treated\nas a separate term in Eq. (3).\nNext, we want to investigate the relativistic spin evo-\nlution of spin-polarized electrons in a magnetic solid. To\nachieve this we need the positive energy, that is, the elec-\ntron solutions that are given by the large component of\nthe Dirac bi-spinor. To arrive at an elucidating formula-\ntion in terms of the spin operator we employ the Foldy-\nWouthuysen transformation approach [35, 36] on the DKS\nequation for the case where an exchange field Bxcis explic-\nitly present (for details, see Ref. [37]). Doing so, one ob-\ntains a Hamiltonian for the electron solutions only, which\nwe expand in orders of 1/c2to select the largest relativistic\ncontributions. This leads to a semi-relativistic, extende d\nPauli Hamiltonian (see Ref. [37]),\nHEP=(p−eA)2\n2m+V−µBσ·B−µBσ·Bxc\neff+eΦ−(p−eA)4\n8m3c2−1\n8m2c2/parenleftbig\np2V/parenrightbig\n−e/planckover2pi12\n8m2c2∇·E\n+i\n4m2c2σ·(pV)×(p−eA)−e/planckover2pi1\n8m2c2σ·{E×(p−eA)−(p−eA)×E}\n+iµB\n4m2c2[(p×Bxc)·(p−eA)]. (4)\nExcept from the last term in Eq. (4), all the appearing relati vistic corrections involving the exchange interaction can be\nadded together giving an effective exchange field [38],\nBxc\neff=Bxc−1\n8m2c2/braceleftBig/bracketleftbig\np2Bxc/bracketrightbig\n+2(pBxc)·(p−eA)+2(p·Bxc)(p−eA)+4[Bxc·(p−eA)](p−eA)/bracerightBig\n≡Bxc+Bxc\ncorr. (5)\nThe Hamiltonian HEPexactly includes all spin-dependent\nrelativistic terms (of the order of 1/c2) and all the terms\ninvolving Bxcand the external electromagnetic fields. We\nemphasize that for our purpose of unveiling the relativisti c\nmechanisms of spin dissipation it is obviously not sufficient\nto work with the conventional Pauli Hamiltonian, which\nonly consists of the five first terms in the nonrelativistic\nlimit. The correct form of all relativistic terms can solely\nbe obtained when one starts from the DKS equation with\nexchange field. We remark that in a previous study Hickey\nand Moodera [17] used a Pauli Hamiltonian different from\nthe above one, without exchange field and without crystal\npotential and thus without the intrinsic spin-orbit intera c-\ntion [the first term in the second line of Eq. (4)].\nThe meaning of the terms in Hamiltonian (4) can bereadily understood, see Ref. [37] for details. The fourth\nterm on the right is a Zeeman-like term due the presence of\nthe relativistically corrected exchange field, which acts a s\nan effective mean field. The ninth term is the one, which\nin a central potential V, gives rise to the conventional form\nof the spin-orbit coupling. The tenth term is a kind of\nspin-orbit interaction but due to the external fields. The\nvery last term is a relativistic correction which depends on\ntheBxcfield but is independent of the spin. As we will\nsee in the following, the terms that are responsible for spin\nrelaxation are the relativistic terms that involve a direct\ncoupling of the spin operator with either the exchange field\nBxcor one of the externally applied fields ( EorA).3\nIII. SPIN EQUATION OF MOTION\nThe spin angular momentum operator is given by S=\n(/planckover2pi1/2)σ. To obtain an equation of motion for the spin op-\nerator we have to evaluate the commutator [S,HEP(t)]. It\nis obvious from the expression of HEPthat only the terms\nwhich are explicitly spin dependent will contribute as oth-\nerwise the commutator vanishes. We can thus extract from\nHEPthe spin Hamiltonian\nHS(t) =H0+Hint\nsoc+Hext\nsoc (6)\nwhere the Zeeman-like fields are added up to an effective\nmagnetic induction,\nH0=−e\nmS·(B+Bxc+Bxc\ncorr)≡ −e\nmS·Beff.(7)\nThe part H0contains the main nonrelativistic contribution,\nall other terms in the spin Hamiltonian HSare of relativis-\ntic origin. The intrinsic spin-orbit coupling is given by th e\nHamiltonian\nHint\nsoc=i\n2/planckover2pi1m2c2S·(pV)×(p−eA). (8)\nThe crystal potential stems from the nuclei-electron and\nelectron-electron interactions and thus should have trans -\nlational symmetry. Consequently, also the intrinsic spin-\norbit Hamiltonian has translational symmetry [39]. If the\nposition of any j-th nucleus is Rj, the electron position is\nr, and the electron position with respect to the nucleus is\nrepresented by rj, then the crystal potential can be rep-\nresented by a sum of atom-centered potentials. Making\nnow in addition the central potential approximation (no\nangular dependence) for each of the atom-centered poten-\ntials, the potential can be written as V(rj) =V(|r−Rj|).\nThe translational symmetry is realized by the fact that\nrj=r−Rj. With the definition of spin-orbit interac-\ntion strength ξ(rj) =1\n2m2c21\nrdV(rj)/dr, and the Coulomb\ngauge,∇·A= 0, for homogeneous magnetic fields, i.e.,\nA= (B×r)/2, this Hamiltonian can further be written as\nHint\nsoc=1\n2m2c21\nrdV\ndrS·L−er\n4m2c2dV\ndrS·B\n+e\n4m2c21\nrdV\ndr(S·r)(r·B)\n=/summationdisplay\njξ(rj)/bracketleftBig\nS·L−e\n2/parenleftBig\nr2S·B−(S·r)(r·B)/parenrightBig/bracketrightBig\n.(9)\nWe note, first, that the full spin-orbit Hamiltonian, Hint\nsoc+\nHext\nsoc, is gauge invariant [40], but for deriving expressions\nwe need to make a choice. The Coulomb gauge is a suit-\nable choice here, yet it can be used exactly only when a\nslowly varying and homogeneous magnetic field is present.\nThis gauge further implies that only the transversal parts\nofEand ofAare retained, the latter being gauge invari-\nant. Doing so, we have thus recovered the “usual” spin-orbit\ncoupling term and other ultra-relativistic terms.\nThe external spin-orbit coupling Hamiltonian is given by\nHext\nsoc=−e\n4m2c2S·{E×(p−eA)−(p−eA)×E},\nwhich has a similar form as Hint\nsoc[Eq. (8)], but contains the\nexternal Maxwell fields instead. Making use of Maxwell’sequation ∇×E=−∂B/∂t, this Hamiltonian can be\nrewritten as\nHext\nsoc=−e\n2m2c2S·(E×p)+ie/planckover2pi1\n4m2c2S·∂B\n∂t\n+e2\n2m2c2S·(E×A). (10)\nThe last term in the Hamiltonian Hext\nsocdescribes the in-\nteraction of the photon spin angular momentum density,\njs=ǫ0(E×A)[41], with the electron spins [40, 42]. A\nrelated interaction energy due to a coupling of the angu-\nlar momentum density of the electromagnetic field with the\nmagnetic moment was proposed recently on phenomenolog-\nical grounds [43]. The relativistic light-spin interactio n in\nthe Hamiltonian (10) adopts thus the form\nHext\nlight−spin=e2\n2m2c2ǫ0S·js. (11)\nThis term, being second order in the external fields can\nbecome important in the strong field regime. As we focus\nin first instance on the damping, we will not consider it in\nthe derivation of the spin damping, but we come back to it\nlater on.\nNow we have the necessary parts of the spin Hamiltonian\nand we are ready to calculate the spin dynamics equations.\nAccording to the definition of magnetization, this quantity\nis given by the expectation value of spin angular momentum\n[44]\nM=/summationdisplay\njgµB\nVTr/braceleftbig\nρSj/bracerightbig\n, (12)\nwhereVis a suitably chosen volume element. The sum-\nmation is taken over all the electrons jand the definition\nof the density matrix is ρ=/summationtext\nipi|ψi/an}b∇acket∇i}ht/an}b∇acketle{tψi|, where the set\nof wave functions |ψi/an}b∇acket∇i}htare in a mixed state and piare the\noccupation numbers. As is customary in spin dynamics\nmodels [12–20, 24, 26, 45] the contribution of the orbital\nangular momentum to the total magnetization has been\nneglected because it is quenched for the common transition\nmetals (e.g., Fe, Ni, Co etc.). The equation of motion of\nthe magnetization is obtained by taking the time deriva-\ntive on both sides of Eq. (12), and using that ∂ρ/∂t= 0\nfor quasiadiabatic processes [46], which gives\n∂M\n∂t=gµB\nV1\ni/planckover2pi1/summationdisplay\njTr/braceleftbig\nρ[Sj,HS(t)]/bracerightbig\n. (13)\nTo obtain the magnetization dynamics we substitute the\nspin Hamiltonian HS(t) =H0+Hint\nsoc+Hext\nsocin the right-\nhand side of Eq. (13) and work out the trace term-by-term.\nBefore presenting the result we consider briefly the ap-\nproximations made in the derivation. Notably, Eq. (13)\nis valid for local processes and will hence provide a local\ndamping mechanism. However, it is known that nonlo-\ncalcontributions to the damping exist (see, e.g., [47–49])\nthat can be caused by spin transport from one region to\nanother [50–52]. Such effects can be treated using the con-\ntinuity equation, ∂ρ/∂t+∇·J= 0, withJthe current\noperator, leading to an additional spin current term (see,\ne.g., [52, 53]). A further remark due at this point concerns4\nthe time dependence of the exchange field. In line with the\nabove, we adopt the adiabatic approximation that is valid\nfor systems not too far from the ground state [54].\nWorking out the commutator, we find that the first or-\nder dynamical equation of motion is given by the mostly\nnonrelativistic part in the spin Hamiltonian, H0. Using\nthe commutation relations for spin angular momentum,\n[Sj,Sk] =i/planckover2pi1ǫjklSl, the first order equation of motion be-\ncomes\n∂M\n∂t/vextendsingle/vextendsingle/vextendsingle0\n=−γM×Beff, (14)\nwhereγ=g|e|/2mis the gyromagnetic ratio and g≈2for\nspin degrees of freedom. Using B=µ0(H+M), the right-\nhand term can be rewritten in the conventional form as\n−γ0M×Heff, whereγ0=µ0γ. This equation provides the\ncommon understanding of the Larmor precessional motion\nof magnetization around an effective magnetic field, with\na distinction that there is a relativistic correction Bxc\ncorrto\nthis field that has not been noted before.\nNext we treat the relativistic spin-orbit effects in the\nmagnetization dynamics. As we will see, these are the ones\nthat lead to local damping, i.e., the spin relaxation mech-\nanisms in a magnetic solid are of relativistic origin [1, 9].\nFirst, we focus on the relativistic intrinsic spin-orbit co u-\npling Hamiltonian Hint\nsocin Eq. (9). Due to the quenching\nof the orbital angular momentum, the first term vanishes.\nThe dynamics due to the remaining two terms in the Hamil-\ntonian is calculated as\n∂M\n∂t/vextendsingle/vextendsingle/vextendsingleint\nsoc=e\n4m2c2/angbracketleftBig\nrdV\ndr/angbracketrightBig\nM×B\n−e\n4m2c2M×/angbracketleftBig\nr1\nrdV\ndr(r·B)/angbracketrightbig\n=e\n2/summationdisplay\nj/bracketleftBig\n/an}b∇acketle{tξ(rj)r2/an}b∇acket∇i}htM×B\n−M×/an}b∇acketle{tξ(rj)r(r·B)/an}b∇acket∇i}ht/bracketrightBig\n. (15)\nThe first term in the dynamics of Eq. (15) can be seen as a\nfurther relativistic correction to the magnetization prec es-\nsion. The second term has a form similar to the first term,\nbut with opposite sign. The terms can be combined, but\nthey do not contribute to any relaxation processes as they\ndo not contain a time variation of the magnetic induction.\nNext we consider the dynamics related to Hext\nsoc. We will\nsee below that it is mainly the relativistic extrinsic spin-\norbit coupling, i.e., the first two terms of Eq. (10), which\ngive rise to dominant local spin relaxation mechanisms in\nmagnetic solids. In addition, we observe here that these\ncorrespond to the transverse spin relaxation. We consider\nhere the long wavelength approximation, where the wave-\nlength of the field is much larger than the size of the system.\nIn other words the GHz/THz electromagnetic field inside\nthe ferromagnetic film is assumed uniform throughout the\nfilm as long as the film thickness is sufficiently small. We\ncan thus use the Coulomb gauge, i.e., A= (B×r)/2. This\ngauge allows us to obtain the explicit time dependence of\nthe Hamiltonian. The transverse electric field in the Hamil-\ntonian is then written as E=1\n2(r×∂B/∂t). Employing\nthe gauge, the first two terms in Eq. (10) can be re-writtenin an explicit, time-dependent form:\nHext\nsoc=ie/planckover2pi1\n4m2c2S·∂B\n∂t/parenleftbigg\n1−(r·p)\ni/planckover2pi1/parenrightbigg\n+e\n4m2c2(S·r)/parenleftbigg∂B\n∂t·p/parenrightbigg\n. (16)\nAt this point it is needed to inspect the hermiticity of\nthe Hamiltonian. It can be shown that the total spin-orbit\nHamiltonian in Eq. (16) is hermitian (see Appendix A),\nhowever, for the individual terms it is different. Writing\ndown the Hamiltonian in component form with the usual\nsummation convention, we obtain\nHext\nsoc=ie/planckover2pi1\n4m2c2Si∂Bi\n∂t/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nanti−hermitian−e\n4m2c2/summationdisplay\ni/negationslash=jSi∂Bi\n∂trjpj\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nnon−hermitian\n+e\n4m2c2/summationdisplay\ni/negationslash=jSiri∂Bj\n∂tpj\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nhermitian. (17)\nPreviously, Hickey and Moodera considered the effect of the\nspin-orbit Hamiltonian on damping, but only obtained the\nfirst two terms in Eq. (10) [17]. They proposed then only\nthe anti-hermitian part of the Hamiltonian as an intrinsic\nsource of Gilbert damping [17]. Anti-hermitian Hamilto-\nnians understandably are always dissipative [55, 56]. Con-\nsequently, their choice of taking the anti-hermitian term\nonly was criticized, given that the full spin-orbit Hamilto -\nnian should be hermitian and that it therefore should not\nexhibit dissipation [55].\nIn our case the total spin-orbit Hamiltonian (16) is man-\nifestly hermitian, yet we will show below that it does give\nrise to spin moment damping. The point is, that even\nwhen the full Hamiltonian is hermitian, it only has this\nproperty when one considers the dynamics of the full sys-\ntem. It is however customary in spin moment dynamics\n[12–20, 24, 26, 45] to integrate out the orbital degree of\nfreedom and other magnetic degrees of freedom (as back-\nground fluctuations of the system) thus restricting the fo-\ncus on the single spin moment dynamics. In the thereby\nrestricted Hilbert space the hermiticity is lost and hence\nthe whole Hamiltonian can contribute to the damping.\nCalculating now the commutation relation [S,Hext\nsoc]and\ntaking the summation of the trace over all electrons, the\nspin moment dynamics adopts the form\n∂M\n∂t/vextendsingle/vextendsingle/vextendsingleext\nsoc=−ie/planckover2pi1\n4m2c2M×∂B\n∂t/parenleftBigg\n1−/angbracketleftbig\nr·p/angbracketrightbig\ni/planckover2pi1/parenrightBigg\n−e\n4m2c2M×/angbracketleftBig\nr/parenleftbigg∂B\n∂t·p/parenrightbigg/angbracketrightBig\n. (18)\nA rewriting of these terms is required to elucidate further\nthe spin relaxation.\nIV. THE DAMPING EQUATIONS\nTo obtain explicit expressions for the damping terms, we\nemploy the general relation between magnetic induction B,5\nmagnetization M, and magnetic field H, given as B=\nµ0(M+H). We take the time derivative on both sides,\n∂B\n∂t=µ0/bracketleftbigg∂M\n∂t+∂H\n∂t/bracketrightbigg\n. (19)\nThis relation is generally valid, also for the stationary ca se,\neven though the magnetization M(t)and magnetic field\nH(t)are time dependent. At this point it is instructive to\nconsider what kinds of magnetic fields H(t)can occur. The\nsimplest case is when at some time t0only a static fieldH0\nis present, then obviously only the first term in Eq. (19)\ncontributes. If the field H(t)is explicitly time dependent,\nwe can distinguish to cases: an ac driven, periodic magnetic\nfield, as is commonly used in measurements, or a more gen-\neral field, for example a magnetic field pulse. In the latter\ncase, one could proceed to derive the spin dynamics by\nkeeping explicitly the term∂H\n∂t. As a result, one obtains a\nLLG-like equation, where however the magnetic field cou-\nples into the damping term. The thus-obtained modified\nLLG equation is given and analyzed further below, in Sect.\nV.\nIn the former case, the effect of the magnetic response\nbecomes apparent when an ac magnetic field is applied.\nFor ferromagnetic materials, where there is a net magneti-\nzation present even in the absence of the applied field, the\nmagnetic susceptibility can be introduced by the definition :\nχ=∂M/∂H. Using a chain rule for the time derivative,\n∂H\n∂t=∂H\n∂M∂M\n∂t, Eq. (19) can be written as\n∂B\n∂t=µ0/parenleftbig\n1+χ−1/parenrightbig\n·∂M\n∂t, (20)\nwhere 1is the3×3identity matrix. This relation has been\nused in the ensuing magnetization dynamics.\nSubstituting Eq. (20) in the first term of Eq. (18), we\nobtain\n∂M(1)\n∂t/vextendsingle/vextendsingle/vextendsingleext\nsoc=−ie/planckover2pi1µ0\n4m2c2M×/bracketleftbigg\n(1+χ−1)·∂M\n∂t/bracketrightbigg/parenleftbigg\n1−/an}b∇acketle{tr·p/an}b∇acket∇i}ht\ni/planckover2pi1/parenrightbigg\n.\n(21)\nThis term can already be recognized to have the form of\nthe Gilbert damping, M×/bracketleftbig\nα·∂M\n∂t/bracketrightbig\n, yet with a tensorial\ndamping constant.\nFor the full damping we have to combine with the second\nterm in Eq. (18), which is rewritten as\n∂M(2)\n∂t/vextendsingle/vextendsingle/vextendsingleext\nsoc=−eµ0\n4m2c2M×/angbracketleftBig\nr/parenleftbigg/bracketleftBig\n(1+χ−1)·∂M\n∂t/bracketrightBig\n·p/parenrightbigg/angbracketrightBig\n.\n(22)\nTo join the terms we proceed with using vector components.\nEquation (21) becomes\n∂M(1)\n∂t/vextendsingle/vextendsingle/vextendsingleext\nsoc=−eµ0\n4m2c2/summationdisplay\nijklnMk/bracketleftbigg\n(1+χ−1)ij∂Mj\n∂t/bracketrightbigg\n×(i/planckover2pi1−/an}b∇acketle{trnpn/an}b∇acket∇i}ht)εkilˆel, (23)\nwithεijkthe Levi-Civita tensor and ˆea unit vector. This\nterm can be written as\n∂M(1)\n∂t/vextendsingle/vextendsingle/vextendsingleext\nsoc=/summationdisplay\nijklMk∂Mj\n∂tΩijεkilˆel, (24)withΩij=−eµ0\n4m2c2/summationtext\nn(i/planckover2pi1−/an}b∇acketle{trnpn/an}b∇acket∇i}ht)(1+χ−1)ij. The sec-\nond term (22) can be written in a similar form, but with\na tensor ∆ij=−eµ0\n4m2c2/summationtext\nn/an}b∇acketle{tripn/an}b∇acket∇i}ht(1+χ−1)nj. Combining\nthese two terms gives the total damping term,\n∂M\n∂t/vextendsingle/vextendsingle/vextendsingleext\nsoc=/summationdisplay\nijklMk/bracketleftBig\nΩij+∆ij/bracketrightBig∂Mj\n∂tεkilˆel,(25)\nwhere it is convenient to define A ij≡Ωij+∆ij,\nAij=−eµ0\n4m2c2/summationdisplay\nn/bracketleftBig\ni/planckover2pi1−/an}b∇acketle{trnpn/an}b∇acket∇i}ht+/an}b∇acketle{trnpi/an}b∇acket∇i}ht/bracketrightBig\n(1+χ−1)ij\n=−eµ0\n8m2c2/summationdisplay\nn,k/bracketleftBig\n/an}b∇acketle{tripk+pkri/an}b∇acket∇i}ht−/an}b∇acketle{trnpn+pnrn/an}b∇acket∇i}htδik/bracketrightBig\n×(1+χ−1)kj.(26)\nNote that a summation over iis not intended in the right-\nhand side expressions. In vector form the spin-orbit damp-\ning term becomes\n∂M\n∂t/vextendsingle/vextendsingle/vextendsingleext\nsoc=M×/bracketleftBig\nA·∂M\n∂t/bracketrightBig\n. (27)\nSummarizing our result, we observe that we have obtained\na damping parameter A ijof Gilbert type that is however\nin its general form not a scalar but a tensor. The tenso-\nrial character of the Gilbert damping was also concluded\nrecently in other investigations [16, 57]. In this form it\naccounts for transversal spin relaxation that conserves th e\nlength of the magnetization, i.e., ∂(M·M)/∂t= 0.\nEvery tensor can be decomposed in a symmetric and an\nanti-symmetric part. Hence, the damping tensor can be\ndecomposed into a scalar ( α) multiplied by the unit matrix,\na symmetric tensor ( I), and an anti-symmetric tensor ( A,\nwithAij=1\n2(Aij−Aji)). The latter tensor can in turn be\nexpressed as Aij=εijkDkwithDbeing a vector. Finally,\nthe damping dynamics can then be written as\n∂M\n∂t/vextendsingle/vextendsingle/vextendsingleext\nsoc=αM×∂M\n∂t+M×/bracketleftBig\nI·∂M\n∂t/bracketrightBig\n+M×/bracketleftBig\nD×∂M\n∂t/bracketrightBig\n. (28)\nThe first term is the conventional Gilbert damping. It orig-\ninates from the decomposition of the symmetric part of the\ntensor into an isotropic Heisenberg-like (α1)contribution\nas well as an anisotropic Ising-like ( I) contribution which\nleads to the second term. Along with that it is not surpris-\ning that the last term implies a Dzyaloshinskii-Moriya-lik e\ncontribution. The anisotropic nature of the Gilbert damp-\ning has been noted before [18, 57], but not the appearance\nof the Dzyaloshinskii-Moriya-like damping. This type of\ndamping could be related to the chiral damping of mag-\nnetic domain walls that was reported recently [58].\nFor the case of a constant, scalar Gilbert damping param-\neter it is straightforward to transform the LLG equation to\nobtain the LL equation with the phenomenological damp-\ning term proposed by Landau and Lifshitz [1]. However,\nthis is no longer the case for tensorial Gilbert damping,\nfor which the transformation is much more involved. The\nspin-dynamics equation in the Landau-Lifshitz form now6\nbecomes (see Appendix B)\n/parenleftbig\nΨ21+G/parenrightbig\n·∂M\n∂t=\n−γ0ΨM×Heff−γ0M×/bracketleftBig\n(α1+I)·(M×Heff)/bracketrightBig\n,(29)\nwhereΨ = 1+ M·Dand the tensor Gis defined through\nG=α2M21−/bracketleftBig\n(M·I·M)−tM2/bracketrightBig\n(α1+I)\n−/parenleftBig\ntM−M·I/parenrightBig\nM·I−M2I2+M/parenleftBig\nM·I2/parenrightBig\n,(30)\nwith the trace, t= Tr( I). In general the trace of such a\nmatrix Iis non-zero, however its value will depend on how\nthe symmetric tensor Asym\nij=1\n2(Aij+Aji) =Iij+αδij\nis decomposed. If the decomposition in Ising and Heisen-\nberg parts is such that the isotropic part is chosen as\nα=1\n3Tr(Asym\nij), then the trace of Iwill vanish, t= 0. Note\nthat the term (15) due to the intrinsic spin-orbit interacti on\nhas been left out, as it is expected to give only a small cor-\nrection to the effective magnetic field. The damping term\nthus adopts the form −γ0M×[Λ·(M×Heff)], similar to\nthe phenomenological damping considered by Landau and\nLifshitz [1], but with damping tensor Λ. A more general\nform of the LL damping as a tensor was already considered\nmuch earlier (see, e.g. [59]), and it is reflected also in our\nderivation. However, a distinction is that here the leading\n∂M/∂tterm on the left-hand side in Eq. (29) is, in its gen-\neral form, multiplied not with a scalar ( 1+α2M2) but with\na tensor which moreover depends on the direction of M.\nIt is worth noting that in the absence of the\nDzyaloshinskii-Moriya and anisotropic relaxation contri bu-\ntions, i.e., setting D=I= 0we retrieve the original LL and\nLLG equations with scalar damping parameters. The va-\nlidity range of our derived equations of spin motion is thus\nlarger than the originally proposed equations of motion. It\nshould also be emphasized that the Dzyaloshinskii-Moriya-\nlike contribution appears in the Gilbert damping, however,\nit does not appear in the damping term of the LL equation\n(29). Instead, it leads to the renormalization of the stan-\ndard dynamical terms in the LL equation as can be seen\nfrom the appearance of the quantity Ψin Eq. (29). We\nlastly note that the here obtained relaxation terms do not\nallow a variation with respect to the coordinates i.e., they\ndo not include effects of spatial dispersion.\nV. DISCUSSION\n1. Analysis of the damping expression\nEquation (26) for the Gilbert damping pertains to the\nrelaxation of spin motion in the presence of spin-orbit in-\nteraction. This damping is of relativistic origin as is ex-\nemplified by its 1/c2dependence. The expression for the\nGilbert tensor is different from that obtained previously\n[17], where only the constant term i/planckover2pi1in the square bracket\nwas found. The new parts /an}b∇acketle{tripj/an}b∇acket∇i}htrelate to how the elec-\ntronic band energies Eνkof Bloch states |νk/an}b∇acket∇i}htdisperse with\nk-space direction. It can be rewritten as (see Appendix C)\n/an}b∇acketle{tripj/an}b∇acket∇i}ht=−i/planckover2pi1\n2m/summationdisplay\nν,ν′,kf(Eνk)−f(Eν′k)\nEνk−Eν′kpi\nνν′pj\nν′ν,(31)wherepνν′≡ /an}b∇acketle{tνk|p|ν′k/an}b∇acket∇i}htandf(Eνk)is the Fermi function.\nThe sum contains interband and intraband contributions.\nThe intraband (Fermi surface) contribution (ν=ν′)can\nbe written as\n/an}b∇acketle{tripj/an}b∇acket∇i}ht=−im\n2/planckover2pi1/summationdisplay\nνk/parenleftbigg∂f\n∂E/parenrightbigg\nEνk/parenleftbigg∂Eνk\n∂ki/parenrightbigg/parenleftbigg∂Eνk\n∂kj/parenrightbigg\n.(32)\nThis expression has a similarity with other previously de-\nrived expressions, as e.g. the breathing Fermi surface mode l\n[8, 24] that has been applied to metallic ferromagnets. The\nexpression for the /an}b∇acketle{tripj/an}b∇acket∇i}htterms has furthermore a form sim-\nilar to that for the conductivity tensor in linear-response\ntheory [60]; it is in particular well-suited for ab initio cal-\nculations. We note further that the influence of electron\ninteraction with quasiparticles can be introduced by repla c-\ningEνk−Eν′kbyEνk−Eν′k+iδ, where the small δgives\na finite relaxation time to the electronic states.\nFor numerical evaluation of the damping tensor the sus-\nceptibility tensor χis furthermore needed, which is in gen-\neral wavevector and frequency dependent, χ(q,ω). Thus,\nalso the Gilbert damping tensor is here a frequency and\nq-dependent quantity, in accordance with recent measure-\nments [32]. Suitable expressions for χhave been considered\npreviously in the context of Gilbert damping [13, 16, 45].\nLinear-response formulations that express χas a spin-spin\ncorrelation function include the Pauli and Van Vleck sus-\nceptibility contributions [61], and expressions for the or -\nbital susceptibility have been derived as well [62]. These\nexpressions are fitting for ab initio calculations of χwithin\na DFT framework. The spin-orbit interaction will have an\nadditional influence on χ, however, unlike the main Gilbert\ndamping contribution which is proportional to the spin-\norbit coupling, this will only be a higher order effect.\nWe can consequently distinguish here two origins for the\ndamping: the first one is related to the terms /an}b∇acketle{tripj/an}b∇acket∇i}ht, which\nrepresent dissipation contributions into the orbital degr ees\nof freedom. The second nature is due to the magnetic sus-\nceptibilityχwhich represents losses through the magnetic\nstructure of the material. Both effects are simultaneously\npresent, and nonzero, for metallic ferromagnets as well as\ninsulators.\nIt is also important to mention that the damping ten-\nsor in the our derivation does not include spin-relaxation\neffects due to interaction of spin-polarized electrons with\nquasiparticles as magnon or phonons or scattering with de-\nfects. Longitudinal spin relaxation due to spin-flip pro-\ncesses caused by electron-phonon scattering have been re-\ncently calculated ab initio for the transition-metal ferro-\nmagnets [63–65], and magnon spin-flip scattering has been\nconsidered as well [66]. Spin angular momentum transfer\ndue to explicit coupling of the spins to the lattice has been\ntreated in several models [67–69]. As mentioned above,\nalthough the spin-lattice dissipation channel is not encom -\npassed in our derivation, an approximate way to include its\ninfluence has been introduced before, by a suitable spec-\ntral broadening of the Bloch electron energies (see, e.g.,\n[24, 70]).\nLastly, we remark that in the present derivation we ob-\ntain only first-order time-derivatives of M(r,t). Second-\norder time-derivatives of M(r,t)have recently been related\nto moment of inertia of the magnetization [71].7\n2. Exchange field and nonlocal contributions\nThus far we have not explicitly discussed the exchange\ninteraction. The influence of the exchange field can be\naccounted for in various levels of approximation, for ex-\nample, within the Heisenberg model or evaluated within\ntime-dependent DFT [72, 73]. In the former, a suitable\nsimplification of the exchange interaction in a magnetic\nsolid is to express it through the Heisenberg Hamiltonian\nHxc=−/summationtext\nα>βJαβSα·Sβ, where the Jαβare exchange\nconstants and Sαis the atomic spin on atom α. Using this\nHamiltonian to express the exchange field leads to Landau-\nLifshitz-Gilbert equations of motion for the dynamics of\natomic moments (see, e.g., [74–76]).\nMore general, the exchange field depends on the spatial\nposition which implies that there can exist an influence of\nspatial nonuniformity of the exchange field on the spin re-\nlaxation. An influence on the dynamics occurring due to\nmagnetization inhomogeneity ( ∇2M) appearing in the ef-\nfective field was already suggested by Landau and Lifshitz\n[1]. Such a term is in fact needed to properly describe\nspin wave dispersions [77]. A nonlocal damping mecha-\nnism due to spatial dispersion of the exchange field was\nproposed by Bar’yakhtar on the basis of phenomenological\nconsiderations such as symmetry arguments and Onsager’s\nrelations [10]. This leads to a modified expression for the\ndamping term in the Landau-Lifshitz-Bar’yakhtar equation\nwhich contains the derivative of the exchange field ∇2Bxc\n[10, 20]. The existence of such nonlocal damping term can\nbe related to the continuity equation connecting the spin\ndensity and spin current; it is important for obtaining the\ncorrect asymptotic behavior of spin wave damping at large\nwavevectors k[20] known for magnetic dielectrics, see [59].\nSuch nonlocal damping is important, too, for describing\nspin current flow in magnetic metallic heterostructures [78 ].\nThese nonlocal damping terms are furthermore related to\nthe earlier proposed magnetization damping effects due to\nspin diffusion [52, 79–81] that have been studied recently\n[82]. As a consequence of the spin current flow the local\nlength of the magnetization is not conserved. In the present\nwork such nonlocal terms are not included since we focus\non the local dissipation and have thus omitted the spin cur-\nrent contribution of the continuity equation. A future full\ntreatment that takes into account both local and nonlocal\nspin dissipation mechanisms would permit to describe mag-\nnetization dynamics and spin transport on an equal footing\nin a broader range of inhomogeneous systems.\n3. General time-dependent magnetic fields\nWhen the driving magnetic field is not an ac harmonic\nfield the dependence of M(r,t)onH(t)will induce a more\ncomplex dynamics. In this case it is possible to derive a\nclosed expression for the spin dynamics by explicitly keep-\ning the term∂H\n∂tin Eq. (19). A similar derivation as pre-\nsented in Sect. IV for the ac driving field leads then to the\nfollowing expression for the magnetization dynamics\n∂M\n∂t=−γ0M×Heff+M×/bracketleftBig\n¯A·/parenleftBig∂M\n∂t+∂H\n∂t/parenrightBig/bracketrightBig\n,(33)where the damping tensor ¯A is given by\n¯Aij=−eµ0\n8m2c2/summationdisplay\nn/bracketleftBig\n/an}b∇acketle{tripj+pjri/an}b∇acket∇i}ht−/an}b∇acketle{trnpn+pnrn/an}b∇acket∇i}htδij/bracketrightBig\n.(34)\nThe time-dependent magnetic field thus leads to a new,\nmodified spin dynamics equation which has, to our knowl-\nedge, not been derived before. The time-derivate of H(t)\nintroduces here an additional torque, M×∂H\n∂t. This field-\nderivative torque might offer new ways to achieve fast mag-\nnetization switching. Consider for example an initially\nsteep magnetic field pulse that thereafter relaxes slowly\nback to its initial value. The derivative of such field will ex -\nert a large but shortly lasting torque on the magnetization,\nwhich could initiate switching. Irradiation of magnetic th in\nfilms with a picosecond THz field pulse was recently shown\nto trigger ultrafast magnetization dynamics [83], and suit -\nable shaping of the THz magnetic field pulse could hence\noffer a route to achieve switching on a picosecond time scale.\n4. The optical spin torque\nThe interaction of the spin moment with the optical spin\nangular moment jsis given by the Hamiltonian Hext\nlight−spin.\nWe note that such relativisitic interaction is important fo r\nrecent attempts to manipulate the magnetization in a ma-\nterial using optical angular momentum, i.e., helicity of th e\nlaser field [40, 84, 85]. This interaction leads to spin dy-\nnamics of the form\n∂M\n∂t/vextendsingle/vextendsingle/vextendsingleext\nlight−spin=−e2\n2m2c2ǫ0M×js, (35)\nwhereM×jsis the optical spin torque exerted by the\noptical angular moment on the spin moment. This equa-\ntion expresses that the spin moment in a material can be\nmanipulated by acting on it with the optical spin angular\nmoment of an external electromagnetic field in the strong\nfield regime.\nVI. CONCLUSIONS\nOn the basis of the relativistic Dirac-Kohn-Sham equa-\ntion we have derived the spin Hamiltonian to describe ad-\nequately the dynamics of electron spins in a solid, tak-\ning into account all the possible spin-related relativisti c\neffects up to the order 1/c2and the exchange field and ex-\nternal electromagnetic fields. From this manifestly hermi-\ntian spin Hamiltonian we have calculated the spin equation\nof motion which adopts the form of the Landau-Lifshitz-\nGilbert equation for applied harmonic fields. For univer-\nsal time-dependent external magnetic fields we obtain a\nmore general dynamics equation which involves the field-\nderivative torque. Our derivation does notably not rely\non phenomenological assumptions but provides a rigorous\ntreatment on the basis of fundamental principles, specifi-\ncally, Dirac theory with all relevant fields included.\nWe have shown the existence of a relativistic correction\nto the precessional motion in the obtained LLG equation\nand have derived an expression for the spin relaxation\nterms of relativistic origin. One of the most prominent8\nresults of the presented article is the derived expression f or\nthe tensorial Gilbert damping, which has been shown to\ncontain an isotropic Gilbert contribution, an anisotropic\nIsing-like contribution, and a chiral, Dzyaloshinskii-\nMoriya-like contribution. Transforming the LLG equation\nto the Landau-Lifshitz equation of motion, we showed\nthat the LLG equation with anisotropic tensorial Gilbert\ndamping cannot trivially be written as a LL equation with\nan anisotropic LL damping term, but an additional matrix\nappears in front of the ∂M/∂tterm. The Dzyaloshinskii-\nMoriya-like contribution serves as a renormalization fact or\nto the common LL dynamical terms. The obtained\nexpression for the Gilbert damping tensor in the case of\na periodic driving field depends on the spin-spin suscep-\ntibility response function along with a term representing\nthe electronic spin damping due to dissipation into the\norbital degrees of freedom. As there exist an on-going\ndiscussion on what the fundamental origin of the Gilbert\ndamping is and how it can accurately be evaluated from\nfirst-principles calculations [28, 30–32], we point out tha t\nthe two components of the derived damping expression\n(spin-spin and current-current response functions) are\nsuitable for future ab initio calculations within the density\nfunctional formalism.\nACKNOWLEDGMENTS\nWe thank B. A. Ivanov, P. Maldonado, A. Aperis, K.\nCarva, and H. Nembach for helpful discussions. We alsothank the anonymous reviewers for valuable comments.\nThis work has been supported by the European Com-\nmunity’s Seventh Framework Programme (FP7/2007-2013)\nunder grant agreement No. 281043, FemtoSpin, the Swedish\nResearch Council (VR), the Knut and Alice Wallenberg\nFoundation (Contract No. 2015.0060), and the Swedish Na-\ntional Infrastructure for Computing (SNIC).\nAppendix A: Hermiticity of Hamiltonian Hext\nsoc\nThe extrinsic spin-orbit Hamiltonian Hext\nsoc, given in Eq.\n(16), can indeed be shown to be hermitian, however its in-\ndividual terms are not all hermitian. Adapting the Einstein\nsummation convention, this Hamiltonian can be written in\ncomponent form as\nHext\nsoc=e\n4m2c2/parenleftBig\ni/planckover2pi1Si∂tBi\n−Si∂tBirjpj+Siri∂tBjpj/parenrightBig\n,(A1)\nwith∂t≡∂/∂t. To demonstrate that it is hermitian, we\ntake the Hermitian conjugate, and rewrite it in a few steps.\n/bracketleftBig\nHext\nsoc/bracketrightBig†\n=e\n4m2c2/parenleftBig\n−i/planckover2pi1Si∂tBi−Si∂tBipjrj+Si∂tBjpjri/parenrightBig\n=e\n4m2c2/parenleftBig\n−i/planckover2pi1Si∂tBi−Si∂tBirjpj+Si∂tBjripj−Si∂tBi(pjrj)+Si∂tBj(pjri)/parenrightBig\n=e\n4m2c2/parenleftBig\n−i/planckover2pi1S·∂tB−(S·∂tB)(r·p)+(S·r)(∂tB·p)−(S·∂tB)(p·r)+S·{(∂tB·p)r}/parenrightBig\n=e\n4m2c2/parenleftBig\n−i/planckover2pi1S·∂tB−(S·∂tB)(r·p)+(S·r)(∂tB·p)+i/planckover2pi1(S·∂tB)(∇·r)−i/planckover2pi1S·{(∂tB·∇)r}/parenrightBig\n=e\n4m2c2/parenleftBig\n−i/planckover2pi1S·∂tB−(S·∂tB)(r·p)+(S·r)(∂tB·p)+3i/planckover2pi1S·∂tB−i/planckover2pi1S·∂tB/parenrightBig\n=e\n4m2c2/parenleftBig\ni/planckover2pi1S·∂tB−(S·∂tB)(r·p)+(S·r)(∂tB·p)/parenrightBig\n=Hext\nsoc. (A2)\nFor the individual terms of the Hamiltonian it is straightfo rward to show their hermitian or non-hermitian character:\nHext\nsoc= =ie/planckover2pi1\n4m2c2Si∂tBi\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nanti−hermitian−e\n4m2c2/summationdisplay\ni/negationslash=jSi∂tBirjpj\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nnon−hermitian+e\n4m2c2/summationdisplay\ni/negationslash=jSiri∂tBjpj\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nhermitian. (A3)\nAs noted before all three terms of the hermitian Hamiltonian contribute to the spin relaxation process.\nAppendix B: From LLG to LL equations of motion\nWe found that the generalized LLG equation of spin dynamics c an be written in the form [see Eq. (27)]\n∂M\n∂t=−γM×Beff+M×/bracketleftBig\nA·∂M\n∂t/bracketrightBig\n. (B1)9\nAs discussed earlier, using the tensor decomposition, one c an also write\n∂M\n∂t=−γM×Beff+αM×∂M\n∂t+M×/bracketleftBig\nI·∂M\n∂t/bracketrightBig\n+M×/bracketleftBig\nD×∂M\n∂t/bracketrightBig\n. (B2)\nThe Dzyaloshinskii-Moriya-like damping terms can be expan ded, using a×(b×c) =b(a·c)−c(a·b), to give M×/bracketleftBig\nD×∂M\n∂t/bracketrightBig\n=−∂M\n∂t(M·D). Since the magnetization length is conserved we therefore h aveM·∂M/∂t= 0. Defining\n(1+M·D) = Ψ, the LLG equation of spin motion reduces to\nΨ∂M\n∂t=−γM×Beff+αM×∂M\n∂t+M×/bracketleftBig\nI·∂M\n∂t/bracketrightBig\n. (B3)\nNote that Ψis both a magnetization and Dzyaloshinskii-Moriya vector d ependent quantity. Next, we have to calculate\nthe second and third terms on the right-hand side of Eq. (B3). Taking a cross product with Mon both sides of the last\nequation gives\nΨM×∂M\n∂t=−γM×(M×Beff)+αM×/parenleftBig\nM×∂M\n∂t/parenrightBig\n+M×/parenleftBig\nM×/bracketleftBig\nI·∂M\n∂t/bracketrightBig/parenrightBig\n=−γM×(M×Beff)−αM2∂M\n∂t−M2/bracketleftBig\nI·∂M\n∂t/bracketrightBig\n+M/parenleftBig\nM·/bracketleftBig\nI·∂M\n∂t/bracketrightBig/parenrightBig\n. (B4)\nSimilarly, to evaluate the last term of Eq. (B3), we take the d ot product with the symmetric part of the tensor, followed\nby a cross product with the magnetization,\nΨM×/bracketleftBig\nI·∂M\n∂t/bracketrightBig\n=−γM×/bracketleftBig\nI·(M×Beff)/bracketrightBig\n+αM×/bracketleftBig\nI·/parenleftBig\nM×∂M\n∂t/parenrightBig/bracketrightBig\n+M×/parenleftBig\nI·/braceleftBig\nM×/bracketleftBig\nI·∂M\n∂t/bracketrightBig/bracerightBig/parenrightBig\n.(B5)\nAt this point we already observe that the first term on the righ t hand side has adopted a form of the LL damping but\nwith a tensor. The second and third terms are treated in the fo llowing. The second term can be written in component\nform as\nαM×/bracketleftBig\nI·/parenleftBig\nM×∂M\n∂t/parenrightBig/bracketrightBig\n=αMlImkMi∂Mj\n∂tεijkεlmnˆen. (B6)\nWe use the following relation for the product of two anti-sym metric Levi-Civita tensors\nεijkεlmn=δil(δjmδkn−δjnδkm)−δim(δjlδkn−δjnδkl)+δin(δjlδkm−δjmδkl), (B7)\nand, defining the trace of the symmetric tensor Tr(I) =t, a little bit of tensor algebra results in\nαM×/bracketleftBig\nI·/parenleftBig\nM×∂M\n∂t/parenrightBig/bracketrightBig\n=αM2/parenleftBig\nI·∂M\n∂t/parenrightBig\n−αtM2∂M\n∂t+α/parenleftBig\nM·I·M/parenrightBig∂M\n∂t−αM/bracketleftBig\nM·/parenleftBig\nI·∂M\n∂t/parenrightBig/bracketrightBig\n.(B8)\nNow we proceed to calculate the last part of Eq. (B5); the comp onents of this term are given by\nM×/parenleftBig\nI·/braceleftBig\nM×/bracketleftBig\nI·∂M\n∂t/bracketrightBig/bracerightBig/parenrightBig\n=MmInlMkIij∂Mj\n∂tεkilεmnoˆeo. (B9)\nUsing once again the relation in Eq. (B7) and expanding in diff erent components we find\nM×/parenleftBig\nI·/braceleftBig\nM×/bracketleftBig\nI·∂M\n∂t/bracketrightBig/bracerightBig/parenrightBig\n=/bracketleftBig\n(M·I·M)−tM2/bracketrightBig/parenleftBig\nI·∂M\n∂t/parenrightBig\n+/parenleftBig\ntM−M·I/parenrightBig/bracketleftBig\nM·/parenleftBig\nI·∂M\n∂t/parenrightBig/bracketrightBig\n+(M·M)/bracketleftBig\nI·/parenleftBig\nI·∂M\n∂t/parenrightBig/bracketrightBig\n−M/bracketleftBig\nM·/braceleftbigg\nI·/parenleftBig\nI·∂M\n∂t/parenrightBig/bracerightbigg/bracketrightBig\n. (B10)\nNow we have the necessary terms to formulate the LL equation o f motion. Taking these together, the LLG dynamics\nof Eq. (B3) can be written as\nΨ2∂M\n∂t=−γΨM×Beff−γM×/bracketleftBig\n(α1+I)·(M×Beff)/bracketrightBig\n−G·∂M\n∂t, (B11)\nwith the general tensorial form of Gwhich is given by\nG=α2M21−/bracketleftBig\n(M·I·M)−tM2/bracketrightBig\n(α1+I)−/parenleftBig\ntM−M·I/parenrightBig\nM·I−M2I2+M/parenleftBig\nM·I2/parenrightBig\n.\nUsingB=µ0(H+M), the transformation from the LLG to the LL equation results i n the form\n/parenleftBig\nΨ21+G/parenrightBig\n·∂M\n∂t=−γ0ΨM×Heff−γ0M×/bracketleftBig\n(α1+I)·(M×Heff)/bracketrightBig\n. (B12)\nAs mentioned before, in general the Landau-Lifshitz dampin g cannot be described by a scalar. We find that in the\ndamping term the effect of the anisotropic Ising-like dampin g is present, while the influence of the Dzyaloshinskii-Mori ya-\nlike damping is accounted for through the renormalizing qua ntityΨ.10\nAppendix C: Expressions for matrix elements\nWe provide here suitable expressions for ab initio calcu-\nlations of the matrix elements /an}b∇acketle{tripj/an}b∇acket∇i}ht. We consider thereto\nthe Bloch states |νk/an}b∇acket∇i}htin a crystal to calculate the expecta-\ntion value\n/an}b∇acketle{tripj/an}b∇acket∇i}ht=/summationdisplay\nν,ν′,k/an}b∇acketle{tνk|ri|ν′k/an}b∇acket∇i}ht/an}b∇acketle{tν′k|pj|νk/an}b∇acket∇i}htf(Eνk),(C1)\nwheref(Eνk)is the Fermi-Dirac function. The momentum\nand position operators are connected through the Ehrenfest\ntheorem, p=im\n/planckover2pi1[H,r], which we employ to obtain matrix\nelements of the position operator\n/an}b∇acketle{tν′k|r|νk/an}b∇acket∇i}ht=−i/planckover2pi1\nm/an}b∇acketle{tν′k|p|νk/an}b∇acket∇i}ht\n(Eν′k−Eνk). 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McConney,4Gail J. Brown,4Brandon M. Howe,4and Nian X. Sun1\n1)Department of Electrical and Computer Engineering, Northeastern University, Boston, MA 02115,\nUSA\n2)Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA\n3)Department of Physics, Northeastern University, Boston, MA 02115, USA\n4)Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright-Patterson AFB, OH 45433,\nUSA\n5)Department of Electrical Engineering, Wright State University, Dayton, OH 45435,\nUSA\n6)Department of Materials Science and Engineering, University of Wisconsin-Madison, WI 53706,\nUSA\n(Dated: April 27, 2018)\nWe experimentally investigate spin-orbit torque and spin pumping in Y 3Fe5O12(YIG)/Pt bilayers with ul-\ntrathin insertion layers at the interface. An insertion layer of Cu suppresses both spin-orbit torque and spin\npumping, whereas an insertion layer of Ni 80Fe20(permalloy, Py) enhances them, in a quantitatively consistent\nmanner with the reciprocity of the two spin transmission processes. However, we observe a large enhance-\nment of Gilbert damping with the insertion of Py that cannot be accounted for solely by spin pumping,\nsuggesting signi\fcant spin-memory loss due to the interfacial magnetic layer. Our \fndings indicate that the\nmagnetization at the YIG-metal interface strongly in\ruences the transmission and depolarization of pure spin\ncurrent.\nThe transmission of pure spin current between a mag-\nnetic insulator and a normal metal is a crucial aspect\nof emerging insulator spintronic devices1,2. Yttrium iron\ngarnet (Y 3Fe5O12, YIG) is an especially promising mag-\nnetic insulator because of its exceptionally low Gilbert\ndamping that allows for e\u000ecient excitation of magne-\ntization dynamics3{5. This magnetic damping can be\nmodi\fed by spin-orbit torque6,7in thin-\flm YIG due\nto absorption of pure spin current8{12, which is gen-\nerated from an electric current in the adjacent metal\n(e.g., Pt) through the spin-Hall e\u000bect13. In the recip-\nrocal process of spin pumping14,15, coherent magnetiza-\ntion dynamics in YIG injects a pure spin current into\nthe metal layer, which can be detected through an en-\nhancement in Gilbert damping16{18or a voltage peak due\nto the inverse spin-Hall e\u000bect19{25. The reciprocity of\nspin-orbit torque and spin pumping is theoretically well\nestablished26. However, while prior reports have shown\nthat various modi\fcations at the YIG-metal interface im-\npact spin pumping (or, more generally, spin transmission\nfrom the YIG to metal layer)17,18,21,22,27{29, how spin-\norbit torque (i.e., spin transmission from the metal to\nthe YIG layer) is a\u000bected by such interfacial modi\fca-\ntions has yet to be reported.\nIn this Letter, we investigate spin-orbit torque and\nspin pumping in the same set of YIG/Pt samples { with\nand without an ultrathin interfacial insertion layer { by\nferromagnetic resonance (FMR) in a microwave cavity.\nThe two spin transmission processes are suppressed with\na nonmagnetic Cu insertion layer and enhanced with a\nmagnetic Ni 80Fe20(permalloy, Py) insertion layer. We\na)Electronic mail: semori@vt.edualso \fnd evidence for substantial spin-memory loss30with\nthe insertion of ultrathin Py. Our \fndings are consistent\nwith the reciprocity of spin-orbit torque and spin pump-\ning, while revealing that the magnetization at the YIG-\nmetal interface has a signi\fcant impact on the transmis-\nsion and scattering of spin current.\nEpitaxial 20-nm thick YIG \flms were grown on\nGd3Ga5O12(111) substrates by pulsed laser deposition\nas reported in Ref. 3. The YIG \flms were transferred\nthrough ambient atmosphere to a separate deposition\nsystem for the growth of the metal overlayers. The YIG\nsamples were sonicated in acetone and ethanol and, after\nintroduction into the deposition chamber, maintained at\n250\u000eC at 50 mTorr O 2for 30 minutes to remove water\nand organics on the surface. The metal overlayers (either\nPt(5 nm), Cu(0.5 nm)/Pt(5 nm), or Py(0.5 nm)/Pt(5\nnm)) were deposited by dc magnetron sputtering at room\ntemperature, base pressure of <\u00182\u000210\u00007Torr, and Ar\nsputtering pressure of 3 mTorr. While the RMS surface\nroughness of the epitaxial YIG \flms is only <\u00180.15 nm\n(consistent with Ref. 3), the nominally 0.5-nm thick Cu\nand Py \\dusting\" layers may not be continuous. Each\nYIG/X/Pt sample (with X = none, Cu, or Py) was pat-\nterned into a 100- \u0016m wide, 1.5-mm long strip by pho-\ntolithography and ion milling. The strip was contacted\nby Cr/Au pads on either end by photolithography, sput-\nter deposition, and lifto\u000b. This sub-mm wide strip ge-\nometry31allows for the use of a cavity electron paramag-\nnetic resonance spectrometer to measure both spin-orbit\ntorque and spin pumping.\nWe \frst demonstrate the transmission of spin current\nfrom the metal layer to the YIG layer through the mea-\nsurement of the damping-like32spin-orbit torque. We\nused a method similar to Refs. 6, 31 where the change inarXiv:1802.03865v3  [cond-mat.mtrl-sci]  29 Apr 20182\n-1.0 0.0 1.00.450.500.55\nJdc (1010 A/m2)\n  W (mT) H||+y\n H||-y\n-1.0 -0.5 0.0 0.5 1.0-0.8-0.40.00.40.8\nYIG/Pt\n  eff (10-4)\nJdc (1010 A/m2)\n-1.0 -0.5 0.0 0.5 1.0-0.8-0.40.00.40.8\nYIG/Cu/Pt\n  eff (10-4)\nJdc (1010 A/m2)\n-1.0 -0.5 0.0 0.5 1.0-0.8-0.40.00.40.8\nYIG/Py/Pt\n  eff (10-4)\nJdc (1010 A/m2)\n238.0 238.5 239.0 239.5H||+y\n  dIFMR/dH (a.u.)\n0H (mT)Jdc= +3 mA\nJdc= -3 mA(a) (b) (c)\n(d) (e) (f)JdchrfH\nSOTPt\nX\nYIGy x\nFigure 1. (a) Schematic of spin-orbit torque (SOT) generated by a dc current Jdcin YIG/(X/)Pt. (b) Jdc-induced modulation\nof FMR spectra in YIG/Pt. The direction of Jdcis as de\fned in (a). (c) Jdc-induced change in FMR linewidth Wwith bias\nmagnetic \feld applied along the + yand -ydirections as de\fned in (a). (d-f) Change in the e\u000bective Gilbert damping parameter\n\u000be\u000bwithJdcfor (d) YIG/Pt, (e) YIG/Cu/Pt, and (f) YIG/Py/Pt. The lines indicate linear \fts to the data.\ndamping is monitored as a function of dc bias current,\nIdc. FMR spectra were measured in a rectangular TE 102\nmicrowave cavity with a nominal excitation power of 10\nmW and several values of Idcin the metallic layer as il-\nlustrated in Fig. 1(a). Each spectrum was \ft with the\nderivative of the sum of symmetric and antisymmetric\nLorentzians (e.g., Fig. 1(b)) to extract the half-width-at-\nhalf-maximum linewidth, W.\nFigure 1(c) shows the variation of WwithIdcunder\nopposite transverse external magnetic \felds, H. The\ndata contain components that are odd and even with\nrespect to Idc, which are due to the spin-orbit torque\nand Joule heating, respectively6,31. The symmetry of the\nspin-Hall spin-orbit torque also gives rise to a component\nofWversusIdcthat is odd with respect to H(Refs. 6{\n8, 31), extracted through \u0001 Wodd(Idc) =fW(Idc;+jHj)\u0000\nW(Idc;\u0000jHj)g=2. We can then obtain the linear change\nin the e\u000bective Gilbert damping parameter due to the\ndc spin-orbit torque, \u0001 \u000be\u000b=j\rj\u0001Wodd=(2\u0019f), where\nj\rj=(2\u0019) = 28 GHz/T and f= 9:55 GHz.\nFrom the linear slope of \u0001 \u000be\u000bover the dc current den-\nsityJdc=Idc=(wtPt) (Fig. 1(d)-(f)), with w= 100\u0016m\nandtPt= 5 nm33, the e\u000bective spin-Hall angle, \u0012e\u000b, can\nbe quanti\fed from7\n\u0012e\u000b=2jej\n~\u0012\nH+Meff\n2\u0013\n\u00160MstYIG\f\f\f\f\u0001\u000be\u000b\nJdc\f\f\f\f;(1)\nwhereMs= 130 kA/m is the saturation magnetization,\nMe\u000b= 190 kA/m is the e\u000bective magnetization including\nthe out-of-plane uniaxial anistropy \feld3, andtYIG= 20\nnm is the thickness of the YIG layer. By \ftting the data\nin Fig. 1(d) with Eq. 1, we arrive at \u0012e\u000b= 0:76\u00060:05%for YIG/Pt.\nWe note that \u0012e\u000bis the product of the intrinsic spin-\nHall angle of Pt, \u0012Pt, and the interfacial spin current\ntransmissivity, T. Assuming that tPtis su\u000eciently larger\nthan the spin di\u000busion length, \u0015Pt, the expression for \u0012e\u000b\nis34{36\n\u0012e\u000b=T\u0012Pt\u00192Ge\u000b\n\"#\u0015Pt\u001aPt\u0012Pt; (2)\nwhereGe\u000b\n\"#is the e\u000bective spin-mixing conductance\n(which includes the spin back\row factor) and \u001aPt\u0019\n4:0\u000210\u00007\n m is the measured resistivity of the Pt layer.\nWith\u0015Pt\u001aPt\u0019(0:6\u00000:8)\u000210\u000015\nm2(Refs. 30, 37, 38),\nwe estimate \u0015Ptto be\u00191.5-2 nm.\nAccording to Eq. 2, the small \u0012e\u000bin our YIG/Pt can\nbe attributed to a reduced T(i.e.,Ge\u000b\n\"#) at the YIG-Pt\ninterface, which may be due to a residual carbon agglom-\neration on the YIG surface39that was not removed by our\ncleaning protocol. In particular, by taking \u0012Pt\u001915\u000030%\nreported from prior spin-orbit torque studies34{36, we ob-\ntain for our YIG/Pt bilayer T\u00190:03\u00000:05, orGe\u000b\n\"#\u0019\n(2\u00005)\u00021013\n\u00001m\u00002, which is an order of magnitude\nlower than the typical values reported for ferromagnetic-\nmetal/normal-metal heterostructures15,34{36, although it\nis comparable to prior reports on YIG/Pt16,20.\nFor YIG/Cu/Pt (Fig. 1(e)), we do not detect a spin-\norbit torque within our experimental resolution, i.e.,\n\u0012e\u000b= 0:01\u00060:10%. Evidently, the Cu dusting layer\nat the YIG-Pt interface suppresses the transmission of\nspin current. By contrast, the Py dusting layer en-\nhances spin transmission from Pt to YIG by \u001940%, with\n\u0012e\u000b= 1:08\u00060:06% derived from the data in Fig. 1(f).\nThe spin-orbit torque experiment thus suggests that the3\ny x\n-3-2-10123-20-1001020\nm0(H-HFMR) (mT)\n  VISH (mV)\n-6-4-20246\nH||-yH||+y YIG/Pt\nVISHW2 (mV mT2)\n-3-2-10123-6-3036\nm0(H-HFMR) (mT)H||-yH||+y YIG/Cu/Pt\n  VISH (mV)\n-0.4-0.20.00.20.4\nVISHW2 (mV mT2)\n-3-2-10123-8-4048\nH||-yH||+y YIG/Py/Pt\n  VISH (mV)\nm0(H-HFMR) (mT)-8-6-4-202468\nVISHW2 (mV mT2)(a) (b) (c) (d)\nhrfH\nFigure 2. (a) Schematic of electrically detected spin pumping in YIG/(X/)Pt. (b-d) Inverse spin-Hall voltage VISHspectra\nmeasured for (b) YIG/Pt, (c) YIG/Cu/Pt, and (d) YIG/Py/Pt. The right vertical axis show VISHscaled by the square of the\nFMR linewidth W, which is proportional to the transmission e\u000eciency of spin current from YIG to Pt.\nnonmagnetic and magnetic insertion layers have opposite\ne\u000bects on spin current transmissivity (Eq. 2).\nIn addition to spin-orbit torque, we show that the mod-\ni\fcation of the YIG-Pt interface equally impacts the re-\nciprocal process of spin pumping. The same sub-mm\nwide YIG/X/Pt strips are measured in the setup identi-\ncal to the spin-orbit torque experiment, except that the\ndc wire leads were connected to a nanovoltmeter, instead\nof a dc current source. As illustrated in Fig. 2(a), FMR\nin the YIG layer pumps a spin current into the Pt layer,\nin which the inverse spin-Hall e\u000bect converts the spin\ncurrent to a charge current that is detected through a\nvoltage peak, VISH, coinciding with FMR. Figure 2(b)-\n(d) shows the VISHspectra obtained at 10 mW of rf ex-\ncitation. The reversal of the voltage polarity with the H\ndirection is consistent with the symmetry of the inverse\nspin-Hall e\u000bect.\nIn the limit of tPtsu\u000eciently larger than \u0015Pt, the re-\nlationship between the peak magnitude of VISHand\u0012e\u000b\nis given by40\njVISHj\u0019h\n2jej\u0012e\u000bfPL\ntPt\u00022; (3)\nwhereL= 1500\u0016m is the length of the sample, P= 1:26\nis the precession ellipticity factor, and \u0002 is the preces-\nsion cone angle. It should be noted that these three\nYIG/X/Pt samples undergo precession at di\u000berent cone\nangles, given by \u0002 = \u00160hrf=W(Refs. 19, 41), since their\nlinewidths Ware di\u000berent. Due to the lack of direct\ncalibration for the microwave \feld amplitude hrfin our\nsetup, the absolute magnitudes of \u0012e\u000bcannot be de-\ntermined accurately from the spin pumping experiment\n(Eq. 3)42.\nNevertheless, we can compare the relative magnitudes\nof\u0012e\u000bamong the three samples. Speci\fcally, we scale\nVISHbyW2(/\u0002\u00002), as shown on the right vertical axis\nof Fig. 2(b)-(d), to quantify the e\u000eciency of spin-current\ntransmission from YIG to Pt. Comparing Fig. 2(c) with\nFig. 2(b), the Cu dusting layer reduces the spin trans-\nmission e\u000eciency ( /VISHW2) by an order of magnitude.\nBy contrast, comparing Fig. 2(d) with Fig. 2(b), the\nPy dusting layer enhnaces the transmission e\u000eciency by\u001940%. This suppression (enhancement) of spin transmis-\nsion with the Cu (Py) insertion layer in the spin pump-\ning experiment quantitatively agrees with the spin-orbit\ntorque experiment, as summarized in Table I. These re-\nsults thus corroborate the reciprocity of the two spin-\ncurrent transmission processes between YIG and Pt.\nWe have revealed that the ultrathin dusting layer of\nnonmagnetic Cu at the YIG-Pt interface suppresses spin\ntransmission, whereas the ferromagnetic Py dusting layer\nenhances it. Our experimental results are qualitatively\nconsistent with the \frst-principles calculations by Jia et\nal.43, which report that the spin-mixing conductance at\nthe YIG-metal interface depends on the interfacial mag-\nnetic moment density. With the ultrathin insertion layer\nof Cu (Py) decreasing (increasing) the interfacial mag-\nnetization, Ge\u000b\n\"#and hence\u0012e\u000bdecrease (increase) as de-\nscribed by Eq. 2. Moreover, the enhancement of spin\ntransmission between YIG and Pt with an ultrathin fer-\nromagnetic insertion layer, quantitatively similar to our\nresults, has been observed in a spin-Seebeck e\u000bect ex-\nperiment by Kikuchi et al.29. We further note that al-\nthough bulk Pt is paramagnetic, it is close to ful\flling the\nStoner criterion such that the direct interface of YIG/Pt\nmay accommodate a higher interfacial magnetic moment\ndensity44,45than YIG/Cu/Pt.\nThe large reduction of spin-orbit torque and spin\npumping with the ultrathin Cu insertion layer may seem\nunexpected, considering that this insertion layer is much\nthinner than the typical spin di\u000busion length of Cu\n(\u0015Cu>100 nm)46. Indeed, prior spin pumping exper-\niments report only a modest decrease (by \u001810%) in spin-\ncurrent transmission between YIG and Pt when the Cu\nspacer thickness is \u00191 nm18,22. However, spin pump-\ning22and spin-Hall magnetoresistance47studies have\nshown that spin transmission decreases by an order-of-\nmagnitude with the insertion of a Cu spacer layer, even\nwhen its thickness (e.g., \u00195 nm) is much smaller than\n\u0015Cu. Other studies also indicate large spin-memory loss\nat the Cu-Pt interface48,49, although we do not observe a\nsigni\fcant increase in spin dissipation (Gilbert damping)\nin YIG/Cu/Pt compared to uncapped YIG, as shown be-\nlow. While further studies are required to understand the\nroles of the Cu spacer layer, one possibility is that spin4\ntransmission is highly sensitive to the nature of the YIG-\nmetal interface, such as the morphology of the ultrathin\nCu layer and the presence of carbon agglomeration39.\nTo gain complementary insight into interfacial spin-\ncurrent transmission, we have examined the enhancement\nof Gilbert damping in YIG/X/Pt strips compared to un-\ncapped YIG \flms. Fig. 3 summarizes the frequency de-\npendence of W, acquired with a broadband FMR setup,\nfrom which the Gilbert damping parameter, \u000b, is quanti-\n\fed. The averaged Gilbert damping parameter for three\nuncapped YIG \flms is \u000b= (4:4\u00060:6)\u000210\u00004, which is\nwithin the range reported by our earlier work3.\nWe observe an increase in \u000bfor each YIG/X/Pt com-\npared to uncapped YIG. Assuming that the damping in-\ncrease is exclusively due to spin pumping, the spin-mixing\nconductance is given by14,15,\nGe\u000b\n\"#=2e2MstYIG\n~2j\rj\u0001\u000b; (4)\nwhere \u0001\u000b(summarized in Table I) is the di\u000berence be-\ntween\u000bof YIG/X/Pt and uncapped YIG. From Eq. 4, we\n\fndGe\u000b\n\"#= (3:3\u00060:5)\u00021013\n\u00001m\u00002for YIG/Pt, which\nis in quantitative agreement with the estimated Ge\u000b\n\"#from\nEq. 2. We also obtain Ge\u000b\n\"#= (0:6\u00060:5)\u00021013\n\u00001m\u00002\nfor YIG/Cu/Pt, which again corroborates the one-order-\nof-magnitude reduction in spin transmission with the ul-\ntrathin Cu insertion layer. Therefore, our experimental\nresults of spin-orbit torque (Fig. 1), electrically detected\nspin pumping (Fig. 2), and Gilbert damping enhance-\nment (Fig. 3) are consistent with each other for YIG/Pt\nand YIG/Cu/Pt.\nThe Gilbert damping enhancement, \u0001 \u000bfor\nYIG/Py/Pt is\u00194 times greater than that for YIG/Pt.\nThis observation is at odds with our \fndings from the\nspin-orbit torque and spin pumping experiments, which\nshow thatGe\u000b\n\"#(i.e., \u0001\u000baccording to Eq. 4) should be\nonly a factor of\u00191.4 greater for YIG/Py/Pt compared\nto YIG/Pt. We thus estimate that only \u001930% of the\ntotal \u0001\u000bis due to spin pumping in YIG/Py/Pt, such\nthat the adjusted value of Ge\u000b\n\"#is\u00195\u00021013\n\u00001m\u00002. The\nremaining\u001970% of \u0001\u000bis likely due to spin-memory loss,\ni.e., spin depolarization by the ultrathin Py layer that\nincreases the Gilbert damping but does not contribute\nto spin-current transmission from YIG to Pt. This large\nspin-memory loss in YIG/Py/Pt is comparable to reports\non ferromagnetic-metal/Pt heterostructures30,49,50.\nIn summary, we have measured the transmission of\nspin current between YIG and Pt thin \flms, separated by\nan interfacial dusting layer of nonmagnetic Cu or mag-\nnetic Py, through FMR-based spin-orbit torque and spin\npumping experiments. Spin transmission decreases by an\norder of magnitude when ultrathin Cu is inserted at the\nYIG-Pt interface and increases by \u001940 % with the inser-\ntion of ultrathin Py. The quantitatively consistent results\nfrom the spin-orbit torque and spin pumping experiments\ncon\frm the reciprocity of these two processes. However,\nwith the Py insertion layer, the Gilbert damping param-\n0 5 10 15051015YIG/Py/Pt\nYIG/Pt\nYIG/Cu/Pt\nYIGW (mT)\nf (GHz)Figure 3. Frequency dependence of half-width-at-half-\nmaximum FMR linewidth, W.\nTable I. Essential extracted parameters - \u0012SOT\ne\u000b: e\u000bective spin-\nHall angle from the spin-orbit torque experiment; VISHW2:\ne\u000eciency of spin transmission from the electrically detected\nspin pumping experiment; Ge\u000b\n\"#: e\u000bective spin-mixing conduc-\ntance from the enhancement in Gilbert damping (YIG/Py/Pt\nadjusted to account for spin-memory loss); \u0001 \u000b: total en-\nhancement of the Gilbert damping parameter.\nYIG/Pt YIG/Cu/Pt YIG/Py/Pt\n\u0012SOT\ne\u000b(%) 0 :76\u00060:05 0:01\u00060:10 1:08\u00060:06\nVISHW2(\u0016V mT2) 4:0\u00060:2 0:35\u00060:02 5:6\u00060:4\nGe\u000b\n\"#(1013\n\u00001m\u00002) 3:3\u00060:5 0:6\u00060:5 \u00195\n\u0001\u000b(10\u00004) 4 :8\u00060:7 0:9\u00060:7 21 \u00061\neter is much larger than expected from spin pumping,\nsuggesting substantial spin-memory loss in YIG/Py/Pt.\nOur \fndings shed light on the roles of interfacial magneti-\nzation in the transmission and depolarization of spin cur-\nrent between a magnetic insulator and a normal metal.\nAcknowledgments: This work is funded by NSF\nERC TANMS 1160504, AFRL through contract FA8650-\n14-C-5706, and by the W.M. 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Silva, (2017), arXiv:1711.07654."    },    {        "title": "1010.0268v2.Ferromagnetic_resonance_study_of_Co_Pd_Co_Ni_multilayers_with_perpendicular_anisotropy_irradiated_with_Helium_ions.pdf",        "content": "Ferromagnetic resonance study of Co/Pd/Co/Ni multilayers with perpendicular\nanisotropy irradiated with Helium ions\nJ-M. L. Beaujour, A. D. Kent,1D. Ravelosona,2and I. Tudosa and E. E. Fullerton3\n1Department of Physics, New York University, 4 Washington Place, New York, NY 10003, USA\n2Institut d'Electronique Fondamentale, UMR CNRS 8622,\nUniversite Paris Sud, 91405 Orsay Cedex, France\n3University of California, San Diego, Center for Magnetic Recording Research, La Jolla, CA 92093-0401, USA\n(Dated: April 24, 2022)\nWe present a ferromagnetic resonance (FMR) study of the e\u000bect of Helium ion irradiation on the\nmagnetic anisotropy, the linewidth and the Gilbert damping of a Co/Ni multilayer coupled to Co/Pd\nbilayers. The perpendicular magnetic anisotropy decreases linearly with He ion \ruence, leading to\na transition to in-plane magnetization at a critical \ruence of 5 \u00021014ions/cm2. We \fnd that the\ndamping is nearly independent of \ruence but the FMR linewidth at \fxed frequency has a maximum\nnear the critical \ruence, indicating that the inhomogeneous broadening of the FMR line is a non-\nmonotonic function of the He ion \ruence. Based on an analysis of the angular dependence of the\nFMR linewidth, the inhomogeneous broadening is associated with spatial variations in the magnitude\nof the perpendicular magnetic anisotropy. These results demonstrate that ion irradiation may be\nused to systematically modify the magnetic anisotropy and distribution of magnetic anisotropy\nparameters of Co/Pd/Co/Ni multilayers for applications and basic physics studies.\nPACS numbers:\nCo/Ni multilayers are of great interest in informa-\ntion technology and in spin-transfer devices because they\ncombine high spin polarization with large perpendicu-\nlar magnetic anisotropy (PMA) [1{3]. Perpendicular\nanisotropy was predicted in Co/Ni multilayers and has\nbeen shown experimentally to be a function of layer com-\nposition and thin \flm growth conditions [4]. Recently\nit has been shown that the coercivity and perpendicu-\nlar anisotropy of a multilayer can be tailored by Helium\nion irradiation [5], making it possible to modify \flms af-\nter growth to tune their magnetic properties. This is\nof great interest for applications and basic physics stud-\nies, as in many cases the perpendicular anisotropy of a\nstructure sets importance device metrics, and ion irradi-\nation o\u000bers the possibility of changing these properties,\nboth locally and globally, after device fabrication. For\ninstance, in spin-transfer magnetic random access mem-\nories (STT-MRAM) the current threshold for switching\nis proportional to the PMA [2, 6].\nLight ion irradiation has been used to vary the mag-\nnetic properties of multilayer \flms in many earlier studies\n[7{11]. For instance, the coercivity of Co/Pt multilayers\nwas found to decrease with ion dose [8]. This behavior\nwas attributed to interface mixing and strain relaxation\nreducing the PMA. Very recently, it was reported that\nthe coercive \feld of Co/Ni multilayers decreases linearly\nwith increasing He+irradiation \ruence up to F= 1015\nions/cm2, suggesting changes in the magnetic anisotropy\nof the \flm [10]. The e\u000bect of ion irradiation on the FMR\nlinewidth has also been studied in Au/Fe multilayer \flms\nwith PMA [11]. The PMA is reduced by He+irradi-\nation and the authors explained this by a reduction of\nthe inhomogeneous contribution to the FMR linewidth.\nIn a recent paper, we presented a FMR study of the\nanisotropy and the linewidth of a Co/Ni multilayer \flmexposed to a relatively high He+irradiation \ruence ( F\n= 1015ions/cm2) [12]. In addition to a strong decrease\nof the PMA, the contribution to the linewidth from spa-\ntial variation of the anisotropy, was reduced compared\nto that of a non irradiated Co/Ni multilayer. Further-\nmore, a correlation between the anisotropy distribution\nand the linewidth broadening from two-magnon scatter-\ning (TMS) mechanism was observed. However, a system-\natic study of the e\u000bect of the He+irradiation on the FMR\nspectra as a function of \ruence has yet to be reported.\nIn this paper, we present a FMR study of a Co/Ni\nmultilayer coupled to Co/Pd bilayers exposed to Helium\nion irradiation of \ruence up to 1015ions/cm2. The PMA\nand the contributions to the FMR linewidth, including\nthose from Gilbert damping ( \u000b), are studied as a function\nof \ruence.\nThe samples had the following layer structure: jj3 Taj1\nPdj0.3 Coj1 Pdj0.14 Coj[0.8 Nij0.14 Co]\u00023j1 Pdj0.3 Coj1\nPdj3 Tajj(layer thickness in nm) and was fabricated by dc\nmagnetron sputtering. The Co/Ni multilayer is embed-\nded between Co/Pd bilayers to enhance the overall PMA\nof the \flm and to have resonance frequencies in which the\nfull angular dependence of the FMR response could be\ninvestigated in a 1 T electromagnet. The substrate was\ncleaved into several pieces that were then exposed to dif-\nferent doses of Helium ion irradiation of energy 20 keV\nwith \ruence in the range 1014\u0014F\u00141015ions/cm2.\nFMR measurements were conducted at room tempera-\nture using a coplanar waveguide (CPW). Details of the\nexperimental setup can be found in [13]. The \feld swept\nCPW transmission signal ( S21) was recorded as a func-\ntion of frequency for dc magnetic \felds normal to the \flm\nplane and as a function of the out-of-plane \feld angle at\n20 GHz. The magnetization density of the \flm at room\ntemperature was measured with a SQUID magnetome-arXiv:1010.0268v2  [cond-mat.mes-hall]  5 Oct 20102\n03 06 09 00.40.60.8K\nK0\n5 1 001020H\n /s615490Hres  (T) 0 \n1 \n2.5 \n F\nield angle /s61542H  (deg)20 GHz 5 \n7.5 \n10HH\n-0.40.00.40.80\n102030f  (GHz)(a) \n(b) /s615490Hres    (T) \n/s61549 0Meff (\nc)  \nF (1014 ions/cm2)K (105 J/m3)21\nFIG. 1: a) Angular dependence of the resonance \feld for dif-\nferent irradiation \ruence ( \u00021014ions/cm2). The solid lines\nare guides to the eye. b) Frequency dependence of the reso-\nnance \feld when the applied \feld is normal to the \flm plane\n(\u001eH= 90o) at selected \ruences. The solid lines are the \fts\nto Eq. 1. The zero frequency intercept gives the e\u000bective de-\nmagnetization \feld, \u00160Me\u000b. c) The second and fourth order\nperpendicular anisotropy constants, K1andK2, versus \ru-\nence.\nter:Ms'4:75\u0002105A/m. Within the measurement\nuncertainty, Msremains unchanged after irradiation.\nFig. 1a shows the out of plane angular dependence\nof the resonance \feld at 20 GHz for di\u000berent \ruences.\nFor the non-irradiated \flm, the resonance \feld when H\nis normal to the \flm plane ( \u001eH= 90o) is smaller than\nthat when the \feld is in the \flm plane ( \u001eH= 0o). This\nshows that the magnetic easy axis is normal to the \flm\nplane. As the \ruence increases, Hresat\u001eH= 90oin-\ncreases whereas that at \u001eH= 0odecreases. In the high\n\ruence range, F>5\u00021014ions/cm2,Hresis the larger\nwhen the \feld is normal to the \flm plane, i.e. the mag-\nnetic easy axis is in the \flm plane. Fig. 1b shows the\nfrequency dependence of the resonance \feld for di\u000berent\n\ruences when the dc \feld is normal to the \flm plane.\nThis data is \ftted to the resonance condition [14]:\nf=1\n2\u0019\r\u00160(Hres\u0000Me\u000b); (1)\nwhere\ris the gyromagnetic ratio. \u00160Me\u000b, the e\u000bec-\ntive easy plane anisotropy, is given by: Me\u000b=Ms\u0000\n2K1=(\u00160Ms), whereK1is the second order anisotropy\nconstant. We \fnd that \u00160Me\u000bis negative at low \ru-\nence which implies that the PMA is su\u000ecient to over-\ncome the demagnetizing energy and hence the easy axis\nis normal to the \flm plane. As the \ruence is further\nincreased,\u00160Me\u000bbecomes positive. These results con-\n\frm that there is a re-orientation of the easy axis, as was\ninferred indirectly through magnetic hysteresis loop mea-\nsurements in Ref. [10]. \u00160Me\u000bchanges sign for \ruence\nbetween 5 and 7.5 \u00021014ions/cm2. Therefore, by expos-\ning the \flm to a speci\fc \ruence, it is posible to engineer\n01000\n1000\n3 06 09 001000\n102030020406080 \n \nF=0Δ\nHα       ΔHinh         ΔHtot  \nF=5/s615490ΔH  (mT)f\n/s61472 ( GHz )  F=10F\nield angle /s61542H  ( deg )HH /s615490ΔH  (mT) \n \nF=7.5H\nFIG. 2: On the left, the linewidth as a function of frequency\nfor the \flm irradiated at 7.5 \u00021014ions/cm2. The solid line\nis a linear \ft to the experimental data. On the right, the\nangular dependence of the linewidth at 20 GHz for a selec-\ntion of \ruences. The solid lines represent the \fts to the total\nlinewidth \u0001 Htot= \u0001H\u000b+\u0001Hinh, , where the intrinsic damp-\ning and the inhomogeneous contribution are represented by\nthe dashed line and the dotted line respectively.\nthe anisotropy so that the PMA \feld just compensates\nthe demagnetization \feld.\nThe second order perpendicular anisotropy constant\nK1decreases linearly with \ruence (Fig. 1c). The \flm\nirradiated at 1015ions/cm2has an anisotropy constant\n40% smaller than that of the non-irradiated \flm. The\n4thorder anisotropy constant K2is determined from\nthe angular dependence of Hresfor magnetization angles\n45\u0014\u001e\u001490o[13].K2is smaller than K1by a factor 10,\nand is nearly independent of \ruence.\nThe FMR linewidth \u00160\u0001Hwhen the dc \feld is ap-\nplied normal the \flm plane was measured as a function\nof frequency. Fig. 2 shows \u00160\u0001Hversusffor the \flm\nirradiated at F= 7:5\u00021014ions/cm2. The linewidth in-\ncreases linearly with frequency, characteristic of Gilbert\ndamping, an intrinsic contribution to the linewidth \u0001 H\u000b\n[15]:\n\u0001H\u000b=4\u0019\u000b\n\u00160\rf: (2)\nFrom a linear \ft to the experimental data, the magnetic\ndamping constant is estimated from the slope of the line:\n\u000b= 0:037\u00060:004. The \flms irradiated at F= 0;1 and\n10\u00021014ions/cm2shows a similar frequency dependence\nof the linewidth and have about the same damping con-\nstant,\u000b\u00190:04. At intermediate \ruence F= 2:5, 5\u00021014\nions/cm2, the linewidth is enhanced and is frequency in-\ndependent, i.e. the linewidth is dominated by an inhomo-\ngeneous contribution, \u0001 Hinh. The angular dependence\nof the linewidth measured at 20 GHz is shown in Fig.\n2 for \flms irradiated at selected \ruences. For the non-\nirradiated \flm and the \flm irradiated at 1015ions/cm2,\nthe linewidth is practically independent of the \feld an-3\n05 1 01 53060901200\n5100.030.04µ0 Δ/s61512/s61472 \n ⊥ \n(mT)F\n  (1014 ions/cm2)ΔK1  ( 105 J/m3 )H\n0.00.10.20.30.4 \nα-dampingF\n (1014 ions/cm2)\nFIG. 3: The \ruence dependence of the linewidth at 20 GHz\nwhen the dc \feld is normal to the \flm plane (squares). The\nsolid circles represent the \ruence dependence of the distribu-\ntion in the PMA constant K1determined from \ftting \u0001 Hvs.\n\u001eH. The inset shows the Gilbert damping constant \u000bas a\nfunction of \ruence.\ngle from about 30oup to 90o. For the \flm irradiated at\n5\u00021014ions/cm2, \u0001His clearly angular dependent and\nshows a minimum at an intermediate \feld angle.\nThe angular dependence of the linewidth was \ft to\na sum of the intrinsic linewidth \u0001 H\u000band an inhomo-\ngeneous contribution \u0001 Hinhfor magnetization angles\n45o\u0014\u001e\u001490o, an angular range in which TMS does\nnot contribute to the linewidth [13]. The inhomogeneous\nlinewidth is given by:\n\u0001Hinh:(\u001eH) =j@Hres=@K 1j\u0001K1+j@Hres=@\u001ej\u0001\u001e;(3)\nwhere \u0001K1is the width of the distribution of anisotropies\nand \u0001\u001eis the distribution of the angles of the magnetic\neasy axis relative to the \flm normal. The computed\nlinewidth contributions are shown for the \flm irradiated\natF= 5\u00021014ions/cm2in Fig. 2. Note that the intrin-\nsic contribution \u0001 H\u000bis practically independent of \feld\nangles, as expected when the angle between the magne-\ntization and the applied \feld is small. For this sample,\nthe maximum angle is about 5oand it is due to the fact\nthat the resonance \feld ( Hres'0:6 T) is much larger\nthan the e\u000bective demagnetization \feld ( Me\u000b'0). Theinhomogeneous contribution from the distribution in the\nanisotropy \feld directions does not signi\fcantly a\u000bect the\n\ft. For the \flm irradiated at the lower and upper \ruence\nrange, the angular dependence of the intrinsic linewidth\nis computed \fxing the value of \u000bto that obtained from\nthe \ft of the frequency dependence of the linewidth. For\nthe other \flms ( F=2.5 and 5\u00021014ions/cm2),\u000bwas\na \ftting parameter.\nThe \ruence dependence of \u0001 K1and the linewidth at\n20 GHz are shown in Fig. 3. The inset shows the Gilbert\ndamping constant as a function of \ruence. The linewidth\nat 20 GHz when the \feld is normal to the \flm plane is\na non monotonic function of \ruence. \u0001 Hincreases as\nthe \ruence increases, reaching a maximum value at F\n\u00195\u00021014ions/cm2. Then, as the \ruence is further in-\ncreased, \u0001Hdecreases and falls slightly below the range\nof values at the lower \ruence range. Interestingly, the\nlarger linewidth is observed just at the \ruence for which\n\u00160Me\u000b= 0. The magnetic damping is practically not\na\u000bected by irradiation within the error bars: \u000b\u00190:04.\nThe distribution of PMA constants, \u0001 K1, shows a similar\n\ruence dependence as the total linewidth, with a maxi-\nmum at F\u00195\u00021014ions/cm2, clearly indicating that\nthis is at the origin of the \ruence dependence of the mea-\nsured linewidth. The distribution in PMA anisotropy is\nalmost zero when the \ruence is above 7 \u00021014ions/cm2.\nThe largest value of \u0001 K1corresponds to variation of K1\nofabout 8%, which is much larger than that of non irradi-\nated \flm and the highly irradiated \flm, \u0001 K1=K1\u00192%\nand 0.3% respectively.\nIn summary, irradiation of Co/Pd/Co/Ni \flms with\nHelium ions leads to clear changes in its magnetic char-\nacteristics, a signi\fcant decrease in magnetic anisotropy\nand a change in the distribution of magnetic anisotropies.\nImportantly, this is achieved without a\u000becting the \flm\nmagnetization density and magnetic damping, which re-\nmain virtually unchanged. It would be of interest to have\na better understanding of the origin of the maximum in\nthe distribution of magnetic anisotropy at the critical\n\ruence, the \ruence needed to produce a reorientation of\nthe magnetic easy axis. Nonetheless, these results clearly\ndemonstrate that ion irradiation may be used to system-\natically tailor the magnetic properties of Co/Pd/Co/Ni\nmultilayers for applications and basic physics studies.\n[1] S. Mangin et al: , Nat. Mater. 5, 210 (2006).\n[2] S. Mangin et al: , Appl. Phys. Lett. 94, 012502 (2009).\n[3] D. Bedau et al: , Appl. Phys. Lett. 96, 022514 (2010).\n[4] G. H. O. Daalderop et al: , Phys. Rev. Lett. 68, 682\n(1992); S. Girod et al: , Appl. Phys. Lett. 94, 262504\n(2009).\n[5] C. Chappert et al: , Science 280, 1919 (1998).\n[6] J. Z. Sun, Phys. Rev. B 62, 570 (2000).\n[7] A. Traverse et al: , Europhysics Letters 8, 633 (1989).\n[8] T. Devolder, Phys. Rev. B 62,5794 (2000).\n[9] J. Fassbender etal: , J. Phys. D: J. Appl. 37, R179 (2004).[10] D. Stanescu et al: , J. Appl. Phys. 103, 07B529 (2008).\n[11] C. Bilzer et al: , J. Appl. Phys. 103, 07B518 (2008).\n[12] J.-M. L. Beaujour et al: , Phys. Rev. B 80, 180415(R)\n(2009).\n[13] J.-M. L. Beaujour etal: , Eur. Phys. J. B 59, 475 (2007).\n[14] S. V. Vonsovskii, Ferromagnetic Resonance (Pergamon,\nOxford, 1966).\n[15] Spin Dynamics in Con\fned Magnetic Structures II\n(edited by B. Hillebrands and K. Ounadjela (Springer,\nHeidelberg, 2002))."    },    {        "title": "0905.4544v2.Hydrodynamic_theory_of_coupled_current_and_magnetization_dynamics_in_spin_textured_ferromagnets.pdf",        "content": "Hydrodynamic theory of coupled current and magnetization dynamics in\nspin-textured ferromagnets\nClement H. Wong and Yaroslav Tserkovnyak\nDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\nWe develop the hydrodynamic theory of collinear spin currents coupled to magnetization dynamics\nin metallic ferromagnets. The collective spin density couples to the spin current through a U(1)\nBerry-phase gauge \feld determined by the local texture and dynamics of the magnetization. We\ndetermine phenomenologically the dissipative corrections to the equation of motion for the electronic\ncurrent, which consist of a dissipative spin-motive force generated by magnetization dynamics and\na magnetic texture-dependent resistivity tensor. The reciprocal dissipative, adiabatic spin torque\non the magnetic texture follows from the Onsager principle. We investigate the e\u000bects of thermal\n\ructuations and \fnd that electronic dynamics contribute to a nonlocal Gilbert damping tensor in\nthe Landau-Lifshitz-Gilbert equation for the magnetization. Several simple examples, including\nmagnetic vortices, helices, and spirals, are analyzed in detail to demonstrate general principles.\nPACS numbers: 72.15.Gd,72.25.-b,75.75.+a\nI. INTRODUCTION\nThe interaction of electrical currents with magnetic\nspin texture in conducting ferromagnets is presently a\nsubject of active research. Topics of interest include\ncurrent-driven magnetic dynamics of solitons such as do-\nmain walls and magnetic vortices,1,2,3,4as well as the\nreciprocal process of voltage generation by magnetic\ndynamics.5,6,7,8,9,10,11,12This line of research has been\nfueled in part by its potential for practical applications\nto magnetic memory and data storage devices.13Funda-\nmental theoretical interest in the subject dates back at\nleast two decades.5,6,14It was recognized early on6that\nin the adiabatic limit for spin dynamics, the conduction\nelectrons interact with the magnetic spin texture via an\ne\u000bective spin-dependent U(1) gauge \feld that is a local\nfunction of the magnetic con\fguration. This gauge \feld,\non the one hand, gives rise to a Lorentz force due to\n\\\fctitious\" electric and magnetic \felds and, on the other\nhand, mediates the so-called spin-transfer torque exerted\nby the conduction electrons on the collective magnetiza-\ntion. An alternative and equivalent view is to consider\nthis force as the result of the Berry phase15accumulated\nby an electron as it propagates through the ferromagnet\nwith its spin aligned with the ferromagnetic exchange\n\feld.8,10,16In the standard phenomenological formalism\nbased on the Landau-Lifshitz-Gilbert (LLG) equation,\nthe low-energy, long-wavelength magnetization dynamics\nare described by collective spin precession in the e\u000bective\nmagnetic \feld, which is coupled to electrical currents via\nthe spin-transfer torques. In the following, we develop\na closed set of nonlinear classical equations governing\ncurrent-magnetization dynamics, much like classical elec-\ntrodynamics, with the LLG equation for the spin-texture\n\\\feld\" in lieu of the Maxwell equations for the electro-\nmagnetic \feld.\nThis electrodynamic analogy readily explains various\ninteresting magnetoelectric phenomena observed recently\nin ferromagnetic metals. Adiabatic charge pumping bymagnetic dynamics17can be understood as the gener-\nation of electrical currents due to the \fctitious electric\n\feld.18In addition, magnetic textures with nontrivial\ntopology exhibit the so-called topological Hall e\u000bect,19,20\nin which the \fctitious magnetic \feld causes a classical\nHall e\u000bect. In contrast to the classical magnetoresis-\ntance, the \rux of the \fctitious magnetic \feld is a topo-\nlogical invariant of the magnetic texture.6\nDissipative processes in current-magnetization dynam-\nics are relatively poorly understood and are of central\ninterest in our theory. Electrical resistivity due to quasi-\none-dimensional (1D) domain walls and spin spirals have\nbeen calculated microscopically.21,22,23More recently, a\nviscous coupling between current and magnetic dynam-\nics which determines the strength of a dissipative spin\ntorque in the LLG equation as well the reciprocal dis-\nsipative spin electromotive force generated by magnetic\ndynamics, called the \\ \fcoe\u000ecient,\"2was also calcu-\nlated in microscopic approaches.3,24,25Generally, such\n\frst-principles calculations are technically di\u000ecult and\nrestricted to simple models. On the other hand, the num-\nber of di\u000berent forms of the dissipative interactions in the\nhydrodynamic limit are in general constrained by sym-\nmetries and the fundamental principles of thermodynam-\nics, and may readily be determined phenomenologically\nin a gradient expansion. Furthermore, classical thermal\n\ructuations may be easily incorporated in the theoretical\nframework of quasistationary nonequilibrium thermody-\nnamics.\nThe principal goal of this paper is to develop a (semi-\nphenomenological) hydrodynamic description of the dis-\nsipative processes in electric \rows coupled to magnetic\nspin texture and dynamics. In Ref. 11, we drew the anal-\nogy between the interaction of electric \rows with quasis-\ntationary magnetization dynamics with the classical the-\nory of magnetohydrodynamics. In our \\spin magnetohy-\ndrodynamics,\" the spin of the itinerant electrons, whose\n\rows are described hydrodynamically, couples to the lo-\ncal magnetization direction, which constitutes the col-\nlective spin-coherent degree of freedom of the electronicarXiv:0905.4544v2  [cond-mat.mes-hall]  16 Nov 20092\n\ruid. In particular, the dissipative \fcoupling between\nthe collective spin dynamics and the itinerant electrons\nis loosely akin to the Landau damping, capturing cer-\ntain kinematic equilibration of the relative motion be-\ntween spin-texture dynamics and electronic \rows. In our\nprevious paper,11we considered a special case of incom-\npressible \rows in a 1D ring to demonstrate the essential\nphysics. In this paper, we establish a general coarse-\ngrained hydrodynamic description of the interaction be-\ntween the electric \rows and textured magnetization in\nthree dimensions, treating the itinerant electron's degrees\nof freedom in a two-component \ruid model (correspond-\ning to the two spin projections of spin-1 =2 electrons along\nthe local collective magnetic order). Our phenomenology\nencompasses all the aforementioned magnetoelectric phe-\nnomena.\nThe paper is organized as follows. In Sec. II, we use a\nLagrangian approach to derive the semiclassical equation\nof motion for itinerant electrons in the adiabatic approx-\nimation for spin dynamics. In Sec. III, we derive the\nbasic conservation laws, including the Landau-Lifshitz\nequation for the magnetization, by coarse-graining the\nsingle-particle equation of motion and the Hamiltonian.\nIn Sec. IV, we phenomenologically construct dissipative\ncouplings, making use of the Onsager reciprocity princi-\nple, and calculate the net dissipation power. In particu-\nlar, we develop an analog of the Navier-Stokes equation\nfor the electronic \ruid, focusing on texture-dependent\ne\u000bects, by making a systematic expansion in nonequi-\nlibrium current and magnetization consistent with sym-\nmetry requirements. In Sec. V, we include the e\u000bects of\nclassical thermal \ructuations by adding Langevin sources\nto the hydrodynamic equations, and arrive at the central\nresult of this paper: A set of coupled stochastic di\u000ber-\nential equations for the electronic density, current, and\nmagnetization, and the associated white-noise correlators\nof thermal noise. In Sec. VI, we apply our results to\nspecial examples of rotating and spinning magnetic tex-\ntures, calculating magnetic texture resistivity and mag-\nnetic dynamics-generated currents for a magnetic spiral\nand a vortex. The paper is summarized in Sec. VII and\nsome additional technical details, including a microscopic\nfoundation for our semiclassical theory, are presented in\nthe appendices.\nII. QUASIPARTICLE ACTION\nIn a ferromagnet, the magnetization is a symmetry-\nbreaking collective dynamical variable that couples to the\nitinerant electrons through the exchange interaction. Be-\nfore developing a general phenomenological framework,\nwe start with a simple microscopic model with Stoner in-\nstability, which will guide us to explicitly construct some\nof the key magnetohydrodynamic ingredients. Within a\nlow-temperature mean-\feld description of short-ranged\nelectron-electron interactions, the electronic action isgiven by (see appendix A for details):\nS=Z\ndtd3r^ y\u0014\ni~@t+~2\n2mer2\u0000\u001e\n2+\u0001\n2m\u0001^\u001b\u0015\n^ :(1)\nHere, \u0001( r;t) is the ferromagnetic exchange splitting,\nm(r;t) is the direction of the dynamical order param-\neter de\fned by ~h^ y^\u001b^ i=2 =\u001asm,\u001asis the local spin\ndensity, and ^ (r;t) is the spinor electron \feld operator.\nFor the short-range repulsion U > 0 discussed in ap-\npendix A, \u0001( r;t) = 2U\u001as(r;t)=~and\u001e(r;t) =U\u001a(r;t),\nwhere\u001a=h^ y^ iis the local particle number density.\nFor electrons, the magnetization Mis in the opposite di-\nrection of the spin density: M=\r\u001asm, where\r <0 is\nthe gyromagnetic ratio. Close to a local equilibrium, the\nmagnetic order parameter describes a ground state con-\nsisting of two spin bands \flled up to the spin-dependent\nFermi surfaces, with the spin orientation de\fned by m.\nWe will focus on soft magnetic modes well below the\nCurie temperature, where only the direction of the mag-\nnetization and spin density are varied, while the \ructu-\nations of the magnitudes are not signi\fcant. The spin\ndensity is given by \u001as=~(\u001a+\u0000\u001a\u0000)=2 and particle den-\nsity by\u001a=\u001a++\u001a\u0000, where\u001a\u0006are the local spin-up/down\nparticle densities along m.\u001ascan be essentially constant\nin the limit of low spin susceptibility.\nStarting with a nonrelativistic many-body Hamilto-\nnian, the action (1) is obtained in a spin-rotationally\ninvariant form. However, this symmetry is broken by\nspin-orbit interactions, whose role we will take into ac-\ncount phenomenologically in the following. When the\nlength scale on which m(r;t) varies is much greater than\nthe ferromagnetic coherence length lc\u0018~vF=\u0001, where\nvFis the Fermi velocity, the relevant physics is captured\nby the adiabatic approximation. In this limit, we start\nby neglecting transitions between the spin bands, treat-\ning the electron's spin projection on the magnetization\nas a good quantum number. (This approximation will\nbe relaxed later, in the presence of microscopic spin-\norbit or magnetic disorder.) We then have two e\u000bec-\ntively distinct species of particles described by a spinor\nwave function ^ 0, which is de\fned by ^ =^U(R)^ 0. Here,\n^U(R) is an SU(2) matrix corresponding to the local spa-\ntial rotationR(r;t) that brings the z-axis to point along\nthe magnetization direction: R(r;t)z=m(r;t), so that\n^Uy(^\u001b\u0001m)^U= ^\u001bz. The projected action then becomes:\nS=Z\ndtZ\nd3r^ 0y\"\n(i~@t+ ^a)\u0000(\u0000i~r\u0000^a)2\n2me\n\u0000\u001e\n2+\u0001\n2^\u001bz\u0015\n^ 0\u0000Z\ndtF[m];(2)\nwhere\nF[m] =A\n2Z\nd3r(@im)2(3)\nis the spin-texture exchange energy (implicitly summing\nover the repeated spatial index i), which comes from the3\nterms quadratic in the gauge \felds that survive the pro-\njection. In the mean-\feld Stoner model, the ferromag-\nnetic exchange sti\u000bness is A=~2\u001a=4me. To broaden our\nscope, we will treat it as a phenomenological constant,\nwhich, for simplicity, is determined by the mean electron\ndensity.26The spin-projected \\\fctitious\" gauge \felds are\ngiven by\na\u001b(r;t) =i~h\u001bj^Uy@t^Uj\u001bi;\na\u001b(r;t) =i~h\u001bj^Uyr^Uj\u001bi: (4)\nChoosing the rotation matrices ^U(m) to depend only on\nthe local magnetic con\fguration, it follows from their\nde\fnition that spin- \u001bgauge potentials have the form:\na\u001b=\u0000@tm\u0001amon\n\u001b(m); a\u001bi=\u0000@im\u0001amon\n\u001b(m);(5)\nwhere amon\n\u001b(m)\u0011 \u0000i~h\u001bj^Uy@m^Uj\u001bi. We show in Ap-\npendix B the well known result (see, e.g., Ref. 27) that\namon\n\u001bis the vector potential (in an arbitrary gauge) of\na magnetic monopole in the parameter space de\fned by\nm:\n@m\u0002amon\n\u001b(m) =q\u001bm; (6)\nwhereq\u001b=\u001b~=2 is the monopole charge (which is ap-\npropriately quantized).\nBy noting that the action (2) is formally identical to\ncharged particles in electromagnetic \feld, we can imme-\ndiately write down the following classical single-particle\nLagrangian for the interaction between the spin- \u001belec-\ntrons and the collective spin texture:\nL\u001b(r;_r;t) =me_r2\n2+_r\u0001a\u001b(r;t) +a\u001b(r;t); (7)\nwhere _ris the spin-\u001belectron (wave-packet) velocity. To\nsimplify our discussion, we are omitting here the spin-\ndependent forces due to the self-consistent \felds \u001e(r;t)\nand \u0001( r;t), which will be easily reinserted at a later\nstage. See Eq. (29).\nThe Euler-Lagrange equation of motion for v=\n_rderived from the single-particle Lagrangian (7),\n(d=dt)(@L\u001b=@_r) =@L\u001b=@r, gives\nme_v=q\u001b(e+v\u0002b): (8)\nThe \fctitious electromagnetic \felds that determine the\nLorentz force are\nq\u001bei=@ia\u001b\u0000@ta\u001bi=q\u001bm\u0001(@tm\u0002@im);\nq\u001bbi=\u000fijk@ja\u001bk=q\u001b\u000fijk\n2m\u0001(@km\u0002@jm):(9)\nThey are conveniently expressed in terms of the tensor\n\feld strength\nq\u001bf\u0016\u0017\u0011@\u0016a\u001b\u0017\u0000@\u0017a\u001b\u0016=q\u001bm\u0001(@\u0017m\u0002@\u0016m) (10)\nbyei=fi0andbi=\u000fijkfjk=2.\u000fijkis the antisymmet-\nric Levi-Civita tensor and we used four-vector notation,de\fning@\u0016= (@t;r) anda\u001b\u0016= (a\u001b;a\u001b). Here and\nhenceforth the convention is to use Latin indices to de-\nnote spatial coordinates and Greek for space-time coor-\ndinates. Repeated Latin indices i;j;k are, furthermore,\nalways implicitly summed over.\nIII. SYMMETRIES AND CONSERVATION\nLAWS\nA. Gauge invariance\nThe Lagrangian describing coupled electron transport\nand collective spin-texture dynamics (disregarding for\nsimplicity the ordinary electromagnetic \felds) is\nL(rp;vp;m;@\u0016m)\n=X\np \nmev2\np\n2+vp\u0001a\u001b+a\u001b!\n\u0000A\n2Z\nd3r(@im)2\n=X\np \nmev2\np\n2+v\u0016\npa\u001b\u0016!\n\u0000A\n2Z\nd3r(@im)2:(11)\nv\u0016\np\u0011(1;vp),vp=_r, and\u001bhere is the spin of indi-\nvidual particles labelled by p. The resulting equations\nof motion satisfy certain basic conservation laws, due to\nspin-dependent gauge freedom, space-time homogeneity,\nand spin isotropicity.\nFirst, let us establish gauge invariance due to an ambi-\nguity in the choice of the spinor rotations ^U(r;t)!^U^U0.\nOur formulation should be invariant under arbitrary di-\nagonal transformations ^U0=e\u0000ifand ^U0=e\u0000ig^\u001bz=2on\nthe rotated fermionic \feld ^ 0, corresponding to gauge\ntransformations of the spin-projected theory:\n\u000ea\u001b\u0016=~@\u0016fand\u000ea\u001b\u0016=\u001b~@\u0016g=2; (12)\nrespectively. The change in the Lagrangian density is\ngiven by\n\u000eL=j\u0016@\u0016fand\u000eL=j\u0016\ns@\u0016g; (13)\nrespectively, where j=j++j\u0000andjs=~(j+\u0000j\u0000)=2\nare the corresponding charge and spin gauge currents.\nThe action S=R\ndtd3rLis gauge invariant, up to sur-\nface terms that do not a\u000bect the equations of motion,\nprovided that the four-divergence of the currents vanish,\nwhich is the conservation of particle number and spin\ndensity:\n_\u001a+r\u0001j= 0;_\u001as+r\u0001js= 0: (14)\n(The second of these conservation laws will be relaxed\nlater.) Here, the number and spin densities along with\nthe associated \rux densities are\n\u001a=X\npnp\u0011\u001a++\u001a\u0000;\nj=X\npnpvp\u0011\u001av; (15)4\nand\n\u001as=X\npq\u001bnp\u0011~\n2(\u001a+\u0000\u001a\u0000);\njs=X\npq\u001bnpvp\u0011\u001asvs; (16)\nwherenp=\u000e(r\u0000rp) and\u001bp=\u0006for spins up and down.\nIn the hydrodynamic limit, the above equations deter-\nmine the average particle velocity vand spin velocity\nvs, which allows us to de\fne four-vectors j\u0016= (\u001a;\u001av)\nandj\u0016\ns= (\u001as;\u001asvs). Microscopically, the local spin-\ndependent currents are de\fned, in the presence of electro-\nmagnetic vector potential aand \fctitious vector potential\na\u001b, by\nme\u001a\u001bv\u001b= Reh y\n\u001b(\u0000i~r\u0000a\u001b\u0000ea) \u001bi; (17)\nwheree<0 is the electron charge.\nB. Angular and linear momenta\nOur Lagrangian (11) contains the dynamics of m(r)\nthat is coupled to the current. In this regard, we note\nthat the time component of the \fctitious gauge poten-\ntial (B4),a\u001b=\u0000~@t'(1\u0000\u001bcos\u0012)=2, is a Wess-Zumino\naction that governs the spin-texture dynamics.4,6,28The\nvariational equation m\u0002\u000emL= 0 gives:\n\u001as(@t+vs\u0001r)m+m\u0002\u000emF= 0: (18)\nTo derive this equation, we used the spin-density con-\ntinuity equation (14) and a gauge-independent identity\nsatis\fed by the \fctitious potentials: their variations with\nrespect to mare given by\n\u000ema\u001b\u0016(m;@\u0016m) =q\u001bm\u0002@\u0016m; (19)\nwhere\n\u000em\u0011@\n@m\u0000X\n\u0016@\u0016@\n@(@\u0016m): (20)\nOne recognizes that Eq. (18) is the Landau-Lifshitz (LL)\nequation, in which the spin density precesses about the\ne\u000bective \feld given explicitly by\nh\u0011\u000emF=\u0000A@2\nim: (21)\nEquation (18) also includes the well-known reactive spin\ntorque:\u001c= (js\u0001r)m,3which is evidently the change\nin the local spin-density vector due to the spin angular\nmomentum carried by the itinerant electrons. One can\nformally absorb this spin torque by de\fning an advective\ntime derivative Dt\u0011@t+vs\u0001r, with respect to the\naverage spin drift velocity vs.\nEquation (18) may be written in a form that explicitly\nexpresses the conservation of angular momentum:27,29\n@t(\u001asmi) +@j\u0005ij= 0; (22)where the angular-momentum stress tensor is de\fned by\n\u0005ij=\u001asvsjmi\u0000A(m\u0002@jm)i: (23)\nNotice that this includes both quasiparticle and collective\ncontributions, which stem respectively from the trans-\nport and equilibrium spin currents.\nThe Lorentz force equation for the electrons, Eq. (8),\nin turn, leads to a continuity equation for the kinetic\nmomentum density.6To see this, let us start with the\nmicroscopic perspective:\n@t(\u001avi) =@tX\npnpvp=X\np( _npvp+np_vp): (24)\nUsing the Lorentz force equation for the second term, we\nhave:\nmeX\npnp_vp=X\npq\u001bnp(ei+\u000fijkbkvpj) =X\npq\u001bnpfi\u0016v\u0016\np\n=\u001asm\u0001(@tm\u0002@im) +\u001asvsjm\u0001(@jm\u0002@im)\n= (@im)\u0001(\u000emF) =\u0000A(@im)\u0001(@2\njm); (25)\nutilizing Eq. (18) to obtain the last line. Coarse-graining\nthe \frst term of Eq. (24), in turn, we \fnd:\nX\np_npvp=\u0000@jX\np\u000e(r\u0000rp)vpivpj!\u0000@jX\n\u001b\u001a\u001bv\u001biv\u001bj:\n(26)\nPutting Eqs. (25) and (26) together, we can \fnally write\nEq. (24) in the form:\nme@t(\u001avi) +@j \nTij+meX\n\u001b\u001a\u001bv\u001biv\u001bj!\n= 0;(27)\nwhere\nTij=A\u0014\n(@im)\u0001(@jm)\u0000\u000eij\n2(@km)2\u0015\n(28)\nis the magnetization stress tensor.6\nA spin-dependent chemical potential ^ \u0016=^K\u00001^\u001agov-\nerned by local density and short-ranged interactions can\nbe trivially incorporated by rede\fning the stress tensor\nas\nTij!Tij+\u000eij\n2^\u001aT^K\u00001^\u001a: (29)\nIn our notation, ^ \u0016= (\u0016+;\u0016\u0000)T, ^\u001a= (\u001a+;\u001a\u0000)Tand ^Kis\na symmetric 2\u00022 compressibility matrix in spin space,\nwhich includes the degeneracy pressure as well as self-\nconsistent exchange and Hartree interactions. In general,\nEq. (29) is valid only for su\u000eciently small deviations from\nthe equilibrium density.\nUsing the continuity equations (14), we can combine\nthe last term of Eq. (27) with the momentum density\nrate of change:\n@t(\u001a\u001bv\u001bi) +@j(\u001a\u001bv\u001biv\u001bj) =\u001a\u001b(@t+v\u001b\u0001r)v\u001bi;(30)5\nwhich casts the momentum density continuity equation\nin the Euler equation form:\nmeX\n\u001b\u001a\u001b(@t+v\u001b\u0001r)v\u001bi+@jTij= 0: (31)\nWe do not expect such advective corrections to @tto\nplay an important role in electronic systems, however.\nThis is in contrast to the advective-like time derivative\nin Eq. (18), which is \frst order in velocity \feld and is\ncrucial for capturing spin-torque physics.\nC. Hydrodynamic free energy\nWe will now turn to the Hamiltonian formulation and\nconstruct the free energy for our magnetohydrodynamic\nvariables. This will subsequently allow us to develop a\nnonequilibrium thermodynamic description. The canon-\nical momenta following from the Lagrangian (11) are\npp\u0011@L\n@vp=mevp+ap;\n\u0019\u0011@L\n@_m=X\npnp@a\u001b\n@_m=X\npnpamon\n\u001b(m): (32)\nNotice that for our translationally-invariant system, the\ntotal linear momentum\nP\u0011X\nppp+Z\nd3r(\u0019\u0001r)m=meX\npvp; (33)\nwhere we have used Eq. (5) to obtain the second equality,\ncoincides with the kinetic momentum (mass current) of\nthe electrons. The latter, in turn, is equivalent to the lin-\near momentum of the original problem of interacting non-\nrelativistic electrons, in the absence of any real or \fcti-\ntious gauge \felds. See appendix A. While Pis conserved\n(as discussed in the previous section and also follows now\nfrom the general principles), the canonical momenta of\nthe electrons and the spin-texture \feld, Eqs. (32), are\nnot conserved separately. As was pointed out by Volovik\nin Ref. 6, this explains anomalous properties of the lin-\near momentum associated with the Wess-Zumino action\nof the spin-texture \feld: This momentum has neither\nspin-rotational nor gauge invariance. The reason is that\nthe spin-texture dynamics de\fne only one piece of the\ntotal momentum, which is associated with the coherent\ndegrees of freedom. Including also the contribution as-\nsociated with the incoherent (quasiparticle) background\nrestores the proper gauge-invariant momentum, P, which\ncorresponds to the generator of the global translation in\nthe microscopic many-body description.\nPerforming a Legendre transformation to Hamiltonianas a function of momenta, we \fnd\nH[rp;pp;m;\u0019] =X\npvp\u0001pp+Z\nd3r_m\u0001\u0019\u0000L\n=X\np(pp\u0000a\u001b)2\n2me+A\n2Z\nd3r(@im)2\n\u0011E+F; (34)\nwhereEis the kinetic energy of electrons and Fis the\nexchange energy of the magnetic order. As could be\nexpected,Eis the familiar single-particle Hamiltonian\ncoupled to an external vector potential. According to\na Hamilton's equation, the velocity is conjugate to the\ncanonical momentum: vp=@H=@ pp. We note that ex-\nplicit dependence on the spin-texture dynamics dropped\nout because of the special property of the gauge \felds:\n_m\u0001@_ma\u001b=a\u001b. Furthermore, according to Eq. (19), we\nhavem\u0002\u000emE= (js\u0001r)m, so the LL Eq. (18) can be\nwritten in terms of the Hamiltonian (34) as11\n\u001as_m+m\u0002\u000emH= 0: (35)\nSo far, we have included in the spin-texture equa-\ntion only the piece coupled to the itinerant electron de-\ngrees of freedom. The purely magnetic part is tedious\nto derive directly and we will include it in the usual LL\nphenomenology.29To this end, we rede\fne\nF[m(r)]!F+F0; (36)\nby adding an additional magnetic free energy F0[m(r)],\nwhich accounts for magnetostatic interactions, crystalline\nanisotropies, coupling to external \felds, as well as energy\nassociated with localized dorforbitals.30Then the to-\ntal free energy (Hamiltonian) is H=E+F, and we in\ngeneral de\fne the e\u000bective magnetic \feld as the thermo-\ndynamic conjugate of m:h\u0011\u000emH. The LL equation\nthen becomes\n%s_m+m\u0002h= 0; (37)\nwhere%sis the total e\u000bective spin density. To enlarge\nthe scope of our phenomenology, we allow the possibility\nthat%s6=\u001as. For example, in the s\u0000dmodel, an extra\nspin density comes from the localized d-orbital electrons.\nMicroscopically, %s@tmterm in the equation of motion\nstems from the Wess-Zumino action generically associ-\nated with the total spin density.\nIn the following, it may sometimes be useful to separate\nout the current-dependent part of the e\u000bective \feld, and\nwrite the purely magnetic part as hm\u0011\u000emF, so that\nh=hm\u0000m\u0002(js\u0001r)m (38)\nand Eq. (37) becomes:\n%s_m+ (js\u0001r)m+m\u0002hm= 0: (39)6\nFor completeness, let is also write the equation of motion\nfor the spin- \u001bacceleration:\nme(@t+v\u001b\u0001r)v\u001bi=q\u001b[m\u0001(@tm\u0002@im)\n+v\u001bjm\u0001(@jm\u0002@im)]\u0000r\u0016\u001b;(40)\nretaining for the moment the advective correction to\nthe time derivative on the left-hand side and reinserting\nthe force due to the spin-dependent chemical potential,\n^\u0016=^K\u00001^\u001a. These equations constitute the coupled re-\nactive equations for our magneto-electric system. The\nHamiltonian (free energy) in terms of the collective vari-\nables is (including the elastic compression piece)\nH[\u001a\u001b;p\u001b;m] =X\n\u001bZ\nd3r\u001a\u001b(p\u001b\u0000a\u001b)2\n2me\n+1\n2Z\nd3r^\u001aT^K\u00001^\u001a+F[m]; (41)\nwhere p\u001b=mev\u001b+a\u001bis the spin-dependent momentum\nthat is locally averaged over individual particles.\nD. Conservation of energy\nSo far, our hydrodynamic equations are reactive, so\nthat the energy (41) must be conserved: P\u0011_H=_E+\n_F= 0. The time derivative of the electronic energy Eis\n_E=Z\nd3rX\n\u001b\u0014\nme\u001a\u001bv\u001b_v\u001b+ _\u001a\u001b\u0012mev2\n\u001b\n2+\u0016\u001b\u0013\u0015\n=Z\nd3rX\n\u001b\u0014\nme\u001a\u001bv\u001bj_v\u001bj\u0000@j(\u001a\u001bv\u001bj)\u0012mev2\n\u001b\n2+\u0016\u001b\u0013\u0015\n=Z\nd3rX\n\u001b\u001a\u001bv\u001bj[me(@t+v\u001b\u0001r)v\u001bj+@j\u0016\u001b]\n=Z\nd3rX\n\u001bq\u001b\u001a\u001bv\u001b\u0001(e+v\u001b\u0002b)\n=Z\nd3rX\n\u001bq\u001b\u001a\u001bv\u001b\u0001e=Z\nd3rjs\u0001e: (42)\nThe change in the spin-texture energy is given, according\nto Eq. (39), by\n_F=Z\nd3r_m\u0001\u000emF=Z\nd3r_m\u0001hm\n=Z\nd3r_m\u0001[%sm\u0002_m+m\u0002(js\u0001r)m)]\n=\u0000Z\nd3rjs\u0001e: (43)\nThe total energy is thus evidently conserved, P= 0.\nWhen we calculate dissipation in the rest of the paper,\nwe will omit these terms which cancel each other. The\ntotal energy \rux density is evidently given by\nQ=X\n\u001b\u001a\u001b\u0012mev2\n\u001b\n2+\u0016\u001b\u0013\nv\u001b: (44)IV. DISSIPATION\nHaving derived from \frst principles the reactive cou-\nplings in our magneto-electric system, summed up in\nEqs. (39)-(41), we will proceed to include the dissipa-\ntive e\u000bects phenomenologically. Let us focus on the lin-\nearized limit of small deviations from equilibrium (which\nmay be spin textured), so that the advective correction\nto the time derivative in the Euler Eq. (40), which is\nquadratic in the velocity \feld, can be omitted. To elimi-\nnate the quasiparticle spin degree of freedom, let us, fur-\nthermore, treat halfmetallic ferromagnets, so that \u001a=\u001a+\nand\u001as=q\u001a, whereq=~=2 is the electron's spin.31From\nEq. (40), the equation of motion for the local (averaged)\ncanonical momentum is:32\n_p=q\n\u001aj\u0002b\u0000r\u0016; (45)\nin a gauge where a\u001b= 0, so that _p=me_v\u0000qe.33\n\u0016=\u001a=K. The Lorentz force due to the applied (real)\nelectromagnetic \felds can be added in the obvious way\nto the right-hand side of Eq. (45). Note that since we\nare now interested in linearized equations close to equi-\nlibrium,\u001ain Eq. (45) can be approximated by its (ho-\nmogeneous) equilibrium value.\nIntroducing relaxation through a phenomenological\ndamping constant (Drude resistivity)\n\r=me\n\u001a\u001c; (46)\nwhere\u001cis the collision time, expressing the \fctitious\nmagnetic \feld in terms of the spin texture, Eq. (45) be-\ncomes:\n_pi=\u0000q\n\u001a(m\u0002@im)\u0001(j\u0001r)m\u0000@i\u0016\u0000\rji: (47)\nAdding the phenomenological Gilbert damping34to\nthe magnetic Eq. (37) gives the Landau-Lifshitz-Gilbert\nequation:\n%s(_m+\u000bm\u0002_m) =h\u0002m; (48)\nwhere\u000bis the damping constant. Eqs. (47) and\n(48), along with the continuity equation, _ \u001a=\u0000r\u0001j,\nare the near-equilibrium thermodynamic equations for\n(\u001a;p;m) and their respective thermodynamic conjugates\n(\u0016;j;h) = (\u000e\u001aH;\u000epH;\u000emH). This system of equations of\nmotion may be written formally as\n@t0\n@\u001a\np\nm1\nA=b\u0000[m(r)]0\n@\u0016\nj\nh1\nA: (49)\nThe matrix ^\u0000 depends on the equilibrium spin texture\nm(r). By the Onsager reciprocity principle, \u0000 ij[m] =\nsisj\u0000ji[\u0000m], wheresi=\u0006if theith variable is even\n(odd) under time reversal.7\nIn the quasistationary description of a nonequilibrium\nthermodynamic system, the entropy S[\u001a;p;m] is for-\nmally regarded as a functional of the instantaneous ther-\nmodynamic variables, and the probability of a given con-\n\fguration is proportional to eS=kB. If the heat conduc-\ntance is high and the temperature Tis uniform and con-\nstant, the instantaneous rate of dissipation P=T_Sis\ngiven by the rate of change in the free energy, P=_H=R\nd3rP:\nP=\u0000\u0016_\u001a\u0000h\u0001_m\u0000j\u0001_p=\u000b%s_m2+\rj2; (50)\nwhere we used Eq. (47) and expressed the e\u000bective \feld\nhas a function of _mby taking m\u0002of Eq. (48):\nh=%sm\u0002_m\u0000\u000b%s_m: (51)\nNotice that the \fctitious magnetic \feld bdoes not con-\ntribute to dissipation because it does not do work.\nSo far, there is no dissipative coupling between the\ncurrent and the spin-texture dynamics, and the macro-\nscopic equations obey the global time-reversal symme-\ntry. However, we know that dissipative couplings ex-\nists due to the misalignment of the electron's spin with\nthe collective spin texture and spin-texture resistivity.3,22\nFollowing Ref. 11, we add these well-known e\u000bects phe-\nnomenologically by making an expansion in the equations\nof motion to linear order in the nonequilibrium quanti-\nties _mandj. To limit the number of terms one can write\ndown, we will only add terms that are spin-rotationally\ninvariant and isotropic in real space (which disregards,\nin particular, such e\u000bects as the angular magnetoresis-\ntance and the anomalous Hall e\u000bect). To second order in\nthe spatial gradients of m, there are only three dissipa-\ntive phenomenological terms with couplings \u0011,\u00110, and\f\nconsistent with the above requirements, which could be\nadded to the right-hand side of Eq. (47).35The momen-\ntum equation becomes:\n_pi=\u0000q\n\u001a(m\u0002@im)\u0001(j\u0001r)m\u0000@i\u0016\u0000\rji\n\u0000\u0011(@km)2ji\u0000\u00110@im\u0001(j\u0001r)m\u0000q\f_m\u0001@im:(52)\nIt is known that the \\ \fterm\" comes from a misalignment\nof the electron spin with the collective spin texture, and\nthe associated dephasing. It is natural to expect thus\nthat the dimensionless parameter \f\u0018~=\u001cs\u0001, where\u001cs\nis a characteristic spin-dephasing time.3The \\\u0011terms\"\nevidently describe texture-dependent resistivity, which\nis anisotropic with respect to the gradients in the spin\ntexture along the local current density. Such term are\nalso naturally expected, in view of the well-known giant-\nmagnetoresistance e\u000bect,36in which noncollinear magne-\ntization results in electrical resistance. The microscopic\norigin of this term is due to spin-texture misalignment,\nwhich modi\fes electron scattering.\nThe total spin-texture-dependent resistivity can be putinto a tensor form:\n\rij[m] =\u000eij\u0002\n\r+\u0011(@km)2\u0003\n+\u00110@im\u0001@jm\n+q\n\u001am\u0001(@im\u0002@jm): (53)\nThe last term due to \fctitious magnetic \feld gives a Hall\nresistivity. Note that ^ \r[m] = ^\rT[\u0000m], consistent with\nthe Onsager theorem. We can \fnally write Eq. (47) as:\n_pi=\u0000\rij[m]jj\u0000@i\u0016\u0000q\f_m\u0001@im: (54)\nAs was shown in Ref. 11, since the Onsager relations\nrequire thatb\u0000[m] =b\u0000[\u0000m]Twithin the current/spin-\ntexture \felds sector, there must be a counterpart to the\n\fterm above in the magnetic equation, which is the well-\nknown dissipative \\ \fspin torque:\"\n%s(_m+\u000bm\u0002_m) =h\u0002m\u0000q\fm\u0002(j\u0001r)m:(55)\nThe total dissipation Pis now given by\nP=\u000b%s_m2+ 2q\f_m\u0001(j\u0001r)m+\u0002\n\r+\u0011(@km)2\u0003\nj2\n+\u00110[(j\u0001r)m]2\n=\u000b%s\u0014\n_m+q\f\n\u000b%s(j\u0001r)m\u00152\n+\u0002\n\r+\u0011(@km)2\u0003\nj2\n+\u0012\n\u00110\u0000(q\f)2\n\u000b%s\u0013\n[(j\u0001r)m]2: (56)\nThe second law of thermodynamics requires the total dis-\nsipation to be positive, which puts some constraints on\nthe allowed values of the phenomenological parameters.\nWe can easily notice, however, that the dissipation (56)\nis guaranteed to be positive-de\fnite if\n\u0011+\u00110\u0015(q\f)2\n\u000b%s; (57)\nwhich may serve as an estimate for the spin-texture re-\nsistivity due to spin dephasing. This is consistent with\nthe microscopic \fndings of Ref. 23.\nV. THERMAL NOISE\nAt \fnite temperature, thermal agitation causes \ruc-\ntuations of the current and spin texture, which are cor-\nrelated due to their coupling. A complete description\nrequires that we supplement the stochastic equations of\nmotion with the correlators for these \ructuations. It\nis convenient to regard these \ructuations as being due\nto the stochastic Langevin \\forces\" ( \u000e\u0016;\u000ej;\u000eh) on the\nright-hand side of Eq. (49). The complete set of \fnite-\ntemperature hydrodynamic equations thus becomes:\n_\u001a=\u0000r\u0001~j;\n_p+q\f_mirmi=\u0000^\r[m]~j\u0000r~\u0016;\n%s(1 +\u000bm\u0002)_m=~h\u0002m\u0000q\fm\u0002(~j\u0001r)m:(58)8\nwhere (~\u0016;~j;~h) = (\u0016+\u000e\u0016;j+\u000ej;h+\u000eh). The simplest\n(while possibly not most realistic) case corresponds to\na highly compressible \ruid, such that K!1 . In this\nlimit,\u0016=\u001a=K!0 and the last two equations com-\npletely decouple from the \frst, continuity equation. In\nthe remainder of this section, we will focus on this special\ncase. The correlations of the stochastic \felds are given\nby the symmetric part of the inverse matrix b\u0007 =\u0000b\u0000\u00001,37\nwhich is found by inverting Eq. (58) (reduced now to a\nsystem of two equations):\n~j=\u0000^\r\u00001(_p+q\f_mirmi);\n~h=%sm\u0002_m\u0000\u000b%s_m\u0000q\f(~j\u0001r)m: (59)\nWriting formally these equations as (after substituting ~j\nfrom the \frst into the second equation)\n\u0012~j\n~h\u0013\n=\u0000b\u0007[m(r)]\u0012\n_p\n_m\u0013\n; (60)\nwe immediately read out for the matrix elements\nb\u0007(r;r0) =b\u0007(r)\u000e(r\u0000r0):\n\u0007ji;ji0(r) =(^\r\u00001)ii0;\n\u0007ji;hi0(r) =q\f(^\r\u00001)ik@kmi0;\n\u0007hi0;ji(r) =\u0000q\f(^\r\u00001)ki@kmi0;\n\u0007hi;hi0(r) =\u000b%s\u000eii0+%s\u000fii0kmk\n\u0000(q\f)2(@kmi)(^\r\u00001)kk0(@k0mi0):(61)\nAccording to the \ructuation-dissipation theorem, we\nsymmetrize b\u0007 to obtain the classical Langevin\ncorrelators:37\nh\u000eji(r;t)\u000eji0(r0;t0)i=T=gii0;\nh\u000eji(r;t)\u000ehi0(r0;t0)i=T=q\fg0\nik@kmi0;\nh\u000ehi(r;t)\u000ehi0(r0;t0)i=T=\u000b%s\u000eii0\n\u0000(q\f)2gkk0(@kmi)(@k0mi0); (62)\nwhereT= 2kBT\u000e(r\u0000r0)\u000e(t\u0000t0) and\n^g= [^\r\u00001+ (^\r\u00001)T]=2;^g0= [^\r\u00001\u0000(^\r\u00001)T]=2 (63)\nare, respectively, the symmetric and antisymmetric parts\nof the conductivity matrix ^ \r\u00001. The short-ranged, \u000e-\nfunction character of the noise correlations in space stems\nfrom the assumption of high electronic compressibility.\nContrast this to the results of Ref. 11 for incompressible\nhydrodynamics. A presence of long-ranged Coulombic\ninteractions and plasma modes would also give rise to\nnonlocal correlations. These are absent in our treatment,\nwhich disregards ordinary electromagnetic phenomena.\nFocusing on the microwave frequencies !characteris-\ntic of ferromagnetic dynamics, it is most interesting to\nconsider the regime where !\u001c\u001c\u00001. This means that\nwe can employ the drift approximation for the \frst of\nEqs. (59):\n_pi=me_vi\u0000qei\u0019\u0000qei=q_m\u0001(m\u0002@im): (64)Substituting this _pin Eq. (59), we can easily \fnd a closed\nstochastic equation for the spin-texture \feld:\n%s(1 +\u000bm\u0002)_m+m\u0002\u001c$_m= (hm+\u000eh)\u0002m;(65)\nwhere we have de\fned the \\spin-torque tensor\"\n\u001c$=q2(^\r\u00001)kk0(m\u0002@km\u0000\f@km)\n\n(m\u0002@k0m+\f@k0m): (66)\nThe antisymmetric piece of this tensor modi\fes the e\u000bec-\ntive gyromagnetic ratio, while the more interesting sym-\nmetric piece determines the additional nonlocal Gilbert\ndamping:\n\u000b$=\u001c$+\u001c$T\n2%s=q2\n%sG$; (67)\nwhere\nG$=gkk0\u0002\n(m\u0002@km)\n(m\u0002@k0m)\u0000\f2@km\n@k0m\u0003\n+\fg0\nkk0[(m\u0002@km)\n@k0m\u0000@km\n(m\u0002@k0m)]:\n(68)\nIn obtaining Eq. (65) from Eqs. (59), we have separated\nthe reactive spin torque out of the e\u000bective \feld: h=\nhm\u0000qm\u0002(j\u0001r)m. (The remaining piece hmthus re\rects\nthe purely magnetic contribution to the e\u000bective \feld.)\nThe total stochastic magnetic \feld entering Eq. (65),\n\u000eh=\u000eh+qm\u0002(\u000ej\u0001r)m; (69)\ncaptures both the usual magnetic Brown noise38\u000eh\nand the Johnson noise spin-torque contribution39\u000ehJ=\nqm\u0002(\u000ej\u0001r)mthat arises due to the substitution j=~j\u0000\u000ej\nin the reactive spin torque q(j\u0001r)m. Using correla-\ntors (62), it is easy to show that the total e\u000bective \feld\n\ructuations \u000ehare consistent with the nonlocal e\u000bec-\ntive Gilbert damping tensor (68), in accordance with the\n\ructuation-dissipation theorem applied directly to the\npurely magnetic Eq. (65).\nTo the leading, quadratic order in spin texture, we can\nreplacegkk0!\u000ekk0=\randg0\nkk0!0 in Eq. (68). This ad-\nditional texture-dependent nonlocal damping (along with\nthe associated magnetic noise) is a second-order e\u000bect,\nphysically corresponding to the backaction of the magne-\ntization dynamics-driven current on the spin texture.11\nIt should be noted that in writing the modi\fed LLG\nequation (55), we did not systematically expand it to\ninclude the most general phenomenological terms up to\nthe second order in spin texture. We have only included\nextra spin-torque terms, which are required by the On-\nsager symmetry with Eq. (52). The second-order Gilbert\ndamping (68) was then obtained by solving Eqs. (52) and\n(55) simultaneously. (Cf. Refs. 11,40.) This means in\nparticular, that this procedure does not capture second-\norder Gilbert damping e\u000bects whose physical origin is\nunrelated to the longitudinal spin-transfer torque physics\nstudied here. One example of that is the transverse spin-\npumping induced damping discussed in Refs. 41.9\nVI. EXAMPLES\nA. Rigidly spinning texture\nTo illustrate the \u0011resistivity terms in the electron's\nequation of motion (52), we \frst consider 1D textures.\nTake, for example, the case of a 1D spin helix m(z)\nalong thezaxis, whose spatial gradient pro\fle is given by\n@zm=\u0014^ z\u0002m, where\u0014is the wave vector of the spatial\nrotation and m?^ z. See Fig. 1. It gives anisotropic re-\nsistivity in the xyplane,r(\u0011)\n?, and along the zdirection,\nr(\u0011)\nk:\nr(\u0011)\n?=\u0011(@zm)2=\u0011\u00142; r(\u0011)\nk= (\u0011+\u00110)\u00142: (70)\nFIG. 1: (Color online) The transverse magnetic helix, @zm=\n\u0014^ z\u0002m, with texture-dependent anisotropic resistivity (70).\nWe assume here translational invariance in the transverse ( xy)\ndirections. Spinning this helix about the vertical zaxis gen-\nerates the dissipative electromotive forces f(\f)\nz, which is spa-\ntially uniform and points everywhere along the zaxis. A\nmagnetic spiral, @zm=\u0014^'\u0002m=\u0014^\u0012, spinning around the z\naxis, on the other hand, produces a purely reactive electromo-\ntive forceez, as discussed in the text, which is oscillatatory\nin space along the zaxis.\nThe \fctitious electric \feld and dissipative \fforce re-\nquire magnetic dynamics. A general texture globally ro-\ntating clockwise in spin space in the xyplane according\nto_m=\u0000!^ z\u0002m(which may be induced by applying a\nmagnetic \feld along the zdirection) generates an electric\n\feld\nei= (m\u0002_m)\u0001@im=\u0000!(m\u0002^ z\u0002m)\u0001@im\n=\u0000!@imz=\u0000!@icos\u0012 (71)and a\fforce\nf(\f)\ni=\u0000\f_m\u0001@im=\f!^ z\u0001(m\u0002@im)\n=\f!sin2\u0012@i'; (72)\nwhere (\u0012;') denote the position-dependent spherical an-\ngles parametrizing the spin texture. The reactive force\n(71) has a simple interpretation of the gradient of the\nBerry-phase15accumulation rate [which is locally deter-\nmined by the solid angle subtended by m(t)]. In the\ncase of the transverse helix discussed above, \u0012=\u0019=2,\n'=\u0014z\u0000!t, so thatez= 0 whilef(\f)\nz=\u0000\f!\u0014 is \fnite.\nAs an example of a dynamical texture that does not\ngenerate f(\f)while producing a \fnite e, consider a spin\nspiral along the zaxis, described by @zm=\u0014^'\u0002m=\u0014^\u0012,\nand rotating in time in the manner described above. It is\nclear geometrically that the change in the spin texture in\ntime is in a direction orthogonal to its gradients in space.\nSpeci\fcally, \u0012=\u0014z,'=\u0000!t, so thatf(\f)\nz= 0 while the\nelectric \feld is oscillatory, ez=!\u0014sin\u0012.\nB. Rotating spin textures\nWe show here that a vortex rotating about its core in\norbital space generates a current circulating around its\ncore, as well as a current going radially with respect to\nthe core. Consider a spin texture with a time depen-\ndence corresponding to the real-space rotation clockwise\nin thexyplane around the origin, such that m(r;t) =\nm(r(t);0) with _r=!^ z\u0002r=!r^\u001e, where we use polar co-\nordinates (r;\u001e) on the plane normal to the zaxis in real\nspace [to be distinguished from the spherical coordinates\n(\u0012;') that parametrize min spin space], we have\n_m= (_r\u0001r)m=!@\u001em: (73)\nForm(r;\u001e) in polar coordinates, the components of the\nelectric \feld are,\ner=!m\u0001(@\u001em\u0002@rm); e\u001e= 0; (74)\nwhile the components of the \fforce are\nf(\f)\nr=\u0000\f!(@rm)\u0001(@\u001em); f(\f)\n\u001e=\u0000\f!(@\u001em)2\nr:(75)\nIn order to \fnd the \fctitious electromagnetic \felds, we\nneed to calculate the following tensors (which depend on\nthe instantaneous spin texture):\nbij\u0011m\u0001(@im\u0002@jm) = sin\u0012(@i\u0012@j'\u0000@j\u0012@i');\ndij\u0011@im\u0001@jm=@i\u0012@j\u0012+ sin2\u0012@i'@j': (76)\nAs an example, consider a vortex centered at the ori-\ngin in thexyplane with winding number 1 and positive\npolarity, as shown in Fig. 2. Its angular coordinates are\ngiven by\n'= (\u001e+!t) +\u0019\n2; \u0012=\u0012(r); (77)10\nwhere\u001e= arg( r) and\u0012is a rotationally invariant func-\ntion such that \u0012!0 asr!0 and\u0012!\u0019=2 asr!1 .\nEvaluating the tensors in equation (76) for this vortex in\npolar coordinates gives drr= (@r\u0012)2,d\u001e\u001e= (sin\u0012=r)2,\ndr\u001e= 0, andbr\u001e=\u0000(@rcos\u0012)=r. The radial electric\n\feld is then given by\ner=\u0000!rbr\u001e=!@rcos\u0012: (78)\nThe\fforce is in the azimuthal direction:\nf(\f)\nr= 0; f(\f)\n\u001e=\u0000\f!rd\u001e\u001e=\u0000\f!sin2\u0012\nr: (79)\nWe can interpret this force as the spin texture \\dragging\"\nthe current along its direction of motion. Notice that the\nforces in Eqs. (78) and (79) are the negative of those in\nEqs. (71) and (72), as they should be for the present case,\nsince the combination of orbital and spin rotations of our\nvortex around its core leaves it invariant, producing no\nforces.\nFIG. 2: Positive-polarity magnetic vortex con\fguration pro-\njected on the xyplane. mhas a positive (out-of-plane) z\ncomponent near the vortex core. Rotating this vortex about\nthe origin in real space generates the current in the xyplane\nshown in Fig. 3.\nThe total resistivity tensor (53) is (in the cylindrical\ncoordinates)\n^\r=\r+\u0011(drr+d\u001e\u001e) +\u00110^d+q\n\u001a^b=\u0012\n\rr\r?\n\u0000\r?\r\u001e\u0013\n;(80)\nwhere\n\rr=\r+ (\u0011+\u00110)(@r\u0012)2+\u0011\u0012sin\u0012\nr\u00132\n;\n\r\u001e=\r+\u0011(@r\u0012)2+ (\u0011+\u00110)\u0012sin\u0012\nr\u00132\n;\n\r?=\u0000q\n\u001a@rcos\u0012\nr: (81)Here, the two diagonal components, \rrand\r\u001e, describe\nthe (dissipative) anisotropic resistivity, while the o\u000b-\ndiagonal component, \r?, captures what is called the\ntopological Hall e\u000bect.19\nIn the drift approximation, Eq. (64), the current-\ndensity \feld j=jr^ r+j\u0012^\u0012is given by\nj= ^\r\u00001q(e+f(\f));\u0012\njr\nj\u001e\u0013\n=q!^\r\u00001\u0012@rcos\u0012\n\u0000\fsin2\u0012=r\u0013\n=\u0000q!sin\u0012\n\rr\r\u001e+\r2\n?\u0012\n\r\u001e\u0000\r?\n\r?\rr\u0013\u0012\n@r\u0012\n\fsin\u0012=r\u0013\n:(82)\nMore explicitly, we may consider a pro\fle \u0012=\u0019(1\u0000\ne\u0000r=a)=2, whereais the radius of the vortex core. The\ncorresponding current (82) is sketched in Fig. 3.\nFIG. 3: We plot here the current in Eq. (82) (all parameters\nset to 1). Near the core, the current spirals inward and charges\nbuild up at the center (which is allowed for our compressible\n\ruid).\nWe note that the \fctitious magnetic \feld \u000fijkbjk=2\npoints everywhere in the zdirection, its total \rux\nthrough the xyplane being given by\nF=Z\nd\u001edr (rbr\u001e) =\u0000Z\nd\u001edr (@\u001e'@rcos\u0012) = 2\u0019:\n(83)\nNote that the integrand is just the Jacobian of the map\nfrom the plane to the sphere de\fned by the spin-texture\n\feld:\n(\u0012(r);'(r)) :R2!S2: (84)\nThis re\rects the fact that the \fctitious magnetic \rux is\ngenerally a topological invariant, corresponding to the \u00192\nhomotopy group of the mapping (84).6,42\nC. Anisotropic resistivity of a 3D spiral\nConsider the texture described by @im=\u0014i^ z\u0002m,\nwhere the spatial rotation stays in the xyplane, but the11\nwave vector\u0014can be in any direction. The spin texture\nforms a transverse helix in the zdirection and a planar\nspiral in the xandydirections. Fig. 4 shows such a\ncon\fguration for \u0014pointing along ( x+y+z)=p\n3. The\n\fctitious magnetic \feld bvanishes, but the anisotropic\nresistivity still depends nontrivially on the spin texture:\n\rij=\u0002\n\r+\u0011(@km)2\u0003\n\u000eij+\u00110@im\u0001@jm\n= (\r+\u0011\u00142)\u000eij+\u00110\u0014i\u0014j; (85)\nwhich, according to j= ^\r\u00001E, would give a transverse\ncurrent signal for an electric \feld applied along the Carte-\nsian axesx,y, orz.\nFIG. 4: (Color online) A set of spin spirals which is topo-\nlogically trivial because r\u0012= 0 (and equivalent to the spin\nhelix, Fig. 1, up to a global real-space rotation), hence the\n\fctitious magnetic \feld b, Eq. (76), is zero. There is, how-\never, an anisotropic texture-dependent resistivity with \fnite\no\u000b-diagonal components, Eq. (85).\nVII. SUMMARY\nWe have developed semi-phenomenologically the hy-\ndrodynamics of spin and charge currents interacting with\ncollective magnetization in metallic ferromagnets, gener-\nalizing the results of Ref. 11 to three dimensions and\ncompressible \rows. Our theory reproduces known re-\nsults such as the spin-motive force generated by mag-\nnetization dynamics and the dissipative spin torque, al-\nbeit from a di\u000berent viewpoint than previous microscopic\napproaches. Among the several new e\u000bects predicted,\nwe \fnd both an isotropic and an anisotropic texture-\ndependent resistivity, Eq. (53), whose contribution to theclassical (topological) Hall e\u000bect should be described on\npar with that of the \fctitious magnetic \feld. By calculat-\ning the dissipation power, we give a lower bound on the\nspin-texture resistivity as required by the second law of\nthermodynamics. We \fnd a more general form, includ-\ning a term of order \f, of the texture-dependent correction\nto nonlocal Gilbert damping, predicted in Ref. 11. See\nEq. (68).\nOur general theory is contained in the stochastic hy-\ndrodynamic equations, Eqs. (58), which we treated in\nthe highly compressible limit. The most general situ-\nation is no doubt at least as rich and complicated as\nthe classical magnetohydrodynamics. A natural exten-\nsion of this work is the inclusion of heat \rows and re-\nlated thermoelectric e\u000bects, which we plan to investigate\nin a future work. Although we mainly focused on the\nhalfmetallic limit in this paper, our theory is in principle\na two-component \ruid model and allows for the inclu-\nsion of a fully dynamical treatment of spin densities and\nassociated \rows.31Finally, our hydrodynamic equations\nbecome amenable to analytic treatments when applied to\nthe important problem of spin-current driven dynamics\nof magnetic solitons, topologically stable objects that can\nbe described by a small number of collective coordinates,\nwhich we will also investigate in future work.\nAcknowledgments\nWe are grateful to Gerrit E. W. Bauer, Arne Brataas,\nAlexey A. Kovalev, and Mathieu Taillefumier for stimu-\nlating discussions. This work was supported in part by\nthe Alfred P. Sloan Foundation and the NSF under Grant\nNo. DMR-0840965.\nAPPENDIX A: MANY-BODY ACTION\nWe can formally start with a many-body action, with\nStoner instability built in due to short-range repulsion\nbetween electrons:25\nS[\u0016 \u001b(r;t); \u001b(r;t)] =Z\nCdtZ\nd3r\n\u0014\n^ +\u0012\ni~@t+~2\n2mer2\u0013\n^ \u0000U\u0016 \"\u0016 # # \"\u0015\n;(A1)\nwhere time truns along the Keldysh contour from \u00001\nto1and back. \u0016 \u001band \u001bare mutually independent\nGrassmann variables parametrizing fermionic coherent\nstates and ^ += (\u0016 \";\u0016 #) and ^ = ( \"; #)T. The four-\nfermion interaction contribution to the action can be de-\ncoupled via Hubbard-Stratonovich transformation, after12\nintroducing auxiliary bosonic \felds \u001eand\u0001:\neiSU=~= exp\u0012\n\u0000i\n~Z\nCdtZ\nd3rU\u0016 \"\u0016 # # \"\u0013\n=Z\nD[\u001e(r;t);\u0001(r;t)] exp\u0012i\n~Z\nCdtZ\nd3r\n\u0014\u001e2\n4U\u0000\u00012\n4U\u0000\u001e\n2^ +^ +\u0001\n2^ +^\u001b^ \u0015\u0013\n:(A2)\nIn obtaining this result, we decomposed the interaction\ninto charge- and spin-density pieces:\n\u0016 \"\u0016 # # \"=1\n4(^ +^ )2\u00001\n4(^ +m\u0001^\u001b^ )2; (A3)\nwhere mis an arbitrary unit vector. It is easy to\nshow thath\u001e(r;t)i=Uh^ +(r;t)^ (r;t)iandh\u0001(r;t)i=\nUh^ +(r;t)^\u001b^ (r;t)i, when properly averaging over the\ncoupled quasiparticle and bosonic \felds.\nThe next step in developing mean-\feld theory is to\ntreat the Hartree potential \u001e(r;t) and Stoner exchange\n\u0001(r;t)\u0011\u0001(r;t)m(r;t) \felds in the saddle-point approx-\nimation. Namely, the e\u000bective bosonic action\nSe\u000b[\u001e(r;t);\u0001(r;t)] =\u0000i~lnZ\nD[^ +;^ ]ei\n~S(^ +;^ ;\u001e;\u0001)\n(A4)\nis minimized, \u000eSe\u000b= 0, in order to \fnd the equations\nof motion for the \felds \u001eand\u0001. In the limit of suf-\n\fciently low electron compressibility and spin suscepti-\nbility, the charge- and spin-density \ructuations are sup-\npressed, de\fning mean-\feld parameters \u0016\u001eand \u0016\u0001. Since\na constant \u0016\u001eonly shifts the overall electrochemical po-\ntential, it is physically inconsequential. Our theory is de-\nsigned to focus on the remaining soft (Goldstone) modes\nassociated with the spin-density director m(r;t), while\n\u001e(r;t) and \u0001( r;t) are in general allowed to \ructuate\nclose to their mean-\feld values \u0016\u001eand \u0016\u0001, respectively.\nThe saddle-point equation of motion for the collective\nspin direction m(r;t) follows from \u000emSe\u000b[m] = 0, after\nintegrating out electronic degrees of freedom. Because\nof the noncommutative matrix structure of the action\n(A2), it is still a nontrivial problem. The problem sim-\npli\fes considerably in the limit of large exchange split-\nting \u0001, where we can project spins on the local magnetic\ndirection m. This lays the ground to the formulation dis-\ncussed in Sec. II, where the collective spin-density \feld\nparametrized by the director m(r;t) interacts with the\nspin-up/down free-electron \feld. The resulting equations\nof motion constitute the self-consistent dynamic Stoner\ntheory of itinerant ferromagnetism.\nIn the remainder of this appendix, we explicitly show\nthat the semiclassical formalism developed in Secs. II-\nIII B is equivalent to a proper \feld-theoretical treatment.\nThe equation of motion for the spin texture follows from\nextremizing the e\u000bective action with respect to variations\ninm. Because of the constraint on the magnitude of m,\nits variation can be expressed as \u000em=\u000e\u0012\u0002m, with\u000e\u0012being an arbitrary in\fnitesimal vector, so that the\nequation of motion is given by m\u0002\u000emSe\u000b= 0:\n0 =m\u0002\u000emSe\u000b\n=1\nZZ\nD[^ +;^ ] (m\u0002\u000emS)ei\n~S[^ +;^ ;\u001e;\u0001]\n=X\n\u001b\u0016(m\u0002\u000ema\u001b\u0016)\u001c@S\n@a\u001b\u0016\u001d\n\u0000m\u0002\u000emF; (A5)\nwhereZ=R\nD[^ +;^ ]ei\n~S[^ +;^ ;\u001e;\u0001]and we have used\nthe path-integral representation of the vacuum expecta-\ntion value. a\u001b\u0016are the spin-dependent gauge potentials\n(4) andFthe spin exchange energy, appearing after we\nproject spin dynamics on the collective \feld \u0001. Equa-\ntion (A5) may be expressed in terms of the hydrody-\nnamic variables of the electrons. De\fning spin-dependent\ncharge and current densities, j\u0016\n\u001b= (\u001a\u001b;j\u001b), by\n\u001a\u001b=\u001c@S\n@a\u001b\u001d\n=h\u0016 \u001b \u001bi;\nj\u001b=\u001c@S\n@a\u001b\u001d\n=1\nmeRe\n\u0016 \u001b(\u0000i~r\u0000a\u001b) \u001b\u000b\n=\u001a\u001bv\u001b;\n(A6)\nEq. (A5) reduces to the Landau-Lifshitz Eq. (18). Min-\nimizing action (A4) with respect to the \u001eand \u0001 \felds\ngives the anticipated self-consistency relations:\n\u001e(r;t) =Uh^ +(r;t)^ (r;t)i=U(\u001a++\u001a\u0000);\n\u0001(r;t) =Uh^ +(r;t)^\u001bz^ (r;t)i=U(\u001a+\u0000\u001a\u0000):(A7)\nAPPENDIX B: THE MONOPOLE GAUGE FIELD\nLet (\u0012;') be the spherical angles of m, the direction\nof the local spin density, and ^ \u001f\u001bbe the spin up/down\n(\u001b=\u0006) spinors given by, up to a phase,\n^\u001f+(\u0012;') =\u0012\ncos\u0012\n2\nei'sin\u0012\n2\u0013\n;\n^\u001f\u0000(\u0012;') = ^\u001f+(\u0019\u0000\u0012;'+\u0019) =\u0012\nsin\u0012\n2\n\u0000ei'cos\u0012\n2\u0013\n:(B1)\nThe spinors are related to the spin-rotation matrix ^U(m)\nby ^\u001f\u001b=^Uj\u001bi. The gauge \feld in mspace, which enters\nEq. (5), is thus given by\namon\n\u001b(\u0012;') =\u0000i~^\u001fy\n\u001b@m^\u001f\u001b=~\n2\u00121\u0000\u001bcos\u0012\nsin\u0012\u0013\n^';(B2)\nwhere we used the gradient on a unit sphere: @m=\n^\u0012@\u0012+^'@'=sin\u0012. The magnetic \feld corresponding to\nthis vector potential [extended to three dimensions by\na(m)!a(\u0012;')=m] is given on the unit sphere by\n@m\u0002amon\n\u001b=@m\u0002(a'^') =m\nsin\u0012@\u0012(sin\u0012a') =\u001b~\n2m:\n(B3)13\nIt follows from Eqs. (5) and (B2) that the spin-dependent\nreal-space gauge \felds are given by\na\u001b\u0016=\u0000~\n2@\u0016'(1\u0000\u001bcos\u0012): (B4)\nNotice that the \u001b=\u0006monopole \feld (B2), as well as the\nabove gauge \felds, are singular on the south/north pole(corresponding to the Dirac string). This is what allows a\nmagnetic \feld with \fnite divergence. Any other choice of\nthe monopole gauge \feld (B2) would correspond to a dif-\nferent choice of the spinors (B1), translating into a gauge\ntransformation of the \felds (B4). This is immediately\nseeing by noticing that amon\n\u001b(m)!amon\n\u001b(m) +@mf\u001b(m)\ncorresponds to a\u001b\u0016(r;t)!a\u001b\u0016(r;t) +@\u0016f\u001b(m(r;t)).\n1L. Berger, J. Appl. Phys. 55, 1954 (1984); M. Kl aui,\nC. A. F. Vaz, J. A. C. Bland, W. Wernsdorfer, G. Faini,\nE. Cambril, and L. J. Heyderman, Appl. Phys. Lett.\n83, 105 (2003); E. Saitoh, H. Miyajima, T. Yamaoka,\nand G. Tatara, Nature 432, 203 (2004); Z. Li and\nS. Zhang, Phys. Rev. Lett. 92, 207203 (2004); S. E. Barnes\nand S. Maekawa, ibid. 95, 107204 (2005); L. Thomas,\nM. Hayashi, X. Jiang, R. Moriya, C. Rettner, and\nS. S. P. Parkin, Nature 443, 197 (2006); M. Yamanouchi,\nD. Chiba, F. Matsukura, T. Dietl, and H. Ohno, Phys.\nRev. Lett. 96, 096601 (2006); M. Yamanouchi, J. Ieda,\nF. Matsukura, S. E. Barnes, S. Maekawa, and H. 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Pitaevskii, Statistical Physics, Part\n2, vol. 9 of Course of Theoretical Physics (Pergamon, Ox-\nford, 1980), 3rd ed.\n30Generally, the phenomenological free energy of a ferromag-\nnet is given by F[m] =A(@im)2=2 +Uani(m)\u0000Mhext\u0001m,\nwhereAis the e\u000bective sti\u000bness, Uani(m) =Uani(\u0000m)\nis an anisotropic potential due to magnetostatic and crys-\ntalline \felds, Mis the equilibrium magnetization, and hext\nis an external magnetic \feld.\n31In the more general two-band model of spin-up and spin-\ndown electrons, we would have to develop a two-\ruid ef-\nfective theory, with spin-\rip scattering between the two\n\ruid components. The phenomenology simpli\fes, however,\nreducing formally to the halfmetallic case, in the long-\nwavelength low-frequency limit: !\u001csf\u001c1 andk\u0015sd\u001c1,14\nwhere!,kare the characteristic frequency, wave number\nof the magnetohydrodynamics and \u001csf,\u0015sd/p\u001csfare the\nspin-\rip time, spin-di\u000busion length. In this limit, it may be\npossible to describe the hydrodynamic state of the system\nby the spin-texture \feld, the charge-density distribution,\nand the charge-current \feld. If any out-of-equilibrium spin\nimbalance decays su\u000eciently fast, therefore, we only need\nto retain a one-\ruid description for the charge \rows. The\nkey phenomenological modi\fcation is then to introduce a\nmaterial-dependent dimensionless \\spin-polarization\" pa-\nrameterp, such that q!pqin the following equations\nof motion. Namely, the e\u000bective charge that couples the\nelectronic particle-number \rux densities jwith the spin-\ntexture gauge \feld is renormalized by p. While in the\nhalfmetallic limit p= 1 and in normal metals p= 0, we\nmay expect some intermediate value in realistic multiple-\nband ferromagnets with fast spin relaxation.\n32To make contact with Ref. 11, de\fne the canonical current\nasJ=H\nCp\u0001dl, for an arbitrary closed curve C. Its equa-\ntion of motion is given by @tJ=qH\nCdl\u0001(v\u0002b). If the\ncurveCcoincides with a quasi-1D wire, then vkdland we\nrecover the reactive equation of Ref. 11: @tJ= 0.\n33Similarly to qthat should generally be viewed as\na material-dependent phenomenological parameters, the\nelectron mass mefrom now on is also an e\u000bective parame-ter, which is not necessarily identical with the free-electron\nmass.\n34T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n35We are not including the ordinary hydrodynamic viscosity\nin our treatment.\n36M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van\nDau, F. Petro\u000b, P. Etienne, G. Creuzet, A. Friederich, and\nJ. Chazelas, Phys. Rev. Lett. 61, 2472 (1988); G. Binasch,\nP. Gr unberg, F. Saurenbach, and W. Zinn, Phys. Rev. B\n39, 4828 (1989).\n37L. D. Landau and E. M. Lifshitz, Statistical Physics, Part\n1, vol. 5 of Course of Theoretical Physics (Pergamon, Ox-\nford, 1980), 3rd ed.\n38W. F. Brown, Phys. Rev. 130, 1677 (1963).\n39J. Foros, A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer,\nPhys. Rev. B 78, 140402(R) (2008).\n40S. Zhang, and S. S.-L. Zhang, Phys. Rev. Lett. 102, 086601\n(2009).\n41E. M. Hankiewicz, G. Vignale, and Y. Tserkovnyak, Phys.\nRev. B 78, 020404(R) (2008); Y. Tserkovnyak, E. M. Han-\nkiewicz, and G. Vignale, ibid.79, 094415 (2009).\n42A. A. Belavin and A. M. Polyakov, JETP Lett. 22, 503\n(1975)."    },    {        "title": "1610.01072v2.Magnetomechanical_coupling_and_ferromagnetic_resonance_in_magnetic_nanoparticles.pdf",        "content": "Magnetomechanical coupling and ferromagnetic resonance in magnetic nanoparticles\nHedyeh Keshtgar,1Simon Streib,2Akashdeep Kamra,3Yaroslav M. Blanter,2and Gerrit E. W. Bauer2, 4\n1Institute for Advanced Studies in Basic Science, 45195 Zanjan, Iran\n2Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands\n3Fachbereich Physik, Universität Konstanz, D-78457 Konstanz, Germany\n4Institute for Materials Research and WPI-AIMR, Tohoku University, Sendai 980-8577, Japan\n(Dated: March 20, 2017)\nWe address the theory of the coupled lattice and magnetization dynamics of freely suspended\nsingle-domain nanoparticles. Magnetic anisotropy generates low-frequency satellite peaks in the\nmicrowave absorption spectrum and a blueshift of the ferromagnetic resonance (FMR) frequency.\nThe low-frequency resonances are very sharp with maxima exceeding that of the FMR, because\ntheir magnetic and mechanical precessions are locked, thereby suppressing the effective Gilbert\ndamping. Magnetic nanoparticles can operate as nearly ideal motors that convert electromagnetic\ninto mechanical energy. The Barnett damping term is essential for obtaining physically meaningful\nresults.\nPACS numbers: 75.10.Hk, 75.80.+q , 75.75.Jn , 76.50.+g\nI. INTRODUCTION\nMagnetic nanoparticles (nanomagnets) are of funda-\nmental interest in physics by forming a link between the\natomic and macroscopic world. Their practical impor-\ntance stems from the tunability of their magnetic prop-\nerties [1], which is employed in patterned media for high\ndensity magnetic data storage applications [2] as well as\nin biomedicine and biotechnology [3–6]. Superparamag-\nnetic particles are used for diagnostics, stirring of liq-\nuids, and magnetic tweezers [7]. The heat generated by\nthe magnetization dynamics under resonance conditions\nis employed for hyperthermia cancer treatment [8–10].\nMolecular based magnets can cross the border from the\nclassical into the quantum regime [11, 12]. The magnetic\nproperties of individual atomic clusters can be studied by\nmolecular beam techniques [13–15].\nEinstein, de Haas, and Barnett [16, 17] established the\nequivalence of magnetic and mechanical angular momen-\ntum of electrons by demonstrating the coupling between\nmagnetization and global rotations. Spin and lattice are\nalso coupled by magnetic anisotropy, induced either by\ndipolar forces or crystalline fields. A quite different in-\nteraction channel is the magnetoelastic coupling between\nlattice waves (phonons) and spin waves (magnons) with\nfinite wave vectors. This magnetoelastic coupling be-\ntween the magnetic order and the underlying crystalline\nlattice has been explored half a century ago by Kittel [18]\nand Comstock [19, 20]. The coupling between spin and\nlattice causes spin relaxation including Gilbert damping\nof the magnetization dynamics [21, 22].\n“Spin mechanics” of thin films and nanostructures en-\ncompasses many phenomena such as the actuation of the\nmagnetization dynamics by ultrasound [23–25], the dy-\nnamics of ferromagnetic cantilevers [26–28], spin current-\ninduced mechanical torques [22, 29], and rotating mag-\nnetic nanostructures [30]. The Barnett effect by rotation\nhas been observed experimentally by nuclear magnetic\nresonance [31]. The coupled dynamics of small magneticspheres has been studied theoretically by Usov and Li-\nubimov [32] and Rusconi and Romero-Isart [33] in clas-\nsical and quantum mechanical regimes, respectively. A\nprecessing single-domain ferromagnetic needle is a sen-\nsitive magnetometer [34], while a diamagnetically levi-\ntated nanomagnet can serve as a sensitive force and in-\nertial sensor [35]. A stabilization of the quantum spin of\nmolecular magnets by coupling to a cantilever has been\npredicted [36, 37] and observed recently [38].\nHere we formulate the dynamics of rigid and single-\ndomain magnetic nanoparticles with emphasis on the\neffects of magnetic anisotropy and shape. We derive\nthe equations of motion of the macrospin and macro-\nlattice vectors that are coupled by magnetic anisotropy\nand Gilbert damping. We obtain the normal modes and\nmicrowave absorption spectra in terms of the linear re-\nsponse to ac magnetic fields. We demonstrate remark-\nable changes in the normal modes of motion that can be\nexcited by microwaves. We predict microwave-activated\nnearly undamped mechanical precession. Anisotropic\nmagnetic nanoparticles are therefore suitable for stud-\nies of non-linearities, chaos, and macroscopic quantum\neffects.\nIn Sec. II we introduce the model of the nanomag-\nnet and give an expression for its energy. In Sec. III we\ndiscuss Hamilton’s equation of motion for the magneti-\nzation of a freely rotating particle, which is identical to\nthe Landau-Lifshitz equation. We then derive the cou-\npled equations of motion of magnetization and lattice in\nSec. IV. Our results for the easy-axis and easy-plane con-\nfigurations are presented in Secs. V and VI. We discuss\nand summarize our results in Secs. VII and VIII. In the\nAppendices A to D we present additional technical de-\ntails and derivations.arXiv:1610.01072v2  [cond-mat.mes-hall]  28 Apr 20172\nz\ny\nxnyb\nxbzb\nθ(a)\nn\nm\nm\nn(b)\n(c)\nFigure 1. (a) Laboratory frame ( x,y,z) and (moving) body\nframe (xb,yb,zb) of a nanomagnet with principal axis nalong\nthezb-axis. The directions of nand magnetization mare\nshown for (b) oblate and (c) prolate spheroids with dipolar\nmagnetic anisotropy.\nII. MACROSPIN MODEL\nWe consider a small isolated nanomagnet that justi-\nfies the macrospin and macrolattice approximations, in\nwhich all internal motion is adiabatically decoupled from\nthe macroscopic degrees of freedom, rendering the mag-\nnetoelastic coupling irrelevant.\nWe focus on non-spherical nanoparticles with mass\ndensity\u001a(r)and tensor of inertia\nI=Z\nd3r\u001a(r)\u0002\n(r\u0001r)^1\u0000r\nr\u0003\n;(2.1)\nwhere ^1is the 3x3 unit matrix. The mechanical proper-\nties of an arbitrarily shaped rigid particle is identical to\nthat of an ellipsoid with a surface that in a coordinate\nsystem defined along the symmetry axes (in which Iis\ndiagonal) reads\n\u0010x\nc\u00112\n+\u0010y\nb\u00112\n+\u0010z\na\u00112\n= 1; (2.2)\nwherea;b;care the shape parameters (principal radii).\nThe volume is V= 4\u0019abc= 3, total mass Q=\u001aV,\nand principal moments of inertia I1=Q\u0000\na2+b2\u0001\n=5;\nI2=Q\u0000\na2+c2\u0001\n=5;I3=Q\u0000\nb2+c2\u0001\n=5. We focus in the\nfollowing on prolate (a > b =c)and oblate (a < b =c)\nspheroids, because this allows analytic solutions of the\ndynamics close to the minimum energy state.\nWe assume that the particle is smaller than the crit-\nical sizedcr\u001836pAKA=(\u00160M2\ns)for magnetic domain\nformation [39], where Ais the exchange constant, KA\nthe anisotropy constant, Msthe saturation magnetiza-\ntion, and\u00160= 4\u0019\u000210\u00007N A\u00002the vacuum permeability.\nFor strong ferromagnets these parameters are typically in\nthe rangeA2[5;30] pJ m\u00001,KA2[10;20000] kJ m\u00003,\nMs2[0:4;1:7] MA m\u00001, leading to dcr2[1;500] nm\n[39]. For a spherical particle of radius Rwith sound\nvelocityv, the lowest phonon mode frequency is approx-imately [40]\n!ph\n2\u0019\u0019v\n4R= 0:25\u0012v=(103m\ns)\nR=nm\u0013\nTHz;(2.3)\nwhile the lowest magnon mode (for bulk dispersion rela-\ntion ~!mag=Dk2)\n!mag\n2\u0019\u0019\u0019D\n8~R2= 0:6\u0012D=(meV nm2)\nR2=nm2\u0013\nTHz;(2.4)\nwhere the spin wave stiffness D= 2g\u0016BA=Msis typically\nof the order meV nm2[39], e.g.,D= 2:81 meV nm2for\niron [41]. We may disregard spin and lattice waves and\nthe effects of their thermal fluctuations when the first\nexcited modes are at sufficiently higher frequencies than\nthat of the total motion (the latter is typically in the\nGHz range) and therefore adiabatically decoupled [33,\n40], i.e. the macrospin and macrolattice model is valid.\nThermal fluctuations of the magnetization with respect\nto the lattice do not play an important role below the\nblocking temperature, TB\u0018KAV=(25kB)[42], where\nkBis the Boltzmann constant. For kBT\u001cVMs\u00160H0,\nthermal fluctuations of the magnetization with respect to\nthe static external magnetic field H0are suppressed.\nUnder the conditions stipulated above the classical dy-\nnamics (disregarding translations of the center of mass)\nis described in terms of the magnetization vector M=\nMsm(withjmj= 1) and the three Euler angles ( \u0012;\u001e; )\nof the crystal orientation direction in terms of the axis\nn(\u0012;\u001e)and a rotation angle  around it (see Appendix A\nfor details). The total energy can be split up into several\ncontributions,\nE=ET+EZ+ED+EK: (2.5)\nET=1\n2\nTI\nis the kinetic energy of the rotational mo-\ntion of the nanomagnet in terms of the angular frequency\nvector \n.EZ=\u0000\u00160VM\u0001Hextis the Zeeman energy\nin a magnetic field Hext.ED=1\n2\u00160VMTDMis the\nmagnetostatic self-energy with particle shape-dependent\ndemagnetization tensor D.EK=K1V(m\u0002n)2is the\n(uniaxial) magnetocrystalline anisotropy energy, assum-\ning that the easy axis is along n, andK1is the material-\ndependent anisotropy constant.\nWe consider an inertial lab frame with origin at the\ncenter of mass and a moving frame with axes fixed in the\nbody. The lab frame is spanned by basis vectors ex,ey,\nez, and the body frame by basis vectors exb,eyb,ezb(see\nFig. 1). The body axes are taken to be the principal axes\nthat diagonalize the tensor of inertia. For spheroids with\nb=cthe inertia and demagnetizing tensors in the body\nframe have the form\nIb=0\n@I?0 0\n0I?0\n0 0I31\nA;Db=0\n@D?0 0\n0D?0\n0 0D31\nA;(2.6)\nwithI?=Q\u0000\na2+b2\u0001\n=5andI3= 2Qb2=5; the elements\nD?andD3for magnetic spheroids are given in [43]. The3\nparticle shape enters the equations of motion via I?,I3,\nand the difference D3\u0000D?, the latter reduces to \u00001=2\nfor a thin needle and 1for a thin disk. When\nE?\u0000Ek=KAV=K1V\u00001\n2\u00160VM2\ns(D3\u0000D?)(2.7)\nis larger than zero, the configuration mknis sta-\nble (“easy axis”); otherwise m?n(“easy plane”).\nThe anisotropy constant KAincludes both magnetocrys-\ntalline and shape anisotropy.\nIII. LANDAU-LIFSHITZ EQUATION\nFor reference we rederive here the classical equation of\nmotion of the magnetization. The magnetization of the\nparticle at rest is related to the angular momentum S=\n\u0000VMsm=\r, where\r= 1:76\u00021011s\u00001T\u00001is (minus) the\ngyromagnetic ratio of the electron. The Poisson bracket\nrelations for angular momentum are\nfS\u000b;S\fg=\u000f\u000b\f\rS\r: (3.1)\nHamilton’s equation of motion reads\nd\ndtS=fS;Hg; (3.2)\nwhereH\u0011Eis the Hamiltonian. We consider a general\nmodel Hamiltonian of a single macrospin coupled to the\nmacrolattice,\nH=X\ni;j;k2N0aijk(n;L)Si\nxSj\nySk\nz; (3.3)\nwhere the coefficients aijk(n;L)may depend on the ori-\nentation nof the lattice and its mechanical angular\nmomentum L=I\n. Since lattice and magnetization\nare different degrees of freedom, the Poisson brackets\nfn;Sg=fL;Sg= 0and thereforefaijk(n;L);Sg= 0.\nWe derive in Appendix B\nfS;Hg=X\ni;j;k2N0aijk(n;L)0\n@iSi\u00001\nxSj\nySk\nz\njSi\nxSj\u00001\nySk\nz\nkSi\nxSj\nySk\u00001\nz1\nA\u0002S;(3.4)\nwhich is the Landau-Lifshitz equation [44],\nd\ndtS=rSHjn;L=const:\u0002S: (3.5)\nIn accordance with Eq. (3.4), the gradient in Eq. (3.5)\nhas to be evaluated for constant nandL.\nThe rotational kinetic energy ET=1\n2\nTI\ndoes\nnot contribute to this equation of motion directly since\nfS;ETg= 0. However, ETis crucial when considering\nthe energy of the nanomagnet under the constraint of\nconserved total angular momentum J=L+S. Minimiz-\ning the energy of the nanomagnet under the constraint\nof constant Jis equivalent to\n~He\u000b=\u00001\n\u00160VMsrmE\f\f\f\f\nJ=const:= 0;(3.6)where the rotational kinetic energy ETcontributes the\nBarnett field\nHB=\u00001\n\u00160VMsrmET\f\f\f\f\nJ=const:=\u0000\n\r\u00160;(3.7)\nwhich gives rise to the Barnett effect (magnetization by\nrotation)[17]. AlthoughtheBarnettfieldappearsherein\nthe effective field ~He\u000bwhen minimizing the energy, it is\nnot part of the effective field He\u000bof the Landau-Lifshitz\nequation,\nHe\u000b=\u00001\n\u00160VMsrmE\f\f\f\f\nn;L=const:;(3.8)\nwhere Lis kept constant instead of J. In the Landau-\nLifshitz-Gilbert equation in the laboratory frame the\nBarnett effect operates by modifying the Gilbert damp-\ning torque as shown below.\nIV. EQUATIONS OF MOTION\nWe now derive the coupled equations of motion of the\nmagnetization mand the Euler angles ( \u001e;\u0012; ). The\nmagnetization dynamics is described by the Landau-\nLifshitz-Gilbert equation [21, 44]\n_m=\u0000\r\u00160m\u0002He\u000b+\u001c(\u000b)\nm; (4.1)\nwhere the effective magnetic field Eq. (3.8) follows from\nthe energy Eq. (2.5),\nHe\u000b=Hext+HD+HK; (4.2)\nand\u001c(\u000b)\nmis the (Gilbert) damping torque. The external\nmagnetic field Hextis the only source of angular mo-\nmentum; all other torques acting on the total angular\nmomentum J=L\u0000VMsm=\rcancel. From\n_J=\u00160VMsm\u0002Hext; (4.3)\nwe obtain the mechanical torque as time-derivative of the\nmechanical angular momentum, which leads to Newton’s\nLaw\n_L=VMs\n\r_m+\u00160VMsm\u0002Hext:(4.4)\nThe dissipation parameterized by the Gilbert constant\n[21] damps the relative motion of magnetization and lat-\ntice. In the body frame of the lattice [30]\n\u001c(\u000b)\nm;b=\u000bmb\u0002_mb; (4.5)\nwherethesubscript bindicatesvectorsinthebodyframe.\nTransformed into the lab frame (see Appendix A)\n\u001c(\u000b)\nm=\u000b[m\u0002_m+m\u0002(m\u0002\n)]:(4.6)\nThis torque is an angular momentum current that flows\nfrom the magnet into lattice [22]. Angular momentum4\n2200 2300 2400 2500\nω/(2π) [MHz]01234567−ωImχxx[1013s−1]\nQf= 3900\nQf= 2900\n40 45 50 55\nω/(2π) [GHz]0.00.51.01.52.0−ωImχxx[1013s−1]\nQf= 50\nFigure 2. Low- and high-frequency resonances in the FMR\nspectrum of an Fe nanosphere of 2 nmdiameter in a static\nmagnetic field of 0:65 Twith Gilbert damping constant \u000b=\n0:01; quality factor Qf=!=(2\u0011).\nis conserved, but the generated heat is assumed to ulti-\nmately be radiated away. In vacuum there is no direct\ndissipation of the rigid mechanical dynamics.\nThe Barnett field \u00160HB=\u0000\n=\renters in the lab\nframe only in the damping term \u001c(\u000b)\nm. To leading order\nin\u000b\n_m\u0019\u0000\r\u00160m\u0002He\u000b\u0000\u000b\r\u0016 0m\u0002[m\u0002(He\u000b+HB)]+O(\u000b2):\n(4.7)\nThe contribution of HBin the damping term causes the\nBarnett effect [17]. We find that this Barnett damping is\nvery significant for the coupled dynamics even though no\nfast lattice rotation is enforced: without Barnett damp-\ning the FMR absorption of the low-frequency modes de-\nscribed below would become negative.\nV. EASY-AXIS CONFIGURATION\nWe first consider an easy-axis configuration ( mknk\nez) in the presence of an external magnetic field with\na large dc component H0along ezand a small trans-\nverse ac component, Hext=\u0000hx(t); hy(t); H 0\u0001T, with\nhx(t)/hy(t)/ei!t:Linearizing the equations of mo-\ntion in terms of small transverse amplitudes, we can solve\n(4.1) and (4.4) analytically to obtain the linear response\ntoh(see Appendix C for the derivation), i.e. the trans-\nverse magnetic susceptibility. Since we find _\nz= 0, we\ndisregard an initial net rotation by setting \nz= 0. Forsmall damping \u000b\u001c1, the normal modes are given by\nthe positive solutions of the equations\n!3\u0007!2!0\u0000!!c!A\u0006!c!A!H= 0;(5.1)\nwhere!H=\r\u00160H0,!A= 2\rKA=Ms,!0=!H+!A, and\n!c=MsV=(\rI?)is the natural mechanical frequency\ngoverned by the spin angular momentum. Note that the\nequivalent negative solutions of Eq. (5.1) have the same\nabsolute values as the positive solutions. We find that\nthe FMR mode !0is blueshifted to !k=!0+\u000e!kwith\n\u000e!k\u0019!2\nA!c\n!2\n0>0; (5.2)\nwhich is significant for small nanomagnets with large sat-\nuration magnetization and low mass density. It is a coun-\nterclockwise precession of mwithnnearly at rest.\nTwo additional low-frequency modes emerge. For !\u001c\n!0;!Awe may disregard the cubic terms in Eq. (5.1) and\nfind\n!l1;2\u0019s\u0012!c!A\n2!0\u00132\n+!H!c!A\n!0\u0006!c!A\n2!0:(5.3)\nAt low frequencies, the magnetization can follow the lat-\ntice nearly adiabatically, so these modes correspond to\nclockwise and counterclockwise precessions of nearly par-\nallel vectors mandn, but with a phase lag that gener-\nates the splitting. The frequency of the clockwise mode\n!l1> !l2(see Fig. 3). Since magnetization and mass\nprecess in unison, the effective Gilbert damping is ex-\npected to be strongly suppressed as observable in FMR\nabsorption spectra as shown below.\nThe absorbed FMR power is (see Appendix D)\nP=\u0000\u00160V\n2!Im\u0000\nh\u0003T\n?\u001fh?\u0001\n; (5.4)\nwhere h?is the ac field normal to the static magnetic\nfieldH0ezand\n\u001f\u000b\f=M\u000b\nh\f\f\f\f\f\nh?=0(5.5)\nis the transverse magnetic susceptibility tensor ( \u000b;\f =\nx;y). The diagonal ( \u001fxx=\u001fyy) and the off-diagonal\ncomponents( \u001fxy=\u0000\u001fyx)bothcontributetotheabsorp-\ntion spectrum near the resonance frequencies, jIm\u001fxxj\u0019\njRe\u001fxyj. For\u000b\u001c1, we find that the sum rule\nZ1\n0d!(\u0000!Im\u001fxx(!))\u0019\u0019\n2!0!M;(5.6)\nwhere!M=\r\u00160Ms, does not depend on !c, meaning\nthat the coupling does not generate oscillator strengths,\nonly redistributes it. Close to a resonance\n\u0000!Im\u001fxx(!)\u0018F\u00112\n(!\u0000!i)2+\u00112;(5.7)5\n10−310−210−1100101\nωH/ωA0.00.51.01.52.02.5angular frequency [1010s−1]\nωl1: clockwise mode\nωl2: countercl. mode\nFigure 3. Low-frequency magnetomechanical modes !l1and\n!l2of an Fe nanosphere of 2 nmdiameter.\nwith integral \u0019\u0011F. For the low-frequency modes the\nmaximumF\u00181\n2!M!2\nA=(\u000b!2\nH)with broadening \u0011\u0018\n1\n2\u000b!c!2\nH=(!A+!H)2; for the FMR mode F\u00181\n2!M=\u000b\nwith\u0011\u0018\u000b!0.\nLetusconsideranironspherewith 2 nmdiameter(a=\nb= 1 nm) under\u00160H0= 0:65 Tor!H=(2\u0019) = 18:2 GHz.\nIts magnetization !M=(2\u0019) = 60:33 GHz, crystalline\nanisotropy !A=(2\u0019) = 29:74 GHz [45], and the mag-\nnetomechanical coupling !c=(2\u0019) = 0:5(nm=a)2GHz.\nThe blocking temperature is TB\u001811(a=nm)3Kand\njEZj=(kBTB)\u001930, while the critical size for domain\nformationdcr\u001820 nm[46, 47]. We adopt a typi-\ncal Gilbert damping constant \u000b= 0:01. The calcu-\nlated FMR spectra close to the three resonances are\nshown in Fig. 2. Both low-frequency resonances are very\nsharp with a peak value up to 3.5 times larger than\nthat of the high-frequency resonance, although the in-\ntegrated intensity ratio is only 0.2 %. Long relaxation\ntimes of low-frequency modes that imply narrow reso-\nnances have been predicted for spherical nanomagnets\n[32]. The blueshift of the high-frequency resonance is\n\u000e!k=(2\u0019)\u00190:2(nm=a)2GHz. In Fig. 3 we plot the low-\nfrequency modes !l1and!l2as a function of !H=!A. For\n!H=!A!0,!l1\u0019!cand!l2!0. The low-frequency\nmodes become degenerate in the limit !H=!A!1.\nIn\"-Fe2O3[48] magnetization is reduced, resulting in\n!M=(2\u0019) = 2:73 GHz and!c=(2\u0019) = 35(nm=a)2MHz.\nFor the single-molecule magnet TbPc 2[38], we esti-\nmate!A=(2\u0019)\u00185 THz[49],!M=(2\u0019)\u001810 GHz,\n!c=(2\u0019)\u0018100 MHz [50], giving access to the strong-\nanisotropy regime with ultra-low effective damping.\nVI. EASY-PLANE CONFIGURATION\nAn easy-plane anisotropy aligns the equilibrium mag-\nnetization normal to the principal axis ( m?n), which is\ntypically caused by the shape anisotropy of pancake-like\noblate spheroids corresponding to !A<0. We choose an\nexternal magnetic field with a static component in the\nplaneH0eyand an ac field along xandz, while the equi-\n284.40 284.45 284.50 284.55 284.60\nω/(2π) [MHz]024681012141618FMR spectrum [1013s−1]\n−ωImχxx\n−ωImχzz\n−ωReχxz\n+ωReχzx\n11.011.512.012.513.013.514.0\nω/(2π) [GHz]0.00.51.01.52.02.53.0FMR spectrum [1013s−1]\n−ωImχxx\n−ωImχzz\n−ωReχxz\n+ωReχzxFigure 4. FMR spectrum of an Fe disk with 15 nmdiameter\nand2 nmthickness in a static magnetic field of 0:25 Twith\nGilbert damping constant \u000b= 0:01.\nlibrium npoints along ez(see Fig. 1(b)). For \u0012\u001c1,\nmy\u00191,nz\u00191, we again obtain analytic solutions for\nmandn(see Appendix C). We find two singularities in\nthe magnetic susceptibility tensor with frequencies (for\n\u000b\u001c1)\n!?\u0019!Hr\n1\u0000!A\n!H\u0000!c!A\n!2\nH; (6.1)\n!l\u0019s\n!2\nH!c!A\n!A!H\u0000!2\nH+!c!A: (6.2)\nSincenxdoes not depend on time there is only one\nlow-frequency mode !l, viz. an oscillation about the x-\naxis of the nanomagnet. Linearization results in _Ly\u0019\nVMs_my=\r\u00190and implies _Ly\u0019I?nx\u00190. The high-\nfrequency resonance !?is blueshifted by \u000e!?\u0018!c. As\nbefore, the lattice hardly moves in the high-frequency\nmode, while at low frequencies the magnetization is\nlocked to the lattice.\nIn Fig. 4 we plot the FMR spectrum of an Fe nan-\nodisk with shape parameters a= 1 nmandb= 7:5 nm\nunder\u00160H0= 0:25 Tor!H=(2\u0019) = 7 GHz . The\ncharacteristic frequencies are !c=(2\u0019) = 17:2 MHzand\n!A=(2\u0019) =\u000014:4 GHz. The blocking temperature with\njEZj=(kBTB)\u001924is now about 300 K:Again, the low-\nfrequency resonance is very sharp and relatively weak.\nThe contribution of Im\u001fxxto the low-frequency reso-\nnance is by a factor of 600 smaller than the dominant\nIm\u001fzzand therefore not visible in the plot.6\nVII. DISCUSSION\nThe examples discussed above safely fulfill all condi-\ntions for the validity of the theory either at reduced tem-\nperatures (T < 11 K, Fe sphere with 2 nmdiameter) or\neven up to room temperature ( 2 nm\u000215 nmFe disk).\nThe levitation of the particle can be achieved in cluster\nbeams [13, 15, 51], in aerosols [52], or by confinement to\na magnetic trap [33, 35, 53]. FMR experiments should\npreferably be carried out in a microwave cavity, e.g., a\ncoplanar wave guide that can also serve as a trap [54].\nMetal oxide nanoparticles, such as \"-Fe2O3[48], have\ncrystal anisotropies of the same order as that of pure\niron but smaller magnetization, which reduces the mag-\nnetomechanical coupling strength, leading to similar re-\nsults for somewhat smaller particles. The strongest\nanisotropies and couplings can be found in single-\nmolecule magnets, e.g., TbPc 2[49], but FMR experi-\nmentshavetobecarriedoutatlowtemperaturesinorder\nto suppress thermal fluctuations.\nOur theory holds for isolated particles at suffi-\nciently low temperatures and disregards quantum ef-\nfects. According to the fluctuation-dissipation theorem\na Gilbert damping is at finite temperatures associated\nwith stochastic fields [55]. A full statistical treatment of\nthe dynamics of magnetic nanoparticles at elevated tem-\nperatures, subject to microwaves, and weakly coupled to\ntheenvironmentisbeyondthescopeofthepresentpaper.\nWhen not suspended in vacuum but in, e.g., a liquid, the\nmechanical motion encounters viscous damping and ad-\nditional random torques acting on the lattice. Vice versa,\nthe liquid in proximity of the particle will be stirred by\nits motion. These effects can be included in principle\nby an additional torque term in Eq. (4.4). The external\ntorque will cause fluctuations in \nzand a temperature\ndependent broadening of the low-frequency resonances.\nMicrowave cavities loaded with thin films or spheres of\nthe high-quality ferrimagnet yttrium iron garnet have re-\nceived recent attention because of the relative ease with\nwhich the (ultra) strong coupling between magnons and\nphotons can be achieved (for references and evidence\nfor coherent magnon-phonon interaction, see [56]). The\nsharp low-frequency modes of free magnetic nanoparti-\nclescoupledtorfcavitymodesat10-100MHzcorrespond\nto co-operativities that are limited only by the quality\nfactor of the cavity. This appears to be a promising\nroute to access non-linear, chaotic, or quantum dynami-\ncal regimes. This technique would work also for magnets\nwith large damping and could break the monopoly of\nyttrium iron garnet for quantum cavity magnonics. Ma-\nterials with a large anisotropy are most attractive by the\nenhanced magnetization-lattice coupling.\nVIII. SUMMARY\nIn conclusion, we discussed the effect of the mag-\nnetomechanical coupling on the dynamics of levitatedsingle-domain spheroidal magnetic nanoparticles, e.g., in\nmolecular cluster beams and aerosols. We predict a blue\nshift of the high-frequency resonance and additional low-\nfrequency satellites in FMR spectra that reflect parti-\ncle shape and material parameters. In the low-frequency\nmodes the nanomagnet precesses together with the mag-\nnetization with strongly reduced effective damping and\nthereby spectral broadening.\nACKNOWLEDGMENTS\nThis work is part of the research program of the Sticht-\ning voor Fundamenteel Onderzoek der Materie (FOM),\nwhich is financially supported by the Nederlandse Or-\nganisatie voor Wetenschappelijk Onderzoek (NWO) as\nwellasJSPSKAKENHIGrantNos. 25247056,25220910,\n26103006. A. K. acknowledges financial support from the\nAlexander v. Humboldt foundation. H. K. would like to\nexpress her gratitude toward her late supervisor Malek\nZareyan for the opportunity to collaborate with the TU\nDelft researchers. S. S. is grateful to Alejandro O. León\nfor insightful discussions.\nAppendix A: Coordinate systems and\ntransformations\nWe derive the coordinate transformation from the lab\nwith basis vectors ex,ey,ezto the body frame exb,eyb,\nezb. The position of the particle is specified by the three\nEuler angles ( \u001e;\u0012; ). These three angles are defined by\nthetransformationmatrixfromthelabtothebodyframe\n(rb=Ar),\nA=0\n@cos sin 0\n\u0000sin cos 0\n0 0 11\nA0\n@1 0 0\n0 cos\u0012sin\u0012\n0\u0000sin\u0012cos\u00121\nA\n\u00020\n@cos\u001esin\u001e0\n\u0000sin\u001ecos\u001e0\n0 0 11\nA: (A1)\nThe main axis nof the particle is given by the local zb-\naxis in the body frame and can be directly obtained via\nthe inverse transformation AT,\nn=0\n@sin\u0012sin\u001e\n\u0000sin\u0012cos\u001e\ncos\u00121\nA: (A2)\nThe angular velocity vector of the rotating particle reads\nin the lab frame\n\n=_ AT0\n@0\n0\n11\nA+_\u00120\n@cos\u001e\u0000sin\u001e0\nsin\u001ecos\u001e0\n0 0 11\nA0\n@1\n0\n01\nA+_\u001e0\n@0\n0\n11\nA\n=0\n@_\u0012cos\u001e+_ sin\u0012sin\u001e\n_\u0012sin\u001e\u0000_ sin\u0012cos\u001e\n_\u001e+_ cos\u00121\nA; (A3)7\nand in the body frame,\n\nb=A\n=0\n@_\u001esin\u0012sin +_\u0012cos \n_\u001esin\u0012cos \u0000_\u0012sin \n_\u001ecos\u0012+_ 1\nA:(A4)\nThe mechanical angular momentum Land the principal\naxisnof the nanomagnet can be related by considering\nthe mechanical angular momentum in the body frame\nLb=Ib\nb: (A5)\nTransforming (A5) to the lab frame and expanding for\nsmall angles \u0012,\nLx\u0019I?d\ndt(\u0012cos\u001e)\u0019\u0000I?_ny;(A6a)\nLy\u0019I?d\ndt(\u0012sin\u001e)\u0019I?_nx; (A6b)\nLz\u0019I3(_\u001e+_ )\u0019I3\nz; (A6c)\nwhich is a valid approximation when \nz=O(\u0012):Fur-\nthermore,nz\u00191and _nz\u00190is consistent with \u0012\u001c1.\nThe Gilbert damping is defined for the relative motion\nof the magnetization with respect to the lattice, i.e. in\nthe rotating frame. The damping in the lab frame is\nobtained by the coordinate transformation\n\u001c(\u000b)\nm=AT\u001c(\u000b)\nm;b=AT(\u000bmb\u0002_mb);(A7)\nwhere mb=Am. Expanding the time derivative\n\u001c(\u000b)\nm=\u000bm\u0002_m+\u000bm\u0002\u0010\nAT_Am\u0011\n:(A8)\nThe angular frequency vector \nis defined by\n_r=\n\u0002r; (A9)\nwhere ris a point in the rotating body, i.e. _rb= 0, and\n_r=_ATrb=_ATAr: (A10)\nUsingd\ndt(ATA) =AT_A+_ATA= 0and comparing\nEqs. (A9) and (A10),\nAT_Ar=r\u0002\n; (A11)\nand therefore\n\u001c(\u000b)\nm=\u000bm\u0002_m+\u000bm\u0002(m\u0002\n):(A12)\nAppendix B: Poisson bracket in Hamilton’s equation\nIn the following, we show how to derive Hamilton’s\nequation of motion (3.4). Using the linearity of the Pois-\nson bracket together with the product rule\nfAB;Cg=AfB;Cg+fA;CgB; (B1)andfaijk(n;L);Sg= 0, we get\nfS;Hg=X\ni;j;k2N0aijk(n;L)\b\nS;Si\nxSj\nySk\nz\t\n:(B2)\nWe only consider the x-component, as the other compo-\nnents can be derived similarly. Using the product rule\n(B1), we may write\n\b\nSx;Si\nxSj\nySk\nz\t\n=Si\nx\b\nSx;Sj\nySk\nz\t\n=Si\nxSj\ny\b\nSx;Sk\nz\t\n+Si\nxSk\nz\b\nSx;Sj\ny\t\n:\n(B3)\nNext, we prove by induction that\n\b\nSx;Sk\nz\t\n=\u0000kSySk\u00001\nz; (B4)\nwhere the base case ( k= 0)\n\b\nSx;S0\nz\t\n= 0 (B5)\nand the inductive step ( k!k+ 1)\n\b\nSx;Sk+1\nz\t\n=Sz\b\nSx;Sk\nz\t\n+Sk\nzfSx;Szg\n=\u0000(k+ 1)SySk\nz (B6)\ncomplete the proof. Similarly, it follows\n\b\nSx;Sj\ny\t\n=jSj\u00001\nySz: (B7)\nSummarizing\n\b\nSx;Si\nxSj\nySk\nz\t\n=jSi\nxSj\u00001\nySk+1\nz\n\u0000kSi\nxSj+1\nySk\u00001\nz; (B8)\nwhich gives with Eq. (B2) the x-component of Eq. (3.4).\nAppendix C: Linearized equations of motion\n1. Easy-axis configuration\nIn the easy-axis case ( mknkez), the linearized equa-\ntions of motion of the magnetization mand mechanical\nangular momentum Lread\n_mx=\u0000!Hmy+!Mhy\nMs\u0000!A(my\u0000ny)\u0000\u000b( _my\u0000_ny);\n(C1a)\n_my=!Hmx\u0000!Mhx\nMs+!A(mx\u0000nx) +\u000b( _mx\u0000_nx);\n(C1b)\n_mz= 0; (C1c)\n_Lx=\u0000I?ny; (C2a)\n_Ly=I?nx; (C2b)\n_Lz=I3_\nz= 0; (C2c)8\n0.0 0.5 1.0 1.5 2.0\nωH/ωA050100150200\nReχxx(ωl1)\nReχxx(ωl2)\n0.0 0.5 1.0 1.5 2.0\nωH/ωA−50000−40000−30000−20000−100000\nImχxx(ωl1)\nImχxx(ωl2)\n0.0 0.5 1.0 1.5 2.0\nωH/ωA−20000−15000−10000−500005000100001500020000\nReχxy(ωl1)\nReχxy(ωl2)\n0.0 0.5 1.0 1.5 2.0\nωH/ωA−200−150−100−50050100150200\nImχxy(ωl1)\nImχxy(ωl2)\nFigure 5. Real and imaginary parts of the magnetic susceptibility tensor \u001f(!)of the low-frequency modes !l1and!l2for an\nFe nanosphere of 2 nmdiameter with Gilbert damping \u000b= 0:01.\nwith\nnx=!2\nN(mx\u0000nx) +\u000b!c( _mx\u0000_nx);(C3a)\nny=!2\nN(my\u0000ny) +\u000b!c( _my\u0000_ny);(C3b)\nnz= 0; (C3c)\nwhere!2\nN=!c!A. Since _\nz= 0and with initial condi-\ntion\nz= 0, there is no net rotation \nz. Introducing the\nchiral modes,\nm\u0006=mx\u0006imy; n\u0006=nx\u0006iny; h\u0006=hx\u0006ihy;(C4)\nwecanwritetheequationsofmotioninthecompactform\n_m\u0006=\u0006i\u0012\n!0m\u0006\u0000!Mh\u0006\nMs\u0000!An\u0006\u0013\n\u0006i\u000b\u0000\n_m\u0006\u0000_n\u0006\u0001\n;\n(C5)\nn\u0006=!2\nN\u0000\nm\u0006\u0000n\u0006\u0001\n+\u000b!c\u0000\n_m\u0006\u0000_n\u0006\u0001\n: (C6)\nFor ac magnetic fields\nh\u0006(t) =h\u0006\n0ei!t; (C7)\nwe solve the equations of motion by the ansatz\nm\u0006(t) =m\u0006\n0ei!t; n\u0006(t) =n\u0006\n0ei!t:(C8)\nThe observables correspond to the real part of the com-\nplexm,n;andh. The susceptibilities are defined\nm\u0006=\u001f\u0006h\u0006=Ms; n\u0006=\u001f\u0006\nnm\u0006;(C9)and read\n\u001f\u0006\nn(!) =!2\nN+i\u000b!!c\n\u0000!2+!2\nN+i\u000b!!c;(C10)\n\u001f\u0006(!) =\u0007!M(\u0000!2+!2\nN+i\u000b!!c)\n\u0002\u0002\n(!\u0007!0\u0007i\u000b!)(\u0000!2+!2\nN+i\u000b!!c)\n\u0006!c(!A+i\u000b!)2\u0003\u00001: (C11)\nClose to a resonance of \u001f\u0006at!ithe absorbed microwave\npower is determined by the contributions\n\u0000!\n2Im\u001f\u0006(!)\u0018F\u0006 (\u0011\u0006)2\n(!\u0000!i)2+ (\u0011\u0006)2;(C12)\nwith\n\u0011\u0006=\u0006\u000b!i\u0000\n!2\ni+!c(\u0006!i\u0000!H)\u0001\n3!2\ni\u00072!i!0\u0000!c!A; (C13)\nF\u0006=1\n2!M(!2\ni\u0000!c!A)\n\u000b(!2\ni+!c(\u0006!i\u0000!H)):(C14)\nNote that for each resonance of \u001f+at!ithere is a cor-\nresponding resonance of \u001f\u0000at\u0000!i.\nThe magnitudes of the x- andy-components of nare\nrelated to mvia the susceptibility \u001f\u0006\nngiven in Eq. (C10).9\nFor high frequencies !we find\u001f\u0006\nn\u00190and for low fre-\nquencies\u001f\u0006\nn\u00191. Therefore, the main axis nis nearly\nstatic for the high-frequency mode, while for the low-\nfrequency modes nstays approximately parallel to m.\nThesusceptibility \u001f\u0006giveninEq.(C11)canberelated\nto the usual magnetic susceptibilities (\u000b;\f=x;y),\n\u001f\u000b\f=M\u000b\nh\f\f\f\f\f\nh?=0: (C15)\nDefining the symmetric and antisymmetric parts of the\nsusceptibility \u001f\u0006,\n\u001f\u0006=\u001fs\u0006\u001fa: (C16)\nwe find the relations\n\u001fxx=\u001fyy=\u001fs; (C17a)\n\u001fxy=\u0000\u001fyx=i\u001fa: (C17b)\nThe magnetization dynamics in terms of the magnetic\nsusceptibility reads\nRe\u0012\nmx(t)\nmy(t)\u0013\n= Re\u0014\u0012\n\u001fxx\u001fxy\n\u0000\u001fxy\u001fxx\u0013\u0012\nhx(t)=Ms\nhy(t)=Ms\u0013\u0015\n;\n(C18)\nwhere\u001fyy=\u001fxxand\u001fyx=\u0000\u001fxy. For linear polariza-\ntionhx(t) =jhxjei!tandhy(t) = 0,\nRe\u0012\nmx(t)\nmy(t)\u0013\n=jhxj\nMs\u0012\nRe\u001fxxcos(!t)\u0000Im\u001fxxsin(!t)\n\u0000Re\u001fxycos(!t) + Im\u001fxysin(!t)\u0013\n:\n(C19)\nAccording to Fig. 5, jRe\u001fxxj;jIm\u001fxyj \u001c j Re\u001fxyj \u0019\njIm\u001fxxj, and Im\u001fxx<0for both low-frequency modes\n!l1and!l2. The direction of the precession depends now\non the sign of Re\u001fxy, which is negative for !l1and posi-\ntive for!l2. The mode !l1is a clockwise precession,\nRe\u0012\nmx(t)\nmy(t)\u0013\n/\u0012\nsin(!l1t)\ncos(!l1t)\u0013\n; (C20)\nwhereas the mode !l2precesses counterclockwise:\nRe\u0012\nmx(t)\nmy(t)\u0013\n/\u0012\nsin(!l2t)\n\u0000cos(!l2t)\u0013\n:(C21)\nNote that\u001f\u0000(!)has a low-frequency peak only at !l1\nand\u001f+(!)only at!l2(for!>0).\n2. Easy-plane configuration\nHere, we consider an equilibrium magnetization nor-\nmal to the principal axis ( m?n) due to the shape\nanisotropy of an oblate spheroid. Linearizing for small\ndeviations from the equilibrium ( \u0012\u001c1,my\u00191,nz\u00191),\nthe equations of motion for the magnetization and me-\nchanical angular momentum read_mx=!Hmz\u0000!Mhz\nMs\u0000!A(mz+ny) +\u000b( _mz+ _ny);\n(C22a)\n_my= 0; (C22b)\n_mz=\u0000!Hmx+!Mhx\nMs\u0000\u000b_mx\u0000\u000b\nz; (C22c)\n_Lx=\u0000I?ny; (C23a)\n_Ly=I?nx; (C23b)\n_Lz=I3_\nz=VMs\n\r(\u0000\u000b_mx\u0000\u000b\nz);(C23c)\nwith\nnx= 0; (C24a)\nny=!2\nN(mz+ny)\u0000\u000b!c( _mz+ _ny);(C24b)\nnz= 0: (C24c)\nIn the presence of ac magnetic fields\nhx(t) =hx;0ei!t; hz(t) =hz;0ei!t;(C25)\nwe use the ansatz\nmx(t) =mx;0ei!t; mz(t) =mz;0ei!t; ny(t) =ny;0ei!t:\n(C26)\nFrom Eq. (C23c)\n\nz=\u0000!I!\u000bmx\n!\u0000i\u000b!I\u0019\u0000\u000b!Imx;(C27)\nwhere!I=VMs=(\rI3)and provided \u000b!Iis sufficiently\nsmaller than all the other relevant frequencies. We ap-\nproximate\u000b\nz=O(\u000b2)\u00190in Eq. (C22c). Due to\nthe reduced symmetry for m?n, we cannot simplify\nthe equations of motion by introducing chiral modes, but\nhave to calculate the Cartesian components of the mag-\nnetic susceptibility tensor \u001fas\n\u001fxx=!M\u0002\n!2(!A\u0000!H)\u0000i\u000b(!3\u0000!!c!H)\u0000!H!2\nN\u0003\n=\u001fd;\n(C28a)\n\u001fzz=\u0000!M(!H+i\u000b!)(!2+!2\nN\u0000i\u000b!c!)=\u001fd;(C28b)\n\u001fxz=i!!M(!2+!2\nN\u0000i\u000b!c!)=\u001fd; (C28c)\n\u001fzx=\u0000\u001fxz; (C28d)\nwhere the denominator\n\u001fd=!4(1 +\u000b2) +i\u000b!3(!A\u0000!c\u00002!H)\n+!2(!A!H\u0000!2\nH+!2\nN\u0000\u000b2!c!H)\n+i\u000b!!H(!c!H\u0000!2\nN)\u0000!2\nH!2\nN:(C29)\nThe singularities in \u001fmark the two resonance frequen-\ncies. For small damping ( \u000b\u001c1)\n!2\n1;2=\u00001\n2(!A!H\u0000!2\nH+!2\nN)\n\u00061\n2q\n(!A!H\u0000!2\nH+!2\nN)2+ 4!2\nH!2\nN:(C30)10\nFrom Eq. (C24b), we obtain the following relation be-\ntween the magnetic and mechanical motion\nny=\u0000!2\nN+i\u000b!c!\n!2+!2\nN\u0000i\u000b!c!mz: (C31)\nFor high frequencies ny\u00190and for low frequencies ny\u0019\n\u0000mz. This implies that for the high frequency mode\n!?=!1we recover the bulk FMR, while in the low-\nfrequency mode !l=!2the magnetization is locked to\nthe lattice.\nAppendix D: FMR absorption\nFMRabsorptionspectraareproportionaltotheenergy\ndissipated in the magnet [25]. The energy density of the\nmagnetic field is given by\nw(t) =1\n2H(t)\u0001B(t); (D1)\nwhere B=\u00160\u001fH. The absorbed microwave power by a\nmagnet of volume Vis\nP(t) =V_w(t) =VH(t)\u0001_B(t): (D2)\nThe average over one cycle T= 2\u0019=!,\nP\u0011hP(t)i=1\nTZT\n0dtP(t); (D3)can be calculated using the identity\n\nRe(Aei!t)\u0001Re(Bei!t)\u000b\n=1\n2Re (A\u0003\u0001B):(D4)\nWhen a monochromatic ac component of the magnetic\nfieldh?is normal to its dc component, the power reads\nP=\u0000\u00160V\n2!Im (h\u0003\n?\u0001M?); (D5)\nwhere M?is the transverse magnetization. When the\nmagnetization and static magnetic field are parallel to\nthe principal axis of the particle, we can write\nP=\u0000\u00160V\n2!h\n(jhxj2+jhyj2)Im\u001fs(!)\n\u00002Im(h\u0003\nxhy)Im\u001fa(!)]; (D6)\nwhere the symmetric and antisymmetric parts of the\nsusceptibility \u001f\u0006Eq. (C11) as defined by Eq. (C16)\nobeythesymmetryrelations Im\u001fs(\u0000!) =\u0000Im\u001fs(!)and\nIm\u001fa(\u0000!) = Im\u001fa(!). 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Adv.\n2, e1501286 (2016)."    },    {        "title": "1408.4861v2.Brownian_motion_of_massive_skyrmions_forced_by_spin_polarized_currents.pdf",        "content": "Brownian motion of massive skyrmions forced by spin polarized currents\nRoberto E. Troncoso1,a)and Alvaro S. Núñez2\n1)Centro para el Desarrollo de la Nanociencia y la Nanotecnología, CEDENNA, Avda. Ecuador 3493,\nSantiago 9170124, Chile\n2)Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Chile, Blanco Encalada 2008,\nSantiago, Chile\nWe report on the thermal effects on the motion of current-driven massive magnetic skyrmions. The reduced\nequation for the motion of skyrmion has the form of a stochastic generalized Thiele’s equation. We propose an\nansatz for the magnetization texture of a non-rigid single skyrmion that depends linearly with the velocity. By\nutilizing this ansatz it is found that the mass of skyrmion is closely related to intrinsic skyrmion parameters,\nsuch as Gilbert damping, skyrmion-charge and dissipative force. We have found an exact expression for the\naverage drift velocity as well as the mean-square velocity of the skyrmion. The longitudinal and transverse\nmobility of skyrmions for small spin-velocity of electrons is also determined and found to be independent of\nthe skyrmion mass.\nIntroduction .- Skyrmions have recently been the focus\nof intense research in spintronics. They are vortex-like\nspin structures that are topologically protected1,2. A\nseries of works report their recent observation in chi-\nral magnets3–8. There is a great interest in their dy-\nnamics due to the potential applications in spintronics\nthat arise from the rather low current densities that\nare necessary to manipulate their location9. Among\nother systems that have been reported hosting skyrmion\ntextures they were observed in bulk magnets MnSi3,4,\nFe1\u0000xCoxSi5,6,10, Mn 1\u0000xFexGe11and FeGe12by means\nof neutron scattering and Lorentz transmission elec-\ntron microscopy. Regarding their dimensions, by the\nproper tuning of external magnetic fields, sizes of the\norder of a few tens of nanometers have been reported.\nSpin transfer torques can be used to manipulate and\neven create isolated skyrmions in thin films as shown\nby numerical simulations13–15. In thin films skyrmions\nhave been observed at low temperatures, however en-\nergy estimates predict the stability of isolated skyrmions\neven at room temperatures16. Under that regime the\nmotion of skyrmions is affected by fluctuating thermal\ntorques that will render their trajectories into stochastic\npaths very much like the Brownian dynamics of a par-\nticle. Proper understanding of such brownian motion is\na very important aspect of skyrmion dynamics. Numer-\nical simulations17,18and experimental results19, suggest\nthat the skyrmion position can be manipulated by expo-\nsure to a thermal gradient and that the skyrmions also\ndisplay a thermal creep motion in a pinning potential20.\nThe thermally activated motion of pinned skyrmions has\nbeen studied in Ref. [21] where it has been reported that\nthermal torques can increase the mobility of skyrmions\nby several orders of magnitude. In this work we present\na study of the random motion of magnetic skyrmions\narising from thermal fluctuations. In our analysis we in-\nclude an assessment of the deformation of the skyrmion\nthat arises from its motion. This deformation induces an\na)Electronic mail: R.E.TroncosoCona@gmail.cominertia-like term into the effective stochastic dynamics of\nthe skyrmion. We present a theory that allows us to es-\ntablish a relation between the fluctuating trajectory of\nthe skyrmion and its effective mass.\nStochastic dynamics .- We begin our analysis from the\nstochastic Landau-Lifschitz-Gilbert (LLG) equation22,23\nthat rules the dynamics of the magnetization direction\n\n. Into this equation we need to include the adiabatic,\ngiven by\u0000vs\u0001r\n, and non-adiabatic, given by \fvs\u0001r\n,\nspin-transfer torques24,25where the strength of the non-\nadiabatic spin-transfer torque is characterized by the pa-\nrameter\f. In those expressions vs=\u0000\u0000\npa3=2eM\u0001\nj\nstands for the spin-velocity of the conduction electrons,\npis the spin polarization of the electric current density\nj,e(>0)the elementary charge, athe lattice constant,\nandMthe magnetization saturation. With those contri-\nbutions the stochastic Landau-Lifshitz-Gilbert equation\nbecomes:\n\u0012@\n@t+vs\u0001r\u0013\n\n=\n\u0002(Heff+h)\n+\u000b\n\u0002\u0012@\n@t+\f\n\u000bvs\u0001r\u0013\n\n;(1)\nwhere Heff=1\n~\u000eE\n\u000e\nis the effective field, with Erepresent-\ning the energy of the system, and \u000bthe Gilbert damping\nconstant. An important aspect of this equation is the in-\nclusion of the white Gaussian fluctuating magnetic field\nh, describing the thermal agitation of the magnetization\nand obeying the fluctuation-dissipation theorem22. The\nstrength of the noise, \u001b= 2\u000bkBTa2=~, is proportional to\nthe thermal energy kBT, the Gilbert damping parameter\n\u000b, and the volume of the finite element grid a2.\nParticle like solutions of the Landau-Lifshitz-Gilbert\nequation, thatrepresentcompactmagnetictexturesmov-\ning as coherent entities with a well defined velocity, have\nknown since long ago. Among other examples we can\nfound the dynamics of domain walls27,28and of Bloch\npoints29,30. The account of the dynamics of skyrmion\ntextures is best handled by the use of the collective\ncoordinates approach. Under this framework the com-\nplex dynamics of the magnetization texture, \n(r;t)isarXiv:1408.4861v2  [cond-mat.mes-hall]  22 Aug 20142\nreduced to the evolution of a small number of degrees\nof freedom given by the skyrmion position and its ve-\nlocity. In this way the magnetization field associated\nto a single-skyrmion moving along the trajectory x(t)\nis represented by a magnetization profile \n(r;t) =\n(r\u0000\nx(t);v(t)). The explicit time-dependence of the magne-\ntization, coming from the dependence on velocity v(t),\nincludes the effects of deformations of the skyrmion31–33.\nThe calculation for the static skyrmion profile, \n0(r),\nhas been addressed elsewhere34, by means of a minimiza-\ntion of the magnetic energy. In this energy the contribu-\ntions from the exchange, perpendicular anisotropy, and\nDzyaloshinskii-Moriya energies must be included. Re-\nplacement of the collective coordinates ansatz and inte-\ngration over space reduce the LLG equation to an equa-\ntion of motion for the collective variables. This equation\nhas the form of a stochastic massive Thiele’s equation\nM_v(t)\u0000g^z\u0002v(t) +\u000bDv(t) =F+\u0011(t):(2)\nNeglecting the contribution of the noise term ( \u0011(t))\nEq. (2) turns into the generalized Thiele’s equation35.\nWe highlight the inertial terms, quantified by the effec-\ntive mass, that correspond to a matrix Mij=Mij+\u0016Mij,\ncomprised by the elements Mij=R\ndr \n\u0001\u0000@\n@xi\u0002@\n@vj\u0001\n,\narising from the conservative dynamics, and \u0016Mij=\n\u000bR\ndr\u0000@\n@xi\u0001@\n@vj\u0001\narising from the dissipative contribu-\ntiontotheLandau-Lifshitz-Gilbertequation. Thesecond\nterm in Eq. (2) describes the Magnus force4exerted by\nthe magnetic texture on the moving skyrmion. In the\ncase of an isolated skyrmion g= 4\u0019WwhereW=\u00001\nstands for the winding number, or skyrmion charge. The\nthird contribution represents the dissipative force which\nis defined through the relation Dij=R\ndr@\n@xi\u0001@\n@xj, that\nbecomesDij=D\u000eijinthe caseofthe highlysymmetrical\ncase of an isolated skyrmion.\nThe dynamics of the skyrmion in Eq. (2) is forced\nby a deterministic term F=\u0000g^z\u0002vs+\fDvs\u0000rV,\nthat contains a contribution from the flowing electrons\nand a force arising from a potential V[x]that reflects\nthe inhomogeneities in the skyrmion‘s path, e.g., mag-\nnetic impurities, local anisotropies or geometric defects.\nWe conclude with the last term of right-hand side of Eq.\n(2), that describes the fluctuating force on the skyrmion.\nThe strength of the Gaussian white noise turns out to\nbe\u001bD, therefore the effective diffusion constant of the\nskyrmion depends not only on the Gilbert damping and\nthe temperature but also on the dissipative parameter\nD. By solving the stochastic Thiele equation Eq. (2),\nfor the homogenous case ( V[x] = 0), we determine the\ntime evolution of both longitudinal and transverse com-\nponents of the velocity of the fluctuating skyrmion (as\nshown in Fig. 1 (a) and (b) respectively) at temperature\nT= 100K. Typical skyrmion speeds of \u00180\u00001m/s are\nreached for spin-velocities on the order of 1m/s. The\nmassive skyrmion dynamics was calculated for a Gilbert-\ndamping parameter \u000b= 0:1, the\fparameter\f= 0:5\u000b,\nthe dissipative force D= 5:577\u0019(from Ref. [13]), and\nwhere the values used for the mass are taken from Ref.[36]. Moreover, it is numerically solved the mean drift\nvelocity, i.e., the steady-state current-induced skyrmion\nmotion, which is displayed in Fig. 1 (c) as a function\nof the spin-velocity vs=vs^xof electrons. In addition,\n0 1 20.01.02.0vxvsHaL\n0 1 2-1.00.01.0\n10-4tt0vyvsHbL\n0 2 4 6 8 100246810\nvs@msD<v>@msDHcL\n<vx>\n<vy>\nFIG. 1. Fluctuating skyrmion velocities in the longitudinal\nand transverse directions, (a) and (b) respectively. The Brow-\nnian dynamics Eq. (2) is solved for a spin velocity of electrons\nvs= 1 m/s along xdirection and for a characteristic time\nscalet0=Mxx\u00196ns (from Ref. [36]). In (c) the average\nvelocities is presented as function of spin-velocity vs=vs^x,\nboth the longitudinal (black line) and transverse (dashed line)\ncomponents. The results are shown for a Gilbert damping\n\u000b= 0:1,\f= 0:5\u000b, and at temperature 100K.\nwe are interested in the probability distribution P[x;v;t]\nassociated to the skyrmion dynamics, which is defined as\nthe probability density that a skyrmion at time t, is in\nthe position xwith a velocity v. The equation of motion\nsatisfied by such distribution is known as Fokker-Planck\nequation and its derivation, as well as its solution in sim-\nple cases, constitutes a standard issue in the analysis of\nstochastic processes37.\nOrigin of the skyrmion mass .- The Brownian skyrmion\nmotion described by Eq. (2) contains as a main ingre-\ndient the inertia term, regarding to Ref. [21], which is\nquantified by the effective mass matrix M. Generally\nspeaking, it is linked to the explicit time-dependence of\nthe magnetization direction. The mass of skyrmions can\nbe determined perturbatively in linear response theory as\nfollows. In the skyrmion center of mass reference frame\nwe see that the magnetization in Eq. (1) is affected by an\nadditional magnetic field \u000eH(r) =\n(r)\u0002(v\u0001r)\n(r).\nWe note that the strength of the effective field is con-\nfined within the perimeter of the skyrmion. However,\nthe effective torque takes a maximum value in the center\nof skyrmion and thus, it suffers a distortion of its shape\ndue to the current-induced motion. This motivates us to\npropose an ansatz for the magnetization texture of a non-\nrigid single skyrmion and its dependence on the velocity\nas\n\n(r;v) =\n0(r) +\u0015\u0018\n0(r)\u0002(v\u0001r)\n0(r);(3)\nwith\n0correspondstothestaticandrigidskyrmiontex-\nture. The deformation of skyrmion size is parameterized3\nby the second term, where \u0015is the dimensionless param-\neter that determines the strength of the velocity induced\ndeformations. In this expression \u0018=~l2\nsk=Ja2withJ\nthe exchange constant, athe lattice constant and lskthe\ncharacteristic skyrmion size. It is worth noting that this\n\u00002\u0000112\u00002\u00001012\nx/lsk0y/lsk\n\u0000101⌦z\n\u00002\u0000112\u00002\u00001012\nx/lsk0y/lsk\n\u0000101\u0000⌦z\nFIG. 2. Top: Pictorial representation of the skyrmion mag-\nnetic texture. The color encodes the out of plane component\nof the magnetization. It changes from being fully aling with\nthe+^zdirection in the center to a complete alignment with\nthe opposite direction in the outer rim. The arrows represent\nthe behavior of the in plane component of the magnetiza-\ntion. For the case displayed those components swirl like a\nvortex. Bottom: Schematic plot for the skyrmion distortion\ngenerated by the motion of the skyrmion. In color we have\nrepresented the out of plane component of the deformation\n\u000e\nz. This contribution is concentrated in the direction trans-\nverse to the motion nearby the perimeter of the skyrmion\n(indicated by a dashed line). The arrows correspond to the\nin plane (\u000e\nx;\u000e\ny).\ncontribution is linear in velocity and conserves the norm\nof the magnetization to leading order in \u0015. In Fig. 2\nwe present schematically the distortion of skyrmions ex-\nerted by the effective field \u000eH. However, without loss of\ngenerality, we assume a motion of the skyrmion along x\ndirection. The deformation in the skyrmion texture con-\nsists of an in plane distortion, that resembles a dipolar\nfield, and an out of plane contribution that is antisym-\nmetric. The nature of the mass of skyrmions can be\ntraced back by using the ansatz given by Eq. (3). By\nreplacing it on the expressions for Mup to linear orderin\u0015, we find that the mass is related both to the dissipa-\ntivematrixandgyrotropictensorby Mxx=Myy=\u0015\u0018D\nandMxy=\u0000Myx=\u0015\u0018\u000bg. We see how the dissipation\nmechanisms encoded by the Gilbert damping \u000bgenerate\nan anti symmetrical contribution to the mass. By com-\nparing our results for Mxxwith earlier theoretical31,32\nand experimental36estimates of the skyrmion mass we\nobtain\u0015\u00180:01. In this case we obtain, using a typical\nskyrmion velocity of 1 m/s in Eq. (3), a characteristic\nstrength of the deformation of the skyrmion in the range\nofj\u000e\nj\u001810\u00003.\nSkyrmion mobility .-Theskyrmiondynamicsisinduced\nby an electric current density via spin-transfer torque\nmechanism. As mentioned previously, the flow of elec-\ntrons exerts a force Fthat drives the skyrmion and its re-\nsponse is described by the stochastic generalized Thiele’s\nequation21. Next, we discuss the role of mass in the dy-\nnamics of skyrmions induced by currents at finite tem-\nperature. The probability distribution for the velocity\nof the skyrmion can be readily found for V(x) = 0by\nsolving the associated Fokker-Planck equation, P[v] =\nNexp\u0000\n\u00001\n2\r(v\u0000\u0016 v)2\u0001\n, where\r=\u0015\u0018\u000b(g2+D2)=\u001bDand\n\u0016 v=\u0016kvs+\u0016?^ z\u0002vs. In this equation \u0016k= (g2+\n\u000b\fD2)=(g2+\u000b2D2)and\u0016?=\u0000(\u000b\u0000\f)gD=(g2+\u000b2D2)\nare the longitudinal and transverse skyrmion mobilities\nrespectively. We are interested in the average drift veloc-\nity, which is determined directly from the probability dis-\ntribution ashvii= \u0016vi. Another relevant quantity as the\nmean-squared velocity is evaluated directly, and is found\ntoobeyhvivji\u0000hviihvji=\u001bD\u0000\n\u0015\u0018\u000b\u0000\ng2+D2\u0001\u0001\u00001\u000eijthat\nrelates the fluctuations on the velocity in terms of tem-\nperature and the skyrmion mass. The mean drift ve-\nlocity scales linearly with the spin-velocity of electrons\nand, unlike the current-induced domain wall dynamics28,\nskyrmions do not exhibit an intrinsic pinning13,14. Note\nthat these values for the spin-velocities correspond to\nelectric current densities on the order of 1010A/m2. As\nwe analytically show, the average velocity of the free\nskyrmion is independent of the mass. However, it is re-\nlated only on intrinsic parameters, such as Gilbert damp-\ning, skyrmion-charge, and dissipative force. Following\nour results, a detailed characterization of the strength of\nvelocity fluctuations can be used to determine the values\nof the skyrmion mass.\nConclusions .- We have investigated the mechanisms by\nwhich thermal fluctuations influence the current-driven\nmassive skyrmion dynamics. Based on the stochas-\ntic Landau-Lifschitz-Gilbert equation we derived the\nLangevin equation for the skyrmion. The equation has\ntheformofastochasticgeneralizedThiele’sequationthat\ndescribes the massive dynamics of a single-skyrmion at\nfinite temperature. We proposed an ansatz for the mag-\nnetization texture of a non-rigid single skyrmion that de-\npends linearly with the velocity. This ansatz has been\nderived based upon an effective field that distorts the\nskyrmion texture. In particular, it implies that the de-\nformation of the skyrmion shape consists of an in plane\ndistortion and an out of plane contribution that is an-4\ntisymmetric. Furthermore, by utilizing this ansatz it is\nfound that the mass of skyrmion is related with intrinsic\nparameters, such as Gilbert damping, skyrmion-charge,\nand dissipative force. This simple results provides a path\nfor a theoretical calculation of the skyrmion mass. We\nhavefoundanexactexpressionfortheaveragedriftveloc-\nity as well as the mean-square velocity of the skyrmion.\nThe longitudinal and transverse mobilities of skyrmions\nfor small spin-velocity of electrons were also determined.\nWe showed that the average velocity of skyrmions, unlike\nthe mean-square velocity, is independent of the mass and\nit varies linearly with the spin-velocity. In future work,\nwe plan to use the formalism developed in this work to\nthe study of the transport of massive skyrmions in disor-\ndered media.\nAcknowledgements .- The authors acknowledge fund-\ning from Proyecto Basal FB0807-CEDENNA, Anillo de\nCiencia y Tecnonología ACT 1117, and by Núcleo Cien-\ntífico Milenio P06022-F.\n1T. H. R. 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Johannes\nGutenberg-Universit at Mainz.\n37H.Risken, TheFokker-PlanckEquation(Springer-Verlag, Berlin,\n1984)."    },    {        "title": "2109.03076v2.Inertial_spin_dynamics_in_epitaxial_cobalt_films.pdf",        "content": "Inertial spin dynamics in epitaxial cobalt \flms\nVivek Unikandanunni,1Rajasekhar Medapalli,2, 3Marco Asa,4Edoardo Albisetti,4\nDaniela Petti,4Riccardo Bertacco,4Eric E. Fullerton,2and Stefano Bonetti1, 5,\u0003\n1Department of Physics, Stockholm University, 10691 Stockholm, Sweden\n2Center for Memory and Recording Research, University of California San Diego, San Diego, CA 92093, USA\n3Department of Physics, Lancaster University, Bailrigg, Lancaster LA1 4YW, United Kingdom\n4Department of Physics, Politecnico di Milano Technical University, Milano, Italy\n5Department of Molecular Sciences and Nanosystems,\nCa' Foscari University of Venice, 30172 Venice, Italy\nWe investigate the spin dynamics driven by terahertz magnetic \felds in epitaxial thin \flms of\ncobalt in its three crystalline phases. The terahertz magnetic \feld generates a torque on the mag-\nnetization which causes it to precess for about 1 ps, with a sub-picosecond temporal lag from the\ndriving force. Then, the magnetization undergoes natural damped THz oscillations at a frequency\ncharacteristic of the crystalline phase. We describe the experimental observations solving the inertial\nLandau-Lifshitz-Gilbert equation. Using the results from the relativistic theory of magnetic inertia,\nwe \fnd that the angular momentum relaxation time \u0011is the only material parameter needed to\ndescribe all the experimental evidence. Our experiments suggest a proportionality between \u0011and\nthe strength of the magneto-crystalline anisotropy.\nThe fundamental understanding of magnetism has\nmuch improved since the \frst experiments on magnetic\nmaterials at femto- and picosecond time scales, the so-\ncalled ultrafast regime. The pioneering work of Beaure-\npaire et al. [1] demonstrated that sub-picosecond magne-\ntization dynamics is possible, against the prediction of\nthe textbook Landau-Lifshitz-Gilbert (LLG) equation.\nThose results have since then been con\frmed in sev-\neral experiments and using both optical [2{12] and X-\nray techniques [13{16], making the \feld of ultrafast mag-\nnetism, and of magnetization dynamics beyond the LLG\nequation, an active \feld of research with possible impli-\ncations for novel data storage technologies.\nAnother recent analysis of the LLG equation showed\nthat from a classical mechanics point of view, the LLG\nequation assumes an unphysical inertial tensor [17], also\nsuggested by Gilbert himself in a footnote almost two\ndecades ago [18]. However, Ciornei et al. [17] re-derived\nthe equation with a realistic inertial tensor which resulted\nin a slightly revised equation known as the inertial LLG\n(iLLG) equation, which predicts the appearance of a nu-\ntation resonance at a frequency much higher than the\nferromagnetic resonance (FMR) one. After that pioneer-\ning work, much theoretical e\u000bort has been performed try-\ning to identify the frequency regime where such nutation\nresonance was to be expected, with predictions varying\nover a few orders of magnitude [19{23]. The experimen-\ntal con\frmation has been achieved only recently and de-\ntected nutation dynamics in the \u00181 THz range [24].\nDespite this novel experimental evidence, and the signif-\nicant theoretical progress to understand the microscopic\norigin of inertia [25{44], a complete picture of how dif-\nferent material parameters a\u000bect the nutation dynamics\nis still missing.\n\u0003stefano.bonetti@fysik.su.seIn this Letter, we provide the \frst experimental\ndata on the dependence of inertial spin dynamics on\na key magnetic property, i.e. the magneto-crystalline\nanisotropy. It has been suggested that spin-orbit cou-\npling is the fundamental interaction needed to derive a\ncorrect inertial tensor from \frst principles. Hence, exper-\niments where the magneto-crystalline anisotropy is well\nde\fned and controlled, are expected to return impor-\ntant insights on this open question. Similar to Ref. [24],\nwe perform THz pump / optical probe time-resolved\nmagneto-optical measurements to trigger and detect in-\nertial spin dynamics. In contrast to the approach of\nRef. [24], where a narrowband (\u0001 f=f\u00190:1) THz source\nwas implemented, we use intense single-cycle terahertz\nradiation [45] whose broad band (\u0001 f=f > 1) allows to\ncover the frequency range where the nutation resonance\nis expected to appear. In addition, due to the impul-\nsive character of the driving pulse, our measurements\nare expected to detect not only the forced response of\nthe system, but also its natural one. We model the mag-\nnetization dynamics solving the inertial LLG equation\nnumerically and we contextualize our results with the\nexisting microscopic theory of magnetic inertia.\nWe choose to investigate three epitaxial cobalt thin\n\flms grown on MgO substrate with face-centered cu-\nbic (fcc), body-centered cubic (bcc), and hexagonal close\npacked (hcp) crystal structures. The fabrication details\nfor the three samples are given in the Supplemental Mate-\nrial [46]. The hcp sample has a strong in-plane magneto-\ncrystalline anisotropy characterized by an easy magne-\ntization axis along the c-direction of the hcp structure\n(which lies in the \flm plane in our \flms), and a hard axis\northogonal to it. For the two cubic crystal structures, the\nanisotropy is still in-plane, but its strength is much re-\nduced, and a hard magnetization direction is not clearly\nidenti\fed [46]. The hcp and fcc samples were grown un-\nder similar deposition conditions and are respectively 10\nnm and 15 nm in thickness, while the bcc sample wasarXiv:2109.03076v2  [cond-mat.mes-hall]  11 Sep 20212\n(c)\nMHTHz\ny\nzx\n𝜃\"#$𝜃%\n(a)(b)\n𝜃\nFIG. 1. (a) Geometry of THz pump-MOKE probe setup. (b)\nFrequency spectrum of terahertz pump pulse (c) Magnetiza-\ntion loops for fcc, bcc, and hcp cobalt measured using the\nlongitudinal MOKE.\ngrown in a di\u000berent laboratory and it has a thickness of\n8 nm. Fig. 1(a) shows the geometry of the single cycle\nTHz pump - optical probe experiment. The magnetiza-\ntionMof the sample is aligned along the x-direction by\nmeans of an external bias \feld jHextj= 100 mT, kept\nconstant during the experiment. The single-cycle THz\npump pulses are generated in the organic crystal OH1\nby the optical recti\fcation of 1300 nm radiation from an\noptical parametric ampli\fer [47]. The pump pulse has a\npeak magnetic \feld of 0.3 T parallel to the y-direction,\nwhich maximizes the torque on the magnetization, and\nimpinges on the sample at an angle of incidence \u0012inc= 45\ndegrees. Fig. 1(b) is the Fourier transform of the electro-\noptical sampling measurement in a a 50 \u0016m-thick GaP\ncrystal [48], used to characterize the THz pulse. It shows\nthat the pump \feld is peaked at around 2 THz, and has\na bandwidth exceeding 1 THz. The magnetization dy-\nnamics is probed using the time-resolved magneto-optical\nKerr e\u000bect (MOKE), in the speci\fc measuring the Kerr\nrotation angle \u0012Kof a nominally 40 fs, 800 nm probe\nbeam, using a balanced detection scheme. All radiation\nis derived from the same ampli\fed laser system ensuring\nintrinsic synchronization, with the relative delay between\nthe beams controlled by a mechanical translation stage,\nand with the pump modulated at a frequency equal to\nhalf the laser repetition rate. Fig. 1(c) shows the easy\naxis magnetization loops for the three samples investi-\ngated in this work. The coercive \feld for the hcp sample\nis about 50 mT, whereas is approximately 30 mT for both\nfcc and bcc samples [46].\nFig. 2 shows the time-resolved MOKE measurements\nof the terahertz-\feld induced dynamics in all three sam-\nples. The plotted traces represent the di\u000berence of the\ndata recorded with magnetic \felds of equal magnitude\nbut opposite polarity, ensuring the magnetic character of\nFIG. 2. Solid symbols: time-resolved Kerr rotation measure-\nments on fcc, bcc and hcp cobalt thin \flms. Dashed line: inte-\ngral of the pump THz magnetic \feld HTHz. Inset: zoomed-in\nmain panel data for t>1:7 ps. The data is shifted vertically\nfor clarity. The continuous lines the best \fts obtained using\nEq. (1).\nthe signal [49]. For all samples, the MOKE response is\ndominated by the coherent precession of the magnetiza-\ntion around the applied THz magnetic \feld HTHz, which\nin fcc and bcc \flms is larger in amplitude than in the\nhcp one. A very small demagnetization, showing up as a\nlingering non-zero average MOKE signal, is also present.\nThe presence of both coherent (precession) and incoher-\nent (demagnetization) e\u000bects in the observed THz-driven\ndynamics is consistent with Ref. [50], where it was also\nobserved that in epitaxial \flms the demagnetization sig-\nnal was negligible. Similar to Refs. [50] and [24], no\ncoherent precession is observed when HTHzkM, since\nin that case the torque acting on the magnetization is\nzero. We present this measurement in the Supplemental\nMaterial [46].\nThe dashed grey line in the same plot is the inte-\ngral ofHTHzover time, obtained numerically from the\nelectro-optic sampling measurement used to character-\nize the THz \feld. Ref. [50] demonstrated that the co-\nherent response of the magnetization to an o\u000b-resonant\nTHz \feld can be obtained simply by integrating HTHz,\nwhich is the solution of the LLG equation for small and\no\u000b-resonant excitations. However, while in Ref. [50] the\ntemporal overlap between the MOKE data and the in-\ntegral ofHTHzwas exact within the experimental error,\nhere we notice a substantial lag between them, approxi-\nmately 200 fs for the fcc and bcc samples, and 400 fs for\nthe hcp one, highlighted by the vertical lines. In other\nwords, the shape of the MOKE response is still consistent\nwith the integral of the THz \feld if properly scaled, how-3\n(b)\n(a)\nFIG. 3. (a) Symbols: time-resolved Kerr signal at t >\n1:7 ps for THz magnetic \feld values of di\u000berent maximum\namplitude. The data is vertically shifted for clarity. Solid\nlines: best \ft obtained using Eq. (1). (b) Symbols: extracted\noscillation amplitude Bas a function of THz magnetic \feld\nand corresponding standard deviation. Dashed line: linear \ft\nto the data with imposed zero o\u000bset.\never its phase is not. This phase shift is particularly dra-\nmatic for the hcp sample, where it looks as if the magne-\ntization precesses in the opposite direction as compared\nto the fcc and bcc samples. We repeated the experiment\non all samples, and the evidence is robust. The sign\nof the magneto-optical coe\u000ecient does not change either\nbetween the di\u000berent samples, as demonstrated by the\nmagneto-optical hysteresis loops in Fig. 1(c). Hence, the\nobserved phase shift is real and it appears to be strongly\ndependent on the crystalline structure of the sample.\nAnother intriguing observation from the data in Fig. 2\nis found in the inset, where we zoom in on the main panel\ndata at temporal delays t>1:7 ps. When the pump \feld\nhas left the sample, a comparatively tiny, yet detectable,\ndamped ringing of the magnetization can be observed.\nWe can \ft such behavior with the phenomenological for-\nmula\n\u0001\u0012K(t) =Ae\u0000t=\u001c1+Be\u0000t=\u001c2sin(2\u0019ft) (1)\nwhere\u001c1is the recovery time of the incoherent demagne-\ntization dynamics, \u001c2is the decay time of the sinusoidal\noscillation, fis the frequency of the oscillations, and A,\nBare the constants describing the the amplitude of the\ndemagnetization and, respectively, of the sinusoidal os-\ncillations. The \ft returns ffcc\u00191:3 THz,fbcc\u00191:4\nTHz andfhcp\u00192:1 THz.\u001c2is found to be approxi-\nmately 0.82, 0.70 and 0.72 ps for the fcc, bcc and, re-\nspectively, hcp samples, corresponding to damping coef-\n\fcients\u000b= 1=!\u001c2which are\u000bfcc\u00190:15,\u000bbcc\u00190:16\nand\u000bhcp\u00190:10.\nBefore discussing these results, we present in Fig. 3\nthe THz \feld dependent measurement on the fcc sample,\nwhich showed the largest signal in Fig. 2. The terahertz\n\feld strength is controlled through the relative orienta-\ntion of a pair of wire-grid polarizers in the THz pump\npath. The second polarizer was kept \fxed in order to\npreserve the polarization of the THz \feld impinging on\nthe sample. The time-resolved MOKE signal is shown inFig. 3(a) for the maximum \feld strength, 75% and 50%\nof it, below which we were at the noise level of our setup.\nWe used again Eq. (1) to \ft the oscillations and to ex-\ntract the amplitudes and recovery times as a function of\nTHz \feld strength. Fig. 3(b) shows the extracted oscil-\nlations amplitude Bas a function of THz \feld strength,\nwhich can be \ftted with a linear function with no o\u000bset.\nThe evidence presented so far is consistent with the\npresence of a sizeable magnetic inertia in crystalline\ncobalt \flms, manifesting itself with a lagging response\nto an external \feld and to the appearance of nutation\noscillations. In order to investigate this hypothesis thor-\noughly, we performed numerical simulations using the in-\nertial LLG equation, written in a slightly di\u000berent form\nthan the one given in Ref. [24]\ndM\ndt=\u0000j\rjM\u0002He\u000b+M\u0002\u0012\n\u000bdM\ndt\u0000\u0011d2M\ndt2\u0013\n;(2)\nwherej\rj/2\u0019= 28 GHz/T is the gyromagnetic ratio,\nHe\u000b= (Hbias+HK)x+HTHz(t)y+Hdzis the e\u000bec-\ntive magnetic \feld which comprises of the external bias\n\feldHbias, the anisotropy \feld HK, the applied THz \feld\nHTHz(t), and the demagnetizing \feld Hd;Msis the sat-\nuration magnetization of the sample, \u000bis the Gilbert\ndamping parameter, and \u0011is the angular momentum re-\nlaxation time de\fned as in Ref. [35], i.e. \u0011=\u000b\u001c. Since\n\u000b\u001c1, the absolute values \u0011are much smaller than \u001cre-\nported in Ref. [24]. The last term on the right hand side\nof Eq. (2) is the nutation term that is present only when\n\u00116= 0. In the following, we solve this equation in the\nmacrospin approximation (i.e. the sample is considered\nas a homogeneous ferromagnet) and using a conventional\nfourth-order Runge-Kutta method.\nIn Fig. 4(a)-(c), we compare the results from these nu-\nmerical simulations to the experimental results in fre-\nquency domain. For both simulations and experiments,\nwe Fourier transform the temporal traces obtained at\ntime delays t >1:7 ps, when the THz pump \feld has\nleft the sample. Including the full temporal trace would\nhide the small features below the broad single-cycle re-\nsponse. In the numerical simulations, we calculate He\u000b\nsolely from experimentally measured quantities found in\nthe Supplemental Material [46] or in previous references\n[51]. This allow us to estimate Hd\u00191:6 T for all sam-\nples, andHK\u00190:8 T for the hcp sample and one order of\nmagnitude smaller for the other two samples. We used\nthe nominal values for HbiasandHTHz. The only two\nfree parameters are then \u0011and\u000b, which can be indepen-\ndently tuned to match the peak frequency and, respec-\ntively, linewidth. Using \u0011fcc= 120 fs,\u0011bcc= 110 fs and\n\u0011hcp= 75 fs we can reproduce the main experimental\npeak frequency, and assuming \u000bfcc= 0:15,\u000bbcc= 0:16\nand\u000bhcp= 0:10 from the \fts using Eq. (1), we can also\nmatch the linewidth of the main peak. No observable\ndi\u000berence was found within 5-10 fs for \u0011and within 0.01\nfor\u000b, giving an approximate 10% relative accuracy. We\nhave performed additional simulations (not shown) and\nwe also observe that the e\u000bective \feld does not a\u000bect4\nFIG. 4. Experimental (solid) and simulated (semi-\ntransparent) Fourier transform of the magnetization dynam-\nics in (a) fcc (blue) (b) bcc (orange) and (c) hcp (green) cobalt\nthin \flms. The full experimental trace is used for the exper-\nimental data in Fig. 2, and the Mzcomponent for the simu-\nlations. (d) Solid lines: simulated response of Mzto a single\ncycle terahertz \feld HTHzin the time-domain using the same\nparameters. Dashed line: integral over time of HTHz.\nthe nutation frequency and linewidth in a noticeable way\nunless it reaches values of the order of a few Tesla.\nWe discuss below the plausibility of these values; as-\nsuming for the time being that they are reasonable, and\nlooking at Fig. 4(d), we obtain the remarkable result that\nthe inertial LLG equation is able to reproduce all the ex-\nperimental evidence of Fig. 2: the presence of a damped\nnutation oscillation and the temporal shift of the coher-\nent magnetization precession. In this small amplitude\nlimit, the inertial LLG also predicts a linear scaling of\nthe coherent precession and of the nutation amplitude\nwith terahertz \feld strength, as shown experimentally in\nFig. 3(b). None of these experimental evidences can be\nreproduced solving the standard LLG equation, proving\nthat the additional inertial term is necessary.\nThe only experimental evidence which is not repro-\nduced by the inertial LLG equation, in the currently\nknown form and in the macrospin approximation, is the\npresence of higher order harmonic peaks in the frequency\nresponse seen in Fig. 4(a)-(c). We can clearly identify\nthe second and third harmonics for the fcc and bcc sam-\nples, and the second harmonic for the hcp one. How-\never, Kikuchi et al. [22] predicted such possibility if the\nthird and other higher order time derivatives of the mag-\nnetization, not included in the standard framework of\nthe inertial LLG model, are considered. We leave thisquestion open to future theoretical and experimental in-\nvestigations, here we simply note that the presence of\nharmonics at integer multiples nof the fundamental fre-\nquency could be consistent with nutation dynamics. It is\nnot consistent with the presence of standing waves across\nthe \flm thickness, which show instead a n2dependence\ndue to con\fnement [52]. We also do not observe any\napparent inverse thickness dependence, which is instead\nexpected in the case of standing waves.\nAs a \fnal control to test the general validity of our\nexperimental results and of the inertial LLG equation,\nwe performed additional measurements using a di\u000berent\nTHz single-cycle pump \feld with a bandwidth extending\nfrom 2 to 4 THz instead, i.e. with negligible overlap with\nthe nutation resonances. This is achieved by replacing\nthe nonlinear crystal generating the THz radiation and\nby adjusting the corresponding pump wavelength, leav-\ning the rest of the setup unchanged. The results are re-\nported in the Supplemental Material [46], and they show\nthat neither THz oscillations nor phase shift of the coher-\nent precession is observed when the pump \feld does not\nmatch the nutation resonance. This is also in agreement\nwith previous measurements done in fcc cobalt driven\nby a THz \feld with similar bandwidth [53]. The inertial\nLLG equation with the same parameters reproduces even\nthese experimental data to an excellent degree, with no\nnutation oscillations nor phase shift observed in this case.\nWe now turn the discussion to the two free parame-\nters in the inertial LLG equation, namely the damping\n\u000band the angular relaxation time \u0011. In order to match\nthe experimental linewidth, we used for all three \flms\na damping parameter which is an order of magnitude\nlarger than the typical FMR Gilbert damping of these\nmaterials [51]. While we do not have a microscopic ex-\nplanation for these large values, we notice that the same\nissue was found in the \frst experimental report of nuta-\ntion in ferromagnets and left as an open question [24].\nOur experiments, which are able to observe the natu-\nral nutation oscillations, allow us to extract the damp-\ning factor directly from the data. In order to estimate\nthe magnitude of \u000bfor the inertial dynamics, a micro-\nscopic theoretical investigation is needed, which is be-\nyond the scope of our work. Our experiments suggest\nthough that a complete inertial LLG equation may con-\ntain either distinct Gilbert and inertial damping coe\u000e-\ncients, or a time-dependent one, in order to fully describe\nthe magnetization dynamics. A time-dependent \u000bcan be\nqualitatively linked to a damping mechanism dominated\nby comparatively stronger electron-phonon scattering at\nsub-picosecond time scales, and weaker spin-lattice relax-\nation at longer time scales [18, 54, 55]. A time-dependent\n\u000bhas also been recently suggested to include all damping\nmechanisms, including time-retardation e\u000bects [56].\nThe most important experimental observation of this\nwork, which is expected to contribute to a microscopic\nunderstanding of inertial dynamics, is the strong depen-\ndence of the nutation frequency on the di\u000berent cobalt\nsamples with di\u000berent magneto-crystalline anisotropy,5\nwhich in turn is dependent on the strength of the spin-\norbit coupling. The relativistic theory of magnetic in-\nertia at ultrafast time scales [29] demonstrated that the\npresence of a \fnite angular momentum relaxation time is\ndue to a spin-orbit coupling e\u000bect of order 1 =c4, whereas\nthe Gilbert damping is of order 1 =c2. These two quan-\ntities are therefore dependent on each other, and it was\nsuggested in Ref. [29] that the ratio \u0011=\u000bshould be a con-\nstant. From our data, we calculate \u0011=\u000b = 746\u000646 fs\nfrom the three cobalt \flms, which is constant within the\naccuracy of our estimates of the two parameters ( \u001810%).\nThis reduces the number of free parameters in the iner-\ntial LLG equation to only one, at least within the same\n3delement, and it further strengthens the interpretation\nof our results in terms of relativistic spin dynamics. We\nnote that the magneto-crystalline anisotropy energy is\nabout one order of magnitude larger in hcp cobalt than\nin the two cubic phases, while the nutation frequency\ndi\u000bers by less than a factor of two among them. This\npreliminary observation suggests that a relation of pro-\nportionality may exist between the frequency of nutation\nand the strength of the magneto-crystalline anisotropy,\nand that it is sub-linear. We also note that fcc and bcc\nphases are energetically close; a small change in lattice\nparameter can induce a so-called Bain transformation be-\ntween them [57]. Hence, it is not too surprising that\nalso their magneto-crystalline anisotropy and nutation\nfrequencies are similar. We anticipate that future works\nwill shed light on how to derive the nutation frequency\nfrom \frst principles or from other magnetic properties of\nthe material.In summary, we measured the temporal evolution of\nterahertz-\feld driven spin dynamics in three epitaxial\ncobalt samples with fcc, bcc and hcp crystal structures.\nWe observed the appearance of THz oscillations with\ndistinct frequencies for the three samples and of a de-\nlayed coherent magnetization response, which could be\nnaturally described in the framework of the inertial LLG\nequation assuming a magnetic damping one order of mag-\nnitude larger than the conventional Gilbert damping at\nFMR frequencies. While surprising, this evidence may\nbe consistent with recent theoretical works suggesting a\ntime-dependent damping coe\u000ecient. We could also esti-\nmate a constant ratio between the angular momentum\nrelaxation time and the measured damping, in agree-\nment with the prediction of the full relativistic theory of\nmagnetic inertia. Finally, we could observe higher har-\nmonics of the nutation oscillations, not described by the\ncurrently accepted inertial LLG equation with temporal\nderivatives up to the second order, but possibly consis-\ntent with a higher order extension of the same equation.\nOur work provides the strongest evidence for inertial spin\ndynamics so far, where all the experimental results can\nbe reproduced with a single free parameter. We envisage\nthat our results will trigger future experimental and the-\noretical investigations towards a deeper microscopic un-\nderstanding of magnetic inertia at ultrafast time scales.\nV.U. and S.B. acknowledge support from the European\nResearch Council, Starting Grant 715452 \\MAGNETIC-\nSPEED-LIMIT\". 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Wegrowe,4 D. Byrne5 \n1Laboratoire de Mathématiques et Physique , Université de Perpignan Via Domitia, F-66860, \nPerpignan, France \n2Department of Electronic and Electrical Engine ering, Trinity College, Dublin 2, Ireland \n3Kotel’nikov Institute of Radi o Engineering and Electronics of the Russian Academy of \nSciences, Vvedenskii Square 1, Fr yazino, Moscow Region, 141120, Russia \n4Laboratoire des Solides Irradiés, Ecole Po lytechnique, 91128 Palaiseau Cedex, France \n5School of Physics, University College  Dublin, Belfield, Dublin 4, Ireland \n \nAbstract  \nThermal fluctuations of nanomagnets driven by  spin-polarized currents are treated via the \nLandau-Lifshitz-Gilbert equation generalized to include both the random thermal noise field \nand the Slonczewski spin-transfer torque term. By averaging this stochastic (Langevin) \nequation over its realizations, the explicit infinite hierarchy of differ ential-recurre nce relations \nfor statistical moments (averaged sp herical harmonics) is derived for arbitrary demagnetizing \nfactors and magnetocrystalline anisotropy for th e generic nanopillar model of a spin-torque \ndevice comprising two ferromagnetic strata re presenting the free and fixed layers and a \nnonmagnetic conducting spacer all sandwiched betw een two ohmic contacts. The influence of \nthermal fluctuations and spin-transfer torques on re levant switching characteristics, such as the \nstationary magnetization, the magnetization revers al time, etc., is calculated by solving the \nhierarchy for wide ranges of temperature, damping, external magnetic field, and spin-\npolarized current indicating new spin-torque eff ects in the thermally assisted magnetization \nreversal comprising several orde rs of magnitude. In particul ar, a pronounced dependence of \nthe switching characteristics on the directions of the external magnetic field and the spin polarization exists.   PACS numbers: 75.60.Jk, 75.75.Jn, 75.76.+j   2I. INTRODUCTION \nOne of the most significant developments in the thermally assisted magnetization \nreversal in nanomagnets sin ce the seminal work of Néel1 and Brown2,3 on the reversal time of \nthe magnetization of single-domain ferromagnetic nanoparticles due to thermal fluctuations \nhas been the discovery of the sp in-transfer torque (STT) effect.4-6 The phenomenon exists \nbecause an electric current with spin polariz ation in a small (nanoscale) ferromagnet may \ntransfer spin angular momentum betw een the current and the magnetization M giving rise to \nthe macroscopic spin-torque on M4-7 so that the latter may be altered by spin polarized \ncurrents.8 Such effects underpin the relativ ely new subject of spintronics,9 where the carrier of \ninformation is the spin state of a ferromagnetic material. Typical practical applications include very-high-speed current-induced magnetization sw itching by (a) reversing the orientation of \nmagnetic bits in high density memory structures  as opposed to the more conventional Oersted \nfield switching\n7,10,11 and (b) using spin polarized curre nts both to genera te and manipulate \nsteady state microwave oscillations with a fr equency proportional to the spin-polarization \ncurrent12,13 via the steady state magnetization preces sion. Essentially both objectives (a) and \n(b) can be achieved because, depending on its si gn, the spin current either enhances or \ndiminishes the effective damping representing  the microscopic degrees of freedom of a \nferromagnetic film8 (cf. Eq. (1) below). The meaning of this,8 considering a bi stable potential, \nis that during a precessional pe riod in a well, the average rate of change of energy E may be \neither negative, positive, or indeed zero. If E< 0 the magnetization is forced to relax into its \nenergy minimum in the well. On the other hand, if E is equal to zero, we have stable \nprecession at constant energy as if the G ilbert damping were absent. Finally if E>0, resulting \nin very large precessional orbits at energies  in the vicinity of the saddle energy, the \nmagnetization is ultimately forced to switch to a new stable position in the other well of the \npotential (see Fig. 2 below).  \nRegarding objective (b) above, a simple treatm ent of the onset of stable precessional \nstates at zero temperature has here given in Re f. 6. There, since the damping torque is small \nand roughly balances the STT wh ile ignoring thermal noise (r epresented by a stochastic \nmagnetic field), perturbation theory  is used to investigate the ons et of precessional states. This \nis accomplished by studying those relatively low- energy phase-space trajectories on which the \nmagnetization is close to its stable direc tions. Thus the equati on of motion of the \nmagnetization (cf. Eq. (1) below) may be linear ized, as is usual in the theory of small \noscillations,14 about a stable direction. This leads to situations, where precessional motion \nexponentially decays for currents less than the cr itical value for the onset of precessional states \n[not to be confused with the switching current in objective (a)6] and exponentially grows for \ncurrents exceeding that value. The phenomenon constitutes a parametric excitation because  3the STT is a function of the orientation of M, representing a time varying modification of a \nsystem parameter and thus may exhibit instability unlike conventional resonance.15 Indeed the \noverall behavior is more or less analogous to  that of a triode vacuum tube oscillator16 whereby \na coil connected to the grid circuit is coupled via mutual inductance to a lightly damped \noscillatory circuit in the anode circuit. Then while the triode  is in operation, the resulting \nfeedback effect is either to decrease or incr ease the effective resistance of the oscillatory \nanode circuit according to the sense in which the coil in the grid is c onnected. If the damping \nis decreased and the mutual coupling is sufficien t, the former may be reduced to zero. Thus an \noscillation once started will persist and will gr ow until limited by the characteristics of the \ntube. A discussion of this limiting behavior in  the spin-torque case is given in Ref. 6. \nNow regarding objective (a) because6 the STT represents a parametric excitation with \nE>0 then bifurcation phenomena due to parametric amplification at energies in the vicinity of \nthe barrier energy may manifest themselves wh ereby a qualitatively different solution for a \nnonlinear system may appear follo wing the variation of some para meter. In our context, this \nbehavior represents crossing of  the potential barrier causing th e magnetization to evolve into \nmore highly damped precessional  states exhibiting ringing osci llations which decay rapidly \n(see Fig. 2). Thus the magnetization is driven into its new stati onary state, where precession is \nprevented due to the sign of the STT which now enhances  rather than reduces the damping. \nHowever, unlike the value of the cr itical current characterizing the onset  of precessional states, \noriginating in the small oscilla tions about a stationa ry orientation of the magnetization, no \nclosed form6 for the switching current (at which the di rection of precession reverses) can be \nderived from simple perturbation theo ry. This is so because unlike the small  oscillations, the \nswitching has its origin in the large  oscillations about the direction of precession  \ncharacterizing the motion near the saddle (barrie r) energy between two stable states. Here the \nprecession is almost metastable and so may easily  be reversed in direction following a small \nchange in the energy (see Fi g. 2). An essentially similar argument was used by Kramers.17 He \nutilized the concept of large osci llations of Brownian particles in a well (at energies near the \nseparatrix energy between the bounded moti on in the well and the unbounded motion outside \nit) in discussing noise-induced es cape for very weak coupling to a thermal bath as explained in \nRef. 18. In the STT application, any results that are available ut ilize Melnikov’s method13, 19 \nfor weakly perturbed Hamiltonian systems that  are periodic in time, where the unperturbed \ntrajectories in phase space may be derived from the energyscape. One of the major benefits of \nhis method is that it establishes6 a clear distinction between the critical currents for the onset \nof precessional states and those for switching. \nAll of the foregoing discussion pertains to  zero temperature, where, for example,6 the \nprecessional states are characterized in the si ngle macrospin approximation by a frequency  4that is a function of the curr ent density, the external magne tic field, the anisotropy, the \ndamping parameter, etc. However, the therma l fluctuations cannot be  ignored due to the \nnanometric  size of STT devices, e.g., leading to mainly noise-induc ed switching at currents far \nless than the critical sw itching current without noise20 a phenomenon corroborated by \nexperiments (e.g., Ref. 21) which demonstrate that STT near room temperature alters \nthermally activated switching processes. At fin ite temperatures, randomness due to the thermal \nmotion of the surroundings is in troduced into the magnetization trajectories, counteracting the \ndamping and giving rise to fluctuations as compared to the zero temperature limit. Furthermore, large fluctuations can cause transitions between metastable states\n6 of the \nmagnetization at currents less than the zero temperature current essentially in the manner \nenvisaged by Kramers.17,18 Here we study the effects of thermal fluctuations in the presence of \nSTT using an adaptation of the Langevin equa tion for the evolution of a single macrospin \nproposed by Brown.2,3 He showed how the noise-induced  magnetization relaxation, i.e., \nreversal of the direction of  precession by crossing over a potential barrier between two \nequilibrium states could be set firmly within the context of the theory of stochastic processes. \nHowever, it should be recalled throughout that unlike in the original  work of Brown and \nNéel1-3 STT devices, due to the injection of the spin -polarized current, inva riably represent an \nopen system in an out-of-equilibrium steady state.  Such behavior is in marked contrast to the \nconventional steady state of nanostructures characterized by the Boltzmann equilibrium \ndistribution that arises when the STTs are omitt ed. Now, the effect of thermal fluctuations, \nusing a modification of the customary Néel-Brown model,1-3 treated here represent a very \nsignificant feature of their operati on. Fluctuations, for example, lead to mainly noise-induced \nswitching at currents far less th an the critical current in th e absence of noi se as well as \nintroducing randomness into the precessional orb its which now exhibit energy-controlled \ndiffusion. Thus the effect of the noise is ge nerally to reduce the current-induced switching \ntime.20 To facilitate our discussi on we first describe the archetypal schematic model of the \nSTT effect.  \nII. MODEL \nThe archetypal model (Fig. 1) of a STT de vice is a nanostructure consisting of two \nmagnetic strata labeled the free and fixed layers, respectively, and a nonmagnetic conducting \nspacer all sandwiched on a pillar between two ohmic contacts.6,13 The fixed and free layers \ndiffer significantly because the fixed layer is pinned12 along its orientation much more \nstrongly than the free one usually because the form er is of a harder magnetic material so that \nthe latter is much easier to manipulate in a magn etic sense. When an el ectric current is passed \nthrough the fixed layer, it become s polarized. The polarized spin current then encounters the \nfree layer and induces a spin torque altering its magnetization so permitting8 a variety of  5dynamical regimes. This phenomenon can lead, in particular, to two distinct magnetization \ndynamics regimes which have been ex tensively verified experimentally,20 viz., steady-state \nprecession and STT-induced reversal of the dire ction of precession governed by the transition \nrate between precessional states . Consequently, one can introdu ce two distinct time scales \nassociated with the magnetization vector M, namely, a slow one, corresponding to reversal of \nthe magnetization over a potential barrier and a fast one, characterized by the precession \nfrequency of the bounded motion in a pot ential well for constant energy.  \nNow in the model under consideration both ferromagnetic layers are assumed to be \nuniformly magnetized (for small ferromagnets the STT may lead to a rotation of the magnetization as a whole, rather than to an ex citation of spin waves; even though the single-\ndomain or coherent rotation approximati on cannot explain all observations of the \nmagnetization dynamics in spin-torque systems, many qualitative features needed to explain \nexperimental data are satisfactorily reproduced\n6,13). Thus in the presence of thermal \nfluctuations, the current-induced magnetization dynamics of M in the free layer is governed \nby the Slonczewski equation (i.e., the Landau-Lifshitz-Gilbert equation3 including the spin-\ntorque)13 augmented by a random magnetic field h with Gaussian white noise properties and \nso becoming a Langevin equation4,6,13,22 \n  ef ST     uu H u h u u . (1) \nThe Gaussian field h has the usual white noise properties  \n \nX e   u Z\nY M  \n \neasy axisH0 \nfixed layerfree layer Jeep \n \nFIG. 1. Geometry of the problem. A STT device consists of two ferromagnetic strata labelled \nthe free and fixed layers, respectively, a nd a normal conducting spacer all sandwiched on a \npillar between two ohmic contacts.13 The fixed layer has a fixed magnetization along the \ndirection Pe. eJ is the spin-polari zed current density, M is the magnetization of the free \nlayer, and 0H is the applied magnetic field.  6(a) (b) (c) \nFIG. 2. (a) Biaxia l anisotropy potential (,)V , Eq. (3). (b) Current-induced trajectory of the \nmagnetization escape. Solid line: numerical soluti on of the deterministic Eq. (1), i.e., omitting \nthe random field h, for a strong spin-polarized current, damping 0.01 , and typical values \nof other model parameters (see Sec. V). Reve rsal of the magnetization from one metastable \nstate to another typi cally occurs after ma ny precessions about the X axis, over saddle points. \nThus having traversed the potential barrier, th e magnetization decays to a new stable direction \nof precession so that in this reverse directi on the current accelerates the decay by increasing \nthe effective damping, as obvious from Eq. (1). (c ) The same as in Fig. 2(b) for a weak spin-\npolarized current and the same  values of other model para meters. Here, only the damped \nprecessions of the magnetization ab out the stable direction exist. \n 12 , 1 2\n0S2() 0 , ( ) ( ) .ii j i jkTht hth t t tvM   (2) \nwhere the indices ,1 , 2 , 3ij  in Kronecker’s delta ,ij and ih correspond to the Cartesian \naxes X,Y,Z of the laboratory coordinate system OXYZ , and ()t is the Dirac-delta function. \nThe overbar means the statistical average over an  ensemble of moments which all have at time \nt the same sharp value of the magnetization M, the sharp values subsequently being regarded \nas random variables. In Eq. (1), 1\nSMuM  is a unit vector in the direction of M, SM is the \nsaturation magnetization,  is the gyromagnetic-type constant,  is a damping parameter \nrepresenting the effect of all th e microscopic degrees of freedom, STu is the STT, \n1\nef 0 S () /MV  Hu  is the effective magnetic field co mprising the anisotropy and external \nfields, and the operator /u indicates the gradient on the su rface of the unit sphere. Here V, \nconstituting a conservative potentia l, is the normalized free energy per unit volume of the free \nlayer which we write in the standard form of superimposed easy-plane and in-plane easy-axis \nanisotropies plus the Zeeman term, viz.,13,22 \n \n22 2\n02\nS\n12 3( , ) cos sin cos\n2c o s s i n s i n s i n c o shM VD     \n    \n   (3) \n 7as shown in Fig. 2a. Thus the potential create s an energyscape with two minima and two \nsaddle points and forces the magnetization to ali gn in a given direction in one or other of the \nenergy minima in the equatorial or XY plane. Here the Z axis is obviously taken as the hard \naxis while the X axis is the easy one;  and  are angular coordinates describing the \norientation of the moment u in the spherical polar coordinate system, 1cos sin   , \n2sin sin   , and 3cos  are the direction cosines of the applied field 0H, \n0S/( 2 ) hH M D is the external field parameter, and /DD is the biaxiality parameter, \nwhere D and D account for both demagnetizing a nd magnetocrystal line anisotropy \neffects.13 The spin-transfer torque term STu is defined as \nST\n0S1\nM uuu , \nwhere  is the non-conservative pote ntial due to the spin-polar ized current given by [4] \n  2\n0Sln 1Pe\nPP\nPpMbJccJ uu e . (4) \nIn Eq. (4) the unit vector Pe identifies the magnetization direct ion in the fixed layer, cf. Fig.1, \neJ is the current density, taken as  positive when the electrons flow from the free into the fixed \nlayer, while 2\n0  /pSJM e d  (e is the electronic charge,  is Planck’s reduced constant, \nand d is the thickness of the free layer). The coefficients Pb and Pc are model dependent and \nare determined by the sp in-polarization factor  01PP4 \n 3/2\n33 / 24\n3(1 ) 16PPbP P,  \n 3\n33 / 2(1 )\n3(1 ) 16PPcP P \nwhere 01 / 2Pb  and 1/3 1Pc  as P increases from 0 to 1. The typical value of pJ for \na 3 nm- thick layer of cobalt, where 6\nS1.4 10 MA m1, is 91.1 10pJA/cm2 (cf. Ref. 13, \np. 237).  \nIn tandem with Eq. (1), we have13 (see Sec. 1.17 of Ref.  23) the Fokker-Planck \nequation (FPE) for the probability density function (,, )Wt  of orientations of u on the unit \nsphere, viz.,  8 \n 11\nN\n11\n2112s i nsin sin\n1sin .sinVV WW vWkT\nvWkTt\nVV W\n    \n  \n\n\n    \n\n              \n        \n\n  (5) \nHere 0 N1\nS() / ( 2 ) vM k T   is the free diffusion time of the magnetic moment, kT is \nthe thermal energy, and v is the free layer volume.  \nBy way of background to the Langevin equation method, during the last decade, \nvarious analytical and numerical approaches to the calculation of the measurable parameters \nfrom the Langevin and Fokker-Planck equations (1 ) and (5) have been ex tensively developed, \ne.g., generalizations (e.g., Refs. 24-26)  of the Kramers escape rate theory17 and stochastic \ndynamics simulations (e.g., Refs. 24, 27-29). For example, the pronounced time separation \nbetween fast precessional and slow energy changes in lightly damped ( <<1) closed phase \nspace trajectories (called Stoner-Wo hlfarth orbits) at energies near  the barrier energy has been \nexploited in Ref. 12 to formulate a FPE for the en ergy distribution essentially similar to that of \nKramers17 for point particles. Regarding the magnetization reversal time it will become \napparent that the effect of the spin-polarized current may be of several orders of magnitude in \nthe very-low-damping limit ( 1, the only relevant case) since the stationary distribution of \norientations of M is no longer the Boltzmann one as it now depends both on the spin \npolarized current eJ and on . The dependence is all the more obvious when one considers, \njust as in Ref. 12, the stationary soluti on of the Fokker-Planck equation for the axially \nsymmetric V and  which arises for uniaxial anis otropy with the easy axis, the \nmagnetization direction in the fixe d layer, and the external field taken as collinear. In this \nspecial case alone the magnetization dynamics are determined by a simple generalized \npotential yielding the stationary distribution in exact closed form as well as an approximate \nexpression for the reversal time. The effective potential 1V comprises that of the \nconservative external and anisotropy fields as well as the nonconservative  one due to the spin-\npolarized current. In general, the existe nce of a nonconservative  effective potential12,13,29 \nallows one to define a current-dependent potenti al barrier between stationary self-oscillatory \nstates (limit cycles) of the magnetization and to  estimate transition rates between these states. \nNow for axial symmetry the approximate solu tion procedure for the smallest nonvanishing \neigenvalue 1 of the Fokker-Planck equation for axially symmetric potentials and for high \nbarrier heights given by Brown2,3 may be used. Here the asymptotic calculation of 1, thus the \nmagnetization reversal time 11/ , may be effected2,3 via the purely mathematical method  9of approximate minimization (stemming from the calculus of variations) for 1 of the axially \nsymmetric Fokker-Planck equation when c onverted to a Sturm-Liouville equation. \nHowever, for nonaxially symmetric problems, as depicted in Fig. 2, no such simple \nasymptotic solution exists because it is impossi ble, by inspection of the nonaxially symmetric \nFokker-Planck Eq. (5) when the STT is included, to derive a simple analytic equation for an \neffective potential. [Such a potential can be calculated only in numerically via the time-\nindependent distribution function;  see Eq. (27) below.] Neverthe less, it is still possible to \ncalculate 1 in numerical fashion, once again yi elding the magnetization reversal time. \nMoreover, it is also possible to calculate the time-independent in -plane component of M, i.e., \nin the X or easy-axis direction 0Xu  as well as the corresponding magnetic susceptibility \n2\n0 02\nXXuu , where the angular brackets 0 mean stationary statistical averaging. The \nmerit of such calculations is that inter alia  they allow one to accura tely assess approximate \nlow-damping solutions for the reversal time based on energy-controlled diffusion.13,29 These \nsolutions all more or less rest on (noting the separation of time scales referred to above) \ntreating both the effects of the stochastic torque s due to the heat bath and the spin torque as \nperturbations of the pr ecessional dynamics of M in the wells of the anisotropy-Zeeman \nenergy potential. The corresponding clos ed phase-space traj ectories are known8,20 as Stoner-\nWohlfarth orbits and steady precession along such an orbit of constant energy, belonging to a \nsphere of radius equal to the saturation magnetization, occurs if  the spin-torque cancels out the \ndissipative torque [cf. Eq. (1)]. The origin of th e orbits of course arises from the two well \nstructure of the anisotropy potential. One should  at this juncture me ntion the treatment of \nreversal of the magnetisation by Apalkov and Visscher.25 Here, for example, the time \nseparation between the fast precessional and sl ow energy change in a lightly damped Stoner-\nWohlfarth orbit at energies near to the barr ier energy is used to formulate a Fokker-Planck \nequation for the probability distribution of the en ergy near the barrier. This method is again \nessentially similar to the approach used by Kramers17,18 in the problem of the very-low-\ndamping noise-activated escape rate from a pot ential well. Moreover, the derivation of the \nFokker-Planck equation in energy-phase variable s for point particles with separable and \nadditive Hamiltonians has been ex tensively discussed by Stratonovich30 and Risken.31 The \ncorresponding magnetic problem (where the Hamilt onian in the absence of spin torque is non-\nseparable) in energy-precession va riables has been discussed by Dunn et al.8 in relation to \ntheir Eq. (1.22), which, on assumi ng rapid equilibration of the pre cession variable, leads to an \nenergy diffusion equation, their Eq. (1.23).  10In this paper, we shall present results for the stationary magnetization 0Xu  and static \nsusceptibility 2\n0 02\nXXuu  in the easy-axis direction and for the reversal time of the \nmagnetization. These results are obtained by extending the general method (based on the \nrelevant Langevin equation) of constructing recu rrence relations for the e volution of statistical \nmoments for arbitrary n onaxially symmetric free energy gi ven in Refs. 23 and 32 to include \nthe STT term and then specializing them to Eq. (3). The recurrence relations can then be solved by matrix continued fraction me thods just as with the zero STT term.\n23 The answers \nwill then constitute benchmarks for approximate  solutions obtained by other methods. Indeed \nthe procedure is entirely analogous to that involving the numerical solutions3 used to test \nasymptotic solutions, based on the Kramers escape rate theory,3,17 for the reversal time in the \nNéel-Brown model when the STT is absent. Notice that if the STT is included its effect may also be described via a modifi cation of the energy barrier in the Néel-Brown model (for \ndetailed discussions see Refs. 13, 24-26, and 33). \nIII. DIFFERENTIAL-RECURRENCE RELATION FOR THE STATISTICAL MOMENTS  \nOur method23,32 is based on first averaging th e appropriate Langevin equation \nincluding the spin-torque term (re garded as a Stratonovitch stocha stic differential equation) for \nthe magnetization evolution, therefore written in terms of the spherical harmonics, over its \nrealizations in the representati on space of polar angles  in an infinitesimally small time starting \nfrom a set of sharp angles, which subsequently  are also regarded as random variables. The \nresult is then averaged over the distribution of these angles ultimately yielding the desired \nrecurrence relation for the observables which are the statistical moments of the system. Here \nthe relevant Langevin equation is the Landau-Lifshitz-Gilbert equation for the evolution of M \nin the free layer as modified by Slonczewski4,13 to include the STT, Eq. (1). As shown in \nAppendix A, Eq. (1) can be wr itten in an equivalent Landa u-Lifshitz form, where the \nprecessional and alignment terms are now clearly delineated, viz. \n \nN0S\n1\n0S\nN2\n.2vVMkT\nvVMkT\n            \n               uu hu\nuu hu\n (6) \nIn the spherical polar coordina te basis shown in Fig. 1, th e vector Langevin equation (6)\nrepresents two coupled nonlin ear stochastic differential equations for the angles  and , \nviz.,23  11 \n11 0S\nNN\n1() () () [ () , () ,] [ () , () ,]22\n[( ) ,( ) , ] [( ) ,( ) , ] ,sin ( )vM vt= h t h t V t t t t t tkT kT\nVt t t t t tt      \n  \n    \n    \n (7) \n \n1\n1 0S\n2\nNN\n1() () 1( ) [( ) ,( ) , ] [( ) ,( ) , ]2 sin ( ) 2 sin ( )\n[( ) ,( ) , ] [( ) ,( ) , ] ,sin ( )ht ht vM vt= V t t t t t tkT t kT t\nVt t t t t tt        \n  \n\n    \n    \n (8) \nwhere the components ()ht and ()ht of the Gaussian random field ()th in the spherical \nbasis are expressed in terms of the components ()Xht , ()Yht , ()Zht  in the Cartesian basis as23 \n () () c o s () c o s () () c o s () s i n () () s i n ()XY Z ht h t t t ht t t ht t     , \n () () s i n () () c o s () .XY ht =h t t ht t      \nEquations (7) and (8) then yield the desire d Langevin equation for the evolution of the \nspherical harmonics ,,lmY34 comprising the orthonormal basis set from which the \nobservables are ultimat ely obtained, viz., \n \n \n ,, ,\n1\n,, 1 0S\nN\n, 11\nN\n, 11\n2() ()() ()2s i n\n1\n2s i n\n11,sin sinlm lm lm\nlm lm\nlm\nlmdY Y Y dd\ndt dt dt\nht ht YY vMht htkT\nY vVVkT\nYVV\n\n\n   \n   \n  \n\n\n \n     \n       \n          (9) \nwhere ,(, )lmY are defined by34 \n ,(2 1)( )!(, ) ( c o s ) ,4( ) !im m\nlm lll mYe Plm  \n *\n,, (1 )m\nlm l mYY , \n()m\nlPx are the associated Legendre functions,34 and the asterisk denotes the complex \nconjugate. \n By averaging this Langevin equation (9) as explained in Sec. 9.2 of Ref. 23 and \nsummarized in Appendix B, we have the ev olution equation of the statistical moments \n,()lmYt  (expected values of the spherical harmonics ,lmY34) for arbitrary anisotropy rendered \nas the differential-recurrence relation viz.,  12 N, , , , , ,() () .l m l m lm lm lmdYt e Y tdt    (10) \nIn Eq. (10) angular brackets m ean statistical averaging and ,, ,lml me are expressed via the \nClebsch–Gordan coefficients ,\n,, ,LM\nlml mC (Ref. 34) as  \n \n  \n\n\n,,, , , , 0\n, ,0 ,\n,, 0 , , 0 , , ,\n,,\n,0\n,0, ,0\n,\n2( 1) 1 (2 1)(2 1)124\n11 1\n22 1\n(2 1)( )!\n() !\n!\n!rsm\nlml m s ll s\nrs rr s\nrs l l l m l m s\nrs\nrs rs\nr\nL\nll\nLs\nLll l le\nll r r l l BAC C\nr\nir r sABrs\nLsCLs \n\n\n\n\n  \n\n\n\n\n\n    \n      \n\n\n\n  \n1\n,, 1\n,, , , 1 , ,1,() 1Ls Ls\nlml m s lm l m slm lmmC s CLsLs\n            \n (11) \nwhere 0s and ,rsA and ,rsB are, respectively, the coefficients of the Fourier series \nexpansions in terms of spherical harmonics  of the (conservative) free energy density V and \nthe nonconservative potential , viz., \n ,, ,,rs rs rsvVAYkT  (12) \n ,, ,.rs rs rsvBYkT  (13) \nEquation (10) represents the e volution of a typical entry in a set of differential-recurrence \nrelations with ,, ,lm l me given by Eq. (11). Only the Fourie r expansions of both potentials in \nterms of ,rsA and ,rsB are needed. In the stationary  state, when the statistical moments are \nindependent of time, Eq. (10) becomes \n ,, , , , 00l m lm lm lseY    , (14) \nThe same result may be obtained, albeit with more labor and in a less transparent manner, \nfrom the Fokker-Planck equation, Eq. (5), by se eking the surface density of magnetic moment \norientations on the unit sphere as3,23,32 \n   ,\n0,*,, ( )l\nlm\nllm\nmlWt Y Y t  \n\n , (15) \nwhere \n   ,0,2\n0() , , s i nlm lmYt Y W t d d    (16) \nby the orthogonality property of the spherical harmonics, viz.,  \n \n11 2 2 1 2 122\n*\n,, , ,\n00(,) (,) s i n .lm l m ll mmYY d d\n         13Equations (10) and (14) have been obtained under th e assumption that the damping parameter \n is independent of M. However, the results may be also generalized to magnetization-\ndependent damping ()M.8,20 \nBoth Eq. (10) and Eq. (14) are valid for an arbitrary  free energy. Here we specialize \nthem to the particular free energy given by Eq. (3). In terms of spherical harmonics, Eq. (3) \ncan be written as \n 2\n,,\n12(,)3r\nrs rs\nrs rvVAYkT \n   , (17) \nwhere the nonzero expansion coefficients ,rsA are given by \n 1,0 3 43Ah ,  \n 1, 1 1 28()3Ah i   , \n 204(1 2 )45A ,  \n 2, 22\n15A , \nand 2\n0S /( ) vM D k T  is an anisotropy (or inverse temperature) parameter. Now, the \npotential , Eq. (4), is in spherical harmonic notation \n 1\n*\n1, 1,\n14ln 1 ( , ) ( , )3PP\nmm P P\nm Pvb cJY YkT c  \n  , (18) \nwhere P and P are the spherical polar coor dinates of the unit vector pe which is the \nmagnetization direction of the fixed layer, and 2\n0S /( )ep Jv M J k T J  is the dimensionless \nspin-polarized current parameter. Next retain ing only the two leading terms in the Taylor \nseries expansion  \n23ln 1 / 2 / 3 ... xx x x   , \nwhich converges fairly well for typical values of the model parameters, we will then have  \n 2\n,,\n0(,)r\nap\nrs rs\nrs rvBYkT\n  , (19) \nwhere the expansion coefficients ,rsB are defined as  \n 3/2\n*2 * *\n0,0 1,0 1,1 1, 14(,)2(, ) (,)9PP\nPP PP PPbcJBY Y Y      ,  14(a) (b) \nFIG. 3. 3D plot of th e nonconservative potentials , , Eq. (18), (a) and ,ap , Eq. \n(19), (b) for typical values of the model parameters ( 0.3P , 6J, 0P, and /2P ).  \n *\n1, 1,4(, )3p\nmm P PbJBY  ,   m=0,1 \n 3/2\n*2 * *\n2,0 1,0 1,1 1, 18(, ) (, ) (, )\n95PP\nPP PP PPbcJBY Y Y     , \n 3/2\n**\n2, 1 1,0 1, 18(, ) (, )\n31 5PP\nPP PPbcJBY Y  , \n 3/2\n*2\n2, 2 1, 18(, )\n33 0PP\nPPbcJBY . \nWe remark that the approximation Eq. (19) accurately reproduces all features of the \nnonconservative potential , Eq. (18) (see Fig. 3) because 2/3 0 . 1 5PPcue  for P  0.4 (\n0.3 0.4P  are typical values for ferromagnetic metals13) and all , , P, and P. \nMoreover, in the calculation of the statistical moments, relaxation time, etc., the \napproximation Eq. (19) yields an accuracy be tter than 5% in the majority of cases.  \nIV. CALCULATION OF OBSERVABLES \nThe general time-dependent Eq. (10) and the time independent Eq. (14), as specialized \nto Eqs. (17) and (19), now yields the 25-term differential-recurrence rela tion for the statistical \nmoments ,()lmYt  governing the dynamics of the magne tization [see Eq. (42) of Appendix \nC]. Equations (10) and (14) may be solved by extending the general matrix continued fraction \nmethods developed in Refs. 23 and 32 to include  the STT. Indeed, we can always transform \nthe moment system, Eq. (10) constituting a multiterm scalar  differential-recurrence relation, \ngoverning the magnetization relaxation into the tridiagonal vector differential-recurrence \nrelation \n N1 1() () () ()nn n n n n ntt tt\n  CQ C Q CQ C. (20) \n 15Here ()ntC  are the column vectors arranged in an appropriate way from the entries ,()lmYt  \ngiven by Eq. (11) and ,nnQQ  are matrices formed from ,, ,.lml me  The explicit equations for \n()ntC  and ,nnQQ  for the free energy Eq. (3) are given ex plicitly in Appendi x C. As shown in \nRef. 23, Chap. 2, the exact matrix continued fraction solution of Eq. (20) for the Laplace \ntransform of 1()tC  is given by \n 1N 1 1 1\n2 2() () ( 0 ) () ( 0 ) ,n\nkk n\nn kss s\n\n\n      C Δ C ΔQC  (21) \nwhere 110() ()stst e d tCC, ()nsΔ  is the matrix continued fr action defined by the recurrence \nrelation \n 1\nN1 1 () () ,nn n n nss s\n    IQ Q Q ΔΔ  (22) \nand I is the unit matrix. Having determined the column vectors 1() ,sC\n2() ,sC … as described \nin Refs. 23 and 32, we then have the relevant observables. In a similar way, we also have the \nsmallest nonvanishing eigenvalue (yielding th e reversal time) from the matrix equation3,23 \n NIS  (23) \nwhere the matrix S is defined via the matrix  continued fractions as \n 1\n11 2 2 1 2 2 (0) (0)     SQ Q Q ΔΔ IQ Q  (24) \nand the prime designates the derivative of 2()sΔ  with respect to Ns (see Ref. 23, Chap. 2, \nSec. 2.11.2). Thus 1 is the smallest nonvanishing eigenvalue of S. \nNow in order to calculate the stationary  characteristics (distribution function, \nmagnetization, etc.) we may replace ,()lmYt  in Eq. (42) of Appendix C by ,0lmY  and set the \ntime derivative equal to zero. Then by exte nding the general matrix continued fraction \nmethods developed in Refs. 23 and 32 to include the STT, we have \n 2,20\n2,2 10\n2, 20\n11 1 1\n21 , 210\n21 , 220\n21 , 2101(0) (0) (0) , ( 1,2,....).\n4nn\nnn\nnn\nnn n n\nnn\nnn\nnnY\nY\nY\nn\nY\nY\nY\n\n \n\n \n \n\n\n\n\n \n\nΔ QΔ QΔ Q\n\n\n (25)  16The out-of-equilibrium  or time-independent stationary distribution 0, W is thus rendered \nvia the Fourier series \n  \n00,0*\n, ,,l\nlm\nlm llm WY Y \n\n . (26) \nNext, by analogy with the Boltzmann distribution, since we expect  the stationary distribution \nto be formally similar to it,13 we may define the effective potential efV via \n 0 ,l n,efVW . (27) \nFurthermore, having determined ,0lmY , we have both the stati onary magnetization in the X \ndirection 0Xu  and the corresponding susceptibility 2\n0 02\nXXuu , viz., \n  1, 1 1,1 00 002sin cos3XuY Y \n  , (28) \n \n 22 22\n00 00\n2\n2 , 2 2 ,2 2 , 0 1 ,1 1 , 102\n00 0 0sin cos sin cos\n24 2 1.15 45 3 3XXuu\nYY Y Y Y   \n \n     (29) \nWe remark that in some ranges of the m odel parameters, e.g., for very low damping \n<0.001, and/or very high potential barriers, V>100, the continued fraction method may \nnot be applicable18,31 because the matrices involved become ill-conditioned, meaning that \nnumerical inversions are no longer possible. \nV.RESULTS \nThroughout the calculations the anisotropy an d spin-polarization parameters will be \ntaken as 0.034 D , 20 , and 0.3 P  just as in Ref. 13. Moreover, the applied field 0H \nand the unit vector Pe identifying the magnetization direction in the fixed layer are taken to lie \nin the equatorial or XY plane, i.e., /2P  and /2 . Thus the orientations of 0H and \nPe in the XY plane are entirely determined by the azimuthal angles  and P, respectively. \nThe values 0P  correspond to the particular  configuration whereby both 0H and Pe \nare directed along the easy ( X-)axis. For 52.2 1011m A s, 300T K, 24~1 0v 3m, \n6\nS1.4 10 M1A m (cobalt), 710eJ2A cm, 910pJ2A cm, and 0.02 , we have the \nfollowing estimates for the principal model parameters \n 5.9 J , 20.2 , 8\nN4.8 10 s.  17Moreover, instead of the free diffusion time N, it will be more convenient to use as the \nnormalizing time 11\n0S N/[ ( ) ] ( 2 ) MD     . The above numerical values yield \n11\n04.8 10 s. \nOnce we have determined the time-i ndependent stationa ry distribution 0, W via \nthe Fourier series, Eq. (26), we  have the effective potential  efV from Eq. (27). A typical \nexample of such calculations is  shown in Figs. 4 and 5. The effective potential comprises a \ndouble-well structure with non-equi valent wells. This energyscape (Fig. 4), as expected on \nintuitive grounds, strongly depends on damping,  external magnetic field magnitude and \norientation, and spin-polarized current. In pa rticular, by varying the magnitude of the spin-\npolarized current, damping, etc., one may alter s ubstantially the effective barriers and thus the \nreversal time (cf. Fig. 5). The stationary averages 0Xu  and 2 2\n0 0XXuu  are calculated \nfrom Eqs. (28) and (29), respectively. In Fi g. 6, we show the consequent dependence of 0Xu  \nand 2 2\n0 0XXuu  on the spin-polarized current via a family  of curves with the spin current as \nthe independent variable, for various values of the damping, external field magnitude ( h) and \norientation (), and the magnetization direction in the fixed layer (P), which are all \nregarded as parameters. In contrast, in Fig. 7 we illustrate the dependence of the magnetisation \nand the susceptibility on the external field parameter h via a family of curves for various \nvalues of the current J, the external field orientation and magnetization direction in the fixed \nlayer ( and P), and inverse temperature parameter . Clearly the switching current SWJ \n(i.e., the current when 0Xu  changes sign corresponding to re versal of the direction of \nprecession or, equivalently, when 2 2\n0 0XXuu  attains its maximum and subsequently \nvanishes) strongly depend on the model parameters , J, h, , , , and P. In particular, \nas both the damping  and external field parameters h increase the value of SWJ rapidly \nincreases [see Fig. 6(a), 6( b), and 7(a)].  Moreover, SWJ may also vary significantly with both \nthe orientation of the external field and the direction of the magnetization of the fixed layer \n(see Fig. 6c and 6d). The half-width of 2 2\n0 0XXuu  and the onset of the slope in 0Xu  \nlargely depend on the damping parameter  [Fig. 6(a)] and the inve rse temperature parameter \n [Fig. 7(d)]: higher values of  (lower temperatures) and smaller values of  correspond \nto a narrower half-width and a more rapid onset of the slope. \n  18 \nFIG. 4. (a) 3D plot of the effective potential ,efV , Eq. (27), in the vicinity of the minima \nfor 6J, 0.02 , 0.15h , 20 , 20 , 0P, and 0.  \n \n   \n\n4\n3\n2Vef (   = 0.02P =  = 0\n1: J = 6\n2: J = 0\n3: J = 6\n4: J = 12\nh = 0.1   = 20  P = 0.3   = 20\n1\n\n(b)3\n2\n1J = 6\n1:  = 0.2\n2:  = 0.02\n3:  = 0.005 (a)\n012301020J = 6 = p = 0Vef ()\nP = 0.3    = 20    = 20    = 0.02\n1: h = 0\n2: h = 0.1\n3: h = 0.2\n123\n01230102030(c)\nJ = 6\n123\n(d) \nFIG. 5. /2 ,efV  vs the azimuthal angle  for various spin polar ized current parameter J = \n6, 0, 6, 12 and 0.02  (a); for various damping  = 0.2, 0.02, 0.005 and J = 6 (b); for \nvarious external field parameter h = 0.0, 0.1, 0.2 and J = 6 (c) and J = 6 (d) ( 0.02 , \n0.15h , 20 , 20 , 0P, and 0). \n 19     h = 0.15 = 20 = 20,  P = 0.3\n4 3 2 1:  = 0.01\n2:  = 0.02\n3:  = 0.05\n4:  = 0.1uX\nJ1(a)\n      P =  /2P = 0 =  /2 = 0\n1 4 3 2\nJu2\nXuX\n\n \n   4\n321: h = 0.05\n2: h = 0.15\n3: h = 0.25\n4: h = 0.40uX = 20 = 20   = 0.02   P= 0.3\nJ1(b)\n   P =  /2P = 0 =  /2 = 0\n143 2\nJu2\nXuX\n\n \n    \n5 = 0.02   h = 0.1 = 20 = 20   P = 0.3\n4 3 21: = 0\n2: =  /4\n3:  =  /2\n4: = 3 /4\n5: = uX\nJ1(c)\n    P= /2P = 0 =  /2\n5 1 4 3 2\nJu2\nXuX\n\n \n   = 0.02   h = 0.1 = 20 = 20   P = 0.3\n43 2\n1: P = 0\n2: P =  /4\n3: P = 3 /4\n4: P = uX\nJ1(d)\n  P= /2 = 0 =  /2\n1432\nJu2\nXuX\n\n \nFIG. 6. 0Xu  and 2 2\n0 0XXuu  vs the dimensionless current parameter J for various values \nof damping  (a), external field parameter h (b), external field orientation in the free layer  \n(c), and spin-polarization orientation P (d).  20    = 0.02 = 20 = 20   P = 0.3\n4 3 2\n1: J = 0\n2: J = 10\n3: J = 20\n4: J = 30uX\nh1(a)\n   P =  /2P = 0 =  /2  =0\n1 4 3 2\nhu2\nXuX\n\n \n   \n51: = 0\n2: =  /6\n3: =  /4\n4: =  /3\n5: = 4321 = 0.02   \nJ = 10\n = 20\n = 20   \nP = 0.3\nhuX(b)\n  \n5P= /2P = 0 =  /2 u2\nXuX\n4 321  = 0.02   \nJ = 10\n = 20\n = 20   \nP = 0.3\nh  \n   \n5\n1:P = 0\n2:P =  /6\n3:P =  /4\n4:P =  /3\n5:P = 4\n321\nhuX(c)\n   5P =  /2= 0 =  /2 u2\nXuX\n4 3 21 = 0.02   \nJ = 10\n = 20\n = 20   \nP = 0.3\nh  \n    5\n43\n2\n11: = 10\n2: = \n3: = \n4: = 25\n5: = 30\nhuX(d)\n  5P= /2P = 0 =  /2,  = 0u2\nXuX\n43 2 1 = 0.02   \nJ = 10\n = 20   \nP = 0.3\nh  \nFIG. 7. 0Xu and 2 2\n0 0XXuu  vs the external field parameter h for various values of the \ncurrent parameter J (a), the external field orientation  (b), the spin-polarization orientation \nP (c), and the inverse temperature parameter  (d). \n Moreover, the smallest nonvanishing eigenvalue 1 of the Fokker-Planck operator \n(inverse of the reversal time of the di rection of precession, i.e., that of the X component of M) \nmay be evaluated from the s ecular Eq. (23). By calculating 1, we also have the dependence \nof the magnetization reversal time 11/  on the spin-polarized current, anisotropy \nparameters, damping, external field magnitude  and orientation in the free layer, and the \nmagnetization direction in the fixed layer. In ge neral, a pronounced dependence of the reversal \ntime on these model parameters exists . Examples are shown in Figs. 8 11. Figure 8 illustrates \nthe damping dependence of  for various values of the current J while Fig. 9 shows the  21temperature dependence of  for various values of the current ( J) and external field ( h) \nparameters. We have also shown for comparison in these figures the reversal time calculated \nfor 0J via escape rate theory for bi axial anisotropy which is given by35 \n 12\nIHD IHD\n12 1 2().2 ( )( )( )AS S\nASA S  (30) \nwhere ()Az is called the depopulation factor, viz. \n 21/4\n21() e x p l n12 1/4z dAz e \n \n\n  , (31) \nIHD\n2 is the escape rate from the shallowe r well 2 to a deeper well 1 given by  \n 21\nIHD 2 2 2 2 2\n2 1\n01() 1 ( 1 ) 4( 1 )2( ) ( 1)hehhh h hh     \n\n           , (32) \n12() ( )hh   , and the dimensionless actions 1S and 2S are given by  \n 21/2 1/212 2 1\n1,2 3/24( 1 )(1 )(1 ) arctan (1 )(1 ) .(1 ) 2hhSh h h h               (33) \nClearly by altering J, the ensuing variation of  may be as much as  several orders of \nmagnitude for very low damping, 1  (Fig. 8). Furthermore,  may greatly exceed or, on \nthe other hand, be much less than the value pertaining to J = 0. Moreover, the increase or \ndecrease in  is entirely governed by the direction of the current, i.e., by the sign of the \nparameter J as expected.3 The temperature dependence of  can be understood via the \neffective potential ef, V, Eq. (27). Clearly, at high barriers, 5, the temperature \ndependence  of  has the customary Arrhenius behavior /( )~efvV k Te, i.e., exponentially \nincreasing with decreasing temperature. The slope of 1()T markedly depends on J, h, , etc. \nbecause the barrier height efV  of the shallow well is strongly influenced by those parameters \n(see Fig. 5). In particular, we  observe that the slope of 1()T significantly decreases with \nincreasing h [Fig. 9(b)] due to a decr ease of the barrier height efV  due to the action of the \nexternal field [see Fig. 5(c) and 5(d) ]. At low barriers, the behavior of 1()T may deviate \nconsiderably from Arrhenius behavior. Figure 10 illustrates the dependence of  on the \ncurrent parameter J for various values of the external field parameter h. Clearly, as J increases \nfrom negative values,  exponentially increases attaining a ma ximum at a critical value of the \nspin-polarized current and then smoothl y switches to exponential decrease as J is further \nincreased through positive values. Such a dependence of  on the applied current implies that \nboth kinds of scaling for the switching time suggested in the literature ,24-26,28 namely, \nSWCJ Je  and 2()SWCJ Je  may be realized for 10SWJJ  and 10SWJJ,  22respectively (where C and SWJ are parameters depending, in general, on h, , , , etc.). \nFigure 11 exemplifies the pronounced dependence of  on the azimuthal angles of the applied \nfield  and the magnetization dire ction in the fixed layer P for various values of J which \nmay comprise several orders  of magnitude [note that () ( 2 )      and \n() ( 2 )P P    ]. Invariably strong STT effects on th e magnetization reversal exist only for \nlow damping, 0.1 , because the magnitude of the STT effects in the magnetization reversal \nis governed by the ratio /J.13 Here, the variation of  with J may be of several orders of \nmagnitude. For 1, however, the STT term in Eq. (1 ) does not influence the reversal \nprocess at all because it is negligible comp ared to the damping and random contributions. \n\n432P =  =  /2\nP =  = 0\n1: J = 0\n2: J = 6\n3: J = 6\n4: J = 12/ \nh =0.15\n =20\n = 20\nP = 0.3\n1\n \nFIG. 8. Reversal time 0/ vs the damping parameter  for various values of the current J \n(solid lines). Asterisks: escape rate formula, Eq.(30). \n  \n/  4321P = =  /2\nP = = 0\n1: J = 6\n2: J = 0\n3: J = 6\n4: J = 12h = 0.1\n = 0.02\n  = 20\nP = 0.3\n(a)\n \n  \n/ \n4321P = =  /2\nP = = 0\n1: h = 0\n2: h = 0.1\n3: h = \n4: h = J = 6\n = 0.02\n  = 20\nP = 0.3\n(b)\n \nFIG. 9. 0/ vs the inverse temperature parameter 1~T for various values of the current \nJ (a) and the external field h (b). Asterisks: escape rate formula, Eq.(30).  23   \n/ \n43 21P = =  /2P = = 0\n1: h = 0\n2: h = 0.05\n3: h = \n4: h =  = 20  = 20\nP = 0.3  = 0.02   \nJ \nFIG. 10. 0/ vs the spin-polarized current parameter J for various values of the external \nfield 0,0.05,0.1,0.2h . \n\n/ \n4321P = =  /2P = 0\n1: J = 6\n2: J = 0\n3: J = 6\n4: J = 12h = 0.1 = 0.02  \n  = 20     = 20  \n              P  = 0.3\n(a)\n \n\n/ \n4321P = =  /2 = 0\n1: J = 6\n2: J = 0\n3: J = 6\n4: J = 12h = 0.1\n = 0.02  \n = 20\n  = 20   \nP = 0.3\nP(b)\n \nFIG. 11. 0/ vs the azimuthal angles  (a) and P (b) for 6, 0, 6, 12J . \n \nSTT effects in the thermally assisted magneti zation reversal have been treated via the \nevolution equation for the stat istical moments yielded by th e Langevin equation rendering \nstationary and nonstationary charac teristics for wide ranges of te mperature, damping, external \nmagnetic field, and spin-polarized current. Variat ion of the latter may alter the reversal time \nby several orders of magnitude conc urring with expe rimental results.11 The virtue of our \nnumerically exact solutions of th e recurrence relations for the relevant statistical moments is \nthat they hold for the most comprehensive fo rmulation of the generi c nanopillar model (Fig. \n1), i.e., for arbitrary directions  of the external field and spin polarization and for arbitrary free \nenergy  density, yielding  the STT switching characteris tics under conditions otherwise \ninaccessible. Thus our results may serve both as  a basis for theoretical investigations and \ninterpretation of a broad range  of STT experiments. Additiona lly, they are essential for the \nfuture development of both escap e rate theory and stochastic dynamics simulations of the  24magnetization reversal time in STT systems, re presenting rigorous benchmark solutions with \nwhich calculations of that time by any othe r method must comply (the procedure being \nentirely analogous to that used to validate such  complementary approaches to fine particle \nmagnetization by exactly calculating the re versal time via the smallest nonvanishing \neigenvalue of the FPE fo r the Néel-Brown model3,36). Finally, we believe that the moment \nmethod may be useful in related problems such  as magnetization reversal of STT devices \ndriven by ac external fields and currents, etc.37 \nACKNOWLEDGEMENTS \nWe thank P.M. Déjardin for helpful conve rsations. One of us, D.B., acknowledges the \nSimSci Structured Ph.D. Progra mme at the University College Dublin for financial support. \nSimSci is funded under the Programme for Resear ch in Third-level Institutions and co-funded \nunder the European Regional Development Fund.  All calculations were performed by the \nLonsdale cluster maintained by the Trinity Ce ntre for High Performance Computing. This \nCluster was funded through grants from the Science Foundation Ireland. \nAPPENDIX A: REDUCTION OF EQ. (1) TO THE LANDAU-LIFSHITZ FORM \nIn order to obtain an explicit equation for u, we rewrite the implicit Eq. (1) in the \nLandau-Lifshitz form. Transposing the u term, we have \n   ef ST.       uu u u H u h   (34) \nOn cross-multiplying vectorially by u in Eq. (34) and using the triple vector product formula \n  ()    uu u uu u u  , (35) \nwe obtain since () 0 uu  \n    ef ST .      uu u u H u h u   (36) \nSubstituting Eq. (36) into Eq. (34) yields the explicit form \n   2\nef ef ST ST(1+ )                 uu H u h u H u h u    \nor equivalently, using Eq. (6), \n N\n0S\nN0S2\n2vVMkT\nvVMkT\n          \n                uu u huu\nuu u huu\n (37) \nwhich reduces to the more easily visualized Eq. (6) because  25           uuu uuu. \nIn the spherical polar co ordinate system shown in Fig. 1, one has  \n  1, 0, 0u ,   =0 , , s i n u  ,  \n 10, ,sinVV V\nu,   \n 10, ,sin       uu. \nAPPENDIX B: DERIVATION OF THE DIFFERENTIAL-RECURRENCE RELATION FOR \nSTATISTICAL MOMENTS, EQ. (10) \nOwing to Eqs. (7) and (8), we can write the Langevin equation (9) in  vector notation as \n \n 1 0\n,, ,\n11\n,[]2\n[] .2S\nlm lm lm\nN\nlm\nNvMYY YkT\nvVV YkT\n\n    \n         hu\nu\n (38) \nHere  denotes the orientation space gradient operator defined as  \n  uu. \nOn averaging the Langevin equation (38) with mu ltiplicative noise as explained in detail in \nRef. 32 and Sec. 9.2 of Ref. 23, we have afte r some algebra the evolution equation of the \nstatistical moment ,()lmYt , viz., \n     \n 11 1\n,, , , ,\nN\n11\n,,1\n22\n1\nsinlm lm lm lm lm\nlm lmdvYY V Y Y V V Ydt kT\nVV YY \n\n  \n     \n            (39) \nwhere the operator 2  is the angular part of the Laplacian, viz., \n 2\n2211sin .sin sin       \nHere we have used23 \n  1 0\n,, , []S\nlm lm lmvMYY YkT     hu  \nand that for any function ,Ft23 \n  ,, , , 2lm lm lm lm FY F Y F Y Y F       .  26We now indicate using the th eory of angular momentum34 how Eq. (39) may be \nwritten as a differential-recurrence relation for the statistical moments. This is accomplished \nby reducing (details in Refs. 32 and 23, chapters  7 and 9) the terms inside the angular braces \non the right-hand side of Eq. (39) to the calculati on of the Fourier coefficients in the expansion \nof a product of spherical harmonics as a sum of spherical harmonics. We begin by expressing \nthe terms within the angular braces on the righ t-hand side of Eq. (39) as functions of the \nangular momentum operators 2ˆ,Lˆ,ZL and ˆL defined as34  \n 2ˆ , L ˆ ,ZLiˆ cot .iiLe i e \n    (40) \nThus we have from Eqs. (39) and (40) the evolution equation  \n       \n      22 2 2\n,, , , ,\nN\n1\n1,1 , ,\n1\n1, 1 , ,11 1 1ˆˆ ˆ ˆ\n22\n3ˆˆ ˆˆ\n22\nˆˆ ˆˆ ,lm lm lm lm lm\nZl m Z l m\nZl m Z l mdYL Y L V Y V L Y Y L Vdt\niY L VL Y L VL Y\nY L VL Y L VL Y\n  \n\n\n   \n\n                   \n     \n     (41) \nwhere we have used the following repr esentations for th e expansions of /( )vV kT V V \nand /( )vk T     in terms of spherical harmonics, viz., \n , 1 ,1\nrs r rs rs VA Y\n\n  ,  , 10 , rs rsr\nrsVA Y\n  , \n , 1 ,1\nrs r rs rsBY\n\n  ,  , 10 , rs rsr\nrsBY\n  . \nThen (see Refs. 23 and 32 for details) the right-hand side of Eq. (41) may ultimately be written \nas a linear combination of averag es of spherical harmonics ,i.e., Eq. (10), because the action of \nthe operators 2 ˆˆˆ,,ZLLL  on ,lmY is34 \n ,,ˆ ,Zlm lmLYm Y 2\n,,ˆ 1,lm lmLYl l Y,, 1ˆ (1 ) ( 1 ) ,lm lmLY l l mm Y     \nand products of spherical harmonics may always be reduced to a sum of  spherical harmonics \nusing the Clebsch-Gordan series, viz.,34 \n22 11\n11 1\n11 21\n21,0 ,\n, 0 ,, 0 , ,, 1\n,, ,\n2(2 1)(2 1)\n4 21ll m mll\nll l m l m\nlm l m l m m\nll lCC llYY Y\nl \n\n\n . \nAPPENDIX C: EXPLICIT FORM OF ()ntC , ,nnQQ , AND ,, ,lml me \nThe general Eq. (10) as specialized to Eqs. (17) and (19) yields the 25-term \ndifferential-recurrence equation for the statistical moments ,,() ()lm lmct Y t , viz.,  27 N , ,2 , 2 ,2 , 1 ,2 , ,2 , 1 ,2 , 2\n,1 , 2 ,1 , 1 ,1 , ,1 , 1 ,1 , 2\n,, 2 ,() () () () () ()\n() () () () ()\n()n m n m nm n m nm n m nm n m nm n m nm\nnm n m nm n m nm n m nm n m nm n m\nnm nm ndc t vc t vc t vc t vc t vc tdt\nwc t wc t wc t wc t wc t\nxc t x   \n    \n   \n    \n\n \n \n, 1 ,, ,, 1 ,, 2\n,1 , 2 ,1 , 1 ,1 , ,1 , 1 ,1 , 2\n,2 , 2 ,2 , 1 ,2 , ,2 , 1() () () ()\n() () () ()\n() () () (mn m n mn m n mn m n mn m\nnm n m nm n m nm n m nm n m nm n m\nnm n m nm n m nm n m nm n mct x c t x ct x ct\nyc yc t yc t yc t yc t\nzc t zc t zc t zc t  \n \n   \n    \n  \n    \n   \n  ,2 , 2 )( ) ,nm n mzc t\n (42) \nwhere the coefficients ,nmx, etc. are given by \n * 3\n,1 0\n2\n*2 * *\n10 11 1 1(1 )(, )23\n2 (1 ) 3 1(, ) (, ) (, ),\n21 23 2 3nm P P P\nPP\nPP PP PPmh nnxi i m b J Y\nbcJ nn mYY Y\nnn \n    \n  \n         \n   \n ,1 2\n** *\n11 1 0 1112\n2( 1 2 )(, ) (, ) (, ) ,62 1 2 3nm\nPP\nPP P P P P Pihxn m n m i\nbcJ mib J Y Y Ynn\n   \n  \n    \n \n    *2\n, 11121 3(, ) ,(2 1)(2 3) 4PP\nnm PPnm nm nm nm bcJxYnn\n       \n \n22\n*\n,3 1 0\n*2 * *\n10 11 1 1(1 ) 1(, )21 23 2 3\n2(, ) (, ) (, ),3P\nnm PP\nPP\nPP PP PPnmm n b Jyi n h Ynn\nmb c JiY Y Y \n         \n  \n \n \n*\n,1 1\n**\n12 1 0 1 112(, )12 32 6\n2( 2 )(, ) (, ) ,23P\nnm P P\nPP\nPP PPnm nm nb JyYnn\nbcJn m nhii Y Y\n     \n\n     \n \n \n    \n *2\n,1 1123(, ) ,43 1 2 3 2PP\nnm P Pnm nm nmnm bcJyi Ynn\n      \n22\n,3 2\n** 2 * *\n10 10 11 1 11(1 )41 2\n(1 ) 2( ,) ( ,) ( ,)( ,) ,33nm\nPP P\nPP PP PP PPnm i mwn hn\nbJn m bcJYi Y Y Y \n          \n   \n \n  \n*\n,1 1 2\n**\n12 1 0 1 11 (1 )(, )41 6\n2( 1 2 ) 1(, ) (, ) ,23P\nnm P P\nPP\nPP PPnmnm bJnwYn\nbcJn m nhii Y Y\n     \n\n  \n    \n\n  28    *2\n,1 1 221 1(, ) ,43 4 1PP\nnm P Pnm nm nmnm bcJwi Yn\n       \n \n22 22\n,\n*2 * *\n10 11 1 1(1 ) (2 )\n23 21 25\n2 1(, ) (, ) (, ),23nm\nPP\nPP PP PPnm n m nznn n\nbcJYY Y        \n       \n\n 22\n**\n,1 0 1 1(1 ) 2 ( 3 ) 22(, ) (, ) ,32 3 2 1 2 5PP\nn m PP PPn m nm nm bcJ nzY Ynn n \n     \n \n *2\n,1 112 ( 3 ) ( 4 )(, ) ,43 2 3 2 1 2 5PP\nnm P Pn m nm nm nm bcJ nzYnn n\n        \n \n22 22\n,\n*2 * *\n10 11 1 1(1 ) ( ) 11\n21 21 23 2\n2(, ) (, ) (, ),3nm\nPP\nPP PP PPnm n m nvnn n\nbcJYY Y\n         \n  \n\n 22\n**\n,1 0 1 12( 1 ) ( ) 22 1(, ) (, )32 1 2 1 2 3PP\nn m PP PPnm nm n m bcJ nvY Ynn n \n   , \n   \n *2\n,1 1321 ( ) 1(, )43 2 1 2 1 2 3PP\nnm P Pnm nm nm nm bcJ nvYnn n\n    . \nIn order to rewrite Eq. (42) in the form of Eq. (20) explicitly, we define ()ntC  as the \ncolumn vectors arranged in an appropriate way from ,()nmct , viz., \n 0()t C0 ,  2,2\n2,2 1\n2, 2\n21 , 21\n21 , 22\n21 , 21()\n()\n()\n()()\n()\n()nn\nnn\nnn\nn\nnn\nnn\nnnct\nct\nct\ntct\nct\nct\n\n \n \n\n\n\n\n\n\nC\n, (n  1), (43) \nwhile  the matrices ,,nnnQQQ  are defined as \n 22\n21 21nn\nn\nnnXWQYX, 22\n21nn\nn\nn\nZYQ0Z, 2\n21 21n\nn\nnn\nV0QWV. (44) \nIn turn, the matrices ,,nnnQQQ  themselves consist of five submatrices lV, lW, lX, lY, and \nlZ of dimensions (2 1) (2 3)ll  , (2 1) (2 1)ll , (2 1) (2 1)ll ,(2 1) (2 3)ll  , and  29(2 1) (2 5)ll  , respectively. The elements of thes e five-diagonal submatrices, which are \nformed from the coefficients occurring in Eq. 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Kalmykov and \nB. Ouari, Phys. Rev B 71, 094410 (2005) \n36 Y. P. Kalmykov, W. T. Coffey , U. Atxitia, O. Chubykalo-Fesenko, P. M. Déjardin, and R. \nW. Chantrell, Phys. Rev. B 82, 024412 (2010). \n37 A. N. Slavin and V. S. Tiberkevich, Phys. Rev. B  72, 092407 (2005); J.-V. Kim, V. S. \nTiberkevich, and A. N. Slavin, Phys. Rev. Lett. 100, 017207 (2008); Phys. Rev. Lett. 100, \n167201 (2008).  "    },    {        "title": "2004.04840v3.Magnetic_Damping_in_Epitaxial_Fe_Alloyed_with_Vanadium_and_Aluminum.pdf",        "content": "1 \n Magnetic Damping in Epitaxial Fe Alloyed with Vanadium and Aluminum  \nDavid A. Smith1, Anish Rai2,3, Youngmin Lim1, Timothy Hartnett4, Arjun Sapkota2,3, Abhishek \nSrivastava2,3, Claudia Mewes2,3, Zijian Jiang1, Michael Clavel5, Mantu K. Hudait5, Dwight D. \nViehland6, Jean J. Heremans1, Prasanna V. Balachandran4,7, Tim Mewes2,3, Satoru Emori1 \n1Department of Physics, Virginia Tech, Blacksburg, VA 24061, U.S.A.  \n2Department of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487, U.S.A.  \n3Center for Materials for Information Technology (MINT), University of Alabama, Tuscaloosa,  \nAL 35487, U.S.A . \n4Department of Material Science and Engineering, University of Virginia,  \nCharlottesville, VA 22904, U.S.A. \n5Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA 24061, \nU.S.A.  \n6Department of Materials Science and Engineering, Virginia Tech, Blacksburg, VA 24061, \nU.S.A.  \n7Department of Mechanical and Aerospace Engineering, University of Virginia,  \nCharlottesville, VA 22904, U.S.A.  \n \n  2 \n To develop low -moment, low -damping metallic ferromagnets for power -efficient spintronic \ndevices, it is crucial to understand how magnetic relaxation is impacted by the addition of \nnonmagnetic elements. Here, we compare  magnetic relaxation in epitaxial Fe films alloyed \nwith light nonmagnetic elements of V and Al. FeV alloys exhibit lower intrinsic damping  \ncompared to pure Fe,  reduced by nearly a factor of 2,  whereas damping in FeAl alloys \nincreases with Al content . Our experimental and computat ional results indicate that \nreducing the density of states at the Fermi level , rather than the average atomic number, \nhas a more significant impact in lowering damping in Fe alloyed with light elements . \nMoreover, FeV is confirmed to exhibit an intrinsic Gi lbert damping parameter of ≃0.001, \namong the lowest ever reported for ferromagnetic metals.  \n \nI. INTRODUCTION  \n The relaxation of magnetization dynamics (e.g., via Gilbert damping) plays important \nroles in many spintronic applications, including those based  on magnetic switching1,2, domain \nwall motion3,4, spin wave propagation5,6, and su perfluid -like spin transport7,8. For devices driven \nby spin -torque  precessional dynamics1,9,10, the critical current density for switching is predicted \nto scale with the produ ct of the Gilbert damping parameter and the saturation magnetization 2,11. \nThus, it is desirable to engineer magnetic materials  that possess both low damping and low \nmoment for energy -efficient operation . While some electrically insulating magnetic oxides have \nbeen considered for certain applications5,12,13, it is essential to engineer low -damping, low -\nmoment metallic  ferromagnets for robust electrical readout via giant magnetoresistance and \ntunnel magnetoresistance. Fe is the elemental ferromagnet with the lowest intrinsic Gilbert \ndamping  parameter ( ≃0.002)14,15, albeit with the highest saturation magnetization ( ≃2.0 T). 3 \n Recent experiments have reported that Gilbert damping can be further reduced by alloy ing Fe \nwith Co (also a ferromagnetic element), with Fe 75Co25 yielding an ultralow intrinsic Gilbert \ndamping parameter of ≃0.00116,17. However, Fe 75Co25 is close to the top of the Slater -Pauling \ncurve , such that its saturation magnetization is greater than that of Fe by approximately 20 %18. \nThere is thus an unmet need to engineer ferromagnetic alloys tha t simultaneously exhibit lower \ndamping and lower moment than Fe.  \n A promising approach towards low -damping, low -moment ferromagnetic metals is to \nintroduce nonmagnetic  elements into Fe . In addition to diluting the magnetic moment, \nnonmagnetic elements int roduced into Fe could influence the spin -orbit coupling strength ξ, \nwhich underlies  spin relaxation  via orbital and electronic degrees of freedom19–21. Simple atomic \nphysics suggests that ξ is related to the average atomic number <Z> of the alloy so that, \nconceivably, damping might be lowered by alloying Fe with lighter (lower -Z) elements. Indeed, \nmotivated by the premise of lowering damping through a reduced  <Z> and presumably ξ, prior \nexperiments have explored Fe thin films alloyed with V20,22,23, Si24, and Al25. However, the \nexperimentally reported damping parameters for these alloys are often a factor of >2 higher22,23 ,25 \nthan the theoretically predicted intrinsic Gilbert damping parameter of ≃0.002 in Fe26 and do not \nexhibit a significant dependence on the alloy composition20,23,24. A possible issue is that the \nreported damping parameters – obtained from the frequency dependence of ferromagnetic \nresonance (FMR) linewidth with the film magnetized in -plane – may include contributions from \nnon-Gilbert relaxation induced by inhomogeneity and defects (e.g., two -magnon scattering)27–36, \nwhich can be affected by the alloying. Therefore, how Gilbert damping in Fe is impacted by \nalloying with low -Z elements remains an open question.  4 \n Here, we investigate the compositiona l dependence of magnetic relaxation at room \ntemperature in  epitaxial thin films of ferromagnetic FeV and FeAl alloys. Both  alloys are \ncrystalline bcc solid solutions  and hence constitute excellent model systems. We employ two \nconfigurations of FMR measurem ents to gain complementary insights: (1) FMR with samples \nmagnetized in the film plane (similar to the prior experiments) to derive the “effective” Gilbert \ndamping parameter, 𝛼𝑒𝑓𝑓𝐼𝑃, which is found to include extrinsic magnetic relaxation due to two -\nmagnon scattering, and (2) FMR with samples magnetized perpendicular to the film plane to \nquantify the intrinsic Gilbert damping parameter, 𝛼𝑖𝑛𝑡, which is free of the two -magnon \nscattering contribution.  \nSince Al ( Z = 13) is a much lighter element  than V ( Z = 23), we might expect lower \nmagnetic relaxation in FeAl than FeV, if the smaller < Z> lowers intrinsic Gilbert damping via \nreduced ξ. Instead, we find a significant decrease in  magnetic relaxation by alloying Fe w ith V – \ni.e., yielding an intrinsic Gilbert damping parameter of ≃0.001, on par with the lowest values \nreported for ferromagnetic metals  – whereas damping in FeAl alloys  increases with Al content . \nThese experimental results , combined with density functi onal theory calculations, point to the \ndensity of states at the Fermi level D(EF) as a plausible dominant factor for the lower (higher) \nGilbert damping in FeV (FeAl). We thus find that incorporating a low -Z element does not \ngenerally lower damping and that, rather, reducing D(EF) is an effective route for lower damping \nin Fe alloyed wi th a nonmagnetic element. Our findings confirm that FeV is an intrinsically \nultralow -damping alloy, as theoretically predicted by Mankovsky et al.26, which also possesses a \nlower saturation magnetization than Fe and FeCo. The combination of low damping and low \nmoment makes FeV a highly promising material for practical metal -based spintronic \napplications.  5 \n II. FILM DEPOSITION AND STRUCTURAL PROPERTIES  \nEpitaxial Fe 100-xVx and Fe 100-xAlx thin films were grown using dc magnetron sputtering \non (001) -oriented MgO substrates. Prior to deposition, the substrates were annealed at 600 oC for \n2 hours37. The base pressure prior to deposition was <  5×10-8 Torr, and all film s were grown with \nan Ar pressure of 3 mTorr. Fe and V (Al) 2” targets were dc co -sputtered to deposit Fe 100-xVx \n(Fe 100-xAlx) films at a substrate temperature of 200 oC. By adjusting the deposition power, we \ntuned the deposition rate of each material (calibrated by X -ray reflectivity) to achieve the desired \natomic percentage x of V (Al). All FeV and FeAl films had a thickness of 25 nm, which is well \nabove the thickness regime where interfacial effects dominate31,38.  The FeV (FeAl) films were \ncapped with 3 -nm-thick V (Al) deposited at room temperature to protect against oxidation, \nyielding a film structure of MgO/Fe 100-xVx(25nm)/V(3nm) or MgO/Fe 100-xAlx(25nm)/Al(3nm).  \n We confirmed the epitaxial bcc structure of our thi n films using high resolution X -ray \ndiffraction. 2θ -ω scans show only the (002) peak of the film and the (002) and (004)  peaks of the \nsubstrate, as shown in Fig ure 1. Rocking curve scans of the film peaks show similar full -width -\nat-half-maximum values of  ≃ 1.3o irrespective of composition .  The epitaxial relation between \nbcc Fe and MgO is well known16,39:  the bcc film crystal is rotated 45o with respect to the \nsubstrate crystal , such that  the [100] axis of the film lies parallel to the [110] axis of the \nsubstrate. The absence of the (001) film peak indicates that our epitaxial FeV and FeAl films are \nsolid sol utions rather than B2 -ordered compounds40. \n  6 \n III. MAGNETIC RELAXATION  \n3.1. In -Plane Ferromagnetic Resonance   \nMany spintronic devices driven by precessional magnetization dynamics are based on in -\nplane magnetized thin films. The equilibrium magnetization also lies in -plane for soft \nferromagnetic thin films dominated by shape anisotropy (i.e., negligible perpendicular magnetic \nanisotropy), as is the case for our epitaxial FeV and FeAl films. We therefore first discuss FMR \nresults w ith films magnetized in -plane. The in -plane FMR results further provide a basis for \ncomparison with previous studies20,22,23,25.  \nSamples were placed with the film side facing a coplanar waveguid e (maximum \nfrequency 50 GHz) and magnetized by an e xternal field H (from a conventional electromagnet, \nmaximum field 1.1 T) along the in -plane [100] and [110] axes of the films.  Here, unless \notherwise stated, we show results for H || [110] of the film.  FMR spectra were acquired via field \nmodulation by sweeping H and fixing the microwave excitation frequency.  \nExemplary  spectra for Fe, Fe 80V20, and Fe 80Al20 are shown in Fig ure 2, where we \ncompare the peak -to-peak linewidths at a microwave excitation frequency of 20 GHz. We see \nthat the linewidth for Fe 80V20 shows a ≃ 25 % reduction compared to Fe. We further note that \nthe linewidth for the Fe 80V20 sample here is a factor of ≃ 2 narrower than that in previously \nreported FeV20; a possible origin of the narrow linewidth is discussed later . In contrast, Fe 80Al20 \nshows an enhancement in linewidth over Fe, which is contrar y to the expectation of lower \nmagnetic relaxation with a lower average atomic number.  \nThe FMR linewidth is generally governed not only by magnetic relaxation, but also by \nbroadening contributions from magnetic inhomogeneities28,41,42. To disentangle the magnetic 7 \n relaxation and inhomogeneous broadening contributions to the linewidth, the typical prescription \nis to fit the frequency f dependence of linewidth ∆𝐻𝑝𝑝𝐼𝑃 with the linear relation41 \n∆𝐻𝑝𝑝𝐼𝑃=∆𝐻0𝐼𝑃+ℎ\n𝑔𝜇𝐵𝜇02\n√3𝛼𝑚𝑒𝑎𝑠𝐼𝑃𝑓, (1) \nwhere h is the Planck constant, 𝜇𝐵 is the Bohr magneton, 𝜇0 is the permeability of free space, \nand 𝑔 is the g-factor obtained from the frequency dependence of the resonance field (see Section \nIV and Supplementa l Material). In Eq. (1), the slope is attributed to viscous magnetic damping, \ncaptured by the measured damping parameter 𝛼𝑚𝑒𝑎𝑠𝐼𝑃, while t he zero -frequency linewidth ∆𝐻0𝐼𝑃 is \nattributed to inhomogeneo us broadening. The fitting with Eq. (1) was carried out for f  10 GHz, \nwhere H was sufficiently large to saturate the films. As is evident from  the results in Fig ure 3, \nFe80V20 has lower linewidths across all frequencies and a slightly lower slope, i.e.,  𝛼𝑚𝑒𝑎𝑠𝐼𝑃. On the \nother hand, Fe 80Al20 shows higher linewidths and a higher slope.  \nThe measured viscous damping includes a small contribution from eddy currents, \nparameter ized by 𝛼𝑒𝑑𝑑𝑦 (Supplemental  Material) , and a contribution due to radiative damping43, \ngiven by 𝛼𝑟𝑎𝑑 (Supplemental Material).  Together these contributions make up ≃20 % of the total \n𝛼𝑚𝑒𝑎𝑠𝐼𝑃 for pure Fe and decrease in magnitude with increasing V or Al content . We subtract these \nto obtain  the effective in -plane Gilbert damping parameter,  \n 𝛼𝑒𝑓𝑓𝐼𝑃=𝛼𝑚𝑒𝑎𝑠𝐼𝑃−𝛼𝑒𝑑𝑑𝑦 − 𝛼𝑟𝑎𝑑.  (2) \nAs shown in Fig ure 4a, 𝛼𝑒𝑓𝑓𝐼𝑃 remains either invariant or slightly decreases in Fe 100-xVx up to x = \n25, whereas we observe a monotonic enhancement of 𝛼𝑒𝑓𝑓𝐼𝑃 with Al content in Figure 4b . These \nresults point to  lower (higher) damping in FeV (FeAl) and suggest a factor other than the average \natomic number governing magnetic relaxation in these alloys.  However, such a conclusion \nassumes that 𝛼𝑒𝑓𝑓𝐼𝑃 is a reliable measure of intrinsic Gilbert damping . In reality, 𝛼𝑒𝑓𝑓𝐼𝑃 may include 8 \n a contribution from defect -induced two -magnon scattering27–31,35,36, a well -known non -Gilbert \nrelaxation mechanism in in -plane magnetized epitaxial films27,32 –34,44. We show in the next \nsubsection that substantial two -magnon scattering  is indeed present in our FeV and FeAl alloy \nthin films.  \n Although Eq. (1) is not necessarily the correct framework for quantifying Gilbert \ndamping in in -plane magnetized thin films, we can gain insight into the quality (homogeneity) of \nthe films from ∆𝐻0𝐼𝑃. For our samples, μ0∆𝐻0𝐼𝑃 is below ≈ 1 mT (see Fig ure 4c,d), which implies  \nhigher film quality for our FeV samples than previously reported20. For example, Fe 73V27 in \nScheck et al.  exhibits μ0∆𝐻0𝐼𝑃 ≃ 2.8 mT20, whereas Fe 75V25 in our study exhibits μ0∆𝐻0𝐼𝑃 ≃ 0.8 \nmT. Although  𝛼𝑒𝑓𝑓𝐼𝑃 is comparable between Scheck et al. and our study, the small ∆𝐻0𝐼𝑃 leads  to \noverall much narrower linewidths  in our FeV films (e.g., as shown in Figs. 2 and 3) . We \nspeculate that the annealing of the MgO substrate prior to film deposition37 – a common practice \nfor molecular beam epitaxy – facilitates high -quality epitaxial film growth  and hence small ∆𝐻0𝐼𝑃 \neven by sputtering.  \n  \n3.2. Out -of-Plane Ferromagnetic Resonance  \nTo quantify intrinsic Gilbert damping, we performed broadband FMR with the film \nmagnetized out -of-plane, which is the configuration that suppresses two -magnon scattering28–31. \nSamples were placed in side a W-band shorted -waveguide spectrometer (frequency range 70 -110 \nGHz) in a superconducting electromagnet that enabled measurements at fields > 4 T. This high \nfield range is well above the shape anisotropy field of ≤2 T for our films and hence sufficient to \ncompletely saturate the film out -of-plane.   9 \n The absence of two -magnon scattering in broadband out -of-plane FMR allows us to \nreliably obtain the measured viscous damping parameter 𝛼𝑚𝑒𝑎𝑠𝑂𝑃 by fitting the linear frequency \ndependence of the linewidth ∆𝐻𝑝𝑝𝑂𝑃, as shown in Figure 5, with \n∆𝐻𝑝𝑝𝑂𝑃=∆𝐻0𝑂𝑃+ℎ\n𝑔𝜇𝐵𝜇02\n√3𝛼𝑚𝑒𝑎𝑠𝑂𝑃𝑓. (3)  \nWe note that the zero -frequency linewidth for the out -of-plane configuration ∆𝐻0𝑂𝑃 (Figure 6c,d) \nis systematically g reater than that for the in -plane configuration ∆𝐻0𝐼𝑃 (Figure 4c,d). Such a trend \nof ∆𝐻0𝑂𝑃>∆𝐻0𝐼𝑃, often seen in epitaxial films15,33,45, may be  explained by the stronger \ncontribution of inhomogeneity to the FMR field when the magnetic precessional orbit is circular, \nas is the case for out -of-plane FMR, compared to the case of the highly elliptical precession in \nin-plane FMR41; however, the detailed mechanisms contributing to the zero -frequency linewidth \nremain  the subject of future work . The larger ∆𝐻0𝑂𝑃 at high V and Al concentrations may be due \nto broader distributions o f anisotropy fields and saturation magnetization, or the presence of a \nsecondary crystal phase that is below the resolution of our X -ray diffraction results.   \nThe absence of two -magnon scattering in out -of-plane FMR allows us to quantify the \nintrinsic Gilbert damping parameter,  \n𝛼𝑖𝑛𝑡=𝛼𝑚𝑒𝑎𝑠𝑂𝑃−𝛼𝑒𝑑𝑑𝑦,  (4) \nby again subtracting the eddy current contribution 𝛼𝑒𝑑𝑑𝑦. Since we utilize a shorted  waveguide, \nthe contribution due to radiative damping does not apply.    \nFrom the compositional dependence of 𝛼𝑖𝑛𝑡 as summarized in Figure 6a1, a reduction in \nintrinsic Gilbert damping is evidenced with V alloying. Our observation is in contrast to the \nprevious experiments on FeV alloys20,22,23 where the reported damping parameters remain >0.002 \n \n1 We were unable to carry out out -of-plane FMR measurements for FeV with x = 20 (Fig. 2(c,d )) as the sample had \nbeen severely damaged during transit.  10 \n and depend weakly on the V concentration. In particular, the observed minimum of 𝛼𝑖𝑛𝑡≃0.001 \nat x ≃ 25-30 is approximately half of the lowest Gilbert damping parameter previously reported \nfor FeV20 and that of pure Fe15. The low 𝛼𝑖𝑛𝑡 here is also comparable to the lowest  damping \nparameters reported for ferromagnetic metals, such as  Fe75Co2516,17 and Heusler compounds46–48. \nMoreover, t he reduced intrinsic damping by alloying Fe w ith V is qualitatively consistent with \nthe computational prediction  by Mankov sky et al.26, as shown by the curve in Figure 6a. Our \nexperimental finding therefore confirms that FeV is indeed an intrinsically ultralow -damping \nferromagnet that possesses a smaller saturation magnetization than Fe.  \nIn contrast to the reduction of 𝛼𝑖𝑛𝑡 observed  in FeV alloys, FeAl shows an increase in \nintrinsic damping with increasing Al concentration, as seen in Figure 6b. Recalling that Al has an \natomic number of Z = 13 that is lower than Z = 23 for V, this trend clashes with the expectation \nthat lower < Z> red uces the intrinsic Gilbert damping through a reduction of the atomic spin -orbit \ncoupling. Thus, we are required to consider an alternative mechanism to explain the higher \n(lower) damping in FeAl (FeV), which we discuss further in Section V.  \n \n3.3. Magnetic Relaxation: Practical Consideration s \nFor both FeV and FeAl alloys, 𝛼𝑖𝑛𝑡 derived from out -of-plane FMR (Figure 6a,b) is \nconsistently lower than 𝛼𝑒𝑓𝑓𝐼𝑃 derived from in -plane FMR (Fig ure 4a,b). Th is discrepancy \nbetween 𝛼𝑖𝑛𝑡 and 𝛼𝑒𝑓𝑓𝐼𝑃 implies  a two-magnon scattering contribution  to magnetic relaxation in \nthe in-plane configuration (Figure 4a,b). For many applications including spin -torque oscillators \nand magnonic devices , it is crucial to minimize magnetic relaxation in in-plane magnetized thin \nfilms. While the in -plane magnetic relaxation ( 𝛼𝑒𝑓𝑓𝐼𝑃≃0.002) is already quite low for the FeV \nalloys shown here, the low intrinsic Gilbert damping ( 𝛼𝑖𝑛𝑡≃0.001) points to the possibility of 11 \n even lower relaxation and narrow er FMR linewidths by minimizing two -magnon scattering and \ninhomogeneous linewidth broadening. Such ultralow magnetic relaxation in FeV alloy thin films \nmay be achieved by optimizing structural properties through growth conditions16 or seed layer \nengineering49.  \nWhile ultralow intrinsic Gilbert damping values have been confirmed  in high -quality \nepitaxial FeV, it would be desirable for device integration to understand how magnetic relaxation  \nin FeV would be impacted by the presence of grain boundaries, i.e. in polycrystalline thin films. \nReports on polycrystalline FeCo49 suggest intrinsic damping values comparable to those  seen in \nepitaxial FeCo16,17. While beyond the scope of this study, our future work will explore the \npossibility of low damping in polycrystalline FeV thin films.  \n \nIV. SPECTROSCOPIC PARAMETERS  \nThe results presented so far reveal that magnetic relaxation is reduced by alloying Fe with \nV, whereas it is increased by alloying Fe with Al. On the other hand, FeV and FeAl alloys \nexhibit similar compositional dependence of the spectroscopic parameters: effective \nmagnetization Meff (here, equivalent to saturation magnetization  Ms), magnetocrystalline \nanisotropy field Hk, and the g-factor 𝑔 – all of which are quantified by fitting the frequency \ndependence of resonance field (Supplemental Material) . As shown in Fig ure 7a, there is a \nsystematic reduction in Meff with increasing concentration of V and Al. We also note in Fig ure 7b \na gradual reduction in magnitude of the in -plane cubic anisotropy. Both of these trends are \nexpected as magnetic Fe atoms are r eplaced with nonmagnetic atoms of V and Al. The reduction \nof Meff by ≃20% in the ultralow -damping Fe 100-xVx alloys with x = 25-30, compared to pure Fe, \nis of particular practical interest. The saturation magnetization of these FeV alloys is on par with 12 \n commonly used soft ferromagnetic alloys (e.g., Ni 80Fe2050, CoFeB51), but the damping parameter \nof FeV is several ti mes lower. Further, w hile FeV and FeCo in the optimal composition window \nshow similarly low intrinsic damping parameters, FeV provides the advantage of lower moment . \nWith the product 𝛼𝑖𝑛𝑡𝑀𝑒𝑓𝑓 approximately proportional to the  critical current densi ty to excite \nprecessional dynamics  by spin torque2,11, FeV is expected to be a superior material platform for \nlow-power spin tronic devices .  \nThe g-factor 𝑔=2(1+𝜇𝐿/𝜇𝑆) is related to the orbital moment 𝜇𝐿 and spin moment 𝜇𝑆; \nthe deviation from the spin -only value of 𝑔= 2.00 provides insight into the strength of spin -orbit \ncoupling ξ52. As seen in Figure 7c, 𝑔 increases  by 1-2% with both V and Al alloying, which \nsuggests that ξ increases slightly with the addition of these low -Z elements.  This finding verifies \nthat < Z> is not necessarily a good predictor of ξ in a solid. Moreover, the higher 𝑔 for FeV is \ninconsistent with the scenario for lower damping linked to a  reduced spin -orbit coupling. Thus, \nspin-orbit coupling alone cannot  explain the observed behavior of Gilbert damping in Fe alloyed \nwith low -Z elements.  \n \nV. DISCUSSION  \nIn contrast to what has been suggested by prior experimental studies20,22 –25, we have \nshown that the reduction of average atomic number by alloying with a light element (e.g., Al in \nthis case) does not generally lower the intrinsic Gilbert dampin g of Fe. A possible source for the \nqualitatively distinct dependencies of damping on V and Al contents is the density of states at the \nFermi level, D(EF): it has been predicted theoretically that the intrinsic Gilbert damping \nparameter is reduced with decr easing D(EF), since D(EF) governs the  availability of  states for \nspin-polarized electrons to scatter into21,26,53 –55. Such a correlation between lower damping and 13 \n smaller D(EF) has been reported by recent experiments on FeCo alloys17,50, FeRh alloys40, CoNi \nalloys56, and Heusler compounds46,48,57. The similarity in the predicted composition dependence \nof the Gilbert damping parameter for FeCo and FeV26 suggests that the low damping of FeV may \nbe correlated with reduced D(EF). However, no prior experiment has corroborated this \ncorrelation for FeV or other alloys of Fe and light elements.  \nWe therefore e xamine whether the lower (higher) damping in FeV (FeAl) compared to Fe \ncan be qualitatively explained  by D(EF).  Utilizing the Quantum ESPRESSO58 package to \nperform density functional theory calculations (details in Supplemental Material) , we calculated \nthe density of states  for Fe, Fe 81.25V18.75, and Fe 81.25Al18.75. It should be recalled that although \nFeV and FeAl films measured experimentally her e are single -crystalline, they are solid solutions \nin which V or Al atoms replace Fe atoms at arbitrary bcc lattice sites.  Therefore, f or each of the \nbinary alloys, we computed 6 distinct atomic configurations  in a 2×2×2 supercell , as shown in \nFigure 8 . The spin -split density of states for each unique atomic configuration is indicated by a \ncurve in Figure 9. Here, D(EF) is the sum of the states for the spin -up and spin -down bands, \naveraged over results from the 6 distinct atomic configurations.  \nAs summari zed in Fig ure 9 and Table 1, FeV has a smaller D(EF) than Fe, whereas FeAl \nhas a larger D(EF). These calculation results confirm a smaller  (larger ) availability of states for \nspin-polarized electrons to scatter into in FeV (FeAl), qualitatively consistent with the lower \n(higher) intrinsic Gilbert damping in FeV (FeAl).   \nWe remark that this correlation between damping and D(EF) is known to hold parti cularly \nwell in the limit of low electronic scattering rates 𝜏−1, where intra band scattering dominates21,54. \nGilmore et al.  have pointed out that at sufficiently high electronic scattering rates, i.e., when \nℏ𝜏−1 is large enough that inter band scattering is substantial, the simple correlation between the 14 \n strength of Gilbert damping and D(EF) breaks down. It is unclear whether our FeV and FeAl \nalloy films at room temperature are in the intraband - or interband -dominated regime. Schoen et \nal. have argued that polycrystalline FeCo alloy films – with higher degree of structural disorder \nand likely higher electronic scattering rates than our epitaxial films – at room temperature are \nstill well within the intraband -dominated regime17. On the other hand, a recent temperature -\ndependent study on epitaxial Fe suggests coexistence of the intraband and interband \ncontributions at room temperature15. A consistent explanation for the observed room -temperature \nintrinsic damping in our alloy films is that the interband contribution depends weakly on alloy \ncomposition; it appears re asonable to conclude that D(EF), primarily through the intraband \ncontribution, governs the difference in intrinsic Gilbert damping among Fe, FeV, and FeAl .  \n \nVI. SUMMARY  \nWe have experimentally in vestigated magnetic relaxation in epitaxial thin films of Fe \nalloyed with low -atomic -number nonmagnetic elements V and Al . We observe a reduction in the \nintrinsic Gilbert damping parameter to 𝛼𝑖𝑛𝑡≃0.001 in FeV films , comparable to the lowest -\ndamping ferromagnetic metals reported to date. In contrast, an increase in damping is observed \nwith the addition of Al, demonstrating that a smaller average atomic number does not necessarily \nlower intrinsic damping in an alloy . Furthermore, our  results on FeV and FeAl cannot be \nexplained by the change  in spin -orbit coupling through alloying . Instead, we conclude that the \ndensity of states at the Fermi level plays a larger role in determining the magnitude of damping \nin Fe alloyed w ith lighter elements. Our work also confirms FeV alloys as promising ultra low-\ndamping , low-moment metallic materials for  practical power -efficient spin -torque  devices.  \n  15 \n Acknowledgements:  \nThis research was funded in part by 4 -VA, a collaborative partnership for advancing the \nCommonwealth of Virginia, as well as by the ICTAS Junior Faculty Program. D.A.S. \nacknowledges support of the Virginia Tech Graduate School Doctoral Assistantship. A. Sapkota \nand C. M . would like to acknowledge support by NSF -CAREER Award No. 1452670,  A.R. and \nT.M. would like to acknowledge support by DARPA TEE Award No. D18AP00011,  and A. \nSrivastava would like to acknowledge support by NASA Award No. CAN80NSSC18M0023.  \n \nWe thank M.D. Stiles for helpful input regarding intrinsic damping mechanisms in alloys.  \n \nThe data that support the findings of this study are available from the corresponding author upon \nreasonable request.  \n \n Number of Spin -Up States  (eV-1) \nat EF Number of Spin -Down States  \n(eV-1) at EF \nFe 10.90 3.44 \nFe81.25V18.75 6.28 ± 1.80  4.61 ± 0.43  \nFe81.25Al18.75 6.81 ± 1.58  10.20 ± 3.03  \nTable 1:  Number of spin -up and spin -down states  at EF. For Fe81.25V18.75 and \nFe81.25Al18.75, the average and standard deviation of values for the 6 distinct atomic \nconfigurations (cf. Figure 8) are shown.    16 \n References:  \n1 A. Brataas, A.D. Kent, and H. Ohno, Nat. Mater. 11, 372 (2012).  \n2 J.Z. Sun, Phys. Rev. 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Phys.Condens. \nMatter 21, 395502 (2009).  \n  21 \n \n15 30 45 60 75 90\nMgO (004) MgO (004) MgO (004)MgO (002) MgO (002) \n MgO (002)BCC\nFe\n(002)\n Log(Intensity) (arb. units)BCC\nFe80V20\n(002)\n \n2q (deg)BCC\nFe80Al20\n(002) \nFigure 1:  (a) 2θ-ω X-ray diffraction scans showing (00 2) and (004)  substrate and (002) film \npeaks for bcc Fe, Fe 80V20, and Fe 80Al20.  \n  22 \n \n-15 -10 -5 0 5 10 15 \n \nFe 2.70 mT\n FMR Signal (arb. units)Fe80V20 2.04 mT\n \nm0(H - HFMR) (mT)Fe80Al203.20 mT \nFigure 2: FMR spectra at f = 20 GHz with the magnetic field H applied in the film plane, fitted \nusing a Lorentzian derivative (solid curve ) for Fe, Fe 80V20 and Fe 80Al20.   23 \n \n0 10 20 30 40 5002468 Fe\n Fe80V20\n Fe80Al20\n Scheck et al.m0DHIP\nPP (mT)\nFrequency (GHz) \nFigure 3:  FMR linewidths versus microwave frequency for the magnetic field applied within the \nplane of the film for three distinct alloys. The solid lines are linear fit s, described by Eq. (1), \nfrom which the effective damping parameter and zero frequency linewidth are determined.  The \ndashed line represents the result for Fe 73V27 from Scheck et al.20 \n  24 \n \n0 10 20 30 40246\n0 10 20 302468\n0 10 20 30 40024\n0 5 10 15 20 25 3001 Fe100-xVx\n Scheck et al.aIP\neff x 103\naIP\neff x 103 Fe100-xAlxm0DHIP\n0 (mT)\nAlloy Composition, x (%)(a) (b)\n(c) (d)m0DHIP\n0 (mT)\nAlloy Composition, x (%) \nFigure 4:  The effective damping parameter 𝛼𝑒𝑓𝑓𝐼𝑃 for (a) Fe 100-xVx and (b) Fe 100-xAlx and zero \nfrequency linewidth 𝜇0Δ𝐻0𝐼𝑃 for (c) Fe 100-xVx and (d) Fe 100-xAlx, obtained from in -plane FMR.  \nThe solid symbols in (a)  and (c) represent results reported by Scheck et al.20 \n  25 \n \n0 20 40 60 80 100 12001020304050\n Fe\n Fe70V30\n Fe70Al30m0DHOP\nPP (mT)\nFrequency (GHz) \nFigure 5:  FMR linewidths versus applied microwave frequency for the magnetic field applied \nperpendicular to the plane of the film for three distinct alloys. The line is a linear fit, described \nby Eq. (3), from which the intrinsic Gilbert damping parameter and zero frequency linewidth are \ndetermined.   \n  26 \n \n0 10 20 30 400123\n0 10 20 300246\n0 10 20 30 4001020\n0 10 20 3001020 Fe100-xVx\n Mankovsky et al.aint x 103 Fe100-xAlxaint x 103(a) (b)\n(c) (d)m0DHOP\n0 (mT)\nAlloy Composition, x (%)\nm0DHOP\n0 (mT)\nAlloy Composition, x (%) \nFigure 6:  The intrinsic Gilbert damping parameter 𝛼𝑖𝑛𝑡 for (a) Fe 100-xVx and (b) Fe 100-xAlx and \nzero frequency linewidth  𝜇0Δ𝐻0𝑂𝑃 for (c) Fe 100-xVx and (d) Fe 100-xAlx, obtained from out -of-\nplane FMR. In (a), the dashed curve  show s the predicted intrinsic damping parameter  computed \nby Mankovsky et al.26 \n  27 \n \n0.81.21.62.02.4\n204060\n0 10 20 30 402.082.102.122.14 Fe\n Fe100-xVx\n Fe100-xAlx \n m0Meff (T)(a)\n |m0Hk| (mT)(b)\n g-factor\nAlloy Composition, x (%)(c) \nFigure 7: (a) Effective magnetization, (b) in -plane cubic anisotropy field, and (c) g-factor versus  \nV and Al concentration.  The solid (open) markers represent data from in -plane (out -of-plane) \nmeasurements . \n  28 \n  \nFigure 8:  The six unique atomic configurations from the supercell program for mimicking the \nFe81.25V18.75  or Fe81.25Al18.75 solid solution.   \n29 \n \n-10010-10010\n-1.0 -0.5 0.0 0.5 1.0-10010 \n (a)\nFe81.25V18.75\n Density of States (eV-1)\n(b)Fe\n \nE - EF (eV)(c)Fe81.25Al18.75 \nFigure 9: Calculated spin-up (positive) and spin -down (negative) densit ies of states for (a) Fe, \n(b) Fe 81.25V18.75 and (c) Fe 81.25Al18.75. Results from the 6 distinct atomic configurations are shown \nin (b,c); the average densities of states at EF for Fe81.25V18.75 and Fe81.25Al18.75 are shown in \nTable  1.   \n "    },    {        "title": "0808.3923v1.Gilbert_Damping_in_Conducting_Ferromagnets_II__Model_Tests_of_the_Torque_Correlation_Formula.pdf",        "content": "arXiv:0808.3923v1  [cond-mat.mtrl-sci]  28 Aug 2008Gilbert Damping in Conducting Ferromagnets II:\nModel Tests of the Torque-Correlation Formula\nIon Garate and Allan MacDonald\nDepartment of Physics, The University of Texas at Austin, Au stin TX 78712\n(Dated: October 29, 2018)\nWe report on a study of Gilbert damping due to particle-hole p air excitations in conducting\nferromagnets. We focus on a toy two-band model and on a four-b and spherical model which provides\nan approximate description of ferromagnetic (Ga,Mn)As. Th ese models are sufficiently simple that\ndisorder-ladder-sum vertex corrections to the long-wavel ength spin-spin response function can be\nsummed to all orders. An important objective of this study is to assess the reliability of practical\napproximate expressions which can be combined with electro nic structure calculations to estimate\nGilbert damping in more complex systems.\nPACS numbers:\nI. INTRODUCTION\nThe key role of the Gilbert parameter αGin current-\ndriven1and precessional2magnetization reversal has led\nto a renewed interest in this important magnetic ma-\nterial parameter. The theoretical foundations which re-\nlate Gilbert damping to the transversespin-spinresponse\nfunction of the ferromagnet have been in place for some\ntime3,4. It has nevertheless been difficult to predict\ntrends as a function of temperature and across mate-\nrials systems, partly because damping depends on the\nstrength and nature of the disorder in a manner that re-\nquires a more detailed characterization than is normally\navailable. Two groups have recently5reported success-\nful applications to transition metal ferromagets of the\ntorque-correlation formula4,5,6forαG. This formula has\nthe important advantage that its application requires\nknowledge only of the band structure, including its spin-\norbit coupling, and of Bloch state lifetimes. The torque-\ncorrelation formula is physically transparent and can be\napplied with relative ease in combination with modern\nspin-density-functional-theory7(SDFT) electronic struc-\nture calculations. In this paper we compare the pre-\ndictions of the torque correlation formula with Kubo-\nformula self-consistent-Born-approximation results for\ntwo different relatively simple model systems, an ar-\ntificial two-band model of a ferromagnet with Rashba\nspin-orbit interactions and a four-band model which cap-\ntures the essential physics of (III,Mn)V ferromagnetic\nsemiconductors8. The self-consistent Born approxima-\ntion theory for αGrequires that ladder-diagram vertex\ncorrections be included in the transverse spin-spin re-\nsponse function. Since the Born approximation is ex-\nact for weak scattering, we can use this comparison to\nassess the reliability of the simpler and more practical\ntorque-correlationformula. Weconcludethat the torque-\ncorrelationformulaisaccuratewhentheGilbertdamping\nis dominated by intra-band excitations of the transition\nmetal Fermi sea, but that it can be inaccurate when it is\ndominated by inter-band excitations.\nOur paper is organized as follows. In Section II we ex-\nplain how we evaluate the transverse spin-spin responsefunction for simple model ferromagnets. Section III dis-\ncusses our result for the two-band Rashba model while\nSection IV summarizes our findings for the four-band\n(III,Mn)V model. We conclude in Section V with a sum-\nmary of our results and recommended best practices for\nthe use of the torque-correlation formula.\nII. GILBERT DAMPING AND TRANSVERSE\nSPIN RESPONSE FUNCTION\nA. Realistic SDFT vs.s-d and p-d models\nWe view the two-band s−dand four band p−dmod-\nels studied in this paper as toy models which capture the\nessential features of metallic magnetism in systems that\nare, at least in principle9, more realistically described\nusing SDFT. The s−dandp−dmodels correspond\nto the limit of ab initio SDFT in which i) the majority\nspind-bands are completely full and the minority spin\nd-bands completely empty, ii) hybridization between s\norpandd-bands is relatively weak, and iii) there is ex-\nchange coupling between dandsorpmoments. In a\nrecent paper we have proposed the following expression\nfor the Gilbert-damping contribution from particle-hole\nexcitations in SDFT bands:\nαG=1\nS0∂ωIm[˜χQP\nx,x] (1)\nwhere ˜χQP\nx,xis a response-function which describes how\nthe quasiparticle bands change in response to a spatially\nsmooth variation in magnetization orientation and S0is\nthe total spin. Specifically,\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(2)\nwhereαandβlabel the xandytransverse spin direc-\ntions and the easy direction for the magnetization is as-\nsummed to be the ˆ zdirection. In Eq.( 2) |i∝an}bracketri}ht,fiandωij\nare Kohn-Sham eigenspinors, Fermi factors, and eigenen-\nergy differences respectively, sαis a spin operator, and2\n∆0(/vector r) is the difference between the majorityspin and mi-\nnority spin exchange-correlation potential. In the s−d\nandp−dmodels ∆ 0(/vector r) is replaced by a phenomeno-\nlogical constant, which we denote by ∆ 0below. With\n∆0(/vector r) replaced by a constant ˜ χQP\nx,xreduces to a standard\nspin-response function for non-interacting quasiparticles\nin a possibly spin-dependent random static external po-\ntential. The evaluation of this quantity, and in particu-\nlar the low-frequency limit in which we are interested, is\nnon-trivial only because disorder plays an essential role.\nB. Disorder Perturbation Theory\nWe start by writing the transverse spin response func-\ntion of a disordered metallic ferromagnet in the Matsub-ara formalism,\n˜χQP\nxx(iω) =−V∆2\n0\nβ/summationdisplay\nωnP(iωn,iωn+iω) (3)\nwhere the minus sign originates from fermionic statistics,\nVis the volume of the system and\nP(iωn,iωn+iω)≡/integraldisplaydDk\n(2π)DΛα,β(iωn,iωn+iω;k)Gβ(iωn+iω,k)sx\nβ,α(k)Gα(iωn,k). (4)\nIn Eq. ( 4) |αk∝an}bracketri}htis a band eigenstate at momentum k,Dis the dimensionality of the system, sx\nα,β(k) =∝an}bracketle{tαk|sx|βk∝an}bracketri}ht\nis the spin-flip matrix element, Λ α,β(k) is its vertex-corrected counterpart (see below), and\nGα(iωn,k) =/bracketleftbigg\niωn+EF−Ek,α+i1\n2τk,αsign(ωn)/bracketrightbigg−1\n. (5)\nWe have included disorder within the Born approximation by incorpora ting a finite lifetime τfor the quasiparticles\nand by allowing for vertex corrections at one of the spin vertices.\nα,kΛβ,kβ,k\nα,ksxβ,k\nk'\nα,kΛ\nα'k''β\n/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0\n/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1\n/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0/0\n/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1/1\n+ =\nFIG. 1: Dyson equation for the renormalized vertex of the tra nsverse spin-spin response function. The dotted line denot es\nimpurity scattering.\nThe vertex function in Eq.( 4) obeys the Dyson equation (Fig. ( 1)):\nΛα,β(iωn,iωn+iω;k) =sx\nα,β(k)+\n+/integraldisplaydDk′\n(2π)Dua(k−k′)sa\nα,α′(k,k′)Gα′(iωn,k′)Λα′,β′(iωn,iωn+iω;k′)Gβ′(iωn+iω,k′)sa\nβ′,β(k′,k),(6)\nwhereua(q)≡naV2a(q)(a= 0,x,y,z),nais the den-\nsity of scatterers, Va(q) is the scattering potential (di-\nmensions: (energy) ×(volume)) and the overline stands\nfor disorder averaging10,11. Ward’s identity requires thatua(q) andτk,αbe related via the Fermi’s golden rule:\n1\nταk= 2π/integraldisplay\nk′ua(k−k′)/summationdisplay\nα′sa\nα,α′sa\nα′,αδ(Ekα−Ek′α′),(7)\nwhere/integraltext\nk≡/integraltext\ndDk/(2π)D. In this paper we restrict\nourselves to spin-independent ( a= 0) disorder and3\nspin-dependent disorder oriented along the equilibrium-\nexchange-field direction( a=z)12. Performing theconventional13integration around the branch cuts of P,\nwe obtain\n˜χQP\nxx(iω) =V∆2\n0/integraldisplay∞\n−∞dǫ\n2πif(ǫ)[P(ǫ+iδ,ǫ+iω)−P(ǫ−iδ,ǫ+iω)+P(ǫ−iω,ǫ+iδ)−P(ǫ−iω,ǫ−iδ)] (8)\nwheref(ǫ) is the Fermi function. Next, we perform an\nanalytical continuation iω→ω+iηand take the imag-\ninary part of the resulting retarded response function.\nAssuming low temperatures, this yields\nαG=∆2\n0\n2πs0{Re[P(−iδ,iδ)]−Re[P(iδ,+iδ)]}\n=∆2\n0\n2πs0Re(PA,R−PR,R) (9)\nwheres0=S0/V,\nPR(A),R=/integraldisplay\nkΛR(A),R\nα,β(k)GR\nβ(0,k)sx\nβ,α(k)GR(A)\nα(0,k)\n(10)\nandGR(A)(0,k) is the retarded (advanced) Green’s func-\ntion at the Fermi energy. The principal difficulty of\nEq.( 9) resides in solving the Dyson equation for the ver-\ntex function. We first discuss our method of solution in\ngeneral terms before turning in Sections III and IV to its\napplication to the s−dandp−dmodels.\nC. Evaluation of Impurity Vertex Corrections for\nMulti-Band Models\nEq.( 6) encodes disorder-induced diffusive correlations\nbetween itinerant carriers, and is an integral equation\nof considerable complexity. Fortunately, it is possible to\ntransform it into a relatively simple algebraic equation,\nprovided that the impurity potentials are short-rangedin\nreal space.Referring back at Eq.( 6) it is clear that the solution of\nthe Dysonequationwouldbe trivialifthevertexfunction\nwasindependent ofmomentum. That is certainlynot the\ncase in general, because the matrix elements of the spin\noperators may be momentum dependent. Yet, for short-\nrange scatterers the entire momentum dependence of the\nvertex matrix elements comes from the eigenstates alone:\nsa\nα,α′(k,k′) =/summationdisplay\nm,m′∝an}bracketle{tαk|m∝an}bracketri}ht∝an}bracketle{tm′|α′k′∝an}bracketri}htsa\nm,m′(11)\nThis property motivates our solution strategy which\ncharacterizes the momentum dependence of the vertex\nfunction by expanding it in terms of the eigenstates of sz\n(sxorsybases would work equally well):\nΛα,β(k) =∝an}bracketle{tαk|Λ|βk∝an}bracketri}ht\n=/summationdisplay\nm,m′∝an}bracketle{tαk|m∝an}bracketri}htΛm,m′∝an}bracketle{tm′|βk∝an}bracketri}ht(12)\nwhere|m∝an}bracketri}htisaneigenstateof sz, witheigenvalue m. Plug-\nging Eqs.( 11) and ( 12) into Eq.( 6) demonstrates that,\nas expected, Λ m,m′isindependent of momentum. After\ncancelling common factors from both sides of the result-\ning expressionand using ∂qua(q) = 0 (a= 0,z)we arrive\nat\nΛR(A),R\nm,m′=sx\nm,m′+/summationdisplay\nl,l′UR(A),R\nm,m′:l,l′ΛR(A),R\nl,l′ (13)\nwhere\nUR(A),R\nm,m′:l,l′≡/parenleftbig\nu0+uzmm′/parenrightbig/integraldisplay\nk∝an}bracketle{tm|αk∝an}bracketri}htGR(A)\nα(0,k)∝an}bracketle{tαk|l∝an}bracketri}ht∝an}bracketle{tl′|βk∝an}bracketri}htGR\nβ(0,k)∝an}bracketle{tβk|m′∝an}bracketri}ht (14)\nEqs. ( 12),(13)and(14)provideasolutionforthevertex\nfunction that is significantly easier to analyse than the\noriginal Dyson equation.III. GILBERT DAMPING FOR A MAGNETIC\n2DEG\nThe first model we consider is a two-dimensional elec-\ntrongas(2DEG)model withferromagnetismandRashba\nspin-orbit interactions. We refer to this as the magnetic4\n2DEG (M2DEG) model. This toy model is almost never\neven approximately realistic14, but a theoretical study\nof its properties will prove useful in a number of ways.\nFirst, it is conducive to a fully analytical evaluation of\nthe Gilbert damping, which will allow us to precisely un-\nderstand the role of different actors. Second, it enables\nus to explain in simple terms why higher order vertex\ncorrections are significant when there is spin-orbit inter-\naction in the band structure. Third, the Gilbert damping\nof a M2DEG has qualitative features similar to those of\n(Ga,Mn)As.\nThe band Hamiltonian of the M2DEG model is\nH=k2\n2m+bk·σ (15)\nwherebk= (−λky,λkx,∆0), ∆0is the difference be-\ntween majority and minority spin exchange-correlation\npotentials, λis the strength of the Rashba SO couplingand/vector σ= 2/vector sis avectorofPaulimatrices. Thecorrespond-\ning eigenvalues and eigenstates are\nE±,k=k2\n2m±/radicalBig\n∆2\n0+λ2k2 (16)\n|αk∝an}bracketri}ht=e−iszφe−isyθ|α∝an}bracketri}ht (17)\nwhere φ=−tan−1(kx/ky) and θ=\ncos−1(∆0//radicalbig\n∆2\n0+λ2k2) are the spinor angles and\nα=±is the band index. It follows that\n∝an}bracketle{tm|α,k∝an}bracketri}ht=∝an}bracketle{tm|e−iszφe−isyθ|α∝an}bracketri}ht\n=e−imφdm,α(θ) (18)\nwheredm,α=∝an}bracketle{tm|e−isyθ|α∝an}bracketri}htis a Wigner function for\nJ=1/2 angular momentum15. With these simple spinors,\nthe azimuthal integral in Eq.( 14) can be performed an-\nalytically to obtain\nUR(A),R\nm,m′:l,l′=δm−m′,l−l′(u0+uzmm′)/summationdisplay\nα,β/integraldisplaydkk\n2πdmαGR(A)\nα(k)dlα(θ)dm′β(θ)GR\nβ(k)dl′β(θ), (19)\nwhere the Kronecker delta reflects the conservation of\nthe angular momentum along z, owing to the azimuthal\nsymmetry of the problem. In Eq.( 19)\ndm,m′=/parenleftbigg\ncos(θ/2)−sin(θ/2)\nsin(θ/2) cos(θ/2)/parenrightbigg\n,(20)\nand the retarded and advanced Green’s functions are\nGR(A)\n+=1\n−ξk−bk+(−)iγ+\nGR(A)\n−=1\n−ξk+bk+(−)iγ−, (21)\nwhereξk=k2−k2\nF\n2m,bk=/radicalbig\n∆2\n0+λ2k2, andγ±is (half)the golden-rulescatteringrate ofthe band quasiparticles.\nIn addition, Eq. ( 13) is readily inverted to yield\nΛR(A),R\n+,+= ΛR(A),R\n−,−= 0\nΛR(A),R\n+,−=1\n21\n1−UR(A),R\n+,−:+,−\nΛR(A),R\n−,+=1\n21\n1−UR(A),R\n−,+:−,+(22)\nIn order to make further progress analytically we as-\nsumethat (∆ 0,λkF,γ)<< EF=k2\nF/2m. It then follows\nthatγ+≃γ−≡γand that γ=πN2Du0+πN2Duz\n4≡\nγ0+γz. Eqs. ( 19) and ( 20) combine to give\nUR,R\n−,+:−,+=UR,R\n+,−:+,−= 0\nUA,R\n−,+:−,+= (γ0−γz)/bracketleftbiggi\n−b+iγcos4/parenleftbiggθ\n2/parenrightbigg\n+i\nb+iγsin4/parenleftbiggθ\n2/parenrightbigg\n+2\nγcos2/parenleftbiggθ\n2/parenrightbigg\nsin2/parenleftbiggθ\n2/parenrightbigg/bracketrightbigg\nUA,R\n+,−:+,−= (UA,R\n−,+:−,+)⋆(23)\nwhereb≃/radicalbig\nλ2k2\nF+∆2\n0and cosθ≃∆0/b. The first\nand second terms in square brackets in Eq.( 23) emerge\nfrom inter-band transitions ( α∝ne}ationslash=βin Eq. ( 19)), while\nthe last term stems from intra-band transitions ( α=β).Amusingly, Uvanishes when the spin-dependent scatter-\ning rate equals the Coulomb scattering rate ( γz=γ0); in\nthis particular instance vertex corrections are completely\nabsent. On the other hand, when γz= 0 and b << γ5\nwe have UA,R\n−,+:−,+≃UA,R\n+,−:+,−≃1, implying that vertex\ncorrections strongly enhance Gilbert damping (recall Eq.\n( 22)). We will discuss the role of vertexcorrectionsmore\nfully below.\n0.00 0.10 0.20 0.30 0.40\n1/(εFτ0)0.0000.0050.0100.015αG∆0=0.3 εF  ;  λ kF = 0 ; u0=3 uz\nno vertex corrections\n1st vertex correction\nall vertex corrections\nFIG. 2: M2DEG : Gilbert damping in the absence of spin-\norbit coupling. When the intrinsic spin-orbit interaction is\nsmall, the 1st vertex correction is sufficient for the evalua-\ntion of Gilbert damping, provided that the ferromagnet’s ex -\nchange splitting is large compared to the lifetime-broaden ing\nof the quasiparticle energies. For more disordered ferroma g-\nnets (EFτ0<5 in this figure) higher order vertex corrections\nbegin to matter. In either case vertex corrections are signi fi-\ncant. In this figure 1 /τ0stands for the scattering rate off spin-\nindependent impurities, defined as a two-band average at the\nFermi energy, and the spin-dependent and spin-independent\nimpurity strengths are chosen to satisfy u0= 3uz.\nAfter evaluating Λ( k) from Eqs. ( 12),( 22)and ( 23),\nthe last step is to compute\nPR(A),R=/integraldisplay\nkΛR(A),R\nα,β(k)sx\nβ,α(k)GR(A)\nα(k)GR\nβ(k).(24)\nSinceweareassumingthattheFermienergyisthelargest\nenergy scale, the integrand in Eq. ( 24) is sharply peaked\nat the Fermi surface, leading to PR,R≃0. In the case of\nspin-independent scatterers ( γz= 0→γ=γ0), tedious\nbut straightforward algebra takes us to\nαG(uz= 0) =N2D∆2\n0\n4s0γ0(λ2k2\nF)(b2+∆2\n0+2γ2\n0)\n(b2+∆2\n0)2+4∆2\n0γ2\n0.(25)\nEq. (29) agrees with results published in the recent\nliterature16. We note that αG(uz= 0) vanishes in the\nabsence of SO interactions, as expected. It is illustrative\nto expand Eq. ( 25) in the b >> γ 0regime:\nαG(uz= 0)≃N2D∆2\n0\n2s0/bracketleftbiggλ2k2\nF\n2(b2+∆2\n0)1\nγ0+λ4k4\nF\n(b2+∆2\n0)3γ0/bracketrightbigg\n(26)\nwhich displays intra-band ( ∼γ−1\n0) and inter-band ( ∼\nγ0) contributions separately. The intra-band damp-\ning is due to the dependence of band eigenenergies on0.00 0.05 0.10 0.15 0.20\n1/(εFτ0)0.00.20.40.60.81.0αG∆0=εF/3  ;  λ kF=1.2 εF  ; u0=3 uz\nno vertex corrections\n1st vertex correction\nall vertex corrections\nFIG.3:M2DEG :Gilbert dampingforstrongSOinteractions\n(λkF= 1.2EF≃4∆0). In this case higher order vertex cor-\nrections matter (up to 20 %) even at low disorder. This sug-\ngests that higher order vertex corrections will be importan t\nin real ferromagnetic semiconductors because their intrin sic\nSO interactions are generally stronger than their exchange\nsplittings.\n0.00 0.10 0.20 0.30 0.40 0.50\n1/(εFτ0)0.0000.0050.0100.0150.020αG∆0=0.3 εF  ;  λkF=∆0/5  ;  u0=3 uz\nintra−band\ninter−band\ntotal\nFIG. 4: M2DEG : Gilbert damping for moderate SO inter-\nactions ( λkF= 0.2∆0). In this case there is a crossover be-\ntween the intra-band dominated and the inter-band domi-\nnated regimes, which gives rise to a non-monotonic depen-\ndence of Gilbert damping on disorder strength. The stronger\nthe intrinsic SO relative to the exchange field, the higher th e\nvalue of disorder at which the crossover occurs. This is why\nthe damping is monotonically increasing with disorder in Fi g.\n( 2) and monotonically decreasing in Fig. ( 3).\nmagnetization orientation, the breathing Fermi surface\neffect4which produces more damping when the band-\nquasiparticles scatter infrequently because the popula-\ntion distribution moves further from equilibrium. The\nintra-band contribution to damping therefore tends to\nscale with the conductivity. For stronger disorder,\nthe inter-band term in which scattering relaxes spin-6\norientations takes over and αGis proportional to the\nresistivity. Insofar as phonon-scattering can be treated\nas elastic, the Gilbert damping will often show a non-\nmonotonic temperature dependence with the intra-band\nmechanism dominating at low-temperatures when the\nconductivity is largeand the inter-band mechanism dom-\ninatingathigh-temperatureswhentheresistivityislarge.\nFor completeness, we also present analytic results for\nthe case γ=γzin theb >> γ zregime:\nαG(u0= 0)≃N2D∆2\n0\n2s0/bracketleftbigg1\nγzλ2k2\nF\n6b2−2∆2\n0+γz3b4+6b2∆2\n0−∆4\n0\n(3b2−∆2\n0)3/bracketrightbigg\n(27)\nThis expression illustrates that spin-orbit (SO) interac-\ntions in the band structure are a necessary condition for\nthe intra-band transition contribution to αG. The in-\nterband contribution survives in absence of SO as long\nas the disorder potential is spin-dependent. Interband\nscatteringis possiblefor spin-dependent disorderbecause\nmajority and minority spin states on the Fermi surface\nare not orthogonal when their potentials are not identi-\ncal. Note incidentally the contrast between Eq.( 26) and\nEq. ( 27): in the former the inter-bandcoefficient is most\nsuppressed at weak intrinsic SO interaction while in the\nlatter it is the intra-band coefficient which gets weakest\nfor small λkF.\nMore general cases relaxing the (∆ 0,λkF,γ)<< E F\nassumption must be studied numerically; the results are\ncollected in Figs. ( 2), ( 3) and ( 4). Fig ( 2) highlights\nthe inadequacy of completely neglecting vertex correc-\ntions in the limit of weak spin-orbit interaction; the in-\nclusion of the the leading order vertex correction largely\nsolves the problem. However, Fig. ( 2) and ( 3) together\nindicate that higher order vertex corrections are notice-\nable when disorder or spin-orbit coupling is strong. In\nthe light of the preceding discussion the monotonic de-\ncay in Fig.( 3) may appear surprising because the inter-\nband contribution presumably increases with γ. Yet,\nthis argument is strictly correct only for weakly spin-\norbit coupled systems, where the crossover betwen inter-\nband and intra-band dominated regimes occurs at low\ndisorder. For strongly spin-orbit coupled systems the\ncrossover may take place at a scattering rate that is (i)\nbeyondexperimentalrelevanceand/or(ii)largerthanthe\nband-splitting, in which case the inter-band contribution\nbehaves much like its intra-band partner, i.e. O(1/γ).\nNon-monotonic behavior is restored when the spin-orbit\nsplitting is weaker, as shown in Fig. ( 4).\nFinally, our analysis opens an opportunity to quan-\ntify the importance of higher order impurity vertex-\ncorrections. Kohno, Shibata and Tatara11claim that the\nbare vertex along with the firstvertex correction fully\ncaptures the Gilbert damping of a ferromagnet, provided\nthat ∆ 0τ >>1. To first order in Uthe vertex function\nis\nΛR(A),R\nm,m′=sx\nm,m′+/summationdisplay\nll′UR(A),R\nm,m′:l,l′sx\nl,l′(28)Takingγ=γzfor simplicity, we indeed get\nlim\nλ→0αG≃Aγ+O(γ2)\nA(1)\nA(∞)= 1 (29)\nwhereA(1) contains the first vertex correction only, and\nA(∞) includes all vertex corrections. However, the state\nof affairs changes after turning on the intrinsic SO inter-\naction, whereupon Eq. ( 29) transforms into\nαG(λ∝ne}ationslash= 0)≃Bγ+C1\nγ\nB(1)\nB(∞)=∆2\n0(3b2−∆2\n0)3(3b2+∆2\n0)\n4b6(3b4+6b2∆2\n0−∆4\n0)\nC(1)\nC(∞)=(b2+∆2\n0)(3b2−∆2\n0)\n4b4(30)\nWhen ∆ 0<< λk F,bothintra-bandand inter-bandratios\nshow a significant deviation from unity17, to which they\nconverge as λ→0. In order to understand this behavior,\nlet us look back at Eq. ( 22). There, we can formally ex-\npand the vertex function as Λ =1\n2/summationtext∞\nn=0Un, where the\nn-th order term stems from the n-th vertex correction.\nFrom Eq. ( 23) we find that when λ= 0,Un∼O(γn)\nand thus n≥2 vertex corrections will not matter for the\nGilbert damping, which is O(γ)18whenEF>> γ. In\ncontrast, when λ∝ne}ationslash= 0 the intra-band term in Eq. ( 23)\nis no longer zero, and consequently allpowers of Ucon-\ntainO(γ0) andO(γ1) terms. In other words, all vertices\ncontribute to O(1/γ) andO(γ) in the Gilbert damping,\nespecially if λkF/∆0is not small. This conclusion should\nprove valid beyond the realm of the M2DEG because it\nrelies only on the mantra “intra-band ∼O(1/γ); inter-\nband∼O(γ)”. Our expectation that higher order vertex\ncorrectionsbe importantin (Ga,Mn)As will be confirmed\nnumerically in the next section.\nIV. GILBERT DAMPING FOR (Ga,Mn)As\n(Ga,Mn)As and other (III,Mn)V ferromagnets are like\ntransition metals in that their magnetism is carried\nmainly by d-orbitals, but unlike transition metals in that\nneither majority nor minority spin d-orbitals are present\nat the Fermi energy. The orbitals at the Fermi energy\nare very similar to the states near the top of the valence\nband states of the host (III,V) semiconductor, although\nthey are of course weakly hybridized with the minority\nand majority spin d-orbitals. For this reason the elec-\ntronic structure of (III,Mn)V ferromagnets is extremely\nsimple and can be described reasonably accurately with\nthe phenomenologicalmodel whichwe employin this sec-\ntion. Becausethe top ofthe valence band in (III,V) semi-\nconductors is split by spin-orbit interactions, spin-orbit\ncoupling plays a dominant role in the bands of these fer-\nromagnets. An important consequence of the strong SO7\ninteraction in the band structure is that diffusive vertex\ncorrections influence αGsignificantly at allorders; this\nis the central idea of this section.\nUsing a p-d mean-field theory model8for the ferro-\nmagnetic groundstate and afour-band sphericalmodel19\nfor the host semiconductor band structure, Ga 1−xMnxAs\nmay be described by\nH=1\n2m/bracketleftbigg/parenleftbigg\nγ1+5\n2γ2/parenrightbigg\nk2−2γ3(k·s)2/bracketrightbigg\n+∆0sz,(31)\nwheresis the spin operator projected onto the J=3/2\ntotal angular momentum subspace at the top of the va-\nlence band and {γ1= 6.98,γ2=γ3= 2.5}are the Lut-\ntinger parameters for the spherical-band approximation\nto GaAs. In addition, ∆ 0=JpdSNMnis the exchange\nfield,Jpd= 55meVnm3is the p-d exchange coupling,\nS= 5/2 is the spin of the Mn ions, NMn= 4x/a3is\nthe density of Mn ions, and a= 0.565nm is the lattice\nconstant of GaAs.\n0.00 0.10 0.20 0.30 0.40\n1/(εFτ0)0.0000.0200.0400.060αGp=0.6 nm−3 (εF=500 meV)  ;  x=0.04  ; u0=3 uz\nno vertex corrections\n1st vertex correction\nall vertex corrections\nFIG. 5: GaMnAs : Higher order vertex corrections make a\nsignificant contributionto Gilbert damping, dueto theprom i-\nnent spin-orbit interaction in the band structure of GaAs.\nxis the Mn fraction, and pis the hole concentration that\ndetermines the Fermi energy EF. In this figure, the spin-\nindependent impurity strength u0was taken to be 3 times\nlarger than the magnetic impurity strength uz. 1/τ0corre-\nsponds to the scattering rate off Coulomb impurities and is\nevaluated as a four-band average at the Fermi energy.\nThe ∆ 0= 0 eigenstates of this model are\n|˜α,k∝an}bracketri}ht=e−iszφe−isyθ|˜α∝an}bracketri}ht (32)\nwhere|˜α∝an}bracketri}htis an eigenstate of szwith eigenvalue ˜ α. Un-\nfortunately, the analytical form of the ∆ 0∝ne}ationslash= 0 eigen-\nstates is unknown. Nevertheless, since the exchange field\npreserves the azimuthal symmetry of the problem, the\nφ-dependence of the full eigenstates |αk∝an}bracketri}htwill be iden-\ntical to that of Eq. ( 32). This observation leads to\nUm,m′:l,l′∝δm−m′,l−l′, which simplifies Eq. ( 14). αG\ncanbe calculatednumericallyfollowingthe stepsdetailed0.00 0.10 0.20 0.30 0.40\n1/(εFτ0)0.0000.0020.0040.0060.0080.010αGp=0.2 nm−3 (εF=240 meV)  ;  x=0.08  ;  u0=3 uz\nintra−band\ninter−band\ntotal\nγ3=γ2=0.5\nFIG. 6: GaMnAs : When the spin-orbit splitting is reduced\n(in this case by reducing the hole density to 0 .2nm−3and\nartificially taking γ3= 0.5), the crossover between inter-\nand intra-band dominated regimes produces a non-monotonic\nshape of the Gilbert damping, much like in Fig. ( 4). When\neitherγ2orpis made larger or xis reduced, we recover the\nmonotonic decay of Fig.( 5).\nin the previous sections; the results are summarized in\nFigs. ( 5) and ( 6). Note that vertex corrections mod-\nerately increase the damping rate, as in the case of a\nM2DEG model with strong spin-orbit interactions. Fig.\n( 5) underlines both the importance of higher order ver-\ntex corrections in (Ga,Mn)As and the monotonic decay\nof the damping as a function of scattering rate. The lat-\nter signals the supremacy of the intra-band contribution\nto damping, accentuated at larger hole concentrations.\nHadtheintrinsicspin-orbitinteractionbeensubstantially\nweaker20,αGwould have traced a non-monotonic curve\nas shown in Fig. ( 6). The degree to which the intraband\nbreathing Fermi surface model effect dominates depends\non the details of the band-structure and can be influ-\nenced by corrections to the spherical model which we\nhaveadoptedheretosimplifythevertex-correctioncalcu-\nlation. The close correspondence between Figs. ( 5)-( 6)\nand Figs. ( 3)-( 4) reveals the success of the M2DEG\nas a versatile gateway for realistic models and justifies\nthe extensive attention devoted to it in this paper and\nelsewhere.\nV. ASSESSMENT OF THE\nTORQUE-CORRELATION FORMULA\nThus far we have evaluated the Gilbert damping for\na M2DEG model and a (Ga,Mn)As model using the\n(bare) spin-flip vertex ∝an}bracketle{tα,k|sx|β,k∝an}bracketri}htand its renormal-\nized counterpart ∝an}bracketle{tα,k|Λ|β,k∝an}bracketri}ht. The vertex corrected re-\nsults are expected to be exact for 1 /τsmall compared\nto the Fermi energy. For practical reasons, state-of-the-\nart band-structure calculations5forgo impurity vertex8\ncorrections altogether and instead employ the torque-\ncorrelation matrix element, which we shall denote as\n∝an}bracketle{tα,k|K|β,k∝an}bracketri}ht(see below for an explicit expression). In\nthis section we compare damping rates calculated using\nsx\nα,βvertices with those calculated using Kα,βvertices.\nWe also compare both results with the exact damping\nrates obtained by using Λ α,β. The ensuing discussion\noverlaps with and extends our recent preprint6.\nWe shall begin by introducing the following identity4:\n∝an}bracketle{tα,k|sx|β,k∝an}bracketri}ht=i∝an}bracketle{tα,k|[sz,sy]|β,k∝an}bracketri}ht\n=i\n∆0(Ek,α−Ek,β)∝an}bracketle{tα,k|sy|β,k∝an}bracketri}ht\n−i\n∆0∝an}bracketle{tα,k|[Hso,sy]|β,k∝an}bracketri}ht.(33)\nIn Eq. ( 33) we have decomposed the mean-field quasi-\nparticle Hamiltonian into a sum of spin-independent, ex-\nchange spin-splitting, and other spin-dependent terms:\nH=Hkin+Hso+Hex, whereHkinis the kinetic (spin-\nindependent) part, Hex= ∆0szis the exchange spin-\nsplitting term and Hsois the piece that contains the in-\ntrinsic spin-orbit interaction. The last term on the right\nhand side of Eq. ( 33) is the torque-correlation matrix\nelement used in band structure computations:\n∝an}bracketle{tα,k|K|β,k∝an}bracketri}ht ≡ −i\n∆0∝an}bracketle{tα,k|[Hso,sy]|β,k∝an}bracketri}ht.(34)\nEq. ( 33) allows us to make a few general remarks on\nthe relation between the spin-flip and torque-correlation\nmatrix elements. For intra-band matrix elements, one\nimmediately finds that sx\nα,α=Kα,αand hence the two\napproaches agree. For inter-band matrix elements the\nagreement between sx\nα,βandKα,βshould be nearly iden-\ntical when the first term in the final form of Eq.( 33)\nis small, i.e.when21(Ek,α−Ek,β)<<∆0. Since this\nrequirement cannot be satisfied in the M2DEG, we ex-\npect that the inter-band contributions from Kandsx\nwill always differ significantly in this model. More typi-\ncalmodels,likethefour-bandmodelfor(Ga,Mn)As, have\nband crossings at a discrete set of k-points, in the neigh-\nborhood of which Kα,β≃sx\nα,β. The relative weight of\nthese crossing points in the overall Gilbert damping de-\npends on a variety of factors. First, in order to make\nan impact they must be located within a shell of thick-\nness 1/τaround the Fermi surface. Second, the contri-\nbution to damping from those special points must out-\nweigh that from the remaining k-points in the shell; this\nmight be the case for instance in materials with weak\nspin-orbit interaction and weak disorder, where the con-\ntribution from the crossing points would go like τ(large)\nwhile the contribution from points far from the cross-\nings would be ∼1/τ(small). Only if these two con-\nditions are fulfilled should one expect good agreement\nbetween the inter-band contribution from spin-flip and\ntorque-correlationformulas. When vertexcorrectionsare\nincluded, of course, the same result should be obtained\nusing either form for the matrix element, since all matrixelements are between essentially degenerate electronic\nstates when disorder is treated non-perturbatively6,16.\n0.00 0.10 0.20 0.30 0.40 0.50\n1/(εFτ)0.0000.0020.0040.0060.0080.010αG∆0=0.8 εF ;  λ kF=0.05 εF  ;  uz =0\nK\nsx\nΛ\nFIG. 7:M2DEG : Comparison of Gilbert damping predicted\nusing spin-flip and torque matrix element formulas, as well a s\nthe exact vertex corrected result. In this figure the intrins ic\nspin-orbit interaction is relatively weak ( λkF= 0.05EF≃\n0.06∆0) and we have taken uz= 0. The torque correla-\ntionformula does notdistinguish between spin-dependenta nd\nspin-independent disorder.\n0.00 0.10 0.20 0.30 0.40 0.50\n1/(εFτ)0.000.050.100.150.20αG∆0=0.1 εF  ;  λ kF=0.5 εF  ;  uz=0\nK\nΛ\nsx\nFIG. 8:M2DEG : Comparison of Gilbert damping predicted\nusing spin-flip and torque matrix element formulas, as well a s\nthe exact vertex corrected result. In this figure the intrins ic\nspin-orbit interaction is relatively strong ( λkF= 0.5EF=\n5∆0) and we have taken uz= 0\nIn the remining part of this section we shall focus on a\nmore quantitative comparison between the different for-\nmulas. For the M2DEG it is straightforward to evaluate\nαGanalytically using Kinstead of sxand neglecting ver-\ntex corrections; we obtain\nαK\nG=N2D∆0\n8s0/bracketleftBigg\nλ2k2\nF\nb2∆0\nγ+/parenleftbiggλ2k2\nF\n∆0b/parenrightbigg2γ∆0\nγ2+b2/bracketrightBigg\n(35)9\nwhere we assumed ( γ,λkF,∆0)<< ǫF. By compar-\ning Eq. ( 35) with the exact expression Eq. ( 25), we\nfind that the intra-band parts are in excellent agreement\nwhen ∆ 0<< λk F, i.e. when vertex corrections are rela-\ntively unimportant. In contrast, the inter-band parts dif-\nfer markedly regardless of the vertex corrections. These\ntrendsarecapturedby Figs. (7) and( 8), which compare\nthe Gilbert damping obtained from sx,Kand Λ matrix\nelements. Fig. ( 7) corresponds to the weak spin-orbit\nlimit, whereitisfoundthatindisorderedferromagnets sx\nmaygrosslyoverestimatetheGilbertdampingbecauseits\ninter-band contributiondoes not vanish even as SO tends\nto zero. As explained in Section III, this flaw may be re-\npaired by adding the leading order impurity vertex cor-\nrection. The torque-correlation formula is free from such\nproblem because Kvanishes identically in absence of SO\ninteraction. Thus the main practical advantage of Kis\nthat it yields a physically sensible result without having\nto resort to vertex corrections. Continuing with Fig.( 7),\nat weak disorder the intra-band contributions dominate\nand therefore sxandKcoincide; even Λ agrees, because\nfor intra-band transitions at weak spin-orbit interaction\nthe vertex corrections are unimportant. Fig. ( 8) cor-\nresponds to the strong spin-orbit case. In this case, at\nlow disorder sxandKagree well with each other, but\ndiffer from the exact result because higher order vertex\ncorrections alter the intra-band part substantially. For a\nsimilar reason, neither sxnorKagree with the exact Λ\nat higher disorder. Based on these model calculations,\nwe do not believe that there are any objective grounds to\nprefer either the Ktorque-correlation or the sxspin-flip\nformula estimate of αGwhen spin-orbit interactions are\nstrong and αGis dominated by inter-band relaxation. A\nprecise estimation of αGunder these circumstances ap-\npears to require that the character of disorder, incud-\ning its spin-dependence, be accounted for reliably and\nthat the vertex-correction Dyson equation be accurately\nsolved. Carrying out this program remains a challenge\nboth because of technical complications in performing\nthe calculation for general band structures and because\ndisorder may not be sufficiently well characterized.\nAnalogous considerations apply for Figs. ( 9) and\n( 10), which show results for the four-band model re-\nlated to (Ga,Mn)As. These figures show results similar\nto those obtained in the strong spin-orbit limit of the\nM2DEG (Fig. 8). Overall, our study indicates that\nthetorque-correlation formula captures the intra-band\ncontributions accurately when the vertex corrections are\nunimportant, while it is less reliable for inter-band con-\ntributions unless the predominant inter-band transitions\nconnect states that are close in energy. The torque-\ncorrelation formula has the practical advantage that it\ncorrectly gives a zero spin relaxation rate when there is\nno spin-orbit coupling in the band structure and spin-\nindependent disorder. The damping it captures derives\nentirely from spin-orbit coupling in the bands. It there-\nfore incorrectly predicts, for example, that the damp-\ning rate vanishes when spin-orbit coupling is absent in0.00 0.10 0.20 0.30 0.40 0.50\n1/(εFτ)0.000.100.200.300.400.50αGp=0.4nm−3  (εF=380 meV) ;  x=0.08 ; uz=0\nsx\nΛ\nK\nFIG.9:GaMnAs : Comparison ofGilbertdampingpredicted\nusing spin-flip and torque matrix element formulas, as well a s\nthe exact vertex corrected result. pis the hole concentration\nthat determines the Fermi energy EFandxis the Mn frac-\ntion. Due to the strong intrinsic SO, this figure shows simila r\nfeatures as Fig.( 8).\n0.00 0.10 0.20 0.30 0.40\n1/(εFτ)0.000.050.100.150.20αGp=0.8nm−3  (εF=605 meV) ; x=0.04 ;  uz=0\nsx\nΛ\nK\nFIG. 10: GaMnAs : Comparison of Gilbert damping pre-\ndicted using spin-flip and torque matrix element formulas, a s\nwell as the exact vertex corrected result. In relation to Fig .\n( 9) the effective spin-orbit interaction is stronger, due to a\nlargerpand a smaller x.\nthe bands and the disorder potential is spin-dependent.\nNevertheless, assuming that the dominant disorder is\nnormally spin-independent, the K-formula may have a\npragmatic edge over the sx-formula in weakly spin-orbit\ncoupled systems. In strongly spin-orbit coupled systems\nthere appears to be little advantage of one formula over\nthe other. We recommend that inter-band and intra-\nband contributions be evaluated separately when αGis\nevaluated using the torque-correlation formula. For the\nintra-band contribution the sxandKlife-time formulas\nare identical. The model calculations reported here sug-10\ngest that vertex corrections to the intra-band contribu-\ntion do not normally have an overwhelming importance.\nWe conclude that αGcan be evaluated relatively reliably\nwhen the intra-band contribution dominates. When the\ninter-band contribution dominates it is important to as-\nsess whether or not the dominant contributions are com-\ning from bands that are nearby in momentum space, or\nequivalently whether or not the matrix elements which\ncontribute originate from pairs of bands that are ener-\ngetically spaced by much less than the exchange spin-\nsplitting at the same wavevector. If the dominant con-\ntributions are from nearby bands, the damping estimate\nshould have the same reliability as the intra-band contri-\nbution. If not, we conclude that the αGestimate should\nbe regarded with caution.\nTo summarize, this article describes an evaluation\nof Gilbert damping for two simple models, a two-\ndimensionalelectron-gasferromagnetmodelwith Rashba\nspin-orbit interactions and a four-band model which pro-\nvides an approximate description of (III, Mn)V of fer-\nromagnetic semiconductors. Our results are exact in\nthe sense that they combine time-dependent mean field\ntheory6with an impurity ladder-sum to all orders, hence\ngiving us leverage to make the following statements.First, previously neglected higher order vertex correc-\ntions become quantitatively significant when the intrin-\nsic spin-orbit interaction is larger than the exchange\nsplitting. Second, strong intrinsic spin-orbit interaction\nleads to the the supremacy of intra-band contributions in\n(Ga,Mn)As, with the corresponding monotonic decay of\nthe Gilbert damping as a function ofdisorder. Third, the\nspin-torque formalism used in ab-initio calculations of\nthe Gilbert damping is quantitatively reliable as long as\nthe intra-band contributions dominate andthe exchange\nfield is weaker than the spin-orbit splitting; if these con-\nditions are not met, the use of the spin-torque matrix\nelement in a life-time approximation formula offers no\nsignificant improvement overthe originalspin-flip matrix\nelement.\nAcknowledgments\nThe authors thank Keith Gilmore and Mark Stiles for\nhelpful discussions and feedback. This work was sup-\nported by the Welch Foundation and by the National\nScience Foundation under grant DMR-0606489.\n1Foranintroductoryreviewsee D.C. RalphandM.D.Stiles,\nJ. Magn. Mag. Mater. 320, 1190 (2008).\n2J.A.C. Bland and B. Heinrich (Eds.), Ultrathin Mag-\nnetic Structures III: Fundamentals of Nanomagnetism\n(Springer-Verlag, New York, 2005).\n3V. Korenman and R. E. Prange, Phys. Rev. B 6, 2769\n(1972).\n4V. Kambersky, Czech J. Phys. B 26, 1366 (1976); V. Kam-\nbersky, Czech J. Phys. B 34, 1111 (1984).\n5K. Gilmore, Y.U.IdzerdaandM.D. Stiles, Phys.Rev.Lett.\n99, 27204 (2007); V. Kambersky, Phys. Rev. B 76, 134416\n(2007).\n6Ion Garate and A.H. MacDonald, arXiv:0808.1373.\n7O. Gunnarsson, J. Phys. F 6, 587 (1976).\n8For reviews see T. Jungwirth et al., Rev. Mod. Phys. 78,\n809 (2006); A.H. MacDonald, P. Schiffer and N. Samarth,\nNature Materials 4, 195 (2005).\n9These simplified models sometimes have the advantage\nthat their parameters can be adjusted phenomenologically\nto fit experiments, compensating for inevitable inaccura-\ncies inab initio electronic structure calculations. This ad-\nvantage makes p−dmodels of (III,Mn)V ferromagnets\nparticularly useful. s−dmodels of transition elements are\nless realistic from the start because they do not account for\nthe minority-spin hybridized s−dbands which are present\nat the Fermi energy.\n10This is not the most general type of disorder for quasi-\nparticles with spin >1/2, but it will be sufficient for the\npurpose of this work.\n11H. Kohno, G. Tatara and J. Shibata, J. Phys. Soc. Japan\n75, 113706 (2006).\n12We assume that the spins of magnetic impurities are frozenalong the staticpart of the exchange field. In reality, the\ndirection of the impurity spins is a dynamical variable that\nis influenced by the magnetization precession.\n13G.D. Mahan, Many-Particle Physics (3rd Ed.), Physics of\nSolids and Liquids Series (2000)\n14A possible exception is the ferromagnetic 2DEG recently\ndiscovered in GaAs/AlGaAs heterostructures with Mn δ-\ndoping; see A. Bove et. al, arXiv:0802.3871v3.\n15J.J. Sakurai, Modern Quantum Mechanics , Addison-\nWesley (1994).\n16E.M. Hankiewicz, G. Vignale and Y. Tserkovnyak, Phys.\nRev. B 75, 174434 (2007). In their case the inter-band\nsplitting in the Green’s function is Ω, while in our case it\nis 2b. In addition, we neglect interactions between band\nquasiparticles.\n17C(1) and C(∞) differ by as much as 25%; the disparity\nbetween B(1) andB(∞) may be even larger.\n18The disorder dependence in αGoriginates not only from\nthe vertex part, but from the Green’s functions as well.\nIt is useful to recall thatR\nGσG−σ∝1/(b+isg(σ)γ) andR\nGσGσ∝1/γ.\n19P. Yu, M. Cardona, Fundamentals of Semiconductors (3rd\nEd.), Springer (2005).\n20Notwithstanding that the four-band model is a SO → ∞\nlimit of the more general six-band model, we shall tune the\neffective spin-orbit strength via p(hole concentration) and\nγ3.\n21Strictly speaking, it is |sx\nα,β|2≃ |Kα,β|2what is needed,\nrather than sx\nα,β≃Kα,β. The former condition is less de-\nmanding, and can occasionally be satisfied when Eα−Eβ\nis of the order of the exchange splitting."    },    {        "title": "2210.00366v1.Nonlinear_features_of_the_superconductor__ferromagnet__superconductor___varphi_0__Josephson_junction_in_ferromagnetic_resonance_region.pdf",        "content": "Nonlinear features of the superconductor{ferromagnet{superconductor '0Josephson\njunction in ferromagnetic resonance region\nAliasghar Janalizadeh1, Ilhom R. Rahmonov2;3;4, Sara A.\nAbdelmoneim5, Yury M. Shukrinov2;3;4, and Mohammad R. Kolahchi1\n1Department of Physics, Institute for Advanced Studies in Basic Sciences (IASBS), P.O. Box 45137-66731, Zanjan, Iran\n2BLTP, JINR, Dubna, Moscow Region, 141980, Russia\n3Dubna State University, Dubna, 141980, Russia\n4Moscow Institute of Physics and Technology, Dolgoprudny, 141700, Moscow Region, Russia\n5Physics department, Meno\fya University, Faculty of Science, 32511, Shebin Elkom,Egypt\n(Dated: October 4, 2022)\nWe demonstrate the manifestations of the nonlinear features in magnetic dynamics and IV-\ncharacteristics of the '0Josephson junction in the ferromagnetic resonance region. We show that\nat small values of system parameters, namely, damping, spin-orbit interaction, and Josephson to\nmagnetic energy ratio, the magnetic dynamics is reduced to the dynamics of the scalar Du\u000eng os-\ncillator, driven by the Josephson oscillations. The role of increasing superconducting current in the\nresonance region is clari\fed. Shifting of the ferromagnetic resonant frequency and the reversal of\nits damping dependence due to nonlinearity are demonstrated by the full Landau-Lifshitz-Gilbert-\nJosephson system of equations, and in its di\u000berent approximations. Finally, we demonstrate the\nnegative di\u000berential resistance in the IV{characteristics, and its correlation with the foldover e\u000bect.\nI. I. INTRODUCTION\nThe coupling of superconducting phase di\u000berence with\nmagnetic moment of ferromagnet in the '0junction leads\nto a number of unique features important for supercon-\nducting spintronics, and modern information technology\n[1{5]. It allows to control the magnetization preces-\nsion by superconducting current and a\u000bects the current{\nvoltage (IV) characteristics by magnetic dynamics in the\nferromagnet, in particular, to create a DC component in\nthe superconducting current [6{8]. A remarkable mani-\nfestation of such coupling is the possibility to stimulate\na magnetization reversal in the ferromagnetic layer by\napplying current pulse through the '0-junction [3, 9{13].\nThere are two features of our Josephson junction that\ncome into play in our study. One is the broken inver-\nsion symmetry in the weak link of the Josephson junc-\ntion, when the link is magnetic, which introduces an ex-\ntra phase in the current|-phase relation, preventing it\nfrom being antisymmetric. Such Josephson junctions are\nnamed'0junctions [1], and examples exist such as MnSi\nand FeGe. Second is the nonlinear property of the system\nthat makes for an anomalous resonance behavior [14].\nWe couple such a Josephson junction to the model\nthat describes the magnetodynamics in thin \flms or\nheterostructure, to form the Landau-Lifshitz-Gilbert-\nJosephson model (LLGJ)[14{16]. It is shown that for\na particular set of parameters, the coupled equations\nreduce to the dynamics of a Du\u000eng oscillator [14].\nThe cubic nonlinearity in this oscillator has applications\nin describing several e\u000bects in other models too [17].\nOne being the resonance e\u000bects in the antiferromagnetic\nbimeron in response to an alternating current, which has\napplications in the detection of weak signals [15, 18, 19].\nThe Gilbert damping term is added phenomenologi-\ncally to the Landau|-Lifshitz model, to reproduce the\ndamping of the precessing magnetic moment. Gilbertdamping is important in modeling other resonance fea-\ntures too, as its temperature dependence a\u000bects them\n[20, 21], and in return in the superconducting correla-\ntions that a\u000bect it [22]. The magnetization precession\nin the ultra thin Co20Fe60B20layer stimulated by mi-\ncrowave voltage under a large angle, needs modeling by\nDu\u000eng oscillator too. This gets help from the so called\nfoldover features, again due to nonlinearity [16, 23, 24].\nThe consequences of the nonlinear nature of the cou-\npled set of LLGJ system of equations in the weak cou-\npling regime was demonstrated recently in Ref. [14]. We\nshowed in this regime, where the Josephson energy is\nsmall compared to the magnetic energy, the '0Joseph-\nson junction is equivalently described by a scalar non-\nlinear Du\u000eng equation. An anomalous dependence of\nthe ferromagnetic resonant frequency (FMR) with the\nincrease of the Gilbert damping was found. We showed\nthat the damped precession of the magnetic moment is\ndynamically driven by the Josephson supercurrent, and\nthe resonance behavior is given by the Du\u000eng spring.\nThe obtained results were based on the numerical simu-\nlations. The role of dc superconducting current, and the\nstate with negative di\u000berential resistance (NDR) in IV-\ncharacteristic were not clari\fed. Also, the e\u000bects of the\nJosephson to magnetic energy ratio and the spin-orbit\ncoupling (SOC) were not investigated at that time.\nIn the present paper, we study the nonlinear aspects\nof the magnetic dynamics and IV-characteristics of the\n'0Josephson junction in the ferromagnetic resonance re-\ngion. We compare description of the anomalous damp-\ning dependence (ADD) exhibited by full LLGJ system\nof equations with approximated equations and demon-\nstrate the Du\u000eng oscillator features in the small param-\neter regime. E\u000bects of the Josephson to magnetic energy\nratio, and the spin-orbit coupling on the ADD, referred\nto earlier as the \u000b-e\u000bect [14] are demonstrated. By de-\nriving the formula which couples the dc superconduct-arXiv:2210.00366v1  [cond-mat.supr-con]  1 Oct 20222\ning current and maximal amplitude of magnetization we\ndiscuss the correlation of superconducting current and\nthe negative di\u000berential resistance in the resonance re-\ngion. Finally, we discuss the experimentally important\nfeatures by emphasizing the details of the magnetization\ndynamics and the IV-characteristics of the '0junction.\nWe have shown that in the limit of small system pa-\nrameters; that is, the Josephson to magnetic energy ra-\ntioG, the damping \u000b, and the spin-orbit coupling r, the\ndynamics is given by the Du\u000eng spring [14]. We focus\non the shift in resonance and the e\u000bects of nonlinear in-\nteractions. We give semi-analytic models to explain our\nresults in various limits.\nThe paper is organized as follows. In Section II we\noutline the theoretical model and discuss the methods\nof calculations. The ferromagnetic resonance and ef-\nfects of system parameters on the anomalous damping\ndependence are considered in Subsection A of Section\nIII. In Subsection B we present analytical description of\nthe dynamics and IV-characteristics of the '0junction\nat small system parameters. Manifestation of the nega-\ntive di\u000berential resistance in IV-characteristics through\nthe foldover e\u000bect is discussed. We compare the de-\nscription of the anomalous damping dependence by full\nLLGJ system of equation with approximated equation,\nand show how the Du\u000eng oscillator captures the non-\nlinearities in the small parameter regime in Subsection\nC. We present results on the critical damping and de-\nrive the formula which couples the dc superconducting\ncurrent and maximal amplitude of magnetization in the\nferromagnetic layer. Finally, in Section IV we concludes\nthe paper.\nII. II. MODELS AND METHOD\nThe following section is closely related to our work\nin [13]. The '0junction [6, 12, 25] that we study is shown\nin Fig.1. The current-phase relation in varphi 0junction\nhas the form Is=Icsin ('\u0000'0), where'0=rMy=M0,\nMydenotes the component of magnetic moment in ^ ydi-\nrection,M0is the modulus of the magnetization. The\nphysics of'0Josephson juncton is determined by system\nof equations which consists of Landau-Lifshits-Gilbert\n(LLG), resistively capacitively shunted junction (RCSJ)\nmodel expression with current-phase relation ( Is) de-\nscribed above, and Josephson relation between phase dif-\nference and voltage.\nThe dynamics of the magnetic moment Mis described\nby the LLG equation [26]\ndM\ndt=\u0000\rM\u0002Heff+\u000b\nM0\u0012\nM\u0002dM\ndt\u0013\n; (1)\nwhere Mis the magnetization vector, \ris the gyromag-\nnetic relation, Heffis the e\u000bective magnetic \feld, \u000bis\nGilbert damping parameter, M0=jMj.\nFigure 1. Schematic view of SFS '0Josephson junction. The\nexternal current applied along x direction, ferromagnetic easy\naxis is along z direction.\nIn order to \fnd the expression for the e\u000bective mag-\nnetic \feld we have used the model developed in Ref.[6],\nwhere it is assumed that the gradient of the spin-orbit\npotential is along the easy axis of magnetization taken to\nbe along ^z. In this case the total energy of the system\ncan be written as\nEtot=\u0000\b0\n2\u0019'I+Es(';' 0) +EM('0); (2)\nwhere'is the phase di\u000berence between the supercon-\nductors across the junction, Iis the external current,\nEs(';' 0) =EJ[1\u0000cos ('\u0000'0)], andEJ= \b 0Ic=2\u0019\nis the Josephson energy. Here \b 0is the \rux quantum,\nIcis the critical current, r=l\u001dso=\u001dFl= 4hL=~\u001dF,L\nis the length of Flayer,his the exchange \feld of the\nFlayer,EM=\u0000KVM2\nz=(2M2\n0), the parameter \u001dso=\u001dF\ncharacterizes a relative strength of spin-orbit interaction,\nKis the anisotropic constant, and Vis the volume of the\nferromagnetic ( F) layer.\nThe e\u000bective \feld for LLG equation is determined by\nHe\u000b=\u00001\nV@Etot\n@M\n=\nF\n\r\u0014\nGrsin\u0012\n'\u0000rMy\nM0\u0013\nby+Mz\nM0bz\u0015\n(3)\nwhere \n F=\rK=M 0is frequency of ferromagnetic reso-\nnance andG=EJ=(KV) determines the ratio of Joseph-\nson energy to magnetic one.\nIn order to describe the full dynamics '0junction the\nLLG equations should be supplemented by the equation\nfor phase di\u000berence ', i.e. equation of RCSJ model for\nbias current and Josephson relation for voltage. Accord-\ning to the extended RCSJ model, which takes into ac-\ncount derivative of '0phase shift, the current \rowing\nthrough the system in underdamped case is determined\nby\nI=~C\n2ed2'\ndt2+~\n2eR\u0014d'\ndt\u0000r\nM0dMy\ndt\u0015\n(4)\n+Icsin\u0012\n'\u0000r\nM0My\u0013\n:\nwhereIis the bias current, CandRare the capacitance\nand resistance of Josephson junction respectively. The3\nJosephson relation for voltage is given by :\n~\n2ed'\ndt=V: (5)\nWe note that in the framework of RCSJ{model the\ndisplacement current is proportional to the \frst deriva-\ntive of voltage (or second derivative of phase di\u000berence).\nFrom the other hand, the magnetization dynamics plays\nrole of the external force and \frst order derivative of '0\nis a source of external current for JJ. This was demon-\nstrated in Ref.[25, 27] where the authors included the \frst\nderivative of '0as the source of the electromotive force.\nVoltage is determined by the phase di\u000berence, and does\nnot depend on '0. From this point of view, in the frame-\nwork of RCSJ model the external current source cannot\nmodify the expression for displacement current. That's\nwhy we do not include the second derivative of varphi 0\nin our model.\nUsing (1), (3), (4) and (5) we can write the system of\nequations, in normalised variables, which describes the\ndynamics of '0junction\n_mx=!F\n1 +\u000b2f\u0000mymz+Grm zsin('\u0000rmy)\n\u0000\u000b[mxm2\nz+Grm xmysin('\u0000rmy)]g;\n_my=!F\n1 +\u000b2fmxmz\n\u0000\u000b[mym2\nz\u0000Gr(m2\nz+m2\nx) sin('\u0000rmy)]g;\n_mz=!F\n1 +\u000b2f\u0000Grm xsin('\u0000rmy)\n\u0000\u000b[Grm ymzsin('\u0000rmy)\u0000mz(m2\nx+m2\ny)]g;\n_V=1\n\fc[I\u0000V+r_my\u0000sin('\u0000rmy)];\n_'=V(6)\nwheremx;y;z =Mx;y;z=M0and satisfy the constraintP\ni=x;y;zm2\ni(t) = 1,\fc= 2eIcCR2=~is McCumber pa-\nrameter. In order to use the same time scale in the\nLLG and RCSJ equations in this system of equations\nwe have normalized time to the !\u00001\nc, where!c=2eIcR\n~,\nand!F= \n F=!cis the normalized frequency of ferro-\nmagnetic resonance \n F=\rK=M 0. Bias current is nor-\nmalized to the critical current Icand voltage V{ to the\nVc=IcR. The system of equations (6), is solved numer-\nically using the fourth-order Runge-Kutta method(see\nRef.[14]).\nIII. III. RESULTS AND DISCUSSION\nA. A. E\u000bect of system parameters on the\nanomalous damping dependence\nADD of the FMR frequency with increasing \u000bwas dis-\ncussed in Ref. [14]. It was found that the resonance\ncurves demonstrate features of Du\u000eng oscillator, re-\n\recting the nonlinear nature of Landau-Lifshitz-Gilbert-\n 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14\n 0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8mymax\nValpha=0.01\nalpha=0.02\nalpha=0.03\nalpha=0.04\nalpha=0.05\nalpha=0.06\nalpha=0.07\nalpha=0.08\nalpha=0.09\nalpha=0.1\n 0 0.1\n 0.5Figure 2. Maximal amplitude of magnetization\nmy\u0000component at each values of bias current and voltage\nalong IV-characteristics of the '0junction in the ferromag-\nnetic resonance region for various \u000b. Inset enlarges the main\nmaximum. Parameters: \fc= 25,G=0.05,r=0.05, !F= 0:5.\nJosephson (LLGJ) system of equations. There is a criti-\ncal damping value at which anomalous dependence comes\ninto play. This critical value depends on the system pa-\nrameters. Here we present the details of such transforma-\ntion from usual to anomalous dependence with variation\nin spin-orbit coupling and ratio of Josephson to magnetic\nenergy.\nTo investigate the e\u000bect of damping, we calculate\nthe maximal amplitude of magnetization component my\ntaken at each value of the bias current based on the\nLLGJ system of equations (6). In Fig.2 we show the\nvoltage dependence of maximal amplitude mmax\nyin the\nferromagnetic resonance region at di\u000berent damping pa-\nrameter and small values of Josephson to magnetic en-\nergy ratio G=0.05 and spin-orbit coupling r= 0:05. We\nfound that the ferromagnetic resonance curves demon-\nstrate the di\u000berent forms. An increase in damping shows\na nonuniform change in the resonant frequency: it is ap-\nproaching the !Finstead of moving away with increase\nin\u000b. We stress that this happens at small Gandr. We\nconsider that such behavior can be explained by the non-\nlinear nature of the LLGJ system of equations. There is\na manifestation of subharmonics of the FMR in Fig.2 at\n!= 1=2;1=3;1=4.\nWe usually expect the resonance peak to move away\nfrom resonance as the \u000bincreases. Figure 2 shows that\nthis normal e\u000bect is accompanied with an anomalous be-\nhaviour as can be seen in the inset to this \fgure, where\nthe resonance peak approaches !Fas\u000bincreases [14].\nThe manifestation of FMR in IV-characteristics of the\n'0junction at three values of damping parameter is\ndemonstrated in Fig. 3. The strong deviation of the\nIV-curve is observing at \u000b= 0:01, which is characteristic4\nFigure 3. Part of the IV characteristic of the '0junction\natG= 0:05;r= 0:05 and di\u000berent values of Gilbert damp-\ning. The numbers show \u000bvalue. Inset shows the total IV-\ncharacteristic and arrow indicates the resonance region\nvalue for many magnetic materials. This fact indicates\nthat ADD can be observed experimentally by measuring\nIV-characteristics in wide interval of the damping param-\neter.\nInteresting features of ADD appear by a variation of\nspin-orbit coupling. As it was demonstrated in Ref.[28],\nan increase in SOC leads to the essential change in IV-\ncharacteristics and magnetization precession in the fer-\nromagnetic resonance region. The nonlinearity is going\nstronger and the state with negative di\u000berential resis-\ntance appears at large SOC.\nFigure 4(a) demonstrates results of numerical simu-\nlations ofmmax\nydependence on \u000bat di\u000berent values of\nSOC parameter r. It shows two speci\fc features of ADD.\nFirst, with an increase in r, the critical value of Vpeakis\ndecreasing (the curve moves away from !F). The sec-\nond important feature is an increasing of \u000bcritwhich is\ndemonstrated in this \fgure by arrows.\nAnother model parameter which a\u000bects the phe-\nnomenon discussed in the present paper is the ratio G\nof Josephson to magnetic energies. Figure 4(b) demon-\nstrates the results of numerical simulations of mmax\nyde-\npendence on \u000bat di\u000berent values of G.\nSimilar to the e\u000bect of r, increasing Galso causes the\nvalue of\u000bcritto increase. By changing the volume of the\nferromagnetic layer, the ferromagnetic energy and con-\nsequently the value of G can be changed [6]. For small\nG, i.e. a situation where the magnetic energy is much\nlarger than the Josephson energy, the magnetic layer re-\nceives less energy, and its amplitude decreases in the y\ndirection, and also the maximum value of the oscillation\nfrequency is closer to the magnetic frequency, !F.\nVpeakα\n0.46 0.47 0.48 0.49 0.500.050.10.150.2\nαcrit\n1 23\nVpeakα\n0.46 0.47 0.48 0.49 0.500.050.10.150.2\nαcrit\n1 23Figure 4. (a) Demonstration of ADD at di\u000berent values of\nSOC parameter ratG= 0:05. Numbers indicate: 1 -\nr= 0:05; 2 -r= 0:1; 3 -r= 0:5; Arrows show critical \u000b\nvalue, corresponded to the reversal in the \u000bdependence (b)\nDemonstration of ADD at di\u000berent values of the Josephson\nto magnetic energy ratio Gatr= 0:05. Numbers indicate: 1\n-G= 0:01; 2 -G= 0:1; 3 -G= 1.\nB. B. Dynamics and IV-characteristics of the '0\njunction at small system parameters\nAs it was discussed in Refs.[6, 29, 30], in case of\nG;r;\u000b<< 1 andmz\u00191, \frst three equations of the sys-\ntem (6) can be simpli\fed. Taking into account '=!Jt\nand neglecting quadratic terms of mxandmy, we get\n(\n_mx=!F[\u0000my+Grsin(!Jt)\u0000\u000bmx]\n_my=!F[mx\u0000\u000bmy];(7)\nThis system of equations can be written as the second\norder di\u000berential equation with respect to my\nmy+ 2\u000b!F_my+!2\nFmy=!2\nFGrsin!Jt: (8)5\nCorresponding solution for myhas the form\nmy(t) =!+\u0000!\u0000\nrsin(!Jt)\u0000\r++\r\u0000\nrcos(!Jt);(9)\nwhere\n!\u0006=Gr2!F\n2!J\u0006!F\n\n\u0006; (10)\nand\n\r\u0006=Gr2!F\n2\u000b!J\n\n\u0006: (11)\nwith \n \u0006= (!J\u0006!F)2+ (\u000b!J)2(see Ref.[6] and corre-\nsponded Erratum[31]).\nWhen the Josephson frequency !Jis approaching the\nferromagnetic one !F,mydemonstrates the damped fer-\nromagnetic resonance. Di\u000berential resistance in the res-\nonance region is decreasing and it is manifested in the\nIV{characteristic as a resonance branch [7].\nTaking into account rmy<<1, we rewrite expression\nfor superconducting current as\nIs(t) = sin(!Jt\u0000rmy(t))\n= sin(!Jt)\u0000rmycos(!Jt) (12)\nUsing solution (9) we can obtain\nIs(t) = sin!Jt\u0000!+\u0000!\u0000\n2sin 2!Jt\n+\r++\r\u0000\n2cos 2!Jt+I0(\u000b) (13)\nwhere\nI0=\r++\r\u0000\n2: (14)\nThis superconducting current explains the appearance\nof the resonance branch in the IV{characteristic. The\ngenerated current I0can be expressed through the am-\nplitude ofmyand SOI parameter r\nI0=r\n2mmax\ny(!J); (15)\nwithmmax\ny(!J) being the frequency response of my.\nAt small model parameters \u000b<<Gr<< 1 of SFS'0\nJosephson junction, the states with a negative di\u000berential\nresistance appear in the IV-characteristics in the FMR\nregion. Due to the nonlinearity, the resonance peak is\nasymmetric. Increasing of nonlinearity leads to the bista-\nbility (foldover e\u000bect). A natural questions appears if the\nstates with a negative di\u000berential resistance are the origin\nof the foldover and ADD. In order to clarify this question,\nwe show in Fig.5 a part of IV{characteristics of '0junc-\ntion together with IV-characteristic of SIS junction in\nthe ferromagnetic resonance region and numerically cal-\nculated superconducting current through this junction.\nThe total IV{characteristics are demonstrated in the in-\nset to this \fgure.\nFigure 5. IV-characteristic of '0and SIS junctions and calcu-\nlated average superconducting current through the '0junc-\ntion.\nWe see the correlation of the foldover e\u000bect in super-\nconducting current (blue) with the NDR part of IV-curve.\nThe peak in the superconducting current and minimum\nof IV-curve are at the same voltage value. So, both ef-\nfects re\rect the nonlinear features of the ferromagnetic\nresonance in '0junction. But in contrast to the foldover\nand ADD e\u000bects, which start manifesting themselves at\nrelatively small deviations from linear case, the nonlin-\nearity in case of the NDR plays a more essential role.\nWe note that in the resonance region for considered\nlimit of model parameters the myamplitude are coupled\nto the value of superconducting current (see Eq.(15)). We\nstress the importance of the performed analysis demon-\nstrating the analytical coupling of time independent su-\nperconducting current and magnetization, re\recting the\nDu\u000eng oscillator features of the '0junction.\nAs it is well known, the states with negative di\u000beren-\ntial resistance appear in IV-characteristics of Josephson\nstructures in di\u000berent physical situations. In particular,\nthe nonlinear superconducting structures being driven\nfar from equilibrium exhibit NDR states [32]. The NDR\nstates plays an essential role in the applications related\nto the THz radiation emission [33]. A detailed expla-\nnation for di\u000berent types of negative di\u000berential resis-\ntance (NDR) in Josephson junction (i.e., N-shaped and\nS-shaped) is introduced in[34]. The authors emphasize\nthat the nonlinear behavior of the Josephson junction\nplays a key role in the NDR feature. In our case, the\nNDR states appear as a result of system's nonlinearity\nat small values of '0junction parameters, such as SOC,\nratio of Josephson to magnetic energy and Gilbert damp-\ning. We demonstrate these e\u000bects here by by presented\nresults of detailed investigation of the NDR state at dif-\nferent system parameters and discuss possibility of their\ncontrol near the ferromagnetic resonance.\nFigure 6 shows the e\u000bect of the spin-orbit coupling on\nthe IV-characteristic at G= 0:05 and\u000b= 0:01. We\nsee the NDR feature which is getting more pronounced\nwith increase in r. A further increase in rleads to the\njump down in voltage and then practically linear growth6\nIV\n0.49 0.495 0.5 0.505 0.510.460.470.480.490.09\n0.15\nIV\n0 0.5 100.20.40.60.811.2r =0.15\nFigure 6. Enlarged parts of the IV-curves in the resonance\nregion at di\u000berent values of SOC parameter rat\u000b= 0:01\nandG= 0:05. The numbers indicate the increasing of rfrom\n0.09 to 0.15 with an increment 0.01. Inset demonstrates total\nIV-characteristic ar r= 0:15\nof IV-characteristic.\nAn interesting question is concerning the e\u000bect of\nGilbert damping. Results of IV-characteristics simula-\ntions in the resonance region at a certain range of the\ndamping parameter \u000batG= 0:05 andr= 0:13 are\nshown in Fig.7(a). In this case, the most pronounced\ncharacteristic appears at \u000b= 0:01. At chosen set of pa-\nrametersG= 0:05 andr= 0:13, the range of \u000bwith\npronounced NDR features is 0 :016\u000b<0:014.\nThe maximal amplitude mmax\nyas a function of volt-\nage is shown in the Fig.7(b). Based on the results, pre-\nsented in Fig.7(a) and Fig.7(b), we came to the important\nconclusion that the foldover e\u000bect (bistability) and NDR\nstate have strong correlations and have the same origin\nrelated to the nonlinearity at small system parameters.\nBut anomalous damping dependence does not show a\none-to-one correlation with either negative di\u000berential re-\nsistance or foldover e\u000bect. The resonance peak positions\nofmmax\nyin bias current Ipeak, and in voltage Vpeakas the\nfunctions of \u000bare demonstrated in Fig.7(c). According\nto our results, we can divide \u000binterval into two regions\n(see Fig.7(c)). Region I introduces the values of \u000bwhere\nthe NDR feature is present, while in the Region II it dis-\nappears. In the Region II the foldover e\u000bect (bistability)\ndisappears as well, but ADD is realized.\nC. C. Du\u000eng oscillator features of '0junction and\ncritical damping\nThe system of equations (6) is nonlinear and very com-\nplex, so in order to provide analytical study of dynamics\nof the'0junction we need to derive an approximated\nequation for some limited values of model parameters.\nIn Ref. [14] was shown that the resonance curves demon-\nstrate features of Du\u000eng oscillator, re\recting the nonlin-\near nature of Landau-Lifshitz-Gilbert-Josephson (LLGJ)\nIV\n0.495 0.5 0.505 0.510.4650.470.4750.480.4850.49\n0.01(a)\n0.018\nIV\n0 0.5 100.20.40.60.811.2\nα= 0.01\nαIpeak, Vpeak\n0.01 0.012 0.014 0.016 0.018 0.020.470.480.490.5Ipeak\nVpeak(c)\nΙΙΙFigure 7. (a) Enlarged parts of the IV-curves at di\u000berent\nvalues of\u000b; (b) Voltage dependence of mmax\nyat di\u000berent \u000b;\n(c)\u000b-dependence of the resonance curve maximum in current\n(Ipeak) and voltage ( Vpeak). The numbers indicate the value\nof\u000bfrom 0.01 to 0.018 in (a) and from 0.01 to 0.02 in (b)\nby an increment 0.001. Results are obtained at r= 0:13, and\nG= 0:05.\nsystem of equations. In this section, we present analyt-\nical approach to describe the nonlinear dynamics of '0-\njunction and compare analytical results followed from ap-\nproximated Du\u000eng equation with numerical simulations\nof total system of equations (6). We show that in the\nlimit of\u000b << G;r << 1, we arrive to the Du\u000eng os-\ncillator. We start by \frst three equations of the (6) for7\nmagnetization components\n_mx\n!F=\u0000mymz+Grm zsin('\u0000rmy)\u0000\u000bmxm2\nz\n_my\n!F=mxmz\u0000\u000bmym2\nz (16)\n_mz\n!F=\u0000Grm xsin('\u0000rmy) +\u000bmz(m2\nx+m2\ny)\nSimplifying this system of equations by the same pro-\ncedure as it was done in Ref.[14] we can write equation\nformyas\nmy+2\u0018\u000b_my+\u00182(1+\u000b2)my\u0000\u00182(1+\u000b2\u0000\u000b4)m3\ny=\u00182\rsin':\n(17)\nFinally, by neglecting the \u000b2and\u000b4terms which are\nmuch smaller than 1, we come to well known Du\u000eng\nequation,\nmy+ 2!F\u000b_my+!2\nFmy\u0000!2\nFm3\ny=!2\nFGrsin': (18)\nIn the small parameter range, this Du\u000eng equation\ncan describe the dynamics of my. We will have the full\ndynamics, once we consider the coupling with the Joseph-\nson equation,\n'+1\n\fc[ _'\u0000r_my+ sin('\u0000rmy)] =1\n\fcI: (19)\nThe system of equations, (18) and (19) can replace the\nLLGJ equations in the limit of G;r<< 1 andG;r>>\u000b .\nTaking into account '=!Jtwe can right analytically\nobtained frequency response for equation (18)\n(mmax\ny)2=\u0000\nGr\u00012\n\u0002\n!2\u00001 +3\n4(mmaxy)2\u00032+\u0000\n2\u000b!\u00012\n(20)\nwhere!=!J=!F. From Eq. (20) we get\n(mmax\ny)6+8\n3(!2\u00001)(mmax\ny)4\n+\u00124\n3\u00132\u0014\n(!2\u00001)2+\u0000\n2\u000b!\u00012\u0015\n(mmax\ny)2\n\u0000\u00124\n3Gr\u00132\n= 0: (21)\nThis equation allows to determine analytically fre-\nquency dependence of the mmax\nyamplitude. To \fnd it\nwe solve the equation (21) by the Newton method. Re-\nsults of analytical calculations (blue dots) corresponded\nto (21) and numerical one (red doted line) corresponded\nto the full system of equation (6) are demonstrated in\nFig.8.\nFigure 8. Numerically (curve 1) and analytically (curve 2)\ncalculated amplitude dependence of my.\nFigure 9. Numerically calculated superconducting current for\nSFS junction (plot 1) and analytical I0(plot 2) and super-\nconducting current for SIS junction (plot 3).\nWe can see that they are close to each other which\nproves the correctness of the chosen approximation.\nBoth curves demonstrate an asymmetric resonance peak,\nwhich is common for Du\u000eng oscillator. When a role of\nthe cubic term is getting larger, we observe a bistability\nof the resonance curve, which is usually called a foldover\ne\u000bect. Note that the foldover e\u000bect can be also achieved\nby the damping decreasing; i.e., by the decreasing of dis-\nsipative term in (18), we can increase the in\ruence of the\ncubic term in this equation.\nThe comparison of analytically and numerically cal-\nculated superconducting current as a function of the\nJosephson frequency is demonstrated in Fig. 9. We note\nthat in our normalization V=!J. We can see the man-\nifestation of the asymmetric resonance peak in the fre-\nquency dependence of superconducting current. So, the\napproximated system of equations 7 re\rects one of the\nmain feature of Du\u000eng oscillator.\nFigure (10) compares anomalous damping dependence\nof the resonance peak of mmax\ny(V) calculated numeri-\ncally according to the full LLGJ system of equations (6)\nwith calculated numerically according to the generalized\nDu\u000eng model (equations (17, 19)). We see that in the\ndamping parameter interval [0.001 { 0.2] the coincidence8\nFigure 10. The \u000b-dependence of the resonance maximum of\nmmax\ny(V) in the damping parameter interval [0.001 { 0.12].\nGreen squares show results calculated numerically according\nto the full system of equations (6), blue circles show results\ncalculated numerically according to the generalized Du\u000eng\nand Josephson equations (17,19). The dashed line connects\nthe symbols to guide eyes. Solid line show analytical \u000b-\ndependence calculated according to the Eq. (22). All calcu-\nlation have been done at \fc= 25, G=0.05, r=0.05, !F= 0:5.\nof the dependences is enough good.\nUsing equation (18) with '=!Jt, we can \fnd (see\nSupplementary materials ??) a relation between posi-\ntion of the resonance peak in mmax\ny(V) dependence and\ndamping\n!peak=s\n1\u00003\u000b2\n2+1\n2r\n(1\u0000\u000b2)2\u000012(Gr\n4\u000b)2(22)\nwhere!peak=!J;peak\n!Fdetermines the position of the res-\nonance peak.\nEquation (22) allows to \fnd the formula for critical\ndamping\u000bcritwhich is an important parameter deter-\nmining the reversal point in damping dependence of the\nresonance peak of mmax\ny(V) .\nTaking into account equation (22) we can write equa-\ntion with respect of Gr=(4\u000b) (See supplementary mate-\nrials??).\n9\u0012Gr\n4\u000bcrit\u00134\n+ 3\u000b2\ncrit(10\u000b2\ncrit\u00001)\u0012Gr\n4\u000bcrit\u00132\n(23)\n\u00002\u000b4\ncrit(\u000b2\ncrit\u00001)2= 0\nUsing approximation 10 \u000b2\ncrit<<1 and\u000b2\ncrit<<1 it\ngives (see Supplementary Materials)\n\u000bcrit\u00191\n2sr\n3\n2Gr (24)\nFigure 11. Numerical calculations according to Eq. (6)\n(squares), analytical according to Eq. (23)(solid line) and\napproximated analytical according to Eq. (24) (dashed line).\nTable 1: A comparison between the numerical and an-\nalytical values of \u000bcrit:at di\u000berent values of Gandr.\nG r Gr\u000bcrit:;numerics \u000bcrit:;analytics\n0.01 0.05 0.0005 0.0100 0.0123\n0.05 0.05 0.0025 0.0300 0.0276\n0.05 0.10 0.0050 0.0400 0.0391\n0.05 0.30 0.0150 0.0700 0.0677\n0.05 0.50 0.0250 0.0900 0.0874\n0.10 0.05 0.0050 0.0391 0.0391\n0.60 0.05 0.0300 0.0950 0.0958\n0.70 0.05 0.0350 0.1000 0.1035\n1.00 0.05 0.0500 0.1200 0.1237\nFigure 11 presents comparison of numerical and ana-\nlytical results \u000bcritversusGr.\nAs we see, it shows a good agreement of numerical\nand analytical results of calculations at small product of\nJosephson to magnetic energy ratio and spin-orbit inter-\naction.\nIV. IV. CONCLUSIONS\nThe understanding of the nonlinear features of\nmagnetization dynamics in superconductor-ferromagnet-\nsuperconductor Josephson junction and their manifesta-\ntion in the IV-characteristics has implications for super-\nconductor spintronics, and modern information technol-\nogy. In'0junctions the nonlinear features can a\u000bect the\ncontrol of magnetization precession by superconducting\ncurrent and external electromagnetic radiation [28].\nHere, using numerical and analytic approaches, we\nhave demonstrated that at small system parameters,9\nnamely, the damping, spin-orbit interaction and Joseph-\nson to magnetic energy ratio in '0junction, magnetic dy-\nnamics is reduced to the dynamics of the scalar Du\u000eng\noscillator, driven by the Josephson oscillations. We have\nclari\fed the role of increasing superconducting current\nin the resonance region leading to the foldover e\u000bect in\nthe ferromagnet magnetization. We have demonstrated\nthe parameter dependence of the anomalous ferromag-\nnetic resonant shifting with anomalous damping depen-\ndence due to nonlinearity of the full LLGJ equation and\nin its di\u000berent approximations. We have derived the an-\nalytical expression for critical damping value. Also, we\ndemonstrated appearance of negative di\u000berential resis-\ntance in the IV-characteristics and the correlation with\noccurrence of the foldover e\u000bect in the magnetization of\nferromagnet.\nWe have stressed that the manifestation of negative\ndi\u000berential resistance is related to the nonlinear features\nof the system[34, 35]. It was demonstrated that in the\nsmall model parameters case the equation for magnetic\nsubsystem takes form of Du\u000eng equation where nonlin-\nearity manifest itself as the cubic term. We have shown\nthat the appearance of negative di\u000berential resistance in\nthe I-V curve is related to the appearance of foldover inthemmax\ny-Vcurve.\nWe believe that the experimentally measured IV-\ncharacteristics of '0junction with manifestations dis-\ncussed in detail in the present paper, would allow close\ninvestigations of its nonlinear features important for su-\nperconductor electronics and spintronics.\nV. SUPPLEMENTARY\nIn supplementary material are presented the details of\ncalculations for Eq.22 and Eq.24.\nVI. FUNDING\nNumerical simulations were funded by Project No. 18-\n71-10095 of the Russian Science Foundation. The pre-\nsented results concerning the calculations of DC super-\nconducting current in the section V are supported by the\nRussian Science Foundation in the framework of project\n22-42-04408. A.J. and M.R.K. are grateful to IASBS for\n\fnancial support.\n[1] Buzdin, A. Physical Review Letters 2008 ,101 (10),\n107005.\n[2] Linder, J., Robinson, J. W. 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P.\nIEEE Transactions on Applied Superconductivity 2014 ,\n24(6), 1{7.\n[35] Nagel, J., Speer, D., Gaber, T., Sterck, A., Eichhorn, R.,\nReimann, P., Ilin, K., Siegel, M., Koelle, D., Kleiner, R.\nPhysical Review Letters 2008 ,100, 217001."    },    {        "title": "1610.04598v2.Nambu_mechanics_for_stochastic_magnetization_dynamics.pdf",        "content": "arXiv:1610.04598v2  [cond-mat.mes-hall]  19 Jan 2017Nambu mechanics for stochastic magnetization\ndynamics\nPascal Thibaudeaua,∗, Thomas Nusslea,b, Stam Nicolisb\naCEA DAM/Le Ripault, BP 16, F-37260, Monts, FRANCE\nbCNRS-Laboratoire de Math´ ematiques et Physique Th´ eoriqu e (UMR 7350), F´ ed´ eration de\nRecherche ”Denis Poisson” (FR2964), D´ epartement de Physi que, Universit´ e de Tours, Parc\nde Grandmont, F-37200, Tours, FRANCE\nAbstract\nThe Landau-Lifshitz-Gilbert (LLG) equation describes the dynamic s of a damped\nmagnetization vector that can be understood as a generalization o f Larmor spin\nprecession. The LLG equation cannot be deduced from the Hamilton ian frame-\nwork, by introducing a coupling to a usual bath, but requires the int roduction of\nadditional constraints. It is shown that these constraints can be formulated ele-\ngantly and consistently in the framework of dissipative Nambu mecha nics. This\nhas many consequences for both the variational principle and for t opological as-\npects of hidden symmetries that control conserved quantities. W e particularly\nstudy how the damping terms of dissipative Nambu mechanics affect t he con-\nsistent interaction of magnetic systems with stochastic reservoir s and derive a\nmaster equation for the magnetization. The proposals are suppor ted by numer-\nical studies using symplectic integrators that preserve the topolo gical structure\nof Nambu equations. These results are compared to computations performed\nby direct sampling of the stochastic equations and by using closure a ssumptions\nfor the moment equations, deduced from the master equation.\nKeywords: Magnetization dynamics, Fokker-Planck equation, magnetic\nordering\n∗Corresponding author\nEmail addresses: pascal.thibaudeau@cea.fr (Pascal Thibaudeau),\nthomas.nussle@cea.fr (Thomas Nussle), stam.nicolis@lmpt.univ-tours.fr (Stam Nicolis)\nPreprint submitted to Elsevier September 18, 20181. Introduction\nIn micromagnetism, the transverse Landau-Lifshitz-Gilbert (LLG ) equation\n(1 +α2)∂si\n∂t=ǫijkωj(s)sk+α(ωi(s)sjsj−ωj(s)sjsi) (1)\ndescribes the dynamics of a magnetization vector s≡M/MswithMsthe sat-\nuration magnetization. This equation can be seen as a generalization of Larmor\nspin precession, for a collection of elementary classical magnets ev olving in an\neffective pulsation ω=−1\n¯hδH\nδs=γBand within a magnetic medium, charac-\nterized by a damping constant αand a gyromagnetic ratio γ[1].His here\nidentified as a scalar functional of the magnetization vector and ca n be consis-\ntently generalized to include spatial derivatives of the magnetizatio n vector [2]\nas well. Spin-transfer torques, that are, nowadays, of particula r practical rele-\nvance [3, 4] can be, also, taken into account in this formalism. In the following,\nwe shall work in units where ¯ h= 1, to simplify notation.\nIt is well known that this equation cannot be derived from a Hamiltonia n\nvariational principle, with the damping effects described by coupling t he magne-\ntization to a bath, by deforming the Poisson bracket of Hamiltonian m echanics,\neven though the Landau–Lifshitz equation itself is Hamiltonian. The r eason is\nthat the damping cannot be described by a “scalar” potential, but b y a “vector”\npotential.\nThis has been made manifest [5] first by an analysis of the quantum ve rsion\nof the Landau-Lifshitz equation for damped spin motion including arb itrary\nspin length, magnetic anisotropy and many interacting quantum spin s. In par-\nticular, this analysis has revealed that the damped spin equation of m otion is\nan example of metriplectic dynamical system [6], an approach which t ries to\nunite symplectic, nondissipative and metric, dissipative dynamics into one com-\nmon mathematical framework. This dissipative system has been see n afterwards\nnothing but a natural combination of semimetric dynamics for the dis sipative\npart and Poisson dynamics for the conservative ones [7]. As a conse quence, this\nprovided a canonical description for any constrained dissipative sy stems through\n2an extension of the concept of Dirac brackets developed originally f or conserva-\ntive constrained Hamiltonian dynamics. Then, this has culminated rec ently by\nobserving the underlying geometrical nature of these brackets a s certain n-ary\ngeneralizations of Lie algebras, commonly encountered in conserva tive Hamilto-\nnian dynamics [8]. However, despite the evident progresses obtaine d, no clear\ndirection emerges for the case of dissipative n-ary generalizations, and even\nno variational principle have been formulated, to date, that incorp orates such\nproperties.\nWhat we shall show in this paper is that it is, however, possible to de-\nscribe the Landau–Lifshitz–Gilbert equation by using the variationa l principle\nof Nambu mechanics and to describe the damping effects as the resu lt of in-\ntroducing dissipation by suitably deforming the Nambu–instead of th e Poisson–\nbracket. In this way we shall find, as a bonus, that it is possible to de duce\nthe relation between longitudinal and transverse damping of the ma gnetization,\nwhen writing the appropriate master equation for the probability de nsity. To\nachieve this in a Hamiltonian formalism requires additional assumptions , whose\nprovenance can, thus, be understood as the result of the prope rties of Nambu\nmechanics. We focus here on the essential points; a fuller account will be pro-\nvided in future work.\nNeglecting damping effects, if one sets H1≡ −ω·sandH2≡s·s/2, eq.(1)\ncan be recast in the form\n∂si\n∂t={si,H1,H2}, (2)\nwhere for any functions A,B,Cofs,\n{A,B,C} ≡ǫijk∂A\n∂si∂B\n∂sj∂C\n∂sk(3)\nis the Nambu-Poisson (NP) bracket, or Nambu bracket, or Nambu t riple bracket,\na skew-symmetric object, obeying both the Leibniz rule and the Fun damental\nIdentity [9, 10]. One can see immediately that both H1andH2are constants of\nmotion, because of the anti-symmetric property of the bracket. This provides the\ngeneralization of Hamiltonian mechanics to phase spaces of arbitrar y dimension;\n3in particular it does not need to be even. This is a way of taking into acc ount\nconstraints and provides a natural framework for describing the magnetization\ndynamics, since the magnetization vector has, in general, three co mponents.\nThe constraints–and the symmetries–can be made manifest, by no ting that\nit is possible to express vectors and vector fields in, at least, two wa ys, that can\nbe understood as special cases of Hodge decomposition.\nFor the three–dimensional case that is of interest here, this mean s that a\nvector field V(s) can be expressed in the “Helmholtz representation” [11] in the\nfollowing way\nVi≡ǫijk∂Ak\n∂sj+∂Φ\n∂si(4)\nwhereAis a vector potential and Φ a scalar potential.\nOn the other hand, this same vector field V(s) can be decomposed according\nto the “Monge representation” [12]\nVi≡∂C1\n∂si+C2∂C3\n∂si(5)\nwhich defines the “Clebsch-Monge potentials”, Ci.\nIf one identifies as the Clebsch–Monge potentials, C2≡H1,C3≡H2and\nC1≡D,\nVi=∂D\n∂si+H1∂H2\n∂si, (6)\nand the vector field V(s)≡˙s, then one immediately finds that eq. (2) takes the\nform\n∂s\n∂t={s,H1,H2}+∇sD (7)\nthat identifies the contribution of the dissipation in this context, as the expected\ngeneralization from usual Hamiltonian mechanics. In the absence of the Gilbert\nterm, dissipation is absent.\nMore generally, the evolution equation for any function, F(s) can be written\nas [13]\n∂F\n∂t={F,H1,H2}+∂D\n∂si∂F\n∂si(8)\nfor a dissipation function D(s).\n4The equivalence between the Helmholtz and the Monge representat ion im-\nplies the existence of freedom of redefinition for the potentials, CiandDand\nAiand Φ. This freedom expresses the symmetry under symplectic tra nsforma-\ntions, that can be interpreted as diffeomorphism transformations , that leave the\nvolume invariant. These have consequences for the equations of m otion.\nFor instance, the dissipation described by the Gilbert term in the Lan dau–\nLifshitz–Gilbert equation (1)\n∂D\n∂si≡α(˜ωi(s)sjsj−˜ωj(s)sjsi) (9)\ncannot be derived from a scalar potential, since the RHS of this expr ession is not\ncurl–free, so the function Don the LHS is not single valued; but it does conserve\nthe norm of the magnetization, i.e. H2. Because of the Gilbert expression,\nbothωandηare rescaled such as ˜ω≡ω/(1 +α2) andη→η/(1 +α2).\nSo there are two questions: (a) Whether it can lead to stochastic e ffects, that\ncan be described in terms of deterministic chaos and/or (b) Whethe r its effects\ncan be described by a bath of “vector potential” excitations. The fi rst case\nwas described, in outline in ref. [14], where the role of an external to rque was\nshown to be instrumental; the second will be discussed in detail in the following\nsections. While, in both cases, a stochastic description, in terms of a probability\ndensity on the space of states is the main tool, it is much easier to pre sent for\nthe case of a bath, than for the case of deterministic chaos, which is much more\nsubtle.\nTherefore, we shall now couple our magnetic moment to a bath of flu ctuating\ndegrees of freedom, that will be described by a stochastic proces s.\n2. Nambu dynamics in a macroscopic bath\nTo this end, one couples linearly the deterministic system such as (8) , to\na stochastic process, i.e. a noise vector, random in time, labelled ηi(t), whose\nlaw of probability is given. This leads to a system of stochastic differen tial\nequations, that can be written in the Langevin form\n∂si\n∂t={si,H1,H2}+∂D\n∂si+eij(s)ηj(t) (10)\n5whereeij(s) can be interpreted as the vielbein on the manifold, defined by the\ndynamical variables, s. It should be noted that it is the vector nature of the\ndynamical variables that implies that the vielbein, must, also, carry in dices.\nWe may note that the additional noise term can be used to “renorma lize”\nthe precession frequency and, thus, mix, non-trivially, with the Gilb ert term.\nThis means that, in the presence of either, the other cannot be ex cluded.\nWhen this vielbein is the identity matrix, eij(s) =δij, the stochastic cou-\npling to the noise is additive, whereas it is multiplicative otherwise. In th at\ncase, if the norm of the spin vector has to remain constant in time, t hen the\ngradient of H2must be orthogonal to the gradient of Dandeij(s)si= 0∀j.\nHowever, it is important to realize that, while the Gilbert dissipation te rm\nis not a gradient, the noise term, described by the vielbein is not so co nstrained.\nFor additive noise, indeed, it is a gradient, while for the case of multiplic ative\nnoise studied by Brown and successors there can be an interesting interference\nbetween the two terms, that is worth studying in more detail, within N ambu\nmechanics, to understand, better, what are the coordinate art ifacts and what\nare the intrinsic features thereof.\nBecause {s(t)}, defined by the eq.(10), becomes a stochastic process, we\ncan define an instantaneous conditional probability distribution Pη(s,t), that\ndepends, on the noise configuration and, also, on the magnetizatio ns0at the\ninitial time and which satisfies a continuity equation in configuration sp ace\n∂Pη(s,t)\n∂t+∂( ˙siPη(s,t)))\n∂si= 0. (11)\nAn equation for /an}b∇acketle{tPη/an}b∇acket∇i}htcan be formed, which becomes an average over all the\npossible realizations of the noise, namely\n∂/an}b∇acketle{tPη/an}b∇acket∇i}ht\n∂t+∂/an}b∇acketle{t˙siPη/an}b∇acket∇i}ht\n∂si= 0, (12)\nonce the distribution law of {η(t)}is provided. It is important to stress here\nthat this implies that the backreaction of the spin degrees of freed om on the\nbath can be neglected–which is by no means obvious. One way to chec k this is\nby showing that no “runaway solutions” appear. This, however, do es not ex-\n6haust all possibilities, that can be found by working with the Langevin equation\ndirectly. For non–trivial vielbeine, however, this is quite involved, so it is useful\nto have an approximate solution in hand.\nTo be specific, we consider a noise, described by the Ornstein-Uhlen beck\nprocess [15] of intensity ∆ and autocorrelation time τ,\n/an}b∇acketle{tηi(t)/an}b∇acket∇i}ht= 0\n/an}b∇acketle{tηi(t)ηj(t′)/an}b∇acket∇i}ht=∆\nτδije−|t−t′|\nτ\nwhere the higher point correlation functions are deduced from Wick ’s theorem\nand which can be shown to become a white noise process, when τ→0. We\nassume that the solution to eq.(12) converges, in the sense of ave rage over-the-\nnoise, to an equilibrium distribution, that is normalizable and, whose co rrelation\nfunctions, also, exist. While this is, of course, not at all obvious to p rove, evi-\ndence can be found by numerical studies, using stochastic integra tion methods\nthat preserve the symplectic structure of the Landau–Lifshitz e quation, even\nunder perturbations (cf. [16] for earlier work).\n2.1. Additive noise\nWalton [17] was one of the first to consider the introduction of an ad ditive\nnoise into an LLG equation and remarked that it may lead to a Fokker- Planck\nequation, without entering into details. To see this more thoroughly and to\nillustrate our strategy, we consider the case of additive noise, i.e. w heneij=\nδijin our framework. By including eq.(10) in (12) and in the limit of white\nnoise, expressions like /an}b∇acketle{tηiPη/an}b∇acket∇i}htmust be defined and can be evaluated by either an\nexpansion of the Shapiro-Loginov formulae of differentiation [18] an d taking the\nlimit ofτ→0, or, directly, by applying the Furutsu-Novikov-Donsker theore m\n[19, 20, 21]. This leads to\n/an}b∇acketle{tηiPη/an}b∇acket∇i}ht=−˜∆∂/an}b∇acketle{tPη/an}b∇acket∇i}ht\n∂si. (13)\n7where ˜∆≡∆/(1 +α2). Using the dampened current vector Ji≡ {si,H1,H2}+\n∂D\n∂si, the (averaged) probability density /an}b∇acketle{tPη/an}b∇acket∇i}htsatisfies the following equation\n∂/an}b∇acketle{tPη/an}b∇acket∇i}ht\n∂t+∂\n∂si(Ji/an}b∇acketle{tPη/an}b∇acket∇i}ht)−˜˜∆∂2/an}b∇acketle{tPη/an}b∇acket∇i}ht\n∂si∂si= 0 (14)\nwhere˜˜∆≡∆/(1 +α2)2and which is of the Fokker-Planck form [22]. This last\npartial differential equation can be solved directly by several nume rical methods,\nincluding a finite-element computer code or can lead to ordinary differ ential\nequations for the moments of s.\nFor example, for the average of the magnetization, one obtains th e evolution\nequation\nd/an}b∇acketle{tsi/an}b∇acket∇i}ht\ndt=−/integraldisplay\ndssi∂/an}b∇acketle{tPη(s,t)/an}b∇acket∇i}ht\n∂t=/an}b∇acketle{tJi/an}b∇acket∇i}ht. (15)\nFor the case of Landau-Lifshitz-Gilbert in a uniform precession field B, we\nobtain the following equations, for the first and second moments,\nd\ndt/an}b∇acketle{tsi/an}b∇acket∇i}ht=ǫijk˜ωj/an}b∇acketle{tsk/an}b∇acket∇i}ht+α[˜ωi/an}b∇acketle{tsjsj/an}b∇acket∇i}ht−˜ωj/an}b∇acketle{tsjsi/an}b∇acket∇i}ht] (16)\nd\ndt/an}b∇acketle{tsisj/an}b∇acket∇i}ht= ˜ωl(ǫilk/an}b∇acketle{tsksj/an}b∇acket∇i}ht+ǫjlk/an}b∇acketle{tsksi/an}b∇acket∇i}ht) +α[˜ωi/an}b∇acketle{tslslsj/an}b∇acket∇i}ht\n+ ˜ωj/an}b∇acketle{tslslsi/an}b∇acket∇i}ht−2˜ωl/an}b∇acketle{tslsisj/an}b∇acket∇i}ht] + 2˜˜∆δij (17)\nwhere ˜ω≡γB/(1 +α2). In order to close consistently these equations, one can\ntruncate the hierarchy of moments; either on the second /an}b∇acketle{t/an}b∇acketle{tsisj/an}b∇acket∇i}ht/an}b∇acket∇i}ht= 0 or third\ncumulants /an}b∇acketle{t/an}b∇acketle{tsisjsk/an}b∇acket∇i}ht/an}b∇acket∇i}ht= 0, i.e.\n/an}b∇acketle{tsisj/an}b∇acket∇i}ht=/an}b∇acketle{tsi/an}b∇acket∇i}ht/an}b∇acketle{tsj/an}b∇acket∇i}ht, (18)\n/an}b∇acketle{tsisjsk/an}b∇acket∇i}ht=/an}b∇acketle{tsisj/an}b∇acket∇i}ht/an}b∇acketle{tsk/an}b∇acket∇i}ht+/an}b∇acketle{tsisk/an}b∇acket∇i}ht/an}b∇acketle{tsj/an}b∇acket∇i}ht+/an}b∇acketle{tsjsk/an}b∇acket∇i}ht/an}b∇acketle{tsi/an}b∇acket∇i}ht\n−2/an}b∇acketle{tsi/an}b∇acket∇i}ht/an}b∇acketle{tsj/an}b∇acket∇i}ht/an}b∇acketle{tsk/an}b∇acket∇i}ht. (19)\nBecause the closure of the hierarchy is related to an expansion in po wers of\n∆, for practical purposes, the validity of eqs.(16,17) is limited to low v alues\nof the coupling to the bath (that describes the fluctuations). For example, if\none sets /an}b∇acketle{t/an}b∇acketle{tsisj/an}b∇acket∇i}ht/an}b∇acket∇i}ht= 0, eq.(16) produces an average spin motion independent of\nvalue that ∆ may take. This is in contradiction with the numerical expe riments\n8performed by the stochastic integration and noise average of eq.( 10) quoted in\nreference [23] and by experiments. This means that it is mandatory to keep\nat least eqs.(16) and (17) together in the numerical evaluation of t he thermal\nbehavior of the dynamics of the average thermal magnetization /an}b∇acketle{ts/an}b∇acket∇i}ht. This was\npreviously observed [24, 25] and circumvented by alternate secon d-order closure\nrelationships, but is not supported by direct numerical experiment s.\nThis can be illustrated by the following figure (1). For this given set of\nFigure 1: Magnetization dynamics of a paramagnetic spin in a constant magnetic field,\nconnected to an additive noise. The upper graphs (a) plot som e of the first–order moments of\nthe averaged magnetization vector over 102realizations of the noise, when the lower graphs\n(b) plot the associated model closed to the third-order cumu lant (eqs.(16)-(17), see text).\nParameters of the simulations : {∆ = 0.13 rad.GHz; α= 0.1;ω= (0,0,18) rad.GHz; timestep\n∆t= 10−4ns}. Initial conditions: s(0) = (1,0,0),/an}bracketle{tsi(0)sj(0)/an}bracketri}ht= 0 but /an}bracketle{ts1(0)s1(0)/an}bracketri}ht= 1.\nparameters, the agreement between the stochastic average an d the effective\nmodel is fairly decent. As expected, for a single noise realization, th e norm\nof the spin vector in an additive stochastic noise cannot be conserv ed during\nthe dynamics, but, by the average-over-the-noise accumulation process, this is\n9observed for very low values of ∆ and very short times. However, t his agreement\nwith the effective equations is lost, when the temperature increase s, because of\nthe perturbative nature of the equations (16-17). Agreement c an, however, be\nrestored by imposing this constraint in the effective equations, for a given order\nin perturbation of ∆, by appropriate modifications of the hierarchic al closing\nrelationships /an}b∇acketle{t/an}b∇acketle{tsisj/an}b∇acket∇i}ht/an}b∇acket∇i}ht=Bij(∆) or/an}b∇acketle{t/an}b∇acketle{tsisjsk/an}b∇acket∇i}ht/an}b∇acket∇i}ht=Cijk(∆).\nIt is of some interest to study the effects of the choice of initial con ditions. In\nparticular, how the relaxation to equilibrium is affected by choosing a c omponent\nof the initial magnetization along the precession axis in the effective m odel, e.g.\ns(0) = (1/√\n2,0,−1/√\n2) and by taking all the initial correlations,\n/an}b∇acketle{tsi(0)sj(0)/an}b∇acket∇i}ht=\n1\n20−1\n2\n0 0 0\n−1\n201\n2\n(20)\nThe results are shown in figure (2).\nBoth in figures (1) and (2), it is observed that the average norm of the spin\nvector increases over time. This can be understood with the above arguments.\nIn general, according to eq.(10) and because Jis a transverse vector,\n(1 +α2)sidsi\ndt=eij(s)siηj(t). (21)\nThis equation describes how the LHS depends on the noise realization ; so the\naverage over the noise can be found by computing the averages of the RHS. The\nsimplest case is that of the additive vielbein, eij(s) =δij. Assuming that the\naverage-over-the noise procedure and the time derivative commu te, we have\nd\ndt/angbracketleftbig\ns2/angbracketrightbig\n=2/an}b∇acketle{tsiηi/an}b∇acket∇i}ht\n1 +α2. (22)\nFor any Gaussian stochastic process, the Furutsu-Novikov-Don sker theorem\nstates that\n/an}b∇acketle{tsi(t)ηi(t)/an}b∇acket∇i}ht=/integraldisplay+∞\n−∞dt′/an}b∇acketle{tηi(t)ηj(t′)/an}b∇acket∇i}ht/angbracketleftbiggδsi(t)\nδηj(t′)/angbracketrightbigg\n. (23)\nIn the most general situation, the functional derivativesδsi(t)\nδηj(t′)can be calculated\n[26], and eq.(23) admits simplifications in the white noise limit. In this limit,\n10-2-1012\n0 1 2 3 4 5\nt (ns)-2-1012\nsxsy\nsz(a)\n(b)\nFigure 2: Magnetization dynamics of a paramagnetic spin in a constant magnetic field, con-\nnected to an additive noise. The upper graphs (a) plot some of the first–order moments of\nthe averaged magnetization vector over 103realizations of the noise, when the lower graphs\n(b) plot the associated model closed to the third-order cumu lant (eqs.(16)-(17), see text).\nParameters of the simulations : {∆ = 0.0655 rad.GHz; α= 0.1;ω= (0,0,18) rad.GHz;\ntimestep ∆ t= 10−4ns,s(0) =/an}bracketle{ts(0)/an}bracketri}ht= (1/√\n2,0,−1/√\n2),/an}bracketle{tsisj/an}bracketri}ht(0) = 0 except for (11)=1/2,\n(13)=(31)=-1/2, (33)=1/2 }.\n11the integration is straightforward and we have\n/angbracketleftbig\ns2(t)/angbracketrightbig\n=s2(0) + 6˜˜∆t, (24)\nwhich is a conventional diffusion regime. It is also worth noticing that w hen\ncomputing the trace of (17), the only term which remains is indeed\nd\ndt/an}b∇acketle{tsisi/an}b∇acket∇i}ht= 6˜˜∆ (25)\nwhich allows our effective model to reproduce exactly the diffusion re gime. Fig-\nure (3) compares the time evolution of the average of the square n orm spin\nvector. Numerical stochastic integration of eq.(10) is tested by in creasing the\n0 1 2 3 4 5\nt (ns)11,522,53\n<|s|2>mean over 103 runs\nmean over 104 runs\ndiffusion regime\nFigure 3: Mean square norm of the spin in the additive white no ise case for the following\nconditions: integration step of 10−4ns; ∆ = 0 .0655 rad.GHz; s(0) = (0 ,1,0);α= 0.1;\nω= (0,0,18) rad.GHz compared to the expected diffusion regime (see te xt).\nsize of the noise sampling and reveals a convergence to the predicte d linear\ndiffusion regime.\n122.2. Multiplicative noise\nBrown [27] was one of the first to propose a non–trivial vielbein, tha t takes\nthe form eij(s) =ǫijksk/(1 +α2) for the LLG equation. We notice, first of\nall, that it is present, even if α= 0, i.e. in the absence of the Gilbert term.\nAlso, that, since the determinant of this matrix [ e] is zero, this vielbein is not\ninvertible. Because of its natural transverse character, this vie lbein preserves the\nnorm of the spin for any realization of the noise, once a dissipation fu nctionD\nis chosen, that has this property. In the white-noise limit, the aver age over-the-\nnoise continuity equation (12) cannot be transformed strictly to a Fokker-Planck\nform. This time\n/an}b∇acketle{tηiPη/an}b∇acket∇i}ht=−˜∆∂\n∂sj(eji/an}b∇acketle{tPη/an}b∇acket∇i}ht), (26)\nwhich is a generalization of the additive situation shown in eq.(13). The conti-\nnuity equation thus becomes\n∂/an}b∇acketle{tPη/an}b∇acket∇i}ht\n∂t+∂\n∂si(Ji/an}b∇acketle{tPη/an}b∇acket∇i}ht)−˜˜∆∂\n∂si/parenleftbigg\neij∂\n∂sk(ekj/an}b∇acketle{tPη/an}b∇acket∇i}ht)/parenrightbigg\n= 0. (27)\nWhat deserves closer attention is, whether, in fact, this equation is invariant\nunder diffeomeorphisms of the manifold [28] defined by the vielbein, o r whether\nit breaks it to a subgroup thereof. This will be presented in future w ork. In the\ncontext of magnetic thermal fluctuations, this continuity equatio n was encoun-\ntered several times in the literature [22, 29], but obtaining it from fir st principles\nis more cumbersome than our latter derivation, a remark already qu oted [18].\nMoreover, our derivation presents the advantage of being easily g eneralizable\nto non-Markovian noise distributions [23, 30, 31], by simply keeping th e partial\nderivative equation on the noise with the continuity equation, and so lving them\ntogether.\nConsequently, the evolution equation for the average magnetizat ion is now\nsupplemented by a term provided by a non constant vielbein and one h as\nd/an}b∇acketle{tsi/an}b∇acket∇i}ht\ndt=/an}b∇acketle{tJi/an}b∇acket∇i}ht+˜˜∆/angbracketleftbigg∂eil\n∂skekl/angbracketrightbigg\n. (28)\nWith the vielbein proposed by Brown and assuming a constant extern al field,\n13one gets\nd/an}b∇acketle{tsi/an}b∇acket∇i}ht\ndt=ǫijk˜ωj/an}b∇acketle{tsk/an}b∇acket∇i}ht+α(˜ωi/an}b∇acketle{tsjsj/an}b∇acket∇i}ht−˜ωj/an}b∇acketle{tsjsi/an}b∇acket∇i}ht)\n−2∆\n(1 +α2)2/an}b∇acketle{tsi/an}b∇acket∇i}ht. (29)\nThis equation highlights both a transverse part, coming from the av erage over\nthe probability current Jand a longitudinal part, coming from the average\nover the extra vielbein term. By imposing, further, the second-or der cumulant\napproximation /an}b∇acketle{t/an}b∇acketle{tsisj/an}b∇acket∇i}ht/an}b∇acket∇i}ht= 0, i.e. “small” fluctuations to keep the distribution of\nsgaussian, a single equation can be obtained, in which a longitudinal rela xation\ntimeτL≡(1 +α2)2/2∆ may be identified.\nThis is illustrated by the content of figure (4). In that case, the ap proxima-\nFigure 4: Magnetization dynamics of a paramagnetic spin in a constant magnetic field, con-\nnected to a multiplicative noise. The upper graphs (a) plot s ome of the first–order moments\nof the averaged magnetization vector over 102realizations of the noise, when the lower graphs\n(b) plot the associated model closed to the third-order cumu lant (eq.(29), see text). Param-\neters of the simulations : {∆ = 0.65 rad.GHz; α= 0.1;ω= (0,0,18) rad.GHz; timestep\n∆t= 10−4ns}. Initial conditions: s(0) = (1,0,0),/an}bracketle{tsi(0)sj(0)/an}bracketri}ht= 0 but /an}bracketle{tsx(0)sx(0)/an}bracketri}ht= 1.\n14tion/an}b∇acketle{t/an}b∇acketle{tsisjsj/an}b∇acket∇i}ht/an}b∇acket∇i}ht= 0 has been retained in order to keep two sets of equations, three\nfor the average magnetization components and nine on the averag e second-order\nmoments, that have been solved simultaneously using an eight-orde r Runge-\nKutta algorithm with variable time-steps. This is the same numerical im ple-\nmentation that has been followed for the studies of the additive nois e, solving\neqs.(16) and (17) simultaneously. We have observed numerically tha t, as ex-\npected, the average second-order moments are symmetrical by an exchange of\ntheir component indices, both for the multiplicative and the additive n oise. In-\nterestingly, by keeping identical the number of random events tak en to evaluate\nthe average of the stochastic magnetization dynamics between th e additive and\nmultiplicative noise, we observe a greater variance in the multiplicative case.\nAs we have done in the additive noise case, we will also investigate briefl y the\nbehavior of this equation under different initial conditions, and in par ticular with\na non vanishing component along the z-axis. This is illustrated by the c ontent\nof figure (5). It is observed that for both figures (4) and (5), th e average spin\nconverges to the same final equilibrium state, which depends ultimat ely on the\nvalue of the noise amplitude, as shown by equation (27).\n3. Discussion\nMagnetic systems describe vector degrees of freedom, whose Ha miltonian\ndynamics implies constraints. These constraints can be naturally ta ken into\naccount within Nambu mechanics, that generalizes Hamiltonian mecha nics to\nphase spaces of odd number of dimensions. In this framework, diss ipation can\nbe described by gradients that are not single–valued and thus do no t define\nscalar baths, but vector baths, that, when coupled to external torques, can lead\nto chaotic dynamics. The vector baths can, also, describe non-tr ivial geometries\nand, in that case, as we have shown by direct numerical study, the stochastic\ndescription leads to a coupling between longitudinal and transverse relaxation.\nThis can be, intuitively, understood within Nambu mechanics, in the fo llowing\nway:\n15-1-0.500.51\n0 1 2 3 4 5\nt (ns)-1-0.500.51\nsxsy\nsz(a)\n(b)\nFigure 5: Magnetization dynamics of a paramagnetic spin in a constant magnetic field, con-\nnected to a multiplicative noise. The upper graphs (a) plot s ome of the first–order moments\nof the averaged magnetization vector over 104realizations of the noise, when the lower graphs\n(b) plot the associated model closed to the third-order cumu lant (eq.(29), see text). Param-\neters of the simulations : {∆ = 0.65 rad.GHz; α= 0.1;ω= (0,0,18) rad.GHz; timestep\n∆t= 10−4ns}. Initial conditions: s(0) =/parenleftbig\n1/√\n2,0,1/√\n2/parenrightbig\n,/an}bracketle{tsi(0)sj(0)/an}bracketri}ht= 0 except for\n/an}bracketle{ts1(0)s1(0)/an}bracketri}ht=/an}bracketle{ts1(0)s3(0)/an}bracketri}ht=/an}bracketle{ts3(0)s3(0)/an}bracketri}ht= 1/2.\n16The dynamics consists in rendering one of the Hamiltonians, H1≡ω·s,\nstochastic, since ωbecomes a stochastic process, as it is sensitive to the noise\nterms–whether these are described by Gilbert dissipation or couplin g to an\nexternal bath. Through the Nambu equations, this dependence is “transferred”\ntoH2≡ ||s||2/2. This is one way of realizing the insights the Nambu approach\nprovides.\nIn practice, we may summarize our numerical results as follows:\nWhen the amplitude of the noise is small, in the context of Langevin-\ndynamics formalism for linear systems and for the numerical modeling ofsmall\nthermal fluctuations in micromagnetic systems, as for a linearized s tochastic\nLLG equation, the rigorous method of Lyberatos, Berkov and Cha ntrell might\nbe thought to apply [32] and be expected to be equivalent to the app roach\npresented here. Because this method expresses the approach t o equilibrium of\nevery moment, separately, however, it is restricted to the limit of s mall fluctua-\ntions around an equilibrium state and, as expected, cannot captur e the transient\nregime of average magnetization dynamics, even for low temperatu re. This is a\nuseful check.\nWe have also investigated the behaviour of this system under differe nt sets of\ninitial conditions as it is well-known and has been thoroughly studied in [ 1] that\nin the multiplicative noise case (where the norm is constant) this syst em can\nshow strong sensitivity to initial conditions and it is possible, using ste reographic\ncoordinates to represent the dynamics of this system in 2D. In our additive noise\ncase however, as the norm of the spin is not conserved, it is not eas y to get long\nrun behavior of our system and in particular equilibrium solutions. Mor eover as\nwe no longer have only two independent components of spin, it is not p ossible\nto obtain a 2D representation of our system and makes it more comp licated to\nstudy maps displaying limit cycles, attractors and so on. Thus under standing\nthe dynamics under different initial conditions would require somethin g more\nand, as it is beyond the scope of this work, will be done elsewhere.\nTherefore, we have focused on studying the effects of the prese nce of an\ninitial longitudinal component and of additional, diagonal, correlation s. No\n17differences have been observed so far.\nAnother issue, that deserves further study, is how the probabilit y density\nof the initial conditions is affected by the stochastic evolution. In th e present\nstudy we have taken the initial probability density to be a δ−function; so it will\nbe of interest to study the evolution of other initial distributions in d etail, in\nparticular, whether the averaging procedures commute–or not. In general, we\nexpect that they won’t. This will be reported in future work.\nFinally, our study can be readily generalized since any vielbein can be ex -\npressed in terms of a diagonal, symmetrical and anti-symmetrical m atrices,\nwhose elements are functions of the dynamical variable s. Because ˙sis a pseu-\ndovector (and we do not consider that this additional property is a cquired by the\nnoise vector), this suggests that the anti-symmetric part of the vielbein should\nbe the “dominant” one. Interestingly, by numerical investigations , it appears\nthat there are no effects, that might depend on the choice of the n oise connection\nfor the stochastic vortex dynamics in two-dimensional easy-plane ferromagnets\n[33], even if it is known that for Hamiltonian dynamics, multiplicative and a d-\nditive noises usually modify the dynamics quite differently, a point that also\ndeserves further study.\nReferences\n[1] Giorgio Bertotti, Isaak D. Mayergoyz, and Claudio Serpico. Nonlinear\nMagnetization Dynamics in Nanosystems . Elsevier, April 2009. Google-\nBooks-ID: QH4ShV3mKmkC.\n[2] Amikam Aharoni. Introduction to the Theory of Ferromagnetism . Claren-\ndon Press, 2000.\n[3] Jacques Miltat, Gon¸ calo Albuquerque, Andr´ e Thiaville, and Caro le Vouille.\nSpin transfer into an inhomogeneous magnetization distribution. Journal\nof Applied Physics , 89(11):6982, 2001.\n[4] Dmitry V. Berkov and Jacques Miltat. Spin-torque driven magnet ization\n18dynamics: Micromagnetic modeling. Journal of Magnetism and Magnetic\nMaterials , 320(7):1238–1259, April 2008.\n[5] Janusz A. Holyst and Lukasz A. Turski. Dissipative dynamics of qu antum\nspin systems. Physical Review A , 45(9):6180–6184, May 1992.\n[6] /suppress Lukasz A. Turski. Dissipative quantum mechanics. Metriplectic d ynamics\nin action. In Zygmunt Petru, Jerzy Przystawa, and Krzysztof Ra pcewicz,\neditors,From Quantum Mechanics to Technology , number 477 in Lecture\nNotes in Physics, pages 347–357. Springer Berlin Heidelberg, 1996.\n[7] Sonnet Q. H. Nguyen and /suppress Lukasz A. Turski. On the Dirac approa ch to\nconstrained dissipative dynamics. Journal of Physics A: Mathematical and\nGeneral , 34(43):9281–9302, November 2001.\n[8] Josi A. de Azc´ arraga and Josi M. Izquierdo. N-ary algebras: A review\nwith applications. Journal of Physics A: Mathematical and Theoretical ,\n43(29):293001, July 2010.\n[9] Yoichiro Nambu. Generalized Hamiltonian Dynamics. Physical Review D ,\n7(8):2405–2412, April 1973.\n[10] Ra´ ul Ib´ a˜ nez, Manuel de Le´ on, Juan C. Marrero, and Dav id Martı´ n de\nDiego. Dynamics of generalized Poisson and Nambu–Poisson bracket s.\nJournal of Mathematical Physics , 38(5):2332, 1997.\n[11] Jerrold E. Marsden and Tudor S. Ratiu. Introduction to Mechanics and\nSymmetry , volume 17 of Texts in Applied Mathematics . Springer New York,\nNew York, NY, 1999.\n[12] Phillip Griffiths and Joseph Harris. Principles of Algebraic Geometry . John\nWiley & Sons, August 2014.\n[13] Minos Axenides and Emmanuel Floratos. Strange attractors in dissipative\nNambu mechanics: Classical and quantum aspects. Journal of High Energy\nPhysics , 2010(4), April 2010.\n19[14] Julien Tranchida, Pascal Thibaudeau, and Stam Nicolis. Quantum Mag-\nnets and Matrix Lorenz Systems. Journal of Physics: Conference Series ,\n574(1):012146, 2015.\n[15] George Eugene Uhlenbeck and Leonard S. Ornstein. On the The ory of the\nBrownian Motion. Physical Review , 36(5):823–841, September 1930.\n[16] Pascal Thibaudeau and David Beaujouan. Thermostatting the atomic spin\ndynamics from controlled demons. Physica A: Statistical Mechanics and\nits Applications , 391(5):1963–1971, March 2012.\n[17] Derek Walton. Rate of transition for single domain particles. Journal of\nMagnetism and Magnetic Materials , 62(2-3):392–396, December 1986.\n[18] V. E. Shapiro and V. M. Loginov. “Formulae of differentiation” an d their\nuse for solving stochastic equations. Physica A: Statistical Mechanics and\nits Applications , 91(3-4):563–574, May 1978.\n[19] Koichi Furutsu. On the statistical theory of electromagnetic waves in a\nfluctuating medium (I). Journal of Research of the National Bureau of\nStandards , 67D:303–323, May 1963.\n[20] Evgenii A. Novikov. Functionals and the Random-force Method in Tur-\nbulence Theory. Soviet Physics Journal of Experimental and Theoretical\nPhysics , 20(5):1290–1294, May 1964.\n[21] Valery I. Klyatskin. Stochastic Equations through the Eye of the Physicist:\nBasic Concepts, Exact Results and Asymptotic Approximatio ns. Elsevier,\nAmsterdam, 1 edition, 2005. OCLC: 255242261.\n[22] Hannes Risken. The Fokker-Planck Equation , volume 18 of Springer Series\nin Synergetics . Springer-Verlag, Berlin, Heidelberg, 1989.\n[23] Julien Tranchida, Pascal Thibaudeau, and Stam Nicolis. Closing th e hier-\narchy for non-Markovian magnetization dynamics. Physica B: Condensed\nMatter , 486:57–59, April 2016.\n20[24] Dmitry A. Garanin. Fokker-Planck and Landau-Lifshitz-Bloch e quations\nfor classical ferromagnets. Physical Review B , 55(5):3050–3057, February\n1997.\n[25] Pui-Wai Ma and Sergei L. Dudarev. Langevin spin dynamics. Physical\nReview B , 83:134418, April 2011.\n[26] Julien Tranchida, Pascal Thibaudeau, and Stam Nicolis. A functio nal\ncalculus for the magnetization dynamics. arXiv:1606.02137 [cond-mat,\nphysics:nlin, physics:physics] , June 2016.\n[27] William Fuller Brown. Thermal Fluctuations of a Single-Domain Partic le.\nPhysical Review , 130(5):1677–1686, June 1963.\n[28] Jean Zinn-Justin. QuantumField Theory and Critical Phenomena . Number\n113 in International series of monographs on physics. Clarendon P ress,\nOxford, 4. ed., reprinted edition, 2011. OCLC: 767915024.\n[29] Jos´ e Luis Garc´ ıa-Palacios and Francisco J. L´ azaro. Langev in-dynamics\nstudy of the dynamical properties of small magnetic particles. Physical\nReview B , 58(22):14937–14958, December 1998.\n[30] Pascal Thibaudeau, Julien Tranchida, and Stam Nicolis. Non-Mar kovian\nMagnetization Dynamics for Uniaxial Nanomagnets. IEEE Transactions\non Magnetics , 52(7):1–4, July 2016.\n[31] Julien Tranchida, Pascal Thibaudeau, and Stam Nicolis. Colored- noise\nmagnetization dynamics: From weakly to strongly correlated noise. IEEE\nTransactions on Magnetics , 52(7):1300504, 2016.\n[32] Andreas Lyberatos, Dmitry V. Berkov, and Roy W. Chantrell. A method\nfor the numerical simulation of the thermal magnetization fluctuat ions in\nmicromagnetics. Journal of Physics: Condensed Matter , 5(47):8911–8920,\nNovember 1993.\n21[33] Till Kamppeter, Franz G. Mertens, Esteban Moro, Angel S´ an chez, and\nA. R. Bishop. Stochastic vortex dynamics in two-dimensional easy- plane\nferromagnets: Multiplicative versus additive noise. Physical Review B ,\n59(17):11349–11357, May 1999.\n22"    },    {        "title": "2212.12673v1.Anatomy_of_ultrafast_quantitative_magneto_acoustics_in_freestanding_nickel_thin_films.pdf",        "content": "arXiv:2212.12673v1  [cond-mat.mtrl-sci]  24 Dec 2022Anatomy of ultrafast quantitative magneto-acoustics in fr eestanding nickel thin films\nAntonia Ghita1, Tudor-Gabriel Mocioi1, Alexey M. Lomonosov2, Jiwan\nKim3, Oleksandr Kovalenko1, Paolo Vavassori4,5, and Vasily V. Temnov1∗\n1LSI, Ecole Polytechnique, CEA/DRF/IRAMIS, CNRS,\nInstitut Polytechnique de Paris, F-91128, Palaiseau, Fran ce\n2B+W Department, Offenburg University of Applied Sciences, 77 652 Offenburg, Germany\n3Department of Physics, Kunsan National University, 54150 K unsan, Korea\n4CIC nanoGUNE BRTA, E-20018 Donostia-San Sebastian, Spain a nd\n5IKERBASQUE, Basque Foundation for Science, E-48013 Bilbao , Spain\n(Dated: December 27, 2022)\nWe revisit the quantitative analysis of the ultrafast magne to-acoustic experiment in a freestand-\ning nickel thin film by Kim and Bigot [1] by applying our recent ly proposed approach of magnetic\nand acoustic eigenmodes decomposition by Vernik et al. [2]. We show that the application of our\nmodeling to the analysis of time-resolved reflectivity meas urements allows for the determination\nof amplitudes and lifetimes of standing perpendicular acou stic phonon resonances with unprece-\ndented accuracy. The acoustic damping is found to scale as ∝ω2for frequencies up to 80 GHz\nand the peak amplitudes reach 10−3. The experimentally measured magnetization dynamics for\ndifferent orientations of an external magnetic field agrees w ell with numerical solutions of magneto-\nelastically driven magnon harmonic oscillators. Symmetry -based selection rules for magnon-phonon\ninteractions predicted by our modeling approach allow for t he unambiguous discrimination between\nspatially uniform and non-uniform modes, as confirmed by com paring the resonantly enhanced\nmagneto-elastic dynamics simultaneously measured on oppo site sides of the film. Moreover, the\nseparation of time scales for (early) rising and (late) decr easing precession amplitudes provide ac-\ncess to magnetic (Gilbert) and acoustic damping parameters in a single measurement.\nPACS numbers: Valid PACS appear here\nI. INTRODUCTION\nSince early experimental studies [3–5], ultrafast\nmagneto-elastic interactions driven by femtosecond light\npulsesareconvenientlydescribedinthetime-domain: the\ndynamics of magnetization driven by single or multiple\nacoustic pulses with picosecond duration are monitored\nusing the magneto-optical pump-probe technique. This\nintuitive picture allows for an elegant description of mag-\nnetization precession amplified by a sequence of acoustic\npulseswithanappropriatetimeintervalviathe magneto-\nacoustic coherent control mechanism [1, 6]. Moreover,\nthe time-domain picture of ultrafast magneto-acoustics\nfacilitates the interpretation of magnetization switching\n[7, 8], where the duration of acoustic pulses is shorter\nthan the period of ferromagnetic resonance (FMR) pre-\ncession.\nAn alternative view on ultrafast magneto-acoustics\nis provided in magneto-optical transient grating exper-\niments [9–12]. Here, the spectrally separated quasi-\nmonochromatic acoustic excitations allow for observing\nresonant amplification of FMR precession induced by\neach acoustic mode. The dependence of FMR frequency\non the external magnetic field makes it possible to tune\nthe FMR precession in resonance with an acoustic mode\nofinterest. Veryrecentlywehaveextended thisapproach\nto ultrafast magneto-acoustic dynamics in free-standing\nthin films and multilayers [2]. Our theoretical approach\n∗vasily.temnov@cnrs.fris based on eigenmodes decomposition of both acoustic\nand magnetization dynamics, which allows for a more in-\nsightful analysis of ultrafast magneto-acoustic dynamics\nexperimentsin termsofresonantmagneto-elasticinterac-\ntions between individual modes of longitudinal acoustic\nphononsand perpendicularstanding spin wave(magnon)\nmodes [13]. For instance, the application of such rigor-\nous theoretical analysis to resonant phonon-magnon in-\nteractions in freestanding multilayer structures predicts\nthe key role of the symmetry of magnetic and acoustic\nmodes in prescribing well-defined selection rules for in-\ndividual phonon-magnon interactions. One of the most\nrelevant conclusions was that in symmetric structures in-\nteractionsbetween magnon and phonon eigenmodes with\ndifferent symmetries were forbidden.\nIn this paper, with the purpose of bench-marking the\npower of our improved approach, we apply it to reinter-\npreting the experimental results by Kim and Bigot [1]\nobtained for a 300 nm freestanding nickel thin film. We\nshow that even for such thick structures, where frequen-\ncies of spatially uniform (FMR) and non-uniform (spin\nwave or magnon) modes cannot be distinguished using\nconventional approaches employed so far, our approach\nenables the detection of their excitation thanks to the\nsymmetry-dependentselectionrulesthatgoverntheirres-\nonant interaction with acoustic modes. The results of\nthis work are multiple: from the one side they demon-\nstrate the ability of our modeling to retrieve fundamen-\ntal parametersgoverningthe complex physics involved in\nultrafast magneto-acousticexperiments with an unprece-\ndented accuracy, from the other side corroboratethat the\nphysical picture embodied in our model is particularly2\nzs(t)Ultrafast \noptical \npump Ultrashort \noptical probe \n    FMR FMR+magnons \n  Time-resolved \nreflectivity, MOKE \n(rotation/ellipticity) Acoustic \n  pulseTime (~ps) \nLength (~ nm) \n4\n3\n2\n1\n0\n-1−∆ψ  (°) \n800 600 400 200 0\nTime delay (ps)-2 -1 01\n∆0 1 (\n R / R4 -)\n  =  65 ° \n 46.5 ° \n 35 ° \n 26 ° \n 15.5 ° (a) \n(b) \nFIG. 1. (a) Schematic picture of the experiment and acous-\ntic pulse propagating inside the sample. The shaded expo-\nnentially decaying functions illustrate the optical penet ration\ndepth of pump and probe pulses, respectively. (b) Experi-\nmental data for reflectivity and Kerr rotation.\ninsightful, for example by highlighting the importance of\nsymmetries in magneto-acoustics.\nII. EXPERIMENT\nFreestanding nickel membranes in the experiment in\nRef. [1] had a thickness L=300 nm and were obtained by\ndepositing Ni on a glass substrate with a layer of sodium\nchloride in between them. The layer was subsequently\ndissolved in water to leave the Ni film stretched on a\nsample holder with a hole. The film was stretched later-\nally by gluing a silver paste around the edges of the film,\nwhich created a static strain in Nickel upon drying out.\nThe nickel thin film was optically excited at the front\nside by a femtosecond pump pulse (400 nm wavelength,45 fs pulse duration, 10 kHz repetition rate, 1.5 mJ/cm2\nfluence), launching pulses of coherent longitudinal acous-\ntic phonons with a duration of a few picoseconds propa-\ngating inside the sample at a constant speed cs=6 nm/ps\n(see Fig. 1(a)). Due to the inverse magnetostrictive ef-\nfect, these acoustic pulses drove the magnetization dy-\nnamics inside the Ni film. Time-delayed probe pulses of\n800nm detected transient changes in the reflectivity and\nmagneto-opticalKerreffect(MOKE)rotationbothatthe\nfront and back sides of the sample. A rotating perma-\nnent magnet positioned on top of the sample produced a\nmagneticfieldwith reportedmagnitude µ0H∼0.4Tat a\nvariable angle ξwith respect to the surface normal. Due\nto the magnetic anisotropy the equilibrium direction of\nmagnetization was non-collinear with the external mag-\nnetic field and made an angle θwith the surface normal.\nFig. 1(b) shows the measured differential reflectivity\n∆R\nRand Kerr rotation ψat the back side of the film for\nfive different orientations of the external magnetic field:\nξ= 15.5◦, 26◦, 35◦, 46.5◦and 65◦. The slowly varying\nthermal background in reflectivity and Kerr rotation sig-\nnals originated from heat diffusion from the front to the\nback side of the film and will be subtracted throughout\nthemanuscriptinordertofacilitatethequantitativecom-\nparison with simulations of rapidly varying elastic and\nmagneto-elastic transients. Complementary Kerr rota-\ntion and reflectivity measurements have been performed\nat the front side ofthe film: these data will be introduced\nand discussed in Fig. 2(a) for reflectivity and Fig. 4(a)\nfor Kerr rotation.\nIII. PHYSICAL MODEL\nExcitation of acoustic and magnetic transients in fer-\nromagnetic nickel with femtosecond laser pulses can\nbe adequately described by the phenomenological two-\ntemperature model (TTM) [14], which governs the phe-\nnomena of ultrafast demagnetization on a deeply sub-\npicosecond time scale [15, 16] and generation of ultra-\nshort acoustic pulses on a picosecond time scale [17]. In\nthe currentpaper, we aregoing to disregardthe transient\nultrafast demagnetization and focus on the interaction\nbetween fs-laser-generated acoustic pulses and magneti-\nzation dynamics. Within the framework of the TTM,\nthe non-equilibrium hot electrons are initially generated\nthrough the absorption of an optical pump pulse within\nits skin depth. Subsequently, they transport energy into\nthe depth of the sample via electron diffusion and heat\nup the cold lattice via electron-phonon scattering. These\ncomplex spatio-temporal dynamics result in the emission\nof picosecond acoustic pulses caused by the thermal ex-\npansionofrapidlyheatedlattice. Incaseofafreestanding\nnickel film, these acoustic pulses generated at the front\nside of the sample, propagate through the film, are re-\nflected at the back Ni/air interface (with a reflection co-\nefficient equal to -1), and keep bouncing back and forth\nbetween these two interfaces before they decay due to\nvarious phonon scattering mechanisms.3\nThe magnetization dynamics induced by such ultra-\nshort acoustic pulses can be adequately described by\na phenomenological approach using magneto-elastically\ndriven Landau-Lifshitz-Gilbert (LLG) equations [3–5].\nAdapted to the experimental geometry in Fig. 1, the\nphenomenological free energy density F=FZ+Fd+\nFex+Fmetakes into account the Zeeman term FZ=\n−µ0M0m·Hdue tothepresenceoftheexternalmagnetic\nfieldH, the anisotropy energy Fd=/parenleftbig1\n2µ0M2\n0+K/parenrightbig\nm2\nz\nconsisting of the thin-film shape anisotropy and the phe-\nnomenological anisotropy constant Kdue to static built-\nin strains in a stretched nickel membrane, the exchange\nenergyFex=1\n2M2\n0/summationtext3\ni=1D/parenleftBig\n∂m\n∂xi/parenrightBig2\nand the magneto-\nelastic energy Fme(t) =b1m2\nzεzz(z,t) (b1≃107J/m3for\nNickel [18]) due to the interaction with an acoustic pulse\nεzz(z,t). The relation between the angle ξof the mag-\nnetic field and θmagnetization at equilibrium is given\nby:\nsin(θ−ξ) =˜M\n2Hsin2θ, (1)\nwhere˜M=M0+2K\nµ0M0andM0is the saturation magne-\ntization for Ni. The length of the magnetization vector\nstays constant in our model (assuming constant temper-\nature), so the magnetization dynamics can be describedwith the unit magnetization vector mand its precession\ns(z,t):\nm=m0+s(z,t), (2)\nwhich can be represented as a sum of magnetic eigen-\nmodes:\ns(z,t) =∞/summationdisplay\nn=0s(n)(t)cos(knz). (3)\nwherekn=πn/Lis the wavevector of the n-th magnetic\neigenmode and free boundary conditions for magnetiza-\ntion dynamics are assumed. The n= 0 magnetic eigen-\nmode with uniform spatial profile corresponds to ferro-\nmagnetic resonance (FMR), while higher-order n≥1\nmodes describe spatially non-uniform modes of exchange\nmagnons.\nIt has been shown [2, 19] that in the linear approxima-\ntion when the acoustic strains are small, the magneto-\nelasticallydriven dynamicsforeach magnonmode satisfy\nthe equation of a damped driven harmonic oscillator,\nd2s(n)\nz\ndt2+2αωnds(n)\nz\ndt+ω2\nns(n)\nz=fn(t),(4)\nwhereαis the Gilbert damping parameter and magnon\neigenfrequencies ωnobey\nωn=γµ0/radicalbigg/parenleftBig\nHcosξ−/parenleftBig\n˜M−˜Dk2n/parenrightBig\ncosθ/parenrightBig2\n+/parenleftBig\nHsinξ+˜Dk2nsinθ/parenrightBig/parenleftBig\nHsinξ+/parenleftBig\n˜M+˜Dk2n/parenrightBig\nsinθ/parenrightBig\n.(5)\nHere˜D=D/(/planckover2pi1γµ0) is the exchange stiffness ( D=\n430meV˚A2from Ref. [13]) and γdenotes the gyro-\nmagnetic ratio.\nThe external magneto-elastic driving force\nfn(t) =Pn(H)/integraldisplayL\n0εzz(z,t)cos(knz)dz, (6)\nis proportional to the overlap integral between the\nmagnon eigenmode with the acoustic strain pulse\nεzz(z,t). For our experimental geometry the prefactor\nPn(H) =µ0γ2b1sin(2θ)/parenleftBig\n˜Dk2\nnsinθ+Hsinξ/parenrightBig\nM0L(7)\nis proportional to the magnetostriction coefficient b1and\ndepends both on the magnitude and orientation of an\nexternal magnetic field H.\nUnderstanding the magneto-elasticdynamics governed\nby Eq. (4) is facilitated by decomposing the acoustic\nstrain pulse in its eigenmodes according to\nεzz(z,t) =∞/summationdisplay\np=1ε(p)\nzz(z)e−γptcos(ωpt+ϕp).(8)We assume acoustic eigenmodes to oscillate at frequen-\nciesωp=cskpand decay with damping constants γp;\nϕpdenote their initial phases. In a freestanding film,\nthe acoustic eigenmodes obey the free boundary condi-\ntionsfortheacousticdisplacement(correspondingtozero\nstrains at both Ni/air interfaces) resulting in\nε(p)\nzz(z) =apsin(kpz), (9)\nwherekp=πp/Lis the wavevector of the p-th acous-\ntic eigenmode. Using the decomposition of the acoustic\nstrain in its respective eigenmodes, the expression of the\nmagneto-elastic driving force becomes\nfn(t) =Pn(H)∞/summationdisplay\np=1Inpape−γptcos(ωpt+ϕp).(10)\nHere we have introduced the overlap integral\nInp=/integraldisplayL\n0cos(knz)sin(kpz)dz (11)\nbetweenthe n-thmagneticand p-thacousticeigenmodes.\nTo sum up this section, after having quantified the\nacoustic strain and decomposed it in its eigenmodes, we4\nTime (ns) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Reflectivity (a.u.) x10-4 \n-2 02\nExperiment \nFit \n100\n200\n300\nStrain x10 -3 \n-2 02Time (ns) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Reflectivity (a.u.) x10-4 \n-2 02\nExperiment \nFit z (nm) (a)\n(b) \n(c) \nFIG. 2. (a) Experimentally measured reflectivity at the fron t side of the Ni film, superimposed with its fit as a superpositi on\nof 10 decaying sinusoidal eigenmodes. (b) Color map showing the reconstituted strain inside the film as a function of posi tion\nand time. (c) Experimental reflectivity measurement at the b ack side of the Ni film, together with its fit.\ncan use equations (4 - 5) and (10 - 11) to simulate the\ntime evolution of the magnetization precession. To solve\nEq. (4) numerically, we use the 4th-order Runge-Kutta\nmethod.\nIV. ANALYSIS OF ACOUSTIC DATA\nThe change in reflectivity measured by the probe pulse\nafter excitation with the pump pulse is related to the\nstrain inside the film through the sensitivity function\nf(z), which dependsonthe opticalconstantsofNi. Their\nrelation (as in [20]) is given by\n∆R(t)\nR= 2Re/parenleftBig∆r(t)\nr/parenrightBig\n=/integraldisplayL\n0εzz(z,t)f(z)dz,(12)\nwhere we used the expression for the sensitivity function\nin the complex notation [17]:\nf(z) =16π\nλRe/parenleftbigg\ni∂˜n\n∂εzz˜n\n˜n2−1ei4π˜n\nλz/parenrightbigg\n.(13)\nHere ˜n= 2.48 + 4.38idenotes the complex index of\nrefraction of Ni at the probe wavelength λ= 800nm\nand its derivative with respect to the applied strain\n∂˜n\n∂εzz= 0.6−1.8i[17] is called the photo-elastic coeffi-\ncient. The spatial dependence of the sensitivity function\nis dominated by the exponential decay ∝exp(−z/δskin)within the optical penetration depth of the probe pulse\nδskin=λ\n4πIm(˜n)= 14.5nm.\nUsing the previous decomposition of strain into eigen-\nmodes, we obtain the following expression for the mea-\nsured reflectivity\n∆R(t)\nR=∞/summationdisplay\np=1apJpe−γptcos(ωpt+ϕp),(14)\nwhere we can interpret Jp=/integraltextL\n0f(z)sin(kpz)dzas the\ndetection integral of the p-th mode. This expression for\nthe transient reflectivity shows that it can be represented\nas a sum of damped harmonic oscillations, suggesting\nthat the Fourier transform of the signal could be useful\nincharacterizingtheacousticeigenmodes. Panels(a)and\n(c) in Fig. 2 show the reflectivity signal at the front and\nback side of the film, while Fig. 3(a) displays the Fourier\ntransformof the backside reflectivity signal. The data al-\nlows us to distinguish ten peaks in the Fourier transform,\nso we carry out the analysis using the first ten acoustic\nmodes.\nTo obtain the amplitudes, lifetimes, and phases of the\nrespective acoustic modes, we performed a non-linear\nleast squares fitting (using the Levenberg-Marquardt al-\ngorithm) of the reflectivity data at the back side with\nEq. (14), where ap,γp, andϕpare taken as fit param-\neters and frequencies ωpare extracted from the Fourier\nspectrum. As an initial guess for the amplitudes ap, we5\n020 40 |fft(Reflectivity)| (a.u.) Experiment \nFi t\n020 40 \nData\nQuadratic fit \nPolynomial fit \n0 20 40 60 80 100\nFrequency (GHz) 00.5110 -3 \nExperiment \nh = 20 nm \nh = 60 nm \nTTM (a)\n(b) \n(c)\nFIG. 3. (a) Fourier Transform of the experimental reflectiv-\nity and its fit, for the backside of the film. (b) Damping as a\nfunction of frequency, as extracted from the fitting procedu re.\nThe result follows a quadratic law to a very good approxima-\ntion. (c) Amplitudes of the acoustic modes, as extracted fro m\nthe fitting procedure. The dashed line represents the acous-\ntic frequency spectrum according to Eq. (5) in Ref. [17] for\nNickel thin films excited by weak fs-pump pulses. , ampli-\ntudes obtained with Eq. (15) assuming exponential heating\nprofiles with h=20 nm and 60 nm are shown for comparison\n(continuous lines).\n.\nused the integrated intensity of each peak, and for damp-\ning constants γp- the width of spectral lines from the\nFourier spectrum. It is important to choose a rather pre-\ncise initial guess for amplitudes and damping constants\nas the algorithm is highly sensitive to them. The results\nof such fitting in Fig. 2(a,c) and Fig. 3(a) appear to be\nin an excellent agreement with experimental data.\nFurthermore, panels (b) and (c) in Fig. 3 show the de-\npendence of damping and amplitudes for the first eight\neigenmodesontheirfrequencies. Thesecond-degreepoly-\nnomial fit of damping γpas a function of frequency shows\nthat the quadratic term dominates. Thus, we can con-\nclude that damping scales quadratically with frequency\nup to around 80GHz. This result is consistent withRef. [21], suggesting that the attenuation mechanism is\ndue to the phonon-phonon scattering.\nThe straightforward attempt to understand the fitted\namplitudes (Fig. 3(c)) within the framework of the TTM\nfailed. Using Eq. (5) and the set of experimental fit pa-\nrameters in the low-fluence excitation regime (pump flu-\nence∼0.01mJ/cm2)inNickelthinfilms[17]resultsinthe\ninitial heat penetrationdepth h=20nm that onlyslightly\nexceeds the optical skin depth of our pump pulses. In\nterms of the acoustic amplitudes the results of the TTM\nare well-approximated by a simplified phenomenological\nmodel assuming an instantaneous heating with an expo-\nnential profile ∝exp(−z/h) giving rise to\nap∝/integraldisplayL\n0e−z/hsin(kpz)dz, (15)\nThe strong disagreement between the theory and the ex-\nperimental data indicates that this modeling cannot be\napplied. In the strong-excitation regime used in this ex-\nperiment the parameters of the two-temperature model\ndisplay strong dependence on the pump fluence [22] re-\nsulting in larger electronic heat capacity and weaker\nelectron-phononcoupling. Botheffectsfavoralargerheat\npenetrationdepthmediatedbyhotelectrondiffusiondur-\ning the increaseelectron-phononrelaxationtime. We can\naccount for this effect by assuming a larger heat penetra-\ntion depth h=60 nm, which provides a better approxi-\nmation to the experimental data. However, it is clear\nthat the discussed theoretical models represent oversim-\nplifications and a further systematic study of the strong\nexcitation regime of picosecond acoustic pulses is neces-\nsary.\nUsing the obtained amplitudes, phases, and damping\nparameters, we can reconstruct the strain inside the film\nas a function of space and time. Figure 2(b) shows an\nevolution of spatial strain that is in accordance with the\nintuitive image of an acoustic echo propagating back and\nforth, undergoing reflections at both ends of the film and\ndamping in time. But, in addition to this intuitive pic-\nture, the eigenmode decomposition also helps explaining\nthe broadening of acoustic echo in time domain, which is\ndue to the frequency-dependent damping.\nV. ANALYSIS OF MAGNETIZATION DYNAMICS\nMagnetization dynamics in the Ni film are analyzed by\nmeasuring the Kerr rotation angle. The depth sensitivity\nfunctionofMOKEbecomesimportantincaseofultrafast\nmagnetization dynamics, varying within the skin depth\noflight dueto the presenceofspatiallynon-uniformhigh-\nfrequency magnons. The relation between the detected\nKerr rotation and the magnetization precession sz(z,t)\ninside the film is given by\n∆ψ(t)\nψs=/integraldisplayL\n0sz(z,t)g(z)dz (16)\nwhereψsis the static Kerr rotation angle, ∆ ψ(t) is its\nchange due to magnetization precession, and g(z) is the6\n0 0.2 0.4 0.6 0.8 \nTime  (ns) (a) Experiment\nFrontside\nBackside \n0 0.2 0.4 0.6 0.8 \nTime (ns) (b) Theory\nFrontside \nBackside m (t) (a.u.) z 46.5 ° \n 35 ° \n 26 ° \n 15.5 °   =  65 ° \nFIG. 4. Comparison of magnetization dynamics at the front\nand back side of the film, as obtained (a) experimentally and\n(b) from our simulations.\ndepth sensitivity function for the polar MOKE [23]:\ng(z) =4π\nλRe/parenleftBig\niQMO˜n2\n1+ ˜n2e−i4π˜n\nλz/parenrightBig\n.(17)\nHere, unlike the acoustical sensitivity function, the\nmagneto-optical response is valued by the complex\nmagneto-optical(Voigt) constant QMO=i˜ǫxy\n˜ǫxx=−(4.9+\n10.5i)×10−3[24].\nUsing our previous decomposition in magnon eigen-\nmodes, we get an expression that ties the dynamics of\nindividual magnon modes to the Kerr rotation:\n∆ψ(t)\nψs=∞/summationdisplay\nn=0˜Jns(n)\nz(t), (18)\nwhere˜Jn=/integraltextL\nz=0g(z)cos(knz)dzisthedetectionintegral\nof then-th magnon mode. Using this expression, we can\nfit the results of our magnetization dynamics simulation\nwith those of the experiment in the next section.\nVI. RESONANT PHONON-MAGNON INTERACTIONS\nThe experimental data for the back- and front side-\nKerr rotation are presented in Fig. 4(a) for different ori-\nentations of the external magnetic field. We simulate\nthe Kerr rotation by solving Eq. (4) for each magnon\nand using the sensitivity function defined in the previ-\nous section to obtain the Kerr rotation from Eq. (18).\nIn order to reach an agreement with experimental datafor all angles ξin Fig. 4, we have used the values for\nthe anisotropy constant K, magnitude of the magnetic\nfieldHand Gilbert damping αas fit parameters. Us-\ning the magnetic field of 0 .3T, the anisotropy constant\nK= 2.05·105J/m3and the Gilbert damping α= 0.04,\nwe achieve the quantitative agreement between the ex-\nperimental data and simulations (Fig. 4). The value of\nthe magnetic field stays within the expected error bar for\na permanent magnet placed on top of the sample. The\nvalue of the Gilbert damping is equal to the one obtained\nin a recent study of ultrafast magnetization dynamics in\nnickel nanomagnets [25]. The simulated Kerr rotation at\nthe front and back sides are represented in Fig. 4(a,b).\nWe observe an excellent agreement between the experi-\nmental data and simulations, except for the initial ther-\nmal excitation of magnetization at the front side, which\nwe did not account for in our model.\nGiven this agreement between experimental data and\nsimulations, we analyze the peculiarities of the magne-\ntization dynamics at different angles. For 15 .5◦, the\nmagnetization dynamics at the back side is in-phase with\nthat at the front-side, while for 65◦they areπ-shifted.\nMoreover,we notice that the magnetizationprecessionat\n15.5 and 65◦lasts longer and is stronger than at other\nangles. While at 15 .5◦and 65◦there are some slowly\nvarying long-lived dynamics, at angles 26, 35, and 46 .5◦\nwe observe weak, somewhat irregular beating patterns.\nA complementary perspective is presented in Fig. 5\nwhich show the reconstructed magnetization dymnamics\ninside the sample (Fig. 5(b-c)) as a function of position\nand time. After the transient regime dies out, the mag-\nnetization profile for 15 .5◦is approximately uniform in\nspace, suggesting that the dynamics is dominated by the\nFMR (n= 0) mode. On the other hand, the dynamics\nat 65◦follow the spatially antisymmetric profile with re-\nspect to the middle of the film, which suggests that in\nthis configuration the n= 1 magnon dominates. This\nconclusion is inline with the observed π-phase shift be-\ntween the data at the front and the backside at 65◦, see\nFig. 4. At an intermediate angle of 35◦, the magnetiza-\ntion dynamics with a much smaller amplitude are mainly\nvisible at early delays times. This suggests that in this\nintermediate regime no magnon modes are resonantlyex-\ncited.\nAll these observations can be explained by a simple\ntheory for a driven harmonic oscillator. The main result\nis that the oscillation amplitude is resonantly enhanced\nwhen the natural frequency (here, that of magnons ωn)\nequalsthe drivingfrequency(in ourcase, thatofphonons\nωp). Away from resonance, the transient regime is char-\nacterized by a beating pattern because of the difference\nbetween the natural and driving frequencies.\nFigure 5(a) shows the amplitude of magnetization pre-\ncession as a function of its frequency and the angle of\nthe external magnetic field. Two bright spots are visible:\none at the point where the frequency of the first phonon\n(p= 1) matches that of FMR ( n= 0) and another\none where the frequency of the second phonon ( p= 2)\nmatches that of the first magnon ( n= 1).7\nFIG. 5. (a) Fourier transform of simulated Kerr rotation as a function of magnetic field angle. The dashed lines indicate t he\nfrequencies of phonons and the continuous white lines are th e dispersion curves of magnons. Vertical cross-sections in to the\nheat map, corresponding to the three angles in panel b, are sh own. (b) Magnetization dynamics inside the film, as a functio n\nof position and time, for three experimental angles. (c) Mag netization profiles, taken at the times indicated in panel (b ) by\ndashed lines.\nFIG. 6. Representation of magnonic eigenmodes (along the\nhorizontal axis) and acoustic eigenmodes (along the axis),\nwith their corresponding overlap integrals.\nHowever, for our symmetric freestanding membrane\nthe overlap integral Inp(Fig. 6) is zero when the acous-\ntic modes possess a different symmetry from that of the\nmagnon modes. This means that symmetric (antisym-\nmetric) acoustic modes will interact only with symmet-ric (antisymmetric) magnon modes, respectively. There-\nfore, the symmetry-based selection rules become as im-\nportant for resonant phonon-magnon interaction as the\npreviously mentionned frequency matching condition.\nThese considerations enable us to identify the driv-\ning forces of the magnetization dynamics observed at the\nthree angles shown in Fig. 5. At 15 .5◦, the frequency\nof the first phonon ( p= 1) matches that of the first few\nmagnonsbutthesymmetryofthemodesallowsonlyeven\nmagnons to be excited. Since the overlap integral decays\nwith increasing magnon number, the dominant magnetic\nmode at 15◦is the FMR ( n= 0). At 65◦, the second\nphonon (p= 2), whose frequency matches the frequen-\ncies of the first few magnons, interacts only with odd\nmagnons. Thus, the dominant mode in this case is the\nfirst magnon ( n= 1). At 35◦, the magnon frequencies\nare between the frequency of the first ( p= 1) and sec-\nond (p= 2) phonon and exhibit no resonant interac-\ntion. Hence, the magnetization precession is significantly\nweaker.\nFigure7illustratesthequantitativeagreementbetween\ntheory and experiment for the two resonantly driven\nmagnetization dynamics. There are two timescales in-\nvolved in such phonon-magnon interactions: excitation\n(the transient regime) and decay (the driven regime).\nWhen the driving frequency is equal to the eigenfre-\nquency, the response in the transient regime follows\n(1−e−t/τexc)cosωpt, whereωpis the frequency of the\nacoustic driving mode and τexcis a characteristic relax-\nation time of magnetization dynamics, which is related\nto Gilbert damping as τexc= (αωn)−1, whereωnis the8Magnetization (degrees) 0 0.5 1 1.5 2-1 01Magnetization dynamics at 15.5° \nτdecay = 1.2 ns τexc  = 0.45 ns\nTime (ns) \n0 0.2 0.4 0.6 0.8 1\nTime (ns) -1 01Magnetization dynamics at 65° \nτdecay = 0.57 ns τexc  = 0.23 ns(a)\n(b) \nFIG. 7. Long-scan magnetization dynamics at the backside of\nthe film, under the magnetic field angles (a) ξ= 15.5◦and (b)\nξ= 65◦. Experimental data is represented by dots, the simu-\nlated curve is a black continuous line, while the dashed curv e\ndenotes the fit envelope A(t). The separation of timescales\nis clearly visible in both graphs: the initial growth govern ed\nby the Gilbert damping is followed by the decay due to the\nacoustic decay.\nfrequency of the dominant magnon. On the other hand,\nthe decay of magnetization is a driven regime when mag-\nnetization dynamics follow the acoustic driving force as\n∝e−t/τdecaycos(ωpt), whereτdecay= 1/γpis the acous-\ntic mode lifetime. Thus, to a good approximation, the\noverall magnetization dynamics fit inside an envelope of\nthe form\nA(t)∝(1−e−t/τexc)e−t/τdecay. (19)\nWe can assign these characteristic relaxation times to\nthe correspondingacoustic and magnetic eigenmodes, re-\nspectively, as shown in Fig. 7. For the magnetization dy-\nnamics at 15 .5◦, the characteristic time of the excitation\nphase isτ(1)\nexc= 0.45ns and the decay time is τ(1)\ndecay=\n1.2ns. We can see a clear correlation between these two\ntimes and the corresponding lifetimes of the acoustic and\nmagnetic eigenmodes,1\nγ(0)\nG= 0.44ns and1\nγ(1)\nac= 1.19ns.\nSimilarly, for the case of magnetization dynamics at 65◦,\nwe extract the excitation time τ(2)\nexc= 0.23ns and a value\nof the decay time τ(2)\ndecay= 0.57ns. Again, this value is\nconsistent with the lifetimes of the first magnon mode\n1\nγ(1)\nG= 0.22ns and of the second acoustic eigenmode,\n1\nγ(2)\nac= 0.56ns.\nThis model also allows for extracting quality factors of\nmagnons and phonon resonances. The magnon quality\nfactorofQG= 13≃1/(2α)doesnotdependonthemode\nnumber, in agreement with previously reported resultson frequency-independent Gilbert damping α[26, 27].\nAcoustic quality factors Q(1)\nac= 37 andQ(2)\nac= 29 are\nslightly different due to the observed nonlinear depen-\ndence of acoustic damping on frequency, converging to\nthe approximate scaling Qac(ω)∝ω−1for higher-order\nacoustic modes. The observed high values of acoustic\nquality factors Qac>QGenable the separation of time-\nscales in the excitation and decay phases in the magne-\ntization dynamics.\nQualitatively similar magnetization dynamics have\nbeen observed earlier in transient grating experiments\n[9]. However, in the latter case the conspicuous decay of\nmagnetization dynamics was explained by the complex\nspatio-temporal dynamics of the magnitude of the mag-\nnetization vector Mz(x,t) on the temperature T(x,t) in\nthe periodically demagnetized nickel film [10, 12]. In this\ncontext, our experimental geometry provides an advan-\ntage of isolating resonant phonon-magnon interactions\nfrom thermal effects and extracting their properties from\nthe same measurement.\nVII. SUMMARY AND CONCLUSIONS\nIn this manuscript, we reported on the quantitative\nanalysis of experimental data by Kim and Bigot in a\nfree-standing nickel thin film [1] based on the decom-\nposition of magnetic and acoustic dynamics in phonon\nand magnon eigenmodes, respectively. The time-domain\nfitting of transient reflectivity data on both sides of the\nnickel film provides frequencies, lifetimes and phases of\nindividual acoustic eigenmodes. The latter is shown\nto drive the magnetization dynamics, to be in quan-\ntitative agreement with time-resolved MOKE measure-\nments. Notably, the comparison of MOKE signals on\nboth sides of the sample evidence the in-phase FMR dy-\nnamics (n= 0, with minor contributions of symmetric\nmagnon eigenmodes n= 2,4,...) induced by the lowest\norder (p=1) symmetric acoustic mode and the opposite-\nsign magnetization oscillation of antisymmetric magnon\nmodes (n= 1,3,...). Being in a quantitative agreement\nwith a simple theoretical model with tabulated material\nparameters, the experimental data clearly evidence the\nresonantly enhanced excitation of nonuniform magnon\nmodes. Moreover, accurate fitting of the magnetization\ndynamics driven by long-lived p= 1 (9.8 GHz) and p= 2\n(19.1 GHz) acoustic modes delivers the correct value for\nmagnetic Gilbert damping α= 0.04, corresponding to\nthe quality factor Qm= 13 for magnon modes. Be-\ning smaller than the quality factors of acoustic modes,\nthis magnetic quality factor assures optimum conditions\nforresonantphonon-magnonexcitation, the phenomenon\nto be further explored in the ultrahigh THz-frequency\nregime [2, 28].9\nACKNOWLEDGMENTS\nThis article is dedicated to the memory of Jean-Yves\nBigot, in whose labs the reported measurements have\nbeen performed. The support of the Physics Depart-\nment of ´Ecole Polytechnique and Institut Polytechnique\ndePariswithintheframeworkofa Projet de Recherche en\nLaboratoire and by the ANR-21-MRS1-0015-01 ”IRON-\nMAG” is gratefully acknowledged. P.V. acknowledgessupport from the Spanish Ministry of Science and In-\nnovation and the European Union under the Maria\nde Maeztu Units of Excellence Programme (CEX2020-\n001038-M) and the project PID2021-123943NB-I00\n(MICINN/FEDER). J.K. acknowledges support from\nBasic Science Research Program through the National\nResearchFoundationofKorea(NRF) fundedbytheMin-\nistryofEducation(2022R1I1A3072023)andbytheMSIT\n(2021R1A4A1031920).\n[1] J.-W. Kim and J.-Y. Bigot,\nPhys. Rev. B 95, 144422 (2017).\n[2] U. Vernik, A. M. Lomonosov, V. S. Vlasov, L. N. Kotov,\nD. A. Kuzmin, I. V. Bychkov, P. Vavassori, and V. V.\nTemnov, Physical Review B 106, 144420 (2022).\n[3] A. V. Scherbakov, A. S. Salasyuk, A. V. Akimov,\nX. Liu, M. Bombeck, C. Br ¨uggemann, D. R. 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Dubowik1 \n1Institute of Molecular Physics, Polish Academy of Sciences, PL -60-179 Poznań, Poland  \n2NanoBioMedical Centre, Adam Mickiewicz University, PL -61-614 Poznań, Poland  \n3Centre for Advanced Technology, Adam Mickiewicz University, PL -61-614 Poznań, Poland  \nElectronic mail: adam.krysztofik@ifmpan.poznan.pl , hubert .glowinski @ifmpan.poznan.pl  \n \nWe show that using maskless photolithography and the lift-off technique patterned \nyttrium iron garnet thin films possessing ultra -low Gilbert damping can be  \naccomplished . The films of the 70 nm thickness we re grown on (001)-oriented \ngadolinium gallium garne t by means of pulsed laser deposition and exhibit high \ncrystalline quality, low surface roughness and effective  magnetization of 127 \nemu/cm3. The Gilbert damping parameter is as low as 5×10−4. The obtained \nstructures have well-defined sharp edges  which  along with good structural and \nmagnetic film properties, pave a path in the fabrication of high -quality magnonic \ncircuits as well as  oxide -based  spintronic devices.  \n \n \nYttrium iron garnet (Y 3Fe5O12, YIG) has become an intensively studied material in recent years due \nto exceptionally low damping of magnetization precession and electrical insulation enabling its \napplication in research on spin -wave propagation1–3, spin-wave based logic devices4–6, spin pumping7, \nand thermally -driven spin caloritronics8. These applications inevitably entail film structurization in \norder to construct complex integrated devices . However, the fabrication of high -quality thin YIG films \nrequires  deposition temperatures over 500 C6,9–18 leading to top -down lithographical approach that is \nion-beam etching  of a previously  deposited plain film where as patterned resist layer serves as a mask.  \nConsequently, this metho d introdu ces crystallographic  defects , imperfections to surface structure  and, \nin the case of YIG films, causes significant  increase of the damping parameter .19–21 Moreover, it does \nnot ensure well-defined  structure edges  for insulators , which play a crucial role in devices utilizing 2 \n edge spin waves22, Goos -Hänchen spin wave shifts23,24 or standing spin waves modes25. On the \ncontrary, t he bottom -up structurization  deals with th ese issues  since it allows  for the film grow th in the \nselect ed, patterned  areas followed by  a removal of the resist layer along with redundant film during \nlift-off process.  Additionally, it reduces the patterning  procedure by one step , that is ion etching , and \nimposes room -temperature deposition  which both are particularly important  whenever  low fabrication  \nbudget is required.  \nIn this letter  we report on ultra -low damping in the bottom -up structured YIG film by means of \ndirect writing photolithography technique.  In our case, t he method allows for structure patterning \nwith 0.6 µm resolution across full writing area . In order to not preclude the lift -off process, the pulsed \nlaser deposition (PLD) was conducted at room temperature and since such as -deposited films are \namorphous19,27 the ex-situ annealing was performed for recrystallization.  Note that post -deposition \nannealing of YIG films  is commonly carried out regardless the substrate temperature  during film \ndeposition6,12,13,28,29. As a reference we  investigated  a plain film which was grown in the same \ndeposition process and underwent the same fabrication procedure except for patterning. Henceforth, \nwe will refer to the structured and the plain film as Sample 1 an d Sample 2, respectively . We \nanticipate that such a procedure may be of potential for fabrication of other magnetic oxide structures \nuseful in spintronics.  \nStructural characteriza tion of both samples was performed by means  of X-Ray Diffraction (XRD). \nAtomic force microscopy (AFM) was applied to investigate  surface morphology and the quality of \nstructure edges.  SQUID magnetometry  provided information on the saturation magnetization  and \nmagnetocrystalline anisotropy field . Using a coplanar waveguide connected to a vector network \nanalyzer , broadband ferromagnetic resonance (VNA -FMR) was performed to determine Gilbert \ndamping parameter  and anisotropy fields . All the experiments were co nducted at the room \ntemperature.  \nThe procedure of samples preparation was as follows. The (001) -oriented gadolinium gallium \ngarnet substrates were ultrasonicated in acetone, trichloroethylene and isopropanol to remove surface \nimpurities. After a 1 minute o f hot plate baking for water evaporation, a positive photoresist was spin -\ncoated onto the substrate (Sample 1). Using maskless photolithography an array of 500  μm x 500  μm \nsquares separated over 500 μm was patterned and the exposed areas were developed. Detailed \nparameters of photolithography process can be found in Ref.26. We chose rather large size of the \nsquares to provide a high signal -to-noise ratio in the latter measurements. Thereafter, plasma etching \nwas performed to remove a residual resist. We would like to emphasize the importance of this step in \nthe fabrication p rocedure as the resist residues may locally affect crystalline structure of a YIG film \ncausing an undesirable increase of overall magnetization damping. Both substrates were then placed in \na high vacuum chamber of 9×10-8 mbar base pressure and a film was d eposited from a stoichiometric \nceramic YIG target under 2×10-4 mbar partial pressure of oxygen. We used a Nd:YAG laser (λ  = 355 \nnm) for the ablation with pulse rate of 2 Hz which yielded 1 nm/min growth rate. The target -to-3 \n substrate distance was approximat ely 50 mm. After the deposition the l ift-off process for the Sample 1  \nwas performed using sonication in acetone  to obtain the expected structures. Subsequently, both \nsamples were annealed in a tube furnace under oxygen atmosphere (p ≈ 1 bar) for 30 minutes  at \n850°C. The heating and cooling rates were about 50  C/min and 10  C/min, respectively.  \n \n \n \nFIG. 1.  (a) XRD θ−2θ plot near the (004) reflection of structured ( Sample 1 ) and plain ( Sample 2 ) YIG film. Blue arrows \nshow clear Laue reflections of the plain film. Insets show schematic illustration of the structured and plain film used in this \nstudy. (b) Height  profile (z(x)) taken from the structured sample (left axis), right shows the differential of the p rofile, clearly \nshowing the slope change.  Inset shows 3D map of the structure’s edge.  \n \n \nThe structure of YIG films was determined by the X -ray diffraction.  Although the as-deposited \nfilms were amorphous, with the annealing treatment they inherited the lattice orientation of the GGG \nsubstrate and recrystallized along [ 001] direction. Figure 1 (a) presents diffraction curves  taken in the \nvicinity of ( 004) Bragg reflection.  The ( 004) reflection position of structured YIG well coincide s with \nthe reflection of the plain film. The 2 θ=28.70 9 corresponds to the cubic lattice constant of 12.428  Å. \nA comparison of this value with lattice parameter of a bulk YIG (12.376 Å) suggest distortion of unit \n4 \n cells due to slight nonstoichiometry.16,30 Both samples exhibit distinct Laue oscillation s depicted by \nthe blue arrows,  indicating  film uniformity and high crystalline order , although the structured film \nshowed lower intensity due to the lower mass of the film . From the oscillation period we estimated \nfilm thickness of 73 nm in agreement with the nominal thickness and the value determined using AFM  \nfor Sample 1 ( Fig. 1 (b)). By measuring the diffraction  in the expanded angle range w e also confirmed \nthat no additional phases like Y 2O3 or Fe 2O3 appeared.  \nThe surface morphology of the structured film was investigated by means of AFM. In Fig. 1 (b) \nprofile of a square’s edge is shown. It should be highlighted that no edge irregularities has formed \nduring lift -off process. The horizontal distance between GGG substrate and the surface of YIG film is \nequal to 170 nm as marked in Fig. 1 (b) by the shaded area. A fitting with Gaussian function to the \nderivative of height profile yields the full width at half maximum of 61 nm. This points to the well -\ndefined struct ure edges achieved with bottom -up structurization. Both samples have smooth and \nuniform surface s. The comparable values of root mean square (RMS) roughness (0.306 nm for Sample \n1 and 0.310 nm for Sample 2) indicate that bottom -up structurization process  did not leave any resist \nresidues. Note that a roughness of a bare GGG substrate before deposition was 0.281 nm, therefore, \nthe surface roughness of YIG is increased merely by 10%.  \n \n \nFIG. 2. Hysteresis loops of structured (Sample 1) and plain (Sample 2) YIG films measured by SQUID \nmagnetometry  along [100] direction at the room temperature . \n \nFigure 2 shows magnetization reversal curves measured along [ 100] direction. For each hysteresis \nloop a paramagnetic contribution arising for the GGG substrates was subtracted. The saturation \nmagnetization 𝑀𝑠 was equal to 117 emu/cm3 and 118.5 emu/cm3 for Sample 1 and 2, respectively . \nBoth hysteresis loops demonstrate in -plane anisotropy. For the (001) -oriented YIG the [ 100] direction \nis a “hard” in -plane axis and the magnetization saturates at 𝐻𝑎 = 65 Oe. This value we identify as \n-100 -75 -50 -25 0 25 50 75 100-1.0-0.50.00.51.0  Sample 1\n Sample 2M / MS\nMagnetic Field (Oe)5 \n magnetocrystalline anisotropy field. The VNA -FMR measurements shown in Fig. 3 (a) confirm these \nresults. Using Kitte l dispersion relation, i.e. frequency 𝑓 dependence of resonance magnetic field 𝐻: \n 𝑓=𝛾\n2𝜋√(𝐻+𝐻𝑎cos 4𝜑)(𝐻+1\n4𝐻𝑎(3+cos 4𝜑)+4𝜋𝑀𝑒𝑓𝑓), (1) \n 4𝜋𝑀𝑒𝑓𝑓=4𝜋𝑀𝑠−𝐻𝑢, (2) \nwe derived 𝐻𝑎 and the effective magnetization 𝑀𝑒𝑓𝑓, both comparable to the values determined using \nSQUID and close to the values of a bulk YIG (see Table I .). Here,  the azimuthal angle 𝜑 defines the \nin-plane orientation of the magnetization direction with respect to the [100] axis of YIG  and  𝛾 is the \ngyromagnetic ratio ( 1.77×107𝐺−1𝑠−1). To better compare the values of 𝐻𝑎 between samples and to \ndetermine if the results are influenced by additional anisotropic contribution arising  from  the squares’ \nshape in the  structured film we performed angular resolved resonance measurements  (inset in Fig. \n3(a)) . The fitting according to Eq. (1) gives |𝐻𝑎| equal to 69.5±0.6 for Sample 1 and 69.74±0.28 for \nSample 2 in agreement with the values derived from 𝑓(𝐻) dependence and better accuracy.  Hence, we \nconclude  that the structurization did not affect the in -plane anisotropy. The deviations of  the derived  \n𝑀𝑠 and 𝐻𝑎 from bulk values can be explained in the framework of Fe vacancy model developed for \nYIG films as a result of nonstoichiometry.13,30 For the experimentally determined 𝑀𝑠 and 𝐻𝑎 the \nmodel yields the chemical unit Y 3Fe4.6O11.4 which closely approximates to  the composition of a \nstoichiometric YIG Y 3Fe5O12. \n \n \nTABLE I.  Key parameters reported for PLD and LPE YIG films.  \n  AFM  SQUID  VNA -FMR  \n Film \nthickness  RMS rough -\nness (nm)  Ms  \n(emu/cm3) Ha  \n(Oe) Field  \norientation  Meff \n(emu/cm3) |Ha|  \n(Oe) Hu \n(Oe) α  \n(× 10-4) ΔH 0  \n(Oe) \nSample 1  70 nm  0.306  117±1  65±5  (100):  \n(110):  \n(001):  125±1  \n126±1  \n129±2  64±1  \n63±1  \n− -101±18  \n-113±18  \n-151±28  5.53±0.13  \n5.24±0.12  \n5.19±0.64  1.45±0.09  \n2.86±0.09  \n2.61±0.34  \nSample 2  70 nm  0.310  118.5±2  65±5  (100):  \n(110):  \n(001):  124±1  \n127±1  \n131±2  62±1  \n65±1  \n− -69±28  \n-107±28  \n-157±36  5.05±0.07  \n5.09±0.09  \n5.02±0.18  0.97±0.05  \n1.28±0.06  \n1.48±0.09  \nLPE-YIG31 106 nm  0.3 143 − (112):  − − − 1.2 0.75 \nLPE-YIG30 120 μm  − 139±2  − (111):  133±2  85±6  76±1  0.3 − \n \n \nAlthough the saturation magnetization  of the films  is decreased by 15% with respect to the bulk \nvalue we can expect similar spin wave dynamics since magnon  propagation does not solely depend on \n𝑀𝑠 but on the effective magnetization or equivalently, on the uniaxial anisotropy field  𝐻𝑢.12 \nSubstitution of 𝑀𝑠 into Eq. (2) gives  average values of  𝐻𝑢 equal to -122 Oe and -111 Oe for Sample 1 \nand 2, respectively  (to determine 𝐻𝑢 from the out -of-plane FMR measurements when H || [001] we 6 \n used the 𝑓=𝛾\n2𝜋(𝐻+𝐻𝑎−4𝜋𝑀𝑒𝑓𝑓) dependence13 to fit the data  and assumed  the value of 𝐻𝑎 from \nangular measurements ). As 𝑀𝑒𝑓𝑓𝑆𝑎𝑚𝑝𝑙𝑒  1,2≈𝑀𝑒𝑓𝑓𝑏𝑢𝑙𝑘, it follows  that the low value of 𝑀𝑠 in room -\ntemperature deposited thin films is “compensated ” by uniaxial anisotropy field.  Note that for bulk YIG \nsaturation magnetization is diminished  by 𝐻𝑢/4𝜋 giving  a lower value of 𝑀𝑒𝑓𝑓 while for Sample 1 \nand 2,  𝑀𝑠 is augmented  by 𝐻𝑢/4𝜋 giving a higher value of 𝑀𝑒𝑓𝑓 (Table I .). The negative sign of \nuniaxial anisotropy field  is typical for PLD -grown YIG films  and originates from preferential \ndistribution of Fe vacancies between different si tes of YIG’s octahedral  sublattice.30 This point s to the \ngrowth -induced anisotropy mechanism while the stress -induced contribution is of ≈10 Oe29 and, as it \ncan be estimated  according to Ref.32, the transition layer at the substrate -film interface due to Gd, Ga, \nY ions diffusion is ca. 1.5 nm thick for the 30 min of annealing treatment.  We argue that the growth -\ninduced anisotropy due to ordering of the magnetic ions is related to the growth condition which in our \nstudy is specific. Namely, it is crystallization of an amorphous material.  \nGilbert damping parameter 𝛼 was obtained by fitting dependence of linewidth 𝛥𝐻 (full width at \nhalf maximum ) on frequency 𝑓 as shown in Fig. 3 (b): \n 𝛥𝐻 =4𝜋𝛼\n𝛾𝑓+𝛥𝐻0, (3) \nwhere  𝛥𝐻0 is a zero -frequency linewidth broadening . The 𝛼 parameter of both samples is nearly the \nsame , 5.32×10−4 for Sample 1 and 5.05×10−4 for Sample 2  on average (see Table I.) . It proves \nthat bottom -up patterning does not compromise magnetization damping. The value of 𝛥𝐻0 \ncontribution is around 1.5 Oe although small variations of 𝛥𝐻0 on 𝜑 can be noticed. Additional \ncomments on  angular dependencies of  𝛥𝐻 can be found in the supplementary material. The derived \nvalues of 𝛼 remain one order of magnitude smaller than for soft ferromagnets like Ni 80Fe2033, CoFeB34 \nor Finemet35, and are comparable to values reported for YIG film s deposited at hi gh temperatures \n(from 1×10−4 up to 9×10−4).6,9,11,14,15,17,18 It should be also highlighted  that 𝛼 constant is \nsignificantly increased in comparison to the bulk YIG made by means of Liquid Phase Epitaxy (LPE) . \nHowever, recently reported LPE-YIG films of  nanometer thickness , suffer from  the increased damping \nas well (Table I.) due to impurity elements  present in the high -temperature solutions used in LPE \ntechnique31. As PLD method allow s for a good  contamination control , we attribute the increase as a \nresult of slight nonstoichiometry determined above with Fe vacancy model .30 Optimization of growth \nconditions , which further improve the film composition may resolve this issue and allow to cross the \n𝛼=1×10−4 limit. We also report that additional annealing of the samples (for 2h) did not influence \ndamping nor it improved the value of 𝐻𝑎 or 𝑀𝑒𝑓𝑓 (within 5% accuracy).  7 \n  \nFIG. 3. (a) Kittel dispersion relation s of the structured (Sample 1) and plain (Sample 2) YIG film.  The i nset \nshows angular dependence of resonance field  revealing perfect fourfold anisotropy for both samples . (b) \nLinewidth dependence on frequency fitted with Eq. (3). The inset shows resonance absorptions peaks with very \nsimilar width (5.3 Oe for Sample 1 and 4.7 Oe for Sample 2  at 10 GHz ). Small differences of the resonance field  \noriginate from different values of 4𝜋𝑀𝑒𝑓𝑓. \n \nIn conclusion , the lift-off patterned YIG films possessing low damping have  been presented. \nAlthough the structurization procedure required deposition at room temperature , the 𝛼 parameter does \nnot diverge from those reported for YIG thin films grown at temperatures above 500 C. Using the \nplain, reference film fabricated along with the structured one, we have shown  that structurization does \nnot significantly affect structural nor magnetic properties  of the films,  i.e. out-of-plane lattice constant, \nsurface roughness, saturation magnetization, anisotropy fields and damping. The structures obtain ed \nwith bottom -up structurization indeed possess sharp , well-defined edges . In particular, o ur findings \nwill help in the development of magnonic and spintronic devices utilizing film boundary effects  and \nlow damping of magnetization precession . \n8 \n  \nSupplementary Material  \nSee supplementary material for the angular dependence of resonance linewidth . \n \nThe research received funding from the European Union Horizon 2020 research and innovation \nprogra mme under the Marie Skłodowska -Curie grant agreement No 644348 (MagIC).  We would like \nto thank Andrzej Musiał for the assistance during film annealing.  \n \n1 H. 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Matter 29, 435803 (2017).  \n35 H. Głowiński, I. Gościańska, A. Krysztofik, J. Barnaś, M. Cecot, P. Kuświk, and J. Dubowik, in 21st Int. Conf. Microwave, \nRadar Wirel. Commun. MIKON 2016  (2016).  \n \n \n "    },    {        "title": "1402.6899v1.On_the_longitudinal_spin_current_induced_by_a_temperature_gradient_in_a_ferromagnetic_insulator.pdf",        "content": "arXiv:1402.6899v1  [cond-mat.mtrl-sci]  27 Feb 2014On the longitudinal spin current induced by a temperature gr adient in a\nferromagnetic insulator\nS. R. Etesami,1,2L. Chotorlishvili,2A. Sukhov,2and J. Berakdar2\n1Max-Planck-Institut f¨ ur Mikrostrukturphysik, 06120 Hal le, Germany\n2Institut f¨ ur Physik, Martin-Luther-Universit¨ at Halle- Wittenberg, 06120 Halle, Germany\n(Dated: March 3, 2022)\nBased on the solution of the stochastic Landau-Lifshitz-Gi lbert equation discretized for a ferro-\nmagnetic chain subject to a uniform temperature gradient, w e present a detailed numerical study\nof the spin dynamics with a focus particularly on finite-size effects. We calculate and analyze the\nnet longitudinal spin current for various temperature grad ients, chain lengths, and external static\nmagnetic fields. In addition, we model an interface formed by a nonuniformly magnetized finite-size\nferromagnetic insulator and a normal metal and inspect the e ffects of enhanced Gilbert damping on\nthe formation of the space-dependent spin current within th e chain. A particular aim of this study\nis the inspection of the spin Seebeck effect beyond the linear response regime. We find that within\nour model the microscopic mechanism of the spin Seebeck curr ent is the magnon accumulation effect\nquantified in terms of the exchange spin torque. According to our results, this effect drives the spin\nSeebeck current even in the absence of a deviation between th e magnon and phonon temperature\nprofiles. Our theoretical findings are in line with the recent ly observed experimental results by M.\nAgrawal et al., Phys. Rev. Lett. 111, 107204 (2013).\nPACS numbers: 85.75.-d, 73.50.Lw, 72.25.Pn, 71.36.+c\nI. INTRODUCTION\nThermal magneto- and electric effects have a long his-\ntory and are the basis for a wide range of contemporary\ndevices. Research activities revived substantially upon\nthe experimental demonstration of the correlation be-\ntween an applied temperature gradient and the observed\nspindynamics, includingaspincurrentalongthetemper-\naturegradientinanopen-circuitmagneticsample,theso-\ncalled spin Seebeck effect (SSE)1. Meanwhile an impres-\nsive body work has accumulated on thermally induced\nspin- and spin-dependent currents1–11(for a dedicated\ndiscussion we referto the topical review12). The SSE was\nobserved not only in metallic ferromagnets (FMs) like\nCo2MnSi or semiconducting FMs, e.g. GaMnAs, (Ref.4),\nbut also in magnetic insulators LaY 2Fe5O12(Ref.5) and\n(Mn, Ze)Fe 2O4(Ref.7). The Seebeck effect is usually\nquantified by the Seebeck coefficient Swhich is defined,\nin a linear response manner, as the ratio of the gener-\nated electric voltage ∆ Vto the temperature difference\n∆T:S=−∆V\n∆T. The magnitude of the Seebeck coeffi-\ncientSdepends on the scattering rate and the density\nof electron states at the Fermi level, and thus it is ma-\nterial dependent variable. In the case of SSE, the spin\nvoltage is formally determined by µ↑−µ↓, whereµ↓(↑)\nare the electrochemical potentials for spin-up and spin-\ndown electrons, respectively. The density of states and\nthe scattering rate for spin-up and spin-down electrons\nare commonly different, which results in various Seebeck\nconstants for the two spin channels. In a metallic magnet\nsubjected to a temperature gradient, one may think of\nthe electrons in different spin channels to generate differ-\nent driving forces, leading to a spin voltage that induces\na nonzero spin current. When a magnetic insulator is\nin contact with a normal metal (NM) and the system issubjected to a thermal gradient, the total spin current\nflowing through the interface is a sum of two oppositely\ndirected currents. The current emitted from the FM into\nthe NM, is commonly identified as a spin pumping cur-\nrentIspand originatesfromthe thermally activatedmag-\nnetization dynamics in the FM, while the other current\nIflis associated with the thermal fluctuations in the NM\nand is known as spin torque13. The competition between\nthe spin pump and the spin torque currents defines the\ndirectionofthetotalspincurrentwhichisproportionalto\nthe thermal gradient applied to the system. The theory\nofthe magnon-drivenSSE5presupposesthat the magnon\ntemperature follows the phonon temperature profile and\ninalinearresponseapproximationprovidesagoodagree-\nment with experiments.\nIn a recent study14the theory of the magnon-driven\nSSE was extended beyond the linear response approxi-\nmation. In particular, it was shown that the nonlinearity\nleads to a saturation of the total spin current and nonlin-\near effects become dominant when the following inequal-\nity holds H0/Tm\nF< kB/(MsV), where H0is the constant\nmagnetic field applied to the system, Tm\nFis the magnon\ntemperature, Msis the saturation magnetization and V\nis the volume of the sample. The macrospin formulation\nofthestochasticLandau-Lifshitz-Gilbert(LLG)equation\nand the Fokker-Planck approach utilized in Ref.14is in-\nappropriate for non-uniformly magnetized samples with\ncharacteristic lengths exceeding several 10 nm. Beyond\nthe macrospin formulation the SSE effect for nonuni-\nformly magnetized samples can be described by intro-\nducing a local magnetization vector15/vector m(/vector r,t). In this\ncase, however, the correspondingFokker-Planckequation\nturns into an integro-differential equation and can only\nbe solved after a linearization16. Recently17, the lon-\ngitudinal SSE was studied in a NM-FM-NM sandwich2\nstructure in the case of a nonuniform magnetization pro-\nfile. The linear regime, however, can not totaly embrace\nnontrivial and affluent physics of the SSE.\nIn the present study we inspect the SSE for a nonuni-\nformlymagnetized finite-size FM-NM interfacesubjected\nto an arbitrary temperature gradient. Our purpose is to\ngo beyond linear response regime which is relevant for\nthe nonlinear magnetization dynamics. It is shown that\nin analogy with the macrospin case14the spin current in\nthe nonlinear regime depends not only on the tempera-\nture gradient, but on the absolute values of the magnon\ntemperature as well. In finite-size non-uniformly magne-\ntized samples, however, the site-dependent temperature\nprofile may lead to new physical important phenomena.\nFor instance, we show that the key issue for the spin cur-\nrent flowing through a nonuniformly magnetized mag-\nnetic insulator is the local exchange spin torque and the\nlocal site-dependent magnon temperature profile, result-\ning in a generic spatial distribution of the steady state\nspin current in a finite chain subject to a uniform tem-\nperaturegradient. The maximalspincurrentispredicted\nto be located at the middle of the chain.\nII. THEORETICAL FRAMEWORK\nFor the description of the transversal magnetiza-\ntion dynamics we consider propagation of the normal-\nized magnetization direction /vector m(/vector r,t) as governed by the\nLaundau-Lifshitz-Gilbert (LLG) equation18,36\n∂/vector m\n∂t=−γ/bracketleftBig\n/vector m×/vectorHeff/bracketrightBig\n+α/bracketleftbigg\n/vector m×∂/vector m\n∂t/bracketrightbigg\n−γ/bracketleftBig\n/vector m×/vectorh(/vector r,t)/bracketrightBig\n,(1)\nwhere the deterministic effective field /vectorHeff=−1\nMSδF\nδ/vector mde-\nrives from the free energy density Fand is augmented by\na Gaussian white-noise random field h(/vector r,t) with a space-\ndependent local intensity and autocorrelation function.\nαis the Gilbert damping, γ= 1.76·1011[1/(Ts)] is the\ngyromagnetic ratio and MSis the saturation magnetiza-\ntion.Freads\nF=1\nV/integraldisplay/bracketleftbiggA\n2|/vector∇m|2+Ea(/vector m)−µ0MS/vectorH0·/vector m/bracketrightbigg\ndV,(2)\nwhere/vectorH0is the external constant magnetic field, Ea(/vector m)\nis the anisotropy energy density and Ais the exchange\nstiffness. Vis the system volume. We employ a dis-\ncretized version of the integro-differential equation (1)\nby defining Ncells with a characteristic length a= /radicalbig\n2A/µ0M2sof the exchange interaction between the\nmagnetic moments. a3= Ω0is the volume of the re-\nspective cell. Assuming negligible variations of /vector m(/vector r,t)\nover a small a, one introduces a magnetization vector\n/vectorMnaveraged over the nth cell/vectorMn=MS\nV/integraltext\nΩ0/vector m(/vector r,t)dVand the total energy density becomes\nε=−/vectorH0·/summationdisplay\nn/vectorMn+K1\nM2\nS/summationdisplay\nn/parenleftbig\nM2\nS−(Mz\nn)2/parenrightbig\n−2A\na2M2\nS/summationdisplay\nn/vectorMn·/vectorMn+1.(3)\n/vectorH0is the external magnetic field and K1is the uniaxial\nanisotropy density (with the easy axis: /vector ez). The effective\nmagnetic field actingon the n-th magnetic moment reads\n/vectorHeff\nn=−∂ε\n∂/vectorMn=/vectorH0+2K1\nM2\nSMz\nn/vector ez\n+2A\na2M2\nS/parenleftBig\n/vectorMn+1+/vectorMn−1/parenrightBig\n.(4)\nThermal activation is introduced by adding to the total\neffectivefieldastochasticfluctuatingmagneticfield /vectorhn(t)\nso that\n/vectorHeff\nn(t) =/vectorH0+/vectorHanis\nn+/vectorHexch\nn+/vectorhn(t).(5)\nHere/vectorHanis\nnis the magnetic anisotropy field, /vectorHexch\nnis the\nexchange field. The random field /vectorhn(t) has a thermal\norigin and simulates the interaction of the magnetization\nwith a thermal heat bath (cf. the review Ref. [19] and\nreferences therein). The site dependence of /vectorhn(t) reflects\ntheexistenceofthelocalnonuniformtemperatureprofile.\nOn the scale ofthe volumeΩ 0the heat bath is considered\nuniform at a constant temperature. The random field is\ncharacterized via the standard statistical properties of\nthe correlation function\n/angb∇acketlefthik(t)/angb∇acket∇ight= 0,\n/angb∇acketlefthik(t)hjl(t+∆t)/angb∇acket∇ight=2kBTiαi\nγMSa3δijδklδ(∆t).(6)\niandjdefine the corresponding sites of the FM-chain\nandk,lcorrespond to the cartesian components of the\nrandom magnetic field, Tiandαiare the site-dependent\nlocal temperature and the dimensionless Gilbert damp-\ning constant, respectively, kB= 1.38·10−23[J/K] is the\nBoltzmann constant.\nIn what follows we employ for the numerical calcu-\nlations the material parameters related to YIG, e.g. as\ntabulated in Ref.6(Table I). Explicitlythe exchangestiff-\nness isA≈10 [pJ/m], the saturation magnetization has\na value of 4 πMS≈106[A/m]. The anisotropy strength\nK1can be derived from the estimate for the frequency\nω0=γ2K1/MS≈10·109[1/s]6. The size of the FM\ncell is estimated from a=/radicalbig\n2A/µ0M2syielding about 20\n[nm]. Fordampingparameterwetakethevalue α= 0.01,\nwhich exceeds the actual YIG value5,6. This is done to\noptimize the numerical procedure in order to obtain rea-\nsonable calculation times. We note that although the\nquasi-equilibrium is assured when tracking the magne-\ntization trajectories on the time scale longer than the\nrelaxation time, the increased αquantitatively alters the3\nFIG. 1: a) Schematics of the FM chain considered in the\ncalculations. b) Suggested alignment for measurements.\nstrength in the correlation function (eq. (6)) and there-\nfore indirectly has an impact on the values of the spin\ncurrent.\nWe focus on a system representing a junction of a FM\ninsulatorand a NM which is schematically shownin FIG.\n1. This illustration mimics the experimental setup for\nmeasuring the longitudinal SSE20, even though the anal-\nysis performed here does not include all the aspects of\nthe experimental setting. The direction of the magnetic\nmoments in the equilibrium is parallel to the FM-NM in-\nterface. Experimentally it was suggested to pick up the\nlongitudinal spin current by means of the inverse spin\nHall effect20. If it is so possible then, the electric field\ngenerated via the inverse spin Hall effect (ISHE) reads− →E=D− →Is×− →σ. Here− →Edenotes electric field related\nto the inverse spin Hall effect,− →Isdefines the spatial di-\nrection of the spin current, and− →σis spin polarization of\nthe electrons in the NM, and Dis the constant. We note,\nhowever, that our study is focused on the spin dynamics\nonly and makes no statements on ISHE.\nIII. DEFINITION OF THE SPIN CURRENT\nFor convenience we rewrite the Gilbert equation with\nthe total energy density (3) in the form suggested in\nRef.17\n∂/vectorSn\n∂t+γ/bracketleftBig\n/vectorSn×/parenleftBig\n/vectorHeff\nn(t)−/vectorHex/parenrightBig/bracketrightBig\n+αγ\nMS/bracketleftBigg\n/vectorSn×∂/vectorSn\n∂t/bracketrightBigg\n+∇·/vectorJ/vector s\nn= 0,\n(7)where/vectorSn=−/vectorMn/γand the expression for the spin cur-\nrent density tensor reads\n∇·/vectorJ/vector s\nn=γ/bracketleftBig\n/vectorSn×/vectorHex\nn/bracketrightBig\n. (8)\nHere\n/vectorQn=−γ[/vectorSn×/vectorHex\nn] (9)\nis the local exchange spin torque.\nFor the particular geometry (FIG. 1) the only nonzero\ncomponents of the spin current tensor are Isxn,Isy\nn,Iszn.\nTaking into account eqs. (4) and (7), we consider a dis-\ncrete version of the gradient operator and for the com-\nponents of the spin current tensor Is\nn=a2Js\nnwe deduce:\nIα\nn=Iα\n0−2Aa\nM2\nSn/summationdisplay\nm=1Mβ\nm(Mγ\nm−1+Mγ\nm+1)εαβγ,(10)\nwhereεαβγis the Levi-Civita antisymmetric tensor,\nGreek indexes define the current components and the\nLatin ones denote sites of the FM-chain. In what fol-\nlows we will utilize eq. (10) for quantifying the spin cur-\nrent in the spin chain. We consider different temperature\ngradients applied to the system taking into account the\ndependence of the magnon temperature on the phonon\ntemperature profile5. Since the temperature in the chain\nis not uniform, we expect a rich dynamics of different\nmagnetic moments /vectorMn. In this case only nonuniform\nsite-dependent spin current Incan fulfil the equation (7).\nIn order to prove this we will consider different configu-\nrations of magnetic fields for systems of different lengths.\nModeling the interface effects between the FM insulator\nand the NM proceeds by invoking the concept of the en-\nhanced Gilbert damping proposed in a recent study21.\nThe increased damping constant in the LLG equation of\nthe last magnetic moment describes losses of the spin\ncurrent due to the interface effect. In order to evaluate\nthe spin current flowing from the NM to the FM insula-\ntor we assume that the dynamics of the last spin in the\ninsulator chain is influenced by the spin torque flowing\nfrom the NM to the magnetic insulator. The magnetic\nanisotropy is considered to have an easy axis5.\nIV. NUMERICAL RESULTS ON ISOLATED\nFERROMAGNETIC INSULATOR CHAIN\nFor the study of thermally activated magnetization\ndynamics we generate from 1000 to 10000 random tra-\njectories for each magnetic moment of the FM-chain.\nAll obtained observables are averaged over the statis-\ntical ensemble of stochastic trajectories. The number\nof realizations depends on the thermal gradient applied\nto the system. For long spin chains (up to 500 mag-\nnetic moments) the calculations are computationally in-\ntensive even for the optimized advanced numerical Heun-\nmethod22, which converges in quadratic mean to the so-\nlution of the LLG equation when interpreted in the sense4\nof Stratonovich23. For the unit cell of the size 20 [nm],\nthe FM-chain of 500 spins is equivalent to the magnetic\ninsulator sample of the width around 10 [ µm]. We make\nsure in our calculations that the magnetization dynam-\nics is calculated on the large time scale exceeding the\nsystem’s relaxation time which can be approximated via\nτrel≈MS/(γ2K1α)≈10 [ns]24.\nA. Role of the local temperature and local spin\nexchange torque\nPrior to studying a realistic finite-size system we con-\nsider a toy model of three coupled magnetic moments.\nOur aim is to better understand the role of local tem-\nperature and local exchange spin torque Qn(eq. (9)) in\nthe formation of the spin current In. Considering eqs.\n(8, 9), we can utilize a recursive relation for the site-\ndependent spin current Inand the local exchange spin\ntorqueIn=In−1+a3\nγQnfor different temperatures of\nthe site in the middle of the chain above T2> Tavand\nbelowT2< Tav. The mean temperature in the system is\nTav=/parenleftbig\nT1+T2+T3/parenrightbig\n/3. The calculations are performed\nfor different values of the site temperatures. We find that\nthe exchange spin torques Qnrelated to magnetic mo-\nmentsMnwith a temperature above the mean temper-\natureTn> Tavhave a positive contribution to the spin\ncurrent in contrast to the exchange spin torques Qmof\nthe on-average-”cold”magnetic momentswith Tm< Tav.\nThis finding hints on the existence of a maximum spin\ncurrent in a finite chain of magnetic moments and/or\nstrong temperature gradient. This means that the site-\ndependent spin current Inincreases if Qn>0 until the\nlocal site temperature drops below the mean tempera-\ntureTn< Tav, in which case the exchange spin torque\nbecomes negative Qn<0 and the spin current decreases.\nIn order to prove that the negative contribution in the\nspin current of the on-average-cold magnetic moments\nis not an artefact of the three magnetic moments only,\nwe studied long spin chains which mimic non-uniformly\nmagnetized magnetic insulators. In the thermodynamic\nlimit for a large number of magnetic moments N≫1\nwe expect to observe a formation of the equilibrium pat-\nterns in the spin current profile correspondingto the zero\nexchange spin torque Qn= 0 between nearest adjacent\nmoments.\nB. Longitudinal spin current\nIn FIG. 2 a dependence of distinct components of the\nspin current on the site is plotted. As inferred from the\nfigure the current is not uniformly distributed along the\nchain. Evidently, the spin current has a maximum in\nthe middle of the chain. The site-dependent spin cur-\nrent is an aftermath of the nonuniform magnon temper-/SolidSquare/SolidSquare\n/SolidSquare\n/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare/Square/Square/Square/Square/Square/Square\n/Square/Square/Square\n/Square/Square\n/Square/Square/Square/Square\n/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square\n/Square/Square/Square/Square/Square/Square\n/Square\n/Square/Square/Square\n/Square/Square/Square/Square\n/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle/SolidSquareInSx\n/SquareInSy\n/SolidCircleInSz\n0 10 20 30 40 5001234\nSitenumber, nSpincurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1\nFIG. 2: Different cartesian components of the statistically\naveraged longitudinal spin current as a function of the site\nnumber. Numerical parameters are ∆ T= 50 [K], α= 0.01\nandH0= 0 [T]. The temperature gradient is defined ∆ T=\nT1−T50, whereT1= 50 [K]. The only nonzero component of\nthe spin current is ISzn. Other two components ISxn, ISy\nnare\nzero because of the uniaxial magnetic anisotropy field which\npreserves XOYsymmetry of the magnetization dynamics.\nature profile applied to the system. This effect was not\nobserved in the single macro spin approximation and is\nonlyrelevantforthe non-uniformlymagnetizedfinite-size\nmagnetic insulator sample. In addition one observesthat\nthe amplitude of the spin current increases with increas-\ningthe thermalgradient. Thisispredictablynatural; less\nsohowever, is the presenceof amaximum ofthe spin cur-\nrent observed in the middle of the chain. We interpret\nthis observation in terms of a collective cumulative av-\neraged influence of the surrounding magnetic moments\non particular magnetic moment. For a linear tempera-\nture gradient, as in FIG. 2, we have ∆ T=T1−TN\naNwhich\nmeans that half of the spins with i < N/2 possess tem-\nperatures abovethe mean temperature of the chain T1/2,\nwhile the other half have temperatures below the mean\ntemperature. Further, the main contributors in the total\nspin current are the hot magnetic moments with temper-\natures above the mean temperature Tn> Tavand with\na positive exchange spin torque Qn>0. While magnetic\nmoments with a temperature below the mean tempera-\ntureTn< Tav,Qn<0absorbthe spincurrentandhavea\nnegativecontribution in the total spin current. This non-\nequivalence of magnetic moments results in a maximum\nof the total spin current in the center of the chain. In\nwhat follows the magnetic moments with temperatures\nhigher than the mean temperature in the chain are re-\nferred to as hot magnetic moments, while the magnetic\nmoments with temperatures lower than Tavwe refer to\nas cold magnetic moments (i.e., our reference tempera-\nture isTav). The idea we are following is that the hot\nmagnetic moments form the total spin current which is\npartly utilized for the activation of the cold magnetic\nmoments. FIG. 3 illustrates the motivation of this state-\nment. The maximum of the spin current (solid circles) is5\n/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidCircleInSz\n/SolidUpTriangle/ScriptA3\nΓQnz\n0 10 20 30 40 5001234\nSitenumber, n1011/LBracket1/HBars/Minus1/RBracket1\nFIG. 3: Z-componentofthestatistically averaged spincurr ent\nISzn(blue solid circles) and the distribution of the exchange\nspin torquea3\nγQz\nn(red solid triangles), both site-dependent.\nDirect correlation between the behavior of the spin current\nand the exchange spin torque can be observed: the change\nof the sign of the exchange spin torque exactly matches the\nmaximum of the spin current.\nobserved in the vicinity of the sites where the exchange\nspin torque term Qnchanges its sign from positive to\nnegative (solid triangles), highlighting the role of the hot\nand cold magnetic moments in finite-size systems. To\nfurther affirm we consider two different temperature pro-\nfiles - linear and exponential - with slightly shifted values\nof the mean temperature (FIG. 4). The dependence of\nthe maximum spin current on the mean temperature is\na quite robust effect and a slight shifts of the mean tem-\nperature to the left lead to a certain shifting of the spin\ncurrent’s maximum. The effect of the nonuniform spin\ncurrent passing through the finite-size magnetic insula-\ntormightbetestedexperimentallyusingtheSSEsetupin\nwhich the spin current’s direction is parallel to the tem-\nperature gradient. One may employ the inverse spin Hall\neffect using FM insulator covered by a stripe of param-\nagnetic metal, e.g. Ptat different sites (cf Ref.20), albeit\nthe chain must be small ( <∼1µm).\nFurthermore, from FIG. 2 we infer that the only\nnonzero component of the spin current is Iz\nn. Due to\nthe uniaxial magnetic anisotropy all orientations of the\nmagnetic moments in the XOYplane are equivalent and\nIx\nn,Iy\nncomponents of the spin current vanish.\nC. Role of boundary conditions\nTo elaborate on the origin of the observed maximum\nof the spin current we inspect the role of boundary con-\nditions. In fact, in spite of employing different bound-\nary conditions for the chain we observe the same effect\n(FIG. 5), from which we can conclude that the effect of/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle\n/Circle/SolidSquareTn/EquΑlLinear\n/CircleTn/EquΑlExponential\n0 10 20 30 40 500.00.51.01.52.02.5\nSitenumber, nSpincurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1\n1 25 502031.234.450\nnTn/LBracket1K/RBracket1Temperature\nprofile\nFIG. 4: Z-component of the statistically averaged spin cur-\nrent for the linear ∆ T=T1−T50and exponential ∆ T(n) =\n50[K]e−(n−1)/50temperature gradients. The slight shift of the\nmean temperature to the left leads to a certain shifting of th e\nmaximum spin current to the left.\nthe cold and hot magnetic moments is inherent to the\nspin dynamics within the chain, which is independent\nfrom the particular choice of the boundary conditions.\nFurthermore, we model the situation with the extended\nregion at the ends of the FM chain (FIG. 6), in which\nthe end temperatures are constant (i.e., one might imag-\nine the heat reservoirs to have finite spatial extensions).\nModeling the ends of the FM-chain with zero temper-\nature gradient by means of the LLG equations is cer-\ntainly an approximation, which can be improved by em-\nploying the Landau-Lifshitz-Bloch equations reported in\nRef.12. It captures, however, the main effects at rela-\ntively low temperatures: the flow of the spin current for\nthe decaying spin density away from the T= const-∆ T-\ninterfaceand anon-zerointegralspin currentfor the sites\n0< n <50 and 150 < n <200. As we see even in the\nfragments of the chain with a zero temperature gradient\nthespincurrentisnotzero. Thereasonisthattheforma-\ntion of the spin current profile is a collective many body\neffect of the interacting magnetic moments. Therefore,\nthe fragment of the chain with nonzero temperature gra-\ndient (sites 50 < n <150) has a significant influence on\nthe formation of the spin current profiles in the left and\nrightregionsofthe chainwhere the temperaturegradient\nvanishes.\nD. Temperature dependence of the longitudinal\nspin current\nInFIG.7thedependenceofthe z-componentoftheav-\neraged longitudinal spin current on the temperature gra-\ndient is shown. The dependence In(∆T) (inset ofFIG. 7)\nis linear and the amplitude of the spin current increases6\n/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle\n/SolidCircle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle/DownTriangle\n/DownTriangle\n/DownTriangle\n/DownTriangle\n/DownTriangle\n/DownTriangle\n/DownTriangle\n/DownTriangle\n/DownTriangle\n/DownTriangle/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square\n/Square\n/Square\n/Square\n/Square\n/Square\n/Square\n/Square\n/Square\n/Square/SolidCircleM0/EquΑlMN/Plus1/EquΑl/LParen10,0,0/RParen1\n/UpTriangleM0/EquΑl/LParen10,0,0/RParen1,MN/Plus1/EquΑl/LParen10,0,Ms/RParen1\n/DownTriangleM0/EquΑl/LParen10,0,Ms/RParen1,MN/Plus1/EquΑl/LParen10,0,0/RParen1\n/SquareM0/EquΑlMN/Plus1/EquΑl/LParen10,0,Ms/RParen1\n0 10 20 30 40 5001234\nSitenumber, nSpinnurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1\nFIG. 5: Effect of different boundary conditions on the aver-\naged spin current. Numerical parameters are ∆ T= 50 [K],\nα= 0.01 andH0= 0 [T]. The temperature gradient is defined\n∆T=T1−T50, whereT1= 50 [K]. Inspite of different bound-\nary conditions we observe the same maximal spin current for\nthe site number corresponding to the mean temperature of\nthe system. Thus, the effect of the cold and hot magnetic\nmoments is independent of the particular choice of boundary\nconditions.\n/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet\n/Bullet\n0 50 100 150 200/Minus1012345\nSitenumber, nSpinnurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1\n1501001502002060100\nnTn/LBracket1K/RBracket1Temperature\nprofile\nFIG. 6: Effect of boundary conditions in the case of different\ntemperature profiles at the boundaries: linear temperature\ngradient (thick curve), constant temperature for 0 < n <50\nand 150 < n <200 (thin curve). Even in the fragments of\nthe chain with zero temperature gradient the spin current is\nnot zero, which results from the formation of the spin cur-\nrent profile as a collective many body effect of the interactin g\nmagnetic moments. Therefore, the fragment of the chain with\nnonzero temperature gradient (sites 50 < n <150) has a sig-\nnificant influence on the formation of the spin current profile s\nin the left and right zero temperature gradient parts of the\nchain.\nwith the temperature gradient. This result is consistent\nwith the experimental facts (Refs.4,5) and our previous\nanalytical estimations obtained via the single macrospin\nmodel14./Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/Circle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/Square/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare\n/SolidSquare/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle\n/UpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n0 10 20 30 40 5001234\nSitenumber, nSpincurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1\n/Circle /Circle/SolidCircle /SolidCircle/Square /Square/SolidSquare /SolidSquare/UpTriangle /UpTriangle/SolidUpTriangle /SolidUpTriangle\n0 5003\n/DifferenceDeltaT/LBracket1K/RBracket1I26/Multiply1011/LBracket1/HBars/Minus1/RBracket1\n/DifferenceDeltaT=50 [K]\n/DifferenceDeltaT=40 [K]\n/DifferenceDeltaT=0 [K]/DifferenceDeltaT=10 [K]/DifferenceDeltaT=20 [K]/DifferenceDeltaT=30 [K]\nFIG. 7: Dependence of the averaged spin current on the\nstrength of the temperature gradient. Numerical parameter s\nareα= 0.01 andH0= 0 [T]. The temperature gradient is\ndefined as ∆ T=T1−T50, whereT1= 50 [K]. The inset shows\nthe averaged spin current for the 26-th site. The maximum\ncurrent increases with elevating the temperature gradient .\nE. Finite-size effects\nFinite-size effects are considered relevant for the ex-\nperimental observations (e.g. Ref.4). In the thermody-\nnamic limit N≫1 we expect the formation of equilib-\nrium patterns in the spin currentprofile correspondingto\nthe zero exchange spin torque Qn= 0 between nearest\nadjacent moments. To address this issue, the spin cur-\nrent for chains of different lengths is shown in FIG. 8. As\nwe see in the case of N= 500 magnetic moments large\npattern of the uniform spin current corresponding to the\nsites 50< n <450 is observed. In order to understand\nsuch a behavior of the spin current for a large system\nsize, we plotted the dependence on the site number of\nthe exchange spin torque Qn(FIG. 9). As we see, the\nexchange spin torque corresponding to the spin current\nplateau is characterizedby largefluctuations aroundzero\nvalue, while nonzero positive (negative) values of the ex-\nchange spin torque Qnobserved at the left (right) edges\ncorrespond to the nonmonotonic left and right wings of\nthe spin torque profile. One may try to interpret the ob-\nservedresults in terms ofthe so called magnon relaxation\nlength (MRL) λm≈2/radicalBig\n(DkBT/¯h2)τmmτmp(Refs.5,6),\nwhereDis the spin-wave stiffness constant and τmm,mp\nare the magnon-magnon, and the magnon-phonon relax-\nation times, respectively. The MRL is a characteristic\nlength which results from the solution of the heat-rate\nequation for the coupled magnon-phonon system5. The\nphysical meaning of λmis an exponential drop of the\nspace distribution of the local magnon temperature for\nthe given external temperature gradient ∆ T. In other\nwords, although the externally applied temperature bias\nis kept constant, the thermal distribution for magnons\nis not necessarily linear. In general, one may suggest a\nsinh(x)-like spatial dependence5and a temperature de-7\npendence λm(T). Estimates of the MRL for the material\nparameters related to YIG (suppl. mater. of Ref.5) and\nTN= 0.2 [K] yield the following λm≈10 [µm]26. As\nseen from FIG. 8 the length starting from which the sat-\nuration of the spin current comes into play as long as\nthe FM-chain exceeds the length 20 [nm] ×100≈2 [µm].\nHowever,werecallthat MRLisawitnessofthe deviation\nbetween the magnon and phonon temperature profiles.\nTherefore, for interpretingthe nonmonotonicpartsofthe\nspin current profile (FIG. 8) in terms of the MRL one has\nto prove the pronounced deviation between magnon and\nphonontemperaturesattheboundaries. Forfurtherclar-\nification we calculate the magnon temperature profile.\nThis can be done self-consistently via the Langevin func-\ntion< Mz\nn>=L/parenleftbig\n< Mz\nn> Hn/kBTm\nn/parenrightbig\n. HereHz\nnis the\nzcomponent of the local magnetic field which depends\non the external magnetic field and the mean values of the\nadjacent magnetic moments < Mz\nn−1>, < Mz\nn+1>(\nsee eq. (4)). As inferred from the FIG. 10 the magnon\ntemperature profile follows the phonon temperature pro-\nfile. Prominent deviation between the phonon and the\nmagnon temperatures is observed only at the beginning\nof the chain and gradually decreases and becomes small\non the MRL scale. Close to the end of the chain the\ntemperature difference becomes almost zero. This means\nthat left nonmonotonic parts of the spin current profile\nFIG. 8 can be interpreted in terms of none-equilibrium\nprocesses. Comparing this result with the exchange spin\ntorque profile (FIG. 9) we see that in this part of the\nspin chain the exchange spin torque is positive. This\nis the reason why the spin current Inis increasing with\nthe site number n. The saturated plateau of the spin\ncurrent shown in FIG. 8 corresponds to the zero ex-\nchange spin torque Qn= 0 (cf. FIG. 9) and the decay\nof the spin Seebeck current Inat the right edge corre-\nsponds to the negative spin exchange torque Qn<0.\nThus, for the formation of the convex spin current profile\nthe key issue is not the difference between magnon and\nphonontemperatures,whichasweseeisprettysmall,but\nthe magnon temperature profile itself. The existence of\nthe hot(cold) magnetic moments with the local magnon\ntemperature up (below) the mean magnon temperature\ngenerates the spin current. This difference in the local\nmagnon temperature of the different magnetic moments\ndrives the spin current in the chain. On the other hand\nany measurement of the spin current done in the vicin-\nity of the right edge of the current profile will demon-\nstrate a non-vanishing spin current in the absence of the\ndeviation between the magnon and phonon temperature\nprofiles. This may serve as an explanation of the re-\ncent experiment25, where a non-vanishing spin current\nwas observed in the absence of the deviation between the\nmagnon and the phonon temperature profiles. We note\nthat zero values of the spin current shown in FIG. 8 is\nthe artefact of isolated magnetic insulator chain. Real\nmeasurement of the spin currents usually involve FM-\ninsulator/NM-interfaces. As will be shown below the in-\nterface effect described by an enhanced Gilbert dampingN=50N=500\nN=200\nN=150\nN=100\n1 100 200 300 400 500012\nSitenumber, nSpincurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1\nFIG. 8: The dependence of the averaged spin current on the\nlength of the FM-chain. Numerical parameters are α= 0.01\nandH0= 0 [K]. The temperature gradient is linear and the\nmaximum temperature is on the left-hand-side of the chain\n(T1= 100 [K]). In all cases the per-site temperature gradient\nis ∆T/N= 0.2 [K].\n0100200300400500/Minus0.10/Minus0.050.000.050.100.150.20\nSitenumber, n/ScriptA3\nΓQnz/Multiply1011/LBracket1/HBars/Minus1/RBracket1\nFIG. 9: The dependence of the exchange spin torque on\nthe site number. Numerical parameters are α= 0.01 and\nH0= 0 [K]. The temperature gradient is linear and the\nmaximum temperature is on the left-hand-side of the chain\n(T1= 100 [K]). The per-site temperature gradient is ∆ T/N=\n0.2 [K]. The exchange spin torque profile consists of three\nparts, the positive part corresponds to the high temperatur e\ndomain and low temperature domain corresponds to the neg-\native exchange spin torque. In the middle of the chain where\nthe spin current is constant, the exchange spin torque fluctu -\nates in the vicinity of the zero value.\nand the spin torque lead to a nonzero spin current at the\ninterfaces which is actually measured in the experiment.\nF. Role of the external magnetic filed ( H0/negationslash= 0)\nIt follows from our calculations that the dependence of\nthe longitudinal spin current on the magnetic field is not8\n/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bu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200 300 400 500020406080100\nSitenumber, nTemperature/LBracket1K/RBracket1\nFIG. 10: The magnon temperature profile (line) formed in the\nsystem. Numerical parameters are α= 0.01 andH0= 0 [K].\nBlue line corresponds to the applied linear phonon tempera-\nture profile. The maximum temperature on the left-hand-side\nof the chain is( T1= 100 [K]). The per-site temperature gradi-\nent is ∆T/N= 0.2 [K]. The maximal deviation between the\nphonon and magnon temperatures is observed only at the left\nedge of chain. The difference between temperatures graduall y\ndecreases and becomes almost zero for the sites with n >400.\ntrivial. Once the external static magnetic field is applied\nperpendicularly to the FM-chain and along the easy axis\nat the same time, we can suppress the spin current at ele-\nvated magnetic fields (FIG. 11). The threshold magnetic\nfield is - asexpected - the strength ofthe anisotropyfield,\ni.e. 2K1/MS∼0.056 [T]. By applying magnetic fields\nmuch higher than 0 .056 [T], the magnetic moments are\nfully aligned along the field direction and hence the X-,\nY-components of the magnetization required to form the\nZ-component of the longitudinal averaged spin current\nvanish.\nIn the case of the magnetic field being applied perpen-\ndicularly to the easy axis, the behavior becomes more\nrich (FIG. 12). In analogy with the situation observed\nin FIG. 11 there are no sizeable changes for the In(∆T)-\ndependence at low static fields. This is the regime where\nthe anisotropy field is dominant. In contrast to the Hz\n0\napplied field, the spin current does not linearly depend\non the strength of the field (inset of FIG. 11), which is\nexplained by the presence of different competing contri-\nbutions in the total energy density and not a simple cor-\nrection of the Z-component of the anisotropy field illus-\ntrated in the previous figure. Surprisingly, the magnetic\nfield oriented along the FM-chain can also suppress the\nappearance of the spin current’s profile. Also in this case\nthe strong magnetic field destroys the formation of the\nmagnetization gradient resulting from the applied tem-\nperature bias./SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle\n/Circle/Circle/Circle/Circle/Circle/Circle\n/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/Square/Square/Square/Square/Square/Square/SolidCircleH0z/EquΑl0/LBracket1T/RBracket1\n/CircleH0z/EquΑl10/Minus2/LBracket1T/RBracket1\n/SolidSquareH0z/EquΑl10/Minus1/LBracket1T/RBracket1\n/SquareH0z/EquΑl1/LBracket1T/RBracket1\n0 10 20 30 40 500123\n/DifferenceDeltaT/LBracket1K/RBracket1Spincurrent I26/Multiply1011/LBracket1/HBars/Minus1/RBracket1\nFIG. 11: Effect of the external magnetic field applied paralle l\nto the easy axis on the averaged spin current. Numerical\nparameters are ∆ T= 50 [K], α= 0.01 andN= 50. The\ntemperature gradient is linear and the maximum temperature\nis on the left-hand-side of the chain.\n/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle\n/Circle/Circle/Circle/Circle/Circle/Circle\n/SolidSquare/SolidSquare/SolidSquare/SolidSquare\n/SolidSquare/SolidSquare /Square/Square/Square/Square/Square/Square/SolidCircleH0x/EquΑl0/LBracket1T/RBracket1\n/CircleH0x/EquΑl10/Minus2/LBracket1T/RBracket1\n/SolidSquareH0x/EquΑl10/Minus1/LBracket1T/RBracket1\n/SquareH0x/EquΑl1/LBracket1T/RBracket1\n0 10 20 30 40 5001234\n/DifferenceDeltaT/LBracket1K/RBracket1Spincurrent I26/Multiply1011/LBracket1/HBars/Minus1/RBracket1\n/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle/DifferenceDeltaT/EquΑl50/LBracket1K/RBracket1\n0.020.040.060.080.1001234\nH0x/LBracket1T/RBracket1I26/Multiply1011/LBracket1/HBars/Minus1/RBracket1\nFIG. 12: Effect of the external magnetic field applied perpen-\ndicularly to the easy axis on the averaged spin current. Nu-\nmerical parameters are ∆ T= 50 [K], α= 0.01 andN= 50.\nThe temperature gradient is linear and the maximum tem-\nperature is on the left-hand-side of the chain.\nV. INTERFACE EFFECTS\nThe experimental setup to detect the spin current\nmight involve a NM adjacent to the spin-current gener-\nating substance, e.g. a FM insulator. This NM converts\nthe injected spin current from the FM to an electric cur-\nrent via ISHE1,5,27. So it is of interest to see the effect\nof the adjacent NM on the generated spin current in the\nconsidered chain. Obviously, the main effects appear in\nthe FM-NM interface. The interface effect can be di-\nvided into two parts which is described in the following\nsubsections.9\nA. Spin pumping and enhanced Gilbert damping\nInmagnetic insulators , chargedynamicsislessrelevant\n(in our model, anyway), and in some cases the dissipa-\ntive losses associated with the magnetization dynamics\nare exceptionally low (e.g. in YIG28α= 6.7×10−5).\nWhen a magnetic insulator is brought in contact with\nanormal metal , magnetization dynamics results in spin\npumping, which in turn causes angular momentum being\npumped to the NM. Because of this nonlocal interaction,\nthe magnetization losses become enhanced21.\nIf we consider the normal metal as a perfect spin\nsinkwhich remains in equilibrium even though spins are\npumped into it (which means there is a rapid spin relax-\nation and no back flow of spin currents to the magnetic\ninsulator), the magnetization dynamics is described by\nthe LLG equation with an additional torque originating\nfrom the FM-insulator/NM interfacial spin pumping21\n∂/vectorM\n∂t=−γ/bracketleftBig\n/vectorM×/vectorHeff/bracketrightBig\n+α\nMS/bracketleftBigg\n/vectorM×∂/vectorM\n∂t/bracketrightBigg\n+/vector τsp,(11)\nwhere\n/vector τsp=γ¯h\n4πM2\nSgeffδ(x−L)/bracketleftBigg\n/vectorM×∂/vectorM\n∂t/bracketrightBigg\n,(12)\nwhereLis the position of the interface, eis the elec-\ntron charge and geffis the real part of the effective\nspin-mixing conductance. In the YIG-Pt bilayer the\nmaximum measured effective spin-mixing conductance is\ngeff= 4.8×1020[m−2] Ref.21. In fact if the spin pumping\ntorque should be completely described, one should add\nanother torque containing the imaginary part of geff29.\nHowever, we omit this imaginary part here because it\nhas been found to be too small at FM-NM interfaces30.\nThe aforementioned spin pumping torque concerns the\ncases that we characterized with /vectorM. In our discrete\nmodel which includes a chain of Nferromagnetic cells,\nwe describe the above phenomena as follows\n∂/vectorMn\n∂t=−γ/bracketleftBig\n/vectorMn×/vectorHeff\nn/bracketrightBig\n+α\nMS/bracketleftBigg\n/vectorMn×∂/vectorMn\n∂t/bracketrightBigg\n+/vector τsp\nn,\n(13)\nwhere\n/vector τsp\nn=γ¯h2\n2ae2M2\nSg⊥δnN/bracketleftbigg\n/vectorMn×∂Mn\n∂t/bracketrightbigg\n,(14)\nwhich means the spin pumping leads to an enhanced\nGilbert damping in the last site\n∆α=γ¯h\n4πaMsgeff. (15)\nAs mentioned, the above enhanced Gilbert damping\ncould solely describe the interfacial effects as long as\nwe treat the adjacent normal metal as a perfect spinsink without any back flow of the spin current from the\nNM17,21. The latter is driven by the accumulated spins\nin the normal metal. If we model the normal metal as\na perfect spin sink for the spin current, spin accumu-\nlation does not build up. This approximation is valid\nwhen the spin-flip relaxation time is very small and so\nit prevents any spin-accumulation build-up. So the spins\ninjected by pumping decayand/orleavethe interfacesuf-\nficiently fast and there won’t be any backscattering into\nthe ferromagnet13,31. We note by passing that in a re-\ncentstudy concerningthis phenomena, it hasbeen shown\nthat spin pumping (and so enhanced Gilbert damping)\ndepends on the transverse mode number and in-plane\nwave vector21.\nB. Spin transfer torque\nIt was independently proposed by Slonczewski32and\nBerger33that the damping torque in the LLG equation\ncould have a negative sign as well, corresponding to a\nnegative sign of α. This means that the magnetization\nvector could move into a final position antiparallel to the\neffective field. In order to achieve this, energy has to be\nsupplied to the FM system to make the angle between\nthe magnetization and the effective field larger. This en-\nergy is thought to be provided by the injection of a spin\ncurrent/vectorIincidentto the FM13,29,34\n/vector τs=−γ\nM2\nSV/bracketleftBig\n/vectorM×/bracketleftBig\n/vectorM×/vectorIinjected/bracketrightBig/bracketrightBig\n,(16)\nwhich describes the dynamics of a monodomain ferro-\nmagnet of volume Vthat is subject to the spin current\n/vectorIincidentand modifies the right-hand side of the LLG\nequation as a source term. In general, a torque-term\nadditional to the Slonczewskis torque (eq. (16)) is also\nallowed29,35\n/vector τsβ=−γ\nMSVβ/bracketleftBig\n/vectorM×/vectorIincident/bracketrightBig\n, (17)\nwhereβgives the relative strength with respect to the\nSlonczewski’s torque (eq. (16)).\nFor the case of a FM-chain, again we assume that the\nabovespin-transfertorques act solelyon the last FM cell.\nC. Numerical results for interface effects\nInordertosimulatetheenhancedGilbertdampingand\nthe spin–transfer torque we assume that they act only on\nthechainend(motivatedbytheiraforementionedorigin).\nSo the dynamics of our FM-chain is described by the\nfollowing LLG18,36equations\n∂/vectorMn\n∂t=−γ\n1+α2/bracketleftBig\n/vectorMn×/vectorHeff\nn/bracketrightBig\n−γα\n(1+α2)MS/bracketleftBig\n/vectorMn×/bracketleftBig\n/vectorMn×/vectorHeff\nn/bracketrightBig/bracketrightBig\n,\nn= 1,...,(N−1),(18)10\nand\n∂/vectorMN\n∂t=−γ\n1+α2\nN/bracketleftBig\n/vectorMN×/vectorHeff\nN/bracketrightBig\n−γαN\n(1+α2\nN)MS/bracketleftBig\n/vectorMN×/bracketleftBig\n/vectorMN×/vectorHeff\nN/bracketrightBig/bracketrightBig\n−γ\nM2\nSa3/bracketleftBig\n/vectorMN×/bracketleftBig\n/vectorMN×/vectorIinjected/bracketrightBig/bracketrightBig\n−γ\nMSa3β/bracketleftBig\n/vectorMN×/vectorIincident/bracketrightBig\n,(19)\nwhereαN=α+γ¯hgeff/(4πaMs).\nEq.(18) and (19) describe the magnetizationdynamics\nin the presence of the interface effects and include both\nspinpump and spintorqueeffects. Results inthe absence\nof the spin torque are presented at the FIG. 13. The en-\nhanced Gilbert damping captures losses of the spin cur-\nrent associated with the interface effect. A nonzero spin\ncurrent corresponding to the last n= 500 spin quantifies\nthe amount of the spin current pumped into the normal\nmetal from the magnetic insulator. However, the con-\nvex profile of the spin current is observed as well in the\npresence ofthe interface effects. The influence ofthe spin\ntorqueonthespincurrentprofileisshowninFIG.14. We\nsee from these results, the large spin torque reduces the\ntotalspincurrentfollowingthroughtheFM-insulato/NM\ninterfaces. The spin torque current is directed opposite\nto the spin pump current and therefore compensates it.\n/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSqu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100 200 300 400 500012\nSitenumber, nSpincurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1\n/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle\n/SmallCircle\n450 475 500012\nnWith enhanced \ndamping, /DifferenceDeltaΑ=0.5\nWithout enhanced \ndamping, /DifferenceDeltaΑ=0.5\nFIG. 13: Statistically averaged spin current in the chain of\nN= 500-sites. Numerical parameters are ∆ T= 100 [K],\nα= 0.01 andH0= 0 [T]. The temperature gradient is lin-\near and the maximum temperature is on the left-hand-side\nof the chain ( T1). The blue curve shows the averaged spin\ncurrent when no enhanced Gilbert damping and no spin–\ntransfer torque is present. The red curve shows the aver-\naged spin current when the enhanced Gilbert damping with\ngeff= 1.14×1022[m−2] is present. The inset shows the aver-\naged spin current of the last fifty sites only./SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare\n/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidS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100 200 300 400 500012\nSitenumber, nSpincurrent/Multiply1011/LBracket1/HBars/Minus1/RBracket1\n/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare/SmallSolidSquare\n/SmallSolidSquare/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle/SmallCircle\n/SmallCircle\n450 475 500012\nnIn both cases /DifferenceDeltaΑ=0.5\nincidentIincident/EquΑl\n1.03/Multiply1015/LParen1/Minus1, 0, 0/RParen1/LBracket2/HBars/Minus1/RBracket2\nIincident/EquΑl\n5.15/Multiply1015/LParen1/Minus1, 0, 0/RParen1/LBracket2/HBars/Minus1/RBracket2\nFIG. 14: Statistically averaged spin current in the chain of\nN= 500 when there are both the enhanced Gilbert damp-\ning and the spin–transfer torque. Numerical parameters are\n∆T= 100 [K], α= 0.01,H0= 0 [T], geff= 1.14×1022[m−2]\nandβ= 0.01. The temperature gradient is linear and the\nmaximum temperature is on the left-hand-side of the chain\n(T1). The blue curve has /vectorIincident= 1×1015(−1,0,0) [¯hs−1]\nand the red curve is with /vectorIincident= 5×1015(−1,0,0) [¯hs−1].\nVI. MECHANISMS OF THE FORMATION OF\nSPIN EXCHANGE TORQUE AND SPIN\nSEEBECK CURRENT\nIn the previous sections we demonstrated the direct\nconnection between the spin Seebeck current profile and\nthe exchange spin torque. Here we consider the mecha-\nnisms of the formation of the exchange spin torque. For\nthis purpose we investigate changes in the magnetization\nprofile associatedwith the changeof the magnontemper-\nature<∆Mz\nn>=< Mz\nn>−< Mz\n0n>, where< Mz\nn>\nis the mean component of the magnetization moment\nfor the case of the applied linear thermal gradient, while\n< Mz\n0n>correspondstothemeanmagnetizationcompo-\nnent in the absence of thermal gradient ∆ T= 0. Quan-\ntity<∆Mz\nn>defines the magnon accumulation as the\ndifferencebetweentherelativeequilibriummagnetization\nprofile and excited one Ref.37and is depicted in FIG. 15.\nWe observe a direct connection between the magnon ac-\ncumulation effect and the exchange spin torque. A pos-\nitive magnon accumulation, meaning an excess of the\nmagnonscomparedtotheequilibriumstateisobservedin\nthe high temperature part of the chain. While in the low\ntemperature part the magnon accumulation is negative\nindicatingalackofmagnonscomparedtotheequilibrium\nstate. The exchange spin torque is positive in the case of\nthe positive magnon accumulation and is negative in the\ncase of the negative magnon accumulation (the exchange\nspin torque vanishes in the equilibrium state). From the\nphysical point of view, the result is comprehensible: the\nspin Seebeck current is generated by the magnon accu-\nmulation, transmitted through the equilibrium part of\nthe chain and partially absorbed in the part of the chain\nwith a negative magnon accumulation.11\n/SolidCircle/SolidCircle/SolidCircle\n/SolidCircle/SolidCircle\n/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle\n/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle\n/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle/SolidCircle\n/SolidCircle/SolidCircle/SolidCircle\n/SolidCircle\n/SolidCircle\n50 100 150 200/Minus4/Minus202\nSitenumber, nMagnon accumulation10/Minus5/Multiply/LParen1m/Minusm0/RParen1\n15010015020020\nnTn/LBracket1K/RBracket1\n050100150200/Minus0.10/Minus0.050.000.050.100.15\nn/ScriptA3\nΓQnz/Multiply1011/LBracket1/HBars/Minus1/RBracket1\nFIG. 15: Site dependence of the exchange spin torque and\nthe magnon accumulation effect. We observe a direct connec-\ntion between magnon accumulation effect and the exchange\nspin torque. A positive magnon accumulation, i.e. excess of\nthe magnons, is observed in the high temperature part of the\nchain. While in the low temperature part magnon accumu-\nlation is negative (lack of magnons compare to the equilib-\nrium state). The exchange spin torque is positive for posi-\ntive magnon accumulation, and negative for negative magnon\naccumulation. The spin Seebeck current is generated by ex-\ncess magnons, transmitted through the equilibrium part of\nthe chain and partially absorbed in the region with magnon\ndrain.\nVII. CONCLUSIONS\nBased on the solution of the stochastic Landau-\nLifshitz-Gilbert equation discretized for a ferromagnetic\nchain in the presence of a temperature gradient formed\nalong the chain, we studied the longitudinal spin See-\nbeck effect with a focus on the space-dependent effects.\nIn particular, we calculated a longitudinal averaged spin\ncurrent as a function of different temperature gradients,\ntemperature gradient strengths, distinct chain lengths\nand differently oriented external static magnetic fields.\nOur particular interest was to explain the mechanisms\nof the formation of the spin Seebeck current beyond the\nlinear response regime. The merit was in pointing out a\nmicroscopicmechanismfortheemergenceofthespinSee-beck current in a finite-size system. We have shown that,\nwithin our model, the microscopic mechanism of the spin\nSeebeck current is the magnon accumulation effect quan-\ntified in terms of the exchange spin torque. We proved\nthat the magnon accumulation effect drives the spin See-\nbeck current even in the absence of significant deviation\nbetween magnon and phonon temperature profiles. Our\ntheoretical findings are in line with recently observed ex-\nperimental results25where non-vanishing spin Seebeck\ncurrent was observed in the absence of a temperature\ndifference between phonon and magnon baths.\nConcerningthe influence ofthe external constant mag-\nnetic fields on the spin Seebeck current we found that\ntheir role is nontrivial: An external static magnetic field\napplied perpendicularly to the FM-chain and along the\neasy axis may suppress the spin current at elevated mag-\nnetic fields (FIG. 11). The threshold magnetic field has a\nstrengthofthe anisotropyfield, i.e. 2 K1/MS∼0.056[T].\nIn the case of the magnetic field applied perpendicu-\nlarly to the easy axis, we observe a more complex be-\nhavior (FIG. 12). In analogy with the situation seen in\nFIG. 11 there are no sizeable changes for the In(∆T)-\ndependence at low static fields. This is the regime where\nthe anisotropy field is dominant. In contrast to the Hz\n0\napplied field, it does not linearly depend on the strength\nof the field (inset of FIG. 11), which is explained by the\npresence of different competing contributions in the total\nenergyand not a simple correctionof the Z-componentof\nthe anisotropy field. Notably, the magnetic field oriented\nalong the FM-chain can also suppress the emergence of\nthe spin current’s profile. Also in this case a strong mag-\nnetic field destroys the formation of the magnetization\ngradient resulting from the applied temperature bias.\nIn addition, we modeled an interface formed by a\nnonuniformly magnetized finite size ferromagnetic insu-\nlator and a normal metal (e.g., YIG-Platinum junction)\nto inspect the effects of the enhanced Gilbert damping\non the formation of space-dependent spin current within\nthe chain.\nVIII. ACKNOWLEDGEMENTS\nThe financial support by the Deutsche Forschungsge-\nmeinschaft (DFG) is gratefully acknowledged.\n1K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae,\nK. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778\n(2008).\n2M. Hatami, G. E. W. Bauer, Q. Zhang, and P. J. Kelly,\nPhys. Rev.B 79, 174426 (2009); A. D.Avery, M. R. Pufall,\nand B. L. Zink, Phys. Rev. Lett. 109, 196602 (2012); C.\nH. Wong, H. T. C. Stoof, and R. A. Duine, Phys. Rev. A\n85, 063613 (2012).\n3S. Bosu, Y. Sakuraba, K. Uchida, K. Saito, T. 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B.89, 024409 (2014)."    },    {        "title": "2002.02686v1.Engineering_Co__2_MnAl__x_Si___1_x___Heusler_compounds_as_a_model_system_to_correlate_spin_polarization__intrinsic_Gilbert_damping_and_ultrafast_demagnetization.pdf",        "content": "1 \n Engineering Co 2MnAl xSi1-x Heusler compounds as a model system to \ncorrelate spin polarization, intrinsic Gilbert damping and ultrafast \ndemagnetization  \nC. Guillemard1,2, W. Zhang1*, G. Malinowski1, C. de Melo1, J. Gorchon1, S. Petit -\nWatelot1, J. Ghan baja1, S. Mangin1, P. Le Fèvre2, F. Bertran2, S. Andrieu1*  \n1 Institut Jean Lamour, UMR CNRS 7198, Université de Lorraine, 54500 Nancy France  \n2 Synchrotron SOLEIL -CNRS, Saint -Aubin, 91192 Gif -sur-Yvette, France  \nAbstract:  \nEngineering of magnetic materials f or developing better spintronic  applications relies on \nthe control of two key parameters: the spin polarization and the Gilbert damping \nresponsible for the spin angular momentum dissipation. Both of them are expected to \naffect the ultrafast magnetization dyna mics occurring on the femtosecond time scale. \nHere, we use engineered Co2MnAl xSi1-x Heusler  compounds to adjust the degree of spin \npolarization P from 60 to 100% and investigate how it correlates with the damping. We \ndemonstrate  experimentally that the damping decreases when increasing the spin \npolarization from 1.1  10-3 for Co 2MnAl with 63% spin polarization  to an ultra -low value \nof 4.10-4 for the half -metal magnet Co 2MnSi. This allows us investigating the relation \nbetween these two parameters and the ultrafast demagnetization time characterizing the \nloss of magnetization occurring after femtosecond laser pulse excitation. The \ndemagnetization time is observed to be inversely proportional to 1 -P and as a consequence \nto the magnetic damping, which can  be attributed to the similarity of the spin angular \nmomentum dissipation processes  responsible for these two effects. Altogether, our high \nquality Heusler  compounds allow controlling the band structure and therefore the channel \nfor spin angular momentum dissipation.  \n \n * corresponding authors :  \nwei.z hang @univ -lorraine.fr  \nstephane.andrieu@univ -lorraine.fr   2 \n I - INTRODUCTION  \nDuring the last decades, extensive magnetic materials research has strived to \nengineer denser, faster and more energy efficient processing and data storage devices. On \nthe one hand, a high spin polarization has been one of the most important ingredients th at \nhave  been seek  [1]. For example, the spin polarization is responsible for a high readout \nsignal in magnetic tunnel junction based devices  [2,3] . Additionally, a high spin \npolarization results in a decrease of the threshold current for magnetization reve rsal by \nspin torques  [4] required for the development of spin-transfer -torque magnetic random \naccess memory devices [5] , for gyrotropic dynamics in spin -torque nano -oscillators [6] \nand for magnetic domain wall motion  [7]. On the other hand, the intrinsic m agnetic \nenergy dissipation during magnetization dynamics, which is determined by the Gilbert \ndamping constant, needs to be low in order to build an energy efficient device. Fortunately, \nspin polarization and damping are usually closely  related  in magnetic materials.  \nNowadays, manipulation of the magnetization on the femtosecond timescale has \nbecome an outstanding challenge since the demonstration of subpicosecond \nmagnetization quenching [8] and magnetization reversal on the picosecond timescale  [9]. \nDespite the theoretical and experimental work that has been reported up to now , the \nrelationship between  the polarization at the Fermi level or the magnetic damping and the \nultrafast demagnetization  excited by femtosecond lasers, remains  unclear  [10-15]. Indeed,  \nnumerous mech anisms have been proposed  but no consensus has yet been reached. In \nparticular, efforts have been undertaken to unify the magnetization dynamics on the \nnanosecond timescale and the ultrafast demagnetization considering that the sp in-flip \nmechanisms involved in both phenomena could be the same  [10-11,16] . Regarding the \ninfluence of the damping on the demagnetization time, different predictions have been \nreported  both experimentally and theoretically . In this situation, the need for engineered \nsamples in which the spin -polarization and magnetic damping are well controlled is of \nutmost importance to unveil their role on the ultrafast magnetization dynamics.  \nHeusler compounds are a notable class of magnetic materials allowing for tunabl e \nspin-polarization and magnetic damping [ 17]. The absence of available electronic states \nin the minority band at the Fermi level leads to very high spin polarization and ultra -low \ndamping due to a strong reduction of spin scattering [ 18-23]. Recently,  ultra-low damping 3 \n coefficient associated with full spin polarization at the Fermi energy  was reported in \nCo2Mn-based Heusler compounds , [22-23]. Among those alloys, Co 2MnSi has the \nsmallest damping down to 4.1 x 10-4 with 100% spin -polarization while Co 2MnAl , which \nis not predicted to be a half -meta llic magnet, has a damping of 1.1 x 10-3 and a spin -\npolarization of 60 %.  \nIn the present work, we used Co 2MnAl xSi1-x quaternary Heusler  compounds \ngrown by Molecular Beam Epitaxy (MBE) to tune the spin -polarization at the Fermi \nenergy . Controlling the amount of Al within the alloys allows tuni ng the spin -polarization \nfrom 60  to 100 % as measured by spin resolved photoemission. We show that  the \nmagnetic  damping parameter for these alloys is among the lowest reported in the literature \nand decreases when the spin -polarization increases. Ultrafast magnetization dynamics \nexperiments were thus performed on these prototype samples. This complete \nexperimental characterization allows us to directly correlat e the ultrafast magnetization \ndynamics to these parameters and comparing our results to the different theory discussed \nabove.  \nThe Co 2MnSi compound grows in the L2 1 structure whereas the Co 2MnAl com pound \ngrows in the B2 phase as shown by STEM -HAADF analysis [22]. Such different \nstructures are directly observable during the growth by Reflexion High Energy Electron \nDiffraction (RHEED ) since the surface lattice is different for bot h compounds. Ind eed, \nhalf streaks are observed along Co 2MnSi [110] azimuth due to the L2 1 chemical ordering \n[24] which is not the case for  Co2MnAl  [22]. The RHEED analysis on Co2MnAl xSi1-x \nfilms with x= 0, ¼ ,½ , ¾ ,1 reveals a regular decrease of these half -streaks intensity with \nx (Figure 1 a). This information that concerns only the surface is confirmed in the entire \nthickness of the films  by using x -ray diffraction. Indeed, the (111) peak typical of the \nchemical ordering in the L2 1 structure clearly decreases and disappears with x ( Figure \n1b).  4 \n  \nFigure 1 :  a) RHEED patterns along [110] showing the progressive vanishing of the half -streaks \n(observed on Co 2MnSi, x=0) at the surface with x. b) Confirmation of the transition from L2 1 to \nB2 chemical ordering in the entire film by the vanishing of the (111) peak and displacement of \n(220) peak with x  as shown by x -ray diffraction. c) Spatial  distribution of both chemical ordering \nin the films deduced from STEM -HAADF experiments: as the L 21 structure is observed in the \nentire Co 2MnSi film (x=0 ), and the B2 one i n Co 2MnAl (x=1 ), a mixing of both structure is clearly \nobserved for x=0.5 . \n \nIn addition, the displacement of the (220) peak with x allows us to extract a linear \nvariation of the lattice constant  (Figure 1b ), as observed in the case of a solid solution. \nThis is an indication that the L2 1 chemical ordering progressively vanishes when \nincreasing the Al substitution rate 𝑥. However, the chemical disorder distribution in the \nfilms cannot be easily determined by using the electron and x -ray diffraction analyses. To \naddress this point, a STEM HAADF analysis has been carried  on the Co 2MnAl ½Si½ films \nwith a comparison with Co 2MnSi and Co 2MnAl. A clear mixing of both structures is \n5 \n observ ed for x=½ where around 50% is L2 1 chemically ordered and 50%, B2, with typical \ndomains size around 10nm along the growth axis (001) and a few nm in the plane of the \nfilm ( Figure 1c). \nThe electronic properties of the Co 2MnAl xSi1-x(001) series were studied using spin -\nresolved photoemission (SR -PES) and  ferromagnetic resonance (FMR). The SR -PES \nspectra were obtained by using the largest slit acceptance of the detector (+/ - 8°) at an \nangle of 8° of the normal axis of the surface. Such geometry allows us to analyze all the \nreciprocal space  as confirmed by similar experiments but performed on similar \npolycrystalline films [23]. Getting the spin-polarization dependence with x using raw SR -\nPES spectra  is however not obvious due to the existence of surface states systematically \nobserved on Co 2MnSi  but also on other Co 2Mn-based Heusler compounds [19, 22-23]. \nTo get the bulk spin polarization, we thus used the S polarization of the photon beam. \nIndeed , we have shown that the surface states are no more detected  due to their symmetry \n[19] without any loss of information on the bulk band structure [ 23]. The corresponding \nSR-PES spectra are shown in figure 2 . As expec ted, we thus obtain a tunable spin \npolarization  at EF from 100% to 63% by substituting Si by Al, as shown in figure 3 . \n \nFigure 2 : spin -resolved photoemission spectra using P photon polarization (left), S photon \npolarization (middle) and resulting spin polarization curves (right) for the Co 2MnAl xSi1-x series,  \n6 \n  The radiofrequency magnetic dynamics of the films were thus studied using \nferromagnetic resonance (FMR) . The magnetic damping coefficient  , the effective \nmagnetic moment Ms (close to the true moment in our films due to very small anisotropy \n– see [ 22]), and the inhomogeneous linewidth f0 were thus extracted from the \nmeasurements performed on the Co 2MnAl xSi1-x(001) series. The results obtained on the \nsame series used for photoemission experiments are shown in table I . As shown in figure \n3, a clear correlation is observed between the spin polarization at EF and the magnetic \ndamping  coefficient , as theoretically expected. An ultra -low  value was obtained for \nCo2MnSi (x= 0) due to the large spin gap [ 22]. By substituting Al by Si, the magnetic \ndamping increase is explained by the decrease of the spin polarization.  \nCo2MnAl xSi1-x Spin polarization  \n(%) Ms  \n(µB/f.u.)    \n(x 10-3) f0  \n(MHz)  g factor  \n(0.01)  \nx = 0  973 5.08 0.460.05 14.3 2.01 \nx = 0.25  903 4.85 0.730.15 21.7 1.99 \nx = 0.5  833 4.85 0.680.15 9 2.01 \nx = 0.75  703 4.8 1.000.05 81.5 2.00 \nx = 1  633 4.32 1.100.05 22 2.01 \nTable 1: data extracted from spin -resolved photoemission and ferromagnetic resonance \nexperiments performed on the Co 2MnAl xSi1-x series.  \n \nFigure 3: -top- spin polarization and magnetic damping dependence with Al content for the \nCo2MnAl xSi1-x series and –bottom - magnetic damping versus spin polarization . The lines are \nguide to the eyes.  \n7 \n In addition, t he magnetization is also observed to decrease with x in agreement with the \nSlater -Pauling description of the valence band electrons in Heusler compounds  [25]. \nIndeed, as a 5 µ B magnetic moment per cell is expected for Co2MnSi (type IV valence \nelectrons), it should decrease to  4 when replacing Si by Al (type III)  as actually observed \n(Table I ). Finally, the FMR susceptibilities reach extremely small inhomogeneous \nlinewidth f0, a proof of the excellent homogeneity of the magnetic properties (hence a \nhigh crystal quality) in our films.  \nFigure 4 (a) shows the ultrafast demagnetization curves measured  on the same  \nCo2MnAl xSi1-x series with a maximum magnetization quenching ~1 5%. The temporal \nchanges of the Kerr signals ∆𝜃𝑘(𝑡) were normalized by the saturation value 𝜃𝑘 just before \nthe pump laser excitation. The time evolution of magnetization on sub -picosecond \ntimescales c an be fitted according to Eq. (2 ) in terms of the three -Temperature M odel \n(3TM)  [26], which describes the energy distribution among electrons, phonons, and spins \nafter laser excitation.  \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)          (2) \nwhere  𝐺(𝑡,𝜏𝐺) represents the convolution product with the Gaussian laser pulse profile, \nG\n is the full width at half maximum (FWHM) of the laser pulses. Θ(𝑡) is the Heavyside  \nfunction . The constant  A1 represents the amplitude of demagnetization obtained after \nequilibrium between the electrons, spins, and phonons is reestablished while A 2 is \nproportional to the initial electron temperature raise .  The two critical time parameters  \n𝜏𝑀,𝜏𝐸 are the ultrafast demagnetization time and magnetization recovery time, \nrespectively. In the low fluence regime, which corresponds to our measurements, 𝜏𝐸 \nbecomes close to  the electron -phonon relaxation time . A unique value of  𝜏𝐸=550 ±\n20 𝑓𝑠 was used for fitt ing the demagnetization curves for all samples. T he ultrafast \ndemagnetization time  𝜏𝑀 decrease s from 380 ±10 fs for Co 2MnSi to 165 ±10 fs for \nCo2MnAl (Figure 4b). The evolution of the demagnetization time with both spin \npolarization P and  Gilbert damping 𝛼 is presented in figure 4c and 4d . A clear linear \nvariation between 1𝜏𝑀⁄ and 1−𝑃 is observed in this series . As the magnetic damping 𝛼 \nis proportional to P here, this means that 1𝜏𝑀⁄ is proportional to 𝛼 too. A similar relation 8 \n between these two par ameters was proposed by  Koopmans  et al.  [10]. However, they also \npredicted an influence of the  Curie temperature . As the Curie temperature in Heusler \ncompounds changes with the number of valence electron s and because the Co 2MnAl xSi1-\nx behave as solid solutions  as indicated by  the lattice spacing variation ( Figure 1b), we \nthus consider a linear decrease of  𝑇𝑐 with x going from 985 K to 697 K as exper imentally \nmeasured for x =0 and x=1, respectively. To test this possi ble influence of  the Curie \ntemperature  on the ultrafast magnetization dynamics , we plot in figure 4d  first the product \n𝜏𝑀.𝛼 and second the product   𝜏𝑀.𝛼.𝑇𝑐(𝑥)𝑇𝑐(𝐶𝑜2𝑀𝑛𝑆𝑖 ) ⁄ . These results demonstrate that \nthe Curie temperature does not influence the ultrafas t demagnetization  in our samples . \n \nFigure 4 : (a) Ultrafast demagnetization curves obtained for different Al  concentration x . The \ncurves have been shifted vertically for sake of clarity. The solid lines represent fitted  curves  \nobtained using Eq. ( 2). (b) Ultrafast demagnetization time as a function of Al content  x, (c) the \ninverse of 𝜏𝑀 as a function of  1-P, P being the spin polarization at E F, and d) test of Koopmans \nmodel with and without taking into account the Curie temperature of the films  (see text).   \n9 \n  One can now compare our experimental results with existing theoretica l models. \nWe first discuss  the dependence between the magnetic damping and the spin polarization. \nUltra -low magnetic damping values are predicted in Half -Metal Magnet (HMM) Heusler \ncompounds and explained by the lack of density of state at the Fermi energy for minority \nspin, or in other words by the full spin polarization  [18,27,28] . Consequently, the \nmagnetic damping is expected to increase when creating some states in the m inority band \nstructure around the Fermi energy that is when decreasing the spin polarization  [28]. If \nwe confirmed in previous experimental works that ultra -low magnetic damping \ncoefficients are actually observed especially on HMM Co2MnSi  and Co 2MnGe [19,2 2-\n23], we could not state any quantitative dependence between the damping values and the \nspin polarization. As prospected, the Co 2MnAl xSi1-x alloys are shown here to be ideal \ncandidates to address this point . This allows us getting  a clear experimental demonstration \nof these theoretical expectations. Furthermore, a linear dependence between the magnetic \ndamping and the spin polarization is obtained. This behavior may be explained by the \nmixing of both L2 1 and B2 phases in the films.  To the best of our knowledge, this \nexperimental result is the first quantitative demonstration of the link between the \nmagnetic damping and spin polarization.  \n Second , the dependence between the magnetic damping and the demagnetization \ntime observed here is a clear opportunity to test the different theoretical explanations \nproposed in the literature  to explain ultrafast dynamics . In the last 15 years, t he influence \nof the damping on the ultrafast dynamics has been explored, both theoreticall y and \nexperimentally. The first type of prediction we want to address is the link between the \ndemagnetization time and the electronic structure via the spin polarization  P. Using a \nbasic approach considering  the Fermi golden rule, several groups [12,13]  proposed that \nthe demagnetization process is linked to the population of minority and majority spin \nstates at E F, leading to a dependence of the spin-scattering rate proportional to 1 -P [13]. \nAs this spin scattering rate is linked to the inverse of the dem agnetization delay time , the  \n𝜏𝑀~(1−𝑃)−1 law was proposed . This law is clearly verified i n our samples series. One \nshould note that this is a strong experimental demonstration since we compare samples \ngrown  in the same conditions , so with the same control of the stoichiometry  and structural \nproperties . 10 \n  However, one point is still not clear since much larger demagnetization times in \nthe picosecond timescale would be expected for large band gap and full spin -polarization. \nIn the case of small band ga p of the order of 0.1 eV, Mann et al [13] showed that  thermal \neffects from the heated electron system lead to  a decrease  of 𝜏𝑀. They calculated a \nreduction of the spin -flip suppression factor from 104 for a gap of 1 eV to 40 for a gap of \n0.3 eV. However,  the band gap of our Co 2MnSi was calculated to be around 0.8 eV with \na Fermi energy  in the middle of the gap  [27,28] . This was corroborated by direct \nmeasurement using SR -PES [19, 22 ]. Therefore, according to their model, we should \nexpect a much longer demagnetization time for Co 2MnSi. However, the largest values \nreported by several groups  [13, 29] all on HMM materials are of the same order of \nmagnitude, i.e. around 350 to 400fs . This probably means that a limitation exists due to \nanother physical reason . One hypothesis should be to consider the 1.5eV photon energy \nwhich is much larger than the spin gap. During the excitation, the electrons occupying the \ntop minority spin valence band can be directly excited into the conduction band. In a \nsimilar way, maj ority spin electrons are excited at energies higher than the spin band gap. \nBoth of these effects may allow for spin flips scattering and only the majority electrons \nexcited within the spin band gap energy range cannot flip their spins. Even if such photon  \nenergy influence is not considered based on the argument that the timescale for photon \nabsorption followed by electronic relaxation is very fast compared to the  magnetic \nrelaxation process [16 ], performing experiments by changing the excitation wavelength  \nto energies below the spin band gap would be very interesting to better understand \nultrafast magnetization dynamics.  \n Concerning the dependence between the demag netization  time and the magnetic \ndamping , different theoretical models have been proposed and two opposite trends were \nobtained; 𝛼 and 𝜏𝑀 being either directly  [15] or inversely [10 ] proportional . From the \nexperimental side, the inverse proportionality between 𝜏𝑀 and 𝛼 proposed by Koopmans \net al.  [10] could not be reproduced by doping a thin Permalloy film with rare -earth atoms \n[14]. However, the introduction of these rare -earth elements strongly modifies the \nmagnetic relaxation properties and could induce different relaxation channels for   𝜏𝑀 and \n𝛼 [30]. Zhang  et al.  performed a similar st udy using thin Co/Ni multilayers  and observed  \na direct proportionality between 𝜏𝑀 and 𝛼 [15]. However, the damping extracted in their 11 \n study should be strongly influenced by the heavy metal Pt capping and seed layers which \nmay induce  strong spin pumping effect during the magnetization precession  [30]. \nFurthermore, they did not take into account the influence of the Curie temperature.  \nTherefore, in these studies, extrinsic effects might influence the magnetization dynamics \nin a different way on both time scales  which makes more complex the comparison \nbetween theory and experiments. Therefore, o ur results offer a nice opportunity to \ndisentangle the se different  effects. According to different studies , the ultrafast \ndemagnetization slows down when approaching  the Curie temperature [ 10,16, 32,33]. In \nother words, a larger difference between the initial temperature and 𝑇𝑐 would lead to a \nfaster demag netization . In our samples,  𝑇𝑐 goes up from Co2MnAl to Co2MnSi, whereas \nthe demagnetization process becomes slower . Therefore,  we conclude that, in the  present  \ncase,  the Curie temperatures of our samples are too high to affect  𝜏𝑀 which only depends \non the intrinsic propertie s of the films,  i.e. Gilbert damping and  spin polarization.  This \nalso clarifies some points reported by  Müller et al. work [ 12]. In their paper, they first \nreported a very fast demagnetization process in Co 2MnSi(110)  and second a slow one in  \nCrO 2 and LaSrMnO 3 films with   𝑇𝑐 values close to room temperature (390 K 360 K  \nrespectively). Therefore,  it is not possible to state whether the very slow demag netization \nprocess  in these compounds is due to a low 𝑇𝑐 or a large spin polarization. Furthermore, \nrecent experimental results demonstrated a large decrease in the spin polarization at the \nFermi level in CrO 2 as function of the temperature, resulting in less than 50% at 300 K \n[34]. In our samples  we disentangle these two effects and the longest demagnetization \ntime is found for Co 2MnSi  (𝜏𝑀=380 𝑓𝑠), a true half -metal magnet with a 0.8 eV spin \ngap and a large 𝑇𝑐. \n In summary, we first demonstrate experimentally that substituting Si by Al in \nCo2MnAl xSi1-x Heusler compounds allows us to get a tunable spin polarization  at E F from \n60% in Co 2MnAl to 100% in Co 2MnSi, indicati ng the transition from metallic  to half \nmetallic behaviors.  Second, a strong correlation between the spin polarization and the \nGilbert magnetic damping  is established in these films . This confirms the theoretical \njustification of ultra -low magnetic damping in Ha lf-Metal -Magnet s as a consequence of \nthe spin gap. Third , the ultrafast spin dynamics  results  also nicely confirm that the spin \ngap is at the origin of the increase of the relaxation time. Our experiments allow us to go 12 \n further by establishing clear relati onships between the spin polarization, the magnetic \ndamping and the demagnetization time. A n inverse relationship between demagnetization \ntime and Gil bert damping is established  in these alloys , which  agrees well with the model \nproposed by Mann et al. [13] and with Koopmans et al. [10] but without considering any  \ninfluence of Curie temperature much larger than room temperature in these films . \nExperimental section  \n Co2MnAl xSi1-x(001) quaternary Heusler  compounds are grown by Molecular \nBeam Epitaxy using an MBE machine e quipped with 24 materials. The s toichiometry is \naccurately controlled during the growth by calibration of the Co, Mn, Si and Al atomic \nfluxes using a quartz microbalance located at the pl ace of the sample. The error on each \nelem ent concentration is less than 1 % [23]. The films are grown directly on MgO(001) \nsubstrates, with the epitaxial relationship [100] (001) MgO // [110] (001) Heusler \ncompound. The thickness is fixed to 20nm.  \n The phot oemission experiments were done at the CASSIOPEE beamline at \nSOLEIL synchrotron source. The films were grown in a MBE connected to the beamline \n(see [19,22,35 ] for details). The SR -PES spectra were obtained by using the largest slit \nacceptance of the detec tor (+/ - 8°) at an angle of 8° of the normal axis of the surface. Such \ngeometry allows us to analyze all the reciprocal space  on similar polycrystalline films \n[23]. \n The radiofrequency magnetic dynamics of the films were thus studied  using \nferromagnetic resonance (FMR). A Vectorial Network Analyzor FMR set -up was used in \nthe perpendicular geometry (see [ 22] for experimental details) where the static magnetic \nfield is applied out of the plane of the film in order to avoid extrinsic bro adening of the \nlinewidth due to the 2 -magnons scattering [ 36,37]. \n Ultrafast magnetization dynamics were  investigat ed using polar time -resolved \nmagneto -optical Kerr (TR -MOKE) experiments. An amplified Ti -sapphire laser \nproducing 35 fs pulses at 800 nm with  a repetition rate of 5 KHz  is used . The pump beam \nis kept at the fundamental mode and is focused down to spot size of ~260  𝜇𝑚 while the \nprobe is frequency doubled to 400 nm and focused to a spot size of ~60 𝜇𝑚. Samples \nwere magnetically saturated alon g the out -of-plane axis by applying a 1T magnetic field.  \n 13 \n Acknowledgement  \nThis work was supported partly by the French PIA project “Lorraine Université \nd’Excellence”, reference ANR -15-IDEX -04-LUE, and by the Agence Nationale de la \nRecherche  (France) under contract no. ANR -17-CE24 -0008 (CHIPMuNCS).  \nReferences  \n[1]  Wolf, S. A. et al.  \n Science 294, 1488 –1495 (2001)  \n[2]  S. Yuasa, T. Nagahama, A. Fukushima, Y. Suzuki  & K. Ando , Nature Mater. 3, 868 -871 \n(2004).  \n[3]  Parkin, S. S. P. et al. N ature Mater. 3, 862 –867 (2004)  \n[4]  M.D. Stiles, J. Miltat, Spin-Transfer Torque and Dynamics .  \n Edited by B. Hillebrands, A. Thiaville, Springer, Berlin, Heidelberg.  \n[5]  A. D. Kent and D. C. Worledge,  \n Nat. Nanotechnol. 10, 187 (2015)  \n[6]  Kiselev, S. I. et al.  \nNature 425, 380 –383 (2003)  \n[7]  O. Boulle, G. Malinowski, and M. Kläui,  \n Mater. Sci. Eng. R Reports 72, 159 (2011)  \n[8]  E. 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Rev. B, 90, 214406 (2014)  \n[36]  K. Lenz, H. Wende, W. Kuch, K. Baberscke, K. Nagy, and A. Jánossy,  \n Phys. Rev. B 73,  144424 (2006)  \n[37]  Kh. Zakeri, J. Lindner, I. Barsukov, R. Meckenstock, M.Farle, U. von Hörsten, H.Wende, \nW. Keune, J. Rocker, S. S. Kalarickal, K. Lenz, W. Kuch, K. Baberschke, Z. Frait,  \n Phys. Rev.  B 76, 104416 (2007)  "    },    {        "title": "0710.2826v2.Ferromagnetic_resonance_study_of_polycrystalline_Fe__1_x_V_x_alloy_thin_films.pdf",        "content": "arXiv:0710.2826v2  [cond-mat.mes-hall]  17 Oct 2007Ferromagnetic resonance study of polycrystalline Fe 1−xVxalloy thin films\nJ-M. L. Beaujour, A. D. Kent\nDepartment of Physics, New York University, 4 Washington Pl ace, New York, NY 10003, USA\nJ. Z. Sun\nIBM T. J. Watson Research Center, Yorktown Heights, NY 10598 , USA\n(Dated: November 18, 2018)\nFerromagnetic resonance has been used to study the magnetic properties and magnetization dy-\nnamics of polycrystalline Fe 1−xVxalloy films with 0 ≤x <0.7. Films were produced by co-\nsputtering from separate Fe and V targets, leading to a compo sition gradient across a Si substrate.\nFMR studies were conducted at room temperature with a broadb and coplanar waveguide at fre-\nquencies up to 50 GHz using the flip-chip method. The effective demagnetization field 4 πMeffand\nthe Gilbert damping parameter αhave been determined as a function of V concentration. The\nresults are compared to those of epitaxial FeV films.\nI. INTRODUCTION\nA decade ago, it was predicted that a spin polar-\nized current from a relatively thick ferromagnet (FM)\ncould be used to switch the magnetization of a thin FM\n[1]. Since then, this effect, known as spin-transfer, has\nbeen demonstrated in spin-valves [2] and magnetic tun-\nnel junctions [3]. In a macrospin model with collinear\nlayer magnetizations, there is a threshold current den-\nsityJcfor an instability necessary for current-induced\nmagnetization switching of the thin FM layer [1, 4]:\nJc=2eαMstf(Hk+2πMs)\n/planckover2pi1η, (1)\nwhereαisthedampingconstant. tfandMsarethethick-\nness and the magnetization density of the free layer, re-\nspectively. Hkis the in-plane uniaxial anisotropy field. η\nis the currentspin-polarization. In orderfor spin-transfer\nto be used in high density memory devices Jcmust be re-\nduced. From Eq. 1 it is seen that this can be achieved by\nemploying materials with low Msandαin spin-transfer\ndevicesor, equivalently materialswith lowGilbert damp-\ning coefficients, G = αMs(gµB//planckover2pi1).\nVery recently, an experimental study of epitaxial FeV\nalloy thin films demonstrated a record low Gilbert damp-\ning coefficient [5]. This material is therefore of interest\nfor spin transfer devices. However, such devices are gen-\nerally composed of polycrystalline layers. Therefore it is\nof interest to examine polycrystalline FeV films to assess\ntheir characteristics and device potential.\nIn this paper, we present a FMR study of thin poly-\ncrystalline Fe 1−xVxalloy films with 0 ≤x <0.7 grown\nby co-sputtering. The FeV layers were embedded be-\ntween two Ta |Cu layers, resulting in the layer structure\n||5 Ta|10 Cu|FeV|5 Cu|10 Ta||, where the numbers are\nlayer thickness in nm. FeV polycrystalline films were\nprepared by dc magnetron sputtering at room tempera-\nture from two separate sources, oriented at a 45oangle\n(Fig. 1a). The substrate, cut from a Silicon wafer with\n100 nm thermal oxide, was 64 mm long and about 5 mm\nwide. The Fe and V deposition rates were found to vary/s49 /s50 /s51 /s52/s45/s50/s45/s49/s48/s40/s98/s41\n/s32/s86/s32/s116/s97/s114/s103/s101/s116\n/s52/s53/s111\n/s32\n/s32/s115/s117/s98/s115/s116/s114/s97/s116/s101/s70/s101/s32/s116/s97/s114/s103/s101/s116/s40/s97/s41\n/s72\n/s114/s101/s115/s65/s98/s115/s111/s114/s112/s116/s105/s111/s110/s32/s108/s105/s110/s101/s32/s32/s40/s97/s117/s41\n/s72\n/s97/s112/s112/s32/s32/s40/s107/s79/s101/s41/s49/s52/s32/s71/s72/s122\n/s32/s32/s32\n/s32/s120 /s61/s48/s46/s51/s55/s120 /s61/s48/s46/s53/s50\n/s120 /s61/s48/s46/s49/s57/s72\n/s115/s117/s98/s115/s116/s114/s97/s116/s101/s32/s104/s111/s108/s100/s101/s114\n/s97/s120/s105/s115\nFIG. 1: a) The co-sputtering setup. b) Typical absorp-\ntion curves at 14 GHz for a selection of ||Ta|Cu|7.5 nm\nFe1−xVx|Cu|Ta||films, with x=0.19, 0.37 and 0.52. The res-\nonance field Hresand the linewidth ∆ Hare indicated.\nlinearly across the wafer. The Fe and V rates were then\nadjusted to produce a film in which xvaries from 0.37\nto 0.66 across the long axis of the wafer. The base pres-\nsure in the UHV chamber was 5 ×10−8Torr and the\nAr pressure was set to 3.5 mTorr. The FeV was 7.5 nm\nin thickness, varying by less than 0.3 % across the sub-\nstrate. An Fe 1−xVxfilm 3 nm thick was also fabricated.\nTo produce films with x <0.30 the rate of the V source\nwas decreased. Finally, pure Fe films with a thickness\ngradient ranging from 7 nm to 13.3 nm were deposited.\nTheFMRmeasurementswerecarriedoutatroomtem-\nperature using a coplanar wave-guide (CPW) and the\nflip-chip method. Details of the experimental setup and\nofthe CPWstructuralcharacteristicscan be found in [6].\nAdc magnetic field, up to 10 kOe, wasapplied in the film\nplane, perpendiculartotheacmagneticfield. Absorption\nlines at frequencies from 2 to 50 GHz were measured by\nmonitoring the relative change in the transmitted signal\nas a function of the applied magnetic field.2\n/s48/s46/s48 /s48/s46/s51 /s48/s46/s54/s48/s53/s49/s48/s49/s53/s50/s48\n/s48/s46/s48 /s48/s46/s51 /s48/s46/s54/s50/s46/s48/s53/s50/s46/s49/s48/s50/s46/s49/s53/s49/s50/s51\n/s120\n/s32/s52 /s77\n/s101/s102/s102/s32/s32/s40/s107/s71/s41/s55/s46/s53/s32/s110/s109/s32/s70/s101\n/s49/s45 /s120/s86\n/s120\n/s51/s32/s110/s109/s32/s70/s101\n/s48/s46/s54/s51/s86\n/s48/s46/s51/s55\n/s49/s50/s46/s57/s32/s110/s109/s32/s70/s101\n/s120/s32\n/s32/s103/s45/s102/s97/s99/s116/s111/s114/s40/s98/s41/s40/s99/s41/s32\n/s32\n/s32/s72\n/s114/s101/s115/s32/s32/s40/s107/s79/s101/s41\n/s49/s49/s32/s71/s72/s122/s40/s97/s41\nFIG. 2: a) The resonance field at 11 GHz versus xand b) the\neffective demagnetization field versus x. The solid line is a\nguide to the eye. c) The Land´ e gfactor as a function of x.\nThe dotted line shows the g-factor value of bulk Fe.\nII. RESULTS\nTypical absorption lines at 14 GHz of selected FeV al-\nloyfilmsareshowninFig. 1b. Thelinesarelorentzianfor\nmost frequencies. At a fixed frequency, the FMR absorp-\ntion decreaseswith increasingV content. The FMR peak\nof a film 7.5 nm thick with x= 0.66 is about 100 times\nsmallerthanthatofapureFeofthesamethickness. This\nis accompaniedby a shift of Hrestowardshigher field val-\nues (Fig. 2a). The effective demagnetization field 4 πMeff\nand the Land´ eg-factorgwere determined by fitting the\nfrequency dependence of the resonance field Hresto the\nKittel formula [7]:\nf2=/parenleftBiggµB\nh/parenrightBig2\nHres(Hres+4πMeff),(2)\nwhere the effective demagnetization field is:\n4πMeff= 4πMs−H⊥. (3)\nNote that in the absence of a perpendicular anisotropy\nfieldH⊥, the effective field would be directly related to\nMs. The dependence of 4 πMeffon V concentration is\nshown in Fig. 2c. As xincreases the effective demagneti-\nzation field decreases dramatically, going from about 16\nkG forx= 0 to 1.1 kG for x= 0.66. Note that the effec-\ntive demagnetization field of the 7.5 nm Fe film is about\n25 % lower than that of bulk Fe (21.5 kG). The 12.9\nnm Fe film exhibits a larger 4 πMeff, which is, however,\nstill lower than 4 πMsof the bulk material. Similarly,\nthe 4πMeffof an Fe 0.63V0.37film is thickness dependent:\ndecreasing with decreasing layer thickness.\nThe Land´ eg-factor increases monotonically with in-\ncreasing V concentration (Fig. 2b). The minimum g-\nfactor is measured for the Fe film: g= 2.11±0.01,\nwhich is slightly larger than the value of bulk material\n(g= 2.09). Note that gof a Fe film 12.9 nm thick is\nequal to that of Fe bulk. However, the g-factor of the/s48 /s50/s48 /s52/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54/s48/s46/s55/s48/s46/s56\n/s48/s46/s48/s48/s46/s51/s48/s46/s54\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s48/s46/s53/s49/s46/s48/s49/s46/s53 /s40/s99/s41/s120 /s32/s61/s32/s48/s46/s53/s50/s120/s32 /s61/s32/s48/s46/s52/s51/s120/s32 /s61/s32/s48/s40/s98/s41/s32/s40/s49/s48/s45/s50\n/s41/s76/s105/s110/s101/s119/s105/s100/s116/s104/s32/s32/s40/s107/s79/s101/s41/s32\n/s32/s72 /s32/s32/s40/s107/s79/s101/s41\n/s72\n/s48/s72\n/s48/s72\n/s49/s52/s32/s32\n/s32/s40/s97/s41\n/s120/s32 /s32/s102/s32/s32/s40/s71/s72/s122/s41\nFIG. 3: a)Frequency dependence of the linewidth for 7.5 nm\nFe1−xVxalloy film with x=0, 0.43 and 0.52. The solid lines\nare the best linear fit of the experimental data. b) ∆ H14, the\nlinewidth at 14 GHz , and ∆ H0are shown as a function of x.\nc) The magnetic damping parameter versus V concentration.\nFe0.63V0.37, does not appear to be thickness dependent:\nthe 3 nm Fe 0.63V0.37layer has about the same gvalue\nthan the 7.5 nm Fe 0.63V0.37layer.\nThe half-power linewidth, ∆ H, was studied as a func-\ntion of the frequency and of the V concentration. Fig.\n3b shows the dependence of the FMR linewidth on xat\n14 GHz. The general trend is that ∆ Hincreases with\nx. However, there are two regimes. For x >0.4, the\nlinewidth depends strongly on x, increasing by a factor\n5 whenxis increased from 0.4 to 0.66. The dependence\nof the linewidth on xis more moderate for the films with\nx <0.4: it increases by about 30 %. For all samples,\nthe linewidth scales linearly with the frequency. A least\nsquare fit of ∆ H(f) gives ∆ H0, the intercept at zero\nfrequency, and the Gilbert damping parameter αwhich\nis proportional to the slope: d∆H/df= (2h/gµB)α[8].\n∆H0is typically associated with an extrinsic contribu-\ntion to the linewidth and related to magnetic and struc-\ntural inhomogeneities in the layer. For two samples with\nthe highest Vanadium concentration, x= 0.60, 0.66, the\nlinewith is dominated by inhomogeneous broadening and\nit wasnot possible to extract α. Asxincreases, ∆ H0and\nαincreases. The damping parameter and ∆ H0remain\npractically unchanged for x≤0.4 and when x >0.4,\nboth the intercept and the slope of ∆ Hversusfincrease\nrapidly.\nIII. DISCUSSION\nSeveral factors can contribute to the dependence of\n4πMeffon the V concentration. First, the decrease of\nthe effective demagnetizationfield canbe associatedwith\nthe reduction of the alloy magnetization density Mssince\nthe Fe content is reduced. In addition, a neutron scat-\ntering study showed that V acquires a magnetic moment3\nantiparallel to the Fe, and that the Fe atom moment de-\ncreases with increasing V concentration [9]. The Curie\ntemperature of Fe 1−xVxdepends on x. In fact, Tcfor\nx=0.65 is near room temperature [10]. It is important to\nmention that a factor that can further decrease 4 πMeff\nis an out-of-plane uniaxial anisotropy field H⊥(Eq. 3).\nIn thin films, the perpendicular anisotropy field is com-\nmonly expressed as H⊥= 2K⊥/(Mst), where K⊥>0\nis the anisotropy constant and tthe ferromagnetic film\nthickness [11]. In this simple picture, it is assumed that\nK⊥is nearly constant over the thickness range of our\nfilms. This anisotropy can be associated with strain\ndue to the lattice mismatch between the FeV alloy and\nthe adjacent Cu layers and/or with an interface contri-\nbution to the magnetic anisotropy. For Fe films with\nt= 7.5 and 12.9 nm, a linear fit of 4 πMeffversus 1/t\ngives 4πMs= 20.2 kG and K⊥= 2.5 erg/cm2. The\nvalue extracted for 4 πMsis in the range of the value of\nFebulk. AsimilaranalysisconductedonFe 0.63V0.37films\nof thickness t=3 and 7.5 nm gives 4 πMs= 12.2 kG and\nK⊥= 0.1 erg/cm2. The result suggests that the surface\nanisotropy constant decreases with increasing x.\nIV. SUMMARY\nThe effective demagnetization field of the polycrys-\ntalline Fe 1−xVxalloy films decreases with increasing xandalmostvanishesfor x≈0.7. AFMRstudyonepitax-\nial films haveshown a similar xdependence of 4 πMeff[5].\nUsing the value of Mscalculated in the analysis above,\nwe estimate the Gilbert damping constant of a 7.5 nm Fe\nlayer and 7.5 nm Fe 0.63V0.37alloy film to be G Fe= 239\nMHz and G FeV= 145 MHz respectively. The decrease of\nthe effective demagnetization field of Fe 1−xVxwith in-\ncreasing xis accompanied by a decrease of the Gilbert\ndamping constant. A similar xdependence of G was\nobserved in epitaxial films [5]. The authors explained\nthe decrease of G by the reduced influence of spin-orbit\ncoupling in lighter ferromagnets. Note that the Gilbert\ndampingofourfilmsislargerthanwhatwasfoundforthe\nepitaxial films (G=57 MHz for epitaxial Fe 8 nm thick).\nWe note that the Fe 0.63V0.37alloy film, which has\n4πMsapproximatly the same as that of Permalloy, has a\nmagnetic damping constant of the same order than that\nof Py layer in a similar layer structure [12]. Hence, with\ntheir low Msandα, polycrystalline FeV alloy films are\npromising materials to be integrated in spin-tranfer de-\nvices.\n[1] J. C. Slonczewski, J. Magn. Magn. Mater. 159(1-2), L1\n(1996) ; L. Berger, Phys. Rev. B 54(12), 9353 (1996).\n[2] see, for example, J. A. Katine et al., Phys. Rev. Lett. 84,\n3149 (2000) ; B. Oezyilmaz et al., Phys. Rev. Lett. 91,\n067203 (2003).\n[3] see, for example, G. D. Fuchs et al., J. Appl. Phys. 85\n(7), 1205 (2004).\n[4] J. Z. Sun, Phys. Rev. B 62(1), 570 (2000).\n[5] C. Scheck et al., Phys. Rev. Lett. 98, 117601 (2007).\n[6] J-M. L. Beaujour et al., Europhys. J. B, DOI:\n10.1140/epjb/e2007-00071-1 (2007).\n[7] C. Kittel in Introduction to Solid State Physics, Ed. 7,\np.505.[8] see, for example, D. L. Mills and S. M. Rezende in Spin\nDynamics in Confined Magnetic Structures II (Eds. B.\nHillebrands andK.Ounadjela), pp.27-58, (Springer, Hei-\ndelberg 2002).\n[9] I. Mirebeau, G. Parette, and J. W. Cable. J. Phys. F:\nMet. Phys. 17, 191 (1987).\n[10] Y. Kakehashi, Phys. Rev. B 32(5), 3035 (1985).\n[11] Y. K. Kim and T. J. Silva, Appl. Phys. Lett. 68, 2885\n(1996).\n[12] S. Mizukami et al., J. Magn. Magn. Mater. 239, 42\n(2002)."    },    {        "title": "1610.01622v1.Finite_dimensional_colored_fluctuation_dissipation_theorem_for_spin_systems.pdf",        "content": "arXiv:1610.01622v1  [cond-mat.stat-mech]  5 Oct 2016Finite-dimensional colored fluctuation-dissipation theo rem for spin systems\nStam Nicolis,1,a)Pascal Thibaudeau,2,b)and Julien Tranchida2,1,c)\n1)CNRS-Laboratoire de Math´ ematiques et Physique Th´ eoriqu e (UMR 7350), F´ ed´ eration de Recherche ”Denis\nPoisson” (FR2964), D´ epartement de Physique, Universit´ e de Tours, Parc de Grandmont, F-37200, Tours,\nFRANCE\n2)CEA DAM/Le Ripault, BP 16, F-37260, Monts, FRANCE\nWhen nano-magnetsare coupled to random external sources, th eir magnetization becomes a random variable,\nwhose properties are defined by an induced probability density, tha t can be reconstructed from its moments,\nusing the Langevin equation, for mapping the noise to the dynamical degrees of freedom. When the spin\ndynamics is discretized in time, a general fluctuation-dissipation the orem, valid for non-Markovian noise, can\nbe established, even when zero modes are present. We discuss the subtleties that arise, when Gilbert damping\nis present and the mapping between noise and spin degrees of freed om is non–linear.\nPACS numbers: 05.40.Ca, 05.10.-a, 75.78.-n\nI. INTRODUCTION\nFor any system, in equilibrium with a bath, the\nfluctuation-dissipation relation (FDR) plays an impor-\ntant role in defining consistently its closure, since it re-\nlates the fluctuations of the subsystem of the dynamical\ndegrees of freedom, that one is, by definition, interested\nin, with the fluctuations of the degrees of freedom that\nare defined as uninteresting and are lumped under the\nterm “dissipation”.\nThe essential reason behind this relation is that, for\nequilibrium situations, it is possible to define a proba-\nbility measure on the space of states, with respect to\nwhich the average values, that enter in the FDR, can\nbe unambiguously computed. So this can be modified,\nif the dynamical degrees of freedom are so affected by\nthe immersion in the bath, that they must be replaced\nby others–the interaction with the bath leads to a phase\ntransition and the equilibrium measure is not unitarily\nequivalent to the measure of the dynamical degrees of\nfreedom, in the absence of the bath.\nWhile it is possible to address these questions by nu-\nmerical simulations, and reconstruct the density that\nway, what has, really, changedin the lastyearsis that ex-\nperiments ofgreatprecision, that probe both issues, have\nbecome possible, particularly in magnetic systems1. It is\nin such a context that the FDR has become of topical\ninterest2–4.\nIn such systems, since the noise affects the magnetic\nfield, that makes the spin precess, it is not additive, but\nmultiplicative. While, already, for additive noise, the\nissue of the “backreaction” of the dynamical degrees of\nfreedom on the bath is quite delicate, for multiplicative\nnoise it becomes even more difficult to evade and must\nbe addressed.\nFurther complications arise when the fluctuations are\ncolored, namely posses finite intrinsic correlation time5,6.\na)Electronic mail: stam.nicolis@lmpt.univ-tours.fr\nb)Electronic mail: pascal.thibaudeau@cea.fr\nc)Electronic mail: julien.tranchida@cea.frIn such a situation, no FDR has been unequivocally ob-\ntained, that relates the intensity of the fluctuations to\nthe damping constant7.\nIn this note, we wish to study these issues in the con-\ntext of magnetic systems placed in random magnetic\nfields, whose distribution can have an auto–correlation\ntime comparable to the time scale defined by the pre-\ncession frequency. The aim of this communication is to\nsketch out a route for establishing a FDR in a quite gen-\neralsetting8, that will be shownto be consistentto previ-\nous results for magnetic systems, obtained in the limit of\nwhite-noise fluctuations, and can be readily adapted be-\nyond this context, especially for explicit calculations. A\nremaining challenge is to obtain the stochastic equation,\nthat defines the mapping between noise and the dynam-\nical degrees of freedom, that are identified with the spin\ncomponents of a nanomagnet, and whose solution does,\nindeed, describe a normalizable density for the spin con-\nfigurations.\nII. GAUSSIAN APPROXIMATION\nIn order to better grasp the issues at stake, we shall\nstart with a finite number of dynamical degrees of free-\ndom,sA\nn. The time index nruns from 0 to N−1 and\nwill be identified with the evolution time instant, in the\ncontinuum limit; the flavor index Aruns from 1 to Nf\nand labels “internal” degrees of freedom–it will label the\ncomponents of the spin. The summation convention on\nrepeated indices is assumed.\nWe assume that these dynamical degrees of freedom\nare immersed in a bath. The bath is described by vari-\nablesηA\nnand is defined by the partition function\nZ=/integraldisplayNf/productdisplay\nA=1N−1/productdisplay\nn=0dηA\nne−1\n2ηA\nnFABDnmηB\nm (1)\nThe matrix Facts on the flavor indices and the ma-\ntrixDon the “target space” indices–that describe the\ninstants in time. The white noise case corresponds to\ntakingDnm=δnm/σ2. The simplest colored noise case2\ncorresponds to taking Dnm=δnm/σ2\nn, with not all the\nσnequal. Furthermore, if it cannot be put in diagonal\nform at all, then it describes higher derivative effects.\nThe average of a functional Fof the variables ηA\nnis\nthen well defined as\n/angbracketleftF/angbracketright=1\nZ/integraldisplayNf/productdisplay\nA=1N−1/productdisplay\nn=0dηA\nnF[η]e−1\n2ηA\nnFABDnmηB\nm(2)\nFrom this expression we may deduce the moments of the\ndegrees of freedom of the bath:\n/angbracketleftbig\nηA\nn/angbracketrightbig\n= 0/angbracketleftbig\nηA\nnηB\nm/angbracketrightbig\n=/bracketleftbig\nF−1/bracketrightbigAB/bracketleftbig\nD−1/bracketrightbig\nnm(3)\nwith the others deduced from Wick’s theorem. What\nwe notice here is that, for non–diagonal matrices, Fand\nD, the degrees of freedom of the bath that have well–\ndefined properties, i.e. the degrees of freedom that are\neigenstates of these matrices, are linear combinations of\ntheηA\nn. So it makes sense to work in that basis. In this\ncontext, the white noise limit corresponds to the case in\nwhichDis the identity matrix–all components have the\nsame relaxation time. The colored noise case, then can\nbe identified as that, where Dis not the identity matrix.\nWhen we immerse a physical system in such a bath it\ncan happen that the eigenbases of the system and of the\nbath do not match.\nThe map between the degrees of freedom of the bath\nand the dynamical degrees of freedom is provided by\na stochastic equation. For instance, one consider the\nLandau-Lifshitz-Gilbert equation ˙s=ω×s+αs×˙s+\nE(s)η, where the vielbein Econtains both an antisym-\nmetric part ×sand at least an additional non-zero diag-\nonal element. Because this vielbein is invertible, we can\nexpressηas a function of s.\nTo illustrate the procedure, we start with the case of\nlinear equations:\nηA\nn=fA\nBCm\nnsB\nm (4)\nAssuming that the matrices are invertible, we obtain the\nchange of variables (we shall study presently what hap-\npens when the matrices have zero modes)\nsA\nn=/bracketleftbig\nf−1/bracketrightbigA\nB/bracketleftbig\nC−1/bracketrightbigm\nnηB\nm (5)\nThe Jacobian is a constant that can be absorbed in the\nnormalizationofthe partition function9, sowe obtain the\npartition function for the dynamical degrees of freedom,\nZ=/integraldisplayNf/productdisplay\nA=1N−1/productdisplay\nn=0dsA\nne−1\n2sA′\nn′fB\nB′FABfA\nA′Cn′\nnDnmCm′\nmsB′\nm′(6)\nthat defines the correlation functions–for the finite-\ndimensional case the moments–of the dynamical degrees\nof freedom. The 1–point function vanishes, /angbracketleftsA\nn/angbracketright= 0,\nwhile the 2–point function is given by the expression\n/angbracketleftbig\nsA\nnsB\nm/angbracketrightbig\n=/bracketleftbig\n[f−1Ff]−1/bracketrightbigAB/bracketleftbig\n[C−1DC]−1/bracketrightbig\nnm(7)This is the FDR for the present case, that relates the\nparameters, fA\nBandCm\nn, of the spin dynamics, with the\nparameters, FABandDnm, of the bath.\nIII. WHEN ZERO MODES ARE RELEVANT\nLet us now consider the case when the matrices fA\nB\nand/orCm\nnhave zero modes, a case that is relevant for\nthe physical system studied in this paper.\nThe zero modes imply, quite simply, that we cannot\nreplace all of the ηA\nnby thesA\nn, since we cannot invert\neq. (4); we can, only, replace the non–zero modes. The\nmatrices fand/orCare not of full rank–but they surely\nhave positive rank, otherwise the stochastic map does\nnot make sense. When we replace the non–zero modes,\nwe shall generate quadratic terms in the sA\nn–but, since\nwe do not replace all of the ηA\nn, on the one hand there\nwill be mixed terms, while there will remain the terms\nquadratic in the ηA\nn, that correspond to the zero modes.\nWhen we integrate over the zero modes, the ηA\nnthat we\ncould not express directly as linear combinations of the\nsA\nn, we shall encounter Gaussian integrals over them that\ncontaintermslinearin thezeromodesandthe sA\nnalready\nreplaced. The result of these Gaussian integrations will\nbequadraticcontributionstothealreadypresent sA\nn, that\nenter in the action with the opposite sign to their coeffi-\ncients. The system will be stable, if these contributions\ndo not completely cancel the existing ones and will lead\nto a modification of the FDR.\nLet us see this in action. We shall take Nf= 3 and\nfA\nB=εA\nBCωC, withωa fixed vector in flavor space. In\nthe magnetic case it will correspond to the fixed part of\nthe precession frequency. We immediately remark that\nfA\nBhas one zeromode, along the vector ω. Since this\nvector is fixed, without loss of generality, we may take it\nto lie along the z−axis:ω= (0,0,ω3).\nThe stochastic equation, eq. (4), takes the form\nη1\nn=ω3Cm\nns2\nm\nη2\nn=−ω3Cm\nns1\nm(8)\nWe may replace these in the partition function for the\nnoise; but we must integrate over η3\nnseparately. We re-\nmark that they do not involve s3\nn, the component of the\ndynamical degrees of freedom, parallel to the precession\nvector.\nIfFAB=δAB, i.e. the spherical symmetry is imposed,\nwe immediately deduce that the integration over η3\nnde-\ncouples from the rest and just gives a contribution to the\nnormalization. The partition function for the dynamical\ndegrees of freedom, s1\nnands2\nn, is given by the expression\nZ=/integraldisplay2/productdisplay\nA′=1N−1/productdisplay\nn=0dsA\nne−(ω3)2\n2sA′\nn′[CDC]n′m′sA′\nm′(9)\nThere’s a subtle point here: the motion of the A′=\n1,2 flavor components is a rotation, with precession3\nfrequency ω3, about the z−axis, so the combination,\n(s1\nm)2+(s2\nm)2should appear–and it does. Therefore we\ndeduce the FDR for this case, that corresponds to Lar-\nmor precession:\n/angbracketleftbig\nsA\nnsB\nm/angbracketrightbig\n=/parenleftbig\nω3/parenrightbig−2/bracketleftbig\nC−1DC/bracketrightbig−1\nnm(10)\nIf the spherical symmetry is not imposed, in flavor space,\ne.g.FAB=κδAB+λAB(1−δAB), we would have had\nterms linear in η3\nn, along with the quadratic terms and\nadditional contributions when we would have integrated\nover the η3\nn.\nIV. BEYOND THE GAUSSIAN APPROXIMATION\nNow let us address the issue of non–linear stochas-\ntic maps, also relevant for the Landau–Lifshitz–Gilbert\nequation. Let us replace eq. (4) by\nηA\nn=fA\n(1)BC(1)m\nnsB\nm+fA\n(2)BCC(2)ml\nnsB\nmsC\nl(11)In this case the Jacobian of the transformation, between\nthe degrees of freedom of the bath and the degrees of\nfreedom that describe the “interesting” dynamics, is not\na constant:\nJAB\nnk(s)≡δηA\nn\nδsB\nk=fA\n(1)BC(1)k\nn+\n+/bracketleftBig\nfA\n(2)BCC(2)kl\nn+fA\n(2)CBC(2)lk\nn/bracketrightBig\nsC\nl(12)\nThismeansthat, ifitispossibletoneglectthezeromodes\nand the concomitant fluctuations in the sign of the de-\nterminant, which is true in perturbation theory, the par-\ntition function for the spin degrees of freedom is given by\nthe expression\nZ=/integraldisplay/bracketleftbig\ndsA\nn/bracketrightbig\ndetJAB\nnk(s)e−1\n2ηA\nn(s)FABDnmηB\nm(s)(13)\nwhere the η(s) are defined by eq. (11). The expression\nin the exponent contains terms that are quadratic and\nquartic in the spin variables. The fluctuation–dissipation\nrelation can then be deduced from the Schwinger–Dyson\nequations9,\nZ−1/integraldisplay/bracketleftbig\ndsA\nn/bracketrightbig∂\n∂sL\nk/braceleftBig\nsA1n1···sAInIdetJAB\nnk(s)e−1\n2ηA\nn(s)FABDnmηB\nm(s)/bracerightBig\n= 0 (14)\nThese relations can be used to generalize eqs. (10) and\nexpress the fact that the spin degrees of freedom are in\nequilibriumwiththebath. Thedeterminantcanbeintro-\nduced into the exponent using anti-commuting variables,\nthat describe the dynamics of the bath9.\nIt should be stressed that, since the η(s) are polynomi-\nals in the spin degrees of freedom, once the determinant\nhas been expressed in terms of anti-commuting fields,\nthere is a finite number of parameters that define the\ndynamics and, thus, enter in the fluctuation–dissipation\nrelation. Indeed, if Dnmis not the identity matrix, which\nmeans that the dynamics is not ultra–local in time, tun-\nneling between configurations implies that the effects of\nthe determinant and it sign will be, inevitably and, thus,\nimplicitly, be generatedby the dynamics, thereforeit suf-\nfices to sample the correlation functions by the action\nof the spin degrees of freedom. The subtleties of the\ndynamics are encoded in the relation between the noise\nfields and the spins, so it is at that point that the zero\nmodes need to be taken into account. There are not any\nissues of principle, involved, however, precisely because\nthe system is consistently closed10. How to sample the\ncorrelation functions will be reported in detail in future\nwork.REFERENCES\n1Markus G. M¨ unzenberg. “Magnetization dynamics: Ferromag -\nnets stirred up”, Nat Mater 9(3):184–185 (2010).\n2Aditi Mitra and A. J. Millis. “Spin dynamics and violation of\nthe fluctuation dissipation theorem in a nonequilibrium ohm ic\nspin-boson model”, Phys. Rev. B72(12), 121102 (2005)\n3Vladimir L. Safonov and H. Neal Bertram, “Fluctuation-\ndissipation considerations and damping models for ferroma gnetic\nmaterials”, Phys. Rev. B71(22):224402 (2005).\n4William T. Coffey and Yuri P. Kalmykov. “Thermal fluctuations\nof magnetic nanoparticles: Fifty years after Brown”, Journal of\nApplied Physics 112(12):121301, (2012).\n5R. Kupferman, G. A. Pavliotis, and A. M. Stuart. “Itˆ o versus\nStratonovich white-noise limits for systems with inertia a nd col-\nored multiplicative noise”, Phys. Rev. E70(3):036120 (2004).\n6Masamichi Nishino and Seiji Miyashita, “Realization of the ther-\nmal equilibrium in inhomogeneous magnetic systems by the\nLandau-Lifshitz-Gilbert equation with stochastic noise, and its\ndynamical aspects”, Phys. Rev. ,91(13):134411 (2015).\n7Marco Baiesi, Christian Maes, and Bram Wynants. “Fluctua-\ntions and Response of Nonequilibrium States”, Phys. Rev. Lett. ,\n103(1):010602 (2009).\n8Camille Aron, Giulio Biroli, and Leticia F Cugliandolo. “Sy mme-\ntries of generating functionals of Langevin processes with colored\nmultiplicative noise”, Journal of Statistical Mechanics: Theory\nand Experiment , 2010(11):P11018 (2010).\n9Jean Zinn-Justin. Quantum Field Theory and Critical Phe-\nnomena. Number 113 in International series of monographs on\nphysics. Clarendon Press, Oxford, 4. ed., reprinted editio n, 2011.\nOCLC: 767915024.\n10S. Nicolis, “How quantum mechanics probes superspace”, con tri-\nbution to SQS15, Dubna. [arXiv:1606.08284 [hep-th]]."    },    {        "title": "1808.07665v3.Reduced_thermal_stability_of_antiferromagnetic_nanostructures.pdf",        "content": "Reduced thermal stability of antiferromagnetic nanostructures\nLevente Rózsa,1,\u0003Severin Selzer,2Tobias Birk,2Unai Atxitia,2, 3,yand Ulrich Nowak2\n1Department of Physics, University of Hamburg, D-20355, Hamburg, Germany\n2Department of Physics, University of Konstanz, D-78457 Konstanz, Germany\n3Freie Universität Berlin, Fachbereich Physik, Arnimallee 14, D-14195 Berlin, Germany\n(Dated: August 28, 2019)\nAntiferromagnetic materials hold promising prospects in novel types of spintronics applications.\nAssessing the stability of antiferromagnetic nanostructures against thermal excitations is a crucial\naspect of designing devices with a high information density. Here we use theoretical calculations\nand numerical simulations to determine the mean switching time of antiferromagnetic nanoparticles\nin the superparamagnetic limit. It is demonstrated that the thermal stability is drastically reduced\ncompared to ferromagnetic particles in the limit of low Gilbert damping, attributed to the exchange\nenhancement of the attempt frequencies. It is discussed how the system parameters have to be\nengineered in order to optimize the switching rates in antiferromagnetic nanoparticles.\nI. INTRODUCTION\nIn the field of spintronics, the storage, transfer and\nprocessing of information is based on the spin magnetic\nmoment of electrons. Conventional spintronic devices are\nmainly based on ferromagnetic (FM) systems. However,\nrecent advances in understanding and controlling anti-\nferromagnetic (AFM) materials have led to an increasing\ninterest in AFM spintronics [1–7]. Possible advantages\nof spintronic devices based on AFM materials include\ntheir lack of stray fields, which normally destroys single-\ndomain states and leads to an interaction between bit\npatterns; the low susceptibility to external fields; and the\nrich choice of new materials, including a variety of AFM\ninsulators. Moreover, AFM spin dynamics are found to\nbe faster than those of FMs [4, 8–10].\nFor many applications, the size of magnetic structures\nwill have to be scaled down to the nanometer regime,\nwhere, eventually, thermal excitations will reduce the\nstability of the magnetic state. In single-domain FM\nnanoparticles this is known as the superparamagnetic\nlimit[11],wherethewholestructurecanbedescribedasa\nsinglemacroscopicmagneticdipole. Besidestheirtechno-\nlogicalrelevance, superparamagneticparticleshavefound\ntheir uses in biomedical applications [12] as well as in\nrock magnetism [13]. Analogously, a single-domain AFM\nnanoparticlemaybedescribedbyamacroscopicNéelvec-\ntor, being the difference of the two sublattice magneti-\nzations. The spontaneous switching of the Néel vector\nunder thermal fluctuations constitutes the superparam-\nagnetic limit in AFMs. In this context, it was shown re-\ncently [14] that thermally activated superparamagnetic\nreversal enhances the current-induced switching rates in\nAFM Hall cross devices. Furthermore, AFM nanoparti-\ncles play an important role in biological molecules such\nas the natural [15] and synthetic forms [16] of the iron-\nstorage protein ferritin, and in the field of geochemistry\n\u0003rozsa.levente@physnet.uni-hamburg.de\nyunai.atxitia@fu-berlin.de[17].\nThe thermal stability of FM nanoparticles has been\nstudied extensively in the past [18–23]. An analytical\nformula for the thermal switching rate in the superpara-\nmagnetic limit was first given by Brown [24] based on\nthe stochastic Landau–Lifshitz–Gilbert equation [25, 26].\nThe mean switching time in AFM nanoparticles has been\ninvestigated in significantly less works so far [27], and the\nanalytical studies [28, 29] have been restricted to the case\nofuncompensatedAFMswithafinitemagnetization. For\ncurrenttechnologicalapplicationsofcompensatedAFMs,\na simple but accurate formula explaining the role of the\ninteraction parameters in the reversal process seems to\nbe lacking.\nHere we theoretically investigate the switching rate in\ncompensated AFM nanoparticles. By deriving an ana-\nlytical expression, it is demonstrated that the coupling\nbetween the Néel vector and the magnetization leads to\nsignificantly faster dynamical processes than in FMs. In\nthe limit of low Gilbert damping, this causes strong os-\ncillations in the Néel vector direction during the reversal\nprocess and an exchange enhancement of the switching\nrate compared to Brown’s formula applicable to FMs.\nThe accuracy of the analytical formula is confirmed by\nspin dynamics simulations. By analyzing the effect of\ndifferent material parameters on the switching rate, the\nadvantages and disadvantages of AFMs over FMs are dis-\ncussed for various applications. Our findings contribute\nto the understanding of thermal effects in AFM nanos-\ntructures, their stability as well as switchability, where\nthe latter is often affected by heating effects due to ap-\nplied currents or laser excitation.\nThe paper is organized as follows. The analytical for-\nmulae for the switching times in uniaxial FM and AFM\nnanoparticles are discussed in Sec. II. The spin dynam-\nics simulations are introduced in Sec. III. The necessary\nconditions for the application of the macrospin model\nto the results of atomistic simulations are detailed in\nSec. IV. The switching times between FMs and AFMs\nare compared in Sec. V, and the results are summarized\nin Sec. VI.arXiv:1808.07665v3  [cond-mat.mes-hall]  27 Aug 20192\nII. ANALYTICAL MODEL\nA. Axially symmetric FM nanoparticle\nFor the analytical investigations we will focus on the\nsimplest example of a magnetic nanoparticle, which\nswitches by coherent rotation between two stable mag-\nnetic states separated by an energy barrier \u0001E, as\nsketched in Fig. 1(a). We will rely on the so-called single-\ndomain approximation, where the total magnetization of\nthe nanoparticle is described by a single magnetic mo-\nment or macrospin in the FM case. This remains valid\nif the particle size stays below the exchange length, cor-\nresponding to the characteristic size of domain walls in\nthe system. The dynamics can be calculated within the\nframework of the macroscopic Landau–Lifshitz–Gilbert\nequation [25, 26]\n_m=\u0000m\u0002\u0012\n\rhm\u0000\u000b_m\nm0\u0013\n; (1)\nwhere\u000bis the Gilbert damping constant, \ris the gyro-\nmagnetic ratio, mthe magnetization of the nanoparti-\ncle,m0the magnitude of the magnetization, and hm=\n\u0000\u000emFthe effective field, where Fis the magnetic free\nenergy of the system. For simplicity, in this work we\nrestrain the discussion to a uniaxial particle, with the\nfree-energy density f=\u0000Ham2\nz=(2m0), whereHa=\n2DzN=(Vm0)is the anisotropy field, Dzis the anisotropy\nenergy of a single spin, Nis the number of spins in the\nnanoparticle and Vis its volume. Equation (1) describes\nthe rotational motion of the macrospin, with its length\nfixed atjmj=m0. In this case, the free energy has two\nminima,mz=m0= 1andmz=m0=\u00001, with the energy\nbarrier between them being \u0001E=Ham0V=2 =DzN.\nThermalactivationallowsthenanoparticletojumpbe-\ntween the free energy minima with a characteristic time\nscale. In the limit of low temperatures, kBT\u001c\u0001E, the\nswitching time for coherent rotation over the barrier was\nderived by Brown [24],\n\u001cfm=1 +\u000b2\n\u000b!\u00001\nar\n\u0019kBT\nDzNeDzN\nkBT: (2)\nThis expression is of the form of the exponential Néel–\nArrhenius law \u001c=\u001c0e\u0001E=k BT, with the energy barrier\n\u0001E=DzNdetermined above. The prefactor \u001c0is called\nthe inverse attempt frequency. Its first factor is related\nto the damping dependence of the switching time, clearly\nwith a minimum at \u000b= 1. The second factor is the\nprecessional time scale of the system, with !a=\rHa=\n\r2DzN=(Vm0). The weak temperature dependence of\nthe prefactor is attributed to the Goldstone mode of the\nsystem at the top of the energy barrier in the axially\nsymmetric free-energy expression.\n(a)\n(b)FIG. 1. Comparison of reversal mechanisms in AFM (left)\nand FM (right) nanoparticles. While the energy barriers \u0001E\nforcoherentrotationareidentical(a), theattemptfrequencies\nstrongly differ caused by the different dynamical properties of\nthe eigenmodes (b). Springs in the AFM case represent that\nenergy may be transferred between anisotropy and exchange\ncontributions, while the latter is not present in the macrospin\ndescription of FMs.\nB. Nonaxially symmetric FM nanoparticles\nEquation (2) is valid for all values of the damping pa-\nrameter\u000b. As pointed out in, e.g., Ref. [30], its simple\nformcanbeattributedtothefactthattheFokker–Planck\nequation derived from the stochastic Landau–Lifshitz–\nGilbert equation simplifies to an ordinary differential\nequation for the polar angle variable cos#=mz=m0.\nIf the rotational symmetry of the system is broken, for\nexample, by a tilted external magnetic field [31], then the\nfreeenergyFmustdescribethecouplingbetweenpolar #\nand azimuthal 'variables, or longitudinal and transver-\nsal degrees of freedom. This transforms the Fokker–\nPlanck equation into a partial differential equation which\nis significantly more difficult to handle.\nFormassiveparticles, escaperatesfromanenergymin-\nimum were first systematically derived by Kramers [32],\nwho differentiated between intermediate-to-high damp-\ning (IHD) and very-low-damping (VLD) limits. For IHD,\nit can be assumed that the system is in thermal equilib-\nrium both close to the energy minimum (min) and in the\nvicinity of the saddle point (sp) which has to be crossed\nduring the escape. The IHD limit of nonaxially symmet-\nric FM nanoparticles was derived by Brown [33], which\nwas later revealed to be [34–39] a special case of Langer’s\n[40] expression for multiple degrees of freedom. Within\nthis description, the Hamiltonian or the free energy is\napproximated by a harmonic expansion around the mini-\nmum and close to the saddle point, while the equations of\nmotion are linearized near the saddle point. The energy\nscale of thermal fluctuations is required to be much lower3\nthan the energy barrier protecting the metastable state,\nleading to an Arrhenius-like formula. Applications to\nmagnetic systems can be found in, e.g., Refs. [34–37, 41].\nThe generalization to an arbitrary number of Goldstone\nmodesaspresentedinEq.(3)belowisbasedonharmonic\ntransition-state theory [42], which differs from Langer’s\ntheory in applying a dynamical prefactor independent of\nthe damping.\nThe switching time \u001cIHDmay be expressed by the for-\nmula\n\u001cIHD=2\u0019\n\u0015+;spVmin\nVsp(2\u0019kBT)Psp\u0000Pmin\n2vuutQ0\njj\"j;spj\nQ0\nj\"j;mineEsp\u0000Emin\nkBT;\n(3)\nwhereEis the energy of the given configuration and \"j\ndenotes the eigenvalues of the harmonic Hamiltonian in\nthe equilibrium state. Ideally, all eigenvalues in the min-\nimum are positive, and there is a single negative eigen-\nvalue (hence the absolute value) in the first-order saddle\npoint, along which direction the transition takes place.\nHowever, the system may possess zero-energy Goldstone\nmodeswhicharetobehandledseparately. Thesemustbe\nleft out of the eigenvalue products, hence the prime nota-\ntion. Each of these will contribute ap2\u0019kBTfactor in-\nstead, with Pdenoting the number of Goldstone modes.\nVis the phase space volume belonging to the Goldstone\nmodes. Finally, \u0015+;spis the single positive eigenvalue of\nthe linearized equations of motion in the saddle point.\nThis determines how fast the system crosses the transi-\ntion state. The derivation of Eq. (2) based on Eq. (3) is\ngiven in Appendix A.\nHowever, in the VLD limit the approximations of\nLanger’s theory break down, since the weak coupling be-\ntween the system and the heat bath encapsulated in the\ndamping parameter is no longer sufficient for ensuring\nthermal equilibrium at higher energy values. In order to\nachieve agreement with the fluctuation–dissipation theo-\nrem, one has to calculate the energy dissipated during a\nsingle precession along the energy contour including the\nsaddle point, and ensure that this is low compared to the\nthermal energy kBTin the VLD case. Such a calcula-\ntion for FM nanoparticles was carried out by Klik and\nGunther [43]. Finally, the missing connection between\nthe VLD and IHD limits, the solution of the so-called\nKramers turnover problem, was derived by Mel’nikov\nand Meshkov [44] for massive particles, and adapted to\nnonaxially symmetric FM nanoparticles by Coffey et al.\n[35, 41]. This can be summarized in the formula\n\u001c=A\u00001\u0012\u000bS\nkBT\u0013\n\u001cIHD; (4)\nwhereA\u0012\u000bS\nkBT\u0013\nis the depopulation factor,\nA(x) =e1\n2\u0019R1\n\u00001ln\u0012\n1\u0000e\u0000x(1\n4+y2)\u0013\n1\n1\n4+y2dy;(5)and\u001cIHDis the switching time in the IHD limit given by\nEq. (3). The validity of the general formula for FMs was\nlater thoroughly confirmed by the numerical solution of\nthe Fokker–Planck equation, spin dynamics simulations\nand experiments; see, e.g., Refs. [29, 30, 39].\nC. Axially symmetric AFM nanoparticle\nFor AFMs, to the best of our knowledge, analytical\nformulae similar to Eq. (2) remain unknown. Only a few\nrecent works have addressed the problem [28, 29]. How-\never, they assumed AFM nanoparticles with uncompen-\nsated magnetic moments, attributed to finite-size effects\nand lattice defects in naturally occurring nanoparticles\n[45, 46]. In this limit, the AFM was effectively described\nas a FM with a very small magnetic moment. In spin-\ntronics applications, it is possible to prepare completely\ncompensated AFM structures, for example by atom ma-\nnipulation as demonstrated in Ref. [27]. The dynamics in\nAFMs are described by coupled equations of motion for\nthe Néel vector and the magnetization [47–50], expected\nto lead to a qualitatively different behavior. Dissipa-\ntive dynamics in two-sublattice AFMs may be derived by\nconsidering two coupled Landau–Lifshitz–Gilbert equa-\ntions (1) for the sublattice magnetizations m1andm2.\nThese are transformed to the dynamical variables of the\nmagnetization m= (m1+m2)=2and the Néel vector\nn= (m1\u0000m2)=2. At low temperature, it is reason-\nable to assume that the Néel vector conserves its length\njnj=m0and only undergoes rotational motion. The\nmagnetization remains perpendicular to the Néel vector,\nn\u0001m= 0, since in a compensated AFM a finite magneti-\nzation may only be formed by canting the two sublattice\nmagnetizations perpendicularly to their original antipar-\nallel orientation. This leads to the equations of motion\n[48]\n_n=\u0000n\u0002\u0012\n\rhm\u0000\u000b_m\nm0\u0013\n; (6)\n_m=\u0000m\u0002\u0012\n\rhm\u0000\u000b_m\nm0\u0013\n\u0000n\u0002\u0012\n\rhn\u0000\u000b_n\nm0\u0013\n;(7)\nwhere hm;n=\u0000\u000em;nFare the effective fields acting on\nthe magnetization and the Néel vector, respectively. The\nfree-energydensityofanaxiallysymmetricsingle-domain\nAFM particle reads f=Hem2=(2m0)\u0000Han2\nz=(2m0),\nwithHe=qJN=(Vm0)being the exchange field de-\nscribing the coupling between the sublattices, where q\nis the number of nearest neighbors and Jthe exchange\nconstant in the corresponding atomistic model. In the\nfollowing, we will assume that qJ \u001dDz, which is true\nfor practically all magnetic materials.\nAlthough the AFM nanoparticle is still axially sym-\nmetric, it fundamentally differs from its FM counter-\npart described in Sec. IIA. As illustrated in Fig. 1(b),\nin AFMs the anisotropy energy assigned to the zcom-\nponent of the Néel vector nzmay transform into the ex-4\nchange energy between the sublattices, leading to a fi-\nnite magnetization m, even in the conservative case. In\ncomparison, the FM particle may only perform a preces-\nsion around the easy axis with a constant polar angle\n#. Consequently, one has to rely on the theory for cou-\npled degrees of freedom, such as in the case of nonaxially\nsymmetric FM systems in Sec. IIB, when deriving the\nswitching time in AFMs. Applying Eq. (4) to this prob-\nlem leads to the expression\n\u001cafm=A\u00001\u0012\u000bS\nkBT\u00131 +\u000b2\n\u000b!\u00001\nafmr\n\u0019kBT\nDzNeDzN\nkBT;(8)\nwith the derivation given in Appendix B.\nIn comparison with Eq. (2), one can observe that\nthe energy barrier \u0001E=DzNbetween the minima at\nnz=m0= 1andnz=m0=\u00001remainsthesameinEq.(8),\nas long as all individual spins rotate coherently during\nswitching. Similarly, the temperature-dependent square-\nroot term attributed to the axial symmetry is preserved.\nOn the other hand, the frequency !ais replaced by !afm,\n!afm=\n\rN\nVm02\n4\u0012\nDz\u00001\n2qJ\u0013\n+s\u00121\n2qJ+Dz\u00132\n+2DzqJ\n\u000b23\n5:\n(9)\nThe transition between the IHD and the VLD limits is\ngoverned by the ratio of the thermal energy kBTand the\nenergy loss per cycle on the contour including the saddle\npoint,\n\u000bS=\u000bN\u001216D2\nz\n3p2DzqJ+ 4p\n2DzqJ\u0013\n;(10)\nfor the derivation see Appendix C.\nIn order to highlight the differences and similarities\nbetween the FM and AFM cases, appropriate asymp-\ntotic expressions are derived. On the one hand, in the\nlimit of high damping \u000b\u001d1, one has!afm\u0019!aand\nA(\u000bS=(kBT))\u00191due to the strong energy dissipation\n\u000bS\u001dkBT, leading to\n\u001cafm;\u000b\u001d1\u0019\u000b!\u00001\nar\n\u0019kBT\nDzNeDzN\nkBT; (11)\nwhich coincides with the asymptotic behavior for FMs,\nEq. (2).\nOn the other hand, significant deviations may be ob-\nserved between the two types of systems in the limit of\nlow damping. For \u000b\u001c1, the characteristic frequency\nof AFMs is \u000b!afm\u0019p2DzqJ=p\nqJ=(2Dz)!a, indi-\ncating that the dynamics are exchange-enhanced com-\npared to FMs. Furthermore, the depopulation factor\nmay be approximated as A(\u000bS=(kBT))\u0019\u000bS=(kBT)\u0019\n\u000b4Np2DzqJ=(kBT)for slow energy dissipation and\nqJ\u001dDz. The VLD limit of Eq. (8) reads\n\u001cafm;\u000b\u001c1\u00191\n\u000bkBT\n4qJN!\u00001\nar\n\u0019kBT\nDzNeDzN\nkBT:(12)In Eq. (12), the switching time is inversely propor-\ntional to the damping parameter, as expected from the\nfluctuation–dissipation theorem [32]. Furthermore, it is\nreduced by a factor of kBT=(4qJN)compared to the ap-\npropriate limit of Eq. (2). The typical value of intrinsic\ndamping in magnetic materials is \u000b= 0:001\u00000:01, e.g.,\n\u000b= 0:0025was determined for Mn 2Au in Ref. [51]. This\nmeans that the switching time in AFMs could be up to\nseveral orders of magnitude shorter than in FMs, which\neffectively means much less thermal stability.\nThe high- and low-damping limits of the AFM switch-\ningtime, definedbyEqs.(11)and(12), maybeconnected\nby the simplified formula\n\u001cafm;asymptotic =kBT\n4qJN+\u000b2\n\u000b!\u00001\nar\n\u0019kBT\nDzNeDzN\nkBT;(13)\nwhich has an analogous form to Eq. (2). This clearly\nexpresses the difference in the behavior between FMs\nand AFMs: while for the former the minimal switch-\ning time is found for \u000bfm,min = 1, for the latter this\nvalue now depends on the system parameters, \u000bafm,min =p\nkBT=(4qJN), being decreased due to the exchange in-\nteraction. Since for high \u000bvalues the switching times in\nFMsandAFMscoincide, whiletheminimumisshiftedto\nlower\u000bvalues in AFMs, this implies that AFM nanopar-\nticles are significantly less resistant against thermal fluc-\ntuations at low damping compared to their FM counter-\nparts. However, note that in the immediate vicinity of\n\u000bafm,min, Eq. (8) is expected to give a more accurate de-\nscription than Eq. (13), since the former includes a more\nprecise interpolation between the VLD and IHD limits\nexactly in this turnover regime.\nIII. SPIN DYNAMICS SIMULATIONS\nTo test the validity of Eqs. (2) and (8), we performed\natomistic spin dynamics simulations. For the description\nof the magnetic system, we introduce the classical atom-\nistic spin Hamiltonian\nH=\u00071\n2X\nhi;jiJSiSj\u0000X\niDzS2\ni;z:(14)\nHere the Sivariables denote unit vectors on a simple cu-\nbic lattice and Jis the Heisenberg exchange interaction\nbetween atoms at nearest-neighbor sites iandj. For the\n\u0000sign in Eq. (14) the ground state is FM, while for the\n+sign it is AFM. Dz>0is the single-ion magnetocrys-\ntalline anisotropy, implying that the ground state of the\nsystem lies along the zdirection.\nThe time evolution of the unit vectors Siis described\nby the Landau–Lifshitz–Gilbert equation,\n(1 +\u000b2)\u0016s_Si=\u0000\rSi\u0002[Hi+\u000b(Si\u0002Hi)];(15)\nwhere\u0016sdenotes the magnetic moment of a single spin\nand\u000bis the Gilbert damping as in the macrospin model.5\nBy including a Langevin thermostat, the equilibrium\nand nonequilibrium thermodynamic properties can be\nobtained in the classical approximation. The effective\nlocal magnetic field at lattice site iis\nHi=\u0000@H\n@Si+\u0018i(t); (16)\nwhereHis given by Eq. (14) in the present case and \u0018i\nis a field-like stochastic process. Here we consider the\nwhite-noise limit [52], with the first and second moments\nh\u0018i(t)i=0;h\u0018i;a(0)\u0018j;b(t)i=2\u000bkBT\u0016s\n\r\u000eij\u000eab\u000e(t);\n(17)\nwhereaandbdenote the Cartesian components.\nIV. CORRESPONDENCE BETWEEN THEORY\nAND SIMULATIONS\nA. Temperature-dependent effective parameters\nFor a direct comparison of Eqs. (2) and (8) with the\nresults of the spin dynamics simulations, it has to be en-\nsuredthat theassumptionswhichtheanalyticalformulae\narebasedonaresatisfiedbytheatomisticmodel. Aslong\nas the linear size of the system is shorter than a charac-\nteristic length scale on the order of the exchange length,\nLe\u0018p\nJ=Dz, it is expected that coherent rotation is\nthe primary mechanism of magnetization reversal in the\nnanoparticles. Above this threshold, the nucleation of\na pair of domain walls becomes energetically favorable\ncompared to the energy barrier which has to be overcome\nby coherent rotation [53–57].\nEven for small particles, one has to take into account\nthat in the atomistic model the thermal fluctuations de-\ncreasem0, the equilibrium length of the magnetization in\nFMsandoftheNéelvectorinAFMs[58]. Inearlierpubli-\ncations for FM systems [59], it was found that the dimen-\nsionless magnetization may be well approximated by the\nphenomenological relation m0V=(N\u0016s) = (1\u0000T=Tc)1=3\nfor 3d Heisenberg models. Furthermore, one has to ac-\ncount for finite-size effects. Small systems such as the\nnanoparticles considered here have a reduced magneti-\nzation compared to the bulk at a given temperature,\nas a result of lower coordination numbers at the sur-\nfaces. For 3d Heisenberg spin models, finite-size-scaling\ntheory provides a value for the apparent Curie temper-\nature as a function of the size L(linear characteristic\nsize of the nanoparticle), Tc(L)=T1\nc= 1\u0000(d0=L)1=\u0017,\nwhere the parameter d0corresponds to the characteristic\nexchange length, and \u0017to the critical exponent. A re-\ncentwork[60]inFMFePtnanoparticlesusingsimilarpa-\nrameters to our simulations has estimated d0= 0:4nm,\nto be compared to a lattice constant of a= 0:38nm,\nand\u0017= 0:856. The critical temperature of the cu-\nbic Heisenberg model in the thermodynamic limit wasfound to be kBT1\nc= 1:443J[61]. In this work we per-\nformed simulations for a cubic nanoparticle composed of\nN= 43= 64spins; therefore, the lateral size is 4 spins,\nmeaningd0=L= 0:4=(0:38\u00024) = 0:238in the finite-\nsize-scaling expression, leading to Tc(L) = 1:173J. We\nfound that the phenomenological relation using this crit-\nical temperature was in agreement with spin dynamics\nsimulations of the dimensionless order parameter at the\ntemperature ranges where coherent reversal is dominant.\nIn the analytical expressions the effect of the re-\nduced order parameter may be considered by assuming\ntemperature-dependent magnetic parameters in Eqs. (2)\nand (8) [58],\nDz=Dz\u0012m0V\nN\u0016s\u00133\n; (18)\nJ=J\u0012m0V\nN\u0016s\u00132\n: (19)\nThe cubic dependence of the anisotropy on the dimen-\nsionless magnetization expressed in Eq. (18) is the result\nof the Callen–Callen theory [62, 63]. The quadratic de-\npendence of the exchange in Eq. (19) may be derived\nfrom the random phase approximation [64].\nFurthermore, the reduced coordination number qat\nthe surface also directly affects Eq. (8). Here we substi-\ntuted the mean value of the number of nearest neighbors:\nfor a nanoparticle composed of N= 64spins in simple\ncubic arrangement, q= 6for the spins inside (23= 8),\nq= 5for the spins at the faces (6\u00022\u00022 = 24),q= 4\nfor the spin at the edges (12\u00022 = 24), andq= 3for the\nspins at the corners (8), thusqavg= 4:5.\nB. Oscillations in the order parameter in the VLD\nlimit of AFM nanoparticles\nA further requirement for an accurate comparison be-\ntween simulations and analytical expressions is that the\nidentified switching events in the simulations have to cor-\nrespond to the reversals described by the theory [39]. For\nuniaxial nanoparticles with easy axis along the zdirec-\ntion, the following criteria may be identified. First, the z\ncomponent of the order parameter mornhas to change\nsign, and thereafter cross a threshold value governed by\nthe equilibrium value m0at the given temperature. Dur-\ning the process, the energy of the particle increases while\ncrossing the energy barrier, before decreasing again when\ncoming to rest in the other energy minimum; see Sup-\nplemental Videos 1 and 2 [65] for an illustration of this\nprocess.\nFor FMs, the sign change of mzis always accompa-\nnied by an increase in the anisotropy energy. On the\nother hand, in AFM nanoparticles the energy can be\ntransformed between the anisotropy contribution of the\nNéel vector and the exchange contribution of the magne-\ntization, meaning that nzmay switch sign even if the\ntotal energy of the system remains constant. In the6\n(a)\n(b)\nFIG. 2. Illustration of the switching events in the AFM\nnanoparticle for (a) low ( \u000b= 0:0005) and (b) intermediate\n(\u000b= 0:1) damping. The other simulation parameters are\nT= 0:6J=k B,Dz= 0:1J, for a cubic nanoparticle consisting\nofN= 43= 64spins. The threshold values for the switching\nare chosen to be\u00060:75hj~nzji, where ~nzis thezcomponent of\nthe dimensionless order parameter and hj~nzjiis the thermal\naverageofitsabsolutevalue. ~navg\nzwasobtainedbyperforming\na moving average on the ~nzdata using a window of width\n\u0001t= 8:8\u0016s=(\rJ).\nlow-damping limit such an oscillatory motion can indeed\nbe observed, where the zcomponent of the Néel vector\nswitches sign and crosses the threshold value many times\nbefore coming to rest in one of the minima, see Fig. 2(a)\nand Supplemental Video 3 [65]. This is analogous to a\nmechanical particle in a double-well potential, where the\nenergy is transformed between the kinetic and potential\nparts during the motion. During these oscillations in the\nNéel vector, the energy of the system is slowly varied\ndue to the weak coupling to the heat bath, meaning that\nthe oscillations take place on a roughly constant energy\nsurface and hence they only represent a single switch-\ning event. For an estimate of the oscillation periods see\nAppendix D.\nTo determine the actual switching events in the low-\ndamping limit in the simulations, we therefore used a\ntime average of the data, where the time window was\nlarger than the period of the fast oscillations of the Néel\n0.0001 0.001 0.01 0.1 1 1010100100010000100000FIG. 3. Damping dependence of the switching time for\nFM and AFM nanoparticles. The system parameters are\nT= 0:6J=k B,Dz= 0:1J,N= 64. Symbols correspond\nto simulations using atomistic spin dynamics methods and\nlines to the analytical formulae Eqs. (2), (8), and (13).\nvector while crossing the energy barrier. As shown in\nFig. 2(a), in the averaged data the zcomponent of the\ndimensionless order parameter only crosses the threshold\nvalue once after its sign change during a single rever-\nsal. In contrast, for intermediate-to-high values of \u000bthe\nenergy fluctuates strongly on the time scale of a single ro-\ntation, and the oscillatory switching is absent as shown\nin Fig. 2(b). In this case, the same number of switching\nevents are registered both with and without the averag-\ning procedure.\nV. COMPARISON OF SWITCHING TIMES\nIn order to validate the damping dependence of the\nswitching time in both FMs and AFMs, we performed\ncomputer simulations by varying the damping value \u000bat\nafixedtemperature T= 0:6J=kB, showninFig.3. Inor-\nder to enable an accurate comparison, the same absolute\nvalue of the exchange interaction Jand the anisotropy\nDz= 0:1Jwas considered during the simulations, per-\nformed for a cubic nanoparticle consisting of N= 64\nspins. As can be seen in the figure, Eq. (2) gives good\nagreement with the simulation results for the FM case,\nwhile Eq. (8) is accurate for the AFM case over the whole\nparameter range. While the switching times are similar\nforhighdamping, theminimalswitchingtimeisfoundfor\nsignificantly lower \u000bvalues in the AFM case, leading to\na reduced thermal stability in the limit of low damping.\nNote that the asymptotic expression Eq. (13) for AFMs,\nwhich has an analogous form to Eq. (2) for FMs, un-\nderestimates the switching time in the turnover regime.\nIn particular, for the present simulation parameters the\nVLD limit, characterized by the relation \u001cafm;\u000b\u001c1/\u000b\u000017\n0.0001 0.001 0.01 0.1 1 10100100010000100000\nFIG. 4. Damping dependence of the switching time\nfor the antiferromagnetic nanoparticle, using the parame-\ntersT= 0:6J=k B,Dz= 0:1J,N= 64. Circles and\nsquares correspond to the same simulation data as in Fig. 3,\nwith and without performing the time averaging. Lines show\nEq. (8), expected to hold for all \u000bvalues, and Eq. (B17) with-\nout the depopulation factor, which is only applicable in the\nintermediate-to-high-damping limit.\nin Eq. (12), is not reached yet for \u000b\u00190:001, and the\nAFM switching time shows a weaker dependence on the\ndamping in this turnover regime.\nFigure 4 illustrates the effect of time averaging of the\nsimulation data on the obtained switching times. Moving\naverages were performed on a time interval of \u0001t= 8:8\n\u0016s=(\rJ). Without performing the time averaging, the\nmean time between sign changes of the zcomponent of\nthe order parameter converges to a constant value at low\ndamping, similarly to the intermediate-to-high-damping\nformula, given by Eq. (B17) in Appendix B. However,\nthis behavior is in contradiction with the fluctuation–\ndissipation theorem. The range in \u000bwhere the time-\naveraging starts to play a significant role in the sim-\nulation data coincides with the interval where the de-\npopulation factor in Eq. (8) becomes important in the\ntheoretical description. This emphasizes the necessity of\ncorrectly determining the switching time in the very-low-\ndamping-limit both in the analytical model as well as in\nthe numerical simulations.\nA further important difference between FMs and\nAFMs, as can be deduced from Eqs. (2) and (8), is\nthat the switching time in AFMs depends on the mi-\ncroscopic exchange interaction J, while this parame-\nter is absent in the single-domain description of FMs.\nThe analytical expressions Eqs. (2) and (8) for differ-\nent values of Jare compared in Fig. 5 as a function\nof temperature, using the parameters Dz= 0:1J0and\n\u000b= 0:0005. For the AFM case simulation results are\nalso presented, confirming the assumed Néel–Arrhenius\nlaw in this parameter range. For the FM case with the\n1 1.5 2 2.510210410610810101012FIG. 5. Dependence of the switching time on the exchange\ninteraction Jfor FM and AFM nanoparticles. The system\nparameters are \u000b= 0:0005,Dz= 0:1J0,N= 64. Sym-\nbols correspond to simulations using atomistic spin dynamics\nmethodsfortheAFMcaseandlinestotheanalyticalformulae\nEqs. (2) and (8).\nsignificantly longer switching times only the analytical\nformula Eq. (2) is shown, which had been confirmed in\nearlier publications [30] and in Fig. 3 here for a differ-\nent damping regime. Note that while Eq. (2) does not\nexplicitly depend on J, the predicted analytical curves\nare still different for J=J0andJ= 10J0, since the\nequilibrium magnetization m0is higher in the latter case\n(cf. Eqs. (18) and (19)). As indicated in the figure, at\nthe lowest temperature where the simulations were per-\nformed (J0=kBT= 2), the ratio \u001cfm=\u001cafmis about 10\ntimes larger for 10times higher exchange interaction, in\nagreement with the VLD damping limits of Eqs. (2) and\n(12).\nVI. CONCLUSION\nIn summary, we investigated the superparamagnetic\nlimit of AFM nanoparticles analytically as well as by\nmeans of computer simulations. The derived analyti-\ncal expression, Eq. (8), for the mean switching time in-\ndicates a drastically reduced thermal stability of AFM\nnanostructures as compared to their FM counterparts\nbecause of the exchange enhancement of the attempt fre-\nquency. The latter is caused by the coupling between the\nanisotropy term connected to the Néel vector and the\nexchange term connected to the magnetization in the\nfree-energy density of single-domain AFMs, which also\ncauses a strong oscillation of the Néel vector direction at\nlow damping values during the switching process.\nThe significantly faster dynamics in AFMs is one of\ntheir main proposed advantages over FMs in spintron-\nics applications [4]. However, this enhanced speed also8\nleads to an increased susceptibility to thermal fluctua-\ntions as demonstrated here; for realistic materials with a\nlow damping value, the switching times of AFMs can be\nexpected to be four to five orders of magnitude shorter\nthan those of FMs, a finding that is in agreement with\na work on antiferromagnetic grains in exchange bias sys-\ntems [66]. Furthermore, the procedures capable of in-\ncreasing the switching times in FMs may be less efficient\nin AFMs. The energy barrier in the Arrhenius expres-\nsions Eqs. (2) and (8), which is the leading term in the\ntemperature dependence, may be increased by choosing\na higher anisotropy value Dz, a larger system size N,\na lower temperature T, or at larger order parameters\nm0achieved by coupling the microscopic spins stronger\nto each other by a higher exchange coupling J. Ac-\ncording to the VLD limit Eq. (12), all of these meth-\nods except increasing the anisotropy lead to a decrease\nin the inverse attempt frequency, meaning that they de-\ncrease the\u001cafm=\u001cfmratio assuming the same system pa-\nrameters. Furthermore, for FMs damping values in the\nrange\u000b= 0:001\u00000:01, typical for materials suggested\nfor spintronic devices [51], will surely fall into the VLD\nregime, where lower \u000bvalues lead to an enhanced switch-\ning time. On the other hand, for AFMs similar values\nmay belong to the turnover region where the lifetime\nis minimal and the dependence on \u000bis weak, around\n\u000bafm,min\u0019p\nkBT=(4qJN). These problems may be\ncircumvented by selecting materials with a high damp-\ning value, where the difference between FM and AFM\nswitching times disappears.\nHowever, fast reversal of the nanoparticles may also be\ndesired in specific applications. Since thermal activation\nfacilitates the current-induced switching in spintronic de-\nvices[14], ahigherattemptfrequencynecessitatesalower\ncurrent density for achieving the same switching rate. In\nmagnetic hyperthermia [12], the reversal of nanoparticles\nis used to provide targeted warming of tissues, which can\nbecome more efficient at higher frequencies. For such\npurposes, AFM nanoparticles may provide advantages\nover their FM counterparts.\nACKNOWLEDGMENTS\nFinancial support for this work at the University of\nKonstanz was provided by the Deutsche Forschungs-\ngemeinschaft via SFB 1214. At the FU Berlin sup-\nport by the Deutsche Forschungsgemeinschaft through\nSFB/TRR 227 \"Ultrafast Spin Dynamics\", Project A08\nis gratefully acknowledged. L.R. would like to acknowl-\nedge the Alexander von Humboldt Foundation and the\nNational Research, Development and Innovation Office\nof Hungary via Project No. K115575 for support.Appendix A: FM switching time in the IHD limit\nHere the switching time of axially symmetric FM\nnanoparticles, given by Eq. (2), is derived based on the\ngeneral expression Eq. (3). The free-energy density is\ngiven byf=\u0000Hem2\nz=(2m0)whereHe= 2DzN=(Vm0),\nand the normalization jmj=m0is assumed. The expres-\nsion has a minimum at mz=m0= 1, and the expansion\nis performed in the small variables mx=m0;my=m0\u001c1.\nThis yields\nFmin=\u0000DzN; (A1)\n\"1;min=\"2;min= 2DzN: (A2)\nThe saddle point is at mx=m0= 1with the expansion\nvariablesmy=m0;mz=m0\u001c1, which results in\nFsp= 0; (A3)\n\"1;sp=\u00002DzN; (A4)\n\"2;sp= 0: (A5)\nNote that\"1;spis negative, corresponding to the un-\nstable mode in the saddle point. The other eigenvalue\n\"2;spdescribes a Goldstone mode, representing the fact\nthat the saddle point can be arbitrarily chosen along the\ncirclem2\nx+m2\ny=m2\n0. The corresponding phase space\nvolume is\nVsp= 2\u0019; (A6)\nthe circumference of the circle.\nThe linearized Landau–Lifshitz–Gilbert equation in\nthe saddle point reads\n@tmy=1\n1 +\u000b2\rN\nVm02Dzmz=1\n1 +\u000b2!amz;(A7)\n@tmz=1\n1 +\u000b2\rN\nVm0\u000b2Dzmz=\u000b\n1 +\u000b2!amz;(A8)\nwith the eigenvalues\n\u00151;sp=\u000b\n1 +\u000b2!a; (A9)\n\u00152;sp= 0; (A10)\nwhere\u0015+;sp=\u00151;spis the single positive eigenvalue.\nSubstituting Eqs. (A1)-(A6) and Eq. (A9) into Eq. (3)\ngives precisely Eq. (2), which in this special case is valid\nfor all values of the damping.\nAppendix B: AFM switching time in the IHD limit\nHere Eq. (8) without the depopulation factor will be\nderived based on Eq. (3). We will use the free-energy\ndensityf=Hem2=(2m0)\u0000Han2\nz=(2m0), withHe=\nqJN=(Vm0)being the exchange field, where qis the\nnumber of nearest neighbors and Jthe exchange con-\nstant in the corresponding atomistic spin model, rescaled\nbyaccountingforthethermallyreducedorderparameter.9\nThe minimum of the free energy Fis at n=m0=\n(0;0;1);m=m0= (0;0;0)with\nFmin=\u0000DzN; (B1)\n\"1;min=\"2;min= 2DzN; (B2)\n\"3;min=\"4;min=qJN: (B3)\nForDz\u001cqJthe saddle point is n=m0=\n(1;0;0);m=m0= (0;0;0), where the expansion yields\nFsp= 0; (B4)\n\"1;sp=\u00002DzN; (B5)\n\"2;sp= 0; (B6)\n\"3;sp=\"4;sp=qJN; (B7)\nHere\"1;spis the unstable mode and \"2;spis the Gold-\nstone mode with\nVsp= 2\u0019: (B8)\nThe linearized equations of motion in the saddle point\nread\n@tmy=\rN\nVm02Dznz\u0000\u000b@tnz; (B9)\n@tmz=\u000b@tny; (B10)\n@tny=\u0000\rN\nVm0qJmz\u0000\u000b@tmz;(B11)\n@tnz=\rN\nVm0qJmy+\u000b@tmy; (B12)\nleading to the eigenvalues\n\u00151;sp=0; (B13)\n\u00152;sp=\u00001\n1 +\u000b2\rN\nVm0\u000bqJ; (B14)\n\u00153;sp=1\n1 +\u000b2\rN\nVm0\"\n\u000b\u0012\nDz\u00001\n2qJ\u0013\n+s\n\u000b2\u00121\n2qJ+Dz\u00132\n+ 2DzqJ#\n;(B15)\n\u00154;sp=\u00001\n1 +\u000b2\rN\nVm0\"\n\u000b\u00121\n2qJ\u0000Dz\u0013\n+s\n\u000b2\u00121\n2qJ+Dz\u00132\n+ 2DzqJ#\n;(B16)\nwhere the positive eigenvalue is \u0015+;sp=\u00153;sp.\nSubstitutingEqs.(B1)-(B8)andEq.(B15)intoEq.(3)\nproduces\n\u001cIHD\nafm=1 +\u000b2\n\u000bVm0\n\rN\u0014\u0012\nDz\u00001\n2qJ\u0013\n+s\u00121\n2qJ+Dz\u00132\n+2DzqJ\n\u000b23\n5\u00001r\n\u0019kBT\nDzNeDzN\nkBT;\n(B17)the intermediate-to-high-damping limit of Eqs. (8) and\n(9). Note that since the eigenvalues \"3;min;\"4;mincancel\nwith\"3;sp;\"4;sp, the difference between the ferromagnetic\nand antiferromagnetic cases only comes from the dynam-\nical prefactor \u0015+;sp, which is exchange-enhanced at low\nand intermediate damping for the latter.\nAppendix C: Energy dissipation per cycle when\npassing through the saddle point\nHere the depopulation factor in Eq. (4) will be cal-\nculated for the AFM particle. The variable Sin the\nargument of Ain Eq. (8) denotes the action of the un-\ndampedmotioncrossingthroughthesaddlepoint. Equa-\ntion(4)expressesthatif \u000bS, theenergydissipatedduring\na single cycle of motion over the saddle point [41, 43], is\nsmall compared to the thermal energy kBT, then it takes\nlonger for the particle to cross the energy barrier since it\ncan no longer be assumed that the equilibrium Maxwell–\nBoltzmann distribution is formed in the region close to\nthe saddle point.\nIn order to calculate this energy dissipation, Eqs. (6)\nand (7) are linearized in \u000bat low damping, yielding\n_n=\u0000\rn\u0002\u0012\nhm+\u000bm\nm0\u0002hm+\u000bn\nm0\u0002hn\u0013\n;(C1)\n_m=\u0000\rm\u0002\u0012\nhm+\u000bm\nm0\u0002hm+\u000bn\nm0\u0002hn\u0013\n\u0000\rn\u0002\u0012\nhn+\u000bn\nm0\u0002hm\u0013\n: (C2)\nThe free energy dissipation per cycle may be written\nas\n\u0000\u0001F=\u000bS=\u0000ZT\n0_Fdt=ZT\n0Z\nhm_m+hn_ndrdt\n=\u000b\rm 0VZT\n0\u0012m\nm0\u0002hm+n\nm0\u0002hn\u00132\n+\u0012n\nm0\u0002hm\u00132\ndt: (C3)\nIntroducing the renormalized variables ^m =\nm=m0,^n=n=m0, and substituting hm =\n\u0000qJN=(Vm0)^m;hn= 2DzN=(Vm0)^nzezfor the\nconsidered system one obtains\n\u000bS=\u000b\rN2\nVm0ZT\n04D2\nz\u0000\n1\u0000^n2\nz\u0001\n^n2\nz+ (qJ)2^m2dt;(C4)\nwith the integral to be evaluated along the trajectory of\nthe undamped motion crossing the saddle point.10\nFor\u000b= 0, Eqs. (C1) and (C2) may be written as\n@t^mx=\u0000\rN\nVm02Dz^ny^nz; (C5)\n@t^my=\rN\nVm02Dz^nx^nz; (C6)\n@t^mz= 0; (C7)\n@t^nx=\rN\nVm0qJ(^ny^mz\u0000^nz^my);(C8)\n@t^ny=\rN\nVm0qJ(^nz^mx\u0000^nx^mz);(C9)\n@t^nz=\rN\nVm0qJ(^nx^my\u0000^ny^mx);(C10)\nfor the axially symmetric AFM nanoparticle. Since the\nconstraintj^nj= 1issatisfiedbythedynamicalequations,\nthe normalized Néel vector may be rewritten in spherical\ncoordinates, (^nx;^ny;^nz) = (sin#cos';sin#sin';cos#).\nFor the variable ^mone has ^n\u0001^m= 0, and itszcompo-\nnent is a constant of motion as expressed by Eq. (C7).\nWithout the damping, the free energy of the system is\nalso conserved during the time evolution,\nF=qJ\n2N^m2\u0000DzNcos2#: (C11)\nUsing the conserved quantities Fand ^mz, Eqs. (C5)-\n(C10) may be expressed as\n@t#=\u0007s\n!2\nF\u0000!2\n0sin2#\u0000!2\n^mz\nsin2#;(C12)\n@t'=\u00001\nsin2#!^mz; (C13)\nwith\n!0=\rN\nVm0p\n2DzqJ; (C14)\n!F=\rN\nVm0s\n2\u0012F\nN+Dz\u0013\nqJ;(C15)\n!^mz=\rN\nVm0qJ^mz: (C16)\nFor the trajectory including the saddle point one has\nF= 0, see Eq. (B4), and ^mz= 0. Equation (C12) may\nbe used to change the parametrization from the time tto\nthe polar angle #, which simplifies Eq. (C4) to\n\u000bS=\u000bNZ2\u0019\n04D2\nzp2DzqJ\u0010\njcos#j\u0000jcos#j3\u0011\n+p\n2DzqJjcos#jd#: (C17)\nEvaluating the integral Eq. (C17) yields Eq. (10).\nAppendix D: Oscillations in the Néel vector based\non the theoretical model\nAsshowninFig.2, significantoscillationsinthe zcom-\nponent of the order parameter were observed in the spin\n00.511.522.533.5\n0 0 .5 1 1 .5 2|F(ω)|\nfrequency ω(γJ/µ s)FIG. 6. Fourier spectrum of the oscillations of the zcom-\nponent of the order parameter from the spin dynamics sim-\nulations. The same simulation parameters were used as for\nFig. 2(a),\u000b= 0:0005,T= 0:6J=k B,Dz= 0:1J,N= 64.\nThe characteristic oscillation frequency from Eq. (C14) is\n!0= 0:66\rJ=\u0016 s.\ndynamics simulations of antiferromagnetic nanoparticles\nat very low damping values. This can be explained by\nthe fact that forF>0where the switching occurs, even\nin the conservative system ^nwill perform full rotations\nduring which its zcomponent changes sign, as described\nbyEq.(C12). For ^mz= 0, theperiodoftheseoscillations\nmay be evaluated in a closed form,\nTF=Z2\u0019\n01q\n!2\nF\u0000!2\n0sin2#d#=4\n!FK\u0012!F\n!0\u0013\n;(D1)\nwithKthe complete elliptic integral of the first kind.\nIt can be seen from Eq. (D1) that the oscillation fre-\nquency will change as the free energy varies due to the\ncoupling to the heat bath. If the thermal fluctuations\nare weak as required for the application of Arrhenius-\nlike expressions such as Eqs. (2) and (8), the free energy\ndoesnotbecomesignificantlyhigherthanitssaddle-point\nvalue during the switching, and in this case the oscilla-\ntion frequencies will be comparable to !0. For example,\n0:01\u0014F=(DzN)\u00140:2yields 0:39\u00142\u0019=(TF!0)\u00140:65.\nThe adiabatic variation of the energy leads to a wide dis-\ntribution of frequency values if the oscillations are inves-\ntigated in Fourier space, as displayed in Fig. 6. Using the\ntemperature-dependent effective parameters described in\nSec. IVA, for the model coefficients in Fig. 6 one obtains\n!0= 0:66\rJ=\u0016 s, roughly corresponding to the peak in\nthe frequency distribution. 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He\nPhysics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong SAR, China\nA global picture of magnetic domain wall (DW) propagation in a nanowire driven by a magnetic\nfield is obtained: A static DW cannot exist in a homogeneous ma gnetic nanowire when an external\nmagnetic field is applied. Thus, a DW must vary with time under a static magnetic field. A moving\nDW must dissipate energy due to the Gilbert damping. As a resu lt, the wire has to release its\nZeeman energy through the DW propagation along the field dire ction. The DW propagation speed\nis proportional tothe energy dissipation rate that is deter mined bythe DW structure. Anoscillatory\nDW motion, either the precession around the wire axis or the b reath of DW width, should lead to\nthe speed oscillation.\nMagnetic domain-wall (DW) propagation in a\nnanowire due to a magnetic field[1, 2, 3, 4, 5] reveals\nmany interesting behaviors of magnetization dynamics.\nFor a tail-to-tail (TT) DW or a head-to-head (HH) DW\n(shown in Fig. 1) in a nanowire with its easy-axis along\nthe wire axis, the DW will propagate in the wire un-\nder an external magnetic field parallel to the wire axis.\nThe propagation speed vof the DW depends on the field\nstrength[3, 4]. There exists a so-called Walker’s break-\ndown field HW[6].vis proportional to the external field\nHforH < H WandH≫HW. The linear regimes\nare characterized by the DW mobility µ≡v/H. Ex-\nperiments showed that vis sensitive to both DW struc-\ntures and wire width[1, 2, 3]. DW velocity vdecreases as\nthe field increases between the two linear H-dependent\nregimes, leading to the so-called negative differential mo-\nbility phenomenon. For H≫HW, the DW velocity,\nwhose time-average is linear in H, oscillates in fact with\ntime [3, 6].\nIII II IMθ=0 θ=πH\nzxy∆\nADW\nFIG. 1: Schematic diagram of a HH DW of width ∆ in a\nmagnetic nanowire of cross-section A. The wire consists of\nthree phases, two domains and one DW. The magnetization\nin domains I and II is along +z-direction ( θ= 0) and -z-\ndirection ( θ=π), respectively. III is the DW region whose\nmagnetization structure could be very complicate. /vectorHis an\nexternal field along +z-direction.\nIt has been known for more than fifty years that\nthe magnetization dynamics is govern by the Landau-\nLifshitz-Gilbert (LLG)[7] equation that is nonlinear\nand can only be solved analytically for some special\nproblems[6, 8]. The field induced domain-wall (DW)propagation in a strictly one-dimensional wire has also\nbeen known for more than thirty years[6], but its exper-\nimental realization in nanowires was only achieved[1, 2,\n3, 4, 5] in recent years when we are capable of fabricating\nvarious nano structures. Although much progress[9, 10]\nhas been made in understanding field-induced DW mo-\ntion, it is still a formidable task to evaluate the DW\npropagation speed in a realistic magnetic nanowire even\nwhen the DW structure is obtained from various means\nlike OOMMF simulator and/or other numerical software\npackages. A global picture about why and how a DW\npropagates in a magnetic nanowire is still lacking.\nIn this report, we present a theory that reveals the\norigin of DW propagation. Firstly, we shall show that\nno static HH (TT) DW is allowed in a homogeneous\nnanowire in the presence of an external magnetic field.\nSecondly, energy conservation requires that the dissi-\npated energy must come from the energy decrease of the\nwire. Thus, the origin of DW propagation is as follows.\nA HH (TT) DW must move under an external field along\nthe wire. The moving DW must dissipate energy because\nof various damping mechanisms. The energy loss should\nbe supplied by the Zeeman energy released from the DW\npropagation. This consideration leads to a general re-\nlationship between DW propagation speed and the DW\nstructure. It is clear that DW speed is proportional to\nthe energy dissipation rate, and one needs to find a way\nto enhance the energy dissipation in order to increase the\npropagation speed. Furthermore, the present theory at-\ntributes a DW velocity oscillation for H≫HWto the\nperiodic motion of the DW, either the precession of the\nDW or oscillation of the DW width.\nIn a magnetic material, magnetic domains are formed\nin order to minimize the stray field energy. A DW that\nseparates two domains is defined by the balance between\nthe exchangeenergy and the magnetic anisotropyenergy.\nThe stray field plays little role in a DW structure. To\ndescribe a HH DW in a magnetic nanowire, let us con-\nsider a wire with its easy-axis along the wire axis (the\nshape anisotropy dominates other magnetic anisotropies\nand makes the easy-axis along the wire when the wire is\nsmall enough) which is chosen as the z-axis as illustrated2\nin Fig. 1. Since the magnitude of the magnetization /vectorM\ndoes not change in the LLG equation[8], the magnetic\nstate of the wire can be conveniently described by the\npolar angle θ(/vector x,t) (angle between /vectorMand the z-axis) and\nthe azimuthal angle φ(/vector x,t). The magnetization energy\nis mainly from the exchange energy and the magnetic\nanisotropy because the stray field energy is negligible in\nthis case. The wire energy can be written in general as\nE=/integraldisplay\nF(θ,φ,/vector∇θ,/vector∇φ)d3/vector x,\nF=f(θ,φ)+J\n2[(/vector∇θ)2+sin2θ(/vector∇φ)2]−MHcosθ,(1)\nwherefis the energydensity due to all kinds ofmagnetic\nanisotropies which has two equal minima at θ= 0 and\nπ(f(θ= 0,φ) =f(θ=π,φ)),J−term is the exchange\nenergy,Mis the magnitude of magnetization, and His\nthe external magnetic field along z-axis. In the absence\nofH, a HH static DW that separates θ= 0 domain and\nθ=πdomain (Fig. 1) can exist in the wire.\nNon-existence of a static HH (TT) DW in a magnetic\nfield-In order to show that no intrinsic static HH DW\nis allowed in the presence of an external field ( H/negationslash= 0),\none only needs to show that following equations have no\nsolution with θ= 0 at far left and θ=πat far right,\nδE\nδθ=J∇2θ−∂f\n∂θ−HMsinθ−Jsinθcosθ(/vector∇φ)2= 0,\nδE\nδφ=J/vector∇·(sin2θ/vector∇φ)−∂f\n∂φ= 0.\n(2)\nMultiply the first equation by ∇θand the second equa-\ntion by∇φ, then add up the two equations. One can\nshow a tensor Tsatisfying ∇·T= 0 with\nT=[f−HMcosθ+J\n2(|∇θ|2+sin2θ|∇φ|2)]1−\nJ(∇θ∇θ+sin2θ∇φ∇φ),\nwhere1is 3×3 unit matrix. A dyadic product ( ∇θ∇θ\nand∇φ∇φ) between the gradient vectors is assumed in\nT. If a HH DW exists with θ= 0 in the far left and\nθ=πin the far right, then it requires −f(0,φ)+HM=\n−f(π,φ)−HMthat holds only for H= 0 since f(0,φ) =\nf(π,φ). In other words, a DW in a nanowire under an\nexternal field must be time dependent that could be ei-\nther a local motion or a propagation along the wire. It\nshould be clear that the above argument is only true for\na HH DW in a homogeneous wire, but not valid with de-\nfect pinning that changes Eq. (2). Static DWs exist in\nfact in the presence of a weak field in reality because of\npinning.\nWhat is the consequence of the non-existence of a\nstatic DW? Generally speaking, a physical system un-\nder a constant driving force will first try a fixed point\nsolution[11]. It goes to other types of more complicatedsolutionsifafixedpointsolutionisnotpossible. Itmeans\nthat a DW has to move when an external magnetic field\nis applied to the DW along the nanowire as shown in\nFig. 1. It is well known[10] that a moving magnetiza-\ntion must dissipate its energy to its environments with a\nrate,dE\ndt=αM\nγ/integraltext+∞\n−∞(d/vector m/dt)2d3/vector x,where/vector mis the unit\nvector of /vectorM,αandγare the Gilbert damping constant\nand gyromagnetic ratio, respectively. Following the simi-\nlar method in Reference 12 for a Stoner particle, one can\nalso show that the energy dissipation rate of a DW is\nrelated to the DW structure as\ndE\ndt=−αγ\n(1+α2)M/integraldisplay+∞\n−∞/parenleftBig\n/vectorM×/vectorHeff/parenrightBig2\nd3/vector x,(3)\nwhere/vectorHeff=−δF\nδ/vectorMis the effective field. In regions I and\nII or inside a static DW, /vectorMis parallel to /vectorHeff. Thus no\nenergy dissipation is possible there. The energy dissipa-\ntion can only occur in the DW region when /vectorMis not\nparallel to /vectorHeff.\nDW propagation and energy dissipation- Foramagnetic\nnanowire in a static magnetic field, the dissipated energy\nmust come from the magnetic energy released from the\nDW propagation. The total energy of the wire equals\nthe sum of the energies of regions I, II, and III (Fig. 1),\nE=EI+EII+EIII.EIincreases while EIIdecreases\nwhen the DW propagates from left to the right along the\nwire. The net energy change of region I plus II due to\nthe DW propagation is\nd(EI+EII)\ndt=−2HMvA, (4)\nwherevis the DW propagating speed, and Ais the cross\nsection of the wire. This is the released Zeeman energy\nstored in the wire. The energy of region III should not\nchange much because the DW width ∆ is defined by the\nbalanceofexchangeenergyand magneticanisotropy,and\nis usually order of 10 ∼100nm. A DW cannot absorb\nor release too much energy, and can at most adjust tem-\nporarily energy dissipation rate. In other words,dEIII\ndt\nis either zero or fluctuates between positive and nega-\ntive values with zero time-average. Since energy release\nfrom the magnetic wire should be equal to the energy\ndissipated (to the environment), one has\n−2HMvA+dEIII\ndt=−αγ\n(1+α2)M/integraldisplay\nIII/parenleftBig\n/vectorM×/vectorHeff/parenrightBig2\nd3/vector x.\n(5)\nor\nv=αγ\n2(1+α2)HA/integraldisplay\nIII/parenleftBig\n/vector m×/vectorHeff/parenrightBig2\nd3/vector x+1\n2HMAdEIII\ndt.\n(6)\nVelocity oscillation- Eq. (6) is our central result that\nrelates the DW velocity to the DW structure. Obviously,\nthe right side of this equation is fully determined by the3\nDW structure. A DW can have two possible types of\nmotion under an external magnetic field. One is that a\nDW behaves like a rigid body propagating along the wire.\nThiscaseoccursoftenatsmallfield, anditisthebasicas-\nsumption in Slonczewski model[9] and Walker’s solution\nforH < H W. Obviously, both energy-dissipation and\nDW energy is time-independent,dEIII\ndt= 0. Thus,and\nthe DW velocity should be a constant. The other case\nis that the DW structure varies with time. For example,\ntheDWmayprecessaroundthewireaxisand/ortheDW\nwidth may breathe periodically. One should expect both\ndEIII\ndtandenergydissipationrateoscillatewith time. Ac-\ncording to Eq. (6), DW velocity will also oscillate. DW\nvelocity should oscillate periodically if only one type of\nDW motion (precession or DW breathing) presents, but\nit could be very irregular if both motions are present and\nthe ratio of their periods is irrational. Indeed, this os-\ncillation was observed in a recent experiment[3]. How\ncan one understand the wire-width dependence of the\nDW velocity? According to Eq. (6), the velocity is a\nfunctional of DW structure which is very sensitive to the\nwire width. For a very narrow wire, only transverse DW\nis possible while a vortex DW is preferred for a wide wire\n(large than DW width). Different vortexes yield differ-\nent values of |/vector m×/vectorHeff|, which in turn results in different\nDW propagation speed.\n/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48 /s49/s48/s48/s48 /s49/s50/s48/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s50/s48/s48/s45/s49/s48/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s32/s32 \n/s68/s87 /s32/s118 /s32\n/s40/s109/s47/s115/s41\n/s72/s32/s40/s79/s101/s41/s32/s118/s40/s109/s47/s115/s41\n/s116/s32/s40/s110/s115/s41/s118/s40/s109/s47/s115/s41/s32\n/s116/s32/s40/s110/s115/s41\nFIG. 2: The time-averaged DW propagation speed versus\nthe applied magnetic field for a biaxial magnetic nanowire\nof cross section 4 nm×20nm. The wire parameters are\nK1=K2= 105J/m3,J= 4.×10−11J/m,M= 106A/m,\nandα= 0.1. Cross are for the calculated velocities from Eq.\n(7), and the open circles are for the simulated average veloc i-\nties. The dashed straight line is the fit to the small H < H W\nresults, and solid curve is the fit to a(H−H0)2/H+b/H.\nInsets: the instantaneous DW speed calculated from Eq. (6)\nforH= 50Oe < H W(left) and H= 1000Oe > H W(right).\nTime averaged velocity is\n¯v=αγ\n2(1+α2)HA/integraldisplay\nIII/parenleftBig\n/vector m×/vectorHeff/parenrightBig2\nd3/vector x,(7)\nwherebardenotestimeaverage. Itsaysthattheaveraged\nvelocity is proportional to the energy dissipation rate. In\norder to show that both Eqs. (6) and (7) are useful in\nevaluating the DW propagation speed from a DW struc-ture. We use OOMMF package to find the DW struc-\ntures and then use Eq. (7) to obtain the averagevelocity.\nFigure 2 is the comparison of such calculated velocities\n(cross) and numerical simulation (open circles with their\nerror bars smaller than the symbol sizes) for a magnetic\nnanowire of cross-section dimension 4 nm×20nmwith\na biaxial magnetic anisotropy f=−K1\n2M2\nz+K2\n2M2\nx.\nThe system parameters are K1=K2= 105J/m3,\nJ= 4.×10−11J/m,M= 106A/m, andα= 0.1. The\ngood overlap between the cross and open circles confirm\nthe correctness of Eq. (7). The ¯ v−Hcurve for H > H W\ncan be fit well by a∆(H−H0)2/H+b/H(see discus-\nsion later). The insets are instantaneous DW propaga-\ntion velocities for both H < H WandH > H W, by Eq.\n(6) from the instantaneous DW structures obtained from\nOOMMF. The left inset is the instantaneous DW speed\natH= 50Oe < H W, reaching its steady value in about\n1ns. The right inset is the instantaneous DW speed at\nH= 1000Oe > H W, showingclearlyanoscillation. They\nconfirm that the theory is capable of capturing all the\nfeatures of DW propagation.\nThe right side of Eq. (7) is positive and non-zero\nsince a time dependent DW requires /vector m×/vectorHeff/negationslash= 0,\nimplying a zero intrinsic critical field for DW propa-\ngation. If the DW keep its static structure, then the\nfirst term in the right side of Eq. (6) shall be pro-\nportional to a∆AH2, where ais a numerical number\nof order of 1 that depends on material parameters and\nthe DW structure. This is because the effective field\ndue to the exchange energy and magnetic anisotropy\nis parallel to /vectorM, and does not contribute to the en-\nergy dissipation. Thus, in this case, v=aαγ∆\n1+α2H\nwithµ=aαγ∆\n1+α2. Consider the Walker’s 1D model[6]\nin which f=−K1\n2M2cos2θ+K2\n2M2sin2θcos2φ,here\nK1andK2describe the easy and hard axes, respec-\ntively. From Walker’s trial function of a DW of width\n∆, lntanθ(z,t)\n2=1\n∆(t)/bracketleftBig\nz−/integraltextt\n0v(τ)dτ/bracketrightBig\nandφ(z,t) =φ(t),\none has (from Eq. (3)) the energy dissipation rate\ndE\ndt=−2αγA∆\n1+α2/bracketleftbig\nK2\n2M3sin2φcos2φ+H2M/bracketrightbig\n,(8)\nand DW energy change rate is\ndEIII\ndt=d\ndt/integraldisplay\nIIIF(θ,φ,/vector∇θ,/vector∇φ)d3/vector x=−4JA·˙∆\n∆2.(9)\nSubstituting Eqs. (8) and (9) into Eq. (6), one can easily\nreproduceWalker’sDWvelocityexpressionfor both H <\nHWand≫HW. For example, for H < H W=αK2M/2\nand ∆ = const., Eq. (6) gives\nv=α∆γ\n1+α2/bracketleftBigg\n1+/parenleftbiggK2Msinφcosφ\nH/parenrightbigg2/bracketrightBigg\nH.(10)\nThis velocity expression is the same as that of the\nSlonczewski model[9] for a one-dimensional wire. In4\nWalker’s analysis, φis fixed by K2andHthrough\nK2Msinφcosφ=H\nα. Using this φin the above ve-\nlocity expression, Walker’s mobility coefficient µ=γ∆\nα\nis recovered. This inverse damping relation is from the\nparticularpotentiallandscape in φ-direction. One should\nexpect different result if the shape of the potential land-\nscape is changed. Thus, this expression should not be\nused to extract the damping constant[1, 3].\nA DW may precess around the wire axis as well as\nbe substantially distorted from its static structure when\nH > H Was it was revealed in Walker’s analysis. Ac-\ncording to the minimum energy dissipation principle[13],\na DW will arrange itself as much as possible to satisfy\nEq. (2). Thus, the distortion is expected to absorb part\nofH. The precession motion shall induce an effective\nfieldg(φ) in the transverse direction, where gdepends\non the magnetic anisotropy in the transverse direction.\nOne may expect /vector m×/vectorHeff≃(H−H0)sinθˆz+sinθg(φ)ˆy,\nwhereH0istheDWdistortionabsorbedpartof H. Using\n|/vector m×/vectorHeff|2= (H−H0)2sin2θ+g2sin2θin Eq. (7), the\nDWpropagatingspeed takesthe followingh-dependence,\nv=aαγ∆(H−H0)2/H(1+α2)+bαγ∆/[H(1+α2)], linear\nin both ∆ and HforH≫H0, but a smaller DW mobil-\nity. This field-dependence is supported by the excellent\nfit in Fig. 2 for H > H W. The reasoning agrees also\nwith the minimum energy dissipation principle[13] since\n|/vector m×Heff|=Hsinθwhen/vectorMforH= 0 is used, and any\nmodification of /vectorMshould only make |/vector m×Heff|smaller.\nThe smaller mobility at H≫HW,H0leads naturally to\na negative differential mobility between H < H Wand\nH≫HW! In other words, the negative differential mo-\nbility is due to the transition of the DW from a high\nenergy dissipation structure to a lower one. This picture\ntells us that one should try to make a DW capable of\ndissipating as much energy as possible if one wants to\nachieve a high DW velocity. This is very different from\nwhat people would believefrom Walker’sspecial mobility\nformula of inverse proportion of the damping constant.\nTo increase the energy dissipation, one may try to re-\nduce defects and surface roughness. The reason is, by\nminimum energy dissipation principle, that defects are\nextra freedoms to lower |/vector m×/vectorHeff|because, in the worst\ncase, defects will not change |/vector m×/vectorHeff|when/vectorMwithout\ndefects are used.\nThe correctness of our central result Eq. (6) depends\nonly on the LLG equation, the general energy expres-\nsion of Eq. (1), and the fact that a static magnetic field\ncan be neither an energy source nor an energy sink of a\nsystem. It does not depend on the details of a DW struc-\nture aslong asthe DWpropagationis induced by a static\nmagnetic field. In this sense, our result is very general\nand robust, and it is applicable to an arbitrary magnetic\nwire. However, it cannot be applied to a time-dependent\nfield orthe current-inducedDWpropagation,atleastnotdirectly. Also, it may be interesting to emphasize that\nthere is no inertial in the DW motion within LLG de-\nscription since this equation contains only the first order\ntime derivative. Thus, there is no concept of mass in this\nformulation.\nIn conclusion, a global view of the field-induced DW\npropagation is provided, and the importance of energy\ndissipation in the DW propagation is revealed. A gen-\neral relationship between the DW velocity and the DW\nstructure is obtained. The result says: no damping, no\nDW propagation along a magnetic wire. It is shown that\nthe intrinsic critical field for a HH DW is zero. This\nzero intrinsic critical field is related to the absence of a\nstatic HH or a TT DW in a magnetic field parallel to\nthe nanowire. Thus, a non-zero critical field can only\ncome from the pinning of defects or surface roughness.\nThe observed negative differential mobility is due to the\ntransition of a DW from a high energy dissipation struc-\nture to a low energy dissipation structure. Furthermore,\nthe DW velocity oscillation is attributed to either the\nDW precession around wire axis or from the DW width\noscillation.\nThis work is supported by Hong Kong UGC/CERG\ngrants (# 603007 and SBI07/08.SC09).\n[1] T. Ono, H. Miyajima, K. Shigeto, K. Mibu, N. Hosoito,\nand T. Shinjo, Science 284, 468 (1999).\n[2] D. Atkinson, D.A. Allwood, G. Xiong, M.D. Cooke, C.\nFaulkner, and R.P. Cowburn, Nat. Mater. 2, 85 (2003).\n[3] G.S.D. Beach, C. Nistor, C. Knutson, M. Tsoi, and J.L.\nErskine, Nat. Mater. 4, 741 (2005); J. Yang, C. Nistor,\nG.S.D. Beach, and J.L. Erskine, Phys. Rev. B 77, 014413\n(2008).\n[4] M. Hayashi, L. Thomas, Y.B. Bazaliy, C. Rettner, R.\nMoriya, X. Jiang, and S.S.P. Parkin, Phys. Rev. Lett.\n96, 197207 (2006).\n[5] G.S.D. Beach, C. Knutson, C. Nistor, M. Tsoi, and J.L.\nErskine, Phys. Rev. Lett. 97, 057203 (2006).\n[6] N.L. Schryer and L.R. Walker, J. of Appl. Physics, 45,\n5406 (1974).\n[7] T. L. Gilbert, Phys. Rev. 100, 1243 (1955).\n[8] Z.Z. Sun and X.R. Wang, Phys. Rev. Lett. 97, 077205\n(2006); X.R. Wang and Z.Z. Sun, ibid98, 077201 (2007).\n[9] A.P.Malozemoff andJ.C. Slonczewski, Magnetic Domain\nWalls in Bubble Material (Academica, New York, 1979).\n[10] A. Thiaville and Y. Nakatani in Spin Dynamics in Con-\nfined Magnetic Structures III Eds. B. Hillebrands and A.\nThiaville, Springer 2002.\n[11] X. R. Wang, and Q. Niu, Phys. Rev. B 59, R12755\n(1999); Z.Z. Sun, H.T. He, J.N. Wang, S.D. Wang, and\nX.R. Wang, Phys. Rev. B 69, 045315 (2004).\n[12] Z.Z. Sun, and X.R. Wang, Phys. Rev. B 71, 174430\n(2005);73, 092416 (2006); 74132401 (2006).\n[13] T. Sun, P. Meakin, and T. Jssang, Phys. Rev. E 51, 5353\n(1995)."    },    {        "title": "1805.11468v1.Gilbert_damping_in_non_collinear_magnetic_system.pdf",        "content": "arXiv:1805.11468v1  [cond-mat.mtrl-sci]  29 May 2018APS/123-QED\nGilbert damping in non-collinear magnetic systems\nS. Mankovsky, S. Wimmer, H. Ebert\nDepartment of Chemistry/Phys. Chemistry, LMU Munich,\nButenandtstrasse 11, D-81377 Munich, Germany\n(Dated: May 30, 2018)\nThe modification of the magnetization dissipation or Gilber t damping caused by an inhomoge-\nneous magnetic structure and expressed in terms of a wave vec tor dependent tensor α(/vector q) is in-\nvestigated by means of linear response theory. A correspond ing expression for α(/vector q) in terms of\nthe electronic Green function has been developed giving in p articular the leading contributions to\nthe Gilbert damping linear and quadratic in q. Numerical results for realistic systems are pre-\nsented that have been obtained by implementing the scheme wi thin the framework of the fully\nrelativistic KKR (Korringa-Kohn-Rostoker) band structur e method. Using the multilayered system\n(Cu/Fe 1−xCox/Pt)nas an example for systems without inversion symmetry we demo nstrate the\noccurrence of non-vanishing linear contributions. For the alloy system bcc Fe 1−xCoxhaving inver-\nsion symmetry, on the other hand, only the quadratic contrib ution is non-zero. As it is shown, this\nquadratic contribution does not vanish even if the spin-orb it coupling is suppressed, i.e. it is a direct\nconsequence of the non-collinear spin configuration.\nPACS numbers: 71.15.-m,71.55.Ak, 75.30.Ds\nI. INTRODUCTION\nThe magnetization dissipation in magnetic materi-\nals is conventionally characterized by means of the\nGilbert damping (GD) tensor αthat enters the Landau-\nLifshitz-Gilbert (LLG) equation [1]. This positive-\ndefinite second-rank tensor depends in general on the\nmagnetization direction. It is well established that in\nthe case of spatially uniformly magnetized ferromagnetic\n(FM) metals two regimes of slow magnetization dynam-\nics can be distinguished, which are governed by differ-\nent mechanisms of dissipation [2–4]: a conductivity-like\nbehaviour occuring in the limiting case of ordered com-\npounds that may be connected to the Fermi breathing\nmechanism and a resistivity-likebehaviourshown by ma-\nterials with appreciable structural, chemical or tempera-\nture induced disorder and connected to a spin-flip scat-\nteringmechanism. Animportantissueisthatbothmech-\nanisms are determined by the spin-orbit coupling in the\nsystem (see e.g. [2, 4, 5]). During the last years, it was\ndemonstrated by variousauthors that first-principles cal-\nculationsforthe GD parameterforcollinearferromagntic\nmaterials allow to cover both regimes without use of any\nphenomenological parameters. In fact, in spite of the dif-\nferences concerning the formulation for the damping pa-\nrameter and the corresponding implementaion [6–8], the\nnumerical results are in generalin rathergood agreement\nwith each other as well as with experiment.\nIn the case of a pronounced non-collinear magnetic\ntexture, e.g. in the case of domain walls or topologi-\ncally nontrivial magnetic configurations like skyrmions,\nthe description of the magnetization dissipation assum-\ning a spatial-invariant tensor αis incomplete, and a non-\nlocal character of GD tensor in such systems has to be\ntaken into account [9–11]. This implies that the dissipa-\ntive torque on the magnetization should be representedby the expression of the following general form [12]:\nτGD= ˆm(/vector r,t)×/integraldisplay\nd3r′α(/vector r−/vector r′)∂\n∂tˆm(/vector r′,t).(1)\nIn the case of a magnetic texture varying slowly in space,\nhowever, an expansion of the damping parameter in\nterms of the magnetization density and its gradients [11]\nis nevertheless appropriate:\nαij=αij+αkl\nijmkml+αklp\nijmk∂\n∂rlmp(2)\n+αklpq\nij∂\n∂rkml∂\n∂rpmq+... ,\nwhere the first term αijstands for the conventional\nisotropic GD and the second term αkl\nijmkmlis associated\nwith the magneto-crystalline anisotropy (MCA). The\nthird so-called chiral term αklp\nijmk∂\n∂rlmpis non-vanishing\nin non-centrosymmetric systems. The important role of\nthis contribution to the damping was demonstrated ex-\nperimentally when investigating the field-driven domain\nwall(DW)motioninasymmetricPt/Co/Pttrilayers[13].\nAs an alternative to the expansion in Eq. (2) one can\ndiscuss the Fourier transform α(/vector q) of the damping pa-\nrametercharacterizinginhomogeneousmagneticsystems,\nwhich enter the spin dynamics equation\n∂\n∂t/vector m(/vector q) =−γ/vector m(/vector q)×/vectorH−/vector m(/vector q)×α(/vector q)∂\n∂t/vector m(/vector q).(3)\nIn this formulation the term linear in qis the first chiral\nterm appearing in the expansion of α(/vector q) in powers of q.\nFurthermore, it is important to note that it is directly\nconnected to the αklp\nijmk∂\n∂rlmpterm in Eq. (2).\nBy applying a gauge field theory, the origin of the\nnon-collinear corrections to the GD can be ascribed to\nthe emergent electromagnetic field created in the time-\ndependent magnetic texture [14, 15]. Such an emergent2\nelectromagneticfieldgivesrisetoaspincurrentwhosedi-\nvergence characterizes the change of the angular momen-\ntum in the system. This allows to discuss the impact of\nnon-collinearity on the GD via a spin-pumping formula-\ntion[9,14,16]. Somedetailsofthephysicsbehind thisef-\nfect depend on the specific propertiesofthe materialcon-\nsidered. Accordingly, different models for magnetisation\ndissipation were discussed in the literature [9, 12, 14, 17–\n19]. Non-centrosymmetric two-dimensional systems for\nwhich the Rashba-like spin-orbit coupling plays an im-\nportant role havereceived special interest in this context.\nThey have been discussed in particular by Akosa et al.\n[19], in order to explain the origin of chiral GD in the\npresence of a chiral magnetic structure.\nThe fourth term on the r.h.s. of Eq. (2) corresponds\nto a quadratic term of an expansion of α(/vector q) with re-\nspect to q. It was investigated for bulk systems with\nnon-magnetic [20] and magnetic [9] impurity atoms, for\nwhich the authors have shown on the basis of model con-\nsideration that it can give a significant correction to the\nhomogeneous GD in the case of weak metallic ferromag-\nnets. In striking contrast to the uniform part of the GD\nthis contribution does not require a non-vanishing spin-\norbit interaction.\nTo our knowledge, only very few ab-initio investiga-\ntions on the Gilbert damping in non-collinear magnetic\nsystems along the lines sketched above have been re-\nported so far in the literature. Yuan et al. [21] calcu-\nlated the in-plane and out-of-plane damping parameters\nin terms of the scattering matrix for permalloy in the\npresence of N´ eel and Bloch domain walls. Freimuth et\nal. [22], discuss the properties of a q-dependent Gilbert\ndamping α(/vector q) calculated for the one-dimensional Rashba\nmodelinthepresenceofthe N´ eel-typenon-collinearmag-\nnetic exchange field, demonstrating different GD for left-\nhanded and right-handed DWs. Here we extend the for-\nmalism developed before to deal with the GD in ferro-\nmagnets [6], to get access to non-collinear system. The\nformalism based on linear response theory allows to ex-\npand the GD parameters with respect to a modulation\nof the magnetization expressed in terms of a wave vector\n/vector q. Correspondingnumerical results will be presented and\ndiscussed.\nII. GILBERT DAMPING FOR\nNON-COLLINEAR MAGNETIZATION\nIn the following we focus on the intrinsic contribution\nto the Gilbert damping, excluding spin current induced\nmagnetizationdissipationwhich occursin the presenceof\nan external electric field. For the considerations on the\nmagnetization dissipation an adiabatic variation of the\nmagnetization in the time and space domain is assumed.\nMoreover, it is assumed that the magnitude of the local\nmagnetic moments is unchanged during a change of the\nmagnetization, i.e. the exchange field should be strong\nenough to separate transverse and longitudinal parts ofthe magnetic susceptibility. With these restrictions, the\nnon-local Gilbert damping can be determined in terms of\nthe spin susceptibility tensor\nχαβ(/vector q,ω) =i1\nV∞/integraldisplay\n0dt∝angbracketleftˆSα(/vector q,t)ˆSα(−/vector q,0)∝angbracketright0ei(ω−δ)t,(4)\nwhereˆSα(/vector q,t) is the /vector q- andt-dependent spin operator\nand reduced units havebeen used ( /planckover2pi1= 1). With this, the\nFourier transformationofthe real-spaceGilbert damping\ncan be represented by the expression [23, 24]\nααβ(/vector q) =γ\nM0Vlim\nω→0∂ℑ[χ−1]αβ(/vector q,ω)\n∂ω.(5)\nHereγ=gµBis the gyromagneticratio, M0=µtotµB/V\nis the equilibrium magnetization and Vis the volume of\nthe system. In order to avoid the calculation of the dy-\nnamical magnetic susceptibility tensor χ(/vector q,ω), which is\nthe Fourier transformed of the real space susceptibility\nχ(/vector r−/vector r′,ω), it is convenient to represent χ(/vector q,ω) in Eq.\n(5), in terms of a correlation function of time deriva-\ntives ofˆS. As˙ˆScorresponds to the torque /vectorT, that may\ninclude non-dissipative and dissipative parts, one may\nconsider instead the torque-torque correlation function\nπ(/vector q,ω) [24–27].\nAssuming the magnetization direction parallelto ˆ zone\nobtains the expression for the Gilbert damping α(/vector q)\nα(/vector q) =γ\nM0Vlim\nω→0∂ℑ[ǫ·π(/vector q,ω)·ǫ]\n∂ω. (6)\nwhereǫ=/bracketleftbigg\n0 1\n−1 0/bracketrightbigg\nis the transverse Levi-Civita tensor.\nThisimpliesthefollowingrelationshipofthe αtensorele-\nments with the elements of the torque-torque correlation\ntensorπ:αxx∼ −πyyandαyy∼ −πxx[24].\nUsing Kubo’s linear response theory in the Matsubara\nrepresentation and taking into account the translational\nsymmetry of a solid the torque-torque correlation func-\ntionπαβ(/vector q,ω) can be expressed by (see, e.g. [28]):\nπαβ(/vector q,iωn) =1\nβ/summationdisplay\npm∝angbracketleftTαG(/vectork+/vector q,iωn+ipm)\nTβG(/vectork,ipm)∝angbracketrightc,(7)\nwhereG(/vectork,ip) is the Matsubara Green function and ∝angbracketleft...∝angbracketrightc\nindicates a configurational average required in the pres-\nence of any disorder (chemical, structural or magnetic)\nin the system. Using a Lehman representation for the\nGreen function [28]\nG(/vectork,ipm) =/integraldisplay+∞\n−∞dE\nπℑG+(/vectork,E)\nipm−E(8)\nwithG+(/vectork,E) the retarded Green function and using the\nrelation\n1\nβ/summationdisplay\npm1\nipm+iωn−E11\nipm−E2=f(E2)−f(E1)\niωn+E2−E13\nfor the sum over the Matsubara poles in Eq. (7), the torque-torq ue correlation function is obtained as:\nπαβ(/vector q,iωn) =1\nΩBZ/integraldisplay\nd3k+∞/integraldisplay\n−∞dE1\nπ+∞/integraldisplay\n−∞dE2\nπTr/angbracketleftbigg\nTαℑG(/vectork,E1)TβℑG(/vectork,E2)f(E2)−f(E1)\niωn+E2−E1/angbracketrightbigg\nc. (9)\nPerfoming finally the analytical continuation iωn→ω+iδone arrives at the expression\nΓαβ(/vector q,ω) =−π\nΩBZ/integraldisplay\nd3k+∞/integraldisplay\n−∞dE1\nπ+∞/integraldisplay\n−∞dE2\nπTr/angbracketleftbigg\nTαℑG(/vectork+/vector q,E1)TβℑG(/vectork,E2)/angbracketrightbigg\nc(f(E2)−f(E1))δ(ω+E2−E1)\n=−π\nΩBZ/integraldisplay\nd3k+∞/integraldisplay\n−∞dE\nπTr/angbracketleftbigg\nTαℑG(/vectork+/vector q,E)TβℑG(/vectork,E+ω)/angbracketrightbigg\nc(f(E)−f(E+ω)) (10)\nfor the imaginary part of the correlation function with Γ αβ(/vector q,ω) =−πℑπαβ(/vector q,ω). Accordingly one gets for the\ndiagonal elements of Gilbert damping tensor the expression\nααα(/vector q) =γ\nM0Vlim\nω→0∂[ǫ·Γ(/vector q,ω)·ǫ]\n∂ω/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nαα\n=γπ\nM0Vlim\nω→0∂\n∂ω1\nΩBZ/integraldisplay\nd3k+∞/integraldisplay\n−∞dE\nπ2(f(E+ω)−f(E))Tr/angbracketleftbigg\nTβℑG(/vectork+/vector q,E)TβℑG(/vectork,E+ω)/angbracketrightbigg\nc\n=γ\nM0V1\nΩBZ/integraldisplay\nd3k+∞/integraldisplay\n−∞dE\nπδ(E−EF)Tr/angbracketleftbigg\nTβℑG(/vectork+/vector q,E)TβℑG(/vectork,E)/angbracketrightbigg\nc\n=1\n4[ααα(/vector q,G+,G+)+ααα(/vector q,G−,G−)−ααα(/vector q,G+,G−)−ααα(/vector q,G−,G+)], (11)\nwhere the index βof the torque operator Tβis related to the index αaccording to Eq. 6, and the auxiliary functions\nααα(/vector q,G±,G±) =γ\nM0Vπ1\nΩBZ/integraldisplay\nd3kTr/angbracketleftbigg\nTβG±(/vectork+/vector q,EF)TβG±(/vectork,EF)/angbracketrightbigg\nc(12)\nexpressed in terms of the retarded and advanced Green function s,G+andG−, respectively.\nTo account properly for the impact of spin-orbit coupling when dealin g with Eqs. (11) and (12) a description of\nthe electronic structure based on the fully relativistic Dirac formalis m is used. Working within the framework of local\nspin density formalism (LSDA) this implies for the Hamiltonian the form [2 9]:\nˆHD=c/vectorα·/vector p+βmc2+V(/vector r)+β/vectorσ·ˆ/vector mBxc(/vector r). (13)\nHereαiandβare the standard Dirac matrices, /vectorσdenotes the vector of relativistic Pauli matrices, /vector pis the relativistic\nmomentum operator [30] and the functions V(/vector r) and/vectorBxc=/vectorσ·ˆ/vector mBxc(/vector r) are the spin-averaged and spin-dependent\nparts, respectively, of the LSDA potential [31] with ˆ/vector mgiving the orientation of the magnetisation.\nWith the Dirac Hamiltonian given by Eq. (13), the torque operator ma y be written as /vectorT=β[/vector σ׈/vector m]Bxc(/vector r).\nFurthermore, the Green functions entering Eqs. (11) and (12) a re determined using the spin-polarized relativistic\nversion of multiple scattering theory [29, 32] with the real space re presentation of the retarded Green function given\nby:\nG+(/vector r,/vector r′,E) =/summationdisplay\nΛΛ′Zn\nΛ(/vector r,E)τnm\nΛΛ′(E)Zm×\nΛ′(/vector r′,E)\n−δnm/summationdisplay\nΛ/bracketleftbig\nZn\nΛ(/vector r,E)Jn×\nΛ′(/vector r′,E)Θ(r′\nn−rn)\n+Jn\nΛ(/vector r,E)Zn×\nΛ′(/vector r′,E)Θ(rn−r′\nn)/bracketrightbig\n. (14)4\nHere/vector r,/vector r′refertoatomiccellscenteredatsites nandm, respectively,where Zn\nΛ(/vector r,E) =ZΛ(/vector rn,E) =ZΛ(/vector r−/vectorRn,E) isa\nfunction centered at the corresponding lattice vector /vectorRn. The four-component wave functions Zn\nΛ(/vector r,E) (Jn\nΛ(/vector r,E)) are\nregular (irregular) solutions to the single-site Dirac equation labeled by the combined quantum numbers Λ = ( κ,µ),\nwithκandµbeing the spin-orbit and magnetic quantum numbers [30]. Finally, τnm\nΛΛ′(E) is the so-called scattering\npath operator that transfers an electronic wave coming in at site minto a wave going out from site nwith all possible\nintermediate scattering events accounted for.\nUsing matrix notation with respect to Λ, this leads to the following exp ression for the auxilary damping parameters\nin Eq. (12):\nααα(/vector q,G±,G±) =γ\nM0Vπ1\nΩBZ/integraldisplay\nd3kTr/angbracketleftbigg\nTβτ(/vectork+/vector q,E±\nF)Tβτ(/vectork,E±\nF)/angbracketrightbigg\nc. (15)\nIn the case of a uniform magnetization, i.e. for q= 0 one obviously gets an expression for the Gilbert damping tensor\nas it was worked out before [7]. Assuming small wave vectors, the te rmτ(/vectork+/vector q,E±\nF) can be expanded w.r.t. /vector qleading\nto the series\nτ(/vectork+/vector q,EF) =τ(/vectork,E)+/summationdisplay\nµ∂τ(/vectork,E)\n∂kµqα+1\n2/summationdisplay\nµν∂τ(/vectork,E)\n∂kµ∂kνqµqν+... (16)\nthat results in a corresponding expansion for the Gilbert damping:\nα(/vector q) =α+/summationdisplay\nµαµqµ+1\n2/summationdisplay\nµναµνqµqν+... (17)\nwith the following expansion coefficients:\nα0±±\nαα=g\nπµtot1\nΩBZTrace/integraldisplay\nd3k/angbracketleftbigg\nTβτ(/vectork,E±\nF)Tβτ(/vectork,E±\nF)/angbracketrightbigg\nc(18)\nαµ±±\nαα=g\nπµtot1\nΩBZTrace/integraldisplay\nd3k/angbracketleftbigg\nTβ∂τ(/vectork,E±\nF)\n∂kµTβτ(/vectork,E±\nF)/angbracketrightbigg\nc(19)\nαµν±±\nαα=g\nπµtot1\n2ΩBZTrace/integraldisplay\nd3k/angbracketleftbigg\nTβ∂2τ(/vectork,E±\nF)\n∂kµ∂kνTβτ(/vectork,E±\nF)/angbracketrightbigg\nc, (20)\nand with the g-factor 2(1+ µorb/µspin) in terms of the spin and orbital moments, µspinandµorb, respectively, and the\ntotal magnetic moment µtot=µspin+µorb. The numerically cumbersome term in Eq. (20), that involves the sec ond\norder derivative of the matrix of /vectork-dependent scattering path operator τ(/vectork,E), can be reformulated by means of an\nintegration by parts:\n1\nΩBZ/integraldisplay\nd3kTβτ(/vectork,EF)Tβ∂2τ(/vectork,EF)\n∂kµ∂kν=/bracketleftBigg/integraldisplay /integraldisplay\ndkβdkγTi\nβτ(/vectork,E)Tj\nβ∂τ(/vectork,E)\n∂kβ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleKα\n2\n−Kα\n2/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\n=0\n−/integraldisplay /integraldisplay /integraldisplay\ndkαdkβdkγTβ∂τ(/vectork,EF)\n∂kµTβ∂τ(/vectork,EF)\n∂kν/bracketrightBigg\n=−1\nΩBZ/integraldisplay\nd3kTβ∂τ(/vectork,EF)\n∂kµTβ∂τ(/vectork,EF)\n∂kν\nleading to the much more convenient expression:\nαµν±±\nαα=−g\n2πµtot/integraldisplay\nd3kTr/angbracketleftbigg\nTβ∂τ(/vectork,E±\nF)\n∂kµTβ∂τ(/vectork,E±\nF)\n∂kν/angbracketrightbigg\nc. (21)\nIII. RESULTS AND DISCUSSIONS\nThe scheme presented above to deal with the Gilbert\ndamping in non-collinear systems has been implementedwithin the SPR-KKR program package [33]. To exam-5\nine the importance of the chiral correction to the Gilbert\ndamping a first application of Eq. (19) has been made\nfor the multilayer system (Cu/Fe 1−xCox/Pt)nseen as a\nnon-centrosymmetricmodelsystem. Thecalculatedzero-\norder (uniform) GD parameter αxxand the correspond-\ning first-order (chiral) αx\nxxcorrection term for /vector q∝bardblˆxare\nplotted in Fig. 1 top and bottom, respectively, as a func-\ntion of the Fe concentration x. Both terms, αxxandαx\nxx,\n0 0.2 0.4 0.6 0.8 100.20.4αxx\n0 0.2 0.4 0.6 0.8 1xCo0123αxxx (a.u.)\nFIG. 1: The Gilbert damping parameters αxx(top) and\nαx\nxx(bottom) calculated for the model multilayer system\n(Cu/Fe 1−xCox/Pt)nusing Eqs. (18) and (19), respectively.\nincrease approaching the pure limits w.r.t. the Fe 1−xCox\nalloy subsystem. In the case of the uniform parame-\nterαxx, this increase is associated with the dominating\nbreathing Fermi-surface damping mechanism. This im-\nplies that the modification of the Fermi surface (FS) in-\nduced by the spin-orbit coupling (SOC) follows the mag-\nnetization direction that slowly varies with time. An ad-\nditional contribution to the GD, having a similar origin,\noccurs for the non-centrosymmertic systems with heli-\nmagnetic structure. In this case, the features of the elec-\ntronicstructure governedby the lackofinversionsymme-\ntry result in a FS modification dependent on the helicity\nof the magnetic structure. This implies a chiral contri-\nbution to the GD which can be associated with the term\nproportional to the gradient of the magnetization. Ob-\nviously, this additional modification of the FS and the\nassociated mechanism for the GD does not show up for\na uniform ferromagnet. As αis caused by the SOC one\ncan expect that it vanishes for vanishing SOC. This was\nindeed demonstrated before [5]. The same holds also for\nαxthat is cased by SOC as well.\nAnother system considered is the ferromagnetic alloy\nsystem bcc Fe 1−xCox. As this system has inversion sym-\nmetry the first-order term αµshould vanish. This expec-\ntation could also be confirmed by calculations that ac-count for the SOC. The next non-vanishing term of the\nexpansion of the GD is the term ∝q2. The correspond-\ning second-order term αxx\nxxis plotted in Fig. 2 (bottom)\ntogether with the zero-order term αxx(top). The bot-\n0 0.1 0.2 0.3 0.4 0.500.511.52αxx× 103\n0 0.1 0.2 0.3 0.4 0.5xCo012αxxxx ((a.u.)2)Fe1-xCox\nFIG. 2: The Gilbert damping terms αxx(top) and αxx\nxx(bot-\ntom) calculated for bcc Fe 1−xCox.\ntom panel shows in addition results for αxx\nxxthat have\nbeen obtained by calculations with the SOC suppressed.\nAs one notes the results for the full SOC and for SOC\nsuppressed are very close to each other. The small dif-\nference between the curves for that reason have to be as-\ncribed to the hybridization of the spin-up and spin-down\nsubsystems due to SOC. As discussed in the literature\n[9, 17, 20] a non-collinear magnetic texture has a corre-\nsponding consequence but a much stronger impact here.\nIn contrastto the GDin uniform FM systemswhereSOC\nisrequiredto breakthe totalspin conservationin the sys-\ntem,αxx\nxxis associated with the spin-pumping effect that\ncan be ascribed to an emergent electric field created in\nthe non-uniform magnetic system. In this case magnetic\ndissipation occurs due to the misalignment of the elec-\ntron spin following the dynamic magnetic profile and the\nmagnetization orientation at each atomic site, leading to\nthe dephasing of electron spins [16]\nIV. SUMMARY\nTo summarize, expressions for corrections to the GD\nofhomogeneoussystems werederived which areexpected\nto contribute in the case of non-collinear magnetic sys-\ntems. The expression for the GD parameter α(/vector q) seen\nas a function of the wave vector /vector qis expanded in powers\nofq. In the limit of weakly varying magnetic textures,\nthis leads to the standard uniform term, α, and the first-\nand second-order corrections, αµandαµν, respectively.\nModel calculations confirmed that a non-vanishing value6\nforαµcan be expected for systems without inversion\nsymmetry. In addition, SOC has been identified as the\nmajor source for this term. The second-order term, on\nthe other hand, may also show up for systems with inver-\nsion symmetry. In this case it was demonstrated by nu-\nmerical work, that SOC plays only a minor role for αµν,\nwhile the non-collinearity of the magnetization plays the\ncentral role.V. ACKNOWLEDGEMENT\nFinancial support by the DFG via SFB 1277 (Emer-\ngente relativistische Effekte in der Kondensierten Ma-\nterie) is gratefully acknowledged.\n[1] T. L. Gilbert, IEEE Transactions on Magnetics 40, 3443\n(2004).\n[2] V. 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Mahan, Many-particle physics , Physics\nof Solids and Liquids (Springer, New York,\n2000), ISBN ISBN 978-1-4757-5714-9, URL\nhttps://openlibrary.org/works/OL4474340W/Many-parti cle_physics .\n[29] H. Ebert, in Electronic Structure and Physical Properties\nof Solids , edited by H. Dreyss´ e (Springer, Berlin, 2000),\nvol. 535 of Lecture Notes in Physics , p. 191.\n[30] M. E. Rose, Relativistic Electron The-\nory(Wiley, New York, 1961), URL\nhttp://openlibrary.org/works/OL3517103W/Relativisti c_electron_theory .\n[31] A. H. MacDonald and S. H. Vosko, J. Phys.\nC: Solid State Phys. 12, 2977 (1979), URL\nhttp://iopscience.iop.org/0022-3719/12/15/007 .\n[32] H. Ebert, J. Braun, D. K¨ odderitzsch, and\nS. Mankovsky, Phys. Rev. B 93, 075145 (2016), URL\nhttp://link.aps.org/doi/10.1103/PhysRevB.93.075145 .\n[33] H. Ebert et al., The Munich\nSPR-KKR package , version 7.7,\nhttp://olymp.cup.uni-muenchen.de/ak/ebert/SPRKKR\n(2017), URL http://olymp.cup.uni-muenchen.de/ak/ebert/SPRKKR ."    },    {        "title": "1804.03743v1.GONG_Catalog_of_Solar_Filament_Oscillations_Near_Solar_Maximum.pdf",        "content": "Draft version April 12, 2018\nPreprint typeset using L ATEX style emulateapj v. 04/17/13\nGONG CATALOG OF SOLAR FILAMENT OSCILLATIONS NEAR SOLAR MAXIMUM\nM. Luna1,2, J. Karpen3, J. L. Ballester4,5, K. Muglach3,6, J. Terradas4,5, T. Kucera3and H. Gilbert3\nDraft version April 12, 2018\nABSTRACT\nWe have catalogued 196 \flament oscillations from the GONG H\u000bnetwork data during several months\nnear the maximum of solar cycle 24 (January - June 2014). Selected examples from the catalog\nare described in detail, along with our statistical analyses of all events. Oscillations were classi\fed\naccording to their velocity amplitude: 106 small-amplitude oscillations (SAOs), with velocities <\n10 km s\u00001, and 90 large-amplitude oscillations (LAOs), with velocities >10 km s\u00001. Both SAOs and\nLAOs are common, with one event of each class every two days on the visible side of the Sun. For\nnearly half of the events we identi\fed their apparent trigger. The period distribution has a mean\nvalue of 58\u000615 min for both types of oscillations. The distribution of the damping time per period\npeaks at\u001c=P = 1:75 and 1:25 for SAOs and LAOs respectively. We con\frmed that LAO damping\nrates depend nonlinearly on the oscillation velocity. The angle between the direction of motion and\nthe \flament spine has a distribution centered at 27\u000efor all \flament types. This angle agrees with the\nobserved direction of \flament-channel magnetic \felds, indicating that most of the catalogued events\nare longitudinal (i.e., undergo \feld-aligned motions). We applied seismology to determine the average\nradius of curvature in the magnetic dips, R\u001989 Mm, and the average minimum magnetic-\feld\nstrength,B\u001916 G. The catalog is available to the community online, and is intended to be expanded\nto cover at least 1 solar cycle.\n1.INTRODUCTION\nFilament oscillations were \frst observed visually\n(Greaves, Newton, & Jackson, reported by Dyson 1930;\nNewton 1935; Bruzek 1951), followed by photographic\nobservations that revealed a signi\fcant relationship with\n\rares (Dodson (1949); Bruzek & Becker (1957) and\nBecker (1958)). Moreton & Ramsey (1960) con\frmed\nthat wave disturbances initiated during the impulsive\nphase of \rares were responsible for triggering prominence\noscillations both near and far from the \rare. Ramsey &\nSmith (1966) determined periods and damping times for\nseveral oscillating \flaments, but did not \fnd any correla-\ntion between the period or damping time and the dimen-\nsions of the \flament, the distance to the associated \rare,\nor its size. In these early observations, some events were\ncalled \\winking \flaments\" because these \flaments were\nvisible inH\u000bwhen they were at rest, but disappeared\nwhile oscillating. Because these observations were made\nwith narrow-band H\u000b\flters, Doppler-shifted absorption\nfrom prominence material traveling at su\u000eciently large\nline-of-sight (LOS) velocities ( >23 km s\u00001) fell outside\nthe 0.5 \u0017A bandpass of the \flter and thus became invisible\ninH\u000b.\nNowadays, thanks to both space- and ground-based\ninstruments, observations of large-amplitude \flament os-\ncillations (LAOs: v>10 km s\u00001) have become common.\n1Instituto de Astrof\u0013 \u0010sica de Canarias, E-38200 La Laguna,\nTenerife, Spain\n2Departamento de Astrof\u0013 \u0010sica, Universidad de La Laguna, E-\n38206 La Laguna, Tenerife, Spain\n3NASA Goddard Space Flight Center, Greenbelt, MD 20771,\nUSA\n4Departament de F\u0013 \u0010sica, Universitat de les Illes Balears\n(UIB), E-07122 Palma de Mallorca, Spain\n5Institute of Applied Computing & Community Code (IAC3),\nUIB, Spain\n6Catholic University of America, Washington, DC 20064, US(The terms \\large\" and \\small\" amplitude are de\fned\nlater in this Section.) The exciters identi\fed thus far in-\nclude Moreton or EIT waves (Eto et al. 2002; Okamoto\net al. 2004; Gilbert et al. 2008; Asai et al. 2012), EUV\nwaves (Liu et al. 2012; Shen et al. 2014a; Xue et al. 2014;\nTakahashi et al. 2015), shock waves (Shen et al. 2014b),\nnearby jets, sub\rares and \rares (Jing et al. 2003, 2006;\nVr\u0014 snak et al. 2007; Li & Zhang 2012), and the eruption\nof the \flament (Isobe & Tripathi 2006; Isobe et al. 2007;\nPouget 2007; Chen et al. 2008; Foullon et al. 2009; Boc-\nchialini et al. 2011).\nMany of the observed \rare-induced LAOs in \flaments\nexhibit motions in di\u000berent directions relative to the ax-\nial magnetic \feld (polarization). For instance, the mate-\nrial can undergo vertical (Eto et al. 2002; Okamoto et al.\n2004; Shen et al. 2014a), horizontal (Kleczek & Kupe-\nrus 1969; Hershaw et al. 2011; Gosain & Foullon 2012;\nLiu et al. 2012; Shen et al. 2014b), or longitudinal (\feld-\naligned) (Jing et al. 2003, 2006; Vr\u0014 snak et al. 2007; Li &\nZhang 2012; Zhang et al. 2012; Luna et al. 2014; Shen\net al. 2014b) motions. Oscillations with a mixed charac-\nter (Gilbert et al. 2008) have also been observed.\nThe \frst theoretical models proposed to explain the\nexcitation, restoring forces, and damping mechanisms of\nlarge-amplitude longitudinal oscillations were purely an-\nalytical (Hyder 1966; Kleczek & Kuperus 1969). One-\ndimensional, hydrodynamic, numerical models have been\nemployed successfully to describe longitudinal oscilla-\ntions (Vr\u0014 snak et al. 2007; Luna & Karpen 2012; Luna\net al. 2012; Zhang et al. 2012, 2013; Ruderman & Luna\n2016; Zhou et al. 2017), while 2D and 3D MHD models\nhave described more completely the features of observed\nlongitudinal and transverse oscillations (Terradas et al.\n2013; Terradas et al. 2015, 2016; Luna et al. 2016).\nSpectroscopic techniques have revealed oscillations\nwith much smaller peak velocities than those of LAOs,arXiv:1804.03743v1  [astro-ph.SR]  10 Apr 20182 Luna et al.\nwith amplitudes from the noise level of 0 :1 km s\u00001to\n10 km s\u00001. Harvey (1969) \frst measured oscillatory pe-\nriods between 1 and 17 min, while later observations\nyielded characteristic periods ranging from a few to 90\nmin. Although the triggering mechanisms of these small-\namplitude oscillations (SAOs) have not been clearly iden-\nti\fed, they are generally believed to be excited by the\nperiodic motions of \flament magnetic \felds driven by\nphotospheric or chromospheric oscillations (see review by\nArregui et al. 2012).\nA variety of approaches has been used to categorize\nand understand \flament oscillations. The simplest are\nbased on a single property, such as the peak velocity\n(e.g., Arregui et al. 2012), the nature of the trigger (e.g.,\nOliver 1999; Oliver & Ballester 2002), or the period (e.g.,\nArregui et al. 2012). The apparent tendency of periods\nto group below 10 min, in the range 10 - 40 min, or 40 -\n90 min (Arregui et al. 2012) led to classi\fcations denoted\nas short-, intermediate- and long-period oscillations, re-\nspectively. Very short-periods below 1 min (Balthasar\net al. 1993), very long-periods above 5 hours (Foullon\net al. 2004; Pouget et al. 2006), and even periods longer\nthan 20 hours (Efremov et al. 2016) have been reported.\nClassi\fcation based only on the period does not re\rect\nthe nature, origin, or exciter of the oscillations, however.\nMore complex schemes have proven to be di\u000ecult to em-\nploy consistently (e.g., Vr\u0014 snak 1993).\nBecause oscillation velocities have been measured from\nthe observable threshold to 100 km s\u00001, the velocity\namplitude alone is not the most de\fnitive criterion by\nwhich oscillation events can be categorized. In spite of\nthese limitations, a widely accepted, velocity-based di-\nvision between small-amplitude and large-amplitude os-\ncillations has proven to be both convenient and physi-\ncally justi\fable. We can relate the observed oscillation\namplitudes to their linear or nonlinear character by con-\nsidering the characteristic Alfv\u0013 en and sound speeds in\nprominences, which are of the order of 100 km s\u00001and\n10 km s\u00001, respectively. Therefore, oscillations with ve-\nlocity amplitudes above 10 km s\u00001exceed the local sound\nspeed, and hence can be considered nonlinear oscilla-\ntions, while smaller velocity amplitudes would be lin-\near. In general, small-amplitude oscillations (SAOs) ex-\nhibit amplitudes below 10 km s\u00001, are not related to\n\rare activity, are local, and can be appropriately ana-\nlyzed or modeled using methods of linear perturbations.\nLAOs are usually associated with energetic events, are of\nglobal character, and as the velocity amplitude is \u001510-\n20 km s\u00001, require a nonlinear approach. As we demon-\nstrate in the present work, however, exceptions to these\n\\rules\" exist.\nProminence seismology aims to determine physical pa-\nrameters that are di\u000ecult to measure by direct means in\nthese magnetized plasma structures. This remote diag-\nnostics method combines observations of oscillations and\nwaves in these structures with theoretical results from\nthe analysis of oscillatory properties of given prominence\nmodels, as \frst suggested by Tandberg-Hanssen (1995).\nThe \frst seismological determinations of magnetic \feld\nstrength in winking \flaments used a simple model of lon-\ngitudinal motions based on a harmonic oscillator (Hyder\n1966; Kleczek & Kuperus 1969). Vr\u0014 snak et al. (2007) an-\nalyzed large-amplitude longitudinal oscillations (LALOs)in a prominence to infer the Alfv\u0013 en speed; assuming the\nmass density of the prominence plasma, they also deter-\nmined the azimuthal and axial magnetic \feld strengths.\nOur theoretical investigation of oscillations in simulated\nprominence threads strengthened the foundations of the\ndamped harmonic oscillator model for LALOs, providing\na basis for applications to observations (Luna & Karpen\n2012; Luna et al. 2012). Subsequent seismological anal-\nyses of LALOs in prominences have derived the radius\nof curvature of dipped \feld lines supporting prominence\nthreads, the minimum magnetic \feld strength, the en-\nergy injected by the triggering jet, and the mass accre-\ntion rate according to the thermal nonequilibrium model\n(Li & Zhang 2012; Bi et al. 2014; Luna et al. 2014, 2016;\nZhang et al. 2017b). Using the same seismological tech-\nniques, we determined the curvature radius of the mag-\nnetic \feld dips and the minimum \feld strength from the\nlargest prominence oscillation ever reported in the lit-\nerature; these results were validated by reconstructing\nthe \flament magnetic \feld from the photospheric \feld in\ncombination with the \rux-rope insertion method (Luna\net al. 2017).\nTo interpret observed prominence LAOs directed\ntransverse to the magnetic \feld, an MHD approach is\nrequired. Some observations of oscillatory behavior have\nbeen interpreted and analyzed as global or standing kink\nmodes (e.g., Hershaw et al. 2011; Liu et al. 2012; Xue\net al. 2014). A theoretical analysis predicted a linear re-\nlationship between the damping time ( \u001c) and the period\n(P) that could be compatible with resonant absorption\nas the damping mechanism (Ruderman & Roberts 2002;\nOfman & Aschwanden 2002; Arregui et al. 2008b). How-\never, this interpretation must be considered with care\nbecause the use of scaling laws to discriminate between\ndamping mechanisms is questionable, at least for res-\nonant absorption (Arregui et al. 2008a). Much work\nremains before the physical models of both longitudi-\nnal and transverse LAOs are su\u000eciently detailed and\ncomprehensive to adequately link theory and simulations\nwith observed prominence motions.\nTo date, all studies of oscillating prominences have\nbeen focused on one or, at most, a few episodes. In\norder to understand this phenomenon thoroughly and\nderive key physical characteristics via seismology of all\ntypes of prominences over the solar cycle, we have be-\ngun to compile a systematic, large data set of oscillation\nevents. Thus far we have identi\fed and analyzed 196\nevents during several months close to the maximum of\nsolar cycle 24, using GONG H\u000bdata. We found that\nLAOs are very common on the Sun (one event every\ntwo days on the visible hemisphere), and that the fre-\nquency of SAOs is similar to that of LAOs, yielding one\nSAO or LAO per day. Our large sample of prominence\noscillations has enabled the \frst statistically signi\fcant\nstudy of \flament oscillations and their pertinent prop-\nerties, including their apparent triggers, damping times,\nperiods, \flament type, \flament dimensions, peak veloci-\nties, directionality with respect to the \flament spine, and\nmaximum displacements. With the information in this\ncatalog, one can derive minimum \feld strength and other\nunobservable characteristics through seismology, and be-\ngin to explore the implications of longitudinal and trans-\nverse oscillations for prominence stability, evolution, and\neruption. We have made the catalog available to theGONG Catalog of Solar Filament Oscillations 3\ncommunity at the following URL: http://www.iac.es/\ngaleria/mluna/pages/gong-catalogue-of-laos.php\nThis paper presents both individual examples of in-\nterest and statistical analyses that explore potential re-\nlationships among the derived parameters. In x2 the\nGONG data used in the catalog are described, while in\nx3 the GONG catalog of prominence oscillations is in-\ntroduced.x4 presents the method used to detect oscil-\nlations and select events for the catalog. The criteria\nused to classify prominence types are introduced in x5.\nx6 explains how we identi\fed the triggering mechanism\nand derived the \flament parameters. x7 andx8 discuss\nthe time-distance approach and analysis methods used\nto characterize the oscillations, respectively. x9 describes\nselected events in detail, while in x10 we present the re-\nsults of our statistical study of \flament oscillations. A\nseismological analysis of selected events comprises x11,\nand the results are summarized in x12. A full list of\nevents and their oscillation parameters is in Appendix\nA. We describe our new method for constructing time-\ndistance diagrams with data from curved slits in Ap-\npendix B.\n2.DESCRIPTION OF THE NSO GONG NETWORK DATA\nNowadays, it is possible to monitor the full Sun nearly\ncontinuously with the space-based Solar Dynamics Ob-\nservatory (SDO; Lemen et al. 2012) or the ground-based\nnetwork of telescopes of the Global Oscillation Network\nGroup (GONG) ( http://gong2.nso.edu ). Continuous\ncoverage of the full Sun is needed for a complete study of\n\flament oscillation events. SDO o\u000bers the best spatial\nresolution and temporal cadence, and the observations\nare independent of the local conditions of the Earth's\natmosphere, in contrast to the GONG telescopes. How-\never, the \flaments and their periodic movements are not\neasy to detect in SDO data. In some situations the os-\ncillation is clear in the GONG H \u000bdata, but it is not\npossible to see the \flament in absorption in the SDO\nEUV images because of foreground emission. In addi-\ntion, the structures seen by SDO are complex and very\ndynamic, making the detection of periodic movements\nvery di\u000ecult. Therefore we use GONG data to perform\nour survey of \flament oscillations. The GONG network\ntelescopes o\u000ber su\u000eciently good spatial resolution and\ntemporal cadence to detect prominence oscillations with\nperiods of a few tens of minutes.\nThe GONG H \u000bimages allow us to identify \flaments\neasily and to follow their motions. We interpreted the\n\flament motions as displacements of the prominence\nplasma in the plane of the sky. However, H\u000bintensity\ndepends on LOS velocities. It is worth to mention that\nexists the possibility that this e\u000bect may produce a dis-\nappearance of parts of the \flament giving the impression\nthat the remaining visible \flament is moving. With the\nH\u000bGONG data we can study the massive set of oscilla-\ntions observed since August 2010, the date when the net-\nwork started to operate. Here we focus on an analysis of\nGONG data from several months close to the maximum\nof solar cycle 24, from 1 January 2014 to 30 June 2014.\nCycle 24 started in 2008 and reached minimum in early\n2010, with a double-peaked maximum in 2013 and 2014.\nThe GONG network telescopes are of identical design\nand construction and are placed around the world at\nthe following locations: Learmonth (L), Udaipur (U), ElTeide (T), Cerro Tololo (C), Big Bear (B) and Mauna\nLoa (M). The telescope locations were selected to fol-\nlow the diurnal motion of the Sun in the sky, in order to\ncollectively ensure full-day coverage (Harvey et al. 1996).\nEach telescope takes data daily, weather permitting, with\nsome temporal overlap of coverage between telescopes.\nThe temporal cadence of the GONG data is 1 min with\na pixel size of\u00181 arcsec. For each data sequence of each\ntelescope we compensate the solar di\u000berential rotation\nusing the drot map.pro solarsoft routine. The reference\ntime to de-rotate the images is the central time of each\ntemporal data sequence for each telescope and day.\n3.GONG CATALOG OF PROMINENCE OSCILLATIONS\nThe objective of our GONG catalog is to completely\ndescribe the oscillations detected in solar \flaments be-\ntween 1 January 2014 and 30 June 2014. The catalog\ncontains information about the properties of the oscil-\nlating \flaments, the apparent triggers of the oscillation,\nand the oscillation parameters. With this information we\nconstruct a comprehensive global picture of the \flament\noscillations close to the maximum of solar cycle 24.\nIn the following sections we describe the methods we\nused to construct the catalog ( x4 tox8). The full results\nof the survey are shown in Tables 1 to 8 in Appendix A.\nThe \frst group (Tables 1 to 4) displays data describing\nthe observations and the \flaments. The \frst column cor-\nresponds to the number of the oscillation event, ordered\nin time starting 1 January 2014. The second column\nlists the telescope where the event is detected (L, U, T,\nC, B, M). The third column lists the central time of the\ntemporal sequence associated with each telescope used\nto analyze the event (see x2). The fourth column shows\nthe averaged position of the \flaments at the reference\ntime (see detailed description in x9). The \ffth column\nindicates the \flament type (AR, IT, QS) described in\nx5. The sixth and seventh columns contain the length,\nL, and width, Wof the \flament measured as described\ninx6. In the eighth column we indicate the possible\ntriggering agent described in x6. The last column shows\nwhether the \flament erupted in the temporal sequence\nanalyzed, indicated by a Y (Yes).\nThe second group, Tables 5 to 8, shows the oscillation\nparameters resulting from the \ftting method described\nin detail inx8 with Equation (2). The columns indicate\n(1) the event number; (2) the initial time of the sequence\nused for the \ft; (3) the angle \u000bbetween the oscillation\ndirection and the \flament spine; (4) the period ( P); (5)\ndamping time ( \u001c); (6) damping time per period ( \u001c=P);\n(7) maximum displacement ( A); and (8) velocity ampli-\ntude (V).\nInx5 tox8 we will use Event 1 from Table 5 as our\nreference event to describe the methodology. Although\nthe \fgures are speci\fc to this event, the results and ex-\nplanations are valid for all events listed in the Tables.\n4.EVENT SELECTION\nOur \frst action was to detect the \flaments that may\noscillate by visual inspection of the GONG H \u000bdata\n(http://gong2.nso.edu ), in which the \flaments are\nseen as dark absorption structures (see Figure 1). The\noscillations were identi\fed as periodic displacements of\na part of the \flament. The GONG webpage shows very\ngood quality movies with full cadence for all six network4 Luna et al.\ntelescopes. We analyzed daily observations from each\ntelescope, and selected data that showed a clear or sus-\npected oscillatory event for in-depth analysis. In this\ninitial inspection we identi\fed 408 potential cases. We\ninitially identi\fed each event to be associated with one\nday and one telescope. For cases where the oscillation\ncontinued at the end of the observing period of the se-\nlected telescope, we did not utilize the subsequent tele-\nscope observations in order to extend the oscillation data.\nIn addition, we checked the data carefully to avoid double\ncounting the same oscillation observed by two telescopes\nwith overlapping data. For cases in which a second os-\ncillation appeared during a given observing period, we\nde\fned a new event with the same location and tele-\nscope as the preceding event and we marked it with an\nasterisk next to the event number.\nOnce we identi\fed the \flaments that might oscillate,\nwe downloaded the reduced H \u000bdata in the form of FITS\n\fles from the GONG server. We de-rotated the images\nin order to compensate for solar rotation and to study\nthe proper motion of the \flaments over the solar surface.\nAll images were de-rotated using a reference time that\ncorresponds to the central time of the temporal sequence\nas described inx2. This de-rotation algorithm only works\non the solar disk, so we discarded events in prominences\nseen at the limb and focused exclusively on \flaments seen\nin absorption on the disk. The coordinates are given in\nthe usual Heliocentric-Cartesian coordinates (Thompson\n2006).\n5.FILAMENT CLASSIFICATION\nIn the catalog we assigned \flament types exclusively\nbased on GONG H \u000bdata, according to the position\nscheme of Engvold (2014), as active region (AR), inter-\nmediate (IT), or quiescent (QS). In Figure 1 we have\nmarked the three types by colored arrows (AR - red, IT\n- green, QS - blue). AR \flaments are located close to\nsunspots and plages with a prominent spine and few or\nno barbs. ITs have one end close to an active region (AR)\nand the other end far from an AR; they exhibit both a\nspine and barbs. The QSes are far from any AR or plage\nregion, with no clear spine. For \flaments whose type was\ndi\u000ecult to determine, we used SDO HMI magnetograms\nto distinguish whether the \flament is close to a strong\nmagnetic \feld or a quiet region. The catalog includes 45\nAR, 99 IT and 52 QS \flaments.\nFollowing this classi\fcation we identify our reference\ncase 1 as intermediate or IT (see Table 1), because the\n\flament has one end located in plage associated with the\nactive region NOAA 11938 and the other end in a quiet\nregion (see Fig. 2).\n6.TRIGGERING AND FILAMENT PARAMETERS\nWe constructed a movie with the FITS data showing\nthe region of interest surrounding each \flament, enabling\nus to identify the most likely triggering agent and to\nstudy the \flament motion. For more than half of the\nevents we could not identify what triggered the oscilla-\ntions, so we left column 4 empty in Tables 1 to 4. Those\ncases for which we found a trigger were marked FLARE\nwhen a sudden H \u000bbrightening was detected nearby just\nbefore the oscillation onset; prominence eruption (PE)\nwhen a nearby \flament erupted before oscillation onset;\nand JET when the trigger was a jet of plasma that hit\nFigure 1. GONG H\u000bimage from the Learmonth telescope illus-\ntrating the 3 types of \flaments, which are seen as dark structures\nagainst the bright chromosphere. The red arrows point to active\nregion (AR) \flaments; green arrows point to intermediate \flaments\n(IT) between ARs; blue arrows point to quiescent (QS) \flaments.\nthe \flament. In one case, 91, we clearly observed a More-\nton Wave (MW) emanating from a \rare and hitting the\n\flament, triggering its oscillation as described in x9.3.\nIn Figure 2 the sequence of events is shown for our\nreference case 1. The trigger was identi\fed as a \rar-\ning region located south of the \flament. In order to\nparametrize the \rare position, we averaged the measured\npositions of several bright regions in the \rare; this aver-\naged position is marked by a red dot in Figure 2(a). This\npanel also shows the equilibrium position of the \flament\nbefore the trigger perturbed the \flament. Panels (b) and\n(c) show the di\u000berence images at the given times with the\nimage shown in (a) subtracted, thus visualizing the dis-\nplacements with respect to the equilibrium con\fguration.\nThe initial motion was in the northwest direction (Fig.\n2(b)), then the motion was reversed to travel toward the\nsoutheast (Fig. 2(c)).\nBecause the \flaments were very dynamic and their\nshapes changed considerably during the observation in-\ntervals, we \frst generated an average image of the region\nof interest, as shown in Figure 3 for case 1. This image\nwas constructed by averaging 10 equally spaced images\nfrom the data sequence of the day and telescope selected\n(i.e., Cerro Tololo or C in this case). From this averaged\nimage we determined the position of the spine following\nthe dark \flament (thick white line in Fig. 3), the length\nof the spine, L, and the average width of the \flament,\nW. We de\fned the width of the \flament at 5 equidistant\npositions along the spine as the length of the 5 segments\nplotted in the \fgure as thin lines. The average width\nand length characterize the size of the \flament, to be\nused later in our statistical study. For this example the\nlength isL= 269 Mm and width W= 15 Mm (see event\n1 in Table 1). Using the positions of the \flament spineGONG Catalog of Solar Filament Oscillations 5\nFigure 2. Temporal sequence of the triggering and oscillations in event 1. (a) H \u000bimage showing the dark \flament in its equilibrium\nposition (12:55 UT, before oscillation onset), outlined by a white contour. The brightening associated with the triggering \rare and the\naveraged \rare position (red dot) are also visible. (b) Base di\u000berence H \u000bimage (13:53 UT - 12:55 UT) showing the northward displacement\nof the \flament. The dark and white regions correspond to negative and positive di\u000berences, respectively. The contour of the equilibrium\n\flament is overplotted in white. (c) Base di\u000berence H \u000bimage (14:33 UT - 12:55 UT) showing the southern displacement of the central\npart of the \flament. In (b) and (c) the orange contour marks the slit used to track the \flament motion.\nwe also obtained the averaged position of the \flament on\nthe solar disk, marked with a cross in the \fgure.\nFigure 3. H\u000bimage of event 1 averaged over 10 equally spaced\ntimes from the full observing period, showing the spine position\n(white line), the spine length ( L), and the average \flament width\n(W). The 5 thin line segments were used to calculate the average\nwidth of the dark band. The orange contour corresponds to the\nslit used to follow the motion and to construct the time-distance\ndiagrams.\u000bmeasures the angle between the direction of motion,\ni.e. the slit, and the \flament spine. In this example \u000b= 16\u000e,\nL= 260 Mm, and W= 15 Mm. The red dot marks the averaged\n\rare position; the white cross is the averaged \flament position on\nthe solar disk.\n7.TIME-DISTANCE DIAGRAMS AND DIRECTION OF THE\nMOTION\nWe used the time-distance approach to analyze the \fla-\nment oscillations. Because many oscillations reported in\nthis work did not follow straight trajectories, but rather\nmoved along curved paths, we could not apply the tech-\nnique described by Luna et al. (2014) based on straight\nslits. In addition, due to the relatively low resolution\nof the GONG data, we needed to generate time-distance\ndiagrams with minimal reduction of the e\u000bective resolu-\ntion in curved slits. To de\fne the curved slit in the H \u000bimages for each event, we tracked the path of the oscilla-\ntions by visual inspection. In order to generate the slit,\nwe \frst traced the motion of the \flament segment with\nthe clearest and largest displacement. The slit, of length\nland width wpixels, was placed lengthwise along the\ncurved path of the motion as described in Appendix B.\nWith the technique shown in the Appendix, we averaged\nthe intensity over the transverse pixels, w, resulting in\nan intensity distribution along l. The time-distance dia-\ngrams display this intensity along the slit as a function of\ntime. Figure 2 shows that the slit matches the trajectory\nof the cool plasma for case 1. The vertical coordinate in\nthe time-distance diagrams (e.g. Fig. 2(a)) corresponds\nto the distance along the slit in Mm, with the origin set\nto coincide approximately with the equilibrium position\nof the \flament. This distance is measured in the plane of\nthe sky inx\u0000ycoordinates (i.e., Heliocentric-Cartesian\ncoordinates); thus the displacements are projections of\nthe actual motions onto the plane of the sky. Similarly,\nthe velocities measured in the time-distance diagrams are\nalso projections, so the actual values are probably larger.\nThe angle \u000bbetween the direction of the oscillatory\nmotion and the \flament spine was measured at the in-\ntersection of the spine curve and the slit (orange curve\nin Fig. 3). This angle (dotted arc) characterizes the po-\nlarization of the oscillations in terms of longitudinal or\ntransverse movements. In this example \u000b= 16\u000e(see Ta-\nble 5), so the oscillation is longitudinal. In case 1, we\nfound that the triggering location was aligned with the\nslit used to track the motion (Fig. 3), suggesting that\nthe perturbation from the \rare followed the same direc-\ntion as the slit to reach the \flament. The angle \u000bis a\nprojection onto the sky plane of the actual angle. The\ndi\u000berence between these angles depends on the \flament's\nposition and orientation on the solar disk.\nUsing the technique explained in Appendix B, we con-\nstructed time-distance diagrams for each event. In the\nresulting time-distance diagram for our reference case 1\n(Fig. 4(a)), the \flament appears as a dark band sur-\nrounded by bright emission from the adjacent chromo-\nspheric plasma, and the \flament oscillations are clearly\nvisible.\nThe slit was traced visually following the motion of the6 Luna et al.\nFigure 4. Oscillation diagnostics of event 1. (a) Time-distance\nH\u000bdiagram. The dark band is the \flament seen in absorption, sur-\nrounded by bright emission from the adjacent chromosphere. The\ntwo dashed lines mark the 1 \u001blevel as discussed in x8.1. (b) Trian-\ngles show the central position of the \flament, s0(t), as a function\nof timetas determined from the Gaussian \ft along the slit using\nEq.(1). The 1 \u001bregion is delimited by two thick dashed lines. The\nthick solid line is the best \ft to the triangles using Eq. (2). The\nblue dashed line is the same \ftted function extrapolated to times\noutside the temporal range used to construct the \ft. (c) The veloc-\nity as a function of time, computed as the time derivative of s0(t).\nThe velocity obtained with the \ftted function (2) is overplotted as\na solid curve.\ncool plasma in the H \u000bimages, introducing a subjective\nfactor in the determination of the slit path. In addi-\ntion, the relatively low spatial resolution of GONG data\ncould produce a misalignment of the slit with the actual\ntrajectory of the cool plasma, reducing the measured dis-\nplacements over the slit and yielding another source of\nerror in the measurements of the displacements. We have\nnot quanti\fed explicitly the error introduced by the mis-\nalignment, however, because we have overestimated the\nerrors in the displacements as discussed in x8.1. Addi-\ntionally, the misalignment introduces an error in \u000b, but it\nis unclear how to assess the uncertainties in this param-\neter. We are currently developing automated techniques\nto track the motion of the \flament, which will enable\nus to quantify and reduce the errors in the displacement\nand\u000b.\n8.OSCILLATION ANALYSIS\nFigure 5. H\u000bintensity along the slit for case 1 at 16:06 UT\n(thin line), corresponding to the blue vertical line in Fig. 4(a).\nAn asterisk marks the intensity minimum. The thick line is the\nGaussian \ft to the observed intensity pro\fle using Eq. (1).\nTo determine the central position of each \flament as\na function of time, we plotted the intensity along the\nslit and \ftted a Gaussian function to the intensity for\neach image of the observing sequence. Because the ab-\nsorption of the \flament depends on the column depth of\ncool plasma along the line of sight (LOS), we assumed\nthat the intensity minimum corresponds to the central\nposition in the direction along the slit. This Gaussian\nfunction also enabled us to determine the uncertainties\nof the oscillatory parameters ( x8.1). We used the gauss-\n\ft.pro IDL routine with a functional form\nI(s) =g0e\u00001\n2\u0010s\u0000s0\ng1\u00112\n+g2+g3s+g4s2; (1)\nwheresis the coordinate along the slit, g0<0 is the in-\ntensity amplitude, s0is the central position of the Gaus-\nsian,g1=\u001bGis the standard deviation, and the remain-\ning terms are the background chromospheric emission.\nFigure 5 shows the intensity along the slit at \u001816:06\nUT (blue vertical line in Figure 4(a)) in the reference\ncase, with the position of the local minimum indicated\nby an asterisk. To avoid errors due to noisy data, we\nconsistently used the central position of the Gaussian\nfunction,s0(t), to track the \flament motion (see Figure\n4(b) for case 1). The measured velocity for all events\nwas derived from the observations by computing the nu-\nmerical derivative of s0(t). The function used to \ft the\noscillation is an exponentially decaying sinusoid, plus a\nthird-order polynomial function to de-trend the proper\nmotions of the \flament:\ny(t) =A0e\u0000A1(t\u0000t0)cos [A2(t\u0000t0) +A3] +\nA4+A5(t\u0000t0) +A6(t\u0000t0)2+A7(t\u0000t0)3;(2)\nwhereAiare the coe\u000ecients of the \ft. Sometimes the\nbeginning of the oscillation is not well described by Equa-\ntion (2), so we performed the \ft in a selected time in-\nterval when the oscillation is clear in the time-distance\ndiagram for each event. In Equation (2), t0is the ini-\ntial time of the \ftted function (column 2, Tables 5 to 8).\nThe \frst few coe\u000ecients of the \ft are associated with the\noscillation in the following way: A0is the \ftted displace-\nment amplitude; A1= 1=\u001c, where\u001cis the damping time;\nA2= 2\u0019=P, wherePis the period; and A3is the initialGONG Catalog of Solar Filament Oscillations 7\nphase. The remaining terms are the coe\u000ecients of the\npolynomial function that \fts the background motion of\nthe \flament. This trend function \flters out motions as-\nsociated with long-period oscillations. Very long periods\nhave been observed in a few oscillating \flaments (Foullon\net al. 2004, 2009; Efremov et al. 2016), but these motions\nare not clear in our data. Hence we focus our attention\non more rapid oscillations.\nFigure 4(c) shows the measured velocity (diamonds)\nand the best \ft obtained from Equation 2(solid curve)\nfor case 1. The \flament remained essentially station-\nary before 13:00 UT, then the velocity increased slowly\nbetween 13:00 and 13:30 UT In the following \u001930 min-\nutes the velocity suddenly jumped up to 60 km s\u00001, re-\nturned to zero at the time of maximum displacement (see\nFig. 4(b)), and increased again in the opposite direction\nto\u000055 km s\u00001. In this phase the acceleration reached\n140 m s\u00002. The velocity does not resemble a sinusoidal\noscillation until after \u001814:00 UT. The measured velocity\nand the \ftted function agree very well.\n8.1. Errors\nSeveral sources of uncertainty in the measured oscil-\nlation parameters are attributable to the relatively low\nspatial resolution of the GONG images, including jitter,\nthe uncertainties in the exposure time, and the above-\nmentioned misalignment of the slit with the cool plasma\ntrajectories. We assumed that the uncertainty in the\noscillation parameters comes mainly from the errors in\ndetermining the position of the \flament along the slit.\nIn Figure 6 we have plotted a time-distance diagram in\nan interval where no oscillation was evident. The aster-\nisks mark the central positions of the Gaussian \ft, s(t).\nWe clearly see a noisy signal in the \fgure, where the\n1\u001bstandard deviation (thin straight lines) around the\nmean value is \u001bnoise = 1:4 Mm. The \fgure also shows\nthe width of the Gaussian \ft, \u001bG, to the intensity (Eq.\n1) as dashed lines, which coincide with the borders of the\ndark band. We consider \u001bGto overestimate the uncer-\ntainty of the central position because this uncertainty is\ncomparable to the \flament width. In fact, \u001bG&3\u001bnoise\nin all analyzed events. Thus, we decided somewhat arbi-\ntrarily to set the uncertainty of the \flament position as\n\u001b= 0:5\u001bG&\u001bnoise(indicated by two solid curves in Fig-\nure 6). This overestimates the error because \u001b > \u001b noise\nand the uncertainty in the position is larger than the\nvariations seen in the \fgure. This error \u001bwill be con-\nsidered the only source of uncertainty in the estimated\noscillation parameters.\nIn Table 5 we have tabulated the best-\ft parameters\nfrom Equation 2 for all oscillation events. Event 1 has a\nperiod ofP= 76\u00061 min, damping time \u001c= 121\u000615\nmin, maximum displacement A= 23\u00062 Mm, and peak\nvelocity amplitude V= 26\u00064 km s\u00001. The ratio\n\u001c=P = 1:6\u00060:2, indicating that the damping was strong\nand very e\u000ecient. Because the velocity amplitude was\nlarger than 10 km s\u00001, event 1 is a LAO. Large velocities\nexceeding 40 km s\u00001occurred early in this event, but this\nphase did not produce a \ft compatible with subsequent\nmotions (Fig. 4(c)). The best \ft derived from Equa-\ntion (2) has a much lower peak value of 26 km s\u00001and\nagrees well with most of the subsequent velocity oscil-\nlations. At the end of the \ftted range, around 18:30\nFigure 6. Time-distance diagram of an interval with no clear\noscillations. The asterisks mark the central positions of the Gaus-\nsian \ft of Eq. (1), g1(t). The standard deviation of these points is\n\u001bnoise; the two thin horizontal lines correspond to \u0006\u001bnoise with re-\nspect to the mean value of these positions. The two dashed curves\nare atg1\u0006\u001bG, containing the dark region in the diagram. We use\n\u001b= 0:5\u001bGas the uncertainty in the \flament position. Here the\nuncertainty region g1\u0006\u001bis delimited by two solid lines.\nUT, the \frst oscillation ceased and a new one appeared.\nThis new oscillation, case 2*, had a di\u000berent phase and\namplitude than case 1, which can be seen by comparing\nthe observed triangles with the blue dashed line from the\n\ft of case 1 in Figure 4(b). Cases 1 and 2* have com-\nparable periods, indicating that they are characteristic\noscillations of the \flament. The direction of the motion\nis o\u000bset by 16\u000efrom the \flament spine (see Fig. 3(b) and\nTable 5), suggesting that the oscillation is longitudinal as\nwell as large-amplitude (i.e., a LALO). This angle is com-\nparable to the typical angle between the magnetic \feld\nand the \flament spine, according to direct measurements\n(Leroy et al. 1983, 1984; E. Tandberg-Hanssen 1995; Tru-\njillo Bueno et al. 2002; Casini et al. 2003; L\u0013 opez Ariste\net al. 2006), suggesting that the oscillation is aligned with\nthe \flament magnetic \feld.\n9.SELECTED CASE STUDIES\nIn the previous Sections we used case 1 as a represen-\ntative oscillation example from our event catalog. Here\nwe describe selected additional events from the catalog\nto illustrate the intriguing variety of behaviors and oscil-\nlation characteristics encountered in our survey.\n9.1. Event 58: LALO triggered by a two-ribbon \rare\nEvent 58 of the survey (Table 5) is a LAO with a peak\nvelocity of 14 km s\u00001, triggered by a two-ribbon \rare that\nstraddled the AR \flament (see Figure 7(a)). This \rare\nproduced an oscillation in which the cool plasma was\ndisplaced initially in the west-east direction from its pre-\n\rare equilibrium position (white contour in Figure 7(a)).\nThen the motion was reversed, so the \flament reached\nmaximum western elongation (Fig. 7(b)), followed by\nanother reversal that produced a peak eastward displace-\nment of smaller amplitude (Fig. 7(c)). The Figure also\ndemonstrates that the slit used to track the motion and\nconstruct the time-distance diagram closely follows the\ntrajectory of the cool \flament plasma. Very clear oscil-\nlations in the H \u000bintensity are evident in Figure 8(a),\ntriggered around 11:50 UT and lasting for more than 78 Luna et al.\nperiods from\u001812:00 to 18:00 UT. Before \rare onset the\ndark \flament band is almost horizontal, so the \flament\nwas almost at rest. In Figure 8(b), the central position\ns(t) and its best \ft from Eq. 2 agree well until \u001816:00\nUT. Thereafter the \ft is more damped than s(t) and a\nslight phase di\u000berence is also evident. After 19:00 UT\nthe measured s(t) is very noisy because the quality of\nthe H\u000bimages was reduced.\nThe \ftted velocity matches the measured velocity well\n(Fig. 8(c)). There were no large velocities during the\ninitial phase associated with the triggering, in contrast\nwith case 1 (Fig. 4(c)). Although the initial velocity\nexceeded 10 km s\u00001, indicating that this event is a LAO,\nin less than a period the velocity fell below this thresh-\nold. The direction of motion was o\u000bset by 32\u000efrom the\n\flament spine, identifying this event as a possible LALO.\n9.2. Event 63: LALO in a large quiescent \flament\nIn case 63 the oscillation occurred in a very large, frag-\nmented quiescent \flament (QS) in the southern hemi-\nsphere. The oscillation appeared in the southern seg-\nment, possibly triggered by the \rare centered at the red\ndot in Figure 9(a). The H \u000b\rare brightened slightly\nbefore the oscillation began at \u001818:26 UT. Part of the\nprominence plasma moved toward the northwest or west\nalong a curved trajectory. Di\u000berence images in Figures\n9(b) and (c) reveal the maximum elongation at two times\nafter the initiation: a dark region along the slits ap-\npeared, \frst on one side and then on the other side of\nthe pre-oscillation position of the \flament (Fig. 9(c)).\nSeveral threads of cool plasma moved in a complex way\nbefore oscillation onset at 20:26 UT (Fig. 10(a)). There-\nafter the oscillation was very clear but only was observed\nfor 2 cycles. Apparently the whole \flament was displaced\nto the southeast before the oscillation, possibly indicat-\ning that the structure of the \flament changed during the\npre-\rare phase. In Figure 10(a) the positional uncer-\ntainty,\u001b, is indicated by two dashed lines. This uncer-\ntainty is very small where the dark band is very narrow.\nInitially some threads moved towards the centroid of the\noscillation, but our tracking code ignored these outliers\nand only followed the motion of the main body, the dark-\nest band at s\u001810 Mm. After 20:39 UT the entire \fla-\nment oscillated, and Equation (2) provided a good \ft to\ns(t) (Fig. 10(b)). The measured velocity amplitude, V,\nreached very large values >50 km s\u00001(Fig. 10(c)) .\nThe results of the \ft are shown in Table 5. The oscil-\nlation had a velocity amplitude of V= 48:5\u00062:4 km s\u00001,\nP= 103\u00061 min, a damping time \u001c= 175\u000612 min, and\n\u001c=P = 1:7\u00060:1 indicating very strong damping. The\nmotion was only \u000b= 2\u000emisaligned with the \flament\nspine, indicating that this event was probably a LALO.\n9.3. Event 91: LALO triggered by a Moreton wave\nEvent 91 occurred in an intermediate \flament (IT) lo-\ncated close to AR NOAA 12017. Around 17:41 UT a\nmajor \rare occurred in the AR followed by a Moreton\nwave, which is visible in Figure 11. This is the only\nevent in the survey for which we could identify a More-\nton wave connected to the \flament oscillation. The wave\nemanated from the \rare region (Fig. 11(a)), hit the \fl-\nament (Fig. 11(b)) then continued to propagate north-\nward (Fig. 11(c)). Once the wave encountered the \fla-ment (Fig. 12(b)), the \flament started to oscillate per-\npendicular to the propagation direction of the wave front.\nAt 18:53 UT (Fig. 12(c)), the motion was reversed and\nthe oscillatory motion was fully established. The time-\ndistance diagram in Figure 13(a) reveals signi\fcant mo-\ntions of the \flament before and after \rare onset. Around\noscillation onset, at 17:47 UT, a white vertical region ap-\npears in Figure 13(a), signalling the arrival of the More-\nton wave at the \flament throughout the slit. The More-\nton wave initially produced complex \flament dynamics,\nbest seen in a movie of the event (not shown here). The\noscillation became distinct around 18:00 UT, with a max-\nimum southeastward displacement of 12 Mm from the\nequilibrium position around 18:22 UT (Fig 13(b)) and a\nsmaller opposing displacement at 18:53 UT (Fig 13(b)).\nInterestingly, when the \flament moved in the northwest\ndirection, it was darker than when the motion was in\nthe opposite direction (Fig. 13(a)). The central posi-\ntion of the dark band oscillated clearly (Fig. 13(b)), and\nwas \ftted very well with Equation (2) after 18:23 UT.\nHowever, between the triggering at 17:47 UT and 18:23\nUT the motion does not \ft the sinusoidal function (blue\ndashed line), probably due to the very complex motions\nprior to and during the passage of the wave through the\n\flament.\nBefore the oscillations were triggered at 17:47 UT the\n\flament reached velocities above 5 km s\u00001, but this mo-\ntion was not periodic (Fig. 13(c)). The oscillation veloc-\nity peaked at 18:00 UT; after \u001818:23 UT the measured\nvelocity was very well \ftted by the derivative of Equation\n(2), yielding a peak amplitude V= 19:3\u00062:3 km s\u00001, a\npeak displacement A= 11\u00061 Mm, period P= 58\u00061\nmin, damping time \u001c= 108\u000612 min, and \u001c=P =\n1:9\u00060:2. The angle between the motion and the \flament\nspine was\u000b= 26\u000e, again consistent with the typical di-\nrect measurements of the orientation of the prominence\nmagnetic \feld relative to the spine (Leroy et al. 1983,\n1984; E. Tandberg-Hanssen 1995; Trujillo Bueno et al.\n2002; Casini et al. 2003; L\u0013 opez Ariste et al. 2006). There-\nfore the motion may be aligned with the local magnetic\n\feld and the event is probably a LALO.\n9.4. Event 31: SALO with a very small velocity\namplitude\nIn the survey we also detected oscillations with very\nsmall amplitudes of only a few km s\u00001, below the LAO\nlower limit of 10 km s\u00001. Event 31 has the smallest ve-\nlocity amplitude of the survey: V= 1:6\u00069 km s\u00001. The\n\flament is an IT and the oscillation was triggered by\na nearby \rare. In this example the displacements were\ntoo small to be visible in a \fgure similar to Figure 2.\nTherefore we refer the reader instead to the movie of\nthis event at the URL: http://www.iac.es/galeria/\nmluna/pages/gong-catalogue-of-laos.php .\nAlthough the displacements are very small in this\nevent, the oscillatory pattern is very clear (Fig. 14(a)).\nBefore 17:00 UT the \flament appears to be moving, but\nclear oscillations started at 17:17 UT and ended at 21:06\nUT. The displacement appears more or less constant\nthroughout the event (Fig. 14(b)). However, the extrap-\nolated oscillation some time before and after the \ftted\noscillation (blue dashed line in Figure 14(b)) does not fol-\nlow the \flament motion, and the oscillation apparently\nended without strong damping. In general, the veloc-GONG Catalog of Solar Filament Oscillations 9\nFigure 7. Temporal sequence of the triggering and oscillations in event 58. Panels and annotations are as in Fig. 2. (a) Pre-oscillation\nH\u000bimage at 11:34 UT. The white contour outlines the equilibrium \flament position at \rare onset (11:24 UT). The triggering two-ribbon\n\rare is not visible in this initial frame. The red dot marks the average position of the \raring region (evident in (b)). (b) Base di\u000berence\nH\u000bimage (12:10 UT - 11:34 UT). The two \rare ribbons appear as bright patches north and south of the \flament. (c) Base di\u000berence H \u000b\nimage (12:33 UT - 11:34 UT).\nFigure 8. Oscillation diagnostics of event 58. Panels and anno-\ntations are as in Fig. 4\nity has a large uncertainty for small-amplitude events.\nIn this case the velocity error is 9 km s\u00001, much greater\nthan the velocity V= 1:6 km s\u00001. Figure 14 exhibits\ndistinct oscillatory motions, so we are probably overesti-\nmating the positional errors (see x8.1). The displacement\nwas very small ( A= 1\u00061 Mm), the oscillation period\nP= 77\u00064 min, and the damping time \u001c= 662\u00061546min, so the damping was very weak with a large uncer-\ntainty (see Table 5). The angle between the slit and the\nspine\u000b= 26\u000e, suggesting that the oscillation was longi-\ntudinal.\n9.5. Events 145 and 146*: double event\nIn these events, two clear oscillations occurred in the\nsame \flament within one observing interval; therefore\nwe labeled them as separate events. In our example the\n\flament is of the IT type with a curved initial structure\n(Fig. 15(a)). Both oscillations involve motions mainly\ntransverse to the \flament spine, with \u000b= 50\u000e. A nearby\n\rare, marked by the red dot in the \fgure, is the most\nlikely trigger. At onset the entire \flament was displaced\nlaterally toward the northeast (Fig. 15(b)), then after a\nhalf period the \flament moved to the other side of its\nequilibrium position (Fig. 15(c)). The slit (red outline)\nis over the region of the \flament that oscillated with the\nlargest displacement, which protruded from the rest of\nthe structure.\nThere were many data gaps, which appear as white\nvertical bands at the beginning of the time-distance dia-\ngram (Fig. 16(a)). The \frst event, 145, oscillated most\nvisibly between 13:31 and 15:33 UT. The oscillation pe-\nriod wasP= 39\u00062 min,V= 14:4\u00066:1 km s\u00001and\n\u001c= 48\u000613 min (see Table 7), indicating very strong\ndamping with \u001c=P = 1:2\u00060:4. Event 146*, which started\nat 17:38 UT and ended at 19:20 UT, had P= 36\u00062 min,\nV= 3:4\u00069:4 and\u001c= 119\u0006146 min with \u001c=P = 3:3\u00064.\nFigure 16(a) leaves the impression that both events were\npart of a long oscillation starting at 13:31 UT and con-\ntinuing throughout the temporal sequence. However, the\nvelocity evolution demonstrates that case 145 ended at\n15:33 UT (Fig. 16(c)), followed by an interval of complex\noscillation. Event 146* started at 17:38 UT with a very\nclear oscillation that was out of phase with previous mo-\ntions. We conclude that repetitive triggering occurred in\nthis \flament.\nIt is interesting that both events had similar periods,\nsuggesting that the \flament oscillated with a character-\nistic frequency of the system (e.g., Hyder (1966)). How-\never, the damping times were very di\u000berent, indicating\nthat either the damping mechanisms were di\u000berent or\nthe damping e\u000eciency changed between events. The \frst10 Luna et al.\nFigure 9. Temporal sequence of the triggering and oscillations in event 63. Panels and annotations are as in Figure 2. Here panels (b)\nand (c) show a smaller region centered at the \flament. (a) Pre-oscillation H \u000bimage at 20:26 UT. (b) Base di\u000berence H \u000bimage (21:04 UT\n- 20:26 UT), showing the initial northwestward displacement of the \flament along the slit. (c) Base di\u000berence H \u000bimage (22:04 UT - 20:26\nUT) showing the subsequent southeastward displacement.\nFigure 10. Oscillation diagnostics of event 63. Panels and anno-\ntations are as in Fig. 4\ncase had a peak velocity \u001814 km s\u00001, much larger than\nthe second event with V\u00183 km s\u00001. Inx10.3 we show\nthat the damping time decreases with Vfor the entire\nset of events. This nonlinear e\u000bect might explain the\ndi\u000berent damping times for cases 145 and 146*. Both\noscillations are very clear in Figure 16, and both \fts ac-cording to Equation (2) are good. Although the second\noscillation continued after 19:20 UT, the extrapolated\n\ftted function (blue dashed line in Fig. 16(b)) does not\nfollow the center of the dark band (triangles) well dur-\ning this interval, probably because the dark band is very\nlight and the signal-to-noise ratio is very low in the time-\ndistance diagram. Event 145 is a transverse LAO with\nV= 14:4 km s\u00001, and 146* is a transverse SAO with\nV= 3:4 km s\u00001(see Figure 16(c)).\n9.6. Event 151 and 152*: double event and ampli\fed\noscillation\nUnlike the double event described in the previous sec-\ntion, the cases presented here exhibit very di\u000berent oscil-\nlation periods: P= 52\u00062 min for case 151 and P= 66\u00063\nmin for case 152*. Figure 17 shows the \frst stages of\nevent 151; the initial motion is southward, followed by a\nreversal toward the north. The direction of motion for\nevent 152* is similar to 151. Both events were apparently\ntriggered by nearby \raring (red dot in Figure 17(a)).\nEvent 151 occurred between 00:00 UT and 3:33 UT\n(Fig. 18(a)). The best-\ft solution to the central position\ntracks the oscillation well, except for a small discrepancy\nin the last period (Fig. 18(b)). The measured and \ftted\nvelocities plotted in panel (c) also agree well. The oscil-\nlation parameters are P= 52\u00062 min,\u001c= 89\u000623 min,\nand\u001c=P = 1:7\u00060:5 (strong damping). This event is a\nSAO withV= 6:8\u00065:1 km s\u00001. In both events \u000b= 36\u000e,\nsuggesting longitudinal oscillations.\nFigure 18 also shows the unusual oscillation of event\n152*, which started at 5:37 UT, increased in amplitude,\nand ended at 8:53 UT. The best \ft agrees well with\ns(t) between 5:37 and 8:15 UT but not after this in-\nterval, suggesting that the plasma motion of this ampli-\n\fed oscillation is more complex than Equation (2) (Fig.\n18(b)). The oscillation parameters are P= 66\u00063 min,\n\u001c=\u0000163\u0006105 min, and \u001c=P =\u00002:5\u00062, indicating\nvery strong ampli\fcation. The large error in the damp-\ning time is probably overestimated, because the \ftted\nfunction agrees very well with the oscillation. The \ftted\nmaximum velocity amplitude is V= 4:0\u00067:8 km s\u00001, but\nthe measured velocity reached a maximum of 14 km s\u00001\nat 08:40 UT. Later H \u000bdata reveals that the oscillation\nceased and the \flament became stationary again. Simi-\nlar behavior was found by Molowny-Horas et al. (1999)GONG Catalog of Solar Filament Oscillations 11\nfor an ampli\fed \flament oscillation.\nIn order to amplify the oscillation the cool plasma must\ngain energy. Recently Zhou et al. (2017) and Zhang et al.\n(2017b) found LALOs with an ampli\fed oscillation fol-\nlowed by a damped phase, which they explain as a beat-\ning phenomenon between two interacting oscillators (see,\nFigure 11. Running di\u000berence H \u000bimages showing the propagat-\ning Moreton wave at selected times. In all panels the \flament equi-\nlibrium position is outlined by a white contour, the slit is marked\nby a red arc, and the averaged \rare position is marked by a red\ndot. (a) 17:46 UT, shortly after the wave was generated at the\n\raring region. (b) 17:48 UT, when the wave (white patch) reached\nthe \flament. (c) 17:51 UT, as the wave (white arc) continued to\nexpand and travel northward.e.g, Luna et al. 2006; Luna et al. 2008). In this scenario\nthe oscillations of two regions of the \flament are cou-\npled: an active oscillator that transfers energy to the\nother part of the \flament (the passive oscillator). The\npassive oscillator gains energy with time so its oscillation\nis ampli\fed, while the active portion loses energy, reduc-\ning its amplitude. The section of the \flament oscillating\nin event 152* should be the passive oscillator because it is\ngaining energy. However, the active oscillator should be\nthe other region of the \flament, but it does not oscillate\nwith a larger amplitude. Therefore this hypothesis does\nnot explain events 151 and 152*. Ballester et al. (2016)\nfound that cooling the prominence plasma could amplify\nits oscillations, but we can't test this hypothesis for lack\nof relevant temperature diagnostics. Alternatively, repet-\nitive nearby \rares could produce both oscillation events\nand possibly amplify the second. The H \u000bdata shows\n\raring activity close to the \flament (Fig. 17(a)). In\nthis situation the ampli\fed oscillation would be driven\nby external forcing, which could explain why the period\nis di\u000berent from that of the non-ampli\fed previous event.\n9.7. Events 107 and 108*: double event with ampli\fed\noscillation and eruption\nThis double event is similar to that discussed in x9.5\n| a damped oscillation (case 107) immediately followed\nby an ampli\fed oscillation (case 108*) | but with a \fnal\neruption. Both events occurred in the same IT \flament,\nand both events were triggered by \raring in an active\nregion north of the \flament (Fig. 19(a)). For the \frst\neventP= 50\u00061 min,V= 6:6\u00062:2 km s\u00001, and\u001c=P =\n3:1\u00060:7; for the second event P= 40\u00063 min,V=\n5:6\u00069:3 km s\u00001, and\u001c=P =\u00002:4\u00062:0. Both events are\nSAOs with \u000b= 20\u000e, denoting longitudinal polarization.\nAs in events 151 and 152*, the periods are di\u000berent.\nSimultaneous with the oscillation onset for event 107, a\nwhite spot appeared north of the \flament (marked with a\nwhite arrow in Figure 19(a)) and continued almost to the\nend of event 107. Because the slit in Figure 19(b,c) passes\nover the white spot, we can also see this brightening in\nthe resulting time-distance diagram (Fig. 20(a)). Base\ndi\u000berence images show the maximum elongation of the\ncool plasma in event 107 (Fig. 19(b)) and for event 108*\n(Fig. 19(c)). At the end of event 108* the prominence\nerupts (\u001823:20 UT, not shown). The white spot that\nappeared north of the \flament appears as a bright region\nat the top of the dark band in Figure 20(a). This bright\nemission apparently followed the motion of the threads\nfrom 18:20 to 21:20 UT (end of event 107).\nAs with events 145 and 146*, the oscillation seems to\nbe continuous between 18:20 UT and 23:26 UT. However,\nthe event 107 and 108* oscillations di\u000ber signi\fcantly in\nphase, period, and damping time. Figure 20(b) shows\nthis discrepancy clearly: one \ft (Eq. (2)) is very good in\nthe \frst event and another is good in the second, except\nat the end of the event when the motion was obviously\na\u000bected by the eruption. The di\u000berences between two\nevents are equally evident in the measured velocities (Fig.\n20(c)). Before 18:20 UT the motions were small and\ndisorganized, while after this time the damped oscillation\nis very clear. At 23:26 UT the period changed and the\noscillation started to grow, ending in an eruption.\nThe main di\u000berence between events 107-108* and the\nampli\fed oscillation of x9.6 is that the \flament erupts.12 Luna et al.\nFigure 12. Temporal sequence of the triggering and oscillations in event 91. Panels and annotations are as in Figure 2. (a) H \u000bimage\nof the \raring region and the \flament at oscillation onset (17:47 UT). The red dot indicates the approximate position of the \rare that\nproduced the Moreton wave that triggered the oscillation. (b) and (c) Running di\u000berence H \u000bimages of a smaller region centered on the\n\flament. The \flament was displaced initially to the southeast (b), then toward the northwest (c).\nFigure 13. Oscillation diagnostics of event 91. Panels and anno-\ntations are as in Fig. 4\nHowever, similar explanations might apply for the am-\npli\fcation. The potential relationship between the am-\npli\fed oscillation and the eruption is an intriguing topic\nfor further study.\n10. STATISTICS\nFigure 14. Oscillation diagnostics of event 31. Panels and anno-\ntations are as in Fig. 4.\nThe catalog consists of 196 oscillation events which we\nfound by analyzing six months of GONG data in cycle 24\n(see Tables 1 to 8). In about 43% of the cases we identi-\n\fed the apparent trigger of the oscillation: 72 events were\ntriggered by \rares, 11 by prominence eruptions, 1 by a\njet, and 1 by a Moreton wave. However, in 111 casesGONG Catalog of Solar Filament Oscillations 13\nFigure 15. Temporal sequence of the triggering and oscillations in event 145. Panels and annotations are as in Fig. 2. In (a) and (b) the\nred dot indicates the approximate position of the \rare that probably triggered the oscillation.\nFigure 16. Oscillation diagnostics of events 145 and 146*. Panels\nare as in Fig. 4 .\nthe triggering agent was not identi\fed. In 9 cases the\n\flament erupted during the temporal range analyzed.\nAs discussed in x1 we classi\fed the oscillations ac-\ncording to their maximum velocity amplitude as SAOs\n(V < 10 km s\u00001) and LAOs ( V > 10 km s\u00001). Of the\n196 oscillation events there are 106 SAOs and 90 LAOs.\nOver the six months of the survey this averages to one\noscillation event per day on the visible solar disk. The\noccurrence rate of one LAO event every two days implies\nthat LAOs are a common phenomena on the Sun, in con-\ntrast to previous statements that LAOs are scarce (e.g.,Tripathi et al. 2009). We also found a similar rate for\nSAOs.\nThe data presented in the catalog enabled us to search\nfor possible dependencies between pairs of \flament and\noscillation parameters: the velocity amplitude ( V), oscil-\nlation period ( P), damping time ( \u001c), damping time per\nperiod (\u001c=P), displacement ( A), and angle between the\nproper motion and the \flament spine ( \u000b). We also com-\nputed the Pearson correlation matrix (Neter et al. 1993)\nusing the IDL subroutine correlate.pro . The matrix ele-\nments are the correlations between pairs of parameters,\nand range from -1 to 1. A linear correlation between two\nparameters yields an associated matrix element close to\n1 (or -1). Although we found that the values of the ma-\ntrix are small, in general, we will discuss those pairs of\nparameters whose correlations or lack thereof are inter-\nesting. Figures 21 to 23 show scatter plots of some pairs\nof these parameters. In Figure 21 the scatter plots of\nthe period, P, vs the other parameters are displayed in\n6 panels (a-f). Figure 22 shows the damping parame-\nters,\u001cor\u001c=P, vsvand\u000b(panels a-e). Figure 23 plots\nseveral parameters vs solar latitude of the \flament. In\nthese scatter plots, the LAOs and SAOs are plotted with\ncircles and squares, respectively.\n10.1. Velocity Amplitude, V\nIn the survey we found velocity amplitudes from a\nfew km s\u00001to 55 km s\u00001(see Tables 5 to 8). His-\ntograms of the velocity distribution for all events and\nthe distribution according to \flament type are plotted\nin Figure 24(a) and (d), respectively. The vertical dot-\nted line separates LAOs ( V > 10 km s\u00001) from SAOs\n(V < 10 km s\u00001). The total number of LAO events de-\ncreases with the velocity amplitude, as expected: more\nenergetic events are less frequent than less energetic ones.\nThe velocity ranges for all \flament types (AR - red, IT -\ngreen and QS - blue) are similar (Figure 24(d)), indicat-\ning that all types of \flaments can support both SAOs and\nLAOs. The velocity distribution for each \flament type\nfollows the same trend as the total distribution except for\nAR \flaments. The apparent rollover in the AR \flament\ndistribution below 5 km s\u00001probably re\rects the di\u000e-\nculty in detecting small \flaments and small-amplitude\nevents by eye, implying that we have underestimated the\nnumber of SAOs.\nThe histograms also do not distinguish two separate14 Luna et al.\nFigure 17. Temporal sequence of the triggering and oscillations in events 151 and 152*. Panels and annotations are as in Fig. 2. In (c)\nthe white arrow points to the part of the prominence that oscillates.\nFigure 18. Oscillation diagnostics of events 151 and 152*. Panels\nare as in Fig. 4 .\npopulations associated with large- and small-amplitude\noscillations, regardless of the choice of LAO threshold\n(i.e., 10 km s\u00001or 20 km s\u00001). Additionally, 32 of the\n106 SAOs were clearly triggered by an identi\fed ener-\ngetic disturbance. These contradict the idea that the\nLAOs and SAOs have a di\u000berent nature and they are trig-\ngered by di\u000berent mechanisms (Oliver & Ballester 2002;\nArregui et al. 2012).\nTheP\u0000Vscatter plot (Figure 21(a)) and the small\ncorrelation P\u0000Vvalue reveal no dependence of the ve-\nlocity on the period, neither for all events nor for dif-ferent \flament types. In contrast, the V\u0000\u000bscatter\nplot (Fig. 22(a)) shows a clear pattern: the Vrange de-\ncreases with \u000b, and theVvalues drop sharply for events\nwith\u000bbeyond 40\u000e. This tendency leads to no LAOs for\n\u000b>65\u000e. The two populations can be also distinguished\nin theA\u0000\u000bscatter plot of Figure 22(b) as we will dis-\ncuss inx10.4. The evident correlation between velocity\namplitude and damping time will be discussed in x10.3.\n10.2. Period,P\nThe period re\rects the restoring force and the under-\nlying physics of the oscillation. The period values range\nfrom 30 to 110 min for the total population, with a mean\nvalue of 58 min, a standard deviation of 15 min, and a\nclear peak centered at \u001858 min (Figure 24(b)). The pe-\nriod distributions for LAOs (striped) and SAOs (shaded)\nhave mean values and standard deviations comparable to\nthose of the Pdistribution for all events. This indicates\nthat SAOs and LAOs are not two distinct populations of\nevents with respect to their periods.\nThe period distributions for the three \flament types\ndo not di\u000ber signi\fcantly from each other or from the\ntotal distribution (Figure 24(e)). For IT \flaments the\nmean period is 56 min \u000614 min; the distribution for AR\n\flaments peaks at 57 \u000616 min; the mean period for QS\n\flaments is 62\u000617 min with long-period tail extending\nto 110 min. If LAOs were nonlinear, as discussed in x1,\nthe period could depend on VorA. However, Figures\n21(a) and 21(b), together with the negligible P\u0000Vand\nP\u0000Acorrelation elements, demonstrate that Pdoes not\ndepend on either VorAfor the catalogued events.\nMany theoretical models of MHD modes in \flaments\npredict a relationship between the oscillation period and\nthe \flament length or width (see review by Arregui et al.\n2012). To test this hypothesis, we plotted the oscillation\nperiod as a function of length Land widthWin Figures\n21(c) and 21(d), respectively. We found no correlation\nbetweenPandLfor all types. Although the period is not\ncorrelated with Wfor AR and IT \flaments, QS \flament\nperiods tend to increase with W. The correlation element\nis relatively large, 0.74, and the linear P\u0000Wrelationship\nis\nPQS= 23:4\u00060:4 + (2:31\u00060:02)WQS; (3)\nwherePQSis in minutes and the errors in the period have\nbeen considered. The general tendency is for wider QS\n\flaments to oscillate with longer periods than narrowerGONG Catalog of Solar Filament Oscillations 15\nFigure 19. Temporal sequence of the triggering and oscillations in events 107 and 108*. Panels and annotations are as in Fig. 2.\nFigure 20. Oscillation diagnostics of events 107 and 108*. Panels\nand annotations are as in Fig. 4 .\nprominences.\nFigure 21(e) shows that, for angles \u000b<70\u000e, the range\nof possible periods generally decreases with \u000b. For\u000b <\n20\u000ethe periods occupy the range from 30 to 110 min,\nwhereas for 20\u000e<\u000b< 40\u000ethe periods range from 30 to\n95 min and for 40\u000e<\u000b< 70\u000ethe range is from 30 to 80\nmin. Only a few cases have \u000b>70\u000e, and some of them\ndo not follow this trend. Pdecreases gradually with \u000b,\nso there is a no clear drop in Pfor\u000b>40\u000eas we found\nforV(seex10.1).\nThe decrease of Pwith\u000b, in conjunction with the\nsharp decrease in Vat\u000b > 40\u000e, suggests a connec-tion with the polarization of the oscillations. Theoretical\nmodeling predicts that oscillations along the \feld have\nlonger periods than transverse oscillations. Wang et al.\n(2016) and Zhang et al. (2017a) observed simultaneous\nlongitudinal and transverse oscillations in a prominence,\nand con\frmed that the transverse oscillation period was\nshorter than the longitudinal period. At this point we are\ntempted to de\fne longitudinal and transverse oscillations\naccording to the V\u0000\u000bresults: longitudinal for \u000b<40\u000e\nand transverse for \u000b>40\u000e. High-resolution observations\nreveal that on-disk \flaments are composed of many nar-\nrow, \feld-aligned threads oriented at a shallow angle to\nthe spine (e.g. Lin et al. 2005), and often are composed\nof segments spaced along a common PIL. That is, the\nspine is not necessarily a coherent, magnetically contin-\nuous structure. Transverse oscillations involve coherent\nmovement of the whole magnetic structure or magnet-\nically linked portions thereof, whereas longitudinal os-\ncillations involve individual thread motions at an angle\nwith the spine. In almost all catalog events, only an small\nfraction of the \flament oscillates, suggesting that the lo-\ncal, rather than global, magnetic \feld is engaged. Fur-\nther study of individual, well-observed events is needed\nto resolve whether \u000bis a reliable marker of the boundary\nbetween transverse and longitudinal events.\nInx9.5 we reported two consecutive oscillations in the\nsame \flament during the same data sequence. The pe-\nriods of both events, PandP\u0003, agreed, suggesting that\nthe common period is the characteristic period of oscilla-\ntion of the structure (Ramsey & Smith 1966). However,\nin the cases described in x9.6 andx9.7,PandP\u0003are\nclearly di\u000berent. In Figure 25 the scatter plot of P\u0003\nvsPis shown for all the double events in the catalog.\nFor several cases the ellipse is inside or close to the re-\ngion ofP\u0003\u0018P\u00065 min (region between the two dotted-\nlines). For these cases, we can reasonably consider the\noscillation as a characteristic of the system. The shaded\nellipses correspond to the double cases with ampli\fed\noscillations, which in some events were probably associ-\nated with \raring activity near the \flament (see, e.g., x9.6\nandx9.7). Thus, these oscillations were probably forced\nand are not characteristic motions. However, more cases\nexhibit signi\fcant di\u000berences between P\u0003andPnot as-\nsociated with oscillation ampli\fcation. In almost all of\nthese cases we found that the substantial period di\u000ber-\nences were associated with recon\fguration of the \flament\nstructure. For example, in cases 174-175* and 186-187*16 Luna et al.\nFigure 21. Scatter plots of period, P, vs: (a) velocity amplitude, V. (b) displacement amplitude, A. (c) Length of the spine, L. (d)\nWidth of the spine, W. (e) Angle between the direction of motion and the spine, \u000b. (f) Damping time per period, j\u001c=Pj. The square\nsymbols are for SAO events ( V < 10 km s\u00001) and circles are for LAOs ( V\u001510 km s\u00001). For greater clarity the error bars are not plotted,\nbut can be found in Tables 1-8. The colors represent the \flament type: active region (AR, red), intermediate (IT, green) and quiescent\n(QS, dark blue). The big black diamonds indicate events with negative values of \u001c=P.\nthe \flament structures change with time, judging from\nthe observed \rows along the slit and movements of the\nequilibrium position of the \flament.\n10.3. Damping,\u001cand\u001c=P\nj\u001c=Pjmeasures the number of oscillations within the\ncharacteristic damping time. The absolute value of\n\u001c=P is considered because \u001cis negative when an os-\ncillation is ampli\fed with time, as discussed in xx9.6\nand 9.7. A large value of j\u001c=Pjindicates weak damp-\ning, while a small ratio indicates strong damping. The\nj\u001c=Pjhistogram (Fig. 24(c)) for all events extends from\n0.6 to 2711 (not shown in the histogram), and peaks\natj\u001c=Pj= 1:25. Most events are strongly damped\n(j\u001c=Pj<3), and a signi\fcant number are very strongly\ndamped (j\u001c=Pj<1). A value ofj\u001c=Pj\u001510 essentially\nsigni\fes an undamped oscillation. In contrast with the V\nandPdistributions considered above, the j\u001c=Pjdistribu-\ntions for SAOs and LAOs clearly di\u000ber: the SAO distri-\nbution is wide, with a peak close to 1.75, while the LAO\ndistribution is narrower with a peak near 1.25 and scat-\ntered points at larger values of j\u001c=Pj. The LAO events\n(V > 10 km s\u00001) are mainly below j\u001c=Pj= 3 while SAOs\ncover a larger range. The distributions for the 3 \flament\ntypes appear similar (Fig. 24(f)) to the total j\u001c=Pjdis-\ntribution.\nFigure 22(c) shows that larger velocity amplitudes are\npositively correlated with stronger damping, which in-\ndicates that the higher-speed oscillations are likely to\nbe nonlinear. The sharp transition in the j\u001cjrange at\nV= 10 km s\u00001divides LAOs from SAOs in Figure 22(a),\nre\recting a distinct boundary between linear and nonlin-\near oscillations. The scatter plot j\u001c=Pj-Vis not shown\nbut resembles that of Figure 22(a) with the same trend:the damping time j\u001c=Pjdecreases as Vincreases.\nZhang et al. (2013) found a nonlinear relationship be-\ntween\u001candVin their simulations of prominence mass\nformation: \u001c\u0018V\u00000:3. This scaling law (solid black line\nin Figure 22(c)) is roughly consistent with observed and\nderived values from our events, suggesting that LAOs\nmay be damped through radiative cooling. In their\nmodel, each \rux tube supporting a cool thread has two\ncoronal segments that connect the thread with the chro-\nmosphere at both footpoints. The oscillations alternately\ncompress and rarefy both segments, heating or cooling\nthe coronal plasma. The combined density and temper-\nature increases raise the radiative losses, thus damping\nthe oscillations. The Zhang et al. (2013) model predicted\nthat this e\u000bect could yield a temperature variation of\nseveral hundred thousand Kelvins, which should be ob-\nservable in some EUV lines. An alternative mechanism\nthat can explain strong damping is the mass accretion as-\nsociated with thermal nonequilibrium (Luna & Karpen\n2012; Ruderman & Luna 2016), when evaporated chro-\nmospheric plasma continually condenses onto the promi-\nnence threads. In this model the damping is not related\ndirectly to the oscillation velocity. However, events with\nlargerVare associated with violent events, which could\nproduce increased evaporation and consequently stronger\ndamping. A combination of mass accretion and radiative\ndamping is also possible.\nj\u001c=Pjand dimensions LandWare uncorrelated (the\ncorresponding correlation elements are close to zero), im-\nplying that the damping is not related to the promi-\nnence size. The building blocks of prominences are cool,\nelongated threads aligned with the magnetic \feld, so\nthe damping process is probably associated with the lo-\ncal magnetic or plasma characteristics and not with theGONG Catalog of Solar Filament Oscillations 17\nFigure 22. Scatter plots of (a) damping time, \u001c, vsV. (b)j\u001c=Pj\nvs\u000b. (c)Vvs\u000b. (d)Avs\u000b. Symbols and colors are as in Fig.\n21.\nglobal dimensions of the \flament. Similarly j\u001c=Pjis un-\ncorrelated with PorA.\nFigure 23. Scatter plots of latitude vs: (a) P. (b)\u000b. (c)A. (d)\nV. Symbols and colors are as in Fig. 21.\nThe\u001c=P-\u000bscatter plot (Fig. 22(d)) shows a decreased\nrange of\u001c=P for\u000b>40\u000e. This behavior is similar to the\nV-\u000b(x10.1) and the A-\u000bscatter plots, as we will discuss18 Luna et al.\nFigure 24. Histograms of the number of events binned by V(\frst column), P(second column), and j\u001c=Pj(third column). In the top\nrow the shaded and striped areas represent SAO and LAO events, respectively, for three properties: V(a),P(b) andj\u001c=Pj(c). In (a)\nthe vertical dashed line indicates the separation between SAOs and LAOs at V= 10 km s\u00001. In (b) and (c) the curve with a white area\nunderneath is the histogram of the total number of events. In the bottom row, histograms of (d) V, (e)P, and (f)j\u001c=Pj, divided according\nto the three types of \flaments: active region (AR, red), intermediate (IT, green) and quiescent (QS, blue) are shown.\nFigure 25. Scatter plot of the two periods in double events in\nthe same \flament. Pis the \frst oscillation period and P\u0003is the\nsubsequent one. The data are shown as ellipses where the vertical\nsemi-axis is the error bar for P\u0003and the horizontal semi-axis is the\nerror bar for P. Shaded ellipses are for double events including one\nampli\fed oscillation, with the relevant event numbers written on\nthe side of each ellipse.\ninx10.4.\nEvents 6;7;38;65;108;134;152;156;and 171 were char-\nacterized by ampli\fed oscillations ( \u001c <0). In Figures 21to 23 these cases are marked by symbols surrounded by a\nbig diamond. Cases 6*, 7, 65, 171 are similar to 108* and\n152*: an ampli\fed oscillation prior to a \flament recon-\n\fguration or eruption. Case 38 is less clear but probably\nis associated with recon\fguration. The ampli\fcation in\ncases 134 and 156 is not evident in the time-distance\ndiagrams, and might be associated with \flament proper\nmotions. These ampli\fed oscillations are very interesting\nand deserve to be studied in greater depth.\n10.4. Displacement, A\nThe maximum displacement of the \flament mass with\nrespect to the equilibrium position during the \ftted os-\ncillation,A, was derived from Equation (2):\nA=MAX (jA0e\u0000A1(t\u0000t0)cos [A2(t\u0000t0) +A3]j):(4)\nThe distributions of Afor SAOs and LAOs di\u000ber sub-\nstantially (Figure 26(a)). For SAOs, the distribution is\nconcentrated at the origin with a large peak in the range\n0-5 Mm, many fewer events between 5-10 Mm, and no\nevents with A > 10 Mm. In contrast, LAO displace-\nments cover a larger range ( A= 0-50 Mm), with a peak\nat 7.5 Mm. The Adistributions for the three \flament\ntypes are similar, with a maximum in the range 0-5 Mm\nand a decreasing number of events for increasing A(Fig.\n26(d)).\nIn theP\u0000Ascatter plot (Fig. 21(b)), SAOs are\nconcentrated at A < 10 Mm while LAOs extend up to\nA= 46 Mm. Note that no events have large Aand\nlowP. Because the velocity amplitude is approximately\nV\u0018A=P, the region of large Aand lowPcorrespondsGONG Catalog of Solar Filament Oscillations 19\nFigure 26. Histograms of the number of events binned by A(\frst column), \u000b(second column), and latitude (third column). Panels and\nannotations are as in Fig. 24.\nto very large Vvalues where no events were found in\nour survey. Recently, we discovered an oscillation event\nwith the largest velocity amplitude reported thus far\n(100 km s\u00001) and a displacement of more than 50 Mm\nLuna et al. (2017), which would \ft in the empty region\nof Figure 21(b).\nIn theA\u0000\u000bscatter plot (Fig. 22(b)) we see that\nthe range of displacements is reduced when \u000bincreases.\nSimilar to the V\u0000\u000bor\u001c=P-\u000bplots,Adrops signi\fcantly\nfor\u000b > 40\u000eand there are no LAOs for \u000b > 65\u000e. This\nsuggests that the oscillation or excitation mechanisms\ndi\u000ber on either side of \u000b= 40\u000e, as discussed in xx10.1\nand 10.3. Figure 23(c) shows that Ais independent of\nthe \flament latitude.\n10.5. Direction of motion \u000b\nThe parameter \u000bis the angle between the direction of\nthe oscillation and the \flament spine ( x9). Within the\ncatalog we found oscillations in any direction from 0\u000e\nto 90\u000e(Fig. 26(b)). The total distribution has a peak\nclose to 18\u000eand a mean value of 27\u000e\u000618\u000e. For LAOs,\nthe maximum is \u001828\u000ewith a mean of 25\u000e\u000614\u000e, while\nfor SAOs the peak also is close to 18\u000eand the mean\nvalue is 29\u000e\u000621\u000e. The number of events decreases for\n\u000b>40\u000eand only SAOs have \u000b>65\u000e, as we found for the\nV\u0000\u000b,j\u001c=Pj-\u000bandA\u0000\u000bscatter plots (xx10.1, 10.3, and\n10.4). Therefore we de\fne two populations of oscillations\nwith respect to \u000b: 163 events with \u000b<40\u000eand 33 with\n\u000b>40\u000e.\nFigure 26(b) shows that LAOs and SAOs have sim-\nilar\u000bdistributions. The mean values are consistent\nwith direct measurements of the angle between the \fl-\nament magnetic \feld and its spine ( \u000b\u001825\u000eon average;\nLeroy et al. 1983, 1984; E. Tandberg-Hanssen 1995; Tru-jillo Bueno et al. 2002; Casini et al. 2003; L\u0013 opez Ariste\net al. 2006). This suggests that most of the oscillations\nin the catalogued events are aligned with the magnetic\n\feld (longitudinal).\nThe\u000bdistribution is clearest for IT events (Fig. 26(e)):\nthe oscillations are aimed in all directions, but the peak\ncoincides with the mean at 25\u000e\u000613\u000e. The\u000bdistribution\nfor AR events has a peak at 37.5\u000ewith a mean value of\n25\u000e\u000614\u000e. Interestingly, there are no oscillations in AR\n\flaments with \u000b>45\u000e, indicating that the motions are\nmainly longitudinal. For QS \flaments the distribution\nhas a maximum around 18\u000eand a mean value at 22\u000e\u0006\n20\u000e. The QS\u000b-distribution covers the entire domain, but\nthe oscillation o\u000bsets are mainly below 40\u000e. In summary,\nthe mean\u000bvalues for all \flament types agree with the\nobserved magnetic-\feld orientation relative to the spine,\nimplying longitudinal polarization, particularly for IT \fl-\naments.\n10.6. Latitude\nProminence oscillations may re\rect the global struc-\nture of the supporting \flament channels, which is intrin-\nsically tied to the large-scale solar magnetic \feld. Figure\n23 displays several oscillation properties | V,P,A, and\n\u000b| as functions of solar latitude in Stonyhurst Helio-\ngraphic Coordinates (Thompson 2006). V,P, and\u000bgen-\nerally display larger ranges of values at speci\fc latitudes.\nFor IT \flaments, these oscillation properties largely oc-\ncupy the region between \u000025\u000eand 0\u000elatitudes (see Fig-\nures 23(a), (b), and (d)). In contrast, AR events exhibit\nlarger ranges of these properties around two latitudes,\n\u000015\u000eand 15\u000e. ForAthis trend is less evident (Figure\n23(c)), but a 2D histogram (not shown) reveals the same\ntrend.20 Luna et al.\nThe latitude distribution (Figure 26(c)) shows that all\nsurvey events were located between 50\u000eand\u000050\u000e, typi-\ncal for solar maximum. However, a substantial fraction of\nevents accumulated around \u000015\u000e, in the southern hemi-\nsphere, regardless of oscillation type (SAO or LAO). In\nFigure 26(f) the latitude distributions for the three \fla-\nment types are shown. The distribution peaks at \u000015\u000e\nand 15\u000efor AR \flaments, at \u000015\u000efor IT \flaments, and\nat\u000025\u000eand 5\u000efor QS \flaments. It is evident From Fig-\nures 26(c) and 26(f) that the regions of a large number\nof events coincide with the regions of large dispersion of\noscillation parameters of Figure 23. This suggests that\nin those latitudes there are more \flaments and more ac-\ntivity triggering oscillations. In this sense, the existence\nof these latitudes is not necessarily showing a latitudinal\ndependence of oscillation parameters or intrinsic charac-\nteristics of the \flaments.\nBashkirtsev & Mashnich (1993) found a smooth, sinu-\nsoidal latitudinal dependence for 30 SAO events observed\nover more than 8 years, with periods of 80 min at \u000020\u000e\nand 20\u000elatitudes and 40 min at 0\u000e. We have not found a\nclear relationship between the periods or other properties\nand the \flament latitude. Their study covered almost a\nsolar cycle, so their latitudinal dependence could be re-\nlated to the well-known migration of \flaments from the\npoles toward the equator during the cycle. To determine\nwhether this potentially profound relationship is solid,\nour catalog would have to be expanded signi\fcantly to\ninclude oscillation events throughout at least 1 solar cy-\ncle.\n11. SEISMOLOGY\nProminence seismology combines observations and the-\noretical modeling to infer hard-to-measure parameters\nsuch as the magnetic \feld (see x1). There are essen-\ntially three driving mechanisms for prominence oscilla-\ntions: gravitational force, pressure imbalance, and mag-\nnetic Lorentz force.\nLongitudinal oscillations are driven by a combination\nof gravity projected along the \feld (pendulum model,\nLuna & Karpen 2012) and gas pressure gradients (slow\nmodes, Joarder & Roberts 1992). In the pendulum\nmodel, the period depends exclusively on the radius of\ncurvature of the dips supporting the cool prominence\nplasma,R. Luna et al. (2012) and Zhang et al. (2013)\ndetermined that gas pressure gradients contribute negli-\ngibly to the restoring force when the radius of curvature\nis much smaller than a limit de\fned by the prominence\ncharacteristics ( R\u001cRlim), whereRlimis\nRlim= 1=4Lt(Lf\u0000Lt)\u0014g=c2\nsc: (5)\nHere\u0014is the temperature contrast between the cool and\nadjacent hot plasmas, Lfis the \feld line length, Ltis the\nthread length, gis the solar gravitational constant, and\ncscis the coronal sound speed. In that case the period is\nP= 2\u0019s\nR\ng: (6)\nAssuming that the magnetic tension in the dipped\npart of the tubes must be larger than the weight of the\nthreads, the minimum magnetic-\feld strength, B, de-\npends on the particle number density of the prominencethread,n, and the period P. In the absence of direct\ndensity measurements, Luna et al. (2014) adopted the\nrange of typical values n= 1010\u00001011cm\u00003as the main\nsource of uncertainty and determined that\nB(G)\u0015(0:28\u00060:15)P(min): (7)\nFor transverse horizontal oscillations, Kleczek & Kupe-\nrus (1969) assumed that the \flament was supported by\na single line-tied magnetic \rux tube, and that the restor-\ning force was supplied by magnetic tension. We assume\nagain thatntakes typical prominence values, and using\ntheir Eq. (9), we \fnd\nB(G) = (5:5\u00063)L(Mm)\nP(min); (8)\nwhereLis the length of the \flament. The uncertainty\nin the numerical coe\u000ecient is associated with the uncer-\ntainty inn.\nWithout additional data analysis and \feld extrapola-\ntion (e.g., Luna et al. 2017), it is di\u000ecult to establish\nwhich catalog events are oscillations parallel or perpen-\ndicular to the magnetic \feld. However, our statistical\nanalysis revealed a clear distinction between oscillations\nwith\u000b < 40\u000eand those with \u000b > 40\u000e(x10). Although\nthe two populations are not necessarily uniquely associ-\nated with di\u000berent oscillation polarizations, for seismol-\nogy purposes we applied the longitudinal model to the\noscillations with \u000b < 40\u000eand the transverse model to\nthe\u000b>40\u000ecases. This is also justi\fed because the two\nmodels predict approximately the same Bfor a given\nevent. We determined BandRfrom Equations (6) and\n(7) for the events with \u000b < 40\u000e(Figure 27(a)). The\nshaded area covers the uncertainties in B. The magnetic\n\feld ranges from 9 to 48 G, andRfrom 25 to 300 Mm.\nThe mean values are B= 16 G and R= 89 Mm. The\nobtained values are consistent with the rare direct mea-\nsurements of prominence magnetic \felds (see review by\nMackay et al. 2010).\nThe magnetic \feld plotted in Figure 27(a) is a lower\nlimit, so we expect larger values to occur. In particular\nthe \feld could be signi\fcantly underestimated for small\nradii of curvature, R. The reason is that the magnetic\ntension is proportional to B2=Rand the weight of the\nprominence is proportional to ng. Thus, assuming sim-\nilarn, theBnecessary to balance the gravity is smaller\nfor smaller Rthan for larger R.\nIn order to check the validity of the pendulum model,\nwe computed Equation 5 and compared it with Rfor all\ncatalog cases. Because we do not have direct measure-\nments ofLfandLt, we usedLandW, the length of the\nspine and width of the \flament. Wis probably compa-\nrable to the thread lengths, but Lis a lower limit on the\nlength of the sheared \feld lines in the \flament channel\nfor\u000b>0.cscis typically\u0018200 km s\u00001and the typical\ntemperature contrast is \u0014= 100. The resulting Rlimis\nlargely greater than Rlim, demonstrating the applicabil-\nity of the pendulum model to the catalog events.\nFigure 27(b) shows the inferred magnetic \feld as a\nfunction of \u000b. The pendulum model (Eq. (7)) is used\nfor events with \u000b<40\u000e, and transverse model (Eq. (8))\nfor\u000b >40\u000e. For longitudinal oscillations ( \u000b <40\u000e) the\nBrange generally decreases with \u000b, reminiscent of the\nbehavior of P. The same trend applies to the transverseGONG Catalog of Solar Filament Oscillations 21\nFigure 27. Seismology diagnostics for longitudinal and transverse\noscillations. (a) The lower limit on Bas a function of Rfor lon-\ngitudinal oscillations. The shaded area corresponds to the uncer-\ntainty range. (b) The estimated magnetic \feld strength for events\nwith longitudinal and transverse oscillations, from Equations (7)\nand (8) respectively. The vertical dot-dashed line indicates the as-\nsumed separation between longitudinal and transverse oscillations.\nSymbols and colors are as in the scatter plots.\noscillations ( \u000b>40\u000e), although some events reach large\nBvalues (38 G). For transverse oscillations the Bval-\nues are consistent with direct measurements (see, e.g.,\nHarvey 1969). Our AR events are all longitudinal, while\nIT and QS events occupy both categories. It is interest-\ning to note that the minimum \feld strengths do not dif-\nfer signi\fcantly among the \flament types, although AR\n\flaments are embedded in higher \feld-strength regions.\nThis lower limit is consistent with direct measurements\nin AR \flaments (Kuckein et al. 2009; Sasso et al. 2010;\nKuckein et al. 2012; Sasso et al. 2014) showing strong\n\felds of up to several hundred Gauss.\n12. SUMMARY AND CONCLUSIONS\nIn this work we have surveyed prominence oscilla-\ntions detected through visual inspection of the GONG\nnetwork H\u000bdata during January - June 2014, provid-\ning an extensive sample of events close to solar maxi-\nmum of cycle 24. We have catalogued a large variety\nof oscillations including strongly damped motions, un-\ndamped oscillations, and ampli\fed oscillations, enabling\nthe \frst statistically signi\fcant study of \flament oscilla-\ntions and their pertinent properties. The \flament and\noscillation parameters are described in the text and Ta-bles; additional information and animations can be found\nin the online catalog: http://www.iac.es/galeria/\nmluna/pages/gong-catalogue-of-laos.php .\nWe have found 196 oscillation events, including 106\nSAOs and 90 LAOs. In 85 cases we have identi\fed the\ntriggering agents of the oscillations as \rares, prominence\neruptions, a jet, and a Moreton wave. For the remaining\n111 events the triggering agent is not identi\fed. The\noccurrence rate of one LAO event every two days implies\nthat LAOs are common phenomena on the Sun, as are\nSAOs.\nWe have parametrized the oscillations by \ftting an ex-\nponentially decaying sinusoid, and statistically the dis-\ntributions and correlations of key physical parameters.\nThe \ftted velocity amplitudes, V, are in the range\n1:6\u000055 km s\u00001, and show a clear tendency to occur\nless frequently with increased V. This indicates that the\nLAOs are less common than SAOs, particularly since we\nprobably underestimated the number of SAOs approach-\ning the small-amplitude limit. The Vrange decreases\nwith\u000b, dropping sharply for events beyond 40\u000e, and\nthere are no LAOs for \u000b>65\u000e.\nThe oscillation periods, P, range from 32 to 110 min.\nSurprisingly, the periods of both LAOs and SAOs have\nwell-de\fned distributions centered at P= 58\u000615 min.\nThis indicates that LAOs and SAOs are not two dis-\ntinct populations of events with respect to their periods.\nFor all three \flament types the mean oscillation period\nis around 1 hour. The Prange decreases with the an-\ngle between the oscillation displacement and the \flament\nspine,\u000b. In general, we have not found strong correla-\ntions between Pand other oscillation parameters.\nThe damping time per period, \u001c=P covers a large\nrange, including some cases with negative values (am-\npli\fcation). The \u001c=P distribution for LAOs peaks at\n1.25, and most of the events exhibit very strong damp-\ning. For SAOs, the range of observed \u001c=Pvalues is wider,\npeaking at 1.75. The three \flament types behave simi-\nlarly. For LAOs \u001cand\u001c=P decrease with V, regardless\nof \flament type, con\frming that LAOs involve nonlin-\near motions with velocity-dependent damping. This is\na very interesting result because the kinetic energy in-\nvolved in large-amplitude oscillations is enormous, due\nto the combination of large thread masses and large ve-\nlocities. Therefore the physical mechanism must be ef-\n\fcient enough to damp the substantial motion in a few\noscillations. Our earlier theoretical studies showed that\nreasonable rates of mass accretion could explain the ob-\nserved damping rates. On the other hand, the observed\nrelation between \u001candVis consistent with the Zhang\net al. (2013) scaling law, \u001c\u0018V\u00000:3, which suggests that\nthe damping is associated with radiative cooling. More\nobservational and theoretical work needs to be done to\nunderstand the damping process more thoroughly.\nFor the catalog events, the direction of the motion with\nrespect to the \flament spine, \u000b, covers all possible angles\nbetween 0\u000eand 90\u000e, and the\u000bdistributions for LAOs\nand SAOs exhibit no clear peak. However, the mean \u000b\nvalue is the same for all three \flament types: 27\u000e, which\nagrees with previous direct measurements of \u000b\u001825\u000eon\naverage (Leroy et al. 1983, 1984; E. Tandberg-Hanssen\n1995; Trujillo Bueno et al. 2002; Casini et al. 2003; L\u0013 opez\nAriste et al. 2006). Thus, most of the oscillation displace-\nments are probably aligned with the \flament magnetic22 Luna et al.\n\felds.\nWe have not found evidence of any relationships be-\ntween the oscillation parameters and the solar latitude,\nin contrast to the \fndings of Bashkirtsev & Mashnich\n(1993). However, their study covered almost a solar cy-\ncle, and their latitudinal dependence could be associated\nwith the well-known migration of \flaments from poles to\nequator. To determine whether this profound relation-\nship is solid, our catalog must be expanded to include\nevents throughout at least 1 solar cycle .\nWe have applied seismological techniques to the en-\ntire catalog. For the longitudinally oscillating cases,\nwe determined the radius of curvature of the magnetic\ndips hosting the prominence, R, and the minimum \feld\nstrength,B, required to support the mass against grav-\nity.R= 25\u0000300 Mm and B= 2\u000038 G with mean values\nofR= 89 Mm and B= 17 G. For transverse oscillations,\nthe magnetic \feld strength derived from the magnetic\nrestoring force yields a wider range of B= 2\u000038 G but\na similar mean value.\nMost of the oscillations are longitudinal, with the mo-\ntion directed along the local magnetic \feld. Surprisingly,\nthe period distributions for both SAOs and LAOs have\na strong peak centered at 58 min, which implies that\nmost solar \flaments share a common structure. Namely,\ntheir structure is composed of dipped \rux tubes with a\nradius of curvature of \u001890 Mm and an angle between\nthe threads and the spine of \u001830\u000e. The magnetic-\feld\nstrength is probably larger than the minimum estimate\nof 16 G. We also found that many SAOs are initiated by\nenergetic disturbances, which contradicts the idea that\nSAOs are exclusively driven by photospheric or chromo-\nspheric waves. On the other hand, Ning et al. (2009) and\nHillier et al. (2013) studied numerous oscillations in small\nprominence features, and found velocities in general be-\nlow 10 km s\u00001and periods of the order of minutes. These\nlocalized versions of SAOs are more consistent with wave\ndriving than our SAOs, which a\u000bect large portions or the\nentire \flament.In future research we will extend the catalog to events\nnear the solar minimum of the same cycle 24, to augment\nour statistics and explore the possibility that oscillation\nparameters and \flament properties evolve during the so-\nlar cycle. We invite the community to utilize this cata-\nlog for other research projects and to aid in expanding\nits contents, in order to advance our understanding of\nthe fundamental structure and evolution of solar promi-\nnences.\nThe Global Oscillation Network Group (GONG) Pro-\ngram is managed by the NSO and operated by AURA,\nInc. under a cooperative agreement with the NSF.\nThe data are acquired by instruments operated by the\nBig Bear Solar Observatory, High Altitude Observa-\ntory, Learmonth Solar Observatory, Udaipur Solar Ob-\nservatory, Instituto de Astrof\u0013 \u0010sica de Canarias, and\nCerro Tololo Interamerican Observatory. The opera-\ntion of Big Bear Solar Observatory is supported by\nNJIT, US NSF AGS-1250818, and NASA NNX13AG14G\ngrants. This paper made use of the IAC Supercom-\nputing facility HTCondor ( http://research.cs.wisc.\nedu/htcondor/ ), partly \fnanced by the Ministry of\nEconomy and Competitiveness with FEDER funds, code\nIACA13-3E-2493. This research also made use of NASA\nAstrophysics Data System.\nThis work was initiated during International Space\nScience Institute (ISSI) team 314 meetings in Bern led\nby M. Luna on \\Large-Amplitude Oscillations in So-\nlar Prominences\". M. Luna acknowledges the support\nby the Spanish Ministry of Economy and Competitive-\nness through project AYA2014-55078-P. H. Gilbert, J.\nKarpen, T. Kucera and K. Muglach acknowledge sup-\nport by the NASA Heliophysics Guest Investigator pro-\ngram. J. Terradas and J. L. Ballester want to thank the\n\fnancial support from MINECO AYA2014-54485-P and\nFEDER Funds, and the Conselleria d'Innovaci\u0013 o, Recerca\ni Turisme del Govern Balear to IAC3.GONG Catalog of Solar Filament Oscillations 23\nAPPENDIX\nEVENT CATALOG\nEvent # Time Tel. Pos. (x,y) Type L (Mm) W (Mm) Trigger Erupts\n1 1-Jan-2014 16:50 C -18, -76 IP 269 15 FLARE\n2* 1-Jan-2014 16:50 C -18, -76 IP 269 15 FLARE\n3 4-Jan-2014 22:04 B -251, 302 IP 172 7 FLARE\n4 5-Jan-2014 17:03 C -399, 48 IP 87 14 FLARE\n5 5-Jan-2014 17:03 C -308, -116 AR 240 13\n6* 5-Jan-2014 17:03 C -308, -116 AR 240 13\n7 5-Jan-2014 17:03 C 730, -430 QS 139 10 PE\n8* 5-Jan-2014 17:03 C 730, -430 QS 139 10 PE\n9 5-Jan-2014 22:11 B 806, -79 IP 206 11\n10 6-Jan-2014 08:59 U 84, 492 QS 44 9 FLARE\n11 6-Jan-2014 07:11 U -132, -261 IP 185 8\n12 6-Jan-2014 16:56 C -268, -466 QS 102 8\n13 6-Jan-2014 16:56 C -184, 57 AR 78 12\n14* 6-Jan-2014 16:56 C -184, 57 AR 78 12\n15 6-Jan-2014 08:59 U 57, 304 IP 119 8 FLARE\n16 7-Jan-2014 09:13 U -335, 78 IP 387 13 FLARE\n17 7-Jan-2014 09:13 U -325, -282 IP 190 7 FLARE\n18 7-Jan-2014 09:13 U 143, -370 IP 445 8\n19 7-Jan-2014 16:58 C -100, -501 QS 143 12 PE\n20 8-Jan-2014 08:59 U -139, -316 IP 313 14 PE\n21 8-Jan-2014 08:59 U 8, -494 IP 187 10\n22* 8-Jan-2014 08:59 U 8, -494 IP 187 10\n23 8-Jan-2014 08:59 U 260, -367 IP 543 9 FLARE\n24 9-Jan-2014 17:10 C 155, -306 IP 287 20 FLARE\n25 9-Jan-2014 17:10 C 242, -505 IP 303 12\n26 10-Jan-2014 14:03 T 433, 80 IP 54 9\n27 11-Jan-2014 16:43 C 815, -393 IP 221 9 PE\n28 15-Jan-2014 20:00 B 502, -217 AR 125 6\n29 16-Jan-2014 07:37 U 268, 125 AR 80 15\n30 16-Jan-2014 19:59 B -624, -298 IP 228 9\n31 24-Jan-2014 17:00 C -311, -385 IP 281 10 FLARE\n32 24-Jan-2014 19:11 M 284, -151 IP 394 14\n33 25-Jan-2014 07:40 U -184, -376 IP 251 8\n34 25-Jan-2014 16:41 C -110, -129 AR 163 10 FLARE\n35* 25-Jan-2014 16:41 C -110, -129 AR 163 10 FLARE\n36 25-Jan-2014 16:41 C 508, -95 IP 527 15 FLARE\n37 26-Jan-2014 08:27 U 850, 99 IP 81 8\n38 27-Jan-2014 16:41 C 335, -142 AR 255 9\n39* 27-Jan-2014 16:41 C 335, -142 AR 255 9\n40 27-Jan-2014 16:41 C 449, -402 IP 209 17\n41 28-Jan-2014 20:23 B 635, -420 IP 247 9\n42 29-Jan-2014 07:23 U 445, -304 IP 82 11\n43 29-Jan-2014 16:49 C 556, -428 IP 251 9\n44 29-Jan-2014 19:59 B -822, 11 IP 182 11 FLARE\n45 30-Jan-2014 07:23 U 702, 33 IP 195 9 PE\n46 31-Jan-2014 15:20 C -557, 16 IP 201 15\n47 1-Feb-2014 07:17 U -404, 58 AR 317 9 FLARE\n48 5-Feb-2014 17:07 C -812, -129 IP 206 10 FLARE Y\n49 5-Feb-2014 17:07 C -212, -269 IP 205 10 FLARE\n50 6-Feb-2014 17:05 C -626, -433 IP 363 13 FLARE\n51 6-Feb-2014 17:05 C -388, -266 IP 231 10 FLARE\n52 7-Feb-2014 13:24 T -506, 243 IP 111 6\n53 8-Feb-2014 20:10 B 428, -271 AR 211 9 FLARE\n54 8-Feb-2014 13:24 T -193, 114 IP 84 16\n55 8-Feb-2014 13:24 T -122, -162 AR 181 8\n56 9-Feb-2014 16:52 C -82, -93 IP 162 11 FLARE\n57 9-Feb-2014 16:52 C -192, -119 IP 109 8 FLARE\n58 9-Feb-2014 16:52 C -390, -197 AR 232 7 FLARE\n59 11-Feb-2014 17:39 C 517, -203 AR 101 9 FLARE\n60 12-Feb-2014 13:28 T 368, 297 AR 184 8 FLARE\nTable 1\nTable of the observation details and parameters of the \flamet for events 1 to 60. In \frst column the asterisk indicates that the oscillation\nis in the same time-distance diagram than in previous case.24 Luna et al.\nEvent # Time Tel. Pos. (x,y) Type L (Mm) W (Mm) Trigger Erupts\n61 12-Feb-2014 19:42 B 571, -105 IP 160 14 FLARE\n62 13-Feb-2014 17:05 C 601, 238 AR 236 6 FLARE\n63 13-Feb-2014 20:04 B 708, -382 QS 721 33 FLARE\n64 14-Feb-2014 13:27 T 723, 248 AR 295 9 FLARE\n65 14-Feb-2014 19:34 B -592, -233 IP 805 8 FLARE\n66 16-Feb-2014 19:51 B -17, -93 AR 132 7 FLARE\n67 17-Feb-2014 07:28 U -110, -202 IP 898 14 FLARE\n68 17-Feb-2014 18:52 B 3, -199 IP 1051 13\n69 19-Feb-2014 19:47 B 380, -215 IP 913 13 FLARE\n70 22-Feb-2014 07:28 U -16, -378 IP 359 9\n71 23-Feb-2014 16:55 C -448, 277 AR 120 12 FLARE\n72 24-Feb-2014 16:59 C 44, -311 IP 219 22\n73 24-Feb-2014 16:59 C 330, -267 IP 200 10\n74 25-Feb-2014 16:38 C 548, -213 IP 447 7\n75 25-Feb-2014 16:38 C 366, -172 AR 114 8\n76 25-Feb-2014 16:38 C 526, 144 QS 46 12 Y\n77 27-Feb-2014 13:22 T -1, 164 IP 209 9 FLARE\n78 27-Feb-2014 16:57 C -203, 350 IP 147 9\n79* 27-Feb-2014 16:57 C -203, 350 IP 147 9\n80 28-Feb-2014 13:22 T 125, -328 QS 813 24\n81 7-Mar-2014 17:16 C 590, -74 IP 437 12 FLARE\n82 12-Mar-2014 07:29 U -567, -237 IP 210 9 FLARE\n83 14-Mar-2014 16:58 C -812, 265 AR 192 8 FLARE\n84 16-Mar-2014 07:22 U -539, -213 IP 831 13 PE\n85 20-Mar-2014 16:59 C -315, -4 IP 242 9\n86 21-Mar-2014 19:21 B 521, 347 AR 335 11 PE\n87 23-Mar-2014 16:54 C 398, 0 IP 320 12 JET\n88 24-Mar-2014 18:57 B 586, -13 IP 152 9\n89* 24-Mar-2014 18:57 B 586, -13 IP 152 9\n90 28-Mar-2014 18:09 B 683, 41 IP 79 18\n91 29-Mar-2014 16:57 C 329, 372 IP 120 9 MW\n92 29-Mar-2014 16:57 C 99, -201 AR 217 6 FLARE\n93 30-Mar-2014 07:10 U 83, -405 IP 95 10\n94 31-Mar-2014 18:49 B 379, -390 IP 289 14 FLARE\n95* 31-Mar-2014 18:49 B 379, -390 IP 289 14 FLARE\n96 7-Apr-2014 18:33 B 522, 416 AR 72 6 FLARE\n97 9-Apr-2014 12:42 T 469, 216 IP 139 9 FLARE\n98 9-Apr-2014 18:30 M 520, 200 AR 111 8 Y\n99 10-Apr-2014 05:07 L 653, -431 QS 458 18\n100 14-Apr-2014 16:44 C 146, -110 IP 248 10 Y\n101 17-Apr-2014 05:01 L -648, -161 IP 531 9 FLARE\n102 17-Apr-2014 05:01 L 297, 447 QS 210 20\n103 17-Apr-2014 16:52 C 5, -324 IP 177 8 PE\n104 17-Apr-2014 16:52 C 731, -150 QS 141 21\n105 19-Apr-2014 04:48 L 602, 466 QS 240 17\n106 19-Apr-2014 16:47 C -185, -125 IP 691 11 FLARE\n107 21-Apr-2014 18:20 B 701, -371 IP 149 7 FLARE Y\n108* 21-Apr-2014 18:20 B 701, -371 IP 149 7 FLARE Y\n109 22-Apr-2014 16:41 C 61, 296 IP 229 6\n110 23-Apr-2014 16:49 C 264, 294 IP 270 7 FLARE\n111 24-Apr-2014 07:05 U -262, -263 AR 197 9\n112 24-Apr-2014 07:05 U 205, 148 IP 289 11\n113 25-Apr-2014 07:10 U 424, 132 IP 264 14\n114 26-Apr-2014 13:27 T 465, -370 IP 800 14 PE\n115 1-May-2014 18:11 B -380, 231 IP 290 8 FLARE\n116 1-May-2014 18:11 B 753, -385 IP 428 11 FLARE\n117 1-May-2014 07:10 U -625, -379 IP 137 12\n118 2-May-2014 07:04 U -296, 190 AR 212 9 FLARE\n119* 2-May-2014 07:04 U -296, 190 AR 212 9 FLARE\n120 2-May-2014 13:42 T 36, -342 IP 173 12\nTable 2\nSame as Table 1 for events 61 to 120.GONG Catalog of Solar Filament Oscillations 25\nEvent # Time Tel. Pos. (x,y) Type L (Mm) W (Mm) Trigger Erupts\n121 5-May-2014 17:22 B 92, -391 IP 291 8\n122 10-May-2014 06:31 U -499, 417 IP 254 14\n123 10-May-2014 13:07 T -181, -581 QS 71 19 PE\n124 11-May-2014 18:06 B -610, -486 QS 395 13 FLARE\n125 12-May-2014 04:11 L -142, 443 QS 117 15\n126 12-May-2014 04:11 L 77, -22 IP 192 12 FLARE\n127 12-May-2014 04:11 L -494, -634 QS 330 10\n128 12-May-2014 07:21 U 281, -102 IP 96 9\n129 12-May-2014 18:01 M -346, 361 IP 164 14\n130 13-May-2014 18:07 B 323, -26 IP 115 9\n131 13-May-2014 18:07 B -86, -464 IP 222 18\n132 14-May-2014 07:22 U 413, -626 QS 149 35\n133 15-May-2014 16:43 C 285, 142 IP 330 25\n134 16-May-2014 06:47 U 354, 197 QS 246 33\n135 16-May-2014 13:07 T -514, 355 IP 142 9 FLARE\n136 17-May-2014 04:54 L 521, -70 IP 202 12\n137 17-May-2014 13:07 T 649, 135 QS 175 14\n138 18-May-2014 19:54 B 455, -96 IP 133 13\n139 23-May-2014 12:58 T -101, 240 IP 228 11\n140 23-May-2014 13:07 T 269, -302 IP 292 9\n141 23-May-2014 19:27 B 658, 146 QS 164 18\n142 23-May-2014 18:10 B -577, -192 IP 329 9 FLARE\n143 26-May-2014 03:39 L 243, -432 IP 603 11 FLARE\n144 26-May-2014 17:08 C 743, -32 QS 97 18\n145 26-May-2014 16:45 C -348, -260 IP 143 10 FLARE\n146* 26-May-2014 16:45 C -348, -260 IP 143 10 FLARE\n147 27-May-2014 04:26 L 280, -233 AR 121 8\n148 27-May-2014 11:24 T 552, -404 IP 796 9\n149 28-May-2014 04:21 L -26, -352 AR 219 7 FLARE\n150 29-May-2014 04:37 L 184, -346 AR 183 8\n151 30-May-2014 05:31 L 371, -353 AR 235 11 FLARE\n152* 30-May-2014 05:31 L 371, -353 AR 235 11 FLARE\n153 30-May-2014 11:42 T 765, -153 IP 152 8 FLARE\n154* 30-May-2014 11:42 T 765, -153 IP 152 8 FLARE\n155 1-Jun-2014 04:36 L 692, -354 AR 123 7 FLARE Y\n156 2-Jun-2014 13:25 T 291, 166 IP 239 19\n157 2-Jun-2014 17:45 B -203, 59 QS 124 19\n158 3-Jun-2014 04:32 L -85, 54 QS 166 15\n159 4-Jun-2014 17:42 B -628, -55 IP 339 10\n160 5-Jun-2014 13:09 T 97, -236 IP 256 9\n161 5-Jun-2014 19:42 M -174, 404 QS 98 12\n162 6-Jun-2014 13:14 T 323, -235 IP 104 12 Y\n163 7-Jun-2014 04:45 L 23, -204 IP 108 15\n164 7-Jun-2014 13:09 T -592, -239 IP 206 9\n165* 7-Jun-2014 13:09 T -592, -239 IP 206 9\n166 8-Jun-2014 13:09 T 3, 205 AR 290 11\n167 8-Jun-2014 17:56 B -345, -224 IP 355 13\n168 9-Jun-2014 13:11 T 242, 212 AR 229 11\n169* 9-Jun-2014 13:11 T 242, 212 AR 229 11\n170 9-Jun-2014 13:11 T 379, 6 IP 326 17\n171 9-Jun-2014 13:11 T 45, -304 AR 166 13\n172* 9-Jun-2014 13:11 T 45, -304 AR 166 13\n173 10-Jun-2014 18:52 M 499, 220 AR 73 17 FLARE Y\n174 12-Jun-2014 04:33 L -328, -575 QS 99 12\n175* 12-Jun-2014 04:33 L -328, -575 QS 99 12\n176 12-Jun-2014 04:46 U 289, -344 IP 176 13\n177* 12-Jun-2014 04:46 U 289, -344 IP 176 13\n178 12-Jun-2014 04:33 L 438, -202 IP 166 11\n179 12-Jun-2014 12:51 T 469, 490 QS 196 13\n180* 12-Jun-2014 12:51 T 469, 490 QS 196 13\nTable 3\nSame as Table 1 for events 121 to 180.26 Luna et al.\nEvent # Time Tel. Pos. (x,y) Type L (Mm) W (Mm) Trigger Erupts\n181 13-Jun-2014 04:30 L -163, -574 QS 125 16\n182 13-Jun-2014 13:11 T 599, -297 IP 606 19\n183 14-Jun-2014 04:40 L 672, -294 IP 288 13\n184 14-Jun-2014 13:10 T 745, 55 IP 23 7\n185 14-Jun-2014 13:10 T -527, 490 QS 58 10 FLARE\n186 15-Jun-2014 17:51 M -345, 512 IP 86 16 FLARE\n187* 15-Jun-2014 17:51 M -345, 512 IP 86 16 FLARE\n188 16-Jun-2014 13:11 T -836, 5 IP 189 15\n189 16-Jun-2014 13:11 T -214, 521 IP 61 14\n190* 16-Jun-2014 13:11 T -214, 521 IP 61 14\n191 17-Jun-2014 17:56 B -334, -510 QS 116 12\n192 17-Jun-2014 13:12 T -757, -7 IP 301 13 FLARE\n193 17-Jun-2014 13:12 T -39, 520 IP 119 12\n194 17-Jun-2014 13:12 T 695, 385 IP 107 7 FLARE\n195 19-Jun-2014 18:44 M 241, -231 IP 131 13\n196 29-Jun-2014 16:50 C 130, -200 IP 691 21\nTable 4\nSame as Table 1 for events 181 to 195.GONG Catalog of Solar Filament Oscillations 27\n# Time Ini. \u000b(\u000e)P(min.)\u001c(min.) \u001c=P A (Mm) V (km s\u00001)\n1 1-Jan-2014 13:50 16 76\u00062 121\u000615 1.6\u00060.2 22\u00062 26.5\u00063.6\n2* 1-Jan-2014 18:56 16 71\u00063 173\u0006107 2.4\u00062. 8\u00063 10.7\u00067.0\n3 4-Jan-2014 20:30 26 57\u00062 105\u000633 1.8\u00060.6 4\u00061 8.0\u00063.5\n4 5-Jan-2014 11:43 9 63\u00064 | | 5\u00063 9.4\u00069.2\n5 5-Jan-2014 12:01 32 52\u00061 | | 1\u00061 2.7\u00066.0\n6* 5-Jan-2014 17:05 32 76\u00063 -115\u000638 -1.5\u00060.6 3\u00061 3.5\u00066.0\n7 5-Jan-2014 10:44 0 63\u00063 -390\u0006545 -6.2\u00069. 2\u00061 3.9\u00066.7\n8* 5-Jan-2014 21:31 0 43\u00063 160\u0006329 3.7\u00068. 5\u00063 10.7\u000611.1\n9 5-Jan-2014 20:13 37 44\u00062 38\u00069 0.9\u00060.2 5\u00062 11.6\u00066.2\n10 6-Jan-2014 06:58 22 68\u00061 617\u0006367 9.0\u00065. 8\u00061 12.4\u00061.9\n11 6-Jan-2014 08:02 23 63\u00062 94\u000623 1.5\u00060.4 4\u00061 5.5\u00063.8\n12 6-Jan-2014 11:07 38 40\u00063 158\u0006141 4.0\u00064. 2\u00061 6.5\u00067.9\n13 6-Jan-2014 12:27 16 61\u00061 118\u000618 1.9\u00060.3 10\u00061 15.4\u00062.1\n14* 6-Jan-2014 16:26 16 56\u00066 475\u00061313 8.5\u000620 3\u00062 5.6\u000611.5\n15 6-Jan-2014 06:45 1 64\u00061 | | 8\u00061 12.8\u00061.7\n16 7-Jan-2014 05:58 14 65\u00062 302\u0006158 4.7\u00063. 4\u00061 6.2\u00063.6\n17 7-Jan-2014 04:04 3 46\u00061 96\u000624 2.1\u00060.5 5\u00061 11.0\u00064.3\n18 7-Jan-2014 04:45 23 50\u00061 46\u00065 0.9\u00060.1 15\u00062 25.5\u00064.7\n19 7-Jan-2014 14:16 26 44\u00061 83\u00068 1.9\u00060.2 14\u00062 31.2\u00064.5\n20 8-Jan-2014 05:47 30 50\u00062 67\u000615 1.4\u00060.3 15\u00063 36.5\u00068.8\n21 8-Jan-2014 04:49 2 43\u00062 100\u000645 2.3\u00061. 4\u00061 9.0\u00064.7\n22* 8-Jan-2014 08:18 2 35\u00062 | | 1\u00061 2.5\u00069.3\n23 8-Jan-2014 03:28 24 59\u00061 102\u000614 1.7\u00060.2 7\u00061 14.7\u00062.9\n24 9-Jan-2014 16:56 40 51\u00062 97\u000629 1.9\u00060.6 5\u00062 9.3\u00065.7\n25 9-Jan-2014 12:31 44 46\u00063 73\u000633 1.6\u00060.8 4\u00062 10.8\u00067.5\n26 10-Jan-2014 11:23 45 43\u00062 80\u000626 1.9\u00060.7 3\u00061 10.2\u00065.5\n27 11-Jan-2014 12:37 24 66\u00063 114\u000651 1.7\u00060.8 2\u00061 2.7\u00067.5\n28 15-Jan-2014 17:38 35 39\u00061 78\u000613 2.0\u00060.3 7\u00062 16.2\u00065.8\n29 16-Jan-2014 08:29 13 53\u00062 232\u0006168 4.4\u00063. 2\u00061 3.1\u00065.4\n30 16-Jan-2014 20:46 12 57\u00062 58\u000610 1.0\u00060.2 9\u00062 18.8\u00066.1\n31 24-Jan-2014 17:17 26 77\u00064 663\u00061546 8.6\u000620 1\u00061 1.6\u00069.5\n32 24-Jan-2014 21:00 21 94\u00062 | | 17\u00062 20.4\u00063.5\n33 25-Jan-2014 07:16 34 72\u00062 94\u000620 1.3\u00060.3 4\u00061 4.4\u00063.0\n34 25-Jan-2014 12:19 37 34\u00063 103\u0006185 3.0\u00066. 1\u00061 3.1\u000612.9\n35* 25-Jan-2014 14:47 37 50\u00062 46\u00066 0.9\u00060.1 9\u00062 25.2\u00066.5\n36 25-Jan-2014 16:06 15 87\u00061 185\u000626 2.1\u00060.3 23\u00062 28.6\u00063.1\n37 26-Jan-2014 08:09 36 74\u00061 343\u0006117 4.6\u00062. 5\u00061 7.8\u00062.7\n38 27-Jan-2014 14:28 27 45\u00063 -189\u0006216 -4.2\u00065. 2\u00061 4.5\u000610.7\n39* 27-Jan-2014 20:02 27 55\u00062 172\u0006112 3.1\u00062. 4\u00061 7.4\u00065.2\n40 27-Jan-2014 16:04 39 52\u00061 327\u0006154 6.3\u00063. 2\u00061 4.4\u00063.8\n41 28-Jan-2014 20:14 29 59\u00063 101\u000635 1.7\u00060.6 6\u00062 11.0\u00065.1\n42 29-Jan-2014 08:23 7 59\u00062 131\u000634 2.2\u00060.6 4\u00061 8.2\u00063.2\n43 29-Jan-2014 19:36 35 52\u00062 424\u0006617 8.2\u000610 2\u00061 4.7\u00068.1\n44 29-Jan-2014 18:35 26 69\u00062 | | 4\u00061 6.5\u00064.2\n45 30-Jan-2014 05:20 28 62\u00061 117\u000614 1.9\u00060.2 6\u00061 11.7\u00062.6\n46 31-Jan-2014 13:40 28 44\u00061 72\u000610 1.6\u00060.2 10\u00062 21.0\u00063.7\n47 1-Feb-2014 07:55 39 59\u000610 159\u0006264 2.7\u00065. 1\u00061 2.4\u000615.4\n48 5-Feb-2014 13:53 26 52\u00062 150\u000658 2.9\u00061. 4\u00062 7.7\u00066.4\n49 5-Feb-2014 20:03 44 54\u00063 | | 2\u00061 3.7\u00068.7\n50 6-Feb-2014 16:45 30 68\u00062 78\u000613 1.2\u00060.2 11\u00062 15.6\u00064.5\n51 6-Feb-2014 17:26 34 60\u00062 105\u000620 1.8\u00060.4 9\u00062 12.2\u00064.2\n52 7-Feb-2014 09:44 30 67\u00062 140\u000629 2.1\u00060.5 8\u00062 13.0\u00063.5\n53 8-Feb-2014 21:28 64 47\u00061 61\u00065 1.3\u00060.01 13\u00061 22.9\u00062.5\n54 8-Feb-2014 10:57 68 36\u00061 194\u0006140 5.3\u00064. 1\u00061 2.8\u00065.9\n55 8-Feb-2014 10:34 25 36\u00061 | | 2\u00061 5.5\u00067.0\n56 9-Feb-2014 14:10 35 63\u00061 78\u00068 1.2\u00060.1 22\u00063 46.2\u00065.0\n57 9-Feb-2014 15:04 42 37\u00061 | | 1\u00061 3.9\u00067.0\n58 9-Feb-2014 11:48 32 47\u00061 82\u000620 1.8\u00060.4 5\u00062 14.0\u00065.0\n59 11-Feb-2014 17:13 35 57\u00061 72\u00066 1.3\u00060.01 14\u00062 25.7\u00063.3\n60 12-Feb-2014 14:19 18 55\u00061 283\u0006170 5.1\u00063. 3\u00061 6.3\u00064.7\nTable 5\nTable of oscillation best \ft parameters for 1 to 60. In \frst column the asterisk indicates that the oscillation is in the same time-distance\ndiagram than in previous case.28 Luna et al.\n# Time Ini. \u000b(\u000e)P(min.)\u001c(min.) \u001c=P A (Mm) V (km s\u00001)\n61 12-Feb-2014 17:45 24 73\u00063 260\u0006144 3.6\u00062. 4\u00061 4.6\u00063.9\n62 13-Feb-2014 18:05 19 48\u00061 118\u000624 2.5\u00060.5 10\u00062 20.9\u00064.2\n63 13-Feb-2014 20:39 2 103\u00061 175\u000612 1.7\u00060.1 47\u00062 48.5\u00062.4\n64 14-Feb-2014 08:38 21 56\u00061 79\u00066 1.4\u00060.1 31\u00062 52.1\u00063.7\n65 14-Feb-2014 20:06 60 63\u00063 -40\u00067 -0.6\u00060.1 10\u00061 17.0\u000611.5\n66 16-Feb-2014 17:26 37 40\u00062 188\u0006255 4.7\u00067. 2\u00062 6.7\u000611.9\n67 17-Feb-2014 02:42 12 92\u00063 122\u000618 1.3\u00060.2 9\u00061 14.9\u00063.1\n68 17-Feb-2014 19:59 25 39\u00063 117\u0006149 3.0\u00064. 3\u00062 8.1\u000611.6\n69 19-Feb-2014 15:38 16 76\u00062 88\u000613 1.2\u00060.2 20\u00063 28.7\u00066.1\n70 22-Feb-2014 09:04 23 51\u00062 51\u000613 1.0\u00060.3 10\u00063 25.7\u00068.2\n71 23-Feb-2014 16:04 26 34\u00061 42\u00068 1.2\u00060.2 11\u00062 34.7\u00066.1\n72 24-Feb-2014 15:03 22 73\u00061 416\u0006127 5.7\u00062. 7\u00061 10.5\u00062.9\n73 24-Feb-2014 19:13 20 54\u00062 155\u000681 2.9\u00062. 5\u00062 9.3\u00067.9\n74 25-Feb-2014 15:23 50 65\u00062 102\u000621 1.6\u00060.3 5\u00061 9.8\u00063.5\n75 25-Feb-2014 15:28 16 65\u00063 80\u000627 1.2\u00060.5 4\u00062 8.9\u00066.5\n76 25-Feb-2014 12:37 15 78\u00062 | | 3\u00061 4.8\u00063.3\n77 27-Feb-2014 09:48 7 57\u00061 75\u00068 1.3\u00060.2 29\u00063 54.6\u00066.3\n78 27-Feb-2014 15:59 34 53\u00062 136\u000690 2.6\u00062. 4\u00062 9.6\u00066.2\n79* 27-Feb-2014 19:41 34 51\u00061 87\u000620 1.7\u00060.4 7\u00062 18.9\u00065.9\n80 28-Feb-2014 12:08 10 109\u00062 312\u0006106 2.9\u00061. 9\u00062 9.6\u00062.8\n81 7-Mar-2014 16:44 21 49\u00061 423\u0006345 8.6\u00067. 4\u00061 9.5\u00063.7\n82 12-Mar-2014 09:02 1 105\u000610 115\u000652 1.1\u00060.6 12\u00063 20.5\u000611.1\n83 14-Mar-2014 13:31 35 70\u00062 89\u000614 1.3\u00060.2 8\u00062 14.4\u00063.0\n84 16-Mar-2014 03:31 37 46\u00061 74\u000615 1.6\u00060.4 11\u00062 26.4\u00065.9\n85 20-Mar-2014 16:02 19 93\u00062 73\u00068 0.8\u00060.09 16\u00062 21.8\u00063.8\n86 21-Mar-2014 17:23 7 61\u00062 89\u000625 1.5\u00060.4 13\u00062 19.6\u00065.1\n87 23-Mar-2014 16:05 22 76\u00061 71\u00065 0.9\u00060.07 16\u00062 29.6\u00063.5\n88 24-Mar-2014 19:17 29 44\u00062 50\u00068 1.2\u00060.2 7\u00061 16.0\u00064.0\n89* 24-Mar-2014 20:57 29 38\u00062 251\u0006269 6.6\u00067. 1\u00061 3.8\u00066.5\n90 28-Mar-2014 17:52 0 81\u00063 135\u000632 1.7\u00060.4 8\u00062 10.8\u00063.9\n91 29-Mar-2014 18:23 26 58\u00061 108\u000612 1.9\u00060.2 11\u00061 19.3\u00062.3\n92 29-Mar-2014 15:50 12 87\u00062 139\u000620 1.6\u00060.3 31\u00063 38.8\u00064.4\n93 30-Mar-2014 03:53 10 59\u00065 | | 8\u00062 15.7\u00069.2\n94 31-Mar-2014 16:11 30 51\u00062 43\u000611 0.8\u00060.2 6\u00062 18.9\u00067.0\n95* 31-Mar-2014 19:38 30 58\u00061 325\u000662 5.6\u00061. 17\u00062 32.5\u00062.5\n96 7-Apr-2014 21:46 19 41\u00061 70\u00069 1.7\u00060.2 10\u00061 24.2\u00062.8\n97 9-Apr-2014 11:21 33 56\u00061 71\u000611 1.3\u00060.2 10\u00062 24.9\u00064.5\n98 9-Apr-2014 18:19 29 51\u00064 103\u0006110 2.0\u00062. 3\u00062 6.8\u000610.6\n99 10-Apr-2014 06:14 20 52\u00061 94\u000613 1.8\u00060.3 8\u00061 13.7\u00062.9\n100 14-Apr-2014 12:16 7 50\u00061 53\u00064 1.0\u00060.09 20\u00062 32.4\u00062.7\n101 17-Apr-2014 05:20 25 52\u00061 83\u00068 1.6\u00060.2 9\u00061 22.8\u00062.8\n102 17-Apr-2014 02:21 3 92\u00062 179\u000635 1.9\u00060.4 8\u00061 9.7\u00062.4\n103 17-Apr-2014 17:41 39 56\u00061 61\u00067 1.1\u00060.1 11\u00062 25.5\u00063.7\n104 17-Apr-2014 18:38 8 73\u00064 | | 1\u00061 2.1\u00067.4\n105 19-Apr-2014 02:57 0 59\u00062 118\u000634 2.0\u00060.6 9\u00062 12.9\u00064.4\n106 19-Apr-2014 12:44 26 65\u00061 114\u000620 1.8\u00060.3 8\u00062 15.0\u00064.3\n107 21-Apr-2014 18:20 20 50\u00061 155\u000634 3.1\u00060.7 4\u00061 6.6\u00062.2\n108* 21-Apr-2014 21:29 20 40\u00063 -97\u000670 -2.4\u00062. 2\u00061 5.6\u00069.3\n109 22-Apr-2014 12:17 11 61\u00061 110\u000619 1.8\u00060.3 5\u00061 8.5\u00063.4\n110 23-Apr-2014 14:32 26 51\u00064 45\u000615 0.9\u00060.3 7\u00062 10.5\u00067.4\n111 24-Apr-2014 04:04 16 43\u00061 232\u000642 5.3\u00060.1 8\u00061 18.5\u00063.3\n112 24-Apr-2014 10:53 21 41\u00062 37\u00066 0.9\u00060.1 13\u00062 27.7\u00066.2\n113 25-Apr-2014 03:17 16 58\u00062 276\u0006189 4.7\u00063. 2\u00061 3.0\u00066.4\n114 26-Apr-2014 12:40 54 47\u00063 55\u000621 1.2\u00060.5 6\u00062 11.0\u00068.7\n115 1-May-2014 17:40 8 70\u00061 78\u00064 1.1\u00060.06 25\u00062 48.4\u00062.8\n116 1-May-2014 16:17 23 52\u00061 218\u0006103 4.2\u00062. 3\u00061 6.4\u00064.9\n117 1-May-2014 03:08 81 52\u00062 125\u000644 2.4\u00060.9 2\u00061 3.3\u00064.8\n118 2-May-2014 02:17 8 76\u00061 377\u0006144 5.0\u00062. 6\u00061 7.0\u00062.4\n119* 2-May-2014 09:08 8 72\u00065 110\u000689 1.5\u00061. 5\u00063 9.3\u00069.8\n120 2-May-2014 11:04 48 50\u00064 51\u000619 1.0\u00060.4 4\u00062 6.1\u00068.6\nTable 6\nSame as Table 5 for events 61 to 120.GONG Catalog of Solar Filament Oscillations 29\n# Time Ini. \u000b(\u000e)P(min.)\u001c(min.) \u001c=P A (Mm) V (km s\u00001)\n121 5-May-2014 16:13 15 35\u00063 26\u00068 0.7\u00060.3 6\u00062 14.2\u00069.1\n122 10-May-2014 08:01 18 81\u00063 198\u0006104 2.4\u00061. 3\u00061 3.7\u00065.8\n123 10-May-2014 09:14 72 69\u00063 119\u000658 1.7\u00060.9 3\u00062 4.6\u00066.5\n124 11-May-2014 18:47 9 64\u00063 46\u00068 0.7\u00060.2 18\u00063 24.0\u00065.8\n125 12-May-2014 02:40 18 65\u00066 86\u000652 1.3\u00060.9 4\u00062 5.3\u00069.1\n126 12-May-2014 01:19 31 88\u00062 212\u000655 2.4\u00060.7 8\u00062 12.3\u00063.3\n127 12-May-2014 06:20 21 32\u00062 34\u00069 1.1\u00060.3 9\u00063 25.0\u00069.1\n128 12-May-2014 08:22 7 55\u00062 123\u000635 2.2\u00060.7 5\u00061 8.4\u00064.0\n129 12-May-2014 01:26 15 43\u00062 50\u000616 1.2\u00060.4 3\u00061 8.0\u00065.8\n130 13-May-2014 18:26 16 50\u00063 72\u000625 1.4\u00060.5 4\u00061 6.8\u00065.6\n131 13-May-2014 14:46 74 77\u00062 185\u000683 2.4\u00061. 2\u00061 2.4\u00063.5\n132 14-May-2014 07:13 87 99\u00065 227\u0006138 2.3\u00061. 4\u00062 3.6\u00065.6\n133 15-May-2014 18:25 44 76\u00062 87\u000612 1.1\u00060.2 8\u00061 12.9\u00062.7\n134 16-May-2014 02:25 35 85\u00065 -224\u0006195 -2.6\u00062. 2\u00061 4.5\u00069.0\n135 16-May-2014 11:49 7 50\u00061 | | 11\u00061 24.7\u00063.2\n136 17-May-2014 02:37 2 59\u00063 218\u0006219 3.7\u00064. 3\u00062 5.1\u00067.4\n137 17-May-2014 10:35 20 58\u00062 78\u000612 1.4\u00060.2 19\u00062 29.5\u00065.1\n138 18-May-2014 18:22 38 57\u00061 49\u00066 0.9\u00060.1 9\u00062 23.5\u00064.3\n139 23-May-2014 15:06 48 56\u00062 179\u0006111 3.2\u00062. 2\u00061 4.1\u00065.4\n140 23-May-2014 14:58 18 48\u00062 82\u000620 1.7\u00060.4 8\u00062 13.8\u00065.2\n141 23-May-2014 14:14 4 48\u00063 213\u0006216 4.4\u00065. 2\u00061 4.0\u00068.5\n142 23-May-2014 18:01 37 46\u00062 32\u00067 0.7\u00060.2 6\u00062 20.1\u00067.2\n143 26-May-2014 02:38 13 62\u00062 53\u00066 0.9\u00060.01 15\u00062 21.2\u00062.8\n144 26-May-2014 17:52 62 53\u00062 180\u0006145 3.4\u00063. 3\u00061 6.9\u00066.2\n145 26-May-2014 13:31 50 39\u00062 48\u000613 1.2\u00060.4 5\u00062 14.4\u00066.1\n146* 26-May-2014 17:38 50 36\u00062 119\u0006146 3.3\u00064. 1\u00061 3.4\u00069.4\n147 27-May-2014 03:16 1 70\u00062 444\u0006385 6.3\u00066. 3\u00061 4.5\u00065.4\n148 27-May-2014 09:16 36 34\u00063 40\u000622 1.2\u00060.7 2\u00061 4.8\u000610.0\n149 28-May-2014 00:34 42 75\u00066 467\u00061263 6.2\u000620 2\u00062 3.5\u00069.2\n150 29-May-2014 01:16 27 42\u00061 86\u000613 2.1\u00060.3 8\u00061 21.2\u00064.3\n151 30-May-2014 00:01 36 52\u00062 89\u000623 1.7\u00060.5 4\u00061 6.8\u00065.1\n152* 30-May-2014 05:37 36 66\u00063 -163\u0006105 -2.5\u00062. 3\u00061 4.0\u00067.8\n153 30-May-2014 11:43 36 45\u000612 | | 1\u00063 4.1\u000630.4\n154* 30-May-2014 13:42 36 37\u00063 49\u000625 1.3\u00060.7 7\u00064 17.2\u000615.4\n155 1-Jun-2014 02:06 32 32\u000613 101\u0006859 3.1\u000630 0\u00062 1.7\u000652.6\n156 2-Jun-2014 07:05 51 73\u00063 -291\u0006236 -4.0\u00063. 2\u00061 3.6\u00065.9\n157 2-Jun-2014 18:26 14 84\u00062 | | 2\u00061 2.4\u00063.9\n158 3-Jun-2014 02:33 19 74\u00067 557\u00062260 7.5\u000630 1\u00061 1.2\u000613.5\n159 4-Jun-2014 17:30 33 79\u00062 155\u000629 2.0\u00060.4 6\u00061 6.6\u00062.6\n160 5-Jun-2014 10:20 50 60\u00061 272\u000673 4.6\u00061. 2\u00061 3.6\u00063.2\n161 5-Jun-2014 19:57 16 49\u00062 51\u000613 1.1\u00060.3 5\u00062 10.3\u00066.1\n162 6-Jun-2014 10:46 2 48\u00063 | | 2\u00061 3.9\u00067.0\n163 7-Jun-2014 00:36 64 45\u00063 33\u00069 0.7\u00060.2 9\u00063 16.2\u000613.1\n164 7-Jun-2014 07:26 42 52\u00063 106\u000662 2.1\u00061. 3\u00061 5.9\u00066.4\n165* 7-Jun-2014 11:26 42 58\u00062 123\u000639 2.1\u00060.7 5\u00062 9.1\u00064.9\n166 8-Jun-2014 10:17 13 56\u00065 46\u000620 0.8\u00060.4 5\u00064 11.3\u000611.9\n167 8-Jun-2014 16:55 29 55\u00063 193\u0006167 3.5\u00063. 2\u00061 3.6\u00068.0\n168 9-Jun-2014 07:32 34 47\u00062 64\u000616 1.3\u00060.4 6\u00062 11.7\u00065.2\n169* 9-Jun-2014 11:55 34 40\u00061 84\u000621 2.1\u00060.5 6\u00061 12.4\u00063.5\n170 9-Jun-2014 14:46 40 58\u00064 | | 1\u00061 2.2\u000617.5\n171 9-Jun-2014 09:34 1 62\u00063 -202\u0006226 -3.2\u00064. 1\u00061 2.6\u00069.3\n172* 9-Jun-2014 13:59 1 57\u00062 280\u0006189 4.9\u00063. 3\u00061 5.5\u00065.3\n173 10-Jun-2014 18:19 35 36\u00061 84\u000620 2.4\u00060.6 6\u00061 17.3\u00064.8\n174 12-Jun-2014 00:01 20 63\u00061 275\u0006118 4.4\u00062. 5\u00061 9.9\u00063.3\n175* 12-Jun-2014 04:31 20 45\u00061 207\u0006134 4.6\u00063. 1\u00061 2.8\u00065.1\n176 12-Jun-2014 01:50 21 67\u00063 | | 3\u00061 5.1\u00065.7\n177* 12-Jun-2014 05:37 21 54\u00063 56\u000635 1.0\u00060.7 13\u00065 15.1\u00069.5\n178 12-Jun-2014 00:30 47 67\u00062 118\u000631 1.8\u00060.5 4\u00061 5.3\u00065.1\n179 12-Jun-2014 13:11 13 52\u00061 57\u00068 1.1\u00060.2 10\u00062 22.1\u00065.0\n180* 12-Jun-2014 14:10 13 51\u00061 142\u000648 2.8\u00060.1 4\u00061 8.3\u00064.1\nTable 7\nSame as Table 5 for events 121 to 180.30 Luna et al.\n# Time Ini. \u000b(\u000e)P(min.)\u001c(min.)\u001c=P A (Mm) V (km s\u00001)\n181 13-Jun-2014 05:38 28 60\u00065 | | 1\u00061 2.3\u000612.9\n182 13-Jun-2014 16:37 2 51\u00063 245\u0006353 4.8\u00067. 1\u00061 3.2\u00068.5\n183 14-Jun-2014 01:44 1 58\u00062 493\u0006673 8.4\u000610 1\u00061 1.7\u00067.4\n184 14-Jun-2014 11:38 81 61\u00064 249\u0006313 4.1\u00065. 1\u00061 2.3\u00068.5\n185 14-Jun-2014 09:46 17 45\u00063 | | 2\u00062 6.3\u000610.0\n186 15-Jun-2014 02:37 9 44\u00066 | | 1\u00061 1.4\u000619.3\n187* 15-Jun-2014 21:04 9 54\u00061 118\u000622 2.2\u00060.4 7\u00061 11.1\u00062.6\n188 16-Jun-2014 11:43 57 63\u00068 237\u0006356 3.7\u00066. 2\u00061 3.9\u000612.6\n189 16-Jun-2014 08:50 88 38\u00069 125\u0006501 3.3\u000610 0\u00061 1.0\u000628.2\n190* 16-Jun-2014 15:59 88 41\u00063 84\u000667 2.1\u00062. 2\u00061 3.7\u000610.1\n191 17-Jun-2014 21:14 20 67\u00066 39\u00069 0.6\u00060.2 4\u00062 4.7\u00067.4\n192 17-Jun-2014 15:50 35 57\u00061 162\u000651 2.9\u00060.9 6\u00062 15.0\u00064.6\n193 17-Jun-2014 09:50 65 42\u00061 246\u0006166 5.8\u00064. 1\u00061 2.8\u00064.5\n194 17-Jun-2014 12:36 27 51\u00062 68\u000616 1.3\u00060.3 4\u00061 9.0\u00064.0\n195 19-Jun-2014 18:10 14 47\u00063 62\u000621 1.3\u00060.5 3\u00061 7.3\u00065.1\n196 29-Jun-2014 16:21 50 56\u00064 61\u000628 1.1\u00060.6 5\u00063 9.4\u000610.6\nTable 8\nSame as Table 5 for events 181 to 195.\nTIME-DISTANCE DIAGRAMS IN CURVED SLITS\nThe GONG network telescopes o\u000ber fairly good spatial resolution of around 1 arcsec per pixel. However, the seeing\nconditions at the network telescope locations often limit the quality of the images, yielding poor e\u000bective spatial\nresolution greater than 1 arcsec. As we discussed in x2, it was necessary to follow the motion of the large-amplitude\ndisplacements with curved paths in order to accurately track the entire motion of the \flament.\nA time-distance diagram is constructed to follow the motion along the path de\fned by the arti\fcial slit. In many\ncases (e.g., Luna et al. 2014), straight slits consisting of rectangles of length land width win pixels are placed\nlengthwise along the path of the motion studied. In order to increase the signal-to-noise ratio, the intensity is averaged\nalong the width w, which essentially projects the intensity onto the axis of the slit. The resulting intensity along the\nslit as a function of time is the time-distance diagram. Using a curved slit is theoretically similar to using a straight\nslit. In a curved slit the projection is de\fned along the normal lines to the curved slit axis. Thus, for each pixel, there\nis a normal line intersecting the slit axis at ( xq,yq) and the distance between the pixel and the slit is d(i.e. between\nthe pixel and ( xq,yq)). A pixel belongs to the slit if d\u0014w=2.\nFigure 28. (a) H\u000bimage of case 1. The \flament is located in the center of the image. The white curve is the axis of the slit, S. (b)\nIsocontours of Dist(i;j) andInt(i;j) (Equation B1) for the image in (a). (c) Close-up view of a region of the slit showing the bins used to\nconstruct the time-distance diagram. All positions inside the bins have B(i;j) =q=constant (Equation B3). The grey gradient highlights\nthe di\u000berent bins, with white corresponding to q= 1 and black to q=Npix.\nIn general an image is described by the 2D function I(x;y), whereIis the intensity in the \flter considered (H \u000bin\nour situation) and xandyare the coordinates of each position in the image. We assume, without loss of generality,\nthat the origin of the coordinates ( x;y) = (0;0) is at the left-bottom boundary of the image. These coordinates take\nentire values of the resolution \u000eof the image, then x=i\u000eandy=j\u000ewhereiandjde\fne the position within the\nimage in pixels. Alternatively, the image can be described in pixels I(i;j).\nWe \frst de\fne a su\u000eciently smooth curve, S, that represents the axis of the slit, by clicking repetitively on the\nimage along the path of the oscillatory motion and \ftting these points with a polynomial function of 4th degree. The\nwhite line in Figure 28(a) shows the curve Sobtained for event 1 of the catalog. We divide this curve into segmentsGONG Catalog of Solar Filament Oscillations 31\nof length\u000ein order to pixelate the curve as the image. The coordinate along the slit axis is then s=\u000eqwhereqis a\none dimensional array with Npix=l=\u000eelements.\nThe time-distance diagram consists of I(t;iq;jq) =hI(t;A)iwhere (iq;jq) is the position of the q-segment of the axis\nof the slit, tis time,Ais an area surrounding ( iq;jq), and theh:::imeans the average of the intensity over A. The\nmain di\u000eculty is how A is de\fned. Some authors just de\fnes a square area centered at ( iq;jq). However, this mixes\nthe intensities from points that are not projected perpendicularly to the slit, and some pixels are projected twice in\nconsecutive segments of the slit. We will de\fne A as the area enclosed by the normal lines between both ends of slit\nsegment,qand within the slit, d\u0014w=2.\nFor this end, we de\fne two matrices, Dist(i;j) is the distance from any point ( i;j) to the closest point along the\nslit, andInd(i;j) is the index of that point along the slit. These are\nDist(i;j) =MIN\u00121\n\u000eq\n(xi\u0000xq)2+ (yj\u0000yq)2\u0013\n(B1)\nInd(i;j) =qmin: (B2)\nwhere the MIN is the minimum over the qindex. We construct these arrays by computing the distancep\n(xi\u0000xq)2+ (yj\u0000yq)2between each pixel of the image, ( i;j), and all the positions over the slit, q. This is equivalent\nto computing the distance, d, between the pixel and the slit axis. However, this way is much more computationally\ne\u000bective. To calculate Ind(i;j) we then \fnd the value of qminthat minimizes the distancep\n(xi\u0000xq)2+ (yj\u0000yq)2.\nThis is the position over the slit where the intensity of the image pixel will be projected. We repeat this process for\nall image pixels and obtain the arrays de\fned by Equations (B1) and (B2). Thus, Dist(i;j) is the distance measured\nin pixels from ( i;j) to the curve S. The closed thick lines in Figure 28(b) are the isolines of the Dist function over the\nimage. We see that each isoline represents the positions of the pixels that are equidistant to the curve segment S, i.e.\nthe slit axis. In the example of Figure 28, the slits has w= 6, which corresponds to the area inside the most internal\nisoline with a distance to Sof 3 pixels. In general we de\fne the slit as the set of pixels ( i;j) that ful\fll the condition\nDist(i;j)\u0014w=2. Thus, to select the pixels inside the slit, we de\fne a masking function\nMask (i;j) =\u001a1;ifDist(i;j)\u0014w=2\n0;ifDist(i;j)>w= 2:(B3)\nThe thin straight lines in Figure 28(b) plot are the isolines of Indmatrix. This isolines coincide with the normal lines\nto the slit axis. We de\fne a new function\nB(i;j) =Mask (i;j)\u0002Ind(i;j): (B4)\nThe values in this array are zero outside the slit and range from 1 to Npixin the slit. In this way we have binned the\nregions of the image that are going to be averaged over the q-position of the slit. We clearly see these bins of constant\nB(i;j) in Figure 28(c), as well as the bins de\fned by the area inside the region formed by the isolines of Dist andInt.\nThen the intensity over the slit, I(q), is the average of the intensity over the bin where B(i;j) =q, that is\nI(q) =hI(B(i;j) =q)i: (B5)\nThis technique can also be used for straight slits to reduce the computational time, because the images do not need\nto be rotated in order to align the x- ory-axis with the direction of the slit. 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Ackermann1, 2and Satoru Emori3,a)\n1)Academy of Integrated Science, Virginia Tech, Blacksburg, VA 24061, USA\n2)Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, USA\n3)Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA\n(Dated: November 21, 2018)\nWe simulate the switching behavior of nanoscale synthetic antiferromagnets (SAFs), inspired by recent\nexperimental progress in spin-orbit-torque switching of crystal antiferromagnets. The SAF consists of two\nferromagnetic thin \flms with in-plane biaxial anisotropy and interlayer exchange coupling. Staggered \feld-like\nRashba spin-orbit torques from the opposite surfaces of the SAF induce a canted net magnetization, which\ntriggers an orthogonal torque that drives 90\u000eswitching of the N\u0013 eel vector. Such dynamics driven by the\n\feld-like spin-orbit torque allows for faster switching with increased Gilbert damping, without a signi\fcant\ndetrimental increase of the threshold switching current density. Our results point to the potential of SAFs as\nmodel systems, based on simple ferromagnetic metals, to mimic antiferromagnetic device physics.\nI. INTRODUCTION\nAntiferromagnets are considered promising material\nplatforms for ultrafast spintronic information-technology\ndevices that are highly stable against external\nmagnetic \felds1{3. Recent experimental studies\nhave demonstrated switching of the antiferromagnetic\norder (N\u0013 eel vector) between two orthogonal states in\nepitaxial antiferromagnetic conductors of CuMnAs4\nand Mn 2Au5. This N\u0013 eel switching is driven by a\ncurrent-induced \\\feld-like\" spin-orbit torque (SOT)\nthat acts locally in opposite directions on the two\nmagnetic sublattices of the antiferromagnet6,7. The key\ningredient for this non-vanishing \feld-like N\u0013 eel SOT is\nthe inversion asymmetry around each magnetic atom\n(i.e., Mn) that is intrinsic to the speci\fc crystal structure\nof CuMnAs and Mn 2Au. However, the synthesis of\nepitaxial CuMnAs and Mn 2Au with the correct crystal\nstructure may not be straightforward, and so far no\nother conductive collinear antiferromagnets with the\ncompatible crystal structure for the \feld-like N\u0013 eel\nSOT have been realized4,5. It has also been shown\nthat epitaxial antiferromagnetic insulator NiO can\nbe switched by a SOT from an adjacent metal with\nstrong spin-orbit coupling (e.g., Pt)8,9. In this case, the\nlimitation may be the relatively small magnetoresistance\nsignal (i.e., spin-Hall magnetoresistance10) to read out\nthe N\u0013 eel vector state. Furthermore, it is generally\ndi\u000ecult to apply conventional laboratory-based\ncharacterization techniques (e.g., magnetometry,\nferromagnetic resonance, magnetic microscopy, etc.) to\nstudy the fundamental properties of antiferromagnets.\nThese points above may constitute a serious obstacle\nto studying and engineering viable materials for\nantiferromagnetic spintronics.\nHere, we study by micromagnetic simulations\nthe switching behavior of synthetic antiferromagnets\n(SAFs)11as a model system analogous to intrinsic\na)Electronic mail: semori@vt.educrystal antiferromagnets. The SAFs consist of two\nin-plane biaxial ferromagnetic metals (FMs) whose\nmagnetizations are locked antiparallel to each other\nby interlayer exchange coupling (e.g., through the\nRuderman-Kittel-Kasuya-Yosida mechanism across a\nnon-ferromagnetic metal such as Cr or Ru)12,13. Such\nbiaxial FMs can be readily synthesized by epitaxial\ngrowth on a cubic single-crystal substrate, e.g., body-\ncentered-cubic Fe on MgO (001) or GaAs (001)14{18.\nThis SAF structure has two orthogonal easy axes in\nthe \flm plane de\fned by cubic magnetocrystalline\nanisotropy. These two digital states, represented by\northogonal N\u0013 eel vector orientations, can be read through\nanisotropic magnetoresistance4,5; e.g., when the N\u0013 eel\nvector is oriented parallel (transverse) to the sense\ncurrent, the SAF exhibits a higher (lower) electrical\nresistance19,20. Switching is achieved when opposite local\n\felds are applied orthogonal to the magnetizations of\nthe two FM layers (Fig. 1), analogous to the opposite\nlocal \felds applied to the two sublattices in CuMnAu and\nMn2Au4,5. In the SAF, the required symmetry breaking\nfor such opposite local \felds occurs at the layer interfaces.\nWe simulate the e\u000bect of the interfacial Rashba spin-\norbit \felds (\feld-like SOTs)21{23arising from the top and\nbottom surfaces of the SAF interfaced with, e.g., an oxide\ncapping layer and substrate24,25. Our study therefore\nFM 1\nFM 2\nx ycurrentBso\nx y\nFigure 1. Schematic of the synthetic antiferromagnet (SAF)\nconsisting of two ferromagnetic metal (FM) layers. The\nSAF has two orthogonal easy axes along the x- andy-\naxes. The current-induced spin-orbit \feld Bsoswitches the\nantiferromagnetic order (N\u0013 eel vector) from the x- toy-axis.arXiv:1811.04094v2  [cond-mat.mes-hall]  22 Nov 20182\nsuggests a possible pathway for simple SAF spintronic\ndevices that inherit some of the switching behavior of\nantiferromagnets.\nAdvantages of SAFs have been reported previously\nfor engineering stable pinned and free layers in spin\nvalves26{28, rapid motion of domain walls29{32, and SOT-\ndriven switching of perpendicular magnetization33{35.\nIn contrast with these prior devices based on 180\u000e\nswitching, we emphasize that our proposed approach is\nbased on 90\u000eswitching. To the best of our knowledge,\nour study is the \frst to numerically examine such\northogonal switching in SAFs, speci\fcally driven by \feld-\nlike SOT. The orthogonal orientation between the initial\nmagnetization and the spin-orbit \feld in each FM layer\n(Fig. 1) maximizes the torque on the magnetization\nand enables rapid switching. This switching scheme\ndriven by the \feld-like torque also allows for faster\nswitching by increasing the Gilbert damping parameter\nwithout an adverse increase of the threshold switching\ncurrent density. Our simulations indicate that SAFs\nwith realistic material parameters are robust against\nup to\u00181 T of external magnetic \feld and can be\nswitched in\u00180.1 ns at a reasonable current density\nof<\u00181011A/m2. We also note that this proposed\ndevice scheme is operated by two orthogonal current\nlines, analogous to four-terminal toggle magnetic random\naccess memories (MRAMs)36. The use of the \feld-\nlike SOT as proposed here, instead of Oersted \felds in\ntoggle MRAMs, may enable alternative scalable memory\ndevices.\nII. MODEL PARAMETERS\nMagnetic switching was simulated using the Mumax3\nmicromagnetics package37. A series of square samples\nwith di\u000berent widths of 26 to 400 nm were studied\nwith a lateral cell size of 2 or 4 nm. Each FM layer\nhad the following \fxed properties: thickness tFM=\n1.5 nm, exchange constant Aex= 20 pJ/m, and the\ncubic anisotropy constant Kc= 30 kJ/m3with the easy\naxes parallel to the square edges. In most simulations,\nwe set the saturation magnetization Msat 1700 kA/m\n(typical value for Fe), the Gilbert damping parameter\n\u000bat 0.01 (typical value for nanometer-thick FMs), and\nthe interlayer exchange coupling energy density Jexat\n\u00000.2 or\u00001 mJ/m2(where the negative sign indicates\nantiferromagnetic interlayer coupling). We note that the\nvalues ofJexused here are similar to those experimentally\nachieved in SAFs consisting of FMs13,32,38,39.\nIII. RESULTS AND DISCUSSION\nA. Stability against global magnetic \feld\nWe \frst compare the stability of the magnetization\nstate against a global external \feld in the SAFs with\nsingle layerJex= -0.2 mJ/m2Jex= -1 mJ/m2(a)\n(b)\nJex= -1 mJ/m2single layer\nsize [nm]Bsw[T]\nBy[nm]myFigure 2. (a) External magnetic \feld Bswrequired to switch\nthe magnetization from the x-axis toy-axis for samples\nof di\u000berent lateral sizes. (b) Equilibrium magnetization\ncomponent myalong they-axis versus external magnetic \feld\nBy. Here the lateral sample size is 52 nm.\ntheir single-layer FM counterparts. With the initial\nmagnetizations set parallel to the x-axis (i.e.,mtop\nx= 1,\nmbot\nx=\u00001, andmtop\ny=mbot\ny= 0), an external\nmagnetic \feld Byalong the + y-direction was applied.\nThe critical switching \feld Bswis de\fned as Byrequired\nto pull the total magnetization, m=1\n2(mtop+mbot),\nto they-direction, i.e., my>0:99. Figure 2(a)\nshows that the single-layer FMs switch at low values\nofBsw<\u00180:01 T, indicating that these samples are\nvulnerable to spontaneous switching from external stray\n\felds. As evidenced by the substantial variation in Bsw{\nas much as an order of magnitude { with lateral size, the\nswitching behavior of the single-layer FMs is also heavily\nimpacted by the device geometry, e.g., due to dipolar\n\felds from the sample edges. A slight variation in the\nshape or edge defects of single-layer in-plane FM devices\ncan lead to a random distribution of switching thresholds.\nThe SAFs show about an order of magnitude greater\nBswthan the single-layer FMs. As shown in Fig. 2(a),\nBswis enhanced with increasing Jex. ForJex=\n\u00001 mJ/m2readily achievable in realistic SAFs13,32,38,39,\nan external \feld of nearly 1 T is required to orient\nthe magnetization along the y-direction. As shown in\nFig. 2(b), while the single-layer FM undergoes abrupt\nswitching at low By, the SAF undergoes a gradual\nmagnetization rotation until the magnetizations of the\ntwo layers are fully oriented along the \feld direction. We\nalso note that Bswonly varies by a factor of \u00192 with\nthe lateral sample dimensions of the SAFs (Fig. 2(a)),\nindicating that the dipolar \felds from the sample edges3\nplay relatively little role. The SAFs are therefore shown\nto be signi\fcantly more stable against disturbances from\nexternal magnetic \felds, and this stability is largely\nindependent of the sample geometry. We emphasize that\nthe stability at \felds of \u00180.1-1 T can be achieved in\nSAFs consisting of simple FMs (e.g., Fe), in contrast\nwith intrinsic crystal antiferromagnetic compounds4,5for\nwhich epitaxial growth is more challenging.\nB. Threshold spin-orbit \feld for switching\nHaving demonstrated the stability of the SAFs, we\ncompute how much spin-orbit \feld is required to switch\nthe antiferromagnetic order in the SAFs between the x-\nandy-axes (e.g., Fig. 1). For example, the magnetization\nof the top (bottom) FM layer, initially oriented along\nthe +x-direction (\u0000x-direction), sees an e\u000bective current-\ninduced \feld pointing along the - y-direction (+ y-\ndirection). When the magnitude of this e\u000bective \feld is\nsu\u000eciently large, the magnetization overcomes the cubic\nanisotropy energy barrier and switches from the x-axis\ntoy-axis. Unlike a global magnetic \feld (Sec. III A)\nthat cants the magnetizations toward the parallel state\nand hence results in a large interlayer antiferromagnetic\nexchange energy penalty, the local spin-orbit \feld rotates\nthe magnetization of each layer while maintaining the\nmostly antiparallel magnetization alignment across the\nlayers. We discuss the details of the switching process in\nSec. III C.\nWe de\fne the threshold Bth\nsoas the e\u000bective local \feld\nrequired to switch the N\u0013 eel vector, l=1\n2(mtop\u0000mbot),\nto they-axis, i.e.,jlyj>0:99. We simulated two cases\nwhere (1) only the top layer sees the spin-orbit \feld (and\nthe bottom layer magnetization is dragged by the top\nlayer magnetization), and (2) the top and bottom layers\nsee the spin-orbit \feld in opposite directions (Fig. 1).\nThese two con\fgurations of the spin-orbit \feld would\narise by enabling an interfacial Rashba \feld-like SOT at\n(1) only the top surface of the SAF and (2) both the top\nand bottom surfaces of the SAF.\nFigure 3 plots the computed Bth\nsoagainst the SAF\nlateral size. Bth\nsois somewhat dependent on the lateral\nsample size, increasing by nearly a factor of 2 when\nthe lateral sample size is decreased from 400 to 26 nm,\nas the mode of switching transitions from incoherent\nto coherent. More importantly, we \fnd a factor of 2\nreduction in Bth\nsowith the current-induced \feld active at\nboth the top and bottom surfaces compared to just one.\nThis \fnding con\frms that the spin-orbit \feld is additive\nand that engineering the Rashba e\u000bect at both surfaces\nwould lead to a more e\u000ecient SAF device. It should\nalso be noted that jJexjdoes not a\u000bect Bth\nso, suggesting\nthat biaxial SAFs can be switched e\u000eciently regardless of\nthe strength of interlayer exchange coupling. Here, since\nthe cubic magnetic anisotropy energy density Kc\u0018104\nJ/m3is signi\fcantly smaller than the interlayer exchange\nenergy densityjJexj=tFM\u0018105\u0000106J/m3, the energy\nBso: 2 layers, Jex= -1 mJ/m2\nBso: 2 layers, Jex= -0.2 mJ/m2Bso: 1 layer, Jex= -1 mJ/m2\nBso: 1 layer, Jex= -0.2 mJ/m2Bso[mT]th\nsize [nm]Figure 3. Threshold current-induced spin-orbit \feld Bth\nso\nrequired to switch the N\u0013 eel vector from the x-axis toy-axis\nfor SAF samples with di\u000berent lateral dimensions.\nbarrier for 90\u000eswitching of the N\u0013 eel vector is mostly\ndetermined by Kcrather thanjJexj. This \fnding is\nconsistent with a prior study of 180\u000eswitching in SAFs,\nwhere the energy barrier is governed by uniaxial magnetic\nanisotropy28. However, we show in the next subsection\n(Sec. III C) that the interlayer exchange coupling can\nin\ruence the switching speed by generating a torque on\nthe N\u0013 eel vector.\nProvided that the spin-orbit \feld arises entirely from\nthe interfacial Rashba-Edelstein e\u000bect, we can estimate\nthe critical threshold current density for switching Jth\nfromBth\nsowith40,41\nJth=\u0016BBth\nsoMs\n\u000bRP; (1)\nwhere\u0016Bis the Bohr magneton, \u000bRis the Rashba\nparameter, and Pis the e\u000bective spin polarization\n(proportional to the exchange interaction between\nthe Rashba-induced spin accumulation and the FM\nmagnetization). For Jthto be comparable to\n<\u00181011A/m2recently reported in antiferromagnetic\nmemory prototypes4,5,8,9, the product \u000bRPwould need\nto be>\u00180:1 eV\u0001\u0017A. This is reasonably achieved with\nRashba parameters similar to those reported in oxide\nsystems24,25,42{46. While the \feld-like SOT has not\nreceived as much attention (compared to the damping-\nlike SOT) for FM-based device applications, an enhanced\ninterfacial Rashba spin-orbit \feld would be a robust\ndriving force to e\u000eciently switch a biaxial SAF memory.\nC. Time-dependence of switching\nFinally, we discuss the mechanism and time\ndependence of SOT-driven switching in the SAFs. In\nthe following, switching is driven by opposite local spin-\nobit \felds acting on the top and bottom layers. The\ninitial orthogonal con\fguration between the spin-orbit\n\feld (e.g., Btop\nsojj\u0000^y,Bbot\nsojj+^y) and the magnetization\n(mtopjj+^x,mbotjj\u0000^x) in each layer maximizes the\ntorque that initiates the switching process. When this4\nmx,mymz(d)\n(e)lz-ly\nlxτdemag\nBdemagBdemagτdemag\nBsoSOT\nmbotτexch\nBexchBexchτexch\nBso(a) (b) (c)\nz\nx ySOT\nmtop\nlx, ly, lzmx, my, mz\ntime [ns]\nFigure 4. (a,b,c) Schematics of the torques due to the (a)\nspin-orbit \feld Bso, (b) demagnetizing \feld Bdemag , and\n(c) interlayer antiferromagnetic exchange \feld Bexch. (d,e)\nTime traces of the (d) N\u0013 eel vector ( lx,ly,lz) and (e) total\nmagnetization ( mx,my,mz) atBso= 6 mT in the 52-nm-\nwide SAF sample with Ms= 1700 kA/m, Jex= -1 mJ/m2,\u000b\n= 0.01.\nSOT (\u0000j\rjmtop\u0002Btop\nso,\u0000j\rjmbot\u0002Bbot\nso) is turned\non, the magnetization is tilted out of the \flm plane in the\nsame direction in both layers (Fig. 4(a)), thereby giving\nrise to a \fnite z-component in the total magnetization\nmz. This out-of-plane canting then yields two torques\nalong\u0000^yin the top layer (+ ^yin the bottom layer):\n(1) a torque due to the out-of-plane demagnetizing\n\feld (Fig. 4(b)) and (2) a torque due to the interlayer\nantiferromagnetic exchange penalty (Fig. 4(c)). These\ndemagnetizing and antiferromagnetic-exchange torques\nhave the same symmetry to drive 90\u000eswitching of the\nN\u0013 eel vector lfrom thex-axis to the y-axis.\nAn exemplary time evolution of the N\u0013 eel vector and\ntotal magnetization is shown in Fig. 4(d,e). The\ninitial rise in mzcon\frms the out-of-plane tilting of the\nmagnetization, while mx;y\u00190 indicates that the in-plane\nmagnetization components remain compensated between\nthe two layers. Moreover, the damped oscillation of l\n(Fig. 4(d)) exhibits a phase o\u000bset of \u0019=2 with respect\ntomz(Fig. 4(e)), i.e., the time rate of change of lis\nmaximized whenjmzjexhibits a maximum. This relationcon\frms that the torque on lis indeed related to the\nmagnetization canting jmzj.\nThe relative contributions of the torques can be tuned\nby varying the saturation magnetization Ms, since a\nsmaller value of Msshould decrease the demagnetizing\ntorque contribution. Figure 5 compares the time-\ndependence of switching for SAFs with Ms= 1700 kA/m\nand 170 kA/m, each with di\u000berent strengths of interlayer\nexchange coupling Jex. For each Ms, the magnitude of\nthe spin-orbit \feld Bsois chosen to be slightly above\nthe threshold for switching Bth\nso. In the case of Ms\n= 1700 kA/m, the switching speed changes only by\na factor of\u00192 whenJexis varied by a factor of 25\n(Fig. 5(a,b)). Evidently, for SAFs consisting of high-\nmoment FMs (e.g., Fe), the out-of-plane demagnetizing\ntorque dominates the switching process, whereas the\nantiferromagnetic-exchange torque plays a relatively\nminor role. We thus \fnd that although SAFs have\nzero net magnetization at equilibrium, their dynamics\ncan be driven predominantly by the demagnetizing \feld\nfrom nonequilibrium magnetization. By contrast, in\nthe case of Ms= 170 kA/m, increasing jJexjresults\nin an order of magnitude faster switching (Fig. 5(c,d)),\nindicating that the antiferromagnetic-exchange torque\nplays a relatively major role when the constituent FMs\nhave low magnetization.\nEnhancing the interlayer exchange coupling is not\nnecessarily an e\u000bective way to speed up switching\nin SAFs because (1) the exchange torque may not\nbe the dominant driving mechanism for switching\nif the constituent FMs have high Msand, (2)\neven if the exchange torque dominates, it would\nbe practically di\u000ecult to increase jJexjwell above\n\u00181 mJ/m2. We therefore explore an alternative method\nof enhancing the switching speed by increasing the\nGilbert damping parameter \u000b, which is experimentally\nmore straightforward (e.g., through alloying the FM\nwith a small concentration of rare-earth metal47).\nThe threshold current density for switching driven\nby the \feld-like torque is not signi\fcantly a\u000bected\nby damping48. This is in contrast with coherent\nswitching of a single-domain in-plane nanomagnet\ndriven by a damping-like torque, where the threshold\ncurrent density is inversely proportional to the damping\nparameter49; since lower damping would prolong\nthe magnetization oscillations before settling along\nthe equilibrium orientation, damping-like-torque-driven\nswitching leads to a trade-o\u000b between reducing the\npower consumption (threshold switching current density)\nand the switching time. Our proposed scheme of\nutilizing the \feld-like torque in the biaxial SAF allows\nfor speeding up switching by increasing the damping\nparameter, without adversely a\u000becting the threshold\nswitching current density.\nFigure 6 shows the in\ruence of the damping parameter\n\u000bon the time evolutions of landmin an SAF. The\noscillations around the y-axis are signi\fcantly suppressed\nat higher values of \u000bin Fig. 6. We note, however, that5\nJex= -5 mJ/m2\nJex= -1 mJ/m2\nJex= -0.2 mJ/m2Ms= 1700 kA/m\n(Bso= 6 mT)Ms= 170 kA/m\n(Bso= 74 mT)(a) (c)\n(d)Jex= -5 mJ/m2\nJex= -1 mJ/m2\nJex= -0.2 mJ/m2\n(b)\ntime [ns]mz -ly\n-ly\nmz\ntime [ns]\nFigure 5. Time traces of the (a,c) N\u0013 eel vector component lyand (b,d) total magnetization component mzfor samples with\n(a,b)Ms= 1700 kA/m and (c,d) Ms= 170 kA/m at di\u000berent strengths of interlayer exchange coupling Jex. The magnitude\nof the spin-orbit \feld Bsois chosen to be slightly above the threshold for switching. The sample width is 52 nm and \u000b= 0:01.\n(a)\n(b)= 0.05\n= 0.01\n= 0.002Ms= 1700 kA/m\n(Bso= 6 mT)\nmz -ly\ntime [ns]\nFigure 6. Time traces of the (a) N\u0013 eel vector component ly\nand (b) total magnetization component mzatBso= 6 mT in\nthe 52-nm-wide SAF sample for with Ms= 1700 kA/m and\nJex= -1 mJ/m2at di\u000berent Gilbert damping parameters \u000b.\nby increasing \u000bfurther to>\u00180:1, the switching process\nbecomes overdamped and is hence slowed down. These\nresults indicate that the switching time is minimized with\na moderately large value of \u000b. The time traces shown in\nFig. 6 are obtained at Bso= 6 mT, which is only slightly\nabove the threshold for switching for the simulated 52-\nnm-wide device (Fig. 3). The switching time can be\ndecreased further with a greater spin-orbit \feld (currentdensity). Our results thus suggest that 90\u000eswitching in\nan SAF device can be accomplished in <\u00180:1 ns at a\nreasonable current density of \u00181011A/m2, provided a\nsu\u000eciently strong interfacial Rashba-Edelstein e\u000bect (as\ndiscussed in Sec. III B). While \u00180.1-ns switching has been\ndemonstrated for SOT-driven perpendicular anisotropy\nmemories, the required current density exceeds 1012\nA/m250. Biaxial SAFs may therefore be an attractive\npower-e\u000ecient alternative to conventional spintronic\nmemory platforms.\nIV. SUMMARY\nWe have demonstrated by micromagnetic\nsimulations that biaxial SAFs { consisting of two\nantiferromagnetically-coupled FMs { are stable against\nlarge external magnetic \felds and can be switched\ne\u000eciently with a \feld-like SOT. Even though SAFs have\nzero net magnetization at equilibrium, the \feld-like SOT\nyields a \fnite nonequilibrium magnetization, which gives\nrise to switching driven mostly by the torque from the\ndemagnetizing \feld, particularly if the SAF consists of\nhigh-moment FMs (e.g., Fe). The 90\u000eswitching scheme\ncan enable fast dynamics, especially when combined with\nmoderately high Gilbert damping. Such SAFs can be\nreadily engineered from simple FMs and are attractive\nmodel systems that mimic some of the dynamics of\nintrinsic crystal antiferromagnets.\nThis work was supported in part by the Luther\nand Alice Hamlett Undergraduate Research Support\nProgram in the Academy of Integrated Science at\nVirginia Tech. 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Ketterson,3Mingzhong Wu,2and Axel Hoffmann1\n1)Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439,\nUSA\n2)Department of Physics, Colorado State University, Fort Col lins, Colorado 80523,\nUSA\n3)Department of Physics and Astronomy, Northwestern Univers ity, Evanston, Illinois 60208,\nUSA\n(Dated: 1 May 2018)\nWe investigated the spin-wave propagation in a micro-structured y ttrium iron garnet waveguide of 40 nm\nthickness. Utilizing spatially-resolved Brillouin light scattering microsc opy, an exponential decay of the spin-\nwave amplitude of (10 .06±0.83)µm was observed. This leads to an estimated Gilbert damping constant\nofα= (8.79±0.73)×10−4, which is larger than damping values obtained through ferromagnet ic resonance\nmeasurements in unstructured films. The theoretically calculated s patial interference of waveguide modes\nwas compared to the spin-wave pattern observed experimentally b y means of Brillouin light scattering spec-\ntroscopy.\nI. INTRODUCTION\nMagnonics is an emerging field of magnetism study-\ning the spin dynamics in micro- and nanostructured\ndevices aiming for the development of new spintron-\nics applications.1–3Up to now, ferromagnetic metals\n(for example, Permalloy and Heusler alloys) have been\nwidely used for the investigation of magnetization dy-\nnamicson the nanoscale.4–10However, the Gilbert damp-\ning of Permalloy is two orders of magnitude higher than\nthat of ferrimagnetic insulator yttrium iron garnet (YIG,\nY3Fe5O12). Recent progress in the growth of YIG films\nallows for the fabrication of low-damping nanometer-\nthick YIG films,11–14which are well-suited for patterning\nof micro-structured YIG devices. This enables investiga-\ntions of spin-wave propagation in plain YIG microstruc-\ntures of sub-100 nm thicknesses which are a step forward\nfor future insulator-based magnonics applications.\nInthiswork,weexperimentallydemonstratespin-wave\npropagation in a micro-structured YIG waveguide of\n40 nm thickness and 4 µm width. By utilizing spatially-\nresolvedBrillouinlightscattering(BLS) microscopy4–7,11\nthe exponential decay length of spin waves is deter-\nmined. The corresponding damping parameter of the\nmicro-structured YIG is estimated and compared to that\ndetermined from ferromagnetic resonance (FMR) mea-\nsurements. Furthermore, we show that different spin-\nwave modes quantized in the direction perpendicular to\nthe waveguide lead to a spatial interference pattern. We\ncompare the experimental results to the theoretically ex-\npected spatial interference of the waveguide modes.\na)Electronic mail: jungfleisch@anl.govII. EXPERIMENT\nFigure1shows a schematic illustration of the sam-\nple layout. The YIG film of 40 nm thickness was de-\nposited by magnetron sputtering on single crystal pol-\nished gadolinium gallium garnet (GGG, Gd 3Ga4O12)\nsubstrates of 500 µm thickness with (111) orientation\nunder high-purity argon atmosphere. The film was sub-\nsequently annealed in-situ at 800◦C for 4 hours under an\noxygenatmosphereof1.12Torr. Themagneticproperties\nofthe unstructured film werecharacterizedby FMR: The\npeak-to-peak linewidth µ0∆Has a function of the exci-\ntation frequency fis depicted in Fig. 2(a). The Gilbert\ndamping parameter αFMRcan be obtained from FMR\nmeasurements using15\n√\n3µ0∆H=2αFMR\n|γ|f+µ0∆H0, (1)\nFIG. 1. (Color online) Schematic illustration of the sample\nlayout. The 4 µm wide yttrium iron garnet waveguide is\nmagnetized transversally by the bias magnetic field H. Spin\nwaves are excited by a shortened coplanar waveguide and the\nspin-wave intensity is detected by means of spatially-reso lved\nBrillouin light scattering microscopy. Colorbar indicate s spin-\nwave intensity.2\nwhereµ0is the vacuum permeability, γis the gyromag-\nnetic ratio, fis the resonance frequency and µ0∆H0\nis the inhomogeneous linewidth broadening. We find a\ndamping parameter of αFMR= (2.77±0.49)×10−4[fit\nshown as a red solid line in Fig. 2(a)]. The resonance\nfield,µ0H, as a function of the excitation frequency fis\nshown in Fig. 2(b). A fit to16\nf=µ0|γ|\n2π/radicalbig\nH(H+Meff) (2)\nyields an effective magnetization Meff= (122 ±\n0.30) kA/m [solid line in Fig. 2(b)]. In a subsequent fab-\nrication process, YIG waveguides of 4 µm width were\npatterned by photo-lithography and ion milling with\nan Ar plasma at 600 V for 5 min. In a last step, a\nshortened coplanar waveguide (CPW) made of Ti/Au\n(3 nm/150 nm) is patterned on top ofthe YIG waveguide\n(see Fig. 1). The shortened end of the CPW has a width\nof 5µm. The Oersted field of an alternating microwave\nsignal applied to the CPW exerts a torque on the mag-\nnetic moments in the YIG and forces them to precess.\nThe bias magnetic field is applied perpendicular to the\nshortaxisofthe waveguide(Fig. 1) providingefficient ex-\ncitation of Damon-Eshbach spin waves. The microwave\npowerPMW= 1 mW is sufficiently small to avoid pos-\nsible perturbations of spin-wave propagation caused by\nnonlinearities.\nIII. DISCUSSIONS\nIn order to detect spin-wave propagation in the YIG\nwaveguide spatially-resolved BLS microscopy with a res-\nolution of 250 nm is employed. To characterize the prop-\nagating spin waves, the BLS intensity was recorded at\ndifferentdistancesfromtheantenna. Aspatially-resolved\nBLS intensity map is shown in Fig. 3(b) at an exemplary\nexcitation frequency of f= 4.19 GHz. Spin waves are\nexcited near the antenna and propagate towards the op-\nposite end of the waveguide. To further analyze the data\nand to minimize the influence of multi-mode propagation\nin the YIG stripe (this will be discussed below), the BLS\nintensity is integrated over the width of the waveguide.\nThe correspondingBLS intensity as a function of the dis-\ntance from the antenna is illustrated in Fig. 3(c). The\ndecayofthespin-waveamplitudecanbedescribedby:7,11\nI(z) =I0e−2z\nλ+b, (3)\nwherezis the distance from the antenna, λis the de-\ncay length of the spin-wave amplitude and bis an offset.\nFrom Fig. 3(b), it is apparent that the data-points fol-\nlow an exponential behavior. A fit according to Eq. ( 3)\nyields the decay length λ. For an excitation frequency\noff= 4.19 GHz we find λ= (10.06±0.83)µm. This\nvalue is larger than decay lengths reported for Permalloy\n(<6µm, see Ref. 8, 21, and 22), but it is smaller thanFIG. 2. (Color online) (a) Ferromagnetic resonance peak-to -\npeaklinewidth µ0∆Has afunctionoftheresonance frequency\nfof the unstructured 40 nm YIG film. The red solid line\nrepresents a fit to Eq. (1). A Gilbert damping parameter\nα= (2.77±0.49)×10−4is determined. (b) Ferromagnetic\nresonance field µ0Has a function of f. Error bars are smaller\nthan the data symbols.\nthe largest decay length found for the Heusler-compound\nCo2Mn0.6Fe0.4Si (8.7 – 16.7µm, see Ref. 7). Pirro et al.\nreported a decay length of 31 µm in thicker YIG waveg-\nuides(100nm) grownbyliquidphaseepitaxy(LPE)with\na 9 nm thick Pt capping layer.11In order to understand\nthis discrepancy between our and their results, one has\nto take into account two facts: (1) State-of-the-art LPE\nfabrication technology can not be employed to grow film\nthicknesses below ∼100 nm. To date, sputtering offers\nan alternative approach to grow sub-100 nm thick YIG\nfilms with a sufficient quality.23(2) Taking into account\nthe spin-wave group velocity vg=∂ω/∂kand the spin-\nwave lifetime τ, the theoretically expected decay length\nλcan be calculated from λ=vg·τ. The groupvelocity vg\ncan be derived directly from the dispersion relation (see\nFig.4). A thinner YIG film has flatter dispersion and a\nsmallervg. Consequently, the expected decay length is\nsmaller for thinner YIG samples and so it is natural that\nthe decay length reported here is shorter than the one\nfound in Ref. 11 for 100 nm thick YIG waveguides.\nWe estimate the groupvelocity from the spin-wavedis-\npersion to be vg= 0.35−0.40µm/ns. Using our exper-\nimentally found decay length, the spin-wave lifetime is\ndetermined to be τ= 29 ns. We can use the BLS-data\nto determine the corresponding Gilbert damping param-\neterαBLS. In case of Damon-Eshbach spin waves, the\ndamping is given by243\nFIG. 3. (Color online) (a) Calculated spatial interference\npattern of the first two odd waveguide modes ( n= 1 and\nn= 3. (b) Spatially-resolved BLS intensity map at an ex-\ncitation frequency f= 4.19 GHz, applied microwave power\nPMW= 1 mW, biasing magnetic field µ0∆H= 83 mT. The\nnumbers 1 – 4 highlight the main features of the interference\npattern. (c) Corresponding BLS intensity integrated over t he\nentire width of the YIG waveguide. An exponential decay of\nthe spin-wave amplitude λ= (10.06±0.83)µm is found.\nαBLS=1\nτ(γµ0Meff\n2+2πf)−1. (4)\nA Gilbert damping parameter of the micro-structured\nYIG waveguide obtained by BLS characterization is\nfound to be αBLS= (8.79±0.73)×10−4, which is a\nfactor of 3 times larger than that determined by FMR in\nthe unstructuredfilm [ αFMR= (2.77±0.49)×10−4]. This\ndifference mightbe attributed to the micro-structuringof\nthe YIG waveguide by Ar ion beam etching. The etch-\ning might enhance the roughness of the edges of the YIG\nwaveguides and the resist processing could have an in-\nfluence on the surface quality17,18which could possibly\nlead to an enhancement of the two-magnon scattering\nprocess.19It would be desirable to perform FMR mea-\nsurements on the YIG waveguide. However, since the\nstructured bar is very small, the FMR signal is van-\nishingly small which makes it difficult to determine the\nGilbert damping in this way.\nWhile the discussion above only considered the BLS-\nintensity integrated over the waveguide width, we will\nfocus now on the spatial interference pattern shown in\nFig.3(b). The spin-wave intensity map can be under-\nstood by taking into account the dispersion relation of\nmagnetostatic spin waves in an in-plane magnetized fer-\nromagnetic thin film (see Fig. 4). Due to the lateral con-\nfinement, thewavevectorisquantizedacrossthewidthof\nthe YIG waveguide, ky=nπ/w, wheren∈N. The wave\nvectorkzalong the long axis of the waveguide ( z-axis)is assumed to be non-quantized. We follow the approach\npresented in Ref. 8. The dynamic magnetization is as-\nsumed to be pinned at the edges of the waveguide which\ncan be considered by introducing an effective width of\nthe waveguide.8,20\nFigure4shows the calculated dispersion relations of\ndifferent spin-wave modes quantized across the width of\nthe strip. The dashed line represents a fixed excitation\nfrequency. At a particular frequency different spin-waves\nmodes with different wave vectors kzare excited simulta-\nneously. Thisleadstothe occurrenceofspatiallyperiodic\ninterference patterns. In the present excitation config-\nuration, only modes with an odd quantization number\nncan be excited ( ndetermines the number of maxima\nacross the width of the waveguide). Since the intensity\nof the dynamic magnetization of these modes decreases\nwith increasing nas 1/n2, we only consider the first two\nodd modes n= 1 and n= 3.\nAccording to Ref. 8 the spatial distribution of the dy-\nnamic magnetization of the n-th mode can be expressed\nas\nmn(y,z)∝sin(nπ\nwy)cos(kn\nzz−2πft+φn),(5)\nwherefis the excitation frequency, kn\nzis the longitudinal\nwave vector of the n-th spin-wave mode and φnis the\nphase.25The spin-waveintensity distribution Inof then-\nth mode can be derived by averaging mn(y,z)2over one\noscillation period 1 /f. The entire interference pattern\ncan be obtained from the same procedure using the sum\nm1(y,z) +1\n3m3(y,z). The factor 1/3 accounts for the\nlower excitation efficiency of the n= 3 mode. Thus, the\nintensity is given by8\nIΣ(y,z)∝sin(π\nwy)2+1\n9sin(3π\nwy)2\n+2\n3sin(π\nwy)sin(3π\nwy)cos(∆kzz+∆φ),(6)\nwhere ∆kz=k3\nz−k1\nzand ∆φ=φ3−φ1. This pattern re-\npeatsperiodically. Thephaseshift φshiftsthe entirepat-\nFIG. 4. (Color online) Dispersion relations the first five\nwaveguide modes of a transversally magnetized YIG stripe.26\nOnly modes with a odd quantization number ncan be excited\n(solid lines). fdenotes the excitation frequency.4\ntern along the z-direction and the wave-vector difference\n∆kz= 0.97 rad/µm can be calculated from the disper-\nsion relation (Fig. 4). The calculated spatial interference\npattern is depicted in Fig. 3(a) using ∆ φ= 0 and taking\ninto account for the exponential decay of the spin-wave\namplitude by multiplying Eq. ( 6) withe−2z/λusing the\nexperimentallydetermined λ= 10.03µm. As is apparent\nfrom Fig. 3(a) and (b) a qualitative agreement between\ncalculation and experiment is found. (The numbers 1 –\n4 highlight the main features of the interference pattern\nin experiment and calculation.) The small difference in\nFig.3(a) and (b) can be explained by considering the\nfact that in the calculation a spin-wave propagation at\nan angle of exactly 90◦(Damon-Eshbach configuration)\nwith respect to the antenna/externalmagnetic field is as-\nsumed. However, in experiment small misalignments of\nthe external magnetic field might lead to a small asym-\nmetry in the interference pattern.\nIV. CONCLUSION\nIn summary, we demonstrated spin-wave excitation\nand propagation in micro-fabricated pure YIG wave-\nguides of 40 nm thickness. BLS-characterization re-\nvealed a decay length of the spin-wave amplitude of\n10µm leading to an estimated Gilbert damping pa-\nrameter of αBLS= (8.79±0.73)×10−4. This value\nis a factor 3 larger than the one determined for the\nunstructured YIG film by means of FMR techniques\n[αFMR= (2.77±0.49)×10−4]. The difference might be\nattributed to micro-structuring using ion beam etching.\nThe observed spatial spin-wave intensity distribution is\nexplained by the simultaneous excitation of the first two\nodd waveguide modes. These findings are important for\nthe development of new nanometer-thick magnon spin-\ntronics applications and devices based on magnetic insu-\nlators.\nV. ACKNOWLEDGMENTS\nWork at Argonne was supported by the U.S. Depart-\nment of Energy, Office of Science, Materials Science and\nEngineering Division. Work at Colorado State Univer-\nsity was supported by the U.S. Army Research Office,\nand the U.S. National Science Foundation. Lithography\nwas carried out at the Center for Nanoscale Materials,\nwhich is supported by DOE, Office of Science, Basic En-\nergy Sciences under ContractNo. DE-AC02-06CH11357.\n1S. Neusser and D. Grundler, Adv. Mater. 21, 2927 (2009).2V.V. Kruglyak, S.O. Demokritov, D. Grundler, J. Phys. 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Reiss, Phys. Rev. B 79, 054417 (2009).\n11P. Pirro, T. Br¨ acher, A.V. Chumak, B. L¨ agel, C. Dubs,\nO. Surzhenko, P. G¨ ornert, B. Leven and B. Hillebrands, Appl .\nPhys. Lett. 104, 012402 (2014).\n12T. Liu, H. Chang, V. Vlaminck, Y. Sun, M. Kabatek, A. Hoff-\nmann, L. Deng, and M. Wu, J. Appl. Phys. 115, 17A501 (2014).\n13O. d’Allivy Kelly, A. Anane, R. Bernard, J. Ben Youssef,\nC. Hahn, A H. Molpeceres, C. Carr´ et´ ero, E. Jacquet, C. Der-\nanlot, P. Bortolotti, R. Lebourgeois, J.-C. Mage, G. de Loub ens,\nO. Klein, V. Cros, and A. Fert, Appl. Phys. Lett. 103, 082408\n(2013).\n14Y. Sun, Y.-Y. Song and M. Wu, Appl. Phys. Lett. 101, 082405\n(2012).\n15S.S.Kalarickal, P.Krivosik,M.Wu, C.E.Patton, M.L.Schne ider,\nP.Kabos, T.J.Silva, andJ.P.Nibarger,J.Appl.Phys. 99,093909\n(2006).\n16A. Azevedo, A.B. Oliveira, F.M de Aguiar, and S.M. Rezende,\nPhys. Rev. B, 62, 5331 (2000).\n17B.J.McMorran, A.C.Cochran, R.K.Dumas, KaiLiu, P.Morrow,\nD.T. Pierce and J. Unguris, J. Appl. Phys. 107, 09D305 (2010).\n18O.D. Roshchupkina, J. Grenzer, T. Strache, J. McCord,\nM. Fritzsche, A. Muecklich, C. Baehtz, and J. Fassbender, J.\nAppl. Phys. 112, 033901 (2012)\n19R. Arias and D.L. Mills, Phys. Rev. B 60, 7395 (1999).\n20K.Yu. Guslienko, S.O. Demokritov, B. Hillebrands, and\nA.N. Slavin, Phys. Rev. B 66, 132402 (2002).\n21M. Madami, S. Bonetti, G. Consolo, S. Tacchi, G. Carlotti,\nG. Gubbiotti, F.B. Mancoff, M.A. Yar, and J. ˚Akerman, Nat.\nNano.6, 635 (2011).\n22P. Pirro, T. Br¨ acher, K. Vogt, Bj¨ orn Obry, H. Schultheiss,\nB. Leven, and B. Hillebrands, Phys. Status Solidi B 238, 2404\n(2011).\n23H. Chang, P. Li, W. Zhang, T Liu, A. Hoffmann, L. Deng, and\nM. Wu, IEEE Magnetic Letters 5, 6700104 (2014).\n24D.D. Stancil and A. Prabhakar, Spin Waves - Theory and Ap-\nplications , (Springer, 2009).\n25T. Schneider, A.A. Serga, T. Neumann, B. Hillebrands, and\nM.P. Kostylev, Phys. Rev. B 77, 2144 (2008).\n26For the calculation of the dispersion relations the followi ng pa-\nrameters have been used: external magnetic field µ0H= 83 mT,\nexchange constant A= 3.6 pJ/m, saturation magnetization\nMS=140 kA/m, effective width of the waveguide weff= 3.5µm,\nYIG-film thickness t= 40 nm."    },    {        "title": "2006.16510v1.Negative_Gilbert_damping_in_cavity_optomagnonics.pdf",        "content": "Negative Gilbert damping in cavity optomagnonics\nYunshan Cao\u0003and Peng Yany\nSchool of Electronic Science and Engineering and State Key Laboratory of Electronic Thin Films and Integrated Devices,\nUniversity of Electronic Science and Technology of China, Chengdu 610054, China\nExceptional point (EP) associated with the parity-time ( PT) symmetry breaking is receiving considerable\nrecent attention by the broad physics community. By introducing balanced gain and loss, it has been realized in\nphotonic, acoustic, and electronic structures. However, the observation of magnonic EP remains elusive. The\nmajor challenge is to experimentally generate the negative Gilbert damping, which was thought to be highly\nunlikely but is demanded by the PT symmetry. In this work, we study the magneto-optical interaction of\ncircularly-polarized lasers with a submicron magnet placed in an optical cavity. We show that the o \u000b-resonant\ncoupling between the driving laser and cavity photon in the far-blue detuning can induce the magnetic gain (or\nnegative damping) exactly of the Gilbert type. A hyperbolic-tangent function ansatz is found to well describe\nthe time-resolved spin switching as the intrinsic magnetization dissipation is overcome. When the optically\npumped magnet interacts with a purely lossy one, we observe a phase transition from the imbalanced to passive\nPTsymmetries by varying the detuning coe \u000ecient. Our findings provide a feasible way to manipulate the sign\nof the magnetic damping parameter and to realize the EP in cavity optomagnonics.\nIntroduction. —One of the most fundamental principles in\nquantum mechanics is that a physical observable should be\ndescribed by a Hermitian operator to guarantee real eigenval-\nues [1]. However, Bender and Boettcher [2] reported a class\nof non-Hermitian Hamiltonians that allow entirely real spec-\ntrum as long as the combined parity ( P) and time (T)-reversal\nsymmetries are respected. By tuning system parameters, both\nthe eigenvalues and eigenstates of the PT-symmetric Hamil-\ntonian simultaneously coalesce [3, 4], giving rise to a non-\nHermitian degeneracy called exceptional point (EP). The na-\nture around the EP that is accompanied by a phase transi-\ntion can trigger many intriguing phenomena, such as unidi-\nrectional invisibility [5, 6], loss-induced laser suppression and\nrevival [7] and optical transparency [8], laser mode selection\n[9], and EP enhanced sensing [10–13]. Over the past decades,\nthe experimental observation of EPs has been realized in a\nbroad field of photonics [14–17], acoustics [18, 19], and elec-\ntronics [20–22]. Very recently, the concept of PTsymmetry\nis attracting significant attention in spintronics and magnonics\n[23–31]. The simplest way to obtain a PT-symmetric system\nconsists in coupling two identical subsystems, one with gain\nand the other with equal amount of loss. The composite sys-\ntem isPT symmetric because space reflection interchanges\nthe subsystems, and time reversal interchanges gain and loss.\nIndeed, aPT-symmetric magnetic structure composed of two\nidentical ferromagnets with balanced gain and loss was first\nproposed by Lee et al. [23] and subsequently investigated by\nYang et al. [27]. One recent breakthrough was made by Liu\net al. [32] who reported EP in passive PT-symmetric devices\nin the form of a trilayer structure with two magnetic layers\nof di\u000berent (positive) Gilbert damping. However, the exper-\nimental observation of genuine PT symmetry for magnons\n(the quanta of spin waves)—as elementary excitations in or-\ndered magnets—is still elusive. The di \u000eculty lies in that the\nGilbert damping can hardly be tuned to be negative [33, 34].\nThe past ten years have witnessed the development and\napplication of spin cavitronics, allowing cavity photons res-\nonantly coupled to magnons with the same microwave fre-quency [35–46]. One recent trend beyond microwaves is the\nrealization of the parametric coupling between optical lasers\nand magnons, that would generate interesting new opportuni-\nties. Tantalizing physics indeed has been demonstrated, such\nas nonreciprocal Brillouin light scattering [47], microwave-\nto-optical converting [48, 49], optical cooling of magnons\n[50], etc. In these studies, considerable interests have been\ndrawn to the scalar properties of magnons, e.g., magnon num-\nber (population), temperature, and chemical potential, which\nis successful to describe the small-angle spin precession. In\ncontrast, their vectorial behavior, i.e., the full time-evolution\nof the magnetic moment driven by optical lasers, remains\nlargely unexplored, with few exceptions [51]. It has been\nshown that a ferromagnetic-to-antiferromangetic phase tran-\nsition may emerge in the vicinity of the magnonic EP [27]. In\nsuch case, the magnetic moment would significantly deviate\nfrom its equilibrium direction, and a vectorial field descrip-\ntion becomes more relevant than a scalar one.\nS\nz\nx y\nCircularly polarized \nlaser beamOptical cavity(a) (b)\n(c)ωlas > ωcav\nωlas‘ωm\nRed-detuningBlue-detuning\nωlas < ωcav\nωlas‘\nωm\nFIG. 1: (a) Schematic illustration of a macrospin Sinteracting\nwith three orthogonally propagating circularly-polarized lasers (red\nbeams) in an optical cavity. O \u000b-resonant coupling between the driv-\ning laser (!las) and the cavity photon ( !cav) mediated by magnons\n(!m\u001c!cav) in the blue (b) and red (c) detuning regimes.\nIn this Letter, we propose to realize the negative GilbertarXiv:2006.16510v1  [cond-mat.mtrl-sci]  30 Jun 20202\ndamping by considering the optomagnonic interaction be-\ntween three orthogonally propagating circularly-polarized\nlasers and a submicron magnet placed in an optical cavity [see\nFig. 1(a)]. By solving the coupled equations of motion and\nintegrating the photon’s degree of freedom, we derive the an-\nalytical formula of the optical torque acting on the macrospin.\nIn the far-blue detuning, we find that the optical torque exactly\ntakes the Gilbert form \u0000\u000bopt\nS˙S\u0002Swith\u000bopt>0 (see below).\nThe total Gilbert damping becomes negative when the intrin-\nsic dissipation is overcome. In such case, a hyperbolic-tangent\nfunction ansatz is found to well describe the time-resolved\nspin switching. We further study the optically pumped spin\ninteracting with a purely lossy one, and observe a phase transi-\ntion from the imbalanced to passive PTsymmetries by vary-\ning the detuning parameter.\nModel. —The proposed setup is schematically plotted in\nFig. 1(a). Three circularly-polarized laser beams propagat-\ning respectively along x;y;zdirections drive the parametric\ncoupling with a macrospin S=(ˆSx;ˆSy;ˆSz) inside the optical\ncavity. The Hamiltonian reads\nH=\u0000~!0ˆSz\u0000~X\nj=x;y;z\u0010\n\u0001j\u0000gjˆSj\u0011\nˆcy\njˆcj+Hdr; (1)\nwhere!0=\rB0is the Larmor frequency around the exter-\nnal magnetic field B0pointing to the negative z-direction with\n\rbeing the gyromagnetic ratio, \u0001j=!las;j\u0000!cavis the de-\ntuning between the laser frequency !las;jand the cavity reso-\nnant frequency !cav, and ˆ cy\nj(ˆcj) is the creation (annihilation)\noperator of the optical cavity photons, with j=x;y;z. The\ncoupling strength gjbetween the spin and optical photon orig-\ninates from the Faraday-induced modification of the electro-\nmagnetic energy in ferromagnets [52]. The last term describes\nthe interaction between the driving laser and the cavity pho-\ntonHdr=i~P\nj(Ajˆcy\nj\u0000h:c:), where Aj=(2\u0014jPj=~!las;j)1=2is\nthe field amplitude, with \u0014jthe laser loss rate and Pjbeing the\ndriving power.\nThe Heisenberg-Langevin equations of motion for coupled\nphotons and spins are expressed as ( o\u0011hˆoi),\n˙cj=(i\u0001j\u0000\u0014j)cj\u0000igjSjcj+Aj; (2a)\n˙Sx=!0Sy+gynySz\u0000gznzSy; (2b)\n˙Sy=\u0000!0Sx\u0000gxnxSz+gznzSx; (2c)\n˙Sz=\u0000gynySx+gxnxSy; (2d)\nwhere nj=hˆcy\njˆcjiis the average photon number in the cav-\nity. Because the spin dynamics usually is much slower than\noptical photons, one can expand the cavity photon operator as\ncj(t)\u0019cj0(t)+cj1(t)+\u0001\u0001\u0001, in orders of ˙Sj. Equation (2a) then\ncan be recast in series\n0=(i\u0001j\u0000\u0014j)cj0\u0000igjSjcj0+Aj; (3a)\n˙cj0=(i\u0001j\u0000\u0014j)(cj0+cj1)\u0000igjSj(cj0+cj1)+Aj;(3b)\nby keeping up to the first-order terms. We can therefore derivethe formula of photon number in the cavity\nnj(t)\u0019 jcj0j2+2Re[ c\u0003\nj0cj1]\n=A2\nj\n(\u0001j\u0000gjSj)2+\u00142\nj\u00004\u0014jA2\njgj(\u0001j\u0000gjSj)\nh\n(\u0001j\u0000gjSj)2+\u00142\nji3˙Sj:(4)\nSubstituting (4) into Eqs. (2b)-(2d), we obtain\n˙S=\u0000\rS\u0002Be\u000b+\u000b\nS(˙S\u0002S)\u0000\fopt\u0002S; (5)\nwhere the e \u000bective magnetic field Be\u000b=\u0000B0ez+Boptin-\ncludes both the external magnetic field and the optically in-\nduced magnetic field\nBopt=X\nj\r\u00001gjA2\nj\n(\u0001j\u0000gjSj)2+\u00142\njej; (6)\nwhich is the zeroth-order of ˙Sj. The second term in the right\nhand side of (5) is the intrinsic Gilbert damping torque, with\nS=jSjthe total spin number and \u000b > 0 being the intrinsic\nGilbert damping constant. The last term in (5) represents the\noptical torque with the anisotropic e \u000bective field\n\fopt=X\nj4\u0014jA2\njg2\nj(\u0001j\u0000gjSj)\nh\n(\u0001j\u0000gjSj)2+\u00142\nji3˙Sjej; (7)\nwhich is linear with the first-order time-derivative of Sj. Be-\nlow, we show that the anisotropic nature of (7) can be smeared\nout under proper conditions.\nNegative Gilbert damping. —To obtain the optical torque of\nexactly the Gilbert form, we make two assumptions: (i) the\nthree laser beams are identical, i.e., Aj=A;gj=g;\u0014j=\u0014;\nand\u0001j= \u0001; (ii) the optomagnonic coupling works in the far\ndetuning regime, i.e., j\u0011j\u001d1 with\u0011= \u0001=(gS), which allows\nus to drop the gjSjterms in Eq. (7). The optically induced\ne\u000bective fields then take the simple form\nBopt=\r\u00001gA2\n\u00012+\u00142X\njej; (8)\nand\n\fopt=\u000bopt\nS˙S;with\u000bopt=4\u0014A2g2S\u0001\n(\u00012+\u00142)3(9)\nbeing the laser-induced magnetic gain or loss that depends the\nsign of the detuning \u0001. Based on the above results, we finally\nobtain the optically modulated spin dynamics\n˙S=\u0000\rS\u0002Be\u000b+\u000be\u000b\nS(˙S\u0002S); (10)\nwith\u000be\u000b=\u0000\u000bopt+\u000b. One can observe that a negative ef-\nfective Gilbert constant ( \u000be\u000b<0) emerges in the far-blue de-\ntuning regime, i.e., 1 < \u0011 < \u0011 c. In case of the red detun-\ning (\u0011 < 0), we have \u000bopt<0, which indicates the enhance-\nment of the magnetic attenuation. In the deep-blue detuning3\nηηηPTηC\nηC=7.11η P (W)P (μWĎ\nr (m)αopt /α Bopt (μT)αeff =0\nBopt =333 μT(a)\n(b)(c)\n(d)\nηPTηCBopt =440 μTαeff =-α\n×\nFIG. 2: Optically induced magnetic gain (a) and magnetic field (b)\nvs. the optical detuning parameter \u0011. (c)\u0011PT(orange) and \u0011C(green)\nas a function of the driving laser power. (d) Radius dependence of\nthe laser power at the compensation point \u0011C=7:11.\nregime (\u0011>\u0011 c), driving lasers can still generate the magnetic\ngain (\u000bopt>0) but cannot compensate the intrinsic dissipa-\ntion, i.e., 0 < \u000b opt< \u000b. Here\u0011cis the critical detuning pa-\nrameter at which the e \u000bective Gilbert damping vanishes. The\nphysics can be understood from the diagram plotted in Figs.\n1(b) and 1(c): In the blue detuning regime ( !las> ! cav), mi-\ncrowave magnons are emitted in the non-resonant interaction\nbetween the driving laser and the cavity photon, representing\na magnetic gain. On the contrary, they are absorbed in the red\ndetuning (!las< ! cav), manifesting a magnon absorption or\ncooling. Below we discuss practical materials and parameters\nto realize this proposal.\nMaterials realizations. —For a ferromagnetic insulator like\nyttrium ion garnet (YIG), the intrinsic Gilbert constant \u000btyp-\nically ranges 10\u00003\u001810\u00005[53–55]. We take \u000b=10\u00004\nin the following calculations. The magneto-optical coupling\nstrength is determined by the Faraday rotation coe \u000ecient\u0012F\nof the materials gS'c\u0012F=p\u000fr, with cthe speed of light and\n\u000frthe relative permittivity (for YIG, we choose \u000fr=15 [56]\nand\u0012F=188\u000e=cm [57]). We thus have gS=2\u0019\u00191 GHz. The\noptical cavity is set at the resonant frequency !cav=2\u0019=100\nTHz with the loss rate \u0014=2\u0019=1 GHz. For a YIG sphere of\nradius r=10 nm and spin density \u001as\u00191028m\u00003, we esti-\nmate the total spin number S=\u001asr3\u0019104and the coupling\nstrength g=2\u0019\u00190:1 MHz. Materials parameters are summa-\nrized in Table I. Because g\u001c\u0014, all interesting physics occurs\nin the weak coupling regime. A negative \u000be\u000bis demanded for\nrealizing thePTsymmetry in magnetic system. Considering\nthe driving laser with a fixed power P=1\u0016W, the e \u000bective\nTABLE I: Parameters for optical cavity and YIG.\n!cav=2\u0019 \u0014= 2\u0019 ! 0=2\u0019 gS=2\u0019 r\u000b\n100 THz 1 GHz 10 GHz 1 GHz 10 nm 10\u00004Gilbert-type magnetic gain is \u000be\u000b=\u0000\u000bat\u0011PT'6:16, and\nthe critical gain-loss point \u000be\u000b=0 occurs at\u0011C'7:11, indeed\nsatisfying the large-detuning condition j\u0011j\u001d1 in deriving (9).\nFigure 2(a) shows the monotonically decreasing dependence\nof the optically induced magnetic gain \u000bopton the detuning pa-\nrameter\u0011. The\u0011-dependence of the optical field is plotted in\nFig. 2(b), showing that it monotonically decreases with the in-\ncreasing of the detuning, too. Enhancing the laser power will\npush the two critical points \u0011Cand\u0011PTinto the deep detuning\nregion, as demonstrated in Fig. 2(c). For a magnetic sphere\nof larger volume (1 \u0016m)3\u0018(1 mm)3that contains a total spin\nnumber S=1010\u00181019with the reduced magneto-optical\ncoupling strength g=2\u0019=10\u00001\u001810\u000010Hz, the required laser\npower then should be 6 \u001815 orders of magnitude higher than\nthe nm-scale sphere case, as shown in Fig. 2(d).\nTime-resolved spin flipping. —To justify the approximation\nadopted in deriving the Gilbert-type magnetic gain, we di-\nrectly simulate the time evolution of the unit spin components\n(sj\u0011Sj=S) based on both Eq. (5) and Eq. (10). Numer-\nical results are, respectively, plotted in Figs. 3(a) and 3(b)\nfor the same detuning parameter \u0011=1:8 (corresponding to\nan e\u000bective magnetic gain \u000be\u000b=\u00000:0453) and!0=2\u0019=10\nGHz. Both figures show that the very presence of the negative\nGilbert damping can flip the spin in a precessional manner,\nwith similar switching curves. The fast Fourier transforma-\ntion (FFT) analysis of the spatiotemporal oscillation of sxalso\nconfirms this point (see the insets). Although the analytical\nform of sz(t) by solving (5) generally is unknown [58, 59], we\nfind an ansatz that can well describe the time-resolved spin\nswitching\nsj sj\nszη=1.8 (αeff =-0.0453 )\n(a) (c)\n(d) (b)szsysx\nττ τ\nηEq. (5)\nEq. (10)τ0=98.9 (a) fitting\n(b) fitting\nTheoryτ0=107.8’’\nτ0=100.4’τp=22.1τp=22.4’’τp=14.9’ tanh(-      )τ-τ0τp\nτ0’\nτ0’’\nTheory\nτp’\nτp’’\nTheory46810121401020 9.779.71\nFrequency (GHz)Frequency (GHz)FFT of s x FFT of s x46810121401020\nFIG. 3: Time evolution of unit spin components ( sx;sy;sz) at de-\ntuning\u0011=1:8 based on Eq. (5) (a) and Eq. (10) (b). Insets\nshow the FFT spectrum of sx. (c) Theoretical fittings of szusing\nthe hyperbolic-tangent ansatz (11) (dashed curves). The solid green\ncurve is the analytical formula without any fitting. (d) Numerical re-\nsults of the \u0011-dependence of the two characteristic times \u001c0and\u001cp,\ncomparing with formula (12) (solid curves).4\nsz(\u001c)'tanh \n\u0000\u001c\u0000\u001c0\n\u001cp!\n; (11)\nwhich is reminiscent of the Walker solution for modeling the\nprofile of 180\u000emagnetic domain wall [60] by replacing the\ntime coordinate \u001cwith the space coordinate x. Here\u001c0is\nthe switching time, \u001cprepresents the life-time of uniform\nmagnons, and \u001c=!0t. From perturbation theory, we derive\nthe analytical form of these two parameters\n\u001cp=\u00001+\u000b2\ne\u000b\n\u000be\u000b;and\u001c0=\u001cptanh\u00001vt\n1\u00004B2\nopt\nB2\ne\u000b:(12)\nFigure 3(c) shows the time evolution of sz. Symbols repre-\nsent the numerical results, dashed curves label the theoretical\nfittings of ansatz (11), and the solid curve is the analytical for-\nmula without fitting. The fitted switching time \u001c0\n0=100:4\n(\u001c00\n0=107:8) and magnon life-time \u001c0\np=14:9 (\u001c00\np=22:4)\nfrom from Eq. (5) [Eq. (10)] compare well with the analyt-\nical formula (12) which gives \u001c0=98:9 and\u001cp=22:1. We\nfurther show that the analytical ansatz agrees excellently with\nnumerical results in a broad range of detuning parameters, as\nplotted in Fig. 3(d).\nPhase transition in spin dimers. —We have shown that un-\nder proper conditions, one can realize the Gilbert-type mag-\nnetic gain which is essential for observing PT-symmetry in\npurely magnetic structures. Next, we consider the optically\npumped spin Sinteracting with a lossy one S0, as shown in\nFig. 4(a). The coupled spin dynamics is described by the\nLandau-Lifshitz-Gilbert equation\n˙s=\u0000\rs\u0002Be\u000b+!exs\u0002s0+\u000be\u000b˙s\u0002s; (13a)\n˙s0=\u0000\rs0\u0002B0\ne\u000b+!exs0\u0002s+\u000b˙s0\u0002s0; (13b)\nwhere s(0)\u0011S(0)=Sis the unit spin vector. Since the optically\ninduced magnetic field is the same order of magnitude with\nthe geomagnetic field (much smaller than B0), it can be safely\nignored. Spin s0is exchange coupled to the optically pumped\nspins, and su \u000bers an intrinsic Gilbert damping. If \u000be\u000b=\u0000\u000b,\nthe two-spin system satisfies the PT-symmetry: Eqs. (13) are\ninvariant in the combined operation of the parity P(s$s0\nandBe\u000b$B0\ne\u000b) and the time reversal T(t!\u0000 t,s!\u0000s,\ns0!\u0000s0,Be\u000b!\u0000Be\u000b, and B0\ne\u000b!\u0000B0\ne\u000b).\nAssuming a harmonic time-dependence for the small-angle\nspin precession sx;y(t)=sx;yei!twithjsx;yj \u001c 1, one can\nsolve the eigenspectrum of Eqs. (13). By tuning the spin-\nspin coupling strength !ex, we observe a transition from exact\nPT phase to the broken PT phase, separated by the EP at\n!c\nex=2\u0019=1 MHz for\u0011=\u0011PT=6:16, as shown in Figs. 4(b)\nand 4(c). Interestingly, the unequal gain and loss, i.e., \u000be\u000b<0\nand\u000be\u000b,\u0000\u000b, leads to an imbalanced parity-time ( IPT )-\nsymmetry. In this region ( \u0011>\u0011 IPT=5:66), the eigenfrequen-\ncies have di \u000berent real parts but share the identical imaginary\none, as plotted in Fig. 4(d). A passive parity-time ( PPT )-\nsymmetry is further identified when \u000be\u000b>0. In such case\nRe[(ω-ω0)/2π] (MHz)(b)(a)\n(d)\n(c)\nωex /2π (MHz)Im[(ω-ω0)/2π] (MHz)\nηη=ηPT\nηPTηPPT ηIPTωex \n2πc\n=1 MHz(e)\nωexs s’\nωex \n2π=1.5 MHzFIG. 4: (a) Spin dimmer consisting of an optically pumped spin sand\na purely lossy one s0. Evolution of eigenfrequencies vs. the exchange\ncoupling (b,c) at the detuning \u0011PT=6:16, and vs. the detuning pa-\nrameter (d,e) at the exchange coupling !ex=2\u0019=1:5 MHz.\n(\u0011 > \u0011 PPT=7:11), the imaginary part of both branches is\nsmaller than their intrinsic damping [see Fig. 4(e)].\nDiscussion. —In the above derivation, we focus on the case\nthat the intrinsic Gilbert damping is isotropic. Our approach\ncan also be generalized to treat the case when the intrinsic\ndamping is anisotrpic [61, 62]. The three propagating lasers\nthen should be accordingly adjusted to match the tensor form\nof the intrinsic magnetic damping, by modulating the driving\npower or the frequency of each beam, for instance. The red-\ndetuning region is appealing to cool magnons to the subtle\nquantum domain. Inspired by PT-symmetric optics [19], we\nenvision a giant enhancement of the magnonic gain and an\nultralow-threshold magnon lasing in a two-cavity system with\nbalanced optical gain and loss, which is an open question for\nfuture study. While the magnonic passive PTsymmetry has\nbeen observed by Liu et al. [32], the exact and imbalanced\nPTphases are still waiting for the experimental discovery.\nConclusion. —To summarize, we have proposed an opto-\nmagnonic method to generate the negative Gilbert damp-\ning in ferromagnets, by studying the parametric dynamics\nof a macrospin coupled with three orthogonally propagating\ncircularly-polarized lasers in an optical cavity. We analyti-\ncally derived the formula of the optical torque on the spin\nand identified the condition for the magnetic gain exactly in\nthe Gilbert form. 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Muniz\nInstituto de F\u0013 \u0010sica, Universidade Federal Fluminense, 24210-346 Niter\u0013 oi, RJ, Brazil.\nWe present a microscopic theory for magnetization relaxation in metallic ferromagnets of\nnanoscopic dimensions that is based on the dynamic spin response matrix in the presence of spin-\norbit coupling. Our approach allows the calculation of the spin excitation damping rate even for\nperfectly crystalline systems, where existing microscopic approaches fail. We demonstrate that the\nrelaxation properties are notcompletely determined by the transverse susceptibility alone, and that\nthe damping rate has a non-negligible frequency dependence in experimentally relevant situations.\nOur results indicate that the standard Landau-Lifshitz-Gilbert phenomenology is not always ap-\npropriate to describe spin dynamics of metallic nanostructure in the presence of strong spin-orbit\ncoupling.\nMagnetization relaxation in metals is at the heart of\nspin current generation and detection processes currently\nunder investigation, many of them candidates to play\nprotagonist roles in innovative spintronic devices. The\nLandau-Lifshitz-Gilbert (LLG) equation is widely used\nto describe the spin dynamic properties of magnetic ma-\nterials [1, 2]. It includes an important system-dependent\nparameter, called the Gilbert damping constant, usually\ndenoted by \u000bG, that regulates the relaxation of the mag-\nnetization towards stability, after it is driven out of equi-\nlibrium. Recently, a lot of e\u000bort has been put into the\ndetermination of this damping rate [2{8], which charac-\nterizes the pumping and absorption of pure spin currents\nin nanostructures that are of great interest in the \feld of\nspintronic. In most of them spin-orbit interaction is sig-\nni\fcant, and responsible for a desirable interplay between\ncharge spin and angular momentum excitations.\nThere is a general agreement between practitioners in\nthe \feld that a proper microscopic theory of magnetiza-\ntion relaxation in metals requires a good description of\nthe electronic structure of the system including spin-\norbit coupling [3{8]. The conventional approach is to\ncombine a realistic electronic structure with some kind of\nadiabatic approximation to derive expressions that can\nbe directly related to the Landau-Lifshitz-Gilbert phe-\nnomenology. This strategy has been employed by Kam-\nbersk\u0013 y [3] and many others since [4{8]. This conven-\ntional approach has important limitations. It neglects the\ncoupling between transverse spin, longitudinal spin and\ncharge excitations (which is an important consequence of\nthe spin-orbit coupling), and incorrectly pedicts the di-\nvergence of the damping parameter for a perfectly crys-\ntalline system. Actually, for ferromagnets that display\nrotation symmetry in spin space, the Goldstone theorem\nensures that any experiment which measures the total\ntransverse magnetic moment of the sample will produce\na resonant response with zero linewidth [9]. In the pres-\nence of spin-orbit interaction, however, this symmetry is\nexplicitly broken, and the resonant spectrum acquires a\n\fnite linewidth [10].\nWe put forward a more fundamental microscopic ap-\nproach to the calculation of the spin dynamics damp-ing rate that takes fully into account the e\u000bects of SOC\non the spectrum of spin excitations of itinerant sys-\ntems. Namely, we consider the coupling of transverse\nspin excitations to longitudinal spin and charge excita-\ntions, induced by the spin-orbit interaction. We calculate\nthe FMR spectrum at \fnite frequencies and arbitrary\nanisotropy values, without employing any adiabatic ap-\nproximation. We will show that those ingredients are es-\nsential to correctly describe the magnetization relaxation\nin very clean metallic ferromagnets of nanoscopic dimen-\nsions, and that the Landau-Lifshitz-Gilbert phenomenol-\nogy fails to capture essential features of the magnetiza-\ntion dynamics in those systems.\nThis letter is organized as follows: we will present\nbrie\ry our formalism, discuss its main features and\npresent numerical results for two model systems that il-\nlustrate common but qualitatively di\u000berent situations.\nGeneral Formalism - The spectrum of spin excitations\nof a ferromagnet can be obtained from the spectral den-\nsity associated with the transverse spin susceptibility\n\u001f+\u0000(l;l0; \n) =Z\ndtei\nthhS+\nl(t);S\u0000\nl0(0)ii; (1)\nwhere\nhhS+\nl(t);S\u0000\nl0(0)ii\u0011\u0000i\u0012(t)h[S+\nl(t);S\u0000\nl0(0)]i; (2)\nand\nS+\nl=X\n\u0016ay\nl\u0016\"al\u0016#: (3)\nThe operator ay\nl\u0016\u001bcreates one electron in the atomic ba-\nsis state\u0016localized at lattice site lwith spin\u001b. Although\nwe are usually interested in \u001f+\u0000(l;l0; \n) as de\fned above,\nits equation of motion involves the orbital-resolved sus-\nceptibility,\n\u001f+\u0000\n\u0016\u0017\u00160\u00170(l;l0;t)\u0011hhay\nl\u0016\"(t)al\u0017#(t);ay\nl0\u00160#(0)al0\u00170\"(0)ii:(4)\nIn the absence of spin-orbit coupling (SOC) and within\nthe random phase approximation (RPA), the equation ofarXiv:1502.02068v1  [cond-mat.mes-hall]  6 Feb 20152\nmotion for \u001f+\u0000\n\u0016\u0017\u00160\u00170(l;l0;t) is closed and \u001f+\u0000(l;l0; \n) can\nbe expressed in the well-known RPA form,\n\u001f+\u0000(\n) = [1 +U\u001f+\u0000\n0(\n)]\u00001\u001f+\u0000\n0(\n) (5)\nwhere\u001f+\u0000\n0(\n) is the mean-\feld (sometimes called non-\ninteracting, or Hartree-Fock) susceptibility. This expres-\nsion is schematic and must be understood as a matrix in\norbital and site indices, in real space, or a wave-vector\ndependent matrix in reciprocal space. The crucial point,\nhowever, is that, in the absence of spin-orbit coupling,\nwithin the RPA, the transverse spin susceptibility is un-\ncoupled from any other susceptibility. This ceases to\nbe true when SOC is included, as we demonstrated in\nref. 10:\u001f+\u0000becomes coupled to three other susceptibil-\nities, namely\n\u001f(2)\n\u0016\u0017\u00160\u00170(l;l0;t)\u0011hhay\nl\u0016\"(t)al\u0017\"(t);ay\nl0\u00160\"(0)al0\u00170\"(0)ii;(6)\n\u001f(3)\n\u0016\u0017\u00160\u00170(l;l0;t)\u0011hhay\nl\u0016#(t)al\u0017#(t);ay\nl0\u00160#(0)al0\u00170#(0)ii;(7)\n\u001f(4)\n\u0016\u0017\u00160\u00170(l;l0;t)\u0011hhay\nl\u0016#(t)al\u0017\"(t);ay\nl0\u00160\"(0)al0\u00170#(0)ii:(8)\nThe system of equations of motion obeyed by these\nfour susceptibilities can be cast into a form strongly re-\nsembling the RPA result by introducing a block-vector\n~ \u001f\u0011(\u001f(1);\u001f(2);\u001f(3);\u001f(4))T, with\u001f(1)\u0011\u001f+\u0000. With\nan equivalent de\fnition for the mean-\feld susceptibili-\nties\u001f(m)\n0we write\n~ \u001f(\n) =~ \u001f0(\n)\u0000\u0003~ \u001f(\n); (9)\nwhere the \\super-matrix\" \u0003 is proportional to the e\u000bec-\ntive Coulomb interaction strength and involves convolu-\ntions of single particle Green functions. Explicit forms\nfor its matrix elements are found in Ref. 10. The nu-\nmerical analysis of the susceptibilities \u001f(2),\u001f(3)and\u001f(4)\nshow that their absolute values are many orders of mag-\nnitude smaller than those of \u001f(1)=\u001f+\u0000. It is, thus,\ntempting to argue that the transverse susceptibility is ap-\nproximately decoupled from \u001f(2),\u001f(3)and\u001f(4)and that\nit can be calculated via the usual RPA expression with\nthe single particle Green functions obtained with spin-\norbit coupling taken into account. This is not a good\napproximation in general, since the matrix elements of \u0003\nthat couple \u001f(1)to the other susceptibilities are far from\nnegligible. Our numerical calculations indicate that they\nare essential to determine correctly the features of the\nFMR mode around the resonance frequency. Thus, the\nbehaviour of \u001f(1)in the presence of spin-orbit coupling\ncannot be inferred from \u001f(1)\n0in the zero-frequency limit\nalone, as it is usually assumed in the literature on the cal-\nculation of the Gilbert damping parameter [3{5, 8, 11].\nNumerical Results - We start the discussion by pre-\nsenting results for the Gilbert constant \u000bGfor unsup-\nported ultrathin Co \flms. Here we determine \u000bGfrom\nthe ratio between the FMR linewidth \u0001\n and the reso-\nnance frequency \n 0. First we turn o\u000b spin-orbit coupling\n0 10 20 30 40 50\nη (meV)00.010.020.030.040.05αGFIG. 1: Gilbert damping constant \u000bGas a function of the\nimaginary part \u0011added to the real energy, for an ultrathin\n\flm of two atomic layers of Co where SOC has been turned\no\u000b. It is clear that \u000bGvanishes as \u0011!0.\nto check the consistency of our approach. Even with\nSOC turned o\u000b we still \fnd a \fnite linewidth for the\nFMR mode. It comes, as we will shortly demonstrate,\nfrom the small imaginary part \u0011that is usually added\nto the energy in the numerical calculations of the sin-\ngle particle Green functions, in order to move their poles\nfrom the real axis. We calculate \u000bGfor various values of\n\u0011and extrapolate to \u0011!0+, as shown in Fig. 1. It is\nclear that lim \u0011!0+\u000bG= 0. Thus, our approach correctly\npredicts that the Gilbert damping constant vanishes in\nthe absence of SOC, as it should. Indeed, it is easy to\nshow [9] that the FMR mode is a stationary state of the\nmean-\feld hamiltonian and, as such, has in\fnite lifetime\nin the limit \u0011!0+. Now we discuss the dependence\nof\u000bGon\u0011for a \fxed, non-zero value of the spin-orbit\ncoupling strength \u0018. We used LCAO parameters appro-\npriate for bulk Co to describe the electronic structure\nof all Co \flms we investigated. The quantitative details\nof the ferromagnetic ground state and excitation spec-\ntra are sensitive to the LCAO parameters used, but their\nqualitative behaviour is very robust to small changes in\nthe electronic structure. Our strategy is to use the same\nset of LCAO parameters for all \flm thicknesses to avoid\nmodi\fcations in \u000bGcoming directly from changes in the\nLCAO parameters. This allows us to focus on geometric\ne\u000bects and on the \u0011-dependence.\nFigure 2 shows the dependence of the Gilbert damping\nconstant\u000bGon the imaginary part \u0011for Co \flms of var-\nious thicknesses. Clearly \u000bGapproaches \fnite values as\n\u0011!0. Cobalt has a small spin-orbit coupling constant.\nWe would like to investigate the e\u000bect of increasing the\nstrength of the SOC on the damping rate. Instead of\narti\fcially increasing \u0018in Co we consider a more realis-\ntic setting where a double layer of Co is attached to a3\n05 10 15 η\n (meV)00.010.020.030.04αG\nFIG. 2: Gilbert damping constant \u000bGas a function of the\nimaginary part \u0011added to the energy, for Co ultra thin \flms\nof various thicknesses: 1 (circles), 2 (squares), 4 (diamonds)\nand 6 (triangles) atomic layers. The strength of the SOC is\n\u0018= 85 meV. The solid lines are guides to the eye.\nnon-magnetic substrate with high SOC parameter, such\nas Pt. This system has a particularly interesting fea-\nture: the magnetization easy axis is perpendicular to the\nplane. However, we found that, for the LCAO param-\neters we employed, the magnetization in-plane is also\na stable con\fguration, with a small magnetocrystalline\nanisotropy. The damping rate, however, is much larger\nin the 2Co/2Pt system than in the unsupported Co \flms.\nThis is a nice example of how the anisotropy energy is\nstrongly in\ruenced by the system's symmetry, but the\ndamping rate is relatively insensitive to it, depending\nstrongly on the intensity of the spin-orbit coupling. It is\nalso an extremely convenient situation to test an assump-\ntion very frequently found in the literature on Gilbert\ndamping, although sometimes not explicitly stated: that\nthe FMR linewidth \u0001\n is linearly dependent on the res-\nonance frequency \n 0and that \u0001\n!0 as \n 0!0. This\nis not an unreasonable hypothesis, considering the weak\nstatic \felds commonly used in FMR experiments and the\nsmallness of the spin-orbit coupling constant, compared\nto other energy scales of a ferromagnet. Our calculations\nfor the Co \flms con\frm that this relationship is approx-\nimately held. In this case, the Gilbert constant \u000bGmay\nbe extracted from the FMR spectrum by simply \ftting\nit to a Lorentzian and is practically \feld-independent.\nHowever, our results for 2Co/2Pt indicate that the FMR\nlinewidth is \fnite as \n 0!0, leading to a signi\fcantly\nfrequency-dependent \u000bG, as shown in Fig. 3. In order to\nillustrate how the determination of a damping parame-\nter is a\u000bected by the \fnite value of \u0001\n as \n 0!0 we\nextracted the linewidths from the calculated spectra for\nthe 2Co/2Pt system by \ftting Lorentzians to our calcu-\nlated spectral densities. The results are shown in Fig. 3.\nOne of its most important consequences is that, if one\nwishes to de\fne a value of \u000bGfor the system above, it\n00.5 1 Ω\n (meV)05e+051e+061.5e+062e+06FMR spectral density01 2 3 4 B (T)\n00.20.40.60.81Ω0 (meV)(a)\n01 2 3 4 B (T)\n0.10.120.140.160.180.2αG0\n0.2 0.4 0.6 0.8 1 Ω\n0 (meV)00.050.1ΔΩ (meV)(b)FIG. 3: a) Spectral densities of the FMR mode for the\n2Co/2Pt system subjected to various static magnetic \felds\n(from -0.3 T to 4 T). The inset shows the resonance frequency\nas a function of the Zeeman \feld B. b) The Gilbert damping\nparameter\u000bGas a function of applied Zeeman \feld B. The\ninset shows the FMR line width as a function of resonance\nfrequency \n 0. The strengths of the SOC are \u0018Co= 85 meV\nand\u0018Pt= 600 meV.\nmust be de\fned as a function of the Zeeman \feld, as is\nillustrated in Fig. 3. In principle this poses a problem\nfor the procedure usually employed to determine FMR\nspectra experimentally, since there the free variable is\nthe Zeeman \feld, not the frequency of the exciting \feld.\nIn Fig. 4 we illustrate this issue by plotting the FMR\nspectral density as a function of the Zeeman \feld for two\n\fxed pumping frequencies, 24 GHz and 54 GHz. The\ncurves have nice Lorentzian shapes, but the values for\nthe Gilbert damping parameter \u000bGextracted from these\ncurves depend on the pumping frequency ( \u000bG= 0:034\nfor \n 0= 0:10 meV and \u000bG= 0:042 for \n 0= 0:22 meV).\nAlso, they do not correspond to any of the values shown\nin Fig. 3b, although the Zeeman \feld values that de-\ntermine the linewidth in Fig. 4 lie within the range of\nZeeman \feld values showed in Fig. 3b. Thus, if \u000bGis\nde\fned as \u0001\n =\n0, its value for a given sample depends4\n-0.4-0.2 0 0.2 B (T)\n05e+051e+061.5e+062e+06FMR spectral density\nFIG. 4: Spectral densitty of the FMR mode for the 2Co/2Pt\nsystem plotted as a function of the Zeeman \feld Bat a\n\fxed pumping frequencies: \u0017p= 24 GHz (squares) and\n\u0017p= 54 GHz (circles). The solid curves are Lorentzian \fts to\nthe calculated points.\non wether the FMR spectrum is obtained in a \fxed fre-\nquency or \fxed Zeeman \feld set ups. Our results also\nimply that the existing expressions for the damping con-\nstant\u000bGare not valid in general, specially for very clean\nsystems with large spin-orbit coupling materials. The\nconventional approaches express \u000bGas the ratio \u0001\n =\n0\nin the \n 0!0 limit. As we have just shown, this limit\ndoes not exist in general, since \u0001\n approaches a \fnite\nvalue as \n 0!0.\nIn experimental papers [12, 13] the FMR linewidth is\nassumed to have a zero-frequency o\u000bset, just as we de-\nscribed. This is usually attributed to extrinsic broad-\nening mechanisms, such as two-magnon scattering [14],\ndue to the combination between inhomogeneities in the\nmagnetic \flms and dipolar interactions. This is certainly\nthe case in systems with small SOC, such as Fe \flms de-\nposited on GaAs or Au [12]. However, we have shown\nthat there can be zero-frequency o\u000bset of intrinsic ori-\ngin if the SOC is large. The e\u000bect of this intrinsic o\u000bset\nshould be easily separated from that of the two-magnon\nscattering mechanism, since the latter is not active when\nthe magnetization is perpendicular to the plane of the\n\flm [14].\nWe would like to remark that Stoner enhancement in\nPt plays a very important role in the determination of the\ndamping rate. We had shown previously [15] that, in the\nabsence of spin-orbit coupling, Stoner enhancement had\na very mild e\u000bect on the damping rate in the Co/Pd(001)\nsystem. In the presence of SOC, however, the e\u000bect can\nbe very large indeed. Both magnetocrystalline anisotropy\nand damping rate are signi\fcantly di\u000berent in the en-\nhanced and non-enhanced cases, as shown in Fig. 5. The\nGilbert parameter is also very di\u000berent in the two cases:\n\u000benh\nG= 0:11, whereas \u000bnon\u0000enh\nG = 0:33. Thus, proper\n01 2 3 Ω\n (meV)02e+054e+056e+058e+05A(Ω)FIG. 5: a) Spectral densities of the FMR mode for the\n2Co/2Pt system with Stoner enhancement in Pt turned on\n(black line) and o\u000b (red line).\ntreatment of Stoner enhancement in substrates like Pd\nan Pt is essential for the correct determination of spin\nrelaxation features.\nWe presented a microscopic approach to the calcu-\nlation of the Gilbert damping parameter \u000bGfor ultra-\nthin metallic magnetic \flms, illustrated by results for\nCo \flms and Co/Pt bilayers. Our approach is based on\nthe evaluation of the dynamic transverse susceptibility in\nthe presence of spin-orbit coupling, taking into account\nrealistic electronic structures and the coupling between\ntransverse spin, longitudinal spin and charge excitations.\nIt predicts \fnite values of \u000bGin the limit of perfectly\ncrystalline \flms, a regime where methods based on the\ntorque correlation formula \fnd a diverging Gilbert damp-\ning parameter. We showed that the coupling between\ntransverse, longitudinal and charge excitations, due to\nspin-orbit coupling, is of fundamental importance for the\ncorrect determination of FMR spectra in metallic sys-\ntems. We have also shown that the damping rate ex-\ntracted from the FMR spectrum for \fxed pumping fre-\nquency di\u000bers considerably from that extracted from the\nFMR spectrum for \fxed Zeeman \feld. In this case the\nGilbert damping parameter \u000bGbecomes frequency de-\npendent, in contrast to what is assumed in the standard\nLandau-Lifshitz-Gilbert phenomenology. Moreover, we\nhave numerical indications that the Gilbert parameter\nis not well de\fned in the limit of vanishing resonance\nfrequency, a fact that is very relevant to calculational\nschemes based on the adiabatic approximation. Inci-\ndentally, Stoner enhancement in materials like Pt and\nPd also plays an important role in the determination of\nFMR frequencies and damping rates. 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B 88, 054423 (2013), URL http://\nlink.aps.org/doi/10.1103/PhysRevB.88.054423 ."    },    {        "title": "2401.12022v1.Damping_Enhanced_Magnon_Transmission.pdf",        "content": "Damping-Enhanced Magnon Transmission\nXiyin Ye,1Ke Xia,2Gerrit E. W. Bauer,3, 4and Tao Yu1,∗\n1School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China\n2School of Physics, Southeast University, Jiangsu 211189, China\n3WPI-AIMR and Institute for Materials Research and CSRN, Tohoku University, Sendai 980-8577, Japan\n4Kavli Institute for Theoretical Sciences, University of the Chinese Academy of Sciences, Beijing 100190, China\n(Dated: January 23, 2024)\nThe inevitable Gilbert damping in magnetization dynamics is usually regarded as detrimental\nto spin transport. Here we demonstrate in a ferromagnetic-insulator–normal-metal heterostructure\nthat the strong momentum dependence and chirality of the eddy-current-induced damping causes\nalso beneficial scattering properties. Here we show that a potential barrier that reflects magnon wave\npackets becomes transparent in the presence of a metallic cap layer, but only in one direction. We\nformulate the unidirectional transmission in terms of a generalized group velocity with an imaginary\ncomponent and the magnon skin effect. This trick to turn presumably harmful dissipation into useful\nfunctionalities should be useful for future quantum magnonic devices.\nIntroduction .—Magnonic devices save power by ex-\nploiting the collective excitations of the magnetic or-\nder, i.e., spin waves or their quanta, magnons, for non-\nreciprocal communication, reprogrammable logics, and\nnon-volatile memory functionalities [1–10]. The possibil-\nity to modulate magnon states and their transport in fer-\nromagnets by normal metals or superconductors brings\nfunctionalities to spintronics [11–14], quantum informa-\ntion [15–21], and topological materials [22, 23]. The pre-\ndiction of inductive magnon frequency shifts by supercon-\nducting gates on magnetic insulators [24–30] have been\nexperimentally confirmed [31]. Normal metals are not\nequally efficient in gating magnons [32–35], but the stray\nfields of magnetically driven “eddy currents” [36–43] sig-\nnificantly brake the magnetization dynamics [36].\nThe intrinsic Gilbert damping seems to be detrimental\nto transport since it suppresses the magnon propagation\nlength. However, in high-quality magnets such as yt-\ntrium iron garnet (YIG) films, this is not such an issue\nsince the magnon mobility is often limited by other scat-\ntering processes such as two-magnon scattering by disor-\nder, and measurements can be carried out in far smaller\nlength scales.\nNatural and artificial potential barriers are impor-\ntant instruments in electronics and magnonics by confin-\ning and controlling the information carriers. They may\nguide magnon transport [31, 44], act as magnonic logic\ngate [45], induce magnon entanglement [18, 46], and help\ndetecting exotic magnon properties [47–50]. In the lin-\near transport regime, the transmission of electrons and\nmagnons through an obstacle has always been assumed\nto be symmetric, i.e., the same for a wave or particle\ncoming from either side.\nIn this Letter, we address the counter-intuitive ef-\nfect that the strong momentum-dependent eddy-current-\ninduced damping by a normal metal overlayer as shown\nin Fig. 1 may help surmount obstacles such as mag-\nnetic inhomogeneities [51], artificial potential barriers\nformed by surface scratches [52], or dc-current carryingwires [46]. Here we focus on the band edges of magnetic\nfilms that are much thinner than the extinction length of\nthe Damon-Eshbach surface states in thick slabs and are\ntherefore not chiral. Instead, the effect therefore origi-\nnates from the Oersted fields generated by the eddy cur-\nrents in the overlayer that act in only half of the recip-\nrocal space [7] and causes magnon accumulations at the\nsample edges or magnon skin effect [8, 9]. The trans-\nmission through a barrier that is small and symmetric\nfor magnons with opposite wave numbers in an uncov-\nered sample becomes unidirectional with the assistance\nof dissipative eddy currents.\nFIG. 1. Ferromagnetic insulator-normal metal heterostruc-\nture. An in-plane external magnetic field H0orients the\nmagnetization at an angle θwith the ˆz-direction. The yellow\nsheet between the normal metal and ferromagnetic insulator\nindicates suppression of the exchange interaction and conven-\ntional spin pumping.\nModel and non-perturbation theory .—We consider the\nferromagnetic insulator (FI)-normal metal (NM) het-\nerostructure with thickness 2 dFanddMand an in-plane\nmagnetic field H0in Fig. 1. The saturated equilib-\nrium magnetization Msmakes an angle θwith the ˆz-\ndirection such that the torques exerted by the external\nand anisotropy fields cancel. For convenience, we set\nθ= 0 in the following discussion and defer results forarXiv:2401.12022v1  [cond-mat.mes-hall]  22 Jan 20242\nfinite θto the Supplemental Material (SM) [53]. We gen-\neralize a previous adiabatic theory [7, 36] to the full elec-\ntrodynamics of the system by self-consistently solving the\nMaxwell equations coupled with the linearized Landau-\nLifshitz (LL) equations and Ohm’s Law. This treatment\nbecomes exact in the limit of an instantaneous response\nof the metal electrons and high-quality ultrathin mag-\nnetic films.\nThe driving force is an externally generated spatiotem-\nporal magnetization dynamics M(r, t) =M(r, ω)e−iωtat\nfrequency ω. According to Maxwell’s theory, the electric\nfieldEobeys the wave equation ∇2E(r, ω)+k2\n0E(r, ω) =\n−iωµ0JM, where the wave number k0=ω√µ0ε0,µ0(ε0)\nis the vacuum permeability (permittivity), and JM=\n∇×Mis the “magnetization current” [54]. Disregarding\nthe intrinsic Gilbert damping, the LL equation\niωM=−µ0γM×Heff[M] (1)\ngoverns the magnetization dynamics in the FI, where γ\nis the gyromagnetic ratio. The effective magnetic field\nHeff[M] =−δF[M]/δM(r), where the free energy Fis\na functional of the magnetization. It includes the static\nfieldH0, the dipolar field Hd, and (in the FI) the ex-\nchange field Hex=αex∇2Mthat depends on the spin-\nwave stiffness αex. In the presence of the NM layer,\nHeff[M] also contains the Oersted magnetic fields gen-\nerated by the “eddy” currents J=σE, where the elec-\ntrical conductivity σis real. This defines a closed self-\nconsistency problem that we solve numerically.\nWe consider a thin FI film with constant Ms=\n(0,0, Ms). The transverse fluctuations M(r, ω) =\n(Mx(k, ω), My(k, ω),0)eik·rwith in-plane wave vectors\nk= (0, ky, kz) are small precessions with iMx(k, ω) =\nakMy(k, ω), where the complex ellipticity akbecomes\nunity for circular motion.\nThe electric-field modes outside the magnet are plane\nwaves with wave numbers km=p\nω2µ0ε0+iωµ0σ,\nwhere σ= 0 in the absence of an NM layer. The continu-\nity of electric and magnetic fields provides the interface\nboundary conditions. The field in the FI\nEη={x,y,z}(−dF⩽x⩽dF)\n=E(0)\nη(−dF⩽x⩽dF) +RkE(0)\nη(x=dF)e−iAk(x−dF)\nis now modified by the reflection coefficient\nRk=\u0000\nA2\nk−B2\nk\u0001\neiBkdM−\u0000\nA2\nk−B2\nk\u0001\ne−iBkdM\n(Ak−Bk)2eiBkdM−(Ak+Bk)2e−iBkdM,(2)\nwhere E(0)is the solution of Eq. (1) inside the FI without\nthe NM cap [53], Ak=p\nk2\n0−k2, and Bk=p\nk2m−k2.\nThe reflection is isotropic and strongly depends on the\nwave vector. Naturally, Rk= 0 when dM= 0. On the\nother hand, when |k|= 0, the electric field cannot escape\nthe FI, since the reflection is total with Rk=−1.A corollary of Maxwell’s equation—Faraday’s Law—\nreads in frequency space iωµ0[Hd(r, t) +M(r, t)] =∇ ×\nE(r, t). When the magnetization of sufficiently thin mag-\nnetic films is uniform, the Zeeman interaction is propor-\ntional to the spatial average Hdover the film thickness.\nReferring to SM for details [53], we find\nHd,x=\u0014\n−Rk\n4A2\nkdFak(e2iAkdF−1)2(−iAkak+ky)\n+i\n2AkdF(e2iAkdF−1)\u0015\nMx≡ζx(k)Mx,\nHd,y=\"\n−Rk\n4iAkdF(e2iAkdF−1)2 \n−ky\niAkak+k2\ny\nA2\nk+ 2!\n+k2\ny\nA2\nk−k2\ny\nA2\nk1\n2iAkdF(e2iAkdF−1)#\nMy≡ζy(k)My.\nBy substitution into the LL equation (1), the spin wave\neigenfrequencies and ellipticities become\nω(k) =µ0γq\n(˜H0−ζx(k)Ms)(˜H0−ζy(k)Ms),(3a)\nak=q\n(˜H0−ζy(k)Ms)/(˜H0−ζx(k)Ms), (3b)\nwhere ˜H0=H0+αexk2Ms. Imω(k)̸= 0 because of the\nJoule heating due to the eddy currents in the cap layer.\nChiral damping and frequency shifts .—The stray elec-\ntric fields of spin waves propagating perpendicular to the\nmagnetization are chiral, i.e., they depend on their prop-\nagation direction by a hand rule. When kz= 0,E=Ezˆz\nis along the equilibrium magnetization and Ez∝My\nis complex only for positive ky. We illustrate the re-\nsults of the self-consistent calculations for dF= 100 nm,\ndM= 500 nm, conductivity σ= 6.0×107(Ω·m)−1\nfor copper at room temperature [55], applied magnetic\nfield µ0H0= 0.02 T, µ0Ms= 0.178 T, the exchange\nstiffness αex= 3×10−16m2for YIG [56], and γ=\n1.77×1011(s·T)−1. The presence of the NM cap lay-\ners shifts the relative phases between the stray electric\nfields and that of the generating spin waves. We focus\nhere on the wave numbers ky=±1µm−1in Fig. 2(a)\n[Fig. 2(b)] at which the electric field is in-phase (out-\nof-phase) with the transverse magnetization Myˆy. The\nresponse to an in-phase (out-of-phase) electric field is dis-\nsipative (reactive). Both components decay in the FI and\nthe vacuum as ∝1/|k|. In the NM, the in-phase compo-\nnent is screened only in the metal region on the scale of\na skin depth λ=p\n2/(ωµ0σ)∼1.5µm at ω= 11 GHz.\nThe out-of-phase electric field, on the other hand, cre-\nates only a reactive response and is therefore symmetric\nabove and below the metallic film. Also in this case the\ndamping is modulated for constant Gilbert damping by\nthe associated spin wave frequency shift in Fig. 2(b), an\neffect that cannot be captured by the adiabatic approxi-\nmation [7, 36].3\nFIG. 2. The system responds strongly to a phase difference\nbetween the spin waves and their wave vector-dependent ac\nelectric stray fields E. ReEcauses damping [(a)] and Im E\na frequency shift [(b)]. Im Ezgoverns the spin wave vector\ndependence of the chiral damping [(c)]. (d) illustrates the\nstrong ky-dependence of the damping of the lowest standing\nspin wave for Cu thicknesses dM={50,100,200,500}nm. (e)\nshows the real and imaginary parts of the reflection coefficient\nRkthat causes the frequency shifts plotted in (f).\nThe chirality of the radiated electric field controls the\nbackaction of the NM layer that modifies the magnon\ndispersion in a chiral fashion. Figure 2(c) illustrates\nthe strong wave vector-dependent damping coefficient\nαeff(k) =|Imωk|/Reωk. Spin waves propagating in the\npositive ˆy-direction decay much faster than those along\nthe negative direction, while the damping for positive\nand negative kzis the same. According to Fig. 2(d), the\ncalculated damping for kz= 0 in Fig. 2(c) increases (de-\ncreases) with the thickness of the Cu (YIG) film. The\nenhancement of the damping saturates for NM thick-\nnesses dN>1/p\nk2+ 1/λ2, depending on the skin depth\n(λ∼1.5µm) and the wave number 1 /kof the electric\nfield. Moreover, the Kittel mode at k= 0 in Fig. 2(e)\nis not affected by the metal at all because the reflection\ncoefficient Rk=−1, which implies that the dynamics\nof the FI and metal fully decouple. Indeed, recent ex-\nperiments do not find a frequency shift of the FMR by a\nsuperconducting overlayer [57, 58]. The additional damp-\ning by eddy currents reported by Ref. [39] is caused bythe width of the exciting coplanar waveguide, a finite-size\neffect that we do not address here.\nThe real part of Rkin Fig. 2(e) causes an in-phase\nOersted magnetic field that chirally shifts the spin wave\nfrequencies by as much as ∼1 GHz, see Fig. 2(f). Refer-\nence [59] indeed reports a frequency shift of perpendicular\nstanding spin wave modes in Bi-YIG films in the presence\nof thin metallic overlayers.\nThe predicted effects differ strongly from those caused\nby spin pumping due to the interface exchange coupling\nαsp= (ℏγ/M sdF)Reg↑↓, where g↑↓is the interfacial spin\nmixing conductance [60]. αspdoes not depend on the\nthickness of the metal and vanishes like 1 /dF. The fre-\nquency shift scales like Im g↑↓/dFand is very small even\nfor very thin magnetic layers. In contrast, the eddy\ncurrent-induced damping is non-monotonic, scaling like\n∝dFwhen 2 kdF≪1, vanishing for much thicker mag-\nnetic layers, and reaching a maximum at dF∼2λ.\nUnidirectional transmission of wave packets through a\npotential barrier .—The transmission of a wave packet im-\npinging from the left or right at a conventional potential\nbarrier is the same [61]. In the presence of a metal cap,\nthis does not hold for magnons in thin magnetic films.\nBefore turning to the potential scattering in this\nmodel, we have to address the effect of the edges. When\nmagnons propagate in the negative direction without\ndamping but decay quickly when propagating in the op-\nposite one, those reflected at the left boundary of the\nsample accumulate, which is a non-Hermitian skin ef-\nfect [62–65]. We substantiate this conclusion by nu-\nmerical calculations for a two-dimensional square lat-\ntice model with ˆ mi= (1 /√\nN)P\nkˆmkeik·ri, where ˆ mk\nis the annihilation operator of magnons with frequency\nωkfrom Eq. (3a) and ilabels the sites and Nis the\nnumber of sites. The Hamiltonian in the real space\nˆH0=P\nijtjiˆm†\njˆmi, where tji= (1/N)P\nkℏωkeik·(rj−ri)is\na hopping amplitude between possibly distant sites iand\njand the summation is over the first Brillouin zone. With\na coarse-grained lattice constant of ay=az= 0.1µm the\nreciprocal lattice vector 2 π/ay,zis much larger than the\nmagnon modes of interest (refer to the SM [53] for de-\ntails). When the frequencies ωkare complex, the Hamil-\ntonian is non-Hermitian, i.e.,tji̸=t∗\nij.\nFigure 3(a) shows the winding path of the real and\nimaginary eigenfrequencies with wave number. In the\ninterval ky= [−25,25]µm−1and an applied magnetic\nfield parallel to the boundary with θ= 0, the complex\ncomponent is hysteretic, indicating localization of modes\nat opposite boundaries. Figure 3(b)-(c) show the average\nspatial distributions W(r) = (1 /Nm)PNm\nl=1|ϕl(r)|2ofNm\nlowest-frequency eigenstates ϕl(r) for ky∈[−1,1]µm−1\nandkz∈[−1,1]µm−1. When the static magnetic field\naligns with the sample boundary z-axis, i.e.θ= 0 in\nFig. 3(b), the magnons tend to accumulate at the left4\nedge. In the antiparallel configurations θ=π[Fig. 3(c)],\nthe magnons aggregate at the right. In the noncollinear\nconfiguration with θ=π/4 [Fig. 3(d)], the maxima shifts\nto the upper-left corner. While Wis an average, we\nalso illustrate the localization of individual low-frequency\nmodes in SM [53].\nFIG. 3. The magnon skin effect caused by chiral damping.\n(a) Complex spectral winding under periodic boundary con-\nditions when kyevolves from −25 to 25 µm−1forθ= 0. (b)-\n(d) corresponds to the edge or corner aggregations of magnon\neigenstates for other magnetic configurations θ∈ {0, π, π/ 4}.\nWe now illustrate the effect of square potential barri-\ners of width dand height u0,ˆV(y) =u0[Θ(y+d/2)−\nΘ(y−d/2)], where Θ( x) is the Heaviside step function,\non the magnon transmission along ˆy(⊥Ms). With in-\ncoming ⟨y|k0⟩=eik0y, the scattered states |ψs⟩obey the\nLippmann-Schwinger formula [66]\n|ψs⟩=|k0⟩+1\niℏ∂t−ˆH0+i0+ˆV|ψs⟩. (4)\nwhere ˆH0=P\nkℏωkˆm†\nkˆmkis the magnon Hamiltonian for\nan extended film. The transmitted waves read\n⟨y|ψs⟩=\u001aT+(k0)eik0y,{y, k0}>0\nT−(k0)eik0y,{y, k0}<0. (5)\nIn the weak scattering limit |u0d| ≪ | ℏvk0|,\nT±(k0) = 1±\u0012iℏvk0\nu0d−vk0\n2|vk0|\u0013−1\n≈1∓iu0d\nℏvk0,(6)\nwhere vk0=∂ωk/∂k|k=k0ˆyis a generalized group ve-\nlocity that dissipation renders complex. The imaginary\npart of the group velocity and transmission amplitudes\ndepend on the direction of the incoming wave:\nD±(k0) =|T±(k0)|2≈1±2Im\u0012u0d\nℏvk0\u0013\n. (7)For example, with u0/ℏ= 30.5 GHz, d= 0.1µm,k0=\n±0.8µm−1,vk0>0= (2.32 + 0 .52i) km/s and vk0<0=\n−(2.64 + 0 .16i) km/s lead to T+(k0>0)≈0.6 while\nT−(k0<0)≈0.9, so even in the weak scattering limit the\nNM cap layer significantly and asymmetrically reduces\nthe transmission probability.\nWe can assess the strong scattering regime with |u0d|≳\n|ℏvk0|by numerical calculations but find dramatic ef-\nfects on the time evolution of a real-space spin-wave\npacket as launched, e.g., by a current pulse in a mi-\ncrowave stripline. We adopt a Gaussian shape Ψ( r,0) =\ne−(r−r0)2/(2η2)eiq0·rcentered at r0with a width η≫ay,z\nthat envelopes a plane wave with wave vector q0and\nˆV(r) = u0f(r) with either f(|y−˜y0|< d) = 1 or\nf(|z−˜z0|< d) = 1, where ˜ y0and ˜z0are the center of the\nbarriers. According to Schr¨ odinger’s equation Ψ( r, t) =\neiˆHt/ℏΨ(r, t= 0) with ˆH=ˆH0+ˆV(r). Numerical results\nin Fig. 4(a) and (b) u0d≪ |ℏvk0|agree with perturbation\ntheory (7) in the weak scattering regime. However, when\n|ℏvk0|≲u0dand|Im(v−k0)| ≪ | Im(vk0)|≲|Re(v±k0)|\nthe transmission and unidirectionality becomes almost\nperfect. Figure 4(c) and (d) show a nearly unidirectional\ntransmission of the wave packet through the potential\nbarrier for the Damon-Eshbach configuration q0⊥Ms;\nit is transparent for spin waves impinging from the left,\nbut opaque for those from the right. In the calculations,\nq0=q(0)\nyˆywith q(0)\ny=±5µm−1andη= 3µm≫d.\nThe potential barrier is peaked with d=ay,z= 0.1µm\nand its height u0/ℏ= 15 GHz is relatively weak (the\nregular on-site energy ∼13 GHz). Also, dM= 50 nm\nanddF= 20 nm. The results are insensitive to the de-\ntailed parameter values (see SM [53]). The red and blue\ncurves are the incident and reflected wave packets, re-\nspectively. When q(0)\ny<0, the barrier does not affect the\nwave packet that propagates freely through the poten-\ntial barrier and accumulates on the left edge [Fig. 4(c)].\nWhen q(0)\ny>0, as shown in Fig. 4(d), the barrier reflects\nthe wave packet nearly completely, which we associate\nagain with the skin effect since these magnons cannot ac-\ncumulate on the right side. The unidirectional transmis-\nsion is therefore a non-local phase-coherent phenomenon\nthat involves the wave function of the entire sample.\nSince we find the skin effect to be crucial, its absence\nin waves propagating in the ˆz-direction must affect the\ntransport over the barrier. Indeed, our calculations in\nFig. 4(e) and (f) find strong reflection for both propaga-\ntion directions, even when reducing the barrier height by\nan order of magnitude to u0= 1.5 GHz (see SM [53]).\nDiscussion and conclusion .—In conclusion, we cal-\nculate the chiral damping, chiral frequency shift, and\nanomalous transport of magnonic modes in ferromag-\nnetic films with NM cap layers beyond the adiabatic ap-\nproximations. We predict anomalous unidirectional spin\ntransport over potential barriers. This effect is rooted\nin the non-Hermitian magnon skin effect and reflects the5\nFIG. 4. Calculated transmissions [(a) and (b)] and time evolu-\ntion of spin-wave packets in the presence of a potential barrier\nat the origin when q0⊥Ms[(c) and (d)] and q0∥Ms[(e)\nand (f)], where Msand the applied magnetic field are parallel\nto the sample edge with θ= 0. The red and blue curves rep-\nresent, respectively, the incident and scattered wave packets\nwith propagation directions indicated by arrows.\nglobal response of the entire system to a local perturba-\ntion. Our predictions are not limited to magnons, but\ncarry over for the propagation of all chiral quasiparticles,\nsuch as surface acoustic waves [67, 68], microwaves in\nloaded waveguides with magnetic insertions [69, 70], or\nchiral waveguides for light [71, 72].\nThis work is financially supported by the National\nKey Research and Development Program of China un-\nder Grant No. 2023YFA1406600, the National Natural\nScience Foundation of China under Grants No. 12374109\nand No. 12088101, the startup grant of Huazhong Uni-\nversity of Science and Technology, as well as JSPS KAK-\nENHI Grants No. 19H00645 and 22H04965.\n∗taoyuphy@hust.edu.cn\n[1] B. Lenk, H. Ulrichs, F. Garbs, and M. M¨ unzenberg,\nThe building blocks of magnonics, Phys. Rep. 507, 107\n(2011).\n[2] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B.Hillebrands, Magnon spintronics, Nat. Phys. 11, 453\n(2015).\n[3] D. Grundler, Nanomagnonics around the corner, Nat.\nNanotechnol. 11, 407 (2016).\n[4] V. E. Demidov, S. 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Vedmedenko,1and Roland Wiesendanger1\n1Department of Physics, University of Hamburg, D-20355 Hamburg, Germany\n(Dated: August 29, 2021)\nDamping mechanisms in magnetic systems determine the lifetime, diffusion and transport prop-\nerties of magnons, domain walls, magnetic vortices, and skyrmions. Based on the phenomenological\nLandau–Lifshitz–Gilbert equation, here the effective damping parameter in noncollinear magnetic\nsystems is determined describing the linewidth in resonance experiments or the decay parameter\nin time-resolved measurements. It is shown how the effective damping can be calculated from the\nelliptic polarization of magnons, arising due to the noncollinear spin arrangement. It is concluded\nthat the effective damping is larger than the Gilbert damping, and it may significantly differ be-\ntween excitation modes. Numerical results for the effective damping are presented for the localized\nmagnons in isolated skyrmions, with parameters based on the Pd/Fe/Ir(111) model-type system.\nSpinwaves(SW)ormagnonsaselementaryexcitations\nof magnetically ordered materials have attracted signifi-\ncant research attention lately. The field of magnonics[1]\nconcerns the creation, propagation and dissipation of\nSWs in nanostructured magnetic materials, where the\ndispersion relations can be adjusted by the system ge-\nometry. A possible alternative for engineering the prop-\nerties of magnons is offered by noncollinear (NC) spin\nstructures[2] instead of collinear ferro- (FM) or antifer-\nromagnets (AFM). SWs are envisaged to act as informa-\ntion carriers, where one can take advantage of their low\nwavelengths compared to electromagnetic waves possess-\ning similar frequencies[3]. Increasing the lifetime and the\nstability of magnons, primarily determined by the relax-\nation processes, is of crucial importance in such applica-\ntions.\nThe Landau–Lifshitz–Gilbert (LLG) equation[4, 5] is\ncommonly applied for the quasiclassical description of\nSWs, where relaxation is encapsulated in the dimen-\nsionless Gilbert damping (GD) parameter α. The life-\ntime of excitations can be identified with the resonance\nlinewidth in frequency-domain measurements such as fer-\nromagnetic resonance (FMR)[6], Brillouin light scatter-\ning (BLS)[7] or broadband microwave response[8], and\nwith the decay speed of the oscillations in time-resolved\n(TR) experiments including magneto-optical Kerr effect\nmicroscopy (TR-MOKE)[9] and scanning transmission x-\nray microscopy (TR-STXM)[10]. Since the linewidth is\nknowntobeproportionaltothefrequencyofthemagnon,\nmeasuring the ratio of these quantities is a widely ap-\nplied method for determining the GD in FMs[3, 6]. An\nadvantage of AFMs in magnonics applications[11, 12] is\ntheir significantly enhanced SW frequencies due to the\nexchange interactions, typically in the THz regime, com-\npared to FMs with GHz frequency excitations. However,\nit is known that the linewidth in AFM resonance is typ-\nically very wide because it scales with a larger effective\ndamping parameter αeffthan the GD α[13].\nThe tuning of the GD can be achieved in magnonic\ncrystals by combining materials with different values of\nα. It was demonstrated in Refs. [14–16] that this leadsto a strongly frequency- and band-dependent αeff, based\non the relative weights of the magnon wave functions in\nthe different materials.\nMagnetic vortices are two-dimensional NC spin config-\nurations in easy-plane FMs with an out-of-plane magne-\ntized core, constrained by nanostructuring them in dot-\norpillar-shapedmagneticsamples. Theexcitationmodes\nofvortices, particularlytheirtranslationalandgyrotropic\nmodes, havebeeninvestigatedusingcollective-coordinate\nmodels[17] based on the Thiele equation[18], linearized\nSW dynamics[19, 20], numerical simulations[21] and ex-\nperimental techniques[22–24]. It was demonstrated theo-\nretically in Ref. [21] that the rotational motion of a rigid\nvortex excited by spin-polarized current displays a larger\nαeffthan the GD; a similar result was obtained based on\ncalculating the energy dissipation[25]. However, due to\nthe unbounded size of vortices, the frequencies as well\nas the relaxation rates sensitively depend on the sample\npreparation, particularly because they are governed by\nthe magnetostatic dipolar interaction.\nIn magnetic skyrmions[26], the magnetic moment di-\nrections wrap the whole unit sphere. In contrast to vor-\ntices, isolated skyrmions need not be confined for stabi-\nlization, and are generally less susceptible to demagneti-\nzation effects[3, 27]. The SW excitations of the skyrmion\nlattice phase have been investigated theoretically[28–30]\nand subsequently measured in bulk systems[3, 8, 31]. It\nwas calculated recently[32] that the magnon resonances\nmeasured via electron scattering in the skyrmion lattice\nphase should broaden due to the NC structure. Calcula-\ntions predicted the presence of different localized modes\nconcentrated on the skyrmion for isolated skyrmions\non a collinear background magnetization[33–35] and for\nskyrmions in confined geometries[20, 36, 37]. From the\nexperimental side, the motion of magnetic bubbles in a\nnanodisk was investigated in Ref. [38], and it was pro-\nposed recently that the gyration frequencies measured in\nIr/Fe/Co/Pt multilayer films is characteristic of a dilute\narray of isolated skyrmions rather than a well-ordered\nskyrmion lattice[6]. However, the lifetime of magnons in\nskyrmionic systems based on the LLG equation is appar-arXiv:1805.01815v2  [cond-mat.mtrl-sci]  15 Oct 20182\nently less explored.\nIt is known that NC spin structures may influence the\nGD via emergent electromagnetic fields[29, 39, 40] or via\nthe modified electronic structure[41, 42]. Besides deter-\nmining the SW relaxation process, the GD also plays\na crucial role in the motion of domain walls[43–45] and\nskyrmions[46–48] driven by electric or thermal gradients,\nboth in the Thiele equation where the skyrmions are\nassumed to be rigid and when internal deformations of\nthe structure are considered. Finally, damping and de-\nformations are also closely connected to the switching\nmechanisms of superparamagnetic particles[49, 50] and\nvortices[51], as well as the lifetime of skyrmions[52–54].\nTheαeffin FMs depends on the sample geometry due\nto the shape anisotropy[13, 55, 56]. It was demonstrated\nin Ref. [56] that αeffis determined by a factor describing\nthe ellipticity of the magnon polarization caused by the\nshape anisotropy. Elliptic precession and GD were also\ninvestigated by considering the excitations of magnetic\nadatomsonanonmagneticsubstrate[57]. Thecalculation\nof the eigenmodes in NC systems, e.g. in Refs. [6, 20, 35],\nalso enables the evaluation of the ellipticity of magnons,\nbut this property apparently has not been connected to\nthe damping so far.\nAlthough different theoretical methods for calculating\nαeffhave been applied to various systems, a general de-\nscriptionapplicabletoallNCstructuresseemstobelack-\ning. Here it is demonstrated within a phenomenological\ndescription of the linearized LLG equation how magnons\nin NC spin structures relax with a higher effective damp-\ning parameter αeffthan the GD. A connection between\nαeffand the ellipticity of magnon polarization forced by\nthe NC spin arrangement is established. The method\nis illustrated by calculating the excitation frequencies\nof isolated skyrmions, considering experimentally deter-\nmined material parameters for the Pd/Fe/Ir(111) model\nsystem[58]. It is demonstrated that the different local-\nized modes display different effective damping parame-\nters, with the breathing mode possessing the highest one.\nThe LLG equation reads\n∂tS=−γ/primeS×Beff−αγ/primeS×/parenleftBig\nS×Beff/parenrightBig\n,(1)\nwithS=S(r)the unit-length vector field describing\nthe spin directions in the system, αthe GD and γ/prime=\n1\n1+α2ge\n2mthe modified gyromagnetic ratio (with gbeing\ntheg-factor of the electrons, ethe elementary charge and\nmthe electron mass). Equation (1) describes the time\nevolution of the spins governed by the effective magnetic\nfieldBeff=−1\nMδH\nδS, withHthe Hamiltonian or free\nenergy of the system in the continuum description and\nMthe saturation magnetization.\nThe spins will follow a damped precession relaxing\nto a local minimum S0ofH, given by the condition\nS0×Beff=0. Note that generally the Hamiltonian rep-\nresents a rugged landscape with several local energy min-\nima, corresponding to e.g. FM, spin spiral and skyrmionlattice phases, or single objects such as vortices or iso-\nlated skyrmions. The excitations can be determined by\nswitching to a local coordinate system[20, 34, 47] with\nthe spins along the zdirection in the local minimum,\n˜S0= (0,0,1), and expanding the Hamiltonian in the\nvariablesβ±=˜Sx±i˜Sy, introduced analogously to spin\nraising and lowering or bosonic creation and annihila-\ntion operators in the quantum mechanical description of\nmagnons[59–61]. The lowest-order approximation is the\nlinearized form of the LLG Eq. (1),\n∂tβ+=γ/prime\nM(i−α)/bracketleftbig\n(D0+Dnr)β++Daβ−/bracketrightbig\n,(2)\n∂tβ−=γ/prime\nM(−i−α)/bracketleftbig\nD†\naβ++ (D0−Dnr)β−/bracketrightbig\n.(3)\nFor details of the derivation see the Supplemental\nMaterial[62]. The term Dnrin Eqs. (2)-(3) is respon-\nsible for the nonreciprocity of the SW spectrum[2]. It\naccounts for the energy difference between magnons\npropagating in opposite directions in in-plane oriented\nultrathin FM films[63, 64] with Dzyaloshinsky–Moriya\ninteraction[65, 66] and the splitting between clockwise\nand counterclockwise modes of a single skyrmion[20].\nHere we will focus on the effects of the anomalous\nterm[34]Da, which couples Eqs. (2)-(3) together. Equa-\ntions (2)-(3) may be rewritten as eigenvalue equations by\nassuming the time dependence\nβ±(r,t) =e−iωktβ±\nk(r). (4)\nForα= 0, the spins will precess around their equilib-\nriumdirection ˜S0. Iftheequationsareuncoupled, the ˜Sx\nand ˜Syvariables describe circular polarization, similarly\nto the Larmor precession of a single spin in an exter-\nnal magnetic field. However, the spins are forced on an\nelliptic path due to the presence of the anomalous terms.\nThe effective damping parameter of mode kis defined\nas\nαk,eff=/vextendsingle/vextendsingle/vextendsingle/vextendsingleImωk\nReωk/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (5)\nwhich is the inverse of the figure of merit introduced in\nRef. [15]. Equation (5) expresses the fact that Im ωk,\nthe linewidth in resonance experiments or decay coeffi-\ncient in time-resolved measurements, is proportional to\nthe excitation frequency Re ωk.\nInterestingly, there is a simple analytic expression con-\nnectingαk,effto the elliptic polarization of the modes at\nα= 0. Forα/lessmuch1, the effective damping may be ex-\npressed as\nαk,eff\nα≈/integraltext/vextendsingle/vextendsingle/vextendsingleβ−(0)\nk(r)/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk(r)/vextendsingle/vextendsingle/vextendsingle2\ndr/integraltext/vextendsingle/vextendsingle/vextendsingleβ−(0)\nk(r)/vextendsingle/vextendsingle/vextendsingle2\n−/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk(r)/vextendsingle/vextendsingle/vextendsingle2\ndr=/integraltext\na2\nk(r) +b2\nk(r)dr/integraltext\n2ak(r)bk(r)dr,\n(6)3\n0.0 0.2 0.4 0.6 0.8 1.00246810\nFIG. 1. Effective damping parameter αk,effas a function of\ninverseaspectratio bk/akofthepolarizationellipse, assuming\nconstantakandbkfunctions in Eq. (6). Insets illustrate the\nprecession for different values of bk/ak.\nwhere the (0)superscript denotes that the eigenvectors\nβ±\nk(r)defined in Eq. (4) were calculated for α= 0, while\nak(r)andbk(r)denote the semimajor and semiminor\naxes of the ellipse the spin variables ˜Sx(r)and ˜Sy(r)\nare precessing on in mode k. Details of the derivation\nare given in the Supplemental Material[62]. Note that\nan analogous expression for the uniform precession mode\nin FMs was derived in Ref. [56]. The main conclusion\nfrom Eq. (6) is that αk,effwill depend on the considered\nSW mode and it is always at least as high as the GD\nα. Although Eq. (6) was obtained in the limit of low\nα, numerical calculations indicate that the αk,eff/αratio\ntends to increase for increasing values of α; see the Sup-\nplementalMaterial[62]foranexample. Theenhancement\nof the damping from Eq. (6) is shown in Fig. 1, with the\nspace-dependent ak(r)andbk(r)replaced by constants\nfor simplicity. It can be seen that for more distorted po-\nlarization ellipses the spins get closer to the equilibrium\ndirectionafterthesamenumberofprecessions, indicating\na faster relaxation.\nSince the appearance of the anomalous terms Dain\nEqs. (2)-(3) forces the spins to precess on an elliptic\npath, it expresses that the system is not axially sym-\nmetric around the local spin directions in the equilib-\nrium state denoted by S0. Such a symmetry breaking\nnaturally occurs in any NC spin structure, implying a\nmode-dependent enhancement of the effective damping\nparameter in NC systems even within the phenomeno-\nlogical description of the LLG equation. Note that the\nNC structure also influences the electronic properties of\nthe system, which can lead to a modification of the GD\nitself, see e.g. Ref. [42].\nIn order to illustrate the enhanced and mode-\ndependent αk,eff, we calculate the magnons in isolated\nchiralskyrmionsinatwo-dimensionalultrathinfilm. Thedensity of the Hamiltonian Hreads[67]\nh=/summationdisplay\nα=x,y,z/bracketleftBig\nA(∇Sα)2/bracketrightBig\n+K(Sz)2−MBSz\n+D(Sz∂xSx−Sx∂xSz+Sz∂ySy−Sy∂ySz),(7)\nwithAthe exchange stiffness, Dthe Dzyaloshinsky–\nMoriya interaction, Kthe anisotropy coefficient, and B\nthe external field.\nIn the following we will assume D>0andB≥0\nwithout the loss of generality, see the Supplemental\nMaterial[62] for discussion. Using cylindrical coordi-\nnates (r,ϕ)in real space and spherical coordinates S=\n(sin Θ cos Φ ,sin Θ sin Φ,cos Θ)in spin space, the equi-\nlibrium profile of the isolated skyrmion will correspond\nto the cylindrically symmetric configuration Θ0(r,ϕ) =\nΘ0(r)andΦ0(r,ϕ) =ϕ, the former satisfying\nA/parenleftbigg\n∂2\nrΘ0+1\nr∂rΘ0−1\nr2sin Θ 0cos Θ 0/parenrightbigg\n+D1\nrsin2Θ0\n+Ksin Θ 0cos Θ 0−1\n2MBsin Θ 0= 0 (8)\nwith the boundary conditions Θ0(0) =π,Θ0(∞) = 0.\nThe operators in Eqs. (2)-(3) take the form (cf.\nRefs. [34, 35, 47] and the Supplemental Material[62])\nD0=−2A/braceleftBigg\n∇2+1\n2/bracketleftbigg\n(∂rΘ0)2−1\nr2/parenleftbig\n3 cos2Θ0−1/parenrightbig\n(∂ϕΦ0)2/bracketrightbigg/bracerightBigg\n−D/parenleftbigg\n∂rΘ0+1\nr3 sin Θ 0cos Θ 0∂ϕΦ0/parenrightbigg\n−K/parenleftbig\n3 cos2Θ0−1/parenrightbig\n+MBcos Θ 0, (9)\nDnr=/parenleftbigg\n4A1\nr2cos Θ 0∂ϕΦ0−2D1\nrsin Θ 0/parenrightbigg\n(−i∂ϕ), (10)\nDa=A/bracketleftbigg\n(∂rΘ0)2−1\nr2sin2Θ0(∂ϕΦ0)2/bracketrightbigg\n+D/parenleftbigg\n∂rΘ0−1\nrsin Θ 0cos Θ 0∂ϕΦ0/parenrightbigg\n+Ksin2Θ0.(11)\nEquation (11) demonstrates that the anomalous terms\nDaresponsible for the enhancement of the effective\ndamping can be attributed primarily to the NC arrange-\nment (∂rΘ0and∂ϕΦ0≡1) and secondarily to the\nspins becoming canted with respect to the global out-\nof-plane symmetry axis ( Θ0∈ {0,π}) of the system.\nTheDnrtermintroducesanonreciprocitybetweenmodes\nwith positive and negative values of the azimuthal quan-\ntum number (−i∂ϕ)→m, preferring clockwise rotat-\ning modes ( m < 0) over counterclockwise rotating ones\n(m > 0) following the sign convention of Refs. [20, 34].\nBecauseD0andDnrdepend onmbutDadoes not, it is\nexpected that the distortion of the SW polarization el-\nlipse and consequently the effective damping will be more\nenhanced for smaller values of |m|.\nThe different modes as a function of external field\nare shown in Fig. 2(a), for the material parameters de-\nscribing the Pd/Fe/Ir(111) system. The FMR mode at4\n0.7 0.8 0.9 1.0 1.1 1.20255075100125150175\n(a)\n0.7 0.8 0.9 1.0 1.1 1.21.01.52.02.53.0\n(b)\nFIG. 2. Localized magnons in the isolated skyrmion, with the\ninteraction parameters corresponding to the Pd/Fe/Ir(111)\nsystem[58]:A = 2.0pJ/m,D =−3.9mJ/m2,K =\n−2.5MJ/m3,M= 1.1MA/m. (a) Magnon frequencies f=\nω/2πforα= 0. Illustrations display the shapes of the excita-\ntion modes visualized on the triangular lattice of Fe magnetic\nmoments, with red and blue colors corresponding to positive\nand negative out-of-plane spin components, respectively. (b)\nEffective damping coefficients αm,eff, calculated from Eq. (6).\nωFMR =γ\nM(MB−2K), describing a collective in-phase\nprecession of the magnetization of the whole sample, sep-\narates the continuum and discrete parts of the spectrum,\nwith the localized excitations of the isolated skyrmion\nlocated below the FMR frequency[34, 35]. We found a\nsingle localized mode for each m∈{0,1,−2,−3,−4,−5}\nvalue, so in the following we will denote the excita-\ntion modes with the azimuthal quantum number. The\nm=−1mode corresponds to the translation of the\nskyrmion on the field-polarized background, which is a\nzero-frequency Goldstone mode of the system and not\nshown in the figure. The m=−2mode tends to zero\naroundB= 0.65T, indicating that isolated skyrmions\nbecome susceptible to elliptic deformations and subse-\nquently cannot be stabilized at lower field values[68].\nThe values of αm,effcalculated from Eq. (6) for the\ndifferent modes are summarized in Fig. 2(b). It is impor-\ntant to note that for a skyrmion stabilized at a selected\n0 20 40 60 80 100-0.04-0.020.000.020.040.06\n-0.03 0.00 0.03-0.030.000.03FIG. 3. Precession of a single spin in the skyrmion in the\nPd/Fe/Ir(111) system in the m= 0andm=−3modes at\nB= 0.75T, from numerical simulations performed at α=\n0.1. Inset shows the elliptic precession paths. From fitting\nthe oscillations with Eq. (4), we obtained |Reωm=0|/2π=\n39.22GHz,|Imωm=0|= 0.0608ps−1,αm=0,eff= 0.25and\n|Reωm=−3|/2π= 40.31GHz,|Imωm=−3|= 0.0276ps−1,\nαm=−3,eff= 0.11.\nfield value, the modes display widely different αm,effval-\nues, with the breathing mode m= 0being typically\ndamped twice as strongly as the FMR mode. The ef-\nfective damping tends to increase for lower field values,\nand decrease for increasing values of |m|, the latter prop-\nerty expected from the m-dependence of Eqs. (9)-(11)\nas discussed above. It is worth noting that the αm,eff\nparameters are not directly related to the skyrmion size.\nWealsoperformedthecalculationsfortheparametersde-\nscribing Ir|Co|Pt multilayers[69], and for the significantly\nlargerskyrmionsinthatsystemweobtainedconsiderably\nsmaller excitation frequencies, but quantitatively similar\neffective damping parameters; details are given in the\nSupplemental Material[62].\nThe different effective damping parameters could pos-\nsibly be determined experimentally by comparing the\nlinewidths of the different excitation modes at a selected\nfield value, or investigating the magnon decay over time.\nAn example for the latter case is shown in Fig. 3, dis-\nplaying the precession of a single spin in the skyrmion,\nobtained from the numerical solution of the LLG Eq. (1)\nwithα= 0.1. AtB= 0.75T, the frequencies of the\nm= 0breathing and m=−3triangular modes are close\nto each other (cf. Fig. 2), but the former decays much\nfaster. Because in the breathing mode the spin is follow-\ning a significantly more distorted elliptic path (inset of\nFig. 3) than in the triangular mode, the different effective\ndamping is also indicated by Eq. (6).\nIn summary, it was demonstrated within the phe-\nnomenological description of the LLG equation that the\neffective damping parameter αeffdepends on the consid-\nered magnon mode. The αeffassumes larger values if5\nthe polarization ellipse is strongly distorted as expressed\nby Eq. (6). Since NC magnetic structures provide an\nanisotropic environment for the spins, leading to a dis-\ntortion of the precession path, they provide a natural\nchoice for realizing different αeffvalues within a single\nsystem. The results of the theory were demonstrated for\nisolated skyrmions with material parameters describing\nthe Pd/Fe/Ir(111) system. 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Vedmedenko,1and Roland Wiesendanger1\n1Department of Physics, University of Hamburg, D-20355 Hamburg, Germany\n(Dated: August 29, 2021)\nIn the Supplemental Material the derivation of the linearized equations of motion and the effective\ndamping parameter are discussed. Details of the numerical determination of the magnon modes in\nthe continuum model and in atomistic spin dynamics simulations are also given.\nS.I. LINEARIZED\nLANDAU–LIFSHITZ–GILBERT EQUATION\nHere we will derive the linearized form of the Landau–\nLifshitz–Gilbert equation given in Eqs. (2)-(3) of the\nmaintextanddiscussthepropertiesofthesolutions. The\ncalculation is similar to the undamped case, discussed in\ndetail in e.g. Refs. [1–3]. Given a spin configuration sat-\nisfying the equilibrium condition\nS0×Beff=0, (S.1)\nthe local coordinate system with ˜S0= (0,0,1)may be\nintroduced, andtheHamiltonianbeexpandedinthevari-\nables ˜Sxand˜Sy. Thelineartermmustdisappearbecause\nthe expansion is carried out around an equilibrium state.\nThe lowest-order nontrivial term is quadratic in the vari-\nables and will be designated as the spin wave Hamilto-\nnian,\nHSW=/integraldisplay\nhSWdr, (S.2)\nhSW=1\n2/bracketleftbig˜Sx˜Sy/bracketrightbig/bracketleftbiggA1A2\nA†\n2A3/bracketrightbigg/bracketleftbigg˜Sx\n˜Sy/bracketrightbigg\n=1\n2/parenleftBig\n˜S⊥/parenrightBigT\nHSW˜S⊥. (S.3)\nThe operator HSWis self-adjoint for arbitrary equi-\nlibrium states. Here we will only consider cases where\nthe equilibrium state is a local energy minimum, mean-\ning thatHSW≥0; the magnon spectrum will only be\nwell-defined in this case. Since hSWis obtained as an\nexpansion of a real-valued energy density around the\nequilibrium state, and the spin variables are also real-\nvalued, fromtheconjugateofEq.(S.3)onegets A1=A∗\n1,\nA2=A∗\n2, andA3=A∗\n3.\nThe form of the Landau–Lifshitz–Gilbert Eq. (1) in\nthe main text may be rewritten in the local coordinates\nby simply replacing Sby˜S0everywhere, including the\ndefinitionoftheeffectivefield Beff. TheharmonicHamil-\ntonianHSWin Eq. (S.2) leads to the linearized equation\nof motion\n∂t˜S⊥=γ/prime\nM(−iσy−α)HSW˜S⊥,(S.4)\n∗rozsa.levente@physnet.uni-hamburg.dewithσy=/bracketleftbigg\n0−i\ni0/bracketrightbigg\nthe Pauli matrix.\nBy replacing ˜S⊥(r,t)→˜S⊥\nk(r)e−iωktas usual, for\nα= 0the eigenvalue equation\nωk˜S⊥\nk=γ\nMσyHSW˜S⊥\nk (S.5)\nis obtained. If HSWhas a strictly positive spectrum,\nthenH−1\n2\nSWexists, and σyHSWhas the same eigenvalues\nasH1\n2\nSWσyH1\n2\nSW. Since the latter is a self-adjoint ma-\ntrix with respect to the standard scalar product on the\nHilbert space, it has a real spectrum, consequently all ωk\neigenvalues are real. Note that the zero modes of HSW,\nwhich commonly occur in the form of Goldstone modes\ndue to the ground state breaking a continuous symme-\ntry of the Hamiltonian, have to be treated separately.\nFinally, we mention that if the spin wave expansion is\nperformed around an equilibrium state which is not a\nlocal energy minimum, the ωkeigenvalues may become\nimaginary, meaning that the linearized Landau–Lifshitz–\nGilbert equation will describe a divergence from the un-\nstable equilibrium state instead of a precession around\nit.\nEquations (2)-(3) in the main text may be obtained\nby introducing the variables β±=˜Sx±i˜Syas described\nthere. The connection between HSWand the operators\nD0,Dnr, andDais given by\nD0=1\n2(A1+A3), (S.6)\nDnr=1\n2i/parenleftBig\nA†\n2−A2/parenrightBig\n, (S.7)\nDa=1\n2/bracketleftBig\nA1−A3+i/parenleftBig\nA†\n2+A2/parenrightBig/bracketrightBig\n.(S.8)\nAn important symmetry property of Eqs. (2)-(3) in\nthe main text is that if (β+,β−) =/parenleftbig\nβ+\nke−iωkt,β−\nke−iωkt/parenrightbig\nis an eigenmode of the equations, then (β+,β−) =/parenleftBig/parenleftbig\nβ−\nk/parenrightbig∗eiω∗\nkt,/parenleftbig\nβ+\nk/parenrightbig∗eiω∗\nkt/parenrightBig\nis another solution. Following\nRefs. [1, 3], this can be attributed to the particle-hole\nsymmetry of the Hamiltonian, which also holds in the\npresence of the damping term. From these two solutions\nmentioned above, the real-valued time evolution of the\nvariables ˜Sx,˜Symay be expressed as\n˜Sx\nk=eImωktcos (ϕ+,k−Reωkt)/vextendsingle/vextendsingleβ+\nk+β−\nk/vextendsingle/vextendsingle,(S.9)\n˜Sy\nk=eImωktsin (ϕ−,k−Reωkt)/vextendsingle/vextendsingleβ+\nk−β−\nk/vextendsingle/vextendsingle,(S.10)arXiv:1805.01815v2  [cond-mat.mtrl-sci]  15 Oct 20182\nwithϕ±,k= arg/parenleftbig\nβ+\nk±β−\nk/parenrightbig\n. As mentioned above, the\nImωkterms are zero in the absence of damping close to\na local energy minimum, and Im ωk<0is implied by\nthe fact that the Landau–Lifshitz–Gilbert equation de-\nscribes energy dissipation, which in the linearized case\ncorresponds to relaxation towards the local energy min-\nimum. In the absence of damping, the spins will precess\non an ellipse defined by the equation\n/parenleftBig\n˜Sx\nk/parenrightBig2\n/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk+β−(0)\nk/vextendsingle/vextendsingle/vextendsingle2\ncos2(ϕ+,k−ϕ−,k)\n+2˜Sx\nk˜Sy\nksin (ϕ+,k−ϕ−,k)/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk−β−(0)\nk/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk+β−(0)\nk/vextendsingle/vextendsingle/vextendsinglecos2(ϕ+,k−ϕ−,k)\n+/parenleftBig\n˜Sy\nk/parenrightBig2\n/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk−β−(0)\nk/vextendsingle/vextendsingle/vextendsingle2\ncos2(ϕ+,k−ϕ−,k)= 1,(S.11)\nwhere the superscript (0)indicatesα= 0. The semima-\njor and semiminor axes of the ellipse akandbkmay be\nexpressed from Eq. (S.11) as\nakbk=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleβ−(0)\nk/vextendsingle/vextendsingle/vextendsingle2\n−/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk/vextendsingle/vextendsingle/vextendsingle2/vextendsingle/vextendsingle/vextendsingle/vextendsingle, (S.12)\na2\nk+b2\nk= 2/parenleftbigg/vextendsingle/vextendsingle/vextendsingleβ−(0)\nk/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk/vextendsingle/vextendsingle/vextendsingle2/parenrightbigg\n.(S.13)\nNote thatβ+\nkandβ−\nk, consequently the parameters of\nthe precessional ellipse akandbk, are functions of the\nspatial position r.\nS.II. CALCULATION OF THE EFFECTIVE\nDAMPING PARAMETER FROM\nPERTURBATION THEORY\nHere we derive the expression for the effective damping\nparameter αeffgiven in Eq. (6) of the main text. By\nintroducingβk=/parenleftbig\nβ+\nk,−β−\nk/parenrightbig\n,\nD=/bracketleftbiggD0+Dnr−Da\n−D†\naD0−Dnr/bracketrightbigg\n,(S.14)\nand using the Pauli matrix σz=/bracketleftbigg\n1 0\n0−1/bracketrightbigg\n, Eqs. (2)-(3)\nin the main text may be rewritten as\n−ωkσzβk=γ/prime\nM(D+iασzD)βk(S.15)\nin the frequency domain. Following standard perturba-\ntion theory, we expand the eigenvalues ωkand the eigen-\nvectorsβkin the parameter α/lessmuch1. For the zeroth-order\nterms one gets\n−ω(0)\nkσzβ(0)\nk=γ\nMDβ(0)\nk, (S.16)\n0.0 0.1 0.2 0.3 0.4 0.50.00.51.01.52.02.5FIG. S1. Effective damping coefficients αm,effof the isolated\nskyrmion in the Pd/Fe/Ir(111) system at B= 1T, calcu-\nlated from the numerical solution of the linearized Landau–\nLifshitz–Gilbert equation (S.15), as a function of the Gilbert\ndamping parameter α.\nwith realω(0)\nkeigenvalues as discussed in Sec. S.I. The\nfirst-order terms read\n−ω(0)\nk/angbracketleftBig\nβ(0)\nk/vextendsingle/vextendsingle/vextendsingleσz/vextendsingle/vextendsingle/vextendsingleβ(1)\nk/angbracketrightBig\n−ω(1)\nk/angbracketleftBig\nβ(0)\nk/vextendsingle/vextendsingle/vextendsingleσz/vextendsingle/vextendsingle/vextendsingleβ(0)\nk/angbracketrightBig\n=γ\nM/angbracketleftBig\nβ(0)\nk/vextendsingle/vextendsingle/vextendsingleD/vextendsingle/vextendsingle/vextendsingleβ(1)\nk/angbracketrightBig\n+iαγ\nM/angbracketleftBig\nβ(0)\nk/vextendsingle/vextendsingle/vextendsingleσzD/vextendsingle/vextendsingle/vextendsingleβ(0)\nk/angbracketrightBig\n,\n(S.17)\nafter taking the scalar product with β(0)\nk. The first terms\non both sides cancel by letting Dact to the left, then\nusing Eq. (S.16) and the fact that the ω(0)\nkare real. By\napplying Eq. (S.16) to the remaining term on the right-\nhand side one obtains\nω(1)\nk=−iαω(0)\nk/integraltext/vextendsingle/vextendsingle/vextendsingleβ−(0)\nk/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk/vextendsingle/vextendsingle/vextendsingle2\ndr/integraltext/vextendsingle/vextendsingle/vextendsingleβ−(0)\nk/vextendsingle/vextendsingle/vextendsingle2\n−/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk/vextendsingle/vextendsingle/vextendsingle2\ndr,(S.18)\nby writing in the definition of the scalar product. By\nusing the definition αk,eff=|Imωk/Reωk|≈/vextendsingle/vextendsingle/vextendsingleω(1)\nk/ω(0)\nk/vextendsingle/vextendsingle/vextendsingle\nand substituting Eqs. (S.12)-(S.13) into Eq. (S.18), one\narrives at Eq. (6) in the main text as long as/vextendsingle/vextendsingle/vextendsingleβ−(0)\nk/vextendsingle/vextendsingle/vextendsingle2\n−\n/vextendsingle/vextendsingle/vextendsingleβ+(0)\nk/vextendsingle/vextendsingle/vextendsingle2\ndoes not change sign under the integral.\nIt is worthwhile to investigate for which values of α\ndoes first-order perturbation theory give a good estimate\nforαk,effcalculated from the exact solution of the lin-\nearized equations of motion, Eq. (S.15). In the materials\nwhere the excitations of isolated skyrmions or skyrmion\nlattices were investigated, significantly different values of\nαhave been found. For example, intrinsic Gilbert damp-\ning parameters of α= 0.02-0.04were determined experi-\nmentallyforbulkchiralmagnetsMnSiandCu 2OSeO 3[4],\nα= 0.28was deduced for FeGe[5], and a total damp-\ning ofαtot= 0.105was obtained for Ir/Fe/Co/Pt mag-\nnetic multilayers[6], where the latter value also includes3\nvarious effects beyond the Landau–Lifshitz–Gilbert de-\nscription. Figure S1 displays the dependence of αm,eff\nonαfor the eigenmodes of the isolated skyrmion in the\nPd/Fe/Ir(111) system, shown in Fig. 2 of the main text.\nMost of the modes show a linear correspondence between\nthe two quantities with different slopes in the displayed\nparameter range, in agreement with Eq. (6) in the main\ntext. For the breathing mode m= 0the convex shape\nof the curve indicates that the effective damping param-\neter becomes relatively even larger than the perturbative\nexpression Eq. (6) as αis increased.\nS.III. EIGENMODES OF THE ISOLATED\nSKYRMION\nHere we discuss the derivation of the skyrmion profile\nEq. (8) and the operators in Eqs. (9)-(11) of the main\ntext. The energy density Eq. (7) in polar coordinates\nreads\nh=A/bracketleftbigg\n(∂rΘ)2+ sin2Θ (∂rΦ)2+1\nr2(∂ϕΘ)2\n+1\nr2sin2Θ (∂ϕΦ)2/bracketrightbigg\n+D/bracketleftbigg\ncos (ϕ−Φ)∂rΘ\n−1\nrsin (ϕ−Φ)∂ϕΘ + sin Θ cos Θ sin ( ϕ−Φ)∂rΦ\n+1\nrsin Θ cos Θ cos ( ϕ−Φ)∂ϕΦ/bracketrightbigg\n+Kcos2Θ−MBcos Θ.\n(S.19)\nThe Landau–Lifshitz–Gilbert Eq. (1) may be rewritten\nas\nsin Θ∂tΘ =γ/primeBΦ+αγ/primesin ΘBΘ,(S.20)\nsin Θ∂tΦ =−γ/primeBΘ+αγ/prime1\nsin ΘBΦ,(S.21)\nwith\nBχ=−1\nMδH\nδχ\n=−1\nM/bracketleftbigg\n−1\nr∂r/parenleftbigg\nr∂h\n∂(∂rχ)/parenrightbigg\n−∂ϕ∂h\n∂(∂ϕχ)+∂h\n∂χ/bracketrightbigg\n,\n(S.22)\nwhereχstands for ΘorΦ. Note that in this form it is\ncommon to redefine BΦto include the 1/sin Θfactor in\nEq. (S.21)[7]. The first variations of Hfrom Eq. (S.19)may be expressed as\nδH\nδΘ=−2A/braceleftbigg\n∇2Θ−sin Θ cos Θ/bracketleftbigg\n(∂rΦ)2+1\nr2(∂ϕΦ)2/bracketrightbigg/bracerightbigg\n−2Ksin Θ cos Θ +MBsin Θ\n−2Dsin2Θ/bracketleftbigg\nsin (ϕ−Φ)∂rΦ + cos (ϕ−Φ)1\nr∂ϕΦ/bracketrightbigg\n,\n(S.23)\nδH\nδΦ=−2A/braceleftbigg\nsin2Θ∇2Φ + sin 2Θ/bracketleftbigg\n∂rΘ∂rΦ +1\nr2∂ϕΘ∂ϕΦ/bracketrightbigg/bracerightbigg\n+ 2Dsin2Θ/bracketleftbigg\nsin (ϕ−Φ)∂rΘ + cos (ϕ−Φ)1\nr∂ϕΘ/bracketrightbigg\n,\n(S.24)\nTheequilibriumconditionEq.(8)inthemaintextmay\nbe obtained by setting ∂tΘ =∂tΦ = 0in Eqs. (S.20)-\n(S.21) and assuming cylindrical symmetry, Θ0(r,ϕ) =\nΘ0(r)and Φ0(r,ϕ) =ϕ. In the main text D>0\nandB≥0were assumed. Choosing D<0switches\nthe helicity of the structure to Φ0=ϕ+π, in which\ncaseDshould be replaced by |D|in Eq. (8). For the\nbackground magnetization pointing in the opposite di-\nrectionB≤0, one obtains the time-reversed solutions\nwith Θ0→π−Θ0,Φ0→Φ0+π,B→−B. Time rever-\nsal also reverses clockwise and counterclockwise rotating\neigenmodes; however, the above transformations do not\ninfluence the magnitudes of the excitation frequencies.\nFinally, we note that the frequencies remain unchanged\neven if the form of the Dzyaloshinsky–Moriya interaction\nin Eq. (S.19), describing Néel-type skyrmions common in\nultrathin films and multilayers, is replaced by an expres-\nsion that prefers Bloch-type skyrmions occurring in bulk\nhelimagnets – see Ref. [3] for details.\nFordeterminingthelinearizedequationsofmotion,one\ncan proceed by switching to the local coordinate system\nas discussed in Sec. S.I and Refs. [1, 3]. Alternatively,\nthey can also directly be derived from Eqs. (S.20)-(S.21)\nby introducing Θ = Θ 0+˜Sx,Φ = Φ 0+1\nsin Θ 0˜Syand\nexpanding around the skyrmion profile from Eq. (8) up\nto first order in ˜Sx,˜Sy– see also Ref. [2]. The operators\nin Eq. (S.3) read\nA1=−2A/parenleftbigg\n∇2−1\nr2cos 2Θ 0(∂ϕΦ0)2/parenrightbigg\n−2D1\nrsin 2Θ 0∂ϕΦ0−2Kcos 2Θ 0+MBcos Θ 0,\n(S.25)\nA2=4A1\nr2cos Θ 0∂ϕΦ0∂ϕ−2D1\nrsin Θ 0∂ϕ,(S.26)\nA3=−2A/braceleftbigg\n∇2+/bracketleftbigg\n(∂rΘ0)2−1\nr2cos2Θ0(∂ϕΦ0)2/bracketrightbigg/bracerightbigg\n−2D/parenleftbigg\n∂rΘ0+1\nrsin Θ 0cos Θ 0∂ϕΦ0/parenrightbigg\n−2Kcos2Θ0+MBcos Θ 0, (S.27)4\nwhich leads directly to Eqs. (9)-(11) in the main text via\nEqs. (S.6)-(S.8).\nThe excitation frequencies of the ferromagnetic state\nmay be determined by setting Θ0≡0in Eqs. (9)-(11) in\nthe main text. In this case, the eigenvalues and eigenvec-\ntors can be calculated analytically[1],\nωk,m=γ/prime\nM(1−iα)/bracketleftbig\n2Ak2−2K+MB/bracketrightbig\n,(S.28)\n/parenleftBig\nβ+\nk,m(r),β−\nk,m(r)/parenrightBig\n= (0,Jm−1(kr)),(S.29)\nwithJm−1theBesselfunction ofthefirstkind, appearing\ndue to the solutions being regular at the origin. Equa-\ntion (S.28) demonstrates that the lowest-frequency exci-\ntation of the background is the ferromagnetic resonance\nfrequencyωFMR =γ\nM(MB−2K)atα= 0. Since the\nanomalous term Dadisappears in the out-of-plane mag-\nnetized ferromagnetic state, all spin waves will be circu-\nlarly polarized, see Eq. (S.29), and the effective damping\nparameterwillalwayscoincidewiththeGilbertdamping.\nRegarding the excitations of the isolated skyrmion, for\nα= 0the linearized equations of motion in Eq. (S.15)\nare real-valued; consequently, β±\nk,m(r)can be chosen to\nbe real-valued. In this case Eqs. (S.9)-(S.10) take the\nform\n˜Sx\nk,m= cos (mϕ−ωk,mt)/parenleftBig\nβ+\nk,m(r) +β−\nk,m(r)/parenrightBig\n,(S.30)\n˜Sy\nk,m= sin (mϕ−ωk,mt)/parenleftBig\nβ+\nk,m(r)−β−\nk,m(r)/parenrightBig\n.(S.31)\nThis means that modes with ωk,m>0form> 0will\nrotate counterclockwise, that is, the contours with con-\nstant ˜Sx\nk,mand ˜Sy\nk,mwill move towards higher values of\nϕastis increased, while the modes with ωk,m>0for\nm < 0will rotate clockwise. Modes with m= 0corre-\nspond to breathing excitations. This sign convention for\nmwas used when designating the localized modes of the\nisolated skyrmion in the main text, and the kindex was\ndropped since only a single mode could be observed be-\nlow the ferromagnetic resonance frequency for each value\nofm.\nS.IV. NUMERICAL SOLUTION OF THE\nEIGENVALUE EQUATIONS\nThe linearized Landau–Lifshitz–Gilbert equation for\nthe isolated skyrmion, Eqs. (2)-(3) with the operators\nEqs.(9)-(11)inthemaintext, weresolvednumericallyby\na finite-difference method. First the equilibrium profile\nwas determined from Eq. (8) using the shooting method\nfor an initial approximation, then obtaining the solution\non a finer grid via finite differences. For the calculationswe used dimensionless parameters (cf. Ref. [8]),\nAdl= 1, (S.32)\nDdl= 1, (S.33)\nKdl=KA\nD2, (S.34)\n(MB)dl=MBA\nD2, (S.35)\nrdl=|D|\nAr, (S.36)\nωdl=MA\nγD2ω. (S.37)\nThe equations were solved in a finite interval for\nrdl∈[0,R], with the boundary conditions Θ0(0) =\nπ,Θ0(R) = 0. For the results presented in Fig. 2 in the\nmain text the value of R= 30was used. It was confirmed\nbymodifying Rthattheskyrmionshapeandthefrequen-\ncies of the localized modes were not significantly affected\nby the boundary conditions. However, the frequencies of\nthe modes above the ferromagnetic resonance frequency\nωFMR =γ\nM(MB−2K)did change as a function of\nR, since these modes are extended over the ferromag-\nnetic background – see Eqs. (S.28)-(S.29). Furthermore,\nin the infinitely extended system the equations of mo-\ntion include a Goldstone mode with/parenleftbig\nβ+\nm=−1,β−\nm=−1/parenrightbig\n=/parenleftbig\n−1\nrsin Θ 0−∂rΘ0,1\nrsin Θ 0−∂rΘ0/parenrightbig\n, corresponding to\nthe translation of the skyrmion on the collinear\nbackground[1]. This mode obtains a finite frequency in\nthe numerical calculations due to the finite value of R\nand describes a slow clockwise gyration of the skyrmion.\nHowever, this frequency is not shown in Fig. 3 of the\nmain text because it is only created by boundary effects.\nIn order to investigate the dependence of the effective\ndamping on the dimensionless parameters, we also per-\nformed the calculations for the parameters describing the\nIr|Co|Pt multilayer system[9]. The results are summa-\nrized in Fig. S2. The Ir|Co|Pt system has a larger di-\nmensionless anisotropy value ( −KIr|Co|Pt\ndl = 0.40) than\nthe Pd/Fe/Ir(111) system ( −KPd/Fe/Ir(111)\ndl = 0.33). Al-\nthough the same localized modes are found in both cases,\nthe frequencies belonging to the m= 0,1,−3,−4,−5\nmodes in Fig. S2 are relatively smaller than in Fig. 2\ncompared to the ferromagnetic resonance frequency at\nthe elliptic instability field where ωm=−2= 0. This\nagrees with the two limiting cases discussed in the lit-\nerature: it was shown in Ref. [1] that for Kdl= 0the\nm= 1,−4,−5modes are still above the ferromagnetic\nresonance frequency at the elliptic instability field, while\nin Ref. [2] it was investigated that all modes become soft\nwithfrequenciesgoingtozeroat (MB)dl= 0inthepoint\n−Kdl=π2\n16≈0.62,belowwhichaspinspiralgroundstate\nis formed in the system. Figure S2(b) demonstrates that\nthe effective damping parameters αm,effare higher at the\nellipticinstabilityfieldinIr|Co|PtthaninPd/Fe/Ir(111),\nshowing an opposite trend compared to the frequencies.\nRegarding the physical units, the stronger exchange\nstiffness combined with the weaker Dzyaloshinsky–5\n0.03 0.04 0.05 0.06 0.07 0.080246810\n(a)\n0.03 0.04 0.05 0.06 0.07 0.081.01.52.02.53.03.5\n(b)\nFIG. S2. Localized magnons in the isolated skyrmion, with\nthe interaction parameters corresponding to the Ir|Co|Pt\nmultilayer system from Ref. [9]: A= 10.0pJ/m,D=\n1.9mJ/m2,K=−0.143MJ/m3,M = 0.96MA/m. The\nanisotropy reflects an effective value including the dipolar in-\nteractions as a demagnetizing term, −K =−K 0−1\n2µ0M2\nwithK0=−0.717MJ/m3. (a) Magnon frequencies f=ω/2π\nforα= 0. Illustrations display the shapes of the excitation\nmodes visualized as the contour plot of the out-of-plane spin\ncomponentsona 1×1nm2grid,withredandbluecolorscorre-\nsponding to positive and negative Szvalues, respectively. (b)\nEffective damping coefficients αm,eff, calculated from Eq. (6)\nin the main text.Moriya interaction and anisotropy in the multilayer sys-\ntem leads to larger skyrmions stabilized at lower field val-\nues and displaying lower excitation frequencies. We note\nthat demagnetization effects were only considered here\nas a shape anisotropy term included in K; it is expected\nthat this should be a relatively good approximation for\nthe Pd/Fe/Ir(111) system with only a monolayer of mag-\nnetic material, but it was suggested recently[6] that the\ndipolar interaction can significantly influence the excita-\ntion frequencies of isolated skyrmions in magnetic multi-\nlayers.\nS.V. SPIN DYNAMICS SIMULATIONS\nFor the spin dynamics simulations displayed in Fig. 3\nin the main text we used an atomistic model Hamiltonian\non a single-layer triangular lattice,\nH=−1\n2/summationdisplay\n/angbracketlefti,j/angbracketrightJSiSj−1\n2/summationdisplay\n/angbracketlefti,j/angbracketrightDij(Si×Sj)−/summationdisplay\niK(Sz\ni)2\n−/summationdisplay\niµBSz\ni, (S.38)\nwith the parameters J= 5.72meV for the Heisenberg\nexchange,D=|Dij|= 1.52meV for the Dzyaloshinsky–\nMoriya interaction, K= 0.4meV for the anisotropy,\nµ= 3µBfor the magnetic moment, and a= 0.271nm\nfor the lattice constant. For the transformation be-\ntween the lattice and continuum parameters in the\nPd/Fe/Ir(111) system see, e.g., Ref. [10]. The simula-\ntionswereperformedbynumericallysolvingtheLandau–\nLifshitz–Gilbert equation on an 128×128lattice with\nperiodic boundary conditions, which was considerably\nlarger than the equilibrium skyrmion size to minimize\nboundary effects. The initial configuration was deter-\nmined by calculating the eigenvectors in the continuum\nmodel and discretizing it on the lattice, as shown in the\ninsets of Fig. 2 in the main text. It was found that such\na configuration was very close to the corresponding exci-\ntation mode of the lattice Hamiltonian Eq. (S.38), simi-\nlarly to the agreement between the continuum and lattice\nequilibrium skyrmion profiles[10].\n[1] C. Schütte and M. Garst, Phys. Rev. B 90, 094423\n(2014).\n[2] V. P. Kravchuk, D. D. Sheka, U. K. Rössler, J. van den\nBrink, andYu.Gaididei, Phys.Rev.B 97, 064403(2018).\n[3] S.-Z. Lin, Phys. Rev. B 96, 014407 (2017).\n[4] T. Schwarze, J. Waizner, M. Garst, A. Bauer, I.\nStasinopoulos, H. Berger, C. Pfleiderer, and D. Grundler,\nNat. Mater. 14, 478 (2015).\n[5] M. Beg, M. Albert, M.-A. Bisotti, D. Cortés-Ortuño, W.\nWang, R. Carey, M. Vousden, O. Hovorka, C. Ciccarelli,\nC. S. Spencer, C. H. Marrows, and H. Fangohr, Phys.\nRev. B95, 014433 (2017).\n[6] B. Satywali, F. Ma, S. He, M. Raju, V. P. Kravchuk,\nM. Garst, A. Soumyanarayanan, and C. Panagopoulos,arXiv:1802.03979 (2018).\n[7] F. Romá, L. F. Cugliandolo, and G. S. Lozano, Phys.\nRev. E90, 023203 (2014).\n[8] A. O. Leonov, T. L. Monchesky, N. Romming, A. Ku-\nbetzka, A. N. Bogdanov, and R. Wiesendanger, New J.\nPhys.18, 065003 (2016).\n[9] C. Moreau-Luchaire, C. Moutafis, N. Reyren, J. Sam-\npaio, C. A. F. Vaz, N. Van Horne, K. Bouzehouane, K.\nGarcia, C. Deranlot, P. Warnicke, P. Wohlhüter, J.-M.\nGeorge, M. Weigand, J. Raabe, V. Cros, and A. Fert,\nNat. Nanotechnol. 11, 444 (2016).\n[10] J. Hagemeister, A. Siemens, L. Rózsa, E. Y. Vedme-\ndenko, and R. Wiesendanger, Phys. Rev. B 97, 174436\n(2018)."    },    {        "title": "1810.10595v4.Nearly_isotropic_spin_pumping_related_Gilbert_damping_in_Pt_Ni___81__Fe___19___Pt.pdf",        "content": "Nearly isotropic spin-pumping related Gilbert damping in Pt/Ni 81Fe19/Pt\nW. Cao,1,\u0003L. Yang,1S. Au\u000bret,2and W.E. Bailey1, 2,y\n1Materials Science and Engineering, Department of Applied Physics and Applied Mathematics,\nColumbia University, New York, New York 10027, USA\n2SPINTEC, Universit \u0013eGrenoble Alpes/CEA/CNRS, F-38000 Grenoble, France\n(Dated: July 12, 2021)\nA recent theory by Chen and Zhang [Phys. Rev. Lett. 114, 126602 (2015)] predicts strongly\nanisotropic damping due to interfacial spin-orbit coupling in ultrathin magnetic \flms. Interfacial\nGilbert-type relaxation, due to the spin pumping e\u000bect, is predicted to be signi\fcantly larger for\nmagnetization oriented parallel to compared with perpendicular to the \flm plane. Here, we have\nmeasured the anisotropy in the Pt/Ni 81Fe19/Pt system via variable-frequency, swept-\feld ferromag-\nnetic resonance (FMR). We \fnd a very small anisotropy of enhanced Gilbert damping with sign\nopposite to the prediction from the Rashba e\u000bect at the FM/Pt interface. The results are contrary\nto the predicted anisotropy and suggest that a mechanism separate from Rashba spin-orbit coupling\ncauses the rapid onset of spin-current absorption in Pt.\nINTRODUCTION\nThe spin-transport properties of Pt have been studied\nintensively. Pt exhibits e\u000ecient, reciprocal conversion\nof charge to spin currents through the spin Hall e\u000bect\n(SHE)[1{4]. It is typically used as detection layer for\nspin current evaluated in novel con\fgurations[5{7]. Even\nso, consensus has not yet been reached on the experi-\nmental parameters which characterize its spin transport.\nThe spin Hall angle of Pt, the spin di\u000busion length of Pt,\nand the spin mixing conductance of Pt at di\u000berent inter-\nfaces di\u000ber by as much as an order of magnitude when\nevaluated by di\u000berent techniques[2, 3, 8{12].\nRecently, Chen and Zhang [13, 14] (hereafter CZ) have\nproposed that interfacial spin-orbit coupling (SOC) is\na missing ingredient which can bring the measurements\ninto greater agreement with each other. Measurements of\nspin-pumping-related damping, particularly, report spin\ndi\u000busion lengths which are much shorter than those es-\ntimated through other techniques[15, 16]. The introduc-\ntion of Rashba SOC at the FM/Pt interface leads to\ninterfacial spin-memory loss, with discontinuous loss of\nspin current incident to the FM/Pt interface. The model\nsuggests that the small saturation length of damping en-\nhancement re\rects an interfacial discontinuity, while the\ninverse spin Hall e\u000bect (ISHE) measurements re\rect the\nbulk absorption in the Pt layer[15, 16].\nThe CZ model predicts a strong anisotropy of the en-\nhanced damping due to spin pumping, as measured in\nferromagnetic resonance (FMR). The damping enhance-\nment for time-averaged magnetization lying in the \flm\nplane ( pc-FMR, or parallel condition) is predicted to be\nsigni\fcantly larger than that for magnetization oriented\nnormal to the \flm plane ( nc-FMR, or normal condition).\nThe predicted anisotropy can be as large as 30%, with\npc-FMR damping exceeding nc-FMR damping, as will be\nshown shortly.\nIn this paper, we have measured the anisotropy of the\nenhanced damping due to the addition of Pt in symmet-ric Pt/Ni 81Fe19(Py)/Pt structures. We \fnd that the\nanisotropy is very weak, less than 5%, and with the op-\nposite sign from that predicted in [13].\nTHEORY\nWe \frst quantify the CZ-model prediction for\nanisotropic damping due to the Rashba e\u000bect at the\nFM/Pt interface. In the theory, the spin-memory loss\nfor spin current polarized perpendicular to the interfa-\ncial plane is always larger than that for spin current po-\nlarized in the interfacial plane. The pumped spin po-\nlarization\u001b=m\u0002_mis always perpendicular to the\ntime-averaged or static magnetization hmit'm. For\nnc-FMR, the polarization \u001bof pumped spin current is\nalways in the interfacial plane, but for pc-FMR, is nearly\nequally in-plane and out-of-plane. A greater damping\nenhancement is predicted in the pccondition than in the\nnccondition, \u0001 \u000bpc>\u0001\u000bnc:\n\u0001\u000bnc=Kh1 + 4\u0011\u0018(tPt)\n1 +\u0018(tPt)i\n(1)\n\u0001\u000bpc=Kh1 + 6\u0011\u0018(tPt)\n1 +\u0018(tPt)+\u0011\n2[1 +\u0018(tPt)]2i\n(2)\n\u0018(tPt) =\u0018(1)\u0002coth(tPt=\u0015sd) (3)\nwhere the constant of proportionality K is the same for\nboth conditions and the dimensionless parameters, \u0011and\n\u0018, are always real and positive. The Rashba parameter\n\u0011= (\u000bRkF=EF)2(4)\nis proportional to the square of the Rashba coe\u000ecient\n\u000bR, de\fned as the strength of the Rashba potential,arXiv:1810.10595v4  [cond-mat.mtrl-sci]  22 Feb 20192\nFIG. 1. Frequency-dependent half-power FMR linewidth\n\u0001H1=2(!) of the reference sample Py(5 nm) (black) and sym-\nmetric trilayer samples Pt(t)/Py(5 nm)/Pt(t) (colored). (a)\npc-FMR measurements. (b) nc-FMR measurements. Solid\nlines are linear \fts to extract Gilbert damping \u000b. (Inset):\ninhomogeneous broadening \u0001 H0inpc-FMR (blue) and nc-\nFMR (red).\nV(r) =\u000bR\u000e(z)(^k\u0002^z)\u0001\u001b, where\u000e(z) is a delta function\nlocalizing the e\u000bect to the interface at z= 0 (\flm plane\nisxy),kFis the Fermi wavenumber, and EFis the Fermi\nenergy. The back\row factor \u0018is a function of Pt layer\nthickness, where the back\row fraction at in\fnitely large\nPt thickness de\fned as \u000f=\u0018(1)=[1 +\u0018(1)].\u000f= 0 (1)\nrefers to zero (complete) back\row of spin current across\nthe interface. \u0015sdis the spin di\u000busion length in the Pt\nlayer.\nTo quantify the anisotropy of the damping, we de\fne\nQ:\nQ\u0011(\u0001\u000bpc\u0000\u0001\u000bnc)=\u0001\u000bnc (5)\nas an anisotropy factor , the fractional di\u000berence be-\ntween the enhanced damping in pc and nc conditions.\nPositive Q (Q >0) is predicted by the CZ model. A\nspin-memory loss \u000efactor of 0.9\u00060.1, corresponding\nto nearly complete relaxation of spin current at the in-\nterface with Pt, was measured through current perpen-\ndicular to plane-magnetoresistance (CPP-GMR)[8] Ac-\ncording to the theory[13, 14], the spin-memory loss can\nbe related to the Rashba parameter by \u000e= 2\u0011, so we\ntake\u0011\u00180:45. The e\u000bect of variable \u0011 < 0:45 will be\nshown in Figure 3. To evaluate the thickness dependent\nback\row\u0018(tPt), we assume \u0015Pt\nsd= 14 nm, which is asso-\nciated with the absorption of the spin current in the bulk\nof Pt layer, as found from CPP-GMR measurements[8]\nand cited in [13]. Note that this \u0015Pt\nsdis longer than that\nused sometimes to \ft FMR data[15, 16]; Rashba interfa-\ncial coupling in the CZ model brings the onset thickness\ndown. The calculated anisotropy factor Q should then\nFIG. 2. Pt thickness dependence of Gilbert damping \u000b=\n\u000b(tPt) inpc-FMR (blue) and nc-FMR (red). \u000b0refers to the\nreference sample ( tPt= 0). (Inset): Damping enhancement\n\u0001\u000b(tPt) =\u000b(tPt)\u0000\u000b0due to the addition of Pt layers in\npc-FMR (blue) and nc-FMR (red). Dashed lines refer to cal-\nculated \u0001\u000bncusing Equation 1 by assuming \u0015Pt\nsd= 14 nm\nand\u000f= 10%. The red dashed line ( \u0011= 0:15) shows a similar\ncurvature with experiments; The black dashed line ( \u0011\u00150:25)\nshows a curvature with the opposite sign.\nbe as large as 0.3, indicating that \u0001 \u000bpcis 30% greater\nthan \u0001\u000bnc(see Results for details).\nEXPERIMENT\nIn this paper, we present measurements of the\nanisotropy of damping in the symmetric Pt( tPt)/Py(5\nnm)/Pt(tPt) system, where \\Py\"=Ni 81Fe19. Because\nthe Py thickness is much thicker than its spin coher-\nence length[17], we expect that spin-pumping-related\ndamping at the two Py/Pt interfaces will sum. The\nfull deposited stack is Ta(5 nm)/Cu(5 nm)/Pt( tPt)/Py(5\nnm)/Pt(tPt)/Al 2O3(3 nm),tPt= 1{10 nm, deposited\nvia DC magnetron sputtering under computer control on\nion-cleaned Si/SiO 2substrates at ambient temperature.\nThe deposition rates were 0.14 nm/s for Py and 0.07\nnm/s for Pt. Heterostructures deposited identically, in\nthe same deposition chamber, have been shown to exhibit\nboth robust spin pumping e\u000bects, as measured through\nFMR linewidth[18, 19], and robust Rashba e\u000bects (in\nCo/Pt), as measured through Kerr microscopy[20, 21].\nThe stack without Pt layers was also deposited as the ref-\nerence sample. The \flms were characterized using vari-\nable frequency FMR on a coplanar waveguide (CPW)\nwith center conductor width of 300 \u0016m. The bias mag-\nnetic \feld was applied both in the \flm plane ( pc) and\nperpendicular to the plane ( nc), as previously shown in\n[22]. The nc-FMR measurements require precise align-\nment of the \feld with respect to the \flm normal. Here,3\nFIG. 3. Anisotropy factor Q for spin-pumping enhanced damping, de\fned in Equation 5. Solid lines are calculations using the\nCZ theory[13], Equations 1{3, for variable Rashba parameter 0 :01\u0014\u0011\u00140:45.\u0015Pt\nsdis set to be 14 nm. Back\row fraction \u000fis\nset to be 10% in (a) and 40% in (b). Black triangles, duplicate in (a) and (b), show the experimental values from Figure 2.\nsamples were aligned by rotation on two axes to maxi-\nmize the resonance \feld at 3 GHz.\nRESULTS AND ANALYSIS\nFigure 1 shows frequency-dependent half-power\nlinewidth \u0001 H1=2(!) in pc- and nc-FMR. The measure-\nments were taken at frequencies from 3 GHz to a cut-o\u000b\nfrequency above which the signal-to-noise ratio becomes\ntoo small for reliable measurement of linewidth. The\ncuto\u000b ranged from 12{14 GHz for the samples with Pt\n(linewidth\u0018200{300 G) to above 20 GHz for tPt= 0.\nSolid lines stand for linear regression of the variable-\nfrequency FMR linewidth \u0001 H1=2= \u0001H0+2\u000b!=\r , where\n\u0001H1=2is the full-width at half-maximum, \u0001 H0is the in-\nhomogeneous broadening, \u000bis the Gilbert damping, !\nis the resonance frequency and \ris the gyromagnetic ra-\ntio. The \fts show good linearity with frequency !=2\u0019for\nall experimental linewidths \u0001 H1=2(!). The inset sum-\nmarizes inhomogeneous broadening \u0001 H0inpc- and nc-\nFMR; its errorbar is \u00182 Oe.\nIn Figure 2, we plot Pt thickness dependence of damp-\ning parameters \u000b(tPt) extracted from the linear \fts in\nFigure 1, for both pc-FMR and nc-FMR measurements.\nStandard deviation errors in the \fts for \u000bare\u00183\u000210\u00004.\nThe Gilbert damping \u000bsaturates quickly as a function\noftPtin both pc and nc conditions, with 90% of the ef-\nfect realized with Pt(3 nm). The inset shows the damp-\ning enhancement \u0001 \u000bdue to the addition of Pt layers\u0001\u000b=\u000b\u0000\u000b0, normalized to the Gilbert damping \u000b0of\nthe reference sample without Pt layers. The Pt thickness\ndependence of \u0001 \u000bmatches our previous study on Py/Pt\nheterostructures[19] reasonably; the saturation value of\n\u0001\u000bPt=Py=Pt is 1.7x larger than that measured for the\nsingle interface \u0001 \u000bPy=Pt [19] (2x expected). The dashed\nlines in the inset refer to calculated \u0001 \u000bncusing Equation\n1 (assuming \u0015Pt\nsd= 14 nm and \u000f= 10%).\u0011= 0:25 shows\na threshold of Pt thickness dependence. When \u0011>0:25,\nthe curvature of \u0001 \u000b(tPt) will have the opposite sign to\nthat observed in experiments, so \u0011= 0:25 is the maxi-\nmum which can qualitatively reproduce the Pt thickness\ndependence of the damping.\nAs shown in Figure 2 inset, the damping enhancement\ndue to the addition of Pt layers is slightly larger in the\nncgeometry than in the pcgeometry: \u0001 \u000bnc>\u0001\u000bpc.\nThis is opposite to the prediction of the model in [13].\nThe anisotropy factor Q\u0011(\u0001\u000bpc\u0000\u0001\u000bnc)=\u0001\u000bncfor the\nmodel (Q>0) and the experiment (Q <0) are shown to-\ngether in Figure 3 (a) and (b). The magnitude of Q\nfor the experiment is also quite small, with -0.05 <Q<0.\nThis very weak anisotropy, or near isotropy, of the spin-\npumping damping is contrary to the prediction in [13],\nand is the central result of our paper.\nThe two panels (a) and (b), which present the same\nexperimental data (triangles), consider di\u000berent model\nparameters, corresponding to negligible back\row ( \u000f=\n0:1, panel a) and moderate back\row ( \u000f= 0:4, panel b)\nfor a range of Rashba couplings 0 :01\u0014\u0011\u00140:45. A spin\ndi\u000busion length \u0015sd= 14 nm for Pt[8] was assumed in all4\ncases.\nThe choice of back\row fraction \u000f= 0:1 or 0:4 and the\nchoice of spin di\u000busion length of Pt \u0015sd= 14 nm follow\nthe CZ paper[13] for better evaluation of their theory.\nFor good spin sinks like Pt, the back\row fraction is usu-\nally quite small. If \u000f= 0, then there will be no spin\nback\row. In this limit, \u0001 \u000bpc, \u0001\u000bncand the Q factor\nwill be independent of Pt thickness.\nIn the case of a short spin di\u000busion length of Pt, e.g.,\n\u0015sd= 3 nm, the anisotropy Q as a function of Pt thick-\nness decreases more quickly for ultrathin Pt, closer to\nour experimental observations. However, we note that\nthe CZ theory requires a long spin di\u000busion length in or-\nder to reconcile di\u000berent experiments, particularly CPP-\nGMR with spin pumping, and is not relevant to evaluate\nthe theory in this limit.\nLeaving apart the question of the sign of Q, we can see\nthat the observed absolute magnitude is lower than that\npredicted for \u0011= 0:05 for small back\row and 0.01 for\nmoderate back\row. According to ref [13], a minimum\nlevel for the theory to describe the system with strong\ninterfacial SOC is \u0011= 0:3.\nDISCUSSION\nHere, we discuss extrinsic e\u000bects which may result in\na discrepancy between the CZ model (Q \u0018+0.3) and our\nexperimental result (-0.05 <Q<0). A possible role of two-\nmagnon scattering[23, 24], known to be an anisotropic\ncontribution to linewidth \u0001 H1=2, must be considered.\nTwo-magnon scattering is present for pc-FMR and nearly\nabsent for nc-FMR. This mechanism does not seem to\nplay an important role in the results presented. It is\ndi\u000ecult to locate a two-magnon scattering contribution\nto linewidth in the pure Py \flm: Figure 1 shows highly\nlinear \u0001H1=2(!), without o\u000bset, over the full range to\n!=2\u0019= 20 GHz, thereby re\recting Gilbert-type damp-\ning. The damping for this \flm is much smaller than\nthat added by the Pt layers. If the introduction of Pt\nadds some two-magnon linewidth, eventually mistaken\nfor intrinsic Gilbert damping \u000b, this could only produce\na measurement of Q >0, which was not observed.\nOne may also ask whether the samples are appropriate\nto test the theory. The \frst question regards sample qual-\nity. The Rashba Hamiltonian models a very abrupt inter-\nface. Samples deposited identically, in the same deposi-\ntion chamber, have exhibited strong Rashba e\u000bects, so we\nexpect the samples to be generally appropriate in terms\nof quality. Intermixing of Pt in Ni 81Fe19(Py)/Pt[25] may\nplay a greater role than it does in Co/Pt[26], although\ndefocused TEM images have shown fairly well-de\fned in-\nterfaces for our samples[27].\nA second question might be about the magnitude of\nthe Rashba parameter \u0011in the materials systems of in-\nterest. Our observation of nearly isotropic damping isconsistent with the theory, within experimental error and\napart from the opposite sign, if the Rashba parameter \u0011is\nvery low and the back\row fraction \u000fis very low. Ab-initio\ncalculations for (epitaxial) Co/Pt in the ref[28] have in-\ndicated\u0011= 0.02{0.03, lower than the values of \u0011\u00180.45\nassumed in [13, 14] to treat interfacial spin-memory loss.\nThe origin of the small, negative Q observed here is un-\nclear. A recent paper has reported that \u0001 \u000bpcis smaller\nthan \u0001\u000bncin the YIG/Pt system via single-frequency,\nvariable-angle measurements[7], which is contrary to the\nCZ model prediction as well. It is also possible that a\nfew monolayers of Pt next to the Py/Pt interfaces are\nmagnetized in the samples[19], and this may have an un-\nknown e\u000bect on the sign, not taken into account in the\ntheory.\nCONCLUSIONS\nIn summary, we have experimentally demonstrated\nthat in Pt/Py/Pt trilayers the interfacial damping at-\ntributed to spin pumping is nearly isotropic, with an\nanisotropy between \flm-parallel and \flm-normal mea-\nsurements of <5%. The nearly isotropic character of the\ne\u000bect is more compatible with conventional descriptions\nof spin pumping than with the Rashba spin-memory loss\nmodel predicted in [13].\nACKNOWLEDGEMENTS\nWe acknowledge support from the US NSF-DMR-\n1411160 and the Nanosciences Foundation, Grenoble.\n\u0003wc2476@columbia.edu\nyweb54@columbia.edu\n[1] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Ap-\nplied Physics Letters 88, 182509 (2006).\n[2] O. Mosendz, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer,\nS. D. Bader, and A. 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Sinclair, Journal of Magnetism and\nMagnetic Materials 134, 173 (1994).\n[27] W. E. Bailey, A. Ghosh, S. Au\u000bret, E. Gautier, U. Ebels,\nF. Wilhelm, and A. Rogalev, Phys. Rev. B 86, 144403\n(2012).\n[28] S. Grytsyuk, A. Belabbes, P. M. Haney, H.-W. Lee, K.-J.\nLee, M. D. Stiles, U. Schwingenschl ogl, and A. Manchon,\nPhys. Rev. B 93, 174421 (2016)."    },    {        "title": "1002.3295v1.Measurement_of_Gilbert_damping_parameters_in_nanoscale_CPP_GMR_spin_valves.pdf",        "content": "Measurement of Gilbert damping paramete rs in nanoscale CPP-GMR spin-valves  \n \nNeil Smith, Matthew J. Carey, and Jeffrey R. Childress. \nSan Jose Research Center \n Hitachi Global Storage Technologies \nSan Jose, CA 95120 \n \nabstract ⎯ In-situ, device level measurement of thermal mag-noise spectral linewidths in  60nm diameter CPP-GMR spin-valve stacks of \nIrMn/ref/Cu/free , with reference and free la yer of similar CoFe/CoFeGe alloy, are used  to simultaneously determine the intrins ic Gilbert damping for \nboth magnetic layers. It is shown that careful alignment at a \"magic-angle\" between free and reference layer static equilibrium  magnetization can \nallow direct measurement of the broadband intrinsic thermal spectr a in the virtual absence of spin-torque effects which otherwi se grossly distort the \nspectral line shapes and require linewidth extrapolations to zer o current (which are nonetheless al so shown to agree well with the direct method). The \nexperimental magic-angle spectra are shown to be in good qualit ative and quantitative agreement with both macros pin calculation s and \nmicromagnetic eigenmode analysis. Despite similar composition and thickness, it is repeatedly found that the IrMn exchange pinn ed reference layer \nhas ten times larger  intrinsic Gilbert damping  than that of the free-layer ) 1 . 0 ( ≈ α ) 01 . 0 ( ≈α .It is argued that the large reference layer damping \nresults from strong, off -resonant coupling to to lossy modes of an IrMn/ref couple,  rather than commonly invoked two-magnon pr ocesses.  \n \n \nI. INTRODUCTION \n     Spin-torque phenomena, in tunneling magnetoresistive (TMR) \nor giant-magnetoresistive (GMR) film stacks lithographically patterned into ~100 nm nanopillars and driven with dc electrical \ncurrents perpendicular to the plane (CPP) of the films have in \nrecent years been the topic of numerous theoretical and \nexperimental papers, both for their novel physics as well as \npotential applications for magnetic memory elements, microwave oscillators, and magnetic field sensors and/or \nmagnetic recording heads.\n1 In all cases, the electrical current \ndensity at which spin-torque instability or oscillation occurs in \nthe constituent magnetic film layers is closely related to the \nmagnetic damping  of these ferromagnetic (FM) films \n     This paper considers the electrical measurement of thermal \nmag-noise spectra to determine intrinsic damping at the device \nlevel  in CPP-GMR spin-valve stacks of sub-100nm dimensions \n(intended for read head applications), which allows simultaneous R-H and transport characterization on the same device. \nCompared to traditional ferromagnetic resonance (FMR) linewidth measurements at the bulk film level, the device-level approach naturally includes finite-size and spin-pumping\n2 effects \ncharacteristic of actual devices, as well as provide immunity to inhomogeneous and/or two-magnon linewidth broadening not relevant to nanoscale devices. Complimentary to spin-torque-\nFMR using ac excitation currents,\n3 broadband thermal excitation \nnaturally excites all modes of the system (with larger, more quantitatively modeled signal amplitudes) and allows \nsimultaneous damping measurement in both reference and free FM layers of the spin-valve, which will be shown to lead to \nsome new and unexpected conclusions. However, spin-torques at \nfinite dc currents can substantially alter the absolute linewidth, and so it is necessary to account for or eliminate this effect in \norder to determine the intrinsic damping. \n                                     1  \nII. PRELIMINARIES AND MAGIC-ANGLES \n \n     Fig. 1a illustrates the basic film stack structure of a \nprospective CPP-GMR spin-valve (SV) read sensor, which apart \nfrom the Cu spacer between free-layer (FL) and reference layer \n(RL), is identical in form to well-known, present day TMR sensors. In addition to the unidirectional exchange coupling \nbetween the IrMn and the pinned-layer (PL), the usual \n\"synthetic-antiferromagnet\" (SAF) structure PL/Ru/RL is meant to increase magnetostatic stability and immunity to field-induced \nrotation of the PL-RL couple, as well as strongly reduce its net \ndemagnetizing field on the FL which otherwise can rotate in \nresponse to signal fields. However, for simplicity in interpreting \nand modeling the spectral and transport data of Sec. III , the \npresent experiment restricts attention to devices with a single RL directly exchange-coupled to IrMn, as shown in Fig. 1b. \n     The simplest practical model for describing the physics of the device of Fig. 1b is a macrospin model that treats the RL unit \nmagnetization as fixed, with only the FL magnetization \nRLˆm\n) (ˆ ) ( ˆFL t tm m ↔  as possibly dynamic in time. As was described \npreviously,4 the linearized  Gilbert equations for small deviations \n) , (z ym m′ ′′ ′=′m  about equilibrium x m′ ↔ˆ ˆ0  can be expressed \nin the primed coordinates as a 2D tensor/matrix equation5: \nmm\nmH\nmmm Hhmmh mm\n′∂∂⋅∂∂⋅∂′∂−⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛⋅ ≡ ′Δ≡⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛−\nγ≡⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛\nγα ≡⋅∂′∂≡′=′⋅′+′⋅ +\nˆ\nˆ ˆ 1 00 1)ˆ () (,0 11 0,1 00 1) (ˆ) ( ) (\neff\n0effFL\nHmV MppGpDt p t HdtdG D\ns\ntt tt tt\n       (1) \ncap\nRL\nIrMnCu\nseedFL\nPL\nIrMn\nseedcap\nRuCu\nRL\n(a)(b)FL\nxz\nFIG. 1. (a) Cartoon of prospective CPP-GMR spin-valve sensor stack, \nanalogous to that used for contemporary TMR read head. (b) Cartoon of simplified spin-valve stack used for present experiments, patterned into ~60nm circular pillars using e-\nbeam lithography.  In (1),  is a 3D Cartesian tensor, m Hˆ/eff∂ ∂ m m′∂ ∂/ˆ  is a 2 3× \ntransformation matrix between 3D unprimed and 2D primed \nvectors (with  its transpose) which depends only on \n, and  is a 3D perturbation field supposed as the origin \nof the deviations . The magnetic moment m mˆ/∂′∂\n0ˆm ) (th\n) (tm′ mΔ is an \narbitrary fixed  value, but  is a natural choice for \nSec. II. Using an explicit Slonczewski6 type expression for the \nspin-torque contribution, the general form for  is  FL) (V M ms → Δ\n)ˆ(effm H\n \nm J P e HHE\nm\neΔ ≡ ⋅ ≡ θ× θ η −∂∂\nΔ−=\n/ ) 2 / ( and , ˆ ˆ cos),ˆ ˆ ( ) (cosˆ1\neffeff\nST FL RLFL RL ST\nh m mm mmH            (2) \n \nfor any free energy function . A positive electron current \ndensity  implies electron flow from the RL to the FL.  is \nthe net spin polarization of th e current inside the Cu spacer. \nOersted-field contributions to )ˆ(mE\neJeffP\neffH  will be neglected here.  \n                                     2      With  in (1), nontrivial solutions  \nrequire s satisfy 0 ) (=thste t−′=′ m m) (\n0 | ) ( | det= + −G D s Httt\n. The value  \nwhen defines the critical onset of spin-torque instability. \nUsing (1), the general criticality  condition is expressible as  crit\ne eJ J≡\n0 Re=s\n \n0 ) (\nt independen=′−′+′+′ α\n∝′ ′ ′ ′ ′ ′ ′ ′4434421 4434421\ne e Jy z z y\nJz z y y H H H H\n-                 (3a) \n) cos ( ) ( 2 ) 1 (2\nSTθ ≡ η −η− ≅′−′′ ′ ′ ′q q qdqdqHH Hy z z y       (3b) \n) ( 2 / ) 1 () (\n) 2 / (2effcrit\nq q dq d qH H\nP emJz z y y\neη − η −′+′ α Δ= ⇒′ ′ ′ ′\nh            (3c) \n \nwhere  is the Gilbert damping. The -scaling of the terms in \nin (3a) follows just from the form of (2). The result in (3b) was \nderived earlier4 in the present approximation of rigid . αeJ\nRLˆm\n     With θ the angle between  and  (at equilibrium), it \nfollows from (3c) that at a \"magic-angle\"  where the \ndenominator vanishes,  and spin-torque effects are \neffectively eliminated from the system at finite . To pursue this point further, explicit results for  will be used from \nthe prototypical case where th e CPP-GMR stack (Fig. 1b) is \napproximately symmetric about th e Cu spacer, which is roughly \nequivalent to the less restrictiv e situation where the RL and FL \nare similar materials with thicknesses that are not small \ncompared the spin-diffusion length. For this quasi-symmetric \ncase, both quasi-ballistic6 and fully diffusive7 transport models \nyield the following simple functional forms:  \nRLˆmFLˆm\nmagicθ\n∞ →crit\neJ\neJ) (cosθ η\n \n) cos 1 () (cos ) (cos) (cos] cos ) 1 ( 1 [ / ) (cos\nmin maxminθ −Γθ η=− ≡ Δ− θ ≡ δ≡ θθ − Γ + + Γ Γ=θ η\nR R RR R Rr     (4) \n \nwhich also relates η to the normalized resistance r 1 0 (≤≤r ) \nwhich is directly measurable experimentally. The transport \nparameter Γ is theoretically related to the Sharvin resistance6,8 \nor mixing conductance8 at the Cu/FL interface, but will be \nestimated via  measurement in Sec III. Using crit\neJ ) (qη  from (4) \nin (3), magicθ  and ) (magicθr vs.  curves are shown in Fig. 2.  Γ\n     The \"magic-angle\" concept also applies to mag-noise power \nspectral density (PSD)  at bias \ncurrent , arising from thermal fluctuations in θ θ Δ = S d dr R I SV2\nbias bias ] ) / ( [\nbiasI θabout \nequilibrium bias angle biasθ . Assuming  in/near the \nfilm plane (FL RL 0ˆ, - m\n=≅′z zˆ ˆplane-normal), and requiring  | , \nit can be shown5 from fluctuation-dissipation arguments that  | | |crit\nbias eI I<\n \n] ) ( [ andwhere) ( ) () (4) ()] ( [ ) ( , ) (4) (\n02 2 2\n022 2 2 21\nz y y z y y z zy z z y z z y yz y z z By yB\nH H H HH H H HH H\nmT kf SG D i H DmT kf S\n′ ′ ′ ′ ′ ′ ′ ′′ ′ ′ ′ ′ ′ ′ ′′ ′ ′ ′\nθ−\n′ ′ θ\n′−′+′+′ α γ = ω Δ′ ′ −′ ′ γ = ωω Δ ω + ω − ωω + ′+′ γ\nΔα γ≅ ⇒+ ω − ′ = ω χ χ ⋅ ⋅ χΔ γ≅tt t t t tt @\n    (5) \n \nComparing (5) with (3), it is se en that the spectral linewidth \nωΔis predicted to be a linear function of  but with ,eJ\n0 /→ ωΔedJ d  when magic bias θ → θ . Since y y z zH H ′ ′ ′ ′′> > ′  \n(due to ~10 kOe out-of-plane demag fields) and z y y yH H ′ ′ ′ ′′> > ′  \n(e.g., for the  measurements in Sec. III), it is only in \nthe linewidth crit\ne eJ J<\nωΔ that the off-diagonal terms y z z yH H ′ ′ ′ ′′ ′,  can \nbe expected to influence . Therefore, measurement of \n with ) (f Sθ\n) (f SV magic bias θ≅ θ  ideally allows direct measurement \nof the natural thermal-equ ilibrium mag-noise spectrum , \nfrom which can be extracted the intrinsic (i.e., -independent) \nGilbert damping constant) (f Sθ\neJ\nα. This is the subject of Sec. III.   9095100105110115120\n  \n 0.20.250.30.350.40.450.5rmagic\n1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6θmagic\n(deg)\nΓFL0 1112m2\nm = +− Γ+ Γ+ c c\nFIG. 2.  Graph of θmagic(blue) and rmagic= rbias(θma g i c) (red) vs. ΓFLas \ndescribed by (4). The equation for cm= cos( θmagic) follows from (3) and \n(4). The red solid squares are measured ( ΓFL, rmagic) from Figs. 3,4 and 6. III. EXPERIMENTAL RESULTS \n \n     The results to be shown below were measured on CPP-GMR-\nspin-valves of stack structure: seed-layers/IrMn (60A)/RL/Cu (30A)/FL/cap layers. The films were fabricated by magnetron \nsputtering onto AlTiC substrates  at room temperature, with \n2mTorr of Ar sputter gas. The bottom contact was a ~1-\nμm thick \nNiFe layer, planarized using ch emical-mechanical polishing. To \nincrease ΔR/R, both the RL and FL were made from \n(CoFe) 70Ge30 magnetic alloys.9 The RL includes a thin CoFe \nbetween IrMn and CoFeGe to help maximize the exchange coupling strength, and both RL and FL include very thin CoFe at \nthe Cu interface.  The resultant  product for the RL and FL \nwere about 0.64 emu/cm\n2. After deposition, SV films were \nannealed for 5hours at 245C in 13kOe applied field to set the \nexchange pinning direction. The IrMn/RL exchange pinning \nstrength of ≈0.75 erg/cm2\n was measured by vibrating sample \nmagnetometry. After annealing, patterned devices with ≈ 60 nm \ndiameter (measured at the FL) were fabricated using e-beam lithography and Ar ion milling. A 0.2\nμm-thick Au layer was \nused as the top contact to devices.  t Ms\n     Fig. 3 illustrates a full measurement sequence. Devices are \nfirst pre-screened to find samp les with approximate ideal in-\nplane δR-H loops (Fig. 3a) for circul ar pillars: non-hysteretic, \nunidirectionally-square loops with  parallel with the \nRL's exchange pinning direction ||H H=\n),ˆ(x+ along with symmetric \nloops about when  is transverse The right-shift in the \n0=H⊥=H H axis).ˆ(-y||H R-δ  loop indicates a large demagnetizing \nfield of ~500 Oe from the RL on the FL.  \n     As shown previously,4 narrow-band \"low\"-frequency  \nmeasurements (eI N-\nMHz) 100 ( = ≡ f PSD N , 1MHz bandwidth) \ncan reveal spin-torque criticality as the  very rapid onset  of \nexcess (1/ f-like) noise when exceeds .  loops \nare measured with  sourced from a continuous sawtooth \ngenerator (2-Hz) which also triggers 1/2 sec sweeps of an \nAgilent-E4440 spectrum analyzer (i n zero-span, averaging mode) \nfor ≈50 cycles. With high sweep repeatability and virtually no \n-hysteresis, this averaging is  sufficient so that after \n(quadratically) subtracting the mean | |eI | |crit\neIeI N-\neI\neI\nHz nV/ 1 ) 0 ( ≈ ≈eI N  \nelectronics noise, the resultant  loops (Fig. 3b) indicate \nstochastic uncertainty eI N-\n. Hz nV/ 1 . 0 < <  \n      With 1 cos±=θ , it readily follows from (3c) and (4) that \n \ncrit critcrit crit\nPAP\n) 0 () (\nI II I\nee\n≡ = θ≡ π = θ− = Γ                           (6) \n \nHence, to estimate Γ,  are measured with applied fields eI N-\nkOe2 . 1 , 45 . 0|| + −≈H  (Fig. 3b),which more than sufficient to \nalign  antiparallel (AP), or parallel (P) to ,respectively \n(see Fig. 3a), thereby reducing possible sensitivity to Oersted \nfield and/or thermal effects. (Reducing  by ~200-300 Oe \ndid not significantly change either  curve.) With  \ndenoting electron  flow from RL  to FL, it is readily found from \n(3) that  and  for the FL. By symmetry, it \nmust follow that and  for spin-torque \ninduced instability of the RL. This sign convention readily \nidentifies these four critical points by inspection of the  \ndata. To account for possible small (thermal) spread in critical \nonset, specific values for the  (excluding ) are defined \nby where the  curves cross the FLˆmRLˆm\n| |||H\neI N- 0>eI\n0crit\nFL AP>-I 0crit\nFL P<-I\n0crit\nRL AP<-I 0crit\nRL P>-I\neI N-\ncrit\neIcrit\nRL P-I\neI N- Hz nV/ 2 . 0 line, which is \neasily distinguished from the mA / Hz nV/ 05 . 0 ~  residual \nmagnetic/thermal background. is estimated in Fig. 3b (and \nrepeatedly in Figs. 4-7) to be ≈ +4.5 mA. Arbitrariness in the \nvalue of  from using the crit\nRL P-I\ncrit\neI Hz nV/ 2 . 0 criterion is thought to \nonly be of minor significance for , due to the rounded \nshape of the AP  curves near this particular critical point, \nwhich may in part explain why  estimated from  is \nfound to be systematically  somewhat larger than crit\nRL AP-I\neI N-\nRL/CuΓeI N-\nCu/FLΓ . \n                                     3      However, the key results here are the 0.1-18 GHz broad-band \n(rms)  spectra (Fig. 3c). They are measured at \ndiscrete dc bias currents with the same Miteq preamp (and in-\nseries bias-T) used for the data, the latter being insitu  gain-) ; PSD(eI f\neI N-H (kOe)-1.5 -1 -0.5 0 0.5 1 1.50246810\n(%)δR\nRRj19.2Ω\n(a)\nFIG. 3. Measurement set for 60nm device. (a) δR-H ||(black) and δR-H ⊥\n(gray) loops at -5mV bias. (b) P-state  N-Ieloops at H| |≈+1.2 kOe (re d), \nand  AP-state N-Ieloops at H ||≈-0.45 kOe ( blue); FL critical currents to \ndeter mine ΓFL(via (6)) enclosed by oval. (c) rms PSD (f, Ie) (normalized to \n1 mA) with Ieas indicated by color. Thin black curves are least-squares fits \nvia (7), fitted values for αFL, αRLlisted on top of graph. M easured rbiasand \napplied field Hlisted inside graph. Field strength and direction (see Fig. 9) \nadjusted to achieve \"magic-angle\".  ±1.5 mA spectra shown, but not fit.02468 1 0 1 2 1 4 1 6 1 80.00.51.01.52.02.5\nnV\nHzPSD\nfrequency (GHz)I  = +0.4, -0.4, +0.6, +0.8, -0.8  mA -0.6, \nH l +750 Oeα  = 0.12, 0.13, 0.11, 0.12, 0.10 R L 0.10, α = 0.011, 0.011 , 0.011 , 0.012, 0.010 F L 0.010, \nrbiasj 0.36normalized \nto 1 mA\n-1.5 mA+1.5 mA\n(c)-5 -4 -3 -2 -1 0 1 2 3 4 500.20.40.60.81\nI  (mA)eH  l +1.2 kOe     | |\nnV\nHzPSD H  l -0.45 kOe  | |\nΓ  l 4(100 MHz) RL\nΓ  l 3.1FL\n(b)calibrated vs. frequency (with ≈50Ω preamp input impedance \nand additionally compensating the present ≈0.7 pF device \ncapacitance) to yield quantitative ly absolute values for these \n (each averaged over ~100 sweeps, with \nsubtracted post-process) . To confirm the real \nexistence of an effective \"magic-angle\", the applied field H was \ncarefully adjusted (by repeated  trial and error) in both amplitude \nand direction  to eliminate as much as possible any real-time \nobserved dependence of the raw  near the FL FMR \npeak (~ 6 GHz) on the polarity  as well as amplitude of  over a \nsufficient range. This procedure was somewhat tedious and \ndelicate, and initial attempts us ing a nominally transverse field \n were empirically found inferior to additionally adjusting the \ndirection of the field, here rotated somewhat toward the pinning direction for the RL. Using a mechanically-positioned permanent magnet as a field source, this field rotation was only \ncrudely estimated at the time to be ~20-30\no (see also Sec. IV). \nWith both H and bias-point \"optimized\" as such, an -\nseries of  were measured, after which the bias-\nresistance , and finally  and  were measured at \na common (low) bias of −10 mV to determine  (as in (4)). ) ; PSD(eI f\n) 0 ; PSD(=eI f\n) ; PSD(eI f\neI\n⊥H\nbiasθeI\n) ; PSD(eI f\nbiasRminRmaxR\nbiasr\n     The key feature of the rms  in Fig. 3c is that \nthese measured spectra (excluding  appear \nessentially independent of both the polarity and magnitude of  \n(after 1mA-normalization), de fining a \"universal\" spectrum \ncurve over the entire 18GHz bandwidth, including the \nunexpectedly wide, low amplitude  RL-FMR peak near 14 GHz \n(more on this below). Because of the relatively large ) ; PSD(eI f\nmA) 5 . 1 + =eI\neI\nHz nV/ 1 ~ ) 0 ; PSD( =eI f  background, these RL peaks were \nnot well discernible during ra w spectrum measurements, and were practically revealed only after electronics background noise subtraction. As suggested in Fig. 3c, eventual breakdown of the \nmagic-angle condition was genera lly found to first occur from \nspin-torque instability of the FL at larger positive .  \neI\n     The spectra Fig. 4 shows the equivalent set of measurements \non a physically different (tho ugh nominally identical) 60-nm \ndevice. They are found to be remarkably alike in all properties to those of Fig. 3, providing additional confirmation that the \"magic-angle\" method can work on real nanoscale structures to \ndirectly obtain the intrinsic   in the absence of of \nspin-torque effects. This appears further confirmed by the close \nagreement of measured  pairs (from data of Figs. 3,4, \nand 6) and the macrospin model predictions described in Fig. 2. ) 0 ; (=θ eI f S\n) , (Cu/FLΓbr\n     To obtain values for linewidth and then damping ω Δ α from \nthe measured , regions of spectra several-GHz wide, \nsurrounding the FL and RL FMR peaks are each nonlinear least-sqaures fitted to the functional form for) ; PSD(eI f\n) 0 ; (=θ eI f S  in (5). In \nparticular, the fitting function is taken to be  \n \nz z z z y yy y z z z z y yz z y y\nV\nH H HH H H HH H\nS f S\n′ ′ ′ ′ ′ ′′ ′ ′ ′ ′ ′ ′ ′′ ′ ′ ′\nπω\n′ ′ α + γ ω → ′′+′ α γ = ω Δ ′ ′ γ = ωω Δ ω + ω − ωω′ ′ + ω ω\n= =\n/ ] 2 / ) ( ) / [( and) ( , with,\n) ( ) (] ) / ( [\n) (\n2\nfit2\npeakfit 02 2 2\n022 2\n02\n0\n0 2\n     (7) \n \nThree fitting parameters are used: ) 0 (0 = = f S SV , fitα, and \npeakω , the latter being already well defined by the data itself. \nThe substitution for y yH ′ ′′ is accurate to order , leaving 2α\nz zH ′ ′′ as yet unknown. With  dominated by out-\nof-plane demagnetizing fields,  depends mostly on the \nproduct y y z zH H ′ ′ ′ ′′> >′\n) (f SV\nz zH ′ ′′ αfit . For simplicity, fixed values  \nand  were used here, based on macrospin \ncalculations that approximately account for device geometry and net  product for FL and RL films. The fitted  \ncurves, and the values obtained for  and  are also \nincluded in Figs. 3c and 4c. These values are notably independent  of (or show no significant trend with) .   kOe 8FL=′′ ′z zH\nkOe, 10RL=′′ ′z zH\nt Ms ) ; PSD(eI f\nFL\nfitαRL\nfitα\neI\n                                     4      Although the  repeatedly found from these data is \na quite typical magnitude for Gilbert damping in CoFe alloys, \nthe extremely large, 10× greater value of  is quite \nnoteworthy, since the RL and FL are not too dissimilar in \nthickness and composition. Although the small amplitude of the \nRL-FMR peaks in Figs. 3-4 (everywhere below the raw 01 . 0FL\nfit≈ α\n1 . 0RL\nfit≈ α\nHz nV/ 1 electronics noise), may suggest a basic unreliability \nin this fitt ed value for , this concern is seemingly dismissed \nby the data of Fig. 5. Measured on a third (nominally identical) \ndevice, an alternative \"extrapolation-method\" was used, in which RL\nfitα-1.5 -1 -0.5 0 0.5 1 1.50246810\n(%)δR\nRRj19.0Ω\nH (kOe)(a)\nFIG. 4. Analogous measurement set for a different (but nominally identical) \n60nm device. as that shown in Fig. 3.  0 2 4 6 8 1 01 21 41 61 80.00.51.01.52.02.5\nfrequency (GHz)I  = +0.5, -0.5, +1.0, -1.5 mA α  = 0.12, 0.13, 0.10, 0.12 R L\n-1.0, 0.12, α = 0.012, 0.011 , 0.013 , 0.013 F L 0.012 , \nrbiasj 0.39nV\nHzPSD\nnormalized \nto 1 mAH l +600 Oe\n(c)-5 -4 -3 -2 -1 0 1 2 3 4 500.20.40.60.81\nI  (mA)eH  l +1.2 kOe     | |\nΓ  l 2.7FLΓ  l 4RLnV\nHzPSD H  l -0.45 kOe  | |\n(100 MHz)\n(b)                                     5 the applied field was purposefully reduced in magnitude (and \nmore transversely aligned than for magic-angle measurements) \nto increase  and thus align  to be more antiparallel to \n. As a result, spin-torque effects at larger negative - will \ndecrease  and concomitantly enhance  RL-FMR peak \namplitude (and visa-versa for the FL), bringing this part of the \nmeasured spectrum above the raw el ectronics noise background.  biasrFLˆm\nRLˆmeI\nω Δ\n     Using the same fitting function from (7), it is now necessary \nto extrapolate the to  (Fig. 5d) in order to \nobtain the intrinsic damping. This method works well in the case \nof the RL since  and the extrapolated ) (RL\nfit eI α 0→eI\n0 | | /RL\nfit< αeI d d 0=eI  \nintercept value of is necessarily larger  than the measured \n, and hence will be (proportionately) less sensitive to \nuncertainty in the estim ated extrapolation slope.  As can be seen \nfrom Fig. 5d, the extrapolated values for intrinsic RLα\n) (RL\nfit eI α\nRLα  are \nvirtually identical to those obtain ed from the data of Figs. 3,4. \nThe extrapolated  is also quite consistent as well. The \nextrapolation data also confirm the expectation (noted earlier \nfollowing (5)) that linewidth  will vary linearly with . FLα\nω ΔeI\n     Comparing with Figs. 3c,4c, the spectra in Fig. 5c illustrate \nthe profound effect of spin-torque on altering the linewidth and peak-height of both FL and RL FMR peaks  even if the system is \nonly moderately misaligned from  the magic-angle condition. By \ncontrast, for other frequencies (where the ωΔ term in the \ndenominator of (5) is unimportant), the 1mA-normalized spectra \nare independent of . Being consistent with (5), this appears to \nverify that this 2nd form of fluctuation-dissipation theorem \nremains valid despite that the system of (1) is not in thermal equilibrium\n10 at nonzero . (Alternatively stated, spin-torques \nlead to an asymmetric  eI\neI\nHt\n, but do not alter the damping tensor \nDt\n in (1)). The α-proportionality in the prefactor of  in \n(5) relatedly shows that the effect of spin-torque on ) (f Sθ\nωΔ is not \nequivalent to additional dampin g (positive or negative) as may \nbe commonly misconstrued. It fu rther indicates that Oersted-\nfield effects, or other -dependent terms in eI Ht\n not contributing \nto ωΔ, are insignificant in this experiment.  \n     Analogous to Figs. 4,5, the data of Figs. 6,7 are measured on \nCPP-GMR-SV stacks differing only by an additional 1-nm thick \nDy cap layer deposited directly on top of the FL. The use of Dy \nin this context (presumed spin-pumping from FL to Dy, but possibly including Dy intermixing near the FL/Dy interface\n11) \nwas found in previous work12 to result in an ~3 × increase in FL-\ndamping, then inferred from the ~3 × increase in measured . \nHere, a more direct measure from the FL FMR linewidth \nindicates a roughly similar,  increase in | |crit\nFLI\n× ≈3 . 2FLα(now using \nsomewhat thicker FL films). This ratio is closely consistent with \nthat inferred from  data measured in this experiment over \na population of devices (see Table 1). Notably, the values found \nfor | |crit\nFLI\nRLα remain virtually the same as before.  \n     Finally, Fig.8 shows results for a \"synthetic-ferrimagnet\" (SF) \nfree-layer of the form FL1/Ru(8A)/FL2. The Ru spacer provides -1.5 -1 -0.5 0 0.5 1 1.50246810\n(%)δR\nRRj19.5Ω\nH (kOe)(a)\nI  (mA)e- 5 - 4 - 3 - 2 - 1 01234500.20.40.60.81\nΓ  l 3.3FLΓ  l 3.7RLnV\nHzPSD\n(100 MHz)\n(b)H  l -0.45 kOe  | |H  l +1.2 kOe     | |\n02468 1 0 1 2 1 4 1 6 1 80.00.51.01.5\nfrequency (GHz)I  =  -1.0, -1.5, -2.5, -3.0 mA -2.0,  \nrbiasj 0.53 H l +500 Oe\nnV\nHzPSD\nnormalized \nto 1 mA\n(c)\nFIG. 5. Measurement set for a different (but nominally identical) 60nm \ndevice as that shown in Figs. 3-4. (c) rms spectra (with least-sqaures fits) \nmeasured at larger r biasand θbi as> θmagic. (d) Ie-dependent values of αfi t(Ie) \nfor FL (red) and RL (blue), with suggested Ie→0 extrapolation lines.0.0 0.5 1.0 1.5 2.0 2.5 3.00.000.050.100.15\nα fitRL\n(d)\n|I  | (mA)eFL-1.5 -1 -0.5 0 0.5 1 1.50246810\nH (kOe)(%)δR\nRRj19.9Ω\n(a)\n-5 -4 -3 -2 -1 0 1 2 3 4 500.20.40.60.81\nI  (mA)eH  l +1.2 kOe     | |\nH  l -0.45 kOe  \nΓ  l 3.2FLΓ  l 4.2RLnVPSD\nHz| |\n(100 MHz)\n(b)\n02468 1 0 1 2 1 4 1 6 1 80.00.51.01.5\nfrequency (GHz)I  = +0.7, -0.7, +1.0 , +1.4, -1.4  mA α  = 0.11, 0.11, 0.11 , 0.11, 0.10 R Lα = 0.027 , 0.026 , 0.027 , 0.027, 0.026 F L\n-1.0, 0.10, 0.026, \nH l +700 Oe\nrnVPSD\nbiasj 0.36 Hz\nnormalized \nto 1 mA\n(c)\nFIG. 6. Analogous measurement set as in Figs.3-4, for (an otherwise \nidentical) device with a 10A Dy cap layer in direct contact with the FL                                     6 an interfacial antiferromagnetic  coupling of . Here, \nFL1 has a thicker CoFeGe layer than used for prior FL films, \nand FL2 is a relatively thin CoFe layer chosen so that \n ≈ 0.64 erg/cm2. Although \nhaving similar static M-H or R-H characteristics to that of the \nsimple FL (of similar net  product) used in earlier \nmeasurements, the transport of the SF-FL in regard to spin-\ntorque effects in particular is fundamentally distinct. The basic \nphysics of this phenomenon was described in detail previously.13 \nIn summary, a spin-torque induced quasi-coresonance between \nthe two natural oscillation modes of the FL1/FL2 couple in the \ncase of negative   and  , can act to transfer \nenergy out of the mode that is destabilized by spin-torque, \nthereby delaying the onset of criticality and substantially \nincreasing . Indeed, the side-by-side comparison of   loops provided in Fig. 8b indicate a nearly 5 × increase  in \n, despite that  remains virtually unchanged.  2erg/cm 0 . 1 ≅\nFL 2 1 FL ) ( ) ( ) (FL t M t M t Ms s s ≅ −\nt Ms\neI 0 ˆ ˆRL 1 FL> ⋅ m m\n| |crit\nFL P-IeI N-\n| |crit\nFL P-I | |crit\nFL AP-I\n     For the SF-FL devices, attempts at finding the magic-angle \nunder similar measurement conditions as used for Figs. 3c,4c, and 6c were not successful, and so  the extrapolation method at \nsimilar \n4 . 0bias≈ r was used instead. To improve accuracy for \nextrapolated-FLα , the data of Fig. 8c include measurements \nfor mA 3 . 0 | | ≤eI  (so that ) for which electronics noise \noverwhelms the signal from the RF FMR peaks. Showing \nexcellent linearity of over a wide -range, the \nextrapolated intrinsiccrit\nFLI Ie<\neI. vsFL\nfitαeI\n01 . 0FL≈ α  is, as expected, unchanged \nfrom before. The same is true for the extrapolated RLα as well. \n     Table 1 summarizes the mean critical voltages   (less \nsensitive to lithographic variations in actual device area) from a \nlarger set of  measurements. The crit\nFL P-I R−\neI- PSD ×≈3 . 2 increase in \n with the use of the Dy-cap is in good agreement with \nthat of the ratio of measured .  | |crit\nFL P-I R\nFLα\n -1.5 -1 -0.5 0 0.5 1 1.50246810\n(%)δR\nRRj19.5Ω\nH (kOe)(a)\n- 5 - 4 - 3 - 2 - 1 01234500.20.40.60.81\nΓ  l 3.3FLΓ  l 4.4RL\nI  (mA)enV\nHzPSD\n(100 MHz)\n(b)H  l -0.45 kOe  | |H  l +1.2 kOe     | |\n0 2 4 6 8 1 01 21 41 61 80.00.51.01.52.0\nfrequency (GHz)I  = -0.7, -1.0,  -1.3,  -1.8, -2.0mA -1.6 , \nnormalized \nto 1 mArbiasj 0.66H l +400 OenV\nHzPSD\n(c)\nFIG. 7. Analogous measurement set as in Fig. 5 for a different (but \nnominally identical) device as that in Fig. 6 with a 10A Dy cap layer..0.0 0.5 1.0 1.5 2.00.000.050.100.15\nα fitRL\n(d)\n|I  | (mA)eFL-1.5 -1 -0.5 0 0.5 1 1.50246810\n(%)δR\nRRj18.2Ω\nH (kOe)(a)\n-5 -4 -3 -2 -1 0 1 2 3 4 500.20.40.60.81\nI  (mA)eH  l +1.2 kOe     | |\nH  l -0.45 kOe  \nnVPSD\nHz| |\n(100 MHz)\n(b)\n0.01.02.03.04.05.06.0I  = +0.15 , -0.15 , +0.3, -0.3,  -1., -2., -2.5 mA \n0 2 4 6 8 10 14 16 18 12-1.5., \nrbiasj 0.41H l +600 OenVPSD\nHz\nnormalized \nto 1 mA\n0 2 4 6 8 1 01 21 41 61 80.00.20.40.60.81.0\nfrequency (GHz)I  = +0.15 , -0.15 , +0.3 , -0.3,  -1., -2., -2.5 mA -1.5., \n(c)\n0.0 0.5 1.0 1.5 2.0 2.50.000.011 FL\n0.026 /  \n0.011 / 0.011 /  αFL0.050.100.15\nα fit\n|I  | (mA)eRL\n(d)\nFIG.  8.  Analogous m ea surem ent set as in Figs.5, for (an otherwise identica l) \ndevic e with a synthetic-ferrimagnet FL (SF-FL) as described in text. (b)  \nincludes for comparison N-Ieloops (in lighter color) from  Fig. 3b ; arrows\nshow SF-FL Icr itfor P-state (red) and AP-state (blue). (c) spectral data and \nfits are repeatedly shown (for clarity) using two different ordinate scales. \n0.12 44.9 !2.0 SF-FL0.11  10.4 !0.1 Control/ αRLstack \n0.11 24.5 !0.5 Dy cap 0.026 /  \n0.011 / 0.011 /  αFL\n0.11 24.5 !0.5 Dy cap\n0.12 44.9 !2.0 SF-FL0.11  10.4 !0.1 Control/ αRLstack (mV)crit\nFLR I −\nTable. 1. Summary of critical voltages (measured over ≈ 8 devices each) \nand damping parameter values α for the present experiment. Estimated \nstatistical uncertainty in the α-values is ~10%.   IV. MICROMAGNETIC MODELLING  \n \n     For more quantitative comparison with experiment than \nafforded by the 1-macrospin model of Sec. II, a 2-macrospin \nmodel equally treating both  and  is now considered \nhere as a simpler, special case of a more general micromagnetic \nmodel to be discussed below. The values , \n, , and will be used \nas simplified, combined representations (of similar thickness and \n) to the actual CoFe/CoFeGe multilayer films used for the \nRL and FL. The magnetic films are geometrically modeled as 60 \nnm squares which (in the macrospin approximation) have zero \nshape anisotropy (like circles), but allow analytical calculation \nof all magnetostatic interactions. The effect of IrMn exchange \npinning on the RL is simply included as a uniform  field \n with measured . \nFirstly, Fig. 9b shows simulated  and  curves \ncomputed assuming , roughly the mean value found \nfrom the data of Sec. III. The agreement with the shape of \nthe measured  is very good (e.g., Figs. 6,7 in particular), \nwhich reflects how remarkably closely these actual devices \nresemble idealized (macrospin) behavior.  RLˆmFLˆm\nemu/cc 950FL=sM\nnm 7FL=t emu/cc 1250RL=sM nm 5RL=t\nt Ms\nx H ˆ] ) /( [RL pin pin t M Js =2erg/cm 75 . 0pin≅ J\n|| bias H r-⊥H r-bias\n2 . 3= Γ\ncrit\nFLI\nH R-\n     Next, Fig. 9d shows simulated PSD curves  computed \n(see Appendix) in the absence of spin-torque  (i.e., ) (f SV\n) 0ST= H , \nbut otherwise assuming typical experimental values R=19Ω, ΔR/R=9%, and T=300K, as well as  and 01 . 0FL= α 1 . 0RL=α , \nso to be compared with the magic-angle spectra of Figs. 3,4. Since (as stated in Sec. III) th e experimental field angle was not \naccurately known, the field angle  was varied systematically \nfor the simulations, and in each case the field-magnitude H was \niterated until Hφ\n37 . 0bias≅ r , approximately matching the mean \nmeasured value. In terms of both absolute values and the ratio of \nFL to RL FMR peak amplitudes, the location of  \n(particularly for the FL), and the magnitude of H (on average \n650-700 Oe from the three magic-angle data in Sec. III), the best \nmatch with experiment clearly occurs with . \nThe agreement, both qualitatively and quantitatively, is again \nremarkable given the simplicit y of the 2-macrospin model. peakf\no o40 30 ≤ φ ≤H\n     Finally, results from a di scretized micromagnetic model are \nshown in Fig. 10. Based on Fig.9, the value  was fixed, \nand H = 685 Oe was determined by iteration until o35= θH\n37 . 0bias≅ r . \nThe equilibrium bias-point magneti zation distribution is shown \n60 nm\nRL FL(a)\n60 nm\nRL FL(a)\nFIG. 10. Micromagnetic model results. (a) cell discretizations with arrow-\nheads showing magnetization orie nta tion when | H|=685 Oe and φH=35o\n(see Fig. 9c). (b) simulated partial rms PSD for first 7 eigenmodes (as \nlabeled) computed individually with αFL=0.01 andαRL=0. 01, other \nparameter values indicated. ( c) simulated total rms PSD with αFL=0.01 and \nαRL=0.01 (green) or αRL=0.1(red or blue); blue curve excludes \ncontribution from 5th(FL) eigenmode at 16 GHz.                                      7 02468 1 0 1 2 1 4 1 6 1 80.00.51.01.52.02.5\nfrequency (GHz)nV\nHzPSD rbiasj 0.37\nΓ =3.2\nα   =0.01FLH   =0STφ   =35HoH=685 Oe\nexclude #5 include # 1-7\nα    =0.01RL\n(c)α    =0.1RL02468 1 0 1 2 1 4 1 6 1 8 2 00.00.51.01.52.02.5\nfrequency (GHz)nV\nHzPSDI=1mA R=19 ΩΔR/R=9%\nΓ =3.2\nα  =0.01  RLT=300K\nH   =0ST(#1)\n(#2)\n(#3)(#4)\n(#5)\n(#6)\n(#7)rbiasj 0.37\nφ   =35HoH=685 Oe\n\"FL\"-mode \"RL\"-mode\nα =0.01  FL\n(b)-1.5 -1 -0.5 0 0.5 1 1.500.20.40.60.81\nH (kOe)δR\nΔRΓ = 3.2\n(b)5 nm3nm7 nm\n60 nm60 nm\nRLFL\n(c)(a)\nH\nx\nzymRL\nmFLφH\n02468 1 0 1 2 1 4 1 6 1 80.00.51.01.52.02.5\nfrequency (GHz)ΔR/R=9% R=19 Ω I=1mA \nφ  =10 , H=802  Oeo\nHφ  =0 , H=895  Oeo\nH\nφ  =20 , H=723  Oeo\nH\nφ  =30 , H=657  Oeo\nH\nφ  =40 , H=605  Oeo\nHα   =0.10RLT=300K\nr H   =0ST j 0.37bias\nnVPSD\nα   =0.01Γ =3.2Hz\nFL\n(d)\nFIG. 9. Two-macrospin model results. (a) cartoon of model geometry.  (b) \nsimulated δR-H loops analogous to data of Figs 3-8c. (c )  cartoon defining \nvector orientations (RL exchange pinned along + x direction). (d) simulated \nrms PSD assuming parameter values indicated, with variable | H| to maintain \na fixed rbiasat each φH( as indicated by color).  in Fig. 10a for this 416 cell model. Estimated values for \nexchange stiffness,  and erg/cm 4 . 1FL μ = A erg/cm 2RL μ=A  \nwere assumed. The simulated spectra in Fig. 10b are shown one \neigenmode at a time  (see Appendix), for the 7 eigenmodes with \npredicted FMR frequencies below 20 GHz (the 8th mode is at \n22.9 GHz). The 1st, 2nd, 5th, and 7th modes involve mostly FL \nmotion, the nearly degenerate 3rd and 4th modes (and the 6th) \nmostly that of the RL. (The amplitudes of  from the 6th or \n7th mode are negligible.). For illustration purposes only, Fig. 10b \nassumed identical damping in each film. ) (f SV\n01 . 0FL FL= α = α\n     For Fig. 10c, the computation of  is more properly \ncomputed using either 6 or all 7 eigenmodes simultaneously, \nwhich includes damping-induced coupling between the modes. \nIncluding higher order modes makes negligible change to \n (but rapidly increases computation time). As \nwas observed earlier, the agreement between simulated and \nmeasured spectra in Figs. 3c,4c is good (with ) (f SV\nGHz) 20 ( <f SV\n1 . 0RL=α ), and \nthe simulations now include the small, secondary FL-peak near  \n8 GHz clearly seen in the measured data (including that of Fig. \n6c), though it is somewhat more pronounced in the model results. \nNotably, the computed spectrum near the RL FMR peak more resembles the measurements after removing the 16-GHz 5\nth \nmode from the calculation, as this (FL) mode does not appear to be physically present in the Figs. 3c,4c spectra. \n     While it is perhaps expected that higher order modes in a \nmicromagnetic simulation assuming perfectly homogeneous \nmagnetic films would show deviations from real devices with finite grain-size, edge-roughness/damage, etc., the situation is \nactually more interesting. Fig. 11 shows measured spectra on yet \nanother device (again, nominally identical to that of Figs. 3-5) in which the experiment was perhaps slightly off from the optimum \nmagic-angle condition, as evidenced by the very small shift in \nthe 6-GHz FL FMR peak position with polarity of . More \nnoteworthy, however, is the clear polarity asymmetry and nonlinear-in-  peak-amplitude (for  in particular) of \nboth the 8-GHz secondary FL mode and a higher order mode \nclose to 15 GHz. (Both resemble typical spin-torque effect at \nangles more antiparallel than the magic-angle condition.) This \nsimilarity in behavior indicates with near certainty that this 15-\nGHz mode is also FL-like in origin, and is thus a demonstration of the \"missing\" 5\nth mode predicted in Fig. 10. (In hindsight, \nthere is now discernible a small but similar 15-GHz peak in the spectra of Fig. 4c). It is worth remembering that the \"magic-\nangle\" argument was based on a simple 1-macrospin model, and so remarkably there appears to be circumstances where this \n\"spin-torque null\" actually does apply simultaneously to both the \nFL and RL, as well as to higher order modes.  \neI\neI 0>eI \nV. DISCUSSION \n \n     In addition to the direct  evidence from the measured spectral \nlinewidth in Figs. 3-8, evidence for large Gilbert damping \nFL RL α> > α  for the RL is also seen in the  data. As ratios \n and  are (from Figs. 3-5 data) both \nroughly ~7, this conclusion is semi-quantitatively consistent with \nthe basic scaling (from (3c)) that . This, as well as the \nsubstantial, 2-3 × variation of  with  in Figs. 5d, and 7d, \nappears to rather conclusively (and expectedly) confirm that \ninhomogeneous broadening is not a factor in the large linewidth-\ninferred values of crit\neI\ncrit crit\nFL P RL P/- -I Icrit crit\nFL AP RL AP /- - I I\nα ∝crit\neI\nRL\nfitαeI\nRLα found in these nanoscale  spin-valves.  \n     Large increases in effective damping of \"bulk\" samples of \nferromagnetic (FM) films in cont act with antiferromagnet (AF) \nexchange pinning layers has been reported previously.14-16 The \nexcess damping was generally attributed to two-magnon scattering processes\n17 arising from an inhomogeneous AF/FM \ninterface. However, the two-magnon description applies to the \ncase where the uniform, ( ,  mode is pumped by a \nexternal rf source to a high excitation ( magnon) level, which \nthen transfers energy via two-magnon scattering into a large  \n(quasi-continuum) number of degenerate 0=k )0ω ≡ ω\n) , 0 (0ω=ω≠k k  \nspin-wave modes, all with low (thermal) excitation levels and \nmutually coupled by the same two-magnon process. In this \ncircumstance, the probability of en ergy transfer back to the \nuniform mode (just one among the degenerate continuum) is \nnegligible, and the resultant one-way flow of energy out of the \nuniform mode resembles that of intrinsic damping to the lattice. \nBy contrast, for the nanoscale  spin-valve device, the relevant \neigenmodes (Fig. 10) are discrete and  generally nondegenerate.  \nin frequency. Even for a coincide ntal case of a quasi-degenerate \npair of modes (e.g., RL modes #3 and #4 in Fig. 10), both modes \nare equally excited to thermal equilibrium levels (as are all \nmodes), and have similar intrinsi c damping rates to the lattice. \nAny additional energy transfer via a two-magnon process should \nflow both ways, making impossible a large (e.g., ~10× ) increase \nin the effective net damping of either mode.  \n0.00.51.01.52.0\nI  = +0.7, -0.7 , +1.0 , +1.4, -1.4 mA \n                                     8     Two alternative hypotheses for large RLα  which are \nessentially independent of device size are 1) large spin-pumping \neffect at the IrMn/RL interface, or 2) strong interfacial exchange coupling at the IrMn/RL resulting in non-resonant  coupling to \nhigh frequency modes in either the RL and/or or the IrMn film.  \nHowever, these two alternatives can be distinguished since the \nexchange coupling strength can be greatly altered without \nnecessarily changing the spin-pumping effect. In particular, \nRLα was very recently measured by conventional FMR methods -1.0, \nnVPSD\nHz\n0 2 4 6 8 1 01 21 41 61 8\nfrequency (GHz)r(normalized \nto 1 mA) j 0.35biasH l +700 Oe\nFIG. 11. The rms PSD measured on a physically diffe rent (but nominally \nidentical) device as that generating the analogous \"magic-angle\" spectra shown in Figs. 3c and 4c. Table. 2. Summary of bulk film FMR measurements18 for reduced film \nstack structure: seed/IrMn( tAF)/Cu( tCu)/RL/Cu(30A)/cap.  Removal of IrMn, \nor alternatively a lack of proper seed layer and/or use of a sufficiently thick \ntCu≈30A can each effectively eliminate exchange pinning strength to RL. 0.013 tAF= tCu= 0          \n(out-of-plane FMR)0.013 tAF=60 A , tCu= 30A0.010 tAF=6 0 A ,  tCu= 0                  \n(no seed layer for IrMn)0.011 tAF= tCu= 0sample type\n0.013 tAF= tCu= 0          \n(out-of-plane FMR)0.013 tAF=60 A , tCu= 30A0.010 tAF=6 0 A ,  tCu= 0                  \n(no seed layer for IrMn)0.011 tAF= tCu= 0sample type ) / (23\nRL ω Δ γ = α d H d\nby Mewes18 on bulk film samples (grown by us with the same \nRL films and IrMn annealing procedure as that of the CPP-\nGMR-SV devices reported herein) of the reduced stack structure: \nseed/IrMn( tAF)/Cu( tCu)/RL/Cu(30A)/cap. For all four cases \ndescribed in Table 2, the exchange coupling was deliberately \nreduced to zero, and the measured  was found to be \nnearly identical to that found here for the FL of similar CoFeGe \ncomposition. However, for the two cases with tAF = 60A, excess \ndamping due to spin-pumping of electrons from RL into IrMn \nshould not have been diminished (e.g., the spin diffusion length \nin Cu is ~100 × greater than tCu ≈ 30A). This would appear to rule \nout the spin-pumping hypothesis. 012 . 0RL≈ α\n    The second hypothesis emphasizes the possibility that the \nenergy loss takes place inside the IrMn, from oscillations excited \nfar off resonance by locally strong interfacial exchange coupling \nto a fluctuating . This local interfacial exchange coupling \n can be much greater than , since the latter reflects a \nsurface average over inhomogeneous spin-alignment (grain-to-\ngrain and/or from atomic roughness) within the IrMn sub-lattice that couples to the RL. Further, though such strong but \ninhomogeneities coupling cannot truly be represented by a \nuniform  acting on the RL, the similarity between \nmeasured and modeled values of ~14 GHz for the \"uniform\" RL \neigenmode has clearly been demonstrated here. Whatever are the natural eigenmodes of the real device, the magic-angle spectrum \nmeasurements of Sec. III reflect the thermal excitation of all \neigenmodes for which \"one-way\" intermodal energy transfer should be precluded by the condition of thermal equilibrium and \nthe orthogonality\n19 of the modes themselves. Hence, without an \nadditional energy sink exclusive  of the RL/FL spin-lattice system, \nthe linewidth of all modes should arguably reflect the intrinsic Gilbert damping of the FL or RL films, which the data of Sec. III \nand Table 2 indicate are roughly equal with . Inclusion \nof IrMn as a combined AF/RL system, would potentially provide \nthat extra energy loss channel for the RL modes. RLˆm\nexJpinJ\npinH\n01 . 0 ~ α\n                                     9      A rough plausibility argument for the latter may be made with a crude AF/FM model in which a 2-sublattice AF film is \ntreated as two ferromagnetic layers (#1 and #2) occupying the same  physical location. Excluding magnetostatic contributions, \nthe free energy/area for this 3-macrospin system is taken to be  \n x m m mx m x m m m\nˆ ˆ ] [ ˆ ˆ] )ˆ ˆ ( ) ˆ ˆ [( ) ( ˆ ˆ ) (\nFM FMAF AF AF\npin 0 2 ex2\n22\n1 212 1\n⋅ − + ⋅ −⋅ + ⋅ − ⋅\nJ J Jt K H t Ms   (8) \n \nFor IrMn with Neel temperature of , the internal AF \nexchange field .20 With K T700N≈\nOe 10 ~ / ~7\nB B AFμNT k H A, 60AF=t  \nAF uniaxial anisotropy is estimated to be .21 A \nrough estimate  for strong  interfacial exchange \nis obtained by equating interface energy erg/cc 10 ~6\nAFK\nFM) / ( 8 ~ex t A J\n2 /2\nexφJ  to the bulk \nexchange energy t A/ 42φ  of a hypothetical, small angle Bloch \nwall ) 2 0 (φ ≤ φ ≤ twisting through the FM film thickness. \nTaking nm 5≈t  and A ~ 10-6 erg/cm yields . \nThe value of  in the last \"field-like\" \nterm  in (8) is more precisely chosen to maintain a constant \neigenfrequency for the FM layer independent  of  or , \nthus accounting for the weaker inhomogeneous coupling averaged over an actual AF/FM interface.  2\nex erg/cm 15 ~ J\n1\nex 0 ] ) ( / 1 / 1 [ ~AF−+ t K J J\nexJAFK\n     As shown in Fig. 12, this crude model can explain a ~10 × \nincrease in the FM linewidth provided  á 5-  and  exJ2erg/cm 10\n1 . 0 05 . 0 ~AF - α . It is worthily noted20 that for the 2-sublattice \nAF, the linewidth ) ) / ( /( 2 /AF AF AF 0 sM K H HK≡ α ≈ ω ω Δ  is \nlarger by a factor of 100 ~ / 2AF KH H  compared to high order \nFM spin-wave modes in cases of comparable α and 0ω (with \nHz 10 ~ 212\n0 AF KH H γ ≈ ω  for the AF). Since the lossy  part \nof the \"low\" frequency susceptibility for FM or AF modes scales \nwith ωΔ, it is suggested that the IrMn layer can effectively sink \nenergy from the ~14 GHz RL mode despite the ~100 × disparity \nin their respective resonant frequencies. S ize-independent  \ndamping mechanisms for FM films exchange-coupled to AF \nlayers such as IrMn are worthy of further, detailed study.  \n0 2 4 6 8 10 12 14 16 18 200.000.010.020.030.040.05\nfrequency (GHz)T=300K\n(GHz)-1/2α   =0.01  FMSθFMJ   =0α   = 0.02,\n          0.05, 0.10  0.01, exAF\nerg\ncm2 J   =10ex\n0 2 4 6 8 10 12 14 16 18 200.000.010.020.030.040.05\nfrequency (GHz)T=300K\nα   =0.01  FMJ   =0exerg 1, 3, 10, 100cm2 J   =ex\nS α   =0.10  θFM\nAF\n(GHz)-1/2(a)\n(b)\nFIG.  12. Simulated rms PSD SθFM(f) for a 3-macrospin model of an AF/FM \ncouple as described via (8) and in the text. The FM film parametrics are the \nsame as used for macrospin RL model in Fig. 9, with  αFM=0.01 and       \nJpin= 0.75 erg/cm2.  (a) varied α AF(denoted by color) with  Jex=10erg/cm2.  \n(b) varied  Jex(denoted by color) with αAF=0.1. The black curve in (a) or \nin (b) corresponds to Jex=0. For AF, Msis taken to be 500 emu/cc. \nACKNOWLEDGMENTS \n \nThe authors wish to acknowledge Jordan Katine for the e-beam \nlithography used to make all the measured devices, and Stefan \nMaat for film growth of alternative CPP spin-valve stacks useful \nfor measurements not included here. The authors wish to thank Tim Mewes (and his student Zachary Burell) for making the \nbulk film FMR measurements on rather short notice. One author \n(NS) would like to thank Thomas Schrefl for a useful suggestion for micromagnetic modeling of an AF film. \n \nAPPENDIX \n \n     As was described in detail elsewhere,22 the generalization of  \n(1) or (5) from a single macrospin to that for an N-cell \nmicromagnetic model takes the form \n \n1)] ( [ ) ( ,2) () ( ) (\n−+ ω − ′ = ω ⋅ ⋅Δ γ≡ ω′=′⋅′+′⋅ +\nG D H D Sh m HmG D\ntt t t ttt trrtrtt\nimT ktdtd\nBχ χ χ@    (A1) \n \nwhere  m′r) (orh′r\n is an  column vector built from the N \n2D vectors , and 1 2×N\nN j... 1=′m H G Dttt\nand , ,  are  matrices \nformed from the  array of 2D tensors N N2 2×\nN N× ,jkDt\n ,jkGt\nand \n Here, and , though .jkHt\njk jk D Dδ =t t\njk jkG Gδ =t t\n.jkHt\nis \nnonlocal  in cell indices j,k  due to the magnetost atic interaction.  \n      The PSD  for any scalar quantity is22 ) (f SQ })ˆ({j Qm\n \nj jj\njN\nk jk jk j QQS f Sm mm\nd d dˆ ˆ, ) ( 2 ) (\n1 , ∂∂⋅∂′∂\n≡′ ′ ⋅ ω ⋅ ′ =∑\n=t\n            (A2) \nThe computations for the PSD of Figs.9, 10 took ) (mrQ  to be \n \n∑\n= ⋅ − Γ + + Γ⋅ − Δ=iN\ni i ii i\niR I\nNQ\n1bias\nFL RLFL RL\nˆ ˆ ) 1 ( 1)ˆ ˆ 1 ( 1\nm mm m \n \naveraged over the  cell pairs at the RL-FL interface. 2 /N Ni=\n     For a symmetric  Ht\n (e.g., the set of eigenvectors ), 0ST= H\nm err←  of the system (A1) can be defined from the following \neigenvalue matrix equation \n \nn n N n ie e H Gr rtt\nω = ⋅ ⋅=−\n2 ... 11) (                       (A3) \n \nThe eigenvectors come in N complex conjugate pairs − +e err, \nwith real eigenfrequencies . With suitably normalized ω ± ,ner \nmatrices  and  are diagonal in the \neigenmode basis .22. The analogue to (A1) becomes  mn mn H δ =n mn mni G ω δ =/\n ∑∑\n′⋅ ≡ ′ ′ ω ′ =ω χ ω χΔ γ≡ ω⋅ ⋅ ≡ ω − δ ω ω − = ω χ\n∗ ∗∗\n′\n′ ′′ ′ ′∗ −\nn mn n n mn m Qn n\nn mn m m mB\nmnn m mn mn n mn\nd d S d f SDmT kSD i\n,,1\n, ) ( 2 ) () ( ) (2) () ( ) / 1 ( )] ( [\nd ee D e\nrrrtr\n     (A4) \n \nThe utility of eigenmodes for computing PSD, e.g, in the \ncomputations of Fig. 10, is that only a small  fraction (e.g., 7 \nrather than 416 eigenvector pairs) need be kept in (A4) (with all \nthe rest simply ignored ) in order to obtain accurate results in \npractical frequency ranges (e.g., GHz). Despite that  \nis (in principle) a full matrix, the reduction in matrix size for the \nmatrix inversion to obtain  at each frequency  more than \nmakes up for the cost of computing the 20<mnD\n) (ω χ\n) , (n nerω  which need be \ndone only once independent of frequency or α-values.  \n \n \nREFERENCES  \n1 Volume 320 (2008) of the Journal of Magnetism and Magnetic \nMaterials offers a series of review articles (with many additional \nreferences therein) covering this field. Some examples include: \n   J. A. Katine and E. E. Fullerton, pp. 1217-1226, \n   J. Z. Sun and D. C. Ralph, pp. 1227-1237, \n   T. J. Silva  and W. H. Rippard, pp. 1260-1271. \n2Y. Tserkovnyak, A. Brataas, and G.  E. W. Bauer, Phys. Rev. B \n  67, 140404(R) (2003). \n3G. D. Fuchs et al. , Appl. Phys. Lett. 91, 062505 (2007). \n4 N. Smith, J. A. Katine, J. R. Childress, and M. J. Carey, \n   IEEE Trans. Magn. 41, 2935 (2005). \n5 As discussed in N. Smith, Phys. Rev. B 80, 064412 (2009), the spin-\npumping effect described in Ref. 2 l eads to additional contributions to \nthe damping tensor-matrix Dt\n in (1) or (A1) which are anisotropic, \nangle-dependent, and nonlocal betw een RL and FL. For simplicity, \nthese are here simply lumped into the Gilbert damping parameter α \nfor either RL or FL. In this c ontext, \"intrinsic \" damping refers \napproximately to that in the linit , but also in the presence of \nthe complete stack structure of the spin-valve device of interest. 0→eI\n6  J. C. Slonczewski, J. Magn. Magn. Mater. 247, 324 (2002). \n7 N. Smith, J. Appl. Phys. 99, 08Q703 (2006). \n8 Y. Tserkovnyak, A. Brataas, G. E. W.  Bauer,  and B. I. Halperin,  \n   Rev. Mod. Phys. 77, 1375 (2005). \n9 S. Maat, M. J. Carey, and J. R. Childress, Appl. Phys. Lett. \n   93, 143505 (2008). \n10 R. A. Duine, A. S. Nunez, J. Sinova, and A. H. MacDonald, \n    Phys. Rev. B 75, 214420 (2007). \n11 G. Woltersdorf, M. Kiessling, G. Meye r, J. U. Thiele, and C. H. Back, \n    Phys. Rev. Lett. 102, 257602 (2009). \n12 S. Maat, N. Smith, M. J. Carey, and J. R. Childress, Appl. Phys. Lett. \n    93 , 103506 (2008). \n13 N. Smith, S. Maat, M. J. Carey, and J. R. Childress, Phys. Rev. Lett. \n    101 , 247205, (2008). \n14 R. D. McMichael, M. D. Stiles, P. J. Chen, and W. F. Egelhoff, Jr., \n    J. Appl. Phys.,  83, 7037 (1998). \n15 S. M. Rezende, A. Azevedo, M. A. Lucena, and F. M. de Aguiar, \n    Phys. Rev. B 63, 214418 (2001). \n16 M. C. Weber, H. Nembach, B. Hillebrands, M. J. Carey, and \n    J. Fassbender, J. Appl. Phys. 99, 08J308 (2006). \n17  R. Arias and D. L. Mills, Phys. Rev. B 60, 7395 (1999). \n                                     10 18 T. Mewes, MINT  Center, U. Alabama (Tuscaloosa). These bulk film \nFMR measurements were made in-plane, except for the last entry in \nTable 2 which was measured in th e out-of-plane configuration which \nshould exclude two-magnon contributions  (Ref. 17). The similarity in \nα -values suggests two-magnon is unim portant here, which in turn \nthen suggests that RL-FL spin-pumping contributions in the present \nCPP-GMR nanopillars spin-valves (footnote (5)) that would not be \npresent in the RL-only bulk samples are also small. Further \ncomparisons of present measuremen ts with additional bulk film FMR \nresults are expected to be addressed in a future publication.  19 A nonzero damping matrix Dt\n can technically cause a coupling of \nnondegenerate eigenmodes if , though this does not \nchange the argument citing this f ootnote. This small effect can be \nseen in Fig. 10c as the slight change in the 6-GHz FL FMR peak \nheight and linewidth when 0≠ ⋅ ⋅∗≠ n n me D ertr\nRLαis grossly varied from 0.01 to 0.1.  \n20 Magnetic Oscillations and Waves , A. G. Gurevich and G. A. Melkov, \nCRC Press (Florida, USA) 1996. Chap. 3. \n21 K. O'Grady, L. E. Ferndandez- Outon, G. Vallejo-Fernandez,  \n    J. Magn. Magn. Mater. 322, 883 (2010). \n22 Appendix A of N. Smith, J. Magn. Magn. Mater.  321, 531 (2009). \n \n \n                                     11 "    },    {        "title": "2301.12797v1.Investigation_of_Ultrafast_Demagnetization_and_Gilbert_Damping_and_their_Correlation_in_Different_Ferromagnetic_Thin_Films_Grown_Under_Identical_Conditions.pdf",        "content": "1  Investigation  of Ultrafast  Demagnetization  and Gilbert  Damping  and their  Correlation  in Different  \nFerromagnetic  Thin  Films  Grown  Under  Identical  Conditions1 \nSuchetana Mukhopadhyay, Sudip Majumder, Surya Narayan Panda, Anjan Barman*  \nDepartment  of Condensed  Matter  and Materials  Physics,  S.N. Bose  National  Center  for Basic  \nSciences,  Block -JD, Sector  III, Salt Lake,  Kolkata  700106,  India  \nEmail:  abarman@bose.res.in  \n \n \nAbstract  \nFollowing the demonstration of laser -induc ed ultrafast demagnetization in ferromagnetic nickel,  several  \ntheoretical  and phenomenological  propositions  have sought  to uncover  its underlying  physics.  In this work \nwe revisit the three temperature model (3TM) and the microscopic three  temperature  model  (M3TM)  to \nperform  a comparative  analysis  of ultrafast  demagnetization  in 20-nm-thick  cobalt,  nickel  and permalloy  \nthin films  measured  using  an all-optical  pump - probe  technique.  In addition  to the ultrafast  dynamics  at \nthe femtosecond  timescales,  the nano second  magnetization  precession  and damping  are recorded  at various  \npump  excitation  fluences  revealing  a fluence -dependent  enhancement  in both the demagnetization  times  \nand the damping  factors.  We confirm  that the Curie  temperature  to magnetic  moment  ratio of a given \nsystem acts as a figure of merit for the demagnetization time, while the demagnetization  times  and damping  \nfactors  show  an apparent  sensitivity  to the density  of states  at the Fermi  level for a given system.  \nFurther, from numerical simulations of the ultrafast demagnetization  based on both the 3TM and the M3TM, \nwe extract the reservoir coupling parameters that best  reproduce  the experimental  data and estimate  the \nvalue  of the spin flip scattering  probability  for each system. We discuss how the f luence -dependence of \ninter-reservoir coupling parameters  so extracted  may reflect  a role played  by nonthermal  electrons  in the \nmagnetization  dynamics  at low laser  fluences.  \n \n1. Introduction  \nOptical  excitation  of a magnetic  material  with a short  and intense  laser pulse  sets in motion  a chain  \nof microscopic  events  that result  in a macroscopic,  measurable  quenching  of the net magnetization.  \nSince  the processing  speed  of magnetic  storage  devices  is limited  by the maximum  speeds  at which  the \nmagnetization  can be manipulated,  the possibility  of controlling  magnetization  at sub-picosecond  timescales  \nby employing  ultrashort  laser pulses  holds  tremendous  potential  for applications  in spin-based  memory  and \nstorage  devices  with ultrafast  processing  speeds  including  laser -induced  opto-magnetism  and all-optical  \nswitching  [1–4] as well as THz spintronic  devices  [5–7]. Meanwhile,  the relaxation  timescales  of the laser -\ninduced  magnetization  precession  in ferromagnetic  thin films  associated  with the magnetic  damping  set \nfundamen tal limits  on magnetization  switching  and data transfer  rates  in spintronic  devices.     \nPicosecond  laser -induced  magnetization  quenching  in ferromagnetic  gadolinium  was reported  by \nVaterlaus  et al. in 1991,  who also reported  for the first time a character istic timescale  of 100±80  ps \n[8] associated  with the magnetization  loss. In this early  work,  the timescale  was identified  with the \ntimescale  of the spin-lattice  interactions  mediating  the disruption of magnetic ordering due to laser \nheating of the lattice,  though later works  contradicted  this interpretation  [9, 10]. In fact, by the mid-\n1990s,  it had begun  to be recognized that finer time resolution was necessary to probe this phenomenon \nto uncover a  fuller  picture  of the associated  relaxation  processes  occu rring  at sub-picosecond  timescales.  \nIn 1996,  Beaurepaire  et al. demonstrated  in a seminal  work  that faster  demagnetization  occurring at sub -\npicosecond timescales could be triggered by femtosecond laser pulses in a  nickel thin film [11].  The \nultrafast demag netization phenomenon went on to be demonstrated  in a wide  array  of ferromagnetic  \nsystems  [12–19], triggering  a flurry  of research  that continues  till date. A few years after the pioneering \nexperiments, Koopmans et al. characterized for the  first time the full magneto -optical response to \nfemtosecond laser pulses in a ferromagnet by  time-resolved measurements of Kerr ellipticity and \nrotation [13].  However, the observation of  nonmagnetic contributions to the Kerr rotation signal \nnaturally led some to question  whether  the ultrafast quenching in response to the laser excitation was \nindeed a magnetic phenomenon  [13, 14]. In 2003, Rhie et al.  used time -resolved photoemission  \n \n1First  submitted  in March  2022  2  spectroscopy to probe the collapse of the 3d exchange s plitting in nickel as an unambiguous signature of a \nphotoinduced demagnetization occurring over 300±70 fs [15]. This was soon followed by the first reports \nof an estimate of the characteristic timescale of femtosecond laser -induced demagnetization derived from \nquantum -mechanical principles [16].  \nOne of the major reasons behind this sustained interest is the need to achieve a complete understanding of \nthe microscopic mechanisms that underlie the ultrafast demagnetization phenomenon that have so far \nremained elusive. Much inquiry has been focused on how the laser -induced loss of magnetic order is \ncompensated for via the transfer of angular momentum of the spin system at the associated timescales. \nOver the years, several mechanisms have been proposed for explai ning the angular momentum \nconservation associated with the ultrafast demagnetization, which may be broadly categorized in two \ndistinct classes. One of these argues in favor of a dominant contribution to the demagnetization arising \nfrom nonlocal transport p rocesses driven by the laser pulse, such as superdiffusive transport of spin -\npolarized hot electrons [20, 21] or heat currents [22]. The other narrative relies on local spin -flip scattering \nprocesses occurring by the collision of excited electrons with imp urities, phonons, and magnons [17, 23, \n24]. In this category, electron -lattice and electron -spin relaxation mechanisms are accepted as the major \ndemagnetization channels in the picoseconds following the laser excitation [25], with several works \nattributing  a pivotal role to the material -specific spin -orbit interaction [23, 25 –27]. On the other hand, it is \npossible to model the demagnetization by adopting a thermal description, considering the laser excitation \nas a “heat” source that excites electrons close to the Fermi level instantaneously to very high energies, \nwhose eventual relaxation processes drive the magnetization loss. The laser excitation energy controlled \nby the applied laser pump fluence has a direct influence on the maximum electron temperatures  reached \nand thus plays a pivotal role in the ultrafast magnetization dynamics that follow as a result.  \nIn this work, employing an all -optical time -resolved magneto -optical Kerr effect (TR -MOKE) technique, \nwe investigate the ultrafast spin dynamics along w ith the nanosecond magnetization precession and \ndamping at different laser pump fluences in three ferromagnetic thin films: 20 -nm-thick cobalt, nickel and \npermalloy grown under uniform deposition conditions. Cobalt and nickel are elementary 3d ferromagneti c \ntransition metals well studied in the literature in both their elementary forms and as constituting elements \nin multilayered structures where their strong spin -orbit coupling mediates interfacial effects such as spin \npumping. Permalloy, an alloy comprise d of approximately 80% nickel and 20% iron, is a prototype \nmaterial for spintronic applications due to several desirable qualities such as low coercivity,   large  \npermeability  and  in  particular,   low  Gilbert  damping,   which  aids  in minimizing the  maximum power \nconsumption of devices and allows spin wave propagation on length scales of the order of device size. The \nsimultaneous investigation of the fluence -dependent modulation of ultrafast demagnetization and damping \nin all the above ferromagnetic thin films is motivated by the primary objectives of exploring the dominant \nmicroscopic processes underlying the demagnetization in these systems, correlating the observed \nmagnetization dynamics to the material properties of each system, and in the process  exploring their \ntunability with laser pump fluence. We set out to achieve this in two ways: (a) by modelling the ultrafast \ndemagnetization at various pump fluences using two theoretical models sharing a common link and (b) by \ncorrelating the laser -induced  changes in ultrafast demagnetization to the changes in the Gilbert damping \nfactor which characterizes the magnetization precession. TR -MOKE is a well established local  and  non -\ninvasive all -optical measurement technique tunable to an extremely fine time resolution limited only by \nthe laser pulse width and is thus well suited to our purpose.  \nFor the detailed analysis of ultrafast demagnetization in our samples, we extract the values of \ndemagnetization time and fast relaxation time for each sample using a p henomenological fitting function \nand record its systematic variation with the pump fluence. We subsequently model the demagnetization \ndata using two well known theoretical models: the phenomenological three -temperature model (3TM) and \nthe microscopic three  temperature model (M3TM), and thereby extract values for the microscopic \nparameters relevant for the demagnetization process and calculate the temporal evolution of the simulated \ntemperatures of the electron, spin and lattice systems within the first few picoseconds of the laser \nexcitation. Both these models assume a thermal picture to explain the initial electron temperature rise and \nsubsequent relaxation but differ significantly in their approach with regards to the treatment of the \nmagnetization. A syst ematic study implementing both models to analyze TR -MOKE data recorded under \nuniform experimental conditions on identically prepared samples is absent in the literature and will prove \ninstructive. Further, a thorough investigation and characterization of t he ultrafast demagnetization, \nmagnetic damping and their intercorrelation in three different systems deposited under identical deposition \nconditions has not been carried out before. For the purposes of a comparative analysis, it is  vital to study \nsamples g rown under the same deposition conditions.  The conductivity of the  substrate  can also directly  \ninfluence  demagnetization  time by promoting  or hindering  ultrafast spin transport processes [20, 28] while \nthe deposition technique determines the structural  properties of the samples that may indirectly affect the 3  demagnetization time.  Moreover,  since the experimentally obtained demagnetization timescale has been \nshown to be quite  sensitive to extrinsic factors such as the laser pulse duration and the spectral ba ndwidth \n[29],  it is useful to measure the demagnetization times for various thin films from experiments  performed  \nin the same  experimental  setup  under  near-identical  experimental  conditions.  \n \n2. Materials  and Methods  \n2.1 Sample  fabrication  \n20 nm -thick cobalt,  nickel and permalloy thin films were deposited by electron beam  evaporation \nunder an average base pressure of 1 ×10−7 Torr and at a very low deposition  rate of 0.2 Å/s chosen to \nachieve uniform deposition.  Each film was deposited on an  insulating  8 mm × 8 mm silicon  substrate  \ncoated  with  285-nm-thick  SiO 2. Subsequently, the films were capped in-situ with a 5 -nm-thick \nprotective gold layer (base  pressure  ~1 × 10−7 Torr,  deposition  rate 0.1 Å/s) to prevent  surface  \noxidation  of the ferromagnetic layer and protect it from possible degradation due to high power laser  \nexposure during the optical pump -probe measurements [ 30, 31]. The thickness of the capping  layer  was \nkept more  than three  times  smaller  than the optical  penetration  depth  of 400 nm light in gold.  After  \nthe deposition  of the capping  layer,  the surface  topography  of the samples  was investigated  by atomic  \nforce  microscopy  (AFM)  using  a Tap190Al -G tapping  mode  AFM  probe as shown in Figure 1 ( a). The \naverage surface roughness  Ra of the cobalt,  nickel and  permalloy  films  were  obtained  as 0.91 nm, 0.36 nm \nand 0.41 nm respectively.  Thus,  reasonably  low surface  roughn ess in the sub-nm range  was obtained  \nwhich  is comparable  for all the samples.  In addition,  the static  magneto -optical  Kerr effect  was used to \nstudy  the magnetic  hysteresis  of the deposited  samples  to confirm  their ferromagnetic  nature  (Figure  \n1(b)). \n \n 2.2 All-optical  measurement  of ultrafast  spin dynamics  \nMeasurement of laser -induced magnetization dynamics is carried out using a TR -MOKE  technique \nin two -color optical pump -probe arrangement using a 400 nm pump beam and 800  nm probe beam \nhaving a pump -probe cr oss-correlation width of ~100 fs [30] shown  schematically in Figure 1( c). A \ntypical TR -MOKE trace is shown in Figure 1( d), comprising  ultrafast  demagnetization  (Regime  I), fast \nremagnetization  (Regime  II) and damped  magnetization precession (Regime III).Th is technique enables \nthe direct observation of the  spin dynamics in the femto - and picosecond time domain. Femtosecond \npulses are generated  by an amplified laser system (Libra, Coherent Inc.)  employing a chirped -pulse \nregenerative  amplifier and a Ti:sapphi re laser oscillator (Coherent Inc.)  pumped by a neodymium -doped  \nyttrium lithium fluoride (Nd -YLF) laser.  The second harmonic of the amplified laser output  \n(wavelength = 400 nm, repetition rate = 1 kHz, pulse width >40 fs), generated through a  lithium  \ntribo rate (LBO)  nonlinear  crystal,  is used  for laser  excitation  of the ferromagnetic  thin films.  The \ntime-delayed fundamental beam (wavelength = 800 nm, repetition rate = 1 kHz, pulse width ~40 fs) is \nused to probe the ensuing magnetization dynamics.  In our  setup, different wavelengths are employed \nfor the pump and the probe pulse to eliminate the  possibility of state blocking effects arising from the \nuse of identical wavelengths for pumping  and probing [32]. A computer -controlled variable delay \ngenerator offers  precise control of the  delay time between pump and probe. Before commencing \nmeasurements on any sample, the  zero delay was carefully estimated by maximizing the transient \nreflectivity signal of a bare  silicon  substrate  placed  adjacent  to the sample  on the same  sample  holder.  \nTR-MOKE  experiments are performed with a non -collinear pump -probe geometry.  The pump beam,  \nfocused to a spot size of ~300 µm, is incident obliquely on the sample while the probe  beam, with a \nspot size of ~100 µm, is incident normal to the sample surface and aligned  to the center of the pump \nspot. The pump -probe spatial overlap on the sample was carefully  maintained. The choice of a \nrelatively smaller spot size of the probe beam as compared to the  pump beam facilitates optical \nalignment and ensures that the probe beam detects the local  magnetization changes from a part of the \nsample uniformly irradiated by the pump.  Before  reflection on the samples, the probe beam is polarized \northogonally to the linearly polarized  pump beam. After reflec tion, the Kerr -rotated probe beam is split \nand one part is fed directly  into a Si -photodiode to measure the time -resolved reflectivity signal.  The \nother part is fed  into an identical photodiode after passing through a Glan -Thompson polarizer adjusted \nto a small  angle  from  extinction  to minimize  optical  artifacts  in the Kerr  rotation  signal.  In this way,  \nsimultaneous  measurement  of the time-resolved  total reflectivity  and Kerr  rotation  signals is possible.  \nAn optical chopper operating at 373 Hz placed in th e path of the pump  beam provides a reference signal \nfor the digital signal processing lock -in amplifiers (Stanford  Research  Systems,  SR830)  which  record  \nthe modulated  signal  in a phase  sensitive  manner.  All experiments  were  carried  out under  ambient  \ncondit ions of temperature  and pressure.  4  3. Results  and Discussion  \n 3.1 Theoretical  models  for ultrafast  demagnetization  \nThe phenomenon  of optically  induced  ultrafast  demagnetization  starts  with the irradiation  of the \nmagnetic  sample  with  a brief  and intense  optical  laser  pulse,  exciting  electrons  momentarily a few \nelectron volts above the Fermi level.  Though the exact sequence of events  following  the initial  excitation  \nis difficult  to trace  due to the highly  nonequilibrium  conditions  created  by it, a qualitative  overview  of \nthe complete  demagnetization  process  is fairly  well   established.  The laser  excitation  generates  a \nnonthermal  pool of excited  electrons  which  thermalize  rapidly  within  several  femtoseconds  via \nelectron -electron  interactions.  Spin -dependent  scatter ing events  taking  place  during  this transient  regime  \nlead to a sharp  drop  in magnetization  observable  around  a few hundred  femtoseconds  in the \nexperimental  Kerr  rotation  signal.  Subsequently,  the thermalized  electrons  may release  their  excess  \nenergy  via a variety  of relaxation  channels,  such  as by excitation  of phonons  or magnons.  This results  in a \npartial  recovery  of the magnetization  beyond  which  heat dissipation  into the environment  promotes  further  \nrecovery  on a longer  timescale.  The 3TM  posits  that the thermodynamics  of the demagnetization  \nphenomenon  can be described  simply  by considering  energy  exchange  between  three  thermal  reservoirs  \n[33],  each of which  is assigned  a temperature:  the electrons  at temperature  Te, the lattice  at temperature  \nTl and the electronic  spin reservoir  at temperature  Ts. Since  the reservoirs  are in thermal  contact  and \nthe overall  process  is adiabatic,  equilibration  of the excited  electrons  with the spin and lattice  reservoirs  \nvia energy  transfer  may be described  by coupled  rate equations  in the following  manner:   \n \n \n \n   (1) \n \n \n \nwhere,  Ce, Cl and Cs denote the specific heats of the electron,  lattice and spin reservoirs  respectively,  \nwhile  Gel, Ges, and Gsl denote  the inter -reservoir  coupling  parameters.  The term  P (t) describes  the action  \nof the laser  pulse  as a source  term driving  the excitation  of the electron  reservoir  to high temperatures.  The \nthermal  diffusion  term  \n  describes  heat dissipation  occurring  via thermal  conduction  \nalong  the sample  thicknes s. Under  this description,  the observed  demagnetization  is attributed  to a \nrise in the spin temperature  Ts occurring  shortly  after  the electron  temperature  rise. The coupling  \nstrengths  between  the electron -spin and electron -lattice  subsystems  qualitatively  determine  the efficiency  \nof energy  transfer  between  them  and hence  influence  the timescales  associated  with the demagnetization \nand fast relaxation.  However, the 3TM is purely phenomenological and does  not explicitly consider any \nmicroscopic mechanisms un derlying the phenomenon it describes.  On the other  hand,  the M3TM  \nproposed  by Koopmans  et al. [23] provides  a “spin -projected”  perspective  [27] to explain  the ultrafast  \ntransfer  of angular  momentum  highlighting  the role of Elliot -Yafet  type ultrafast  spin-flip scattering  in \nthe demagnetization  process.  The initial  excitation  by the laser  pulse  disturbs  the electronic  subsystem  \nfrom  equilibrium  which  leads  to an imbalance  in spin-up and spin-down  scattering  rates,  resulting  in the \nobserved  loss of magnetic  order.  The process  is mediated  by spin-orbit  interactions  leading  to the \nformation  of hot spots  in the band  structure  where  spin-up and spin-down  channels  are intermixed.  An \nelectron  scattered  into these  hot spots  via a phonon - or impurity -mediated  scatteri ng will flip its spin with  \na finite  probability.  The individual  scattering  events  are characterized  by a parameter asf across  the  \nsample,  identified  with the probability  of spin-flip due to  electron -phonon  scattering.  The magnitude  \nof this parameter  will directly  depend  on the extent of  spin-orbit  coupling and  hence  is expected  to be \ncomparable  in materials with  similar  spin-orbit  coupling  strengths.  The M3TM  retains  the coupled  rate \nequations  for the electron  and lattice  temperatures,  similar  to the thermal description  provided  by the \n3TM.  However  the fundamental  difference  from  the 3TM  is that in the framework  of the M3TM,  the \nspin bath is formed  by a collection  of two-level  systems  obeying  Boltzmann  statistics.  Instead  of \nassigning  a temperature  to the spin bath,  the normalized  magnetization  is directly  calculated  from  the \nassociated  exchange  splitting.  The rate of change  of the magnetization,  derived  analytically \nconsidering an Elliot -Yafet scattering -driven demagnetization, is parametrized by  asf and c oupled to the \nelectron and the lattice subsystem temperatures. The assignment of a  characteristic temperature to the spin \nsubsystem is replaced in the M3TM by an evolution  equation  for the magnetization:   \n 5  \n        (2) \n \nThe quantity  R is a mate rial-specific  factor  which  influences  the demagnetization  rate and is \nproportional  to asfTc2/μat where  TC is the material Curie temperature and µat is the atomic magnetic \nmoment. Though the two models  differ in their approaches, one can immediately discern certain \nsimilarities in their domains of  validity.  Both models are appropriate only when th e nonlocal \nmechanisms driving ultrafast  demagnetization such as superdiffusive spin transport can be neglected. \nWe note here that it  has been  reported  that spin transport  is not a major  contributor  to the ultrafast  \ndemagnetization in transition metals [34] . We nevertheless use an insulating SiO 2-coated Si  substrate \nfor our samples to minimize spin transport effects such that analysis with the local  models described \nabove should suffice in our case. Any additional contributions arising from  the gold capping  layer  \nwould  be uniform  across  all samples  investigated  and therefore  unlikely  to impact the main results of our \ncomparative study.  Moreover, since the thickness of the  capping layer is much smaller than the \npenetration depth of both 400 nm and 800 nm light  in gold [35], the pump excitation fully penetrates \ndown to the magnetic layer ensuring that the  effect of direct laser excitation of the ferromagnet is \nprobed in our case.  Thus, we set up  numerical calculations based on the models described above in \norder  to extract microscopic  information  from  the experimentally  obtained  demagnetization  traces.  \n 3.2 Ultrafast  demagnetization  in cobalt,  nickel,  and permalloy  thin films  \nWe proceed by performing time -resolved measurements of the polar magneto -optical Kerr  effect in \nthe cobalt, nickel and permalloy thin film samples as a function of the laser fluence.  Measurements are \ncarried out under a strong enough external magnetic field kept constant at  around 2 kOe tilted at a small \nangle from the sample plane to saturate  the magnetization of  the samples. The pump fluence is varied \nbetween 0.8 -8.7 mJ/cm2 by varying the power of the  pump pulse.   The results are presented in Figure 2.   \nTo ascertain that the measured Kerr  signal  reflects  the true magnetization  dynamics  without any \nspurious  contribution  from  optical effects triggered in the initial stages of laser excitation, we also \nexamine the transient  reflectivity signal for each sample. The fluence dependent variation in the \nreflectivity can be  found in Figure S1 of the  Supplementary Materials, demonstrating that at any given \nfluence  the amplitude  of the reflectivity  signal  is negligibly  small  compared  to that of the  Kerr  rotation.  \nWe nevertheless restrict ourselves to the low fluence regime to avoid nonlinear  effects a nd sample \ndamage.  For all our experiments, the probe fluence is kept constant at a  value about half that of the \nlowest pump fluence used to prevent additional contribution to  the spin dynamics by probe excitation.  \nAs seen in Figure 2, the ultrafast demagne tization  completes within 1 ps for all three samples \nconsidered which is followed by a fast recovery of  the magnetization,  all observed  within  the \nexperimental  time window  of 4 ps. These  experimental traces clearly exhibit the “Type -I” or “one -step” \ndemagn etization expected for  transition metal thin films at room temperature and under low -to-\nmoderate pump fluence  [23]. The amplitude of the maximum quenching of the Kerr rotation signal \nincreases with the  laser  fluence,  allowing  us to rule out nonlinear  effec ts [36].  Closer  inspection  of the \ntraces  also reveals an increase of the time taken to demagnetize the samples with increasing fluence for  \nall three  samples.  To quantify  this increase,  we fit our demagnetization  traces  to a phenomenological  \nexpression  based on the 3TM  and valid  in the low laser  fluence  regime  [37]:   \n \n(3) \n \nwhere  Θ(t) is the Heaviside  step function,  δ(t) is the Dirac  delta  function  and Γ(t) is the \nGaussian laser pulse.  The constant A1 represents the value of the normalized  magnetization after \nremagnetization has completed and equilibrium between the electron,  spin and lattice  reservoirs  has \nbeen  re-established.  A2 is proportional  to the initial  rise in electron  temperature  and hence  to the \nmaximum  magnetization  quenching.  A3 represents  the magnitude of state -filling effects present during \nthe onset of the demagnetiz ation response,  which is negligible in our case.  τM and τE are the \ndemagnetization time and fast relaxation  time,  respectively.  Prior  to the fitting,  all the experimental  \ntraces  were  normalized  by hysteresis measurements of the Kerr rotation signal under the saturating \nmagnetic field in the  absence of la ser excitation.  We find that within the range of fluence values \nconsidered,  permalloy  exhibits  the largest  magnetization  quenching  of 54.6%,  followed  by a 23.7%  \nquenching achieved in nickel, while the magnetization of cobalt the least, only about 8% for  the largest  \napplied  fluence.  The demagnetization  occurs  at a characteristic  timescale  of 230-280 fs for cobalt, 160 -6  210 fs for nickel, and 220 -250 fs for permalloy, increasing with the  laser fluence.  This effect can be \nattributed to enhanced spin -fluctuation s at elevated spin  temperatures  for  higher  fluences  [38].      \nAt  a  fluence  of  4.8  mJ/cm2,   the  extracted  demagnetization  times  are 276.6  ± 3.41 fs for cobalt,  \n187.3  ± 2.89 fs for nickel  and 236.8  ± 2.45 fs for permalloy.  The timescale  for the magnetization  \nrecovery  τE also increases  with increasing pump fluence. The variation of these characteristic \ntimescales with laser fluence is  shown  in Figure  3. These  fluence -dependent  trends  in τM and τE hint at a \nspin-flip process -dominated ultrafast demagnetization in our stu died systems [23, 39, 40]. The values  of \nτM extracted  from  our experiments  lie within  the typical  range  of 100-300 fs consistent  with \nprevious reports of the ultrafast demagnetization times in these metals [17, 23], and are  too large  to \nrepresent  a superdi ffusive  transport -driven  demagnetization  [41].  \n For the 3TM and M3TM simulations, we choose a laser pump term given by \nproportional to  the pump fluence F and following a Gaussian temporal profile.  \nThe maximum rise of the  electron temperature a nd thus also the extent of demagnetization depends \nsensitively on this  term which is hence adjusted to reproduce the maximum quenching observed \nexperimentally.  We use a pulse width τp = 100 fs determined by the pump -probe cross -correlation in all  \ncalculations.  Intrinsic  to both the models  we consider  is the assumption  that electron  thermalization occurs \nextremely fast.  The thermal  diffusion term  can be neglected in our case since th e thicknesses of the films \nwe study are kept slightly greater than the optical penetration depth of 400 nm pump beam in  those films.  \nThis ensures uniform heating of the films in the vertical direction while also  avoiding laser penetration \ninto the substrat e in which case heat dissipation into the substrate  would have to be taken into account. \nBesides, the timescales associated with heat dissipation  are generally  tens to hundreds  of picoseconds,  \nmuch  longer  than the demagnetization  and fast relaxation  times,  and hence  unlikely  to significantly  \ninfluence  our observations  at these  timescales.  Since both models we consider are thermal in their \napproach, choosing correct  values for the reservoir specific heats is vital for a proper simulation of the \ndemagnetizati on. For the electronic  specific  heat Ce, we assume  a linear  dependence  on the electronic  \ntemperature Ce(Te) = γTe  derived from the Sommerfeld free -electron approximation where γ is determined \nby the electronic density of states at the Fermi level [42].  The value of γ for permalloy is approximated as \na weighted average of the individual γ values of nickel and iron  in permalloy. The lattice specific heat Cl is \ncalculated at each value of the lattice temperature  according to the following relation derived from Debye \ntheory:  \n(4) \n \nwhere NA is the  Avogadro’s  number,  kB is the Boltzmann’s  constant  and θD  is  the  Debye  temperature.  \nFinally,  we fix the spin specific  heat Cs to its value  at room  temperature  for the 3TM  calculations,  \nobtained  by subtracting  the electronic  and lattice  contributions  from  the experimental values of the total \nspecific heat found in the literature [43].  Considering a  spin-temperature -dependent form of Cs was not \nfound to significantly affect our conclusions  as described in Section IV of the Supplementary Materials.  \nThe fixed parameter set used in  our calculations have been listed in Table 1. To relate the experimental \ndemagnetization to  the temperature  of the spin subsystem  under  the 3TM  framework,  the spin \ntemperature  Ts is  mapped to the magnetization of the system via the Weiss’ mean field theory [44], \nwhich is  then fitted with the experimental magnetization traces to obtain the empirical inter -reservoir  \ncoupling parameters Gel and Ges consistent with the observed dynamics.  We neglect the  spin-lattice \ncoupling parameter Gsl for the 3TM simulations since in ferromagnetic transitio n metals the energy \ntransfer between electrons and lattice is far greater than that between  lattice  and spins  [45].  \n \n Table 1 . Fixed parameter set used in the calculations. Literature values have been used for all parameters listed   \n[30, 31, 34, 35].  \n \n For the 3TM simulations, we proceeded to extract Gel and Ges by first fitting the  demagnetization \ndata at the lowe st fluence to the model. However, fitting the higher fluence  data using identical values \nof the coupling parameters as extracted at the lowest fluence did  not result in a good match to our \nexperimental results.  The coupling parameters extracted  from the lo w fluence data led to an \noverestimation of the demagnetization time at the higher  fluences. It was seen that a 5 -10% increase in Sampl e Tc (K) θD (K) γ (Jm-3K-2) Cs (Jm-3K-1) at (B) \nNi 627  450  1065  3.07 × 105 0.62  \nCo 1388  445  714  1.59 × 105 1.72  \nPy 860  454  992  2.67 × 105 1.00  7  Ges from its value at its adjacent lower fluence  value rectifies the overestimation of the demagnetization \ntime.  On the other h and, the remagnetization dynamics is most sensitive to the Gel parameter so that the \noverall dynamics  is best reproduced  only  by adjusting  both  Gel and Ges. As shown  in Figure  4, the \nresulting  fit shows excellent agreement with the experimental data.  This exercise reveals the crucial role  \nplayed by the electron -spin relaxation channels in determining the timescale associated with  the initial \ndemagnetization while the magnetization recovery is primarily mediated by the  electron -lattice \ninteraction.  We also f ind that the mismatch between model and experiment  can be resolved by \nconsidering an increasing trend of Gel and Ges with pump fluence arising  from a faster demagnetization \nprocess for the same percentage quenching as compared to the  model predictions with in the studied \nfluence range. The values of the microscopic parameters  extracted from the least -squares fits with their \ncorresponding error bounds can be found in  Supplementary Tables S1 -S3. Since the exact values of the \ncoupling parameters extracted  from  the fits naturally  depend  on the values  chosen  for the fixed  \nparameters,  the interpretation  of the results  from  these  fits is best limited  to a comparative  one. \nFor the M3TM  simulations,  the demagnetization  traces  are fitted  directly  to Equation  2 yielding  \nGel and asf as fit parameters.  In this case,  asf plays  a role  in  determining  the  maximum extent and the \nassociated timescale of the demagnetization via the scaling factor R while Gel continues to influence \nmainly the magnetization recovery process.  How ever, the  demagnetization  time is less sensitive  to \nchanges  in asf than it is to Ges in the 3TM  case.  This results in somewhat higher values of Gel and a \nsharper rise with pump fluence than those  extracted  from  the 3TM  simulations,  in order  to compensate  \nfor the overestimation  of demagnetization time that results from the model if Gel  at the lowest fluence is \nused for all  the fits.  We have obtained an asf of ~0.02 for cobalt,  ~0.05-0.06 for nickel, and  ~0.03-0.06 \nfor permalloy.  The value of asf we have ext racted for nickel is an order of  magnitude lower than the value \nasf = 0.185 first reported by Koopmans et al. [23] but quite  close to the value of 0.08 reported by Roth et \nal. [39].  This discrepancy is expected, as the  artificially high value of 0.185 aros e due to an \noverestimation of the electronic specific heat in  the original  work,  avoided  here by considering  \nexperimentally  determined  γ values  reported  in the literature.  The observation  that asf [Co] <asf [Ni] is \nconsistent  with previous  reports  [23, 40]. \nWith  regard  to thermal  treatments  of ultrafast  demagnetization  like the 3TM  and M3TM,  there is \nsome contention centered on the role o f the nonthermal electrons generated within the  first 50-100 \nfemtoseconds  after laser excitation  [9, 48, 49]. Within  this time frame,  the excited  hot electrons  have not \nyet thermalized  and their energies  cannot  be described  by a Fermi -Dirac  distribution.  However,  both \nthe 3TM  and the M3TM  completely  neglect  the finite  electron  thermalization  time and consider  only the \nenergy  exchange  between  the thermalized  electron  population  with the lattice  and/or  the spin subsystem  as the \ninitial  nonequilibrium  distribu tion is assumed  to last for much  shorter  durations  (≳ 10 fs) than the \ntimescale  over which  demagnetization  typically  occurs  (≳ 100 fs). However,  strictly  speaking,  this \nassumption  is valid  only when  the thermalization  proceeds  very rapidly  such as can be ensured with  high \nfluence  excitations  [50].  In the low  fluence  regime,  electron thermalization  is slower,  hence  the influence  \nof nonthermal  electrons  on the measured  dynamics  may become  more  important  with decreasing  pump  \nfluence.  The increasing  of Gel with fluence  deduced  from our calculations may be an indication of a \ndecrease in the efficiency of electron -lattice  interactions  at low laser  fluences  when  nonthermal  electrons  \nprevail.  However,  the specific  values of Gel and Ges values we have obtained for th e different systems \nremain comparable to  values  previously  reported  for these  metals  [11, 23, 37]. \nAfter completion of the fitting routine,  we set up simulations using the optimized values of  the free \nparameters to obtain the temporal evolution of Te, Tl and Ts as a function of the  pump -probe delay.   The \ncalculated profiles of Te, Tl and Ts for four different values of the  pump fluence are presented in Figure \n5 for nickel, showing good agreement between the  results predicted by the 3TM and the M3TM. Simil ar \nprofiles for cobalt and permalloy are  given in Figs.  S2 and S3 of the Supplementary Materials.  As the \npump fluence is increased,  laser excitation deposits a greater amount of energy in the electronic \nreservoir which leads to  a proportionate increase in the initial electron temperature rise.  However, the \nmaximum  electron temperature reached for the highest applied pump fluence remains well below the  \nFermi  temperature  for the respective  material,  ensuring  the assumption  of the linear  temperature  \ndependence  of the electronic  specific  heat remains  valid  [42].  Given  the fundamentally different origins \nof the 3TM and M3TM, the close agreement between the  temperature profiles calculated using the two \nmodels and among the extracted values of the  electron -lattice coupling parameter Gel is remarkable. \nThis observation reflects that, given the  strength of the laser excitation determining the initial \ntemperature excursion of the electron  system,  the Gel  parameter common to both the 3TM and the \nM3TM framework plays a  pivotal  role in determining  the profile  of the ensuing  magnetization  changes.  8  This  also  explains why Gel is seen to follow a similar fluence -dependent trend for a given system in \nboth models.  While the electron temperature is coupled to the spin temperat ure via the Ges parameter  in \nthe 3TM,  it enters  into the evolution  equation  for the magnetization  via  Equation 2 in the M3TM case. \nFrom Figure 5, a qualitative difference in the temporal profiles  of the electron,  spin and lattice \ntemperatures is also appa rent.  Referring to the figures  corresponding  to the 3TM,  while  Te exhibits  a \nsharp  peak  within  the first few hundred  femtoseconds, Ts exhibits a shorter and broader peak at slightly \nlonger time delays.  The delayed peak in Ts occurs when excited electrons t ransfer energy to the spin \ndegrees of  freedom  [51] through  scattering  events  mediated  by the coupling  parameter  Ges, which  \nqualitatively determines the maximum spin temperature value [11].  These scattering events  can result in \nthe generation of magnons, St oner excitations and spin fluctuations which may  play a part in the \nobserved demagnetization [52].  Lastly, the lattice temperature gradually  increases over the first few \npicoseconds after excitation, reaching an equilibrium value at  which  it stays  nearly  constant  over \nseveral  picoseconds.  \nIn Figure  6 we present  a comparison  between  the experimental  demagnetization  curves  and calculated \nelectron temperature profiles for each of our three samples. Under the same applied  pump  fluence,  different  \namplitudes  of magnetization  quenching  and electron  temperature  rise are observed. It is clearly seen that \nnickel quenches more strongly than cobalt, an observation  which  can be explained  by band  structure  \neffects  and a more  efficient  laser -induced  electronic  excitation i n nickel owing to its much lower Curie \ntemperature [34, 40, 53], reflected in the  M3TM as a higher extracted value of the spin -flip probability  \nasf  in nickel.   On the other  hand, a lower electronic specific heat capacity in cobalt as compared to \nnickel [42] predictably  leads  to a much  greater  rise in its electron  temperature  for the same  pump  fluence.  \nIn the case of permalloy,  alloying  with iron can reduce  the electronic  density  of states  at the Fermi  level  \n as compared to nickel, decreasing γ. Because o f the larger electron temperatures reached as a  result, a \nvalue of asf comparable to that of nickel is sufficient to drive a much larger reduction  in the \nmagnetization.  Permalloy also shows a somewhat larger demagnetization time at a  given pump fluence \nas compared to nickel.  Since a standard TR -MOKE optical pump -probe  measurement  cannot  be used  to \nprobe  the demagnetization  response  in alloys  in an element -specific manner, interpreting this finding is \ndifficult. It has been reported previously  using element -specific measurements in a nickel -iron alloy that \nthe demagnetization of nickel  proceeds  faster  than that of iron [54].  Our observation  of a slower  \ndemagnetization  in permalloy may be reflective of this result.  Finally, the demagnetization times for the \nthree systems  at any given  pump  fluence  is seen to sensitively  depend  on the ratio  of TC/µat, \nconfirming  its role as a ‘figure  of merit’  (FOM)  for the demagnetization  time [23].  As shown  in \nFigure 8 ( a), the demagnetization proceeds slower for a lower value  of the FOM. Thus at a  given \nincident laser pump fluence, the demagnetization times are found to correlate with the  values  of this \nquantifying  parameter,  rather  than the material  Curie  temperature  directly.  \n \n 3.3 Precessional  dynamics  and correlation  of Gilbert damping  parameter  with ultrafast  demagnetization  \ntime \nTo gain further  insight  into the microscopic  mechanism  underlying  the ultrafast  \ndemagnetization,  we study also the precessional magnetization dynamics in our thin film  systems. A \ntypical time -resol ved Kerr rotation data showing the different temporal regimes of  laser -induced \nmagnetization dynamics is given in Figure 1( d). Regime I and II represent the  ultrafast  demagnetization  \nand magnetization  recovery  processes  respectively,  while  the oscillatory  signal  of regime  III represents  \nthe magnetization  precession  with the Gilbert  damping  characteristic  of the ferromagnetic  material.  To \nextract  the Gilbert  damping  factor  α, the background -subtracted oscillatory Kerr signal is first fitted \nwith a damped sinusoidal  function  given  by:  \n \n  \n (5) \n   \nwhere  f is the precessional frequency, τ is the relaxation time, and φ is the initial phase of oscillation. \nThe depende nce of f on applied bias magnetic field H is fitted to the Kittel formula relevant for \nferromagnetic systems:  \n \n \n (6) \n \nwhere  γ0 = gµB/ℏ is  the  gyromagnetic  ratio  and  Meff  is the effective  saturation  magnetization  \nof the probed  volume.  From the Kittel  fit, the values  of g and Meff can be extracted  as fit \nparameters.  While the value  of g is found  to be 2 ± 0.1 for all the samples,  the extracted  values  9  of Meff are 1280.2  ± 1.69 emu/cm3 for cobalt,  400.5  ± 11.46  emu/cm3 for nickel  and 737.1  ± 5.32 \nemu/cm3 for permalloy  (Section  V of the Supplementary  Materials).  \nSubsequently,  the effective  damping  factor  α is calculated  depending  on the extracted  value  of τ and Meff \nas:  \n \n(7) \n \nAlthough  the timescale  of precessional  dynamics  and magnetic  damping  in ferromagnetic  thin films differ  \nby many orders of magnitude,  similar underlying  phenomena may oft en determine  the characteristics of \neither process.  Through several independent investigations relating these  two distinct processes, the \ncorrelation between the demagnetization time τM and the Gilbert  damping  factor  α emerged  as a critical  \nindicator  of the dominant  microscopic  contribution  to the ultrafast  demagnetization.  Koopmans  et al. in \n2005  analytically  derived  an inversely  proportional  relationship  between  the Gilbert  damping  factor  and \nultrafast  demagnetization  time based  on quantum  mechanical  considerations  [16].  However,  the validity  of \nthe model  could  not be confirmed  experimentally  when  τM and α were  tuned  by transition  metal  and rare \nearth  doping  [18].  Later  it was proposed  that the Gilbert  damping  parameter  and ultrafast  \ndemagnetization  time can be either  directly  or inversely  proportional  to each other  depending on the \npredominant micr oscopic mechanism contributing to the magnetic damping  [55].  A proportional  dependence  \nwould  suggest  a dominating  conductivity -like or “breathing  Fermi  surface”  contribution  to the damping  \ndue to the scattering  of intraband  electrons  and holes,  while  an inverse  dependence  may arise in materials  \nwith a dominant  resistivity -like damping  arising  from  inter-band  scattering  processes.  Zhang  et al. later \nsuggested  that an inverse  correlation  between  Gilbert  damping  factor  and ultrafast  demagnetization  time \nmay arise in the presence  of spin transport,  which  provides  an additional  relaxation  channel  resulting  in an \nenhancement  of the magnetic  damping  along  with an acceleration  of the demagnetization  process  at the \nfemtosecond  timescale  [56].  In cases  where  local  spin-flip scattering  processes  are expected  to dominate  the \ndemagnetization  process,  the breathing  Fermi  surface  model  is valid  predicting  a proportional  relationship  \nbetween  τM and α. \nTo study  the correlation  between  ultrafast  demagnetization  and damping  in a system,  both τM and α \nhave to be systematically  varied.  For a given  material,  the exciting  pump  fluence  can be used to \nmodulate the demagnetization time.  As discussed in the previous section, we  observe  an increment  of the \ndemagnetization  time with the pump  fluence  for the systems  under  investigation.  An enhancement  in α \nis also observed  as the pump  fluence  is increased  from  0.8 mJ/cm2 to 8.7 mJ/cm2 for each sample,  \nassoci ated with a corresponding  decrease  in the precessional  relaxation  time.  Within  the experimental  \nfluence  range,  α is found  to be enhanced from 0.0089 to 0.0099 for cobalt, from 0.0346 to 0.0379 for nickel \nand from 0.0106 to  0.0119  for permalloy.  This enhanc ement  of the damping  can be attributed  to the \ntransient  rise in the system temperature as the laser pump fluence is increased [57].  As a laser -induced  \nincrease in electron temperature also drives the demagnetization process at the femtosecond  timescale,  one \ncan qualitatively  correlate  the demagnetization  time with the magnetic  damping at various excitation \nenergies even for a single sample.  Clearly,  a direct correlation  between demagnetization time and the Gilbert \ndamping factor is obtained in our samples b y measuring the magnetization dynamics at various pump \nfluences, as shown in Figure 7 for the  nickel thin film.  Thus, we infer that similar relaxation processes likely \ndrive the laser induced  transient  magnetization  dynamics  at the two different  timescales .   At a low \nfluence  value  of 2.1 mJ/cm2, the extracted  value  of α is the highest  for the nickel  film,  and relatively  \nlow for cobalt  and permalloy.  The observed  trend  in Figure  8 (b) can be directly  correlated  with the \nvalues  of the Sommerfeld  coefficient  γ given  in Table  1. This observation  is in accordance  with \nKamber sky’s  spin flip scattering  theory  [58],  wherein  α is directly  proportional  to the density  of states  at \nthe Fermi  level,  which  in turn governs  the value  of γ. \n \n \n4. Summary  \nTo summarize,  we have investigated  fluence -dependent  ultrafast  demagnetization  in identi cally -grown  \ncobalt,  nickel  and permalloy  thin films  using  an all-optical  time-resolved  magneto -optical Kerr effect \ntechnique.  We study both the femtosecond laser -induced ultrafast  demagnetization  and the magnetization  \nprecession  and damping  in each of our samples  for various  pump  excitation  fluences.  Using  a \nphenomenological  expression,  we extract  the values  of demagnetization  time and fast relaxation  time 10  for each of our samples  to show  that they exhibit  an increasing  trend  with applied  laser fluence  consi stent  \nwith a spin-flip scattering -dominated  demagnetization  in our samples.  Two distinct  ultrafast  demagnetization  \nmodels  can each reproduce  our experimental  demagnetization  traces  remarkably  well.  By directly  fitting  \nour experimental  data to numerical  solutions  of these  models,  we have extracted  values  for the electron -\nlattice  and electron -spin coupling  parameters  and an empirical  estimate  of the spin flip scattering  \nprobability  for each of our samples  and demonstrated  the fluence -dependent  variation  in calculated  \nelectron,  spin,  and lattice  temperatures  as a function  of the pump -probe  delay.  We conjecture  that the \nincreasing  trend  of the extracted  electron -lattice  coupling  parameter  with fluence  may indicate  a decrease  \nin the efficiency  of electron -lattice  interactions  in the low laser fluences  when  nonthermal  electrons may \nplay a role in influencing the magnetization dynamics.  We have compared the  experimental  and theoretical  \nresults  obtained  from  each of the systems  under  investigation  and confirmed  that the ratio of the Curie  \ntemperature  to the atomic  magnetic  moment  acts as a figure  of merit  sensitive  to the demagnetization  time \nfor these  systems.  Our study  demonstrates the mutual agreement between the phenomenological three \ntemperature model  and the microscopic  three  temperature  model  for the systems  we study  and highlights  the \npossibility  of a fluence -dependent  enhancement  in the inter-reservoir  interactions  in the low laser  fluence  \nregime.  Finally,  we have reported  from  our experiments  a direct  correla tion between  the ultrafast  \ndemagnetization  time and the enhancement  of the magnetic  damping  that transiently appears as a result of \nthe laser excitation, reflecting that both phenomena are  sensitive  to similar  underlying  microscopic  \nprocesses.  \n \n5. Acknowledge ment  \nAB gratefully acknowledges the financial support from the S. N. Bose National Centre for  Basic \nSciences (SNBNCBS) under Project No.  SNB/AB/18 -19/211 and Department of Science  and \nTechnology  (DST),  Govt.  of  India  under  Grant  No.  DST/NM/TUE/QM -3/2019-1C-SNB.  SM \nacknowledges  DST,  India  for financial  support  from  INSPIRE  fellowship,  while  SM and SNP  \nacknowledge  SNBNCBS  for senior  research  fellowship.  \n \nReferences  \n[1] C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, T. Rasing,  All-\noptical  magnetic  recording  with  circularly  polarized  light,  Phys.  Rev.  Lett.  99 (2007)  047601.  \n[2] T. A. Ostler,  J. Barker,  R. F. L. Evans,  R. W. Chantrell,  U. Atxitia,  O. Chubykalo - \nFesenko,  S. El Moussaoui,  L. Le Guyader,  E. Mengotti,  L. J. Heyderman,  F. Nolting,  \nA. Tsukamoto,  A. Itoh,  D. Afanasiev,  B. A. Ivanov,  A. M. Kalashnikova,  K. Vahaplar,  \nJ. Mentink,  A. Kirilyuk,  T. 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The downward arrows represent \nincreased pump fluence.  The symbols  represent  experimental  data points  and the solid  lines  represent  \nfit to Equation  3. 16  \n \n \nFigure 3:  Variation of ( a) demagnetization time τM and ( b) fast relaxation time τE with the pump fluence \nfor 20-nm-thick cobalt, nickel and permalloy films. Both timescales show an increasing trend with \nincreasing pump  fluence.  \n \n \n \n \nFigure 4:  Sensitivity of the demagnetization traces to the various adju stable fit parameters.  \nSymbols  represent  data points  and solid  lines  represent  fits to the 3TM.  Data  shown  is for the \ncobalt  thin film.  For F =0.8  mJ/cm2, Gel=8.2  × 1017 Js−1m−3K−1 and Ges=6.9  × 1016 Js−1m−3K−1; \nfor F =2.1  mJ/cm2, Gel=1.1  × 1018 Js−1m−3K−1 and Ges=8 × 1017 Js−1m−3K−1. \n17   \n \n \nFigure  5: Temporal  evolution  of the reservoir  temperatures  of nickel  Te (blue),  Tl (red),  and Ts (green)  \ncalculated  using  the (a) phenomenological  three  temperature  model  and (b) the microscopic  three  \ntemperature  model.  \n \n \n \n \n \n \n \n \nFigure  6: Comparison  between  (a) the experimental  demagnetization  and (b) the simulated  electron  \ntemperature  profiles  for cobalt,  nickel  and permalloy  at a pump  fluence  of 4.8 mJ/cm2. \n18   \n \n \nFigure  7: (a) Fluence -dependent  variation  of precessional  dynamics  for the nickel  thin film.  \nSymbols  are experimental data points and the solid lines are fits to Equation 5.  (b) Variation of f \nwith pump fluence.  (c) Variation  of α with pump  fluence.  (d) Direct  correlation  between  τM and \nα. \n \n \n \n \n \n \n \n \nFigure 8:  (a) Inverse variation of the demagnetization time with the ratio of Curie temperature to \natomic  magnetic moment and ( b) inverse correlation of the demagnetization time with ma gnetic \ndamping across three  different  materials.  \n19   \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nSupplementary  Materials  S1  I. Transient  reflectivity  measurements  \nTime -resolved  transient  reflectivity  signal  is recorded  by TR-MOKE  measurements  \nsimultaneously with the Kerr rotation signal  quantifying the magnetic response.At any  given \nfluence, the maximum amplitude of the transient reflectivity change is about a  factor of 100 \nsmaller as compared to that of the Kerr rotation.  This validates that the  Kerr rotation signal \nreflects the genuine  magnetic response with negligible nonmagnetic  contribution.  \n \n \nFigure S1:  Time -resolved transient reflectivity measured by TR -MOKE for cobalt, nickel, and \npermalloy.  The upward  arrows  represent  increased  pump  fluence.  \n \n \n \nII. Calculation of electron, spin, an d lattice temperature profiles  \nThe calculated  temporal  evolution  of the reservoir  temperatures  for the 20-nm-thick  cobalt  and \npermalloy  thin films  are represented  in Fig. S2 and Fig. S3, respectively.  The electron  \ntemperature  shows  a rapid  and dramatic  increase  within  one picosecond  of the laser excitation,  \nbeyond  which  it decays  to an equilibrium  value  over a few picoseconds.  Over  this period,  the lattice  \nsubsystem  is gradually  heated  above  the equilibrium  temperature  while  the heating  of the spin \nsubsyste m directly  leads  to the demagnetization.  The maximum  electron,  spin and lattice  temperatures  \nincrease  with increasing  laser  fluence.  \nS2  \n \n \nFigure  S2: Electron,  lattice,  and spin temperature  profiles  for the cobalt  thin film calculated  \nusing  (a) the three tem perature model and (b) the microscopic three temperature model for \nvarious laser fluences  (blue:  Te, dashed,  red: Tl and dotted,  green:  Ts). \n \n \nFigure S3: Electron, lattice, and spin temperature profiles for the permalloy thin film calculated \nusing (a)  the three  temperature  model  and (b) the microscopic  three  temperature  model  for various  \nlaser  fluences).  \n \nIII. Extraction  of inter -reservoir  coupling  constants  \nThe least -squares fitting routines set up based on the three temperature model and the  microscopic  \nthree  temperature  model  return  the values  of the inter-reservoir  coupling  \nS3  constants  that best reproduce  the experimental  demagnetization  traces.  The microscopic  three  \ntemperature  model  additionally  returns  the value  of spin-flip probability asf associated with  the \nspin-flip scattering events which directly lead to the  magnetization quenching.  The extracted \nvalues of these microscopic parameters are  given in Tables S1 -S3. Fig. S4 shows a comparison of \nthe values of the electron -lattice  coupling parameter Gel as extracted from the fits to the three \ntemperature model and  the microscopic three temperature model.  For reasons discussed in the \nmain text, the  spin-lattice coupling parameter Gsl is taken to be zero for the three temperature \nmodel  simulations.  Fig. S5 shows  that,  when  Gsl  is  set  equal  to  3  ×  1016 Jm−1s−1m−3K−1 as \nconsidered in Phys.  Rev.  Lett.  76, 4250 (1996), the simulated trace  and calculated electron \ntemperature profile are nearly indistinguishable from the case in  which  Gsl is neglected.  \n \nFigure  S4: Values  of the electron -lattice  coupling  parameter  Gel extracted  from  fitting  to the three  \ntemperature  model  and microscopic  three  temperature  model.  \n \n \n \n \nFigure  S5:  (a)  Results  of three  temperature  modelling  and (b) calculated  reservoir  temperature  \nprofiles  for two values  of Gsl. \n \nPump fluence \n(mJ/cm2) Gel[3TM]  (×1018) \n(Js-1m-3K-1) Gel[M3TM]  (×1018) \n(Js-1m-3K-1) Ges(×1017) \n(Js-1m-3K-1) asf (×10-1) \n0.8 0.82±0.081  0.85±0.097  0.69±0.037  0.19±0.023  \n2.1 1.08±0.063  1.18±0.105  0.8±0.029  0.16±0.014  \n3.5 1.26±0.057  1.48±0.106  0.91±0.028  0.16±0.009  \n4.8 1.2±0.037  1.57±0.096  0.97±0.015  0.17±0.008  \n6.0 1.21±0.042  1.49±0.072  1.02±0.022  0.17±0.008  \n7.3 1.56±0.076  1.98±0.087  1.15±0.041  0.16±0.004  \n8.7 1.65±0.045  2.18±0.066  1.28±0.036  0.17±0.003  \nTabl e S1:  Microscopic parameters extracted from the 3TM and M3TM s imulations for cobalt.  \n \n \n \nS4   \nPump fluence \n(mJ/cm2) Gel[3TM]  (×1018) \n(Js-1m-3K-1) Gel[M3TM]  (×1018) \n(Js-1m-3K-1) Ges(×1017) \n(Js-1m-3K-1) asf (×10-1) \n0.8  1.29±0.154  1.41±0.172  2.1±0.182  0.53±0.03  \n2.1 1.52±0.061  1.74±0.077  2.84±0.087  0.55±0.012  \n3.5 1.53±0.0 51 1.77±0.069  2.93±0.076  0.56±0.011  \n4.8 1.65±0.047  1.9±0.06  2.86±0.064  0.51±0.008  \n6.0 1.49±0.036  1.73±0.039  2.74±0.048  0.57±0.007  \n7.3 1.58±0.028  1.88±0.033  2.87±0.038  0.59±0.006  \n8.7 1.58±0.048  1.77±0.039  2.77±0.062  0.62±0.008  \nTabl e S2:  Microscopic pa rameters extracted from the 3TM and M3TM simulations for nickel . \n \n \nPump fluence \n(mJ/cm2) Gel[3TM]  (×1018) \n(Js-1m-3K-1) Gel[M3TM]  (×1018) \n(Js-1m-3K-1) Ges(×1017) \n(Js-1m-3K-1) asf (×10-1) \n0.8 1.62±0.144  1.86±0.184  2.16±0.145  0.31±0.016  \n2.1 1.79±0.074  2.14± 0.084  2.34±0.073  0.34±0.007  \n3.5 2.05±0.081  2.46±0.065  2.65±0.084  0.39±0.005  \n4.8 2.08±0.068  2.55±0.038  2.81±0.071  0.45±0.006  \n6.0 2.02±0.053  2.5±0.065  2.72±0.058  0.49±0.007  \n7.3 2.07±0.083  2.67±0.052  2.87±0.089  0.56±0.009  \n8.7 2.11±0.084  2.71±0.069  2.87±0 .086 0.64±0.01  \nTabl e S1:  Microscopic parameters extracted from the 3TM and M3TM simulations for permalloy . \n \nIV. Temperature dependence of spin specific heat  \nConsidering the theory of Manchon et al. ( Phys.  Rev.  B 85, 064408 (2012)) as adopted  in Sci. \nRep. 10:6355 (2020), the specific heat Cs of the spin subsystem in the 3TM  framework  can be \ncalculated  as a function  of the spin temperature  Ts as: \n \n      (S1) \n \nFig. S6 below  shows  the trends  in the inter-reservoir  coupling  parameters  for two scenar ios: \nconsidering the  room temperature  value  of Cs(T0) or an explicit temperature  (and hence  time)  \ndependence  of Cs. \n \n \nFigure  S6: Fluence -dependent  variation  of (a) Gel and (b) Ges considering  two different  forms  \nfor the spin specific  heat Cs. \n \n \n \nS5  ∼ V. Bias fiel d dependence of precessional frequency and Gilbert damping  \nThe bias  magnetic  field-dependent variation of precessional dynamics in cobalt,  nickel and permalloy  \nis shown in Fig.  S7 - Fig. S9 for a pump fluence of 4.8 mJ/cm2. The oscillatory Kerr  signal is \nfitted with Eq. 5 of the main text to extract the precessional frequency f and relaxation time τ.   \nThe bias field dependence of the extracted frequency,  shown in Fig.  S7(b) for cobalt, is fitted to the \nKittel formula (Eq. 6) to extract the effective saturation  magnetization  Meff . Once  Meff have  been  \nderived,  the Gilbert  damping  factor  α is calcula ted using Eq. 7 of the main text. The extracted α \ndoes not show any systematic  variation  with  bias field.  \n \n \n \nFigure  S7: (a) Bias field -dependent  variation  of precessional  dynamics  for the cobalt  thin film.  \nSymbols  are experimental  data points  and the solid lines  are fits to Eq. 5 of the main  text. (b) \nVariation  of f with H. The solid  red line is a fit to the Kittel  formula.  (c) Variation  of τ with \nH. (d) Variation  of α with H. The solid  line is a guide  to the eye. \n \nVI. Pump fluence dependence of precessional frequency and Gilbert \ndamping  \nThe variation of precessional dynamics in cobalt, nickel and permalloy with the laser  pump \nfluence is shown in Fig. S10 for the cobalt and in Fig. S11 for the permalloy thin  film at a \nsaturating  field  of   2 kOe.  The oscilla tory Kerr  signal  is fitted  with Eq.  5 of the main  text to \nextract  the precessional  frequency  f and relaxation  time τ. With  increasing  pump  fluence,  the \nprecessional  frequency  decreases  and the magnetic  damping  is enhanced.  A direct  correlation  \nbetween  the demagnetization  time τM and α is observed.  S6  \n \n \nFigure  S8: (a) Bias field -dependent  variation  of precessional  dynami cs for the nickel  thin film.  \nSymbols  are experimental  data points  and the solid  lines  are fits to Eq. 5 of the main  text. (b) \nVariation  of f with H. The solid  red line is a fit to the Kittel  formula.  (c) Variation  of τ with \nH. (d) Variation  of α with H. The solid  line is a guide  to the eye. \n \n \nFigure  S9: (a) Bias field -dependent  variation  of precessional  dynamics  for the  permalloy  thin  \nfilm.  Symbols  are experimental  data points  and the solid  lines  are fits to Eq. 5 of the main  text. \n(b) Variation  of f with H. The solid  red line is a fit to the Kittel  formula.  (c) Variation  of τ  \nwith H.  (d) Variation  of α with H. The solid  line is a guide  to the eye. \nS7  \n \n \nFigure S10: (a) Fluence -dependent variation of precessional dynamics for the cobalt thin film. \nSymbo ls are experimental data points and the solid lines are fits to Eq.  5 of the main text.  (b) \nVariation of f with pump  fluence.  (c) Variation  of α with pump  fluence.  (d) Direct  correlation  \nbetween  τM and α. \n \n \n \nFigure S11:  (a) Fluence -dependent variation of precessional dynamics for the permalloy thin film.  \nSymbols  are experimental  data points  and the solid  lines  are fits to Eq. 5 of the main  text. (b) \nVariation  of f with pump  fluence.  (c) Variation  of α with  pump  fluence.  (d) Direct  correlation  \nbetween  τM and α. \n"    },    {        "title": "2105.11193v3.Spin_pumping_of_two_dimensional_electron_gas_with_Rashba_and_Dresselhaus_spin_orbit_interactions.pdf",        "content": "Spin pumping of two-dimensional electron gas with Rashba and Dresselhaus\nspin-orbit interactions\nM. Yama1, M. Tatsuno1, T. Kato1, and M. Matsuo2;3;4;5\n1Institute for Solid State Physics,\nThe University of Tokyo, Kashiwa 277-8581, Japan\n2Kavli Institute for Theoretical Sciences,\nUniversity of Chinese Academy of Sciences,\nBeijing 100190, China\n3CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences,\nBeijing 100190, China\n4Advanced Science Research Center,\nJapan Atomic Energy Agency, Tokai 319-1195, Japan\n5RIKEN Center for Emergent Matter Science (CEMS),\nWako, Saitama 351-0198, Japan\n(Dated: August 10, 2021)\nWe theoretically consider spin pumping in a junction between a ferromagnetic insulator (FI) and\na two-dimensional electron gas (2DEG) in which the Rashba and Dresselhaus spin-orbit interactions\ncoexist. Using second-order perturbation theory, we derive an increase in linewidth in the case of\nan interfacial exchange coupling in a ferromagnetic resonance (FMR) experiment. We clarify how\nthe enhancement of Gilbert damping depends on the resonant frequency and spin orientation of the\nFI. We show that this setup of an FMR experiment can provide information on the spin texture of\n2DEG at the Fermi surface.\nI. INTRODUCTION\nSpin pumping has been studied intensively in the \feld\nof spintronics as a versatile method to generate spin cur-\nrent using magnetization dynamics1,2. Spin pumping is\nused for injecting spins from a ferromagnet into an ad-\njacent material in various heterojunction systems, such\nas ferromagnetic metal/normal metal (NM) junctions3{6\nand ferromagnetic insulator (FI)/NM junctions7. As the\nbackaction of the spin injection, the Gilbert damping of\nthe ferromagnet is modulated. Therefore, spin pumping\ncan be used as a probe of nonequilibrium spin states of\nmaterials in the sense that information on spin transport\ndue to an adjacent material is re\rected in the modulation\nof the Gilbert damping.\nAn attractive strategy is spin pumping of semiconduc-\ntor microstructures because highly developed semicon-\nductor technologies can be utilized8,9. In particular, a\ntwo-dimensional electron gas (2DEG) in a semiconduc-\ntor heterostructure is an easily controlled physical sys-\ntem that has been used in spintronics devices10{13. A\n2DEG system has two types of spin-orbit interaction,\nRashba14,15and Dresselhaus16,17. By combining these\ntwo interactions, we can control the spin polarization of\nconduction electrons in a way that is dependent on the\npropagation direction18. In fact, electron transport re-\n\recting the spin texture on the Fermi surface has been\nobserved through, e.g., the Aharonov-Casher e\u000bect19{24\nand the persistent spin helix state25{31.\nIn our work, we focus on spin pumping of a 2DEG\nin semiconductor heterostructures. It is remarkable that\nthere already exist experimental studies on injection of\nspins into two-dimensional electron systems with theRashba spin-orbit interaction in surfaces32,33, atomic\nlayer materials34,35, transition oxides36,37, and bulk semi-\nconductors38,39. However, to the best of our knowledge,\nspin pumping in a junction system composed of an FI and\na 2DEG in which the Rashba and Dresselhaus spin-orbit\ninteractions coexist has not been studied yet.\nIn this study, we consider a planar junction composed\nof an FI and a 2DEG in a semiconductor heterostructure\n(see Fig. 1). By taking both Rashba and Dresselhaus\nspin-orbit interactions in 2DEG into account, we theoret-\nically show an enhancement in Gilbert damping in the FI\nwithin the second-order perturbation with respect to in-\nterfacial exchange coupling40{45. We investigate how the\nlinewidth increase in a ferromagnetic resonance (FMR)\nexperiment depends on the resonant frequency and the\n2DEGFerromagneticinsulator (FI)Semiconductorheterostructure\nmicrowave\nFIG. 1. Schematic picture of junction composed of a fer-\nromagnetic insulator (FI) and two-dimensional electron gas\n(2DEG) in a semiconductor heterostructure. Spin current is\ninduced at the interface under microwave irradiation, whose\nfrequency is chosen to be near the frequency of the spin pre-\ncession of the FI.arXiv:2105.11193v3  [cond-mat.mes-hall]  7 Aug 20212\nspin orientation of the FI. We show that a spin pump-\ning measurement can directly detect the complex spin\ntexture of a 2DEG with spin-orbit interactions.\nThe rest of this work is organized as follows. In Sec. II,\nwe theoretically show an enhancement in Gilbert damp-\ning due to the exchange interaction with a 2DEG within\nsecond-order perturbation. In Sec. III, we show how the\nenhancement of the Gilbert damping depends on the spin\norientation of the FI. We also discuss the experimental\nrelevance of our result. Finally, we summarize our results\nin Sec. IV. In the subsequent four appendices, we give a\ndetailed derivation of the equations in Sec. II.\nII. FORMULATION\nIn this section, we analytically formulate the damping\nrate of the spin precession in a FMR experiment when\nthe FI is coupled with a 2DEG through an interfacial\ninteraction. First, we introduce the model of the 2DEG\nand the FI in Sec. II A and Sec. II B. Next, we consider\nthe interfacial coupling between the 2DEG and the FI in\nSec. II C. Finally, we calculate the second-order perturba-\ntion with respect to the interfacial coupling in Sec. II D.\nThroughout this paper, we will use the laboratory coor-\ndinates shown in Fig. 1; the xyplane is parallel to the\n2DEG, while the z-axis is perpendicular to the junction\narea.\nA. Two-dimensional electron gas\nWe consider a 2DEG with both Rashba and Dressel-\nhaus interactions. The Hamiltonian for the kinetic en-\nergy and the spin-orbit interactions is given as\nHkin=X\nk(cy\nk\"cy\nk#)^hk\u0010ck\"\nck#\u0011\n; (1)\n^hk=\u0010~2(k2\nx+k2\ny)\n2m\u0003\u0000\u0016\u0011\n^I+\u000b(ky\u001bx\u0000kx\u001by)\n+\f(kx\u001bx\u0000ky\u001by); (2)\nwhere ^hkis a 2\u00022 matrix,\u001ba(a=x;y;z ) are the Pauli\nmatrices, ^Iis an identity matrix, and ck\u001bis the annihila-\ntion operator of conduction electrons with wavenumber\nk= (kx;ky) andz-component of a spin, \u001b(=\";#). The\n\frst term of ^hkdescribes the kinetic energy of an electron\nwith chemical potential \u0016and e\u000bective mass m\u0003. The\nsecond and third terms of ^hkdescribe the Rashba and\nDresselhaus spin-orbit interactions, respectively, and \u000b\nand\fdenote the amplitudes of the respective spin-orbit\ninteractions.\nHereafter, we assume that the Fermi energy is much\nlarger than the other energy scales, such as the tem-\nperature and the amplitude of the spin-orbit interac-\ntions. The energy dispersion is then approximated as\n\u0018k\u0011~2k2=2m\u0003\u0000\u0016'vF(jkj\u0000kF), wherekFand\nvFare the Fermi wavenumber and the Fermi velocity,\nrespectively. We also approximate the spin-orbit in-\nteraction using ( kx;ky)'(kFcos';kFsin'), where'\nis the azimuth angle of the conduction electrons. Ac-\ncordingly, the Hamiltonian matrix can be rewritten as\nFIG. 2. Schematic picture of the e\u000bective magnetic \feld\nhe\u000b(') that acts on the spins of conduction electrons.\n^hk=\u0018k^I\u0000he\u000b\u0001\u001b, where\nhe\u000b'kF(\u0000\u000bsin'\u0000\fcos';\u000bcos'+\fsin';0);(3)\nis the e\u000bective magnetic \feld that acts on the spin of\na conduction electron propagating in the direction of '.\nNote that the e\u000bective magnetic \feld he\u000bcharacterizes\nthe spin texture of conduction electrons as a function of\nthe azimuth angle of the electron wavenumber. Fig. 2 (a)\nand (b) shows pro\fles of the e\u000bective magnetic \feld on\nthe Fermi surface46for\u000b=\f= 1 and 3. The spin texture\nis simpli\fed for the special case of \u000b=\f= 1; the direction\nof the e\u000bective \feld is always parallel to the straight line\nky=\u0000kx.\nBy diagonalizing ^hk, we obtain the spin-dependent elec-\ntron dispersion,\nE\u0006\nk=\u0018k\u0006he\u000b('); (4)\nhe\u000b(')\u0011jhe\u000b(')j'kFp\n\u000b2+\f2+ 2\u000b\fsin 2': (5)\nNote that 2 he\u000b(') corresponds to the spin-splitting en-\nergy for a conduction electron propagating at the azimuth\nangle'. When\u000b=\f(\u000b=\u0000\f), the spin-splitting en-\nergy vanishes at '= 3\u0019=4 ('=\u0019=4).\nGreen's function at \fnite temperature is de\fned as a\n2\u00022 matrix ^g(k;\u001c), whose elements are\ng\u001b\u001b0(k;\u001c) =\u0000~\u00001hck\u001b(\u001c)cy\nk\u001b0i; (6)\nfor 0< \u001c < ~\f, whereck\u001b(\u001c) =eHkin\u001c=~ck\u001be\u0000Hkin\u001c=~\nis the Heisenberg representation for the imaginary time\nevolution and \fis inverse temperature. Note that there\nare o\u000b-diagonal components in the Green's function that\nare due to spin-\rip processes by the spin-orbit interac-\ntions. The Fourier transform of the Green's function is\nde\fned as\n^g(k;i!n) =Z~\f\n0d\u001cei!n\u001c^g(k;\u001c); (7)\nwhere!n=\u0019(2n+ 1)=~\fare the Matsubara frequen-\ncies for fermions. For the Hamiltonian Hkin, the Green's\nfunction is calculated as\n^g(k;i!n) =(i~!n\u0000\u0018k)^I\u0000he\u000b\u0001\u001b\n(i~!n\u0000E+\nk)(i~!n\u0000E\u0000\nk): (8)3\nFIG. 3. Relation between laboratory coordinates ( x;y;z ) and\nmagnetization-\fxed coordinates ( x0;y0;z0) for FI.\nWe also consider the e\u000bect of impurity scattering in a\n2DEG by the Hamiltonian,\nVimp=uX\ni2impX\n\u001b\ty(Ri)\t(Ri) (9)\n\t(r) =1p\nAX\nkck\u001beik\u0001r; (10)\nwherei2imp indicates the impurity site, uis the\nstrength of the impurity potential, Ais the junction area,\nandRiis the position of the impurity site. The total\nHamiltonian for the 2DEG is thus H2DEG =Hkin+Vimp.\nThrough second-order perturbation with respect to the\nimpurity potential, Green's function is calculated as\n^g(k;i!n) =(i~!n\u0000\u0018k+i\u0000sgn(!n)=2)^I\u0000he\u000b\u0001\u001bQ\n\u0017=\u0006(i~!n\u0000E\u0017\nk+i\u0000sgn(!n)=2);\n(11)\nwhere \u0000 = 2 \u0019niu2D(\u000fF) is level broadening and D(\u000fF) =\nkF=(2\u0019~vF) is the density of states per spin per unit vol-\nume. For a detailed derivation, see Appendix A.\nB. Ferromagnetic insulator\nWe assume that the magnetization of the FI is in the\nxyplane parallel to the 2DEG. We de\fne the direction\nof the in-plane ordered spin measured from the x-axis as\n\u0012. Accordingly, the expectation value of the ordered spin\nin FI is expressed as\nhSii= (hSx\nii;hSy\nii;hSz\nii) = (S0cos\u0012;S0sin\u0012;0);(12)\nwhereiis the index of a localized spin site, and S0\nis the amplitude of the average spin per site. To sim-\nplify the subsequent calculation, we introduce a new co-\nordinate (x0;y0;z0), for which the direction of the or-\ndered spin is \fxed to the x0-axis, as shown in Fig. 3.\nThe average spin is expressed in this new coordinate as\n(hSx0\nii;hSy0\nii;hSz0\nii) = (S0;0;0). The spin operators for\nthe new coordinates are related to the ones for the orig-\ninal coordinates as\nSx0\ni= cos\u0012Sx\ni+ sin\u0012Sy\ni; (13)\nSy0\ni=\u0000sin\u0012Sx\ni+ cos\u0012Sy\ni; (14)\nSz0\ni=Sz\ni: (15)The Hamiltonian of the FI is written in the new coordi-\nnates as\nHFI=X\nhi;jiJij(Sx0\niSx0\nj+Sy0\niSy0\nj+Sz0\niSz0\nj)\n\u0000~\rhdcX\niSx0\ni; (16)\nwherehi;jiindicates a pair of nearest-neighbor sites, Jij\nis the ferromagnetic exchange coupling, \r(<0) is the\ngyromagnetic ratio, and hdcis the static magnetic \feld.\nWe assume hdc<0, for which the direction of the spin\n(the magnetization) becomes + x0(\u0000x0).\nWe further assume that the temperature is much lower\nthan the magnetic transition temperature. As well, we\nassumeS0\u001d1, to which the spin-wave approximation\ncan be applied. The Hamiltonian of the FI can then be\nrewritten as\nHFI=X\nk~!kby\nkbk; (17)\n~!k=Dk2+~\rhdc; (18)\nwherebkis the annihilation operator of a magnon, and\nDis spin sti\u000bness. A detailed derivation is given in Ap-\npendix B.\nWe de\fne the imaginary-time spin correlation function\nof the FI as\nG(k;\u001c) =\u00001\n~hSx0+\nk(\u001c);Sx0\u0000\nk(0)i; (19)\nG(k;i!n) =Z~\f\n0d\u001cG(k;\u001c)ei!n\u001c; (20)\nwhere!n= 2\u0019n=~\fare the Matsubara frequencies for\nbosons. This spin correlation function can be calculated\nwithin the spin-wave approximation as\nG(k;i!n) =2S0=~\ni!n\u0000!k+i\u000bGj!nj: (21)\nHere, we have introduced a phenomenological dimen-\nsionless parameter \u000bGthat describes the strength of the\nGilbert damping.\nC. Interfacial exchange interaction\nWe consider an interfacial exchange interaction be-\ntween the FI and the 2DEG with the Hamiltonian,\nHint=X\nk(TkSx0+\nksx0\u0000\nk+T\u0003\nkSx0\u0000\nksx0+\nk): (22)\nwhereTkis an exchange interaction at the interface, and\nSx0\u0006\nk=Sy0\nk\u0006iSz0\nkare creation and annihilation opera-\ntors of the localized spins in the FI. Here, the interface\nis assumed to be su\u000eciently \rat so that the momentum\nof spin excitation is conserved. The e\u000bective \feld (the\nexchange bias) which acts on the conduction electrons\nvia the interfacial coupling is also assumed to be much\nsmaller than the e\u000bective magnetic \feld he\u000b. The spin4\nladder operators for conduction electrons, sx0\u0006\nk, are de-\n\fned as follows. First, we de\fne the spin operators in\nthe magnetization-\fxed coordinates as\nsx0\nk= cos\u0012sx\nk+ sin\u0012sy\nk; (23)\nsy0\nk=\u0000sin\u0012sx\nk+ cos\u0012sy\nk; (24)\nsz0\nk=sz\nk; (25)\nusing the spin operators in the original coordinates de-\n\fned as\nsa\nk=X\n\u001b\u001b0X\nk0cy\nk0\u001b(\u001ba)\u001b\u001b0ck0+k\u001b;(a=x;y;z ):(26)\nHere,\u001baare the Pauli matrices. We de\fne the spin ladder\noperators as\nsx0+\nk\u0011sy0\nk+isz0\nk; (27)\nsx0\u0000\nk\u0011(sx0+\nk)y=sy0\n\u0000k\u0000isz0\n\u0000k: (28)\nUsing Eqs. (23)-(26), these ladder operators are rewritten\nas\nsx0\u0006\nk=1\n2X\n\u001b;\u001b0X\nk0cy\nk0\u001b(^\u001bx0\u0006)\u001b\u001b0ck0\u0006k\u001b0; (29)\n^\u001bx0\u0006=\u0000sin\u0012\u001bx+ cos\u0012\u001by\u0006i\u001bz: (30)\nD. Second-order perturbation\nLet us consider a second-order perturbation with re-\nspect to the interfacial exchange interaction Hintto the\nbulk system described by H2DEG +HFI. The spin corre-\nlation function is written in the form,\nG(k;i!n) =1\n(G0(k;i!n))\u00001\u0000\u0006(k;i!n); (31)\n\u0006(k;i!n) =jTkj2\n4\fX\nk0;i!mTr\"\n^\u001bx0\u0000^g(k0;i!m)\n\u0002^\u001bx0+^g(k0+k;i!m+i!n)#\n; (32)\nwhereG0(k;i!n) is the unperturbed part of the Green's\nfunction given in Eq. (21), and \u0006( k;i!n) is the self-\nenergy due to the interfacial exchange coupling. Since a\nuniform spin precession is induced in FMR experiments,\nwe only calculate the self-energy for k=0. To simplify\nthe notation, we will rewrite Green's function ^ g(k;i!n)\nas\n^g(k;i!n) =A(i!n)^I\u0000he\u000b\u0001\u001b\nD(i!n); (33)\nA(i!n) =i~!n\u0000\u0018k+i\u0000sgn(!n)=2; (34)\nD(i!n) =Y\n\u0017=\u0006(i~!n\u0000E\u0017\nk+i\u0000sgn(!n)=2): (35)By a straightforward calculation using the algebra of the\nPauli matrices, we obtain\n\u0006(q=0;i!n)\n=jT0j2\n\fX\nk;i!mA\u0000he\u000b\u0001^m\nDA0+he\u000b\u0001^m\nD0; (36)\nwhere ^m= (cos\u0012;sin\u0012;0) indicates the spin orientation\nof the FI,A=A(i!m),A0=A(i!m+i!n),D=D(i!m),\nandD0=D(i!m+i!n) (for the detailed derivation, see\nAppendix C). For further calculation, we evaluate the\nsum with respect to the Matsubara frequency !mby us-\ning the standard procedure based on the contour integral.\nThen, we obtain the retarded component by analytic con-\ntinuation:i!n!!+i\u000e(the details of the calculation\nare in Appendix D).\nWe will focus on the increase in the FMR linewidth due\nto the exchange coupling with the 2DEG and consider\nonly the imaginary part of the self-energy. Using the\nunperturbed spin correlation function Eq. (21), we obtain\nthe retarded component of the spin (magnon) correlation\nfunction as\nGR(q=0;!)'2S0=~\n!\u0000!q=0+i(\u000bG+\u000e\u000bG)!; (37)\n\u000e\u000bG(!)\u0011\u00002S0\n~!Im \u0006R(q=0;!): (38)\nFor\u000bG+\u000e\u000bG\u001c1 that holds for a standard setup of\nspin pumping, the linewidth of the ferromagnetic reso-\nnance is su\u000eciently small, allowing us to replace !with\nthe resonance frequency !q=0to express the enhance-\nment of the damping constant in Eq. (38). Hereafter,\nthe resonance frequency is simply written as \n( \u0011!q=0).\nThen, Im \u0006R(q=0;\n) corresponds to the increase in\nthe linewidth in the FMR experiment. Analytic calcula-\ntion of Im \u0006( q=0;\n) gives a correction for the Gilbert\ndamping as\n\u000e\u000bG(\n)'\u00002S0\n~\nIm \u0006R(q=0;\n)\n=\u000bG;0X\n\u0017;\u00170=\u00061Z2\u0019\n0d'\n2\u0019F\u0000\n~\n + (\u0017\u0000\u00170)he\u000b\u0001\n\u00021\u0000\u0017^he\u000b(')\u0001^m\n21 +\u00170^he\u000b\u0001^m\n2;(39)\nF(x) =\u0000=\u0019\u00010\n(x=\u00010)2+ (\u0000=\u00010)2; (40)\nwhere'describes the propagation direction of conduc-\ntion electrons, \u000bG;0= 2\u0019S0jT0j2AD(\u000fF)=\u00010is a dimen-\nsionless parameter, Ais the junction area, and ^he\u000b(') =\nhe\u000b(')=he\u000b(') is a unit direction vector of the e\u000bective\nmagnetic \feld. Here, we have introduced the unit of en-\nergy \u0001 0, which is the amplitude of the Dresselhaus spin-\norbit interaction, kF\f. The enhancement of the Gilbert5\ndamping can be separated into three parts:\n\u000e\u000bG=\u000e\u000bG;1+\u000e\u000bG;2+\u000e\u000bG;3 (41)\n\u000e\u000bG;1=\u000bG;0Z2\u0019\n0d'\n2\u0019F(~\n)1\u0000(^he\u000b\u0001^m)2\n2; (42)\n\u000e\u000bG;2=\u000bG;0Z2\u0019\n0d'\n2\u0019F(~\n\u00002he\u000b)(1 + ^he\u000b\u0001^m)2\n4;\n(43)\n\u000e\u000bG;3=\u000bG;0Z2\u0019\n0d'\n2\u0019F(~\n + 2he\u000b)(1\u0000^he\u000b\u0001^m)2\n4;\n(44)\nThese expressions indicate the physical mechanism of the\nenhancement of the Gilbert damping as follows. The\n\frst contribution \u000e\u000bG;1comes from elastic spin \ripping\nof conduction electrons caused by the transverse com-\nponent of the e\u000bective magnetic \feld he\u000bvia the inter-\nfacial exchange coupling. In fact, \u000e\u000bG;1vanishes when\nhe\u000bis parallel or anti-parallel to the magnetization of FI,\n^m. Since this process is elastic, the frequency-dependent\npart is just a Lorentzian form F(~\n), which has a peak\nat \n = 0. The second contribution \u000e\u000bG;2originates from\nthe magnon absorption process. This is a dynamical pro-\ncess as indicated by the peak shift of the Lorentzian form;\nthe peak of F(~\n\u00002he\u000b(')) is shifted to \n = 2 he\u000b(')=~,\nat which the magnon energy coincides with the spin-\nsplitting energy gap of conduction electrons. The second\ncontribution takes a maximum when ^he\u000bis parallel to\n^m. This is consistent with the fact that a spin of a con-\nduction electron is converted from a low-energy state ^he\u000b\nto a higher state \u0000^he\u000bby receiving a spin carried by a\nmagnon, that is in the direction of \u0000^m. We should also\nnote that the second contribution vanishes when ^he\u000bis\nanti-parallel to ^m. The third contribution \u000e\u000bG;3comes\nfrom the magnon emission process. This process is usu-\nally not important in FMR experiments, because the fre-\nquency \n is taken as positive.\nIII. RESULTS\nA. Enhancement of Gilbert damping\nFig. 4 illustrates the enhancement of the Gilbert damp-\ning,\u000e\u000bG, as a function of the resonant frequency \n for\n\u0000=\u00010= 0:5. In particular, Fig. 4 (a) and (b) shows the\nenhancement for \u000b=\f = 1 and 3, respectively. We note\nthat\u000e\u000bGdepends on the resonant frequency in contrast\nwith the Gilbert damping coe\u000ecient for the bulk FI, \u000bG.\nIn both cases, the enhancement \u000e\u000bGclearly depends on\nthe spin orientation of the FI. A peak at \n = 0 is caused\nby the static part \u000e\u000bG;1, while the structure at a \fnite\n\n comes from the magnon-absorption process described\nby\u000e\u000bG;2. Note that the broad structure indicated by\n\u000e\u000bG;2is produced by the variation in the energy split-\nting 2he\u000b(') when the azimuth angle 'changes from 0\nto 2\u0019. The range of 2 he\u000b(') is obtained from Eq. (5) as\n0\u00142he\u000b\u00144\u00010for\u000b=\f= 1 and 4\u0001 0\u00142he\u000b\u00148\u00010for\n\u000b=\f= 3. This variation in the spin splitting corresponds\nto the range of the broad structure at \fnite \n shown in\nFIG. 4. Enhancement of Gilbert damping, \u000e\u000bG, due to cou-\npling to 2DEG plotted as a function of FMR resonance fre-\nquency \n. (a) \u000b=\f= 1. (b)\u000b=\f= 3.\nFig. 4 (a) and (b).\nWhen the azimuth angle of the spin polarization in the\nFI,\u0012, is varied, the enhancement of the Gilbert damping\nis modi\fed, as shown in Fig. 4. Its general properties are\nsummarized as follows. First, \u000e\u000bGis independent of the\nspin orientation of the FI for \u000b= 0 or\f= 0. Second,\nthe result for \u000b<0 is related to that for \u000b>0 through\n\u000e\u000bG(\u000b;\u0012;\n) =\u000e\u000bG(\u0000\u000b;\u0012\u0000\u0019=2;\n): (45)\nThird,\u000e\u000bGhas symmetry relations with respect to \u0012as\n\u000e\u000bG(\u0012;\n) =\u000e\u000bG(\u0012+\u0019;\n) =\u000e\u000bG(\u0019=2\u0000\u0012;\n):(46)\nTo see the spin-orientation dependence in more detail, we\nshow a contour plot of \u000e\u000bGas a function of \u0012and \n in\nFig. 5. For \u000b= 0,\u000e\u000bGis independent of \u0012(Fig. 5 (a)).\nThe same result is obtained for \f= 0 after replacing\n\u00010withkF\u000b. The spin-orientation dependence becomes\nstrongest for \u000b=\f= 1 (Fig. 5 (b)). For this special case,\nthe direction of ^he\u000bis \fxed:\n^he\u000b=\u0006(1=p\n2;\u00001=p\n2;0): (47)\nFor the spin texture of this special case, see Fig. 2 (a).\nTherefore, the \u0012-dependent part in Eqs. (42)-(44) can be6\nFIG. 5. Contour plot of enhancement of Gilbert damping in 2DEG for (a) \u000b=\f = 0, (b)\u000b=\f = 1, and (c) \u000b=\f = 3. The\nhorizontal axis is the FMR frequency \n and the vertical axis is the azimuth angle of the spin orientation of the FI, \u0012.\ntaken out of the integrals:\n\u000e\u000bG;1/1\u0000(^he\u000b\u0001^m)2\n2; (48)\n\u000e\u000bG;2;\u000e\u000bG;3/1 + ( ^he\u000b\u0001^m)2\n4; (49)\nwhere we have used the fact that the term proportional\nto^he\u000b\u0001^mvanishes after the integration with respect to\n'. From this expression, the spin-orientation dependence\nshown in Fig. 5 (b) can be explained as follows. The peak\nat \n = 0 that is caused by \u000e\u000bG;1takes a maximum (a\nminimum) when ^he\u000b?^m(^he\u000bk^m) or equivalently\nwhen\u0012=\u0019=4;5\u0019=4 (\u0012= 3\u0019=4;7\u0019=4). This observation\nsupports the conclusion that the enhancement in Gilbert\ndamping at \n = 0 is induced by the transverse com-\nponent of the e\u000bective magnetic \feld he\u000b. In contrast,\nthe broad structure at \fnite frequencies in the range of\n0\u0014~\n\u00144\u00010, that is caused by \u000e\u000bG;2, takes a max-\nimum (a minimum) when ^he\u000bk^m= 0 ( ^he\u000b?^m).\nThis is consistent with the fact that this contribution\ncomes from the magnon absorption accompanying spin-\n\rips of the conduction electrons. Fig. 5 (c) shows the\nspin-orientation dependence for \u000b=\f = 3. Although the\n\u0012dependence cannot be expressed in a simple form for\n\u000b=\f= 3, the qualitative features are the same as in the\ncase of\u000b=\f= 1, as indicated by comparing Fig. 5 (b) and\n(c) except that the \fnite-frequency bread structure shifts\ntoward the high-frequency region 4\u0001 0\u0014~\n\u00148\u00010.\nB. Relevance to experiments\nOur results indicate that the spin-orientation depen-\ndent provides information on spin-orbit interactions in\n2DEG, in which both the Rashba and Dresselhaus spin-\norbit interactions coexist. Let us estimate a necessary\ncondition for observation of the present result. For\nGaAs/AlGaAs heterostructures47, the magnitude of the\nspin-orbit interactions is given as \u000b\u0018\f\u00184 meV\u0001\u0017A,\nleading to \u0001 0=kF\f\u00180:10 meV for the electron den-\nsity 5\u00021011cm\u00002. Because the FMR frequency for YIG\nunder a magnetic \feld of 1 T is about ~\n = 0:06 meV,\nthe ratio ~\n=\u00010is of order 1. This indicates that both\nthe elastic contribution \u000e\u000bG;1and the magnon absorp-\ntion contribution \u000e\u000bG;2can be observed experimentally\nusing a magnetic \feld of a few tesla. Note that theRashba spin-orbit interaction can be controlled by ap-\nplying an electric \feld to the sample. The amplitude of\nthe spin-orbit interactions depends on the aspects of bulk\nsemiconductors as well as on sample fabrication consid-\nerations. For example, in asymmetric InAs heterostruc-\ntures48,49, the magnitude of the Rashba spin-orbit inter-\naction is about \u000b\u0018400 meV\u0001\u0017A, leading to kF\u000b\u001814 meV\nfor the electron density 1012cm\u00002. In this case, the de-\npendence of the spin-orientation of FI is governed by the\nelastic contribution \u000e\u000bG;1. However, by using symmet-\nric InAs heterostructures50, it is possible to reduce the\nmagnitude of the Rashba spin-orbit interaction down to\nthe same order as in GaAs/AlGaAs heterostructures. In\nsuch heterostructures, we can also observe the contribu-\ntion from magnon absorption, \u000e\u000bG;2.\nIV. SUMMARY\nWe theoretically investigated spin pumping from a fer-\nromagnetic insulator (FI) into a two-dimensional gas\n(2DEG) with both Rashba and Dresselhaus spin-orbit\ninteractions. We considered the interfacial coupling\nthrough the tunnel Hamiltonian in which the momentum\nof spin excitation is conserved and derived an increase in\nthe linewidth in a ferromagnetic resonance (FMR) exper-\niment that is induced by the 2DEG within a second-order\nperturbation with respect to the interfacial coupling. We\nfound that there are three processes that enhance the\nGilbert damping: (a) an elastic process, (b) a magnon\nabsorption process, and (c) a magnon emission process.\nThe elastic process is induced by spin-\rips through the\ntransverse component of the e\u000bective magnetic \feld felt\nby conduction electrons that originate from the spin-orbit\ninteraction in the 2DEG. This elastic process is dominant\nwhen the FMR frequency is su\u000eciently low compared\nwith the energy scale of the spin-orbit interaction. In\ncontrast, the magnon absorption/emission process is a\ndynamical one that changes the number of magnons in\nthe FI and a\u000bects the Gilbert damping when the FMR\nfrequency is comparable to the spin splitting energy by\nspin-orbit coupling in the 2DEG. We discussed how these\nthree processes of enhancing the Gilbert damping depend\non the spin orientation in the FI. We also showed that\nour results can be detected in an FMR experiment using\na GaAs/AlGaAs heterostructure under a magnetic \feld7\nFIG. 6. Feynman diagram of second-order perturbation with\nrespect to the impurity potential.\nof a few tesla. Our work provides a helpful experimental\nmethod for the detection of spin texture of conduction\nelectrons at the Fermi surface.\nACKNOWLEDGMENTS\nWe would like to thank Dr. Y. Ominato for fruit-\nful discussions. T. K. acknowledges support from the\nJapan Society for the Promotion of Science (JSPS KAK-\nENHI Grant No. JP20K03831). M. M. is \fnancially sup-\nported by a Grant-in-Aid for Scienti\fc Research (Grants\nNo. JP20H01863 and No. JP21H04565) from MEXT,\nJapan.\nA. IMPURITY SCATTERING\nIn this study, we consider the e\u000bect of impurity scat-\ntering within a second-order perturbation with respect to\nan impurity potential by taking a random average. This\napproximation corresponds to the Born approximation,\nwhose diagram is shown in Fig. 6. In this approximation,\nthe temperature Green's function is written as\n(^g(k;i!n))\u00001= (^g0(k;i!n))\u00001\u0000^\u0000(i!n); (A1)\n^\u0000(i!n) =niu2Zd2k\n(2\u0019)2^g0(k;i!n); (A2)\nwhereniis the number of impurity sites. We assume that\nthe scattering rate is much smaller than the bandwidth\nof the conduction electrons. Accordingly, the self-energy\nis calculated as\n^\u0000(i!n) =\u0000iniu2kF\n2vFsgn(!n)^I\u0011\u0000i\u0000\n2sgn(!n)^I;(A3)\nwhere \u0000 denotes the impurity scattering rate. Using\nthe Dyson equation (A1), the retarded component of the\nGreen's function is obtained as Eq. (11).\nB. SPIN-WAVE APPROXIMATION\nWe derive the Hamiltonian within the spin-wave ap-\nproximation by using the Holstein-Primakov transforma-\ntion. ForS0\u001d1, it is written as\nSx0\u0000\ni'p\n2S0by\ni; (B1)\nSx0+\ni'p\n2S0bi; (B2)\nSx0\ni=S0\u0000by\nibi; (B3)wherebi(by\ni) is an annihilation (creation) operator de-\n\fned at site i. We replace the spin operators with these\nboson operators and take the Fourier transform,\nbi=1pNFX\nkeik\u0001ribk; (B4)\nwhereNFis the number of unit cells in the FI. The\nHamiltonian of the FI is modi\fed into Eqs. (17) and (18).\nWhen we consider the cubic lattice model with only the\nnearest-neighbor exchange coupling J, the dispersion is\ngiven as ~!k=~!(0)\nk+~\rhdc, where\n~!(0)\nk= 2JS0(3\u0000cos(kxa)\u0000cos(kya)\u0000cos(kza))\n'JS0a2k2: (B5)\nandais a lattice constant of the FI. The last equation is\nthe long-wavelength approximation.\nC. DERIVATION OF EQUATION (36)\nHere, we derive Eq. (36). We rewrite Green's function\nof the conduction electrons as\n^g(k;i!n) =1\nD(i!n)h\nA(i!n)^I+b\u0001\u001bi\n; (C1)\nwherea= (\u0000sin\u0012;cos\u0012;i) andb=\u0000he\u000b. Then, the\ntrace in Eq. (32) is rewritten as\nI\u0011Trh\n^\u001bx0\u0000^g(k;i!m)^\u001bx0+^g(k;i!m+i!n)i\n=1\nDD0Trh\na\u0003\u0001\u001b(A^I+b\u0001\u001b)a\u0001\u001b(A0^I+b\u0001\u001b)i\n:(C2)\nUsing the identity,\n(a\u0001\u001b)(b\u0001\u001b) = (a\u0001b)^I+i(a\u0002b)\u0001\u001b; (C3)\nTr [\u001ba] = 0; Tr [^I] = 2; (C4)\na straightforward calculation gives\nI=2\nDD0h\nAA0a\u0003\u0001a+iA0(a\u0003\u0002b)\u0001a+iAa\u0003\u0001(a\u0002b)\n\u0000(a\u0003\u0002b)\u0001(a\u0002b) + (a\u0003\u0001b)(a\u0001b)i\n: (C5)\nWe obtain Eq. (36) by substituting the explicit forms of\naandb.\nD. ANALYTIC CONTINUATION\nHere, we perform the summation in the self-energy by\nusing analytic continuation. Using the identities,\nA\nD=1\n2X\n\u0017=\u00061\ni~!m\u0000E\u0017\nk+i\u0000=2 sgn(!m); (D1)\nhe\u000b\u0001^m\nD=1\n2X\n\u0017=\u0006\u0017^he\u000b\u0001^m\ni~!m\u0000E\u0017\nk+i\u0000=2 sgn(!m);(D2)8\nFIG. 7. Contour on the complex plane.\nand the counterparts for A0andD0, the self-energy is\nrewritten as\n\u0006(0;i!n) =jT0j2\n4X\nkX\n\u0017=\u0006X\n\u00170=\u0006(1\u0000\u0017^he\u000b\u0001^m)\n\u0002(1 +\u00170^he\u000b\u0001^m)Ik\u0017\u00170;(D3)\nIk\u0017\u00170=1\n\fX\ni!m1\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);(D4)\nwhere ^he\u000b=he\u000b=he\u000b. By using analytic continuation,\nIk\u0017\u00170can be expressed as a contour integral,\nIk\u0017\u00170=\u0000Z\nCdz\n2\u0019if(z)1\nz\u0000E\u0017\nk+i\u0000=2 sgn(Imz)\n\u00021\nz+i~!n\u0000E\u00170\nk+i\u0000=2 sgn(Imz+!n);(D5)\nwheref(z) = (e\fz+1)\u00001andCis a contour surrounding\nthe poles of f(z).\nWe modify the contour Cto be a sum of C1,C2,C3,\nandC4, as shown in Fig. 7, and change the integration\nvariable to z=E+i\u0011forC1,z=E\u0000i\u0011forC2,z=\nE\u0000i!n+i\u0011forC3, andz=E\u0000i!n\u0000i\u0011forC4, where\n\u0011is a positive in\fnitesimal. Then, we obtain\nIk\u0017\u00170=\u0000ZdE\n2\u0019if(E)\n\u0002\"\n\u0000i\u0000\n(E\u0000E\u0017\nk)2+ (\u0000=2)2\u00021\nE+i~!n\u0000E\u00170\nk+i\u0000=2\n+1\nE\u0000i~!n\u0000E\u0017\nk\u0000i\u0000=2\u0002\u0000i\u0000\n(E\u0000E\u00170\nk)2+ (\u0000=2)2#\n:\n(D6)By changing the variable to E0=E\u0000E\u0017\nkfor the \frst\nterm and to E0=\u0000(E\u0000E\u00170\nk) for the second term, we\nobtain\nIk\u0017\u00170=\u0000ZdE0\n2\u0019i\u0000i\u0000\nE02+ (\u0000=2)2\n\u0002\"\nf(E0+E\u0017\nk)\u0000f(\u0000E0+E\u00170\nk)\nE0+i~!n+E\u0017\nk\u0000E\u00170\nk+i\u0000=2#\n:(D7)\nThe summation with respect to the wavenumber can be\nreplaced with an integral,\n1\nAX\nkIk\u0017\u00170'D(\u000fF)Z1\n\u00001d\u0018Z2\u0019\n0d'\n2\u0019Ik\u0017\u00170; (D8)\nwhereAis the junction area and \u0018\u0011\u0018k. Using the\nintegral formulas,\nZ1\n\u00001d\u0018(f(E0+E\u0017\nk)\u0000f(\u0000E0+E\u00170\nk))\n=\u0000(2E0+E\u0017\nk\u0000E\u00170\nk); (D9)\nZ1\n\u00001dx\n2\u0019a2\nx2+ (a=2)2x+b=2\n(x+b+c)2+ (a=2)2\n=\u0000ac\n(b+c)2+a2;(a>0); (D10)\nwe \fnally obtain\nIm \u0006R(0;!)\n=\u0000jT0j2AD(\u000fF)\n4X\n\u0017;\u00170Z2\u0019\n0d'\n2\u0019(1\u0000\u0017^he\u000b(')\u0001^m)\n\u0002(1 +\u00170^he\u000b(')\u0001^m)\u0000~!\n(~!+E\u0017\nk\u0000E\u00170\nk)2+ \u00002:\n(D11)\nNote that the \fnal result does not depend on the temper-\nature. This feature emerges when the density of states\nfor conduction electrons is approximated as being con-\nstant near the Fermi energy. In general, one can derive a\nsmall temperature-dependent correction by using a Som-\nmerfeld expansion that takes into account the energy de-\npendence of the density of states.9\n1Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n2Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. 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Phys. 3, 650 (2007)."    },    {        "title": "1010.1537v1.Power_optimization_for_domain_wall_motion_in_ferromagnetic_nanowires.pdf",        "content": "arXiv:1010.1537v1  [cond-mat.mes-hall]  7 Oct 2010Power optimization for domain wall motion in ferromagnetic nanowires\nO. A. Tretiakov,1Y. Liu,1and Ar. Abanov1\nDepartment of Physics & Astronomy, Texas A&M University, Co llege Station, Texas 77843-4242,\nUSA\n(Dated: October 7, 2010)\nThe current mediated domain-wall dynamics in a thin ferromagnetic w ire is investigated. We derive the\neffective equations of motion of the domain wall. They are used to stu dy the possibility to optimize the power\nsupplied by electric current for the motion of domain walls in a nanowire . We show that a certain resonant\ntime-dependent current moving a domain wall can significantly reduc e the Joule heating in the wire, and\nthus it can lead to a novel proposal for the most energy efficient me mory devices. We discuss how Gilbert\ndamping, non-adiabatic spin transfer torque, and the presence o f Dzyaloshinskii-Moriya interaction can effect\nthis power optimization.\nIntroduction. Due to its direct relevance to future\nmemory and logic devices, the dynamics of domain walls\n(DW) in magnetic nanowires has become recently a very\npopulartopic.1–3Therearemainlytwogoalswhichscien-\ntists try to achieve in this field. One goal is to move the\ndomain walls with higher velocity in order to make faster\nmemory or computer logic. The other one is inspired by\nthe modern trend of energy conservation and concerns a\npower optimization of the domain-wall devices.\nGenerally, the domain walls can be manipulated\nwhether by a magnetic field3,4or electric current.1,5Al-\nthough the latter method is preferred for industrial ap-\nplications due to the difficulty with the application of\nmagnetic fields locally to small wires. For this reason, we\nconsider in this paper the current induced domain-wall\ndynamics. We make a proposal on how to optimize the\npowerfor the DWmotionby meansofreducingthe losses\non Joule heating in ferromagnetic nanowires.6Moreover,\nbecause the averaged over time (often called drift) veloc-\nity of a DW generally increases with applied current, we\nalso address the first goal. Namely, our proposal allows\nto move the DWs with higher current densities without\nburning the wire by the excessive heat and thus archive\nhigher drift velocities of DWs. The central idea of this\nproposal is to employ resonant time-dependent current\nto move DWs, where the period of the current pulses is\nrelated to the periodic motion of DW internal degrees of\nfreedom.\nThe schematic view of a domain wall in a narrow fer-\nromagnetic wire is shown in Fig. 1. These DWs are char-\nFIG. 1. (color online) A schematic view of a current-driven\ndomain wall in a ferromagnetic wire. The DW width is ∆.acterized by their width ∆ which is mainly determined\nby exchange interaction and anisotropy along the wire λ.\nAnother important quantity is the transverse anisotropy\nacrossthewire K, whichgovernsthepinningofthetrans-\nverse component of the DW magnetization. When no\ncurrent is applied to the wire it leads to two degenerate\npositions of the transverse magnetization component of\nthe wall: as shown in Fig. 1 and anti-parallel to it.\nTo describe the dynamics of DW in a thin wire we\nderived the effective equations of motion from general-\nized Landau-Lifshitz-Gilbert7,8(LLG) equation with the\ncurrentJ,\n˙S=S×Heff−J∂S\n∂z+βJS×∂S\n∂z+αS×˙S,(1)\nwhereSis magnetization unit vector, Heff=δH/δSis\nthe effective magnetic field given by the Hamiltonian H\nof the system, βis non-adiabatic spin torque constant,\nandαis Gilbert damping constant. The derivation of\nthe effective equations of motion is based on the fact\nthat in thin ferromagnetic wires the static DWs are rigid\ntopologically constrained spin-textures. Therefore, for\nnot too strong drive, their dynamics can be described\nin terms of only a few collective coordinates associated\nwith the DW degrees of freedom.9In very thin wires,\nthere are two collective coordinates corresponding to two\nsoftest modes of the DW motion: the DW position along\nthe wire z0and the magnetization angle φin the DW\naround the wire axis. All other degrees of freedom are\ngapped by strong anisotropic energy along the wire.\nBy applying the orthogonality condition to LLG, one\ncan obtain the equations of motion for the two DW soft-\nest modes, z0(t) andφ(t),10\n˙z0=AJ+B[J−jcsin(2φ)], (2)\n˙φ=C[J−jcsin(2φ)], (3)\nwhereJ(t) is a time-dependent current. The co-\nefficients A,B,C, and critical current jccan be\nevaluated for a particular model in terms of α,β\nand other microscopic parameters. Following Ref. 10,\nfor the model with Dzyaloshinskii-Moriya interaction\n(DMI) one can find A=β/α,B= (α−β)(1 +\nαΓ∆)/[α(1 +α2)],C= (α−β)∆/[(1 +α2)∆2\n0], and2\nFIG. 2. (color online) DW motion characteristics for dc cur-\nrents. (a) Drift velocity Vdof DW as a function of current J\nforB >0 andB <0, see Eq. (2). The slope at J < jcis given\nbyA, whereas at J≫jcit isA+B. (b) Power of Ohmic\nlossespdc(Vd/Vc) =J2/j2\ncas a function of drift velocity Vd.\nForB <0 the power has a discontinuity at Vd/Vc= 1.\njc= (αK∆/|α−β|)[πΓ∆/sinh(πΓ∆)], where Jexis ex-\nchange constant, Dis DMI constant, and Γ = D/Jex.\nAlso, ∆ = ∆ 0//radicalbig\n1−Γ2∆2\n0where ∆ 0is the DW width in\nthe absence of DMI.\nAlternatively, Eqs. (2) and (3) can be obtained in a\nmore general framework by means of symmetry argu-\nments. We note that because of the translational invari-\nance ˙z0and˙φcannot depend on z0. Furthermore, to the\nfirst order in small transverse anisotropy K,˙φand ˙z0are\nproportional to the first harmonic sin(2 φ). Then the ex-\npansion in small current Jup to a linear in Jorder gives\nEqs. (2) and (3). In this case the coefficients A,B,C,\nandjchave to be determined directly from experimental\nmeasurements.11,12\nFor the dc current applied to the wire the DW dy-\nnamics governed by Eqs. (2) and (3) can be obtained\nexplicitly.10ForJ < j candA/negationslash= 0 the DW only\nmoves along the wire and is tilted on angle φ0from\nthe transverse-anisotropy easy axis given by condition\nsin(2φ0) =J/jc. Thedriftvelocityis Vd=/angbracketleft˙z0(J)/angbracketright=AJ,\nsee Eq. (2). Therefore, the linear slope of Vd(J) belowjc\ngives constant A, see Fig. 2 (a). The value of jcis deter-\nmined as the endpoint of this linear regime. At J=jc\nthe magnetization angle becomes perpendicular to the\neasy axis, φ0=π/2. ForJ > jcthe DW both moves and\nrotates, and Eqs. (2) and (3) give Vd=AJ+B/radicalbig\nJ2−j2c,\nso that the slope of Vd(J) at large JgivesA+B.\nPower optimization. The largestlossesin the nanowire\nwith a DW are the Ohmic losses of the current. In gen-\neral, the influence of the DW on the resistance is negli-\ngible and therefore we can assume that the resistance of\nthe wire is constant with time. Then the time-averaged\npower of Ohmic losses is proportional to /angbracketleftJ2(t)/angbracketright. Since\nthe resistance is almost constant, in this paper we will\ncalculate P=/angbracketleftJ2(t)/angbracketrightand loosely call it the power of\nOhmic losses. Our goal is to minimize the Ohmic losses\nwhile keeping the DW moving with a given constant drift\nvelocity.\nFor the following it will be convenient to introduce the\ndimensionless variables for time, drift velocity, current,\npower, and the ratio of slopes of Vd(J) at large and smallcurrents,\nτ=Cjct, vd=Vd\nVc, j=J\njc, p=P\nj2c, a=A+B\nA.\n(4)\nAlthough we note that in the special case of α=β,\nit can be shown that C=B= 0 and one cannot use\ndimensionless variables (4). However, in this case the\nDW dynamics is trivial:13the DW does not rotate φ=\n0,πand moves with the velocity ˙ z0=J.\nFirst, we consider the case of dc current and the power\nas a function of drift velocity. For vd<1 we find pdc=\nv2\nd. For currents above jcthe power pdc(vd) =j2is given\nintermsofdriftvelocity vd=j+(B/A)/radicalbig\nj2−1asshown\nin Fig. 2 (b). The poweris quadraticin vd, and for B <0\nit has a discontinuity at vd= 1.\nIn general, the DW motion has some period Tand\ncurrentj(τ) must be a periodic function with the same\nTto minimize the Ohmic losses. Measuring the angle\nfrom the hard axis instead of easy axis and scaling it\nby 2, i.e, 2 φ=θ−π/2, we can write the dimensionless\ncurrent drift velocity as6\nj(τ) =˙θ/2−cosθ, vd=a\n2/angbracketleft˙θ/angbracketright−/angbracketleftcosθ/angbracketright,(5)\nwhere˙θ=∂θ/∂τ.\nTo minimize the power of Ohmic losses we need to find\nthe minimum of /angbracketleftj2(τ)/angbracketrightat fixedvd,\np=/angbracketleftBig\n(˙θ/2−cosθ)2−2ρ(a˙θ/2−cosθ−vd)/angbracketrightBig\n,(6)\nwhere we use a Lagrange multiplier 2 ρto account for the\nconstraint given by vdfrom Eq. (5). Power (6) can be\nconsidered as an effective action for a particle in a peri-\nodic potential U, and its minimization gives the equation\nof motion ¨θ/2 =−∂U/∂θwhich in turn can be reduced\nto\n˙θ=±2/radicalbig\nd−U(θ,ρ), U(θ,ρ) =−cos2θ−2ρcosθ.\n(7)\nwheredis an arbitrary constant. Since changing ρ→ −ρ\ninUof Eq. (7) is equivalent to changing θ→π+θ, below\nwe can consider only positive ρ.\nEq. (7) shows that there are two different regimes: 1)\nthe bounded regime where d <max[U(θ,ρ)] in which\ncaseθis bounded, and the particle oscillates in potential\nwellU(θ), see inset of Fig. 3 (a); and 2) the rotational\nregime where d >max[U(θ,ρ)] with freely rotating mag-\nnetization in the DW.\nIn the bounded regime the particle moves between the\ntwo turning points −θ0andθ0given by d=U(±θ0,ρ).\nSinceθis a bounded function /angbracketleft˙θ/angbracketright= 0 and vd=−/angbracketleftcosθ/angbracketright.\nOne can show6that in this regime the power of Ohmic\nlosses is minimal for dc current, i.e., p=v2\nd.\nIn the rotational regime the term in Eq. (5) with /angbracketleft˙θ/angbracketright\nshould be kept because θis not bounded. The equation\nof motion is the same as for a nonlinear oscillator.6Using3\n0 00 0.2 0.4 0.6 0.8 1 1.2 1.402468\n1-2-\n--\n01\n10\nFIG. 3. (color online) (a) Minimal power of Ohmic losses\np=/angbracketleftJ2/angbracketright/j2\ncas a function of drift velocity Vdshown by solid\nline fora= 0.5. The dashed line depicts pfor dc current. The\ninset shows the potential U(θ) in which a “particle” is moving\nin the bounded (pendulum-like) and unbounded (rotational)\nregimes. A sketch of /angbracketleftJ2/angbracketright(Vd) shown by solid line in (b) for\nβ≫α(a≪1) and (c) for β≪α(a≫1).\nthe minimization condition ∂p/∂ρ|vd= 0 one finds\n/integraldisplayπ\n−π/radicalbig\nd−U(θ,ρ)dθ= 2πaρ. (8)\nThis equation defines the relationship between dandρ.\nThe results for the minimal power of Ohmic losses\np(vd) are presented in Fig. 3. For a >1 there is a crit-\nical velocity vrc<1, such that at vd< vrcthe power\nof Ohmic losses is p=v2\nd=pdc. Above vrcone can\nminimize the Ohmic losses by moving DW with resonant\ncurrent pulses. Right above vrcthere is a certain rangeof\nvdwherep= 2ρ0vd−ρ2\n0withρ0(a)<1 given by Eq. (8)\nwithd=ρ2. The critical velocity is found as vrc=ρ0(a).\nFora <1, see e.g. Fig. 3 (a), we find that vrc= 1,\nwhereas at vd>1 minimal power pis significantly lower\nthanpdc. Immediately above vd= 1 we find that there\nis a range of vdwherepis linear in vd. At large vdthe\nminimal power is always smaller than pdc, the difference\nbetween them then approaches pdc−p= (1−1/a)2/2.\nWe note that even in the limiting cases of the systems\nwith weak ( β≪α) or strong ( β≫α) non-adiabatic spin\ntransfer torque, see Fig. 3 (b) and (c), where the power\nof Ohmic losses is high for dc currents, the optimized ac\ncurrent gives dramatic reduction in heating power thus\ngreatly expanding the range of materials which can be\nused for spintronic devices.1,3We also note that DMI\nsuppresses critical current jcand affects parameter a.\nForvd< vrcthe optimal current coincides with the dc\ncurrent, above vrcthe resonant current j(t) is plotted in\nFig. 4 for a= 2 and two different velocities vd. Atvd>\nvrcthe current’s maximum jmaxincreases from 2 −vrc\nat small enough vd<∼1 up to jmax≈vd/aatvd≫\n1. The current’s minimum increases monotonically from0 10 20 30 40 50 600123\nFIG. 4. (color online) Resonant time-dependent current J(τ)\nwithτ=Cjctfor drift velocities vd= 0.5 (dashed line) and\nvd= 4.5 (solid line) for a= 2.\nsmall positive values jmin=vrcatvd∼1 up tojmin=\njmax−2|1−a|/aatvd≫1. Atvd<∼1 (fora >1) the\ntime between the current picks decreases with increasing\nvelocity as T≃(πa−2arcsinvrc)/(vd−vrc), whereas the\npick’s width is given by ≈1.3//radicalbig\n(1−vrc). Therefore, at\nsmallvd−vrcthe picks are widely separated, then as vd\nincreases the time between the picks decreases. At vd≫\n1 the optimal current has a large constant component\nand small-amplitude ac modulations on top of it.\nConclusions. We have studied the current driven DW\ndynamicsinthinferromagneticwires. 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Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, and\nT. Shinjo, Phys. Rev. Lett. 92, 077205 (2004).\n6O. A. Tretiakov, Y. Liu, and Ar. Abanov, Phys. Rev. Lett., in\npress; arXiv:1006.0725.\n7Z. Li and S. Zhang, Phys. Rev. Lett. 92, 207203 (2004).\n8A. Thiaville et al., Europhys. Lett. 69, 990 (2005).\n9O. A. Tretiakov, D. Clarke, G.-W. Chern, Y. B. Bazaliy, and\nO. Tchernyshyov, Phys. Rev. Lett. 100, 127204 (2008); D. J.\nClarke, O. A. Tretiakov, G.-W. Chern, Y. B. Bazaliy, and\nO. Tchernyshyov, Phys. Rev. B 78, 134412 (2008).\n10O. A. Tretiakov and Ar. Abanov, Phys. Rev. Lett. 105, 157201\n(2010).\n11S. A. Yang, G. S. D. Beach, C. Knutson,\nD. Xiao, Q. Niu, M. Tsoi, and J. L. Erskine,\nPhys. Rev. Lett. 102, 067201 (2009).\n12Y. Liu, O. Tretiakov, and Ar. Abanov, (unpublished).\n13S. E. Barnes and S. Maekawa,\nPhys. Rev. Lett. 95, 107204 (2005)."    },    {        "title": "2404.00845v1.Harnessing_Interlayer_Magnetic_Coupling_for_Efficient__Field_Free_Current_Induced_Magnetization_Switching_in_a_Magnetic_Insulator.pdf",        "content": " \n 1 Harnessing Interlayer Magnetic Coupling for Efficient, Field -Free  \nCurrent -Induced Magnetization Switching in a Magnetic Insulator  \n \nLeran Wang1, Alejandro O. Leon2*, Wenqing He 3, Zhongyu Liang1,  \nXiaohan L i3, Xiaoxiao Fang1, Wenyun Yang1, Licong Peng4,  \nJinbo Yang1,5*, Caihua Wan3, Gerrit E. W. Bauer6,7, Zhaochu Luo1,5* \n1State Key Laboratory of Artificial Microstructure and Mesoscopic Physics, Institute of Condensed \nMatter Physics and Materials, School of Physics, Peking University, 100871 Beijing, China.  \n2Departamento de Física, Facultad de Ciencias Naturales, Matemática y del Medio Ambiente, \nUniversidad Tecnológica Metropolitana, Las Palmeras 3360, Ñuñoa 780- 0003, Santiago, Chile.  \n3Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, University of Chinese \nAcademy of Sciences, Chinese Academy of Sciences, 100190 Beijing, China.  \n4School of Materials Science and Engineering, Peking University, 100871 Beijing, China.  \n5Beijing Key Laboratory for Magnetoelectric Materials and Devices, 100871 Beijing, China  \n6Kavli Institute for Theoretical Sciences, University of the Chinese Academy of Sciences, 100864 \nBeijing, China.  \n7Advanced Institute for Materials Research (AIMR), Tohoku University, 980- 8576 Sendai, Japan.  \n \n*Correspondence to: zhaochu.luo@pku.edu.cn (Z.Luo); aleonv@utem.cl  (A.L.);  \njbyang@pku.edu.cn  (J.Y .)  \n \nKeywords: Magnetic coupling; magnetic insulator; spin- orbit torques; spin -mixing \nconductance; spin -Hall magnetoresistance  \nAbstract: Owing to the unique features of low Gilbert damping, long spin -diffusion lengths \nand zero Ohmic losses, magnetic insulators are pro mising candidate materials for next -\ngeneration spintronic applications. However, due to the localized magnetic moments and the \ncomplex metal -oxide interface between magnetic insulators and heavy metals, spin -functional \nDzyaloshinskii –Moriya interactions or  spin Hall and Edelstein effects are weak, which \ndiminishes the performance of these typical building blocks for spintronic devices. Here, we \nexploit the exchange coupling between metallic and insulating magnets for efficient electrical \nmanipulation of heavy metal/magnetic insulator heterostructures. By inserting a thin Co layer, \nwe enhance the spin -orbit torque efficiency by more than 20 times, which significantly reduces \nthe switching current density. Moreover, we demonstrate field -free current -induced \nmagnetization switching caused by a symmetry- breaking non -collinear magnetic texture. Our \nwork launches magnetic insulators as an alternative platform of low -power spintronic devices.    \n 2 1. Introduction  \nEngineering the magnetic coupling plays a crucial role in determining the functionalities \nof spintronic devices. The interlayer Ruderman –Kittel–Kasuya –Yosida (RKKY) interaction[1] \nand exchange -bias effect[2] stabilize the magnetic reference layer[3,4] in the magnetic tunnel \njunctions of non- volatile magnetoresistive random -access memories (MRAM), while the \nintralayer dipolar and chiral couplings are essential for scalable spin logic[5-7] and neuromorphic \ncomputing devices[8-10]. Research on the physics and applications of these couplings focusses \non electric conductors such as magnetic/ non-magnetic and ferro magnetic/antiferromagnetic \nmetal bilayers, but  much of the cor responding phenomenology in magnetic insulators remains \nto be explored. Taking advantages of low Gilbert damping[11,12], long spin- diffusion lengths[13-\n15] and zero Ohmic losses, magnetic insulators, particularly the iron garnets, are promising \ncandidate materials for low -power and high -speed spintronic applications. Pioneering studies \non magnetic metal/magnetic insulator bilayers demonstrated injection and modulation of spin \nwaves[16-20], but due to the complexity of polycrystalline metal- oxide interfaces[21-23] a full \nunderstanding of the underlying mechanisms remains elusive.  \nHere, we report experiments on a high- quality magnetic heterostructure comprising of an \nultrathin Co film on an epitaxial terbium iron garnet (Tb 3Fe5O12, TbIG ) layer, which shows a \nstrong ferromagnetic exchange coupling of ~69.0 μ J/m2. By growing a heavy metal layer on \ntop of the Co/TbIG bilayer, we electrically detect and manipulate the TbIG magnetization via \nthe direct and inverse spin Hall and/or Edelstein effects. The interlayer exchange coupling \nsignificantly enhances the interfac ial spin -mixing conductance, leading to a large spin -Hall \nmagnetoresistance (SMR) and a high spin -orbit torque (SOT) efficiency. Moreover, the non-\ncollinear magnetization of the Co  layer breaks the mirror symmetry of the device, thereby \nfacilitating current -induced switching of the TbIG magnetization without external magnetic \nfields. Our work demonstrates an interplay between magnetic coupling and SOTs that can help to design reliable and efficient magnetic memory devices and pave a new pathway for magnetic \ninsulator spintronics.  \n2. Structural and magnetic properties of Pt/TbIG   \n 3 The insulating rare -earth iron garnet TbIG was epitaxially grown on (111) -oriented single -\ncrystalline gadolinium gallium garnet (Gd 3Ga5O12, GGG) substrates by pulsed laser deposition \n(PLD) (see Method s). The structure of TbIG is constructed with three magnetic sublattices: an \noctahedral Fe, a tetrahedral F e, and a dodecahedral Tb sublattice[24] (Figure 1a ). TbIG films \nwith various thicknesses ranging from 5 to 53 nm were deposited for different experiments. The \n5 nm -thick TbIG was used for the electrical tr ansport measurement, whereas the thick TbIG \nyielding a stronger structural signal was used for X -ray diffraction (XRD) analysis that gives \ninsight into the microstructure and strain state. As shown in Figure 1b, we present the symmetric \nXRD scans of a 53 n m-thick TbIG around the (444) peak of GGG substrate. The rocking curve \ngives a full -width at half -maximum (FWHM) of ~0.02 ° (see Supplementary Information S1), \nindicating a high- quality single crystallinity. We can further extract the lattice cons tant of d 444 \n= 0.1815 nm for the epitaxial TbIG  film which is slightly larger than that (~0.1795 nm) in a \nTbIG bulk[25], implying a strained state originating from the TbIG/GGG interface[26,27]. We then \nconducted the magnetization measurements using vibrating- sample magnetometer (VSM) and \nmagneto -optical Kerr microscopy (MOKE) to study the magnetic properties of TbIG films \n(Figure 1c). The magnetization hysteresis loops with out -of-plane magnetic fields reveal a \nstrong perpendicular magnetic anisotropy and a small saturation magnetization ( MS ~23 emu/cc ) \nas a result of the partial compensation of sublattice magnetizations.  \nWe sputtered a 5 nm -thick Pt layer on TbIG (5 nm)/GGG with a low deposition power (20 \nW) to mitigate the bombardment damage of Pt atoms on the TbIG surface. The cross- sectional \nhigh- resolution transmission electron microscopy (HR -TEM) image confirms the cl ean and \nsharp interface in the heterostructure that is p rerequisite to efficiently manipulate the \nmagnetization by spin currents (Figure 1d). The Pt/TbIG heterostructure was then patterned \ninto Hall bar structures with the channel width of 10 µ m using UV p hotolithography combined \nwith the ion milling process. We measured the electrical transport with out -of-plane magnetic \nfields and observed an anomalous Hall resistance of - 1.3 mΩ at room temperature (Figure 1e).  \nBy varying the measurement temperature, the coercivity exhibits a peak at a critical temperature \nat which the polarity of the anomalous Hall resistance is reversed, indicating magnetic moment compensation in TbIG around 270 K  (T\ncomp) (Figure 1f). Since the sign of the anomalous Hall \nresistance depends on the direction of the magnetic moment of  the tetrahedral Fe  sublattice[28],  \n 4 the sign is reversed  when the temperature varies across the compensation state due to the fact \nthat the alignment relation ship between the magnetic field and the magnetic moment of the \ntetrahedral Fe sublattice reverses at T comp (Figure 1e).  \n3. Spin Hall magnetoresistance and magnetic coupling in Pt/Co/TbIG  \nTo exploit the rich interfacial magnetic effect of heavy metal/metallic magnet such as high \nspin-mixing conductance and large Dzyaloshinskii –Moriya interaction (DMI), we inserted a \nthin Co layer ( tCo = 0.3, 0.6 and 0.9 nm as calculated by multiplying the deposition rate and \ntime) between the Pt and TbIG layers by DC magnetron sputtering ( Figure 2 a). With the \ninsertion of Co layers, a linear Hall resistance with positive slope emerges on top of the square -\nlike anomalous Hall loop, indicating an in- plane magnetic anisotropy of the Co layer (Figure \n2b). The Co magnetization gets fully perpendicular at  large out -of-plane magnetic fields (Figure \n2c). Interestingly, the magnitude of the square -like anomalous Hall loop is significantly \nenhanced for thicker Co films, accompanying a change of sign of the Hall resistance from \nnegative to positive (Figure 2d). Moreover, the anomalous Hall loop in Pt/Co ( 0.9 nm)/TbIG of \n52.0 mΩ is more than an order of magnitude larger than that in Pt/TbIG ( -1.3 mΩ) with the \nsame TbIG thickness. An ultrathin Co insertion therefore offers a sensitive method to \nelectrically detect the magnetization direction of insulators.  \nTwo mechanisms may cause the large anomalous Hall loop. On one hand, the exchange \ncoupling between Co and TbIG may force a non- collinear alignment of the Co and TbIG \nmagnetizations such that the out -of-plane component of the Co magnetization contributes to  \nthe anomalous Hall effect. In th is scenario, the longitudinal ( Rxx) and transverse Hall ( Rxy) \nresistance can be written as:  \n𝑅𝑅𝑥𝑥𝑥𝑥=𝑅𝑅0+∆𝑅𝑅AMR sin2𝜃𝜃Cosin2𝜑𝜑Co,                      (1)  \n𝑅𝑅𝑥𝑥𝑥𝑥=−∆𝑅𝑅AMR sin2𝜃𝜃Cosin2𝜑𝜑Co+𝑅𝑅AHECo𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃Co+𝑅𝑅OHE𝐻𝐻𝑧𝑧,             (2)  \nwhere R0 and ΔRAMR represent the magnetization -independent longitudinal resistance and the \nanisotropic magnetoresistance (AMR), respectively. 𝑅𝑅AHECo (ROHE) is the anomalous (ordinary) \nHall resistance. θ Co (φCo) is the angle between Co magnetization and z-axis ( y-axis) as defined \nin Figure 2a. We disregard the very small magnetic- proximity effect in Pt (see Supplementary \nInformation S2). The second mechanism would follow the spin Hall magnetoresistance (SMR)  \n 5 scenario as widely used to extract spin transport parameters in heavy metal/magnetic insulator \nbilayers[29,30] such as the interfacial spin -mixing conductance G. In such measurements, an \nelectric current is applied to the heavy metal layer and due to the spin Hall effect, it can generate a transverse spin current that will either be transmitted or reflected at the heavy metal/magnetic \ninsulator interface depending on the relative orientations between the spin current polarization \nσ and the magnetization m . The ratio of spin transmission/reflection at the interface will further \nmodulate the electric current in the heavy metal layer due to the inverse spin Hall effe ct. This \nleads to the presence of a resistance change that has a distinct angle dependence. In the high \nfield limit, the magnetizations of Co and TbIG are aligned along the field direction with angle θ (φ) with the z -axis ( y-axis). The longitudinal ( R\nxx) and transverse Hall ( Rxy) resistance then \nread[29,30]:  \n𝑅𝑅𝑥𝑥𝑥𝑥=𝑅𝑅0−∆𝑅𝑅SMR sin2𝜃𝜃cos2𝜑𝜑,                     (3) \n𝑅𝑅𝑥𝑥𝑥𝑥=−∆𝑅𝑅SMR sin2𝜃𝜃sin2𝜑𝜑+𝑅𝑅AHESMRcos𝜃𝜃+𝑅𝑅OHE𝐻𝐻𝑧𝑧,            (4)  \nwhere ΔRSMR and 𝑅𝑅AHESMR are the SMR and the SMR -induced anomalous Hall resistances of the \nTbIG/Co interfaces adjacent to Pt, respectively, which are a function of the Co thickness.  \nNote that the transverse SMR and the SMR -induced anomalous Hall effect (AHE) have \nthe same symmetry as the AMR -induced transverse (planar Hall) resistance and anomalous Hall \nresistance in Co, respectively. Hence, we cannot quantitively distinguish the cont ribution of \nanomalous Hall resistance from the SMR -induced AHE and conventional AHE by the \ntransverse resistance only (see Supplementary Information  S3). By contrast, the contributions \nof the AMR and SMR to the longitudinal resistance changes distinctly de pend on the \nmagnetization angles and can be resolved by angular -dependent measurements. As shown in \nFigures 2e -2g, we measured Rxx with a rotating magnetic field of 60 kOe in zx, zy and xy planes. \nAccording to the symmetry of SMR and AMR, a fit to the angular -dependent  data in zy plane \nyields ΔRSMR = 61.1, 87.7, 225.9 and 555.7 m Ω for different Co thickness of tCo=0, 0.3, 0.6 and \n0.9 nm, respectively, while a fit to the angular -dependent  data in zx  plane yields Δ RAMR = -0.9, \n1.4, 13.2 and 72.6 m Ω. The sizeable ΔRSMR implies a substantial absorption of the spin currents \nby the heavy metal/magnets interface.  \nInterfacial spin transmission/reflection can be quantified by the complex spin -mixing \nconductance G with a real ( Gr) and an imaginary ( Gi) part[29,30]. Gr is related to the damping - \n 6 like torques acting upon m, which is proportional to m ×(m×σ) and manifests itself as a \nlongitudinal SMR resistance Δ RSMR. On the other hand, the imaginary part G i is associated with \na field -like torque, which is proportional to m×σ and contributes a Hall resistance 𝑅𝑅AHESMR in Pt. \nThe experimental values of ΔRSMR and 𝑅𝑅AHESMR, and assuming spin Hall angle θ Pt = 0.08 and spin \ndiffusion length λPt =1.4 nm for Pt[30,32], we can estimate Gr = 4.3×1014 Ω-1m-2 and Gi = -1.2×1013 \nΩ-1m-2, which is consistent with reported values for Pt/magnetic garnet bilayers[30-32]. The \ninsertion of the Co layer greatly enhances the spin -mixing conductance up to one order of \nmagnitude to a G r for tCo of 0.6 and 0.9 nm of 3.9 ×1015 Ω-1m-2 and -5.8×1015 Ω-1m-2, respectively, \nmuch larger than that in simple Pt/TbIG or Pt/Co heterostructures[33], implying that the \nexchange coupling plays a critical role in the spin -dependent transport.  \nTo unravel the effect of the Co spacer, we recorded the hysteresis loop by sweeping the in -\nplane magnetic field. In Pt/TbIG, the Hall resistance exhibits the usual shape for a perpendicular \nmagnetization under an in -plane magnetic field ( Figure 3a ). By fitting the in -plane hysteresis \nloop, we can obtain the effective magnetic anisotropy field HK = 8 kOe and magnetic anisotropy \nenergy Ean = HKMS/2 = 9.2× 103 J m-3 (see Supplementary Information  S6). In contrast, Pt/Co \n(0.9 nm)/TbIG  shows a distinctly different shape with a sharp peak at small magnetic fields and \nsaturation at a higher magnetic field  (Figure 3b). This curve  contains information about the \nexchange coupling parameter J  between Co and TbIG. The magnetic energy per unit area of \nPt/Co/TbIG can be written as:  \n𝐸𝐸=−𝐽𝐽𝒎𝒎�Co∙𝒎𝒎�TbIG−𝐾𝐾TbIG𝑡𝑡TbIG𝒎𝒎�TbIG ,𝑧𝑧2−𝐾𝐾Co𝑡𝑡Co𝒎𝒎�Co,𝑧𝑧2−𝑯𝑯∙(𝒎𝒎Co+𝒎𝒎TbIG ),  (5)  \nwhere KCo(TbIG)  is the magnetic anisotropy coefficient and 𝒎𝒎�Co(TbIG )  is the unit vector of the \nmagnetization 𝒎𝒎Co(TbIG )  of Co (TbIG) . At zero magnetic field, the Co and TbIG \nmagnetization unit vectors 𝒎𝒎�Co and 𝒎𝒎�TbIG  are in -plane and out-of-plane , respectively . \nMinimizing Eq. (5) for H =0 and leading order in the interface exchange J  leads to:  \n𝑚𝑚�Co,𝑧𝑧=±𝐽𝐽(2|𝐾𝐾Co|𝑡𝑡Co) ⁄ ,                               \n𝑚𝑚�TbIG ,𝑥𝑥=𝐽𝐽(2|𝐾𝐾TbIG |𝑡𝑡TbIG ) ⁄ ,                         (6)   \nwhere the sign of ± holds for up - and downward TbIG  magnetization (see Supplement S4). A \nrelatively weak in -plane magnetic field overcomes  the exchange coupling and pulls the Co \nmagnetization into the plane, lead ing to a sharp decrease of the Hall resistance around zero \nmagnetic field (Figure 3c). Due to its large perpendicular magnetic anisotropy,  the TbIG  \n 7 magnetization persists to be perpendicular up to higher in- plane magnetic fields. The gradual \npull into the plane is accompanied by a reduced and ultimately vanishing Hall resistance. The \neffective magnetic anisotropy field H K = 53.0 kOe of TbIG in Co (0.9 nm)/TbIG is even larger \nthan that in pure TbIG (see Figure 3b). It has been reported that the interfaces of Pt/Co and \nCo/oxide can give large interfacial perpendicular magnetic anisotropy due to spin- orbit \ncoupling[34]. Owing to the interlayer exchange co upling and proximity effect, the perpendicular \nmagnetic anisotropy of TbIG may get enhancement from Pt/Co interfaces.  We can then deduce \na Hall resistance caused by the out -of-plane tilt of the Co magnetization ( 𝑅𝑅AHECo = 71.0 m Ω), \nwhich is larger than the high -field TbIG/Co SMR ( 𝑅𝑅AHESMR = -19.0 mΩ). We hence estimate G i \nin Pt/Co (0.9 nm)/TbIG  to be -7.2×1014 Ω-1 m-2, which is 60 times larger than that in Pt/TbIG. \nFrom the ratio of 𝑅𝑅AHECo at zero and that at the saturation out -of-plane magnetic fields (Figure \n2c) the competition between Co/TbIG exchange coupling and magnetic anisotropies leads to a \ntilt angle of the Co magnetization of 83.2 ° at zero magnetic fields, i.e. pulled out of the plane \nby 6.8° . Substituting the observed magnetic anisotropies into the macrospin model, we arrive \nat an  exchange coupling strength between Co and TbIG of J = 69.0 μ J m-2 (see Supplementary \nInformation S4 ).  \n4. Efficient current -induced magnetization switching  \nThe enhanced spin -mixing conductance implies efficient SOTs in current -biased \nPt/Co/TbIG structures. We measured the SOT efficiency by recording magnetic hysteresis loops as a function of a charge current bias and in -plane and out -of-plane magnetic fields,  a common \ntechnique for both metallic and insulating magnet/heavy metal bilayers\n[35-37]. As shown in \nFigure 4a , we can measure the damping -like SOTs by the current -induced shifts of the \nhysteresis loops since they give rise to out -of-plane effective fields H eff that act on the DMI -\nstabilized Néel -type domain walls[36]. We may define an SOT efficiency χ as: \n𝜒𝜒=𝐻𝐻eff/𝑗𝑗.                                 (7)  \nRepresentative hysteresis loops of Pt/Co (0.6 nm)/TbIG with H x = 300 Oe and j = ±4.3×1010 \nAm-2 are shown in Figure 4b. Changing the direction of the applied currents cause opposite \nshifts of the hysteresis loops as expected for a current -induced Heff caused by damping -like \ntorques. The hysteresis loops shifts increase with an in -plane magnetic field and saturate at a  \n 8 critical value that is governed by the DMI[32,33,35] (Figure 4c). In Pt/TbIG, the SOT efficiency \nsaturates to 0.9 ×10-14 TA-1m2 at Hx = 340 Oe. We can estimate the modulus of the effective DMI \nconstant |𝐷𝐷|=𝜇𝜇0𝑀𝑀sΔ|𝐻𝐻𝑥𝑥| from the saturation field H x, where  μ0 is the vacuum permeability \nand Δ = (A/Ku)1/2 the width of the domain wall. Here, A = 2.3 pJ m-1 is the exchange stiffness \nand Ku is the anisotropy energy obtained from the experiment , leading to |DPt/TbIG | =8.6 μJ m-2 \nconsistent with other  Pt/magnetic garnet bilayers[35,36,38 -41],. Interestingly, the insertion of a thin \nCo layer between TbIG and Pt significantly enhances the SOT efficiency as well as the effective \nDMI. For Pt/Co (0.6 nm)/TbIG 𝜒𝜒=24.0× 10-14 TA-1m2 at Hx = 500 Oe, more than an order of \nmagnitude larger than that of other heavy metal/magnetic insulator systems[35,42].  \nThe model of magnetization reversal by the motion and annihilation of Neel domain walls \nimplies that the spin -orbit torque is most efficient at the domain wall center (see Figure 4a) \nwhere the magnetizations of Co and TbIG are strictly parallel. We may the refore carry over the \nanalysis of the SMR in the collinear limit of high magnetic fields to understand the observed \nenhancement of the damping- like torque. Here the Co capping layer enhances the effective spin \nmixing conductance G r from a small value for p ure TbIG to the large one of Co. Since Gr \nmeasures the absorption of the transverse spin current by the ferromagnet, this result directly \nexplains the enhanced spin transfer torque. The interface exchange coupling subsequently \ncommunicates the torque to the TbIG which leads to the motion of the e ntire domain wall. The \nCo overlayer ensures non -chiral domain walls required for the magnetization reversal. An \ninteresting subject for future research is an analysis of the Co -thickness dependence of the spin \ntransfer, since partial trapping of the spins into quantum wells[43,44] and the associated multiple \nscattering at the TbIG/Co interface could additionally increase 𝐻𝐻eff.  \nSince large damping -like torques in Pt/Co/TbIG should reduce the critical currents that \nswitch the magnetization, we measure the changes in the transverse resistance under current \npulses with a duration of 0.2 ms and a DC current of up to 1 mA corresponding to a current \ndensity of 2.5 ×1010 Am-2 in the same direction (see Supplementary Information S5). In the \npresence of an in -plane magnetic field of 400 Oe  along the current direction, pulses with an \namplitude up to j  = 6.1×1010 Am-2 change the sign of the t ransverse resistance which indicates \na complete TbIG magnetization switching (Figure 4d). Reversing the direction of the in- plane \nmagnetic field leads to an anti -clockwise hysteresis loop (Figure 4e), which agrees with the  \n 9 sign of the spin Hall angle of Pt[45]. Increasing the in- plane magnetic fields reduces the energy \nbarrier of magnetization reversal and the critical switching current density of T bIG/Co (0.6 \nnm)/P t from 6.9×1010 to 5.0×1010 Am-2 (Figure 4f) half of that in Pt/TbIG (1.2 ×1011 Am-2 at H x \n= 204 Oe). The suppression of the critical switching current density is less than expected from \nthe enhanced SOT efficiency between Pt/Co/TbIG and Pt/TbIG deduced above, which suggests that Joule heating plays a role in the magnetization switching at high  current densities\n[46,47].  \n5. Field-free magnetization switching  \nCurrent -induced switching of perpendicular magnetization without the need to apply \nmagnetic fields is highly desirable in high -density magnetic memories. To this end, various \nschemes have been proposed in metallic systems, e.g., by breaking the mirror symm etry in \nasymmetric lateral designs[48,49] and non- collinear magnetic alignment[50,51]. These invoke \ncomplex  devices incorporating multiple exotic materials that might be difficult to realize with \nmagnetic insulators. Here we may take advantage of the exchange coupling between Co and TbIG that leads to a non- collinear magnetic texture that allows switching of a perpendicular \nmagnetization by electric currents without the assistance of magnetic fields ( Figure 5a ).  \nAs in Pt/Co (0.6 nm)/TbIG, we observed a full current -induced switching of magnetization \nin Pt/Co (0.9 nm)/TbIG  with the same switching polarity cycle as function of the in- plane \nmagnetic field sweeps (Figures 5b -5c). Taking the device out of the ele ctromagnet set -up, where \nthe residual magnetic field is less than 1.0 Oe, does not deteriorate the switching performance \n(Figures 5d -5e). The switching polarity is determined by the history of the applied in -plane \nmagnetic fields. When initialized with a p ositiv e (negative) in -plane magnetic field of 1 kOe, \nthe electric current can switch the magnetization with anti -clockwise (clockwise) polarity. To \nverify the reliability of current -induced field- free switching, we repeated measurements by \napplying alternating c urrent pulses of ± 1.43×10\n11 Am-2 in the absence of magnetic fields (Figure \n5f). The Hall resistance jumps between the resistance levels corresponding to up and down \nmagnetizations after every pulse. The switching polarity reverses when the direction of the  pre-\nset in -plane magnetic field changes from positive to negative, which is consistent with the \ncurrent -driven hysteresis loops.  \nThe SOT efficiency measurement in Pt/Co (0.9 nm)/TbIG  led to several interesting  \n 10 observations (Figure 5g). First, in contrast to Pt/Co (0.6 nm)/TbIG, the magnitude of SOT \nefficiency in Pt/Co (0.9 nm)/TbIG saturates at a low magnetic field, implying the existence of \nan effective in -plane magnetic field due to the coupling with the in- plane magnetized Co layer. \nSecond, the saturation SOT efficiency ( 2.6×10-14 TA-1m2) in Pt/Co (0.9 nm)/TbIG is lower than \nthat ( 24.0× 10-14 TA-1m2) in Pt/Co (0.6 nm)/TbIG, though the spin-mixing conductance in Pt/Co \n(0.9 nm)/TbIG is higher than that in Pt/Co (0.6 nm)/TbIG. However, a thicker Co layer also \ndissipates more spin currents, leading to a reduced SOT efficiency. Therefore, there is a trade -\noff of Co thickness to obtain the  optimum SOT efficiency. The SOT efficiency can be further \nenhanced by optimizing the thickness of TbIG[52] and tuning its magnetization compensation \nstate[53]. Moreover, there is a substantial  non-zero SOT efficiency at zero magnetic fields and \nits sign depe nds on the history of sweeping in- plane magnetic fields, supp orting the \nperformance of field -free magnetization switching.  \n6. Conclusion  \nIn summary, we  exploit the magnetic coupling between metallic and insulating magnets, \nto efficiently manipulate the magnetism in an insulator with perpendicular magnetization. The itinerant conduction electrons in metallic magnets provide rich and strong spin -related \ninterfacial effects in magnetic trilayers. By harnessing the magnetic coupling between metallic \nand insulating magnets, these interfacial effects can be imprinted into magnetic insulators, \nallowing for efficient electrical de tection and manipulation of magne tism. In addition, coating  \nthe interface with  a few metallic magnetic atoms significantly enhances the SOT efficiency as \nwell as the DMI. Furthermore, we demonstrate the performance of field -free current -induced \nmagnetization switching that results from symmetry -breaking non -collinear magnetic textures, \npaving the way for scalable magnetic memory devices. Therefore, our work offers a new avenue to engineer  efficient  spin memory and logic devices based on magnetic insulators.  \n \n  \n 11 Methods  \nGrowth of TbIG films : TbIG thin films were deposited on 5 mm × 5 mm Ga 3Gd5O12 (111) single -\nside-polished substrates via pulsed laser deposition (PLD) at a laser fluence of ~1.4 Jcm-2, and a \ntarget -to-substrate distance of ~6 cm. During deposition, the substrate temperature was heated to \n800°C and the oxygen pressure was 30 mtorr. After deposition, annealing process was performed to \npromote the epitaxial growth of TbIG films with high quality and the cooling rate of the chamber \nwas 20 °C min-1. Epitaxial growth of the films was confirmed via a high- resolution X -ray diffraction \n2θ scan of the (444) reflection. The thickness of thick TbIG films was determined by X -ray \nreflectometry, whereas the thickness of thin TbIG films (~5 nm) was calculated by the number of \nlaser pulses.  \nDevice fabrication and measurement : After the deposition of the TbIG thin film via PLD, metallic \nlayers (such as Pt and Co/P t) were deposited by DC magnetron sputtering at room temperature with \na base pressure <5 ×  10-8 torr. The deposition rate was 0.026 nms-1 for P t and 0.011 nms-1 for C o. \nThe thickness of Co and Pt layers was calculated according to the deposition rate. These multilayers \nwere patterned into Hall bars using a combination of UV photolithography and Ar ion milling \ntechnique with lateral dimensions of 10 µ m × 35 µm (width ×  length). The magnetic properties of \nTbIG films  were  measured by superconducting quantum interference device (SQUID) \nmagnetometer. 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Commun. 9, 3612 (2018).  \n53. H. Wu, Y. Xu, P. Deng, Q. Pan, S. A. Razavi, K. Wong, L. Huang, B. Dai, Q. Shao, G. Yu, X. Han, J.- C. Rojas -Sánchez, S. Mangin, K. L. Wang, Spin- Orbit Torque Switching \nof a Nearly Compensated Ferrimagnet by Topological Surface States. Adv. Mater.  31, \n1901681 (2019).  \n \n \n   \n 16 Figures and figure captions  \n \nFigure 1. Structural and magnetic properties of epitaxial TbIG films.  (a) Crystal structure \nof TbIG. The red, grey and blue balls represent the Tb, Fe and O atoms, respectively. Schematics \nof magnetic sublattices of TbIG illustrating the compensation state when the temperature is \nabove  or below the compensation temperature Tcomp, are shown . (b) X-ray diffraction patterns \n(2θ scan) of a 53 -nm-thick TbIG film grown on a (111) -oriented GGG substrate with peaks \nassigned to TbIG and GGG. ( c) Magnetization as a function of out -of-plane magnetic field Hz \nobtained by SQUID VSM (red curves) and MOKE (black curves) mea surement. ( d) Cross -\nsectional HR -TEM image of the T bIG/P t heterostructure on a GGG substrate. Scale bar: 10 nm. \n(e) AHE hysteresis loops measured at different temperatures in Pt/T bIG. (f) Measured \ncoercivity as a function of temperature measured in Pt/T bIG.  \n  \n \n 17  \nFigure 2. Transport properties of Pt/Co/TbIG heterostructures.  (a) Schematics of \nPt/Co/TbIG/GGG multilayer (not on scale) and optical image of a TbIG Hall bar device in a \nthree -dimensional rendering of the measurement setup with white s cale bar of 20 µm.  The \ncoordinate system and magnetic field angles are indicated. ( b) AHE hysteresis loops measured \nin Pt/Co/T bIG for different thicknesses of the Co layer. ( c) AHE hysteresis loops measured in \nPt/Co (0.9 nm)/T bIG up to large out -of-plane magnetic fields. The AHE hysteresis loops at \nsmall magnetic fields are shown in the inset. ( d) Magnitude of the anomalous Hall loops as a \nfunction of thickness of the Co layer. ( e-g) Angular -dependent longitudinal resistance in \nPt/Co/T bIG with an applied ma gnetic field of 60 kOe rotating in the xy, xz  and yz planes.  \n  \n \n 18  \nFigure 3. AHE hysteresis loops with in -plane magnetic fields. AHE hysteresis loops as a \nfunction of in- plane magnetic fields in ( a) Pt/TbIG and ( b) Pt/Co (0.9 nm)/TbIG. A fit to the \ndata according to the experimental parameters and the model of magnetic coupling is indicated. \n(c) Schematics illustrating the magnetization when the in -plane magnetic field is swept from \nnegative to positive in P t/Co (0.9 nm)/TbIG.  \n  \n \n 19  \nFigure 4. Efficient current -induced SOTs in Pt/Co (0.6 nm)/TbIG.  (a) Current -induced \ndomain- wall motion (domain expansion) with an in- plane magnetic field Hx used to realign \ndomain- wall moments. ( b) AHE measurement with currents of ±4.3×1010 Am-2 under an in-\nplane magnetic field Hx of 300 Oe. The horizontal shift of the hysteresis loops corresponds to \nthe current -induced effective field H eff. (c) SOT efficiency calculated from horizontal shifts for \ndifferent current densities at different in -plane magnetic fields in Pt/ TbIG and Pt/Co (0.6 \nnm)/TbIG. ( d) and ( e) Current -induced hysteresis loops with in- plane magnetic fields of H x = \n±400 Oe in Pt/Co ( 0.6 nm)/TbIG. The switching polarity is indicated by the arrows. ( f) Critical \nswitching current densit ies as a function of in -plane magnetic fields in Pt/ TbIG and Pt/Co ( 0.6 \nnm)/TbIG.  \n  \n \n 20  \nFigure 5. Field -free magnetization switching in Pt/Co (0.9 nm)/TbIG. (a) Mirror symmetry \nof the degenerate P t/Co/TbIG  magnetization configurations underlying field- free \nmagnetization switching. ( b) and ( c) dc current -induced hysteresis loops with in- plane magnetic \nfields of H x = ±140 Oe in Pt/Co ( 0.9 nm)/TbIG. ( d) and ( e): Current -induced hysteresis loops \nin the absence of magnetic fields in Pt/Co ( 0.9 nm)/TbIG after the pre -set with positive/negative \nin-plane magnetic fields of 1 kOe. ( f) Current -induced magnetization switching with 0.2 ms -\nlong current pulses of ±1.4×1011 Am-2 in the absence of an external magnetic field after the pre-\nset with positive/negative in -plane magnetic fields of 1 kOe. The magnetic field -induced \nhysteresis loops are shown in left, giving the resistance reference for magnetizations pointing \nup and down. ( g) SOT efficiency as a function of in -plane magnetic fields in Pt/Co (0.9 \nnm)/TbIG.   \n \n"    },    {        "title": "1308.3787v1.Thickness_and_power_dependence_of_the_spin_pumping_effect_in_Y3Fe5O12_Pt_heterostructures_measured_by_the_inverse_spin_Hall_effect.pdf",        "content": "arXiv:1308.3787v1  [cond-mat.mes-hall]  17 Aug 2013Thickness and power dependence of the spin-pumping effect in Y3Fe5O12/Pt\nheterostructures measured by the inverse spin Hall effect\nM. B. Jungfleisch,1,∗A. V. Chumak,1A. Kehlberger,2V. Lauer,1\nD. H. Kim,3M. C. Onbasli,3C. A. Ross,3M. Kl¨ aui,2and B. Hillebrands1\n1Fachbereich Physik and Landesforschungszentrum OPTIMAS,\nTechnische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany\n2Institute of Physics, Johannes Gutenberg-University Main z, 55099 Mainz, Germany\n3Department of Materials Science and Engineering, MIT, Camb ridge, MA 02139, USA\n(Dated: September 17, 2018)\nThe dependence of the spin-pumpingeffect on the yttrium iron garnet (Y 3Fe5O12, YIG) thickness\ndetected by the inverse spin Hall effect (ISHE)has been inves tigated quantitatively. Due to the spin-\npumping effect driven by the magnetization precession in the ferrimagnetic insulator Y 3Fe5O12film\na spin-polarized electron current is injected into the Pt la yer. This spin current is transformed into\nelectrical charge current by means of the ISHE. An increase o f the ISHE-voltage with increasing film\nthickness is observed and compared to the theoretically exp ected behavior. The effective damping\nparameter of the YIG/Pt samples is found to be enhanced with d ecreasing Y 3Fe5O12film thickness.\nThe investigated samples exhibit a spin mixing conductance ofg↑↓\neff= (7.43±0.36)×1018m−2and\na spin Hall angle of θISHE= 0.009±0.0008. Furthermore, the influence of nonlinear effects on the\ngenerated voltage and on the Gilbert damping parameter at hi gh excitation powers are revealed. It\nis shown that for small YIG film thicknesses a broadening of th e linewidth due to nonlinear effects at\nhighexcitation powers is suppressedbecause of alack ofnon linear multi-magnon scatteringchannels.\nWe have found that the variation of the spin-pumping efficienc y for thick YIG samples exhibiting\npronounced nonlinear effects is much smaller than the nonlin ear enhancement of the damping.\nI. INTRODUCTION\nThe generation and detection of spin currents have at-\ntracted much attention in the field of spintronics.1,2An\neffective method for detecting magnonic spin currents is\nthe combination of spin pumping and the inverse spin\nHall effect (ISHE). Spin pumping refers to the genera-\ntion of spin-polarized electron currents in a normal metal\nfrom the magnetization precession in an attached mag-\nnetic material.3,4These spin-polarized electron currents\nare transformed into conventional charge currents by the\nISHE, which allows for a convenient electric detection of\nspin-wave spin currents.5–7\nAfter the discovery of the spin-pumping effect in fer-\nrimagnetic insulator (yttrium iron garnet, Y 3Fe5O12,\nYIG)/non-magnetic metal (platinum, Pt) heterosystems\nby Kajiwara et al.7, there was rapidly emerging inter-\nest in the investigation of these structures.6–13Since\nY3Fe5O12is an insulator with a bandgap of 2.85 eV14no\ndirect injection of a spin-polarized electron current into\nthe Pt layer is possible. Thus, spin pumping in YIG/Pt\nstructures can only be realized by exchange interaction\nbetween conduction electrons in the Pt layer and local-\nized electrons in the YIG film.\nSpin pumping into the Pt layer transfers spin angular\nmomentum from the YIG film thus reducing the mag-\nnetization in the YIG. This angular momentum transfer\nresults in turn in an enhancement of the Gilbert damp-\ning of the magnetization precession. The magnitude of\nthe transfer of angular momentum is independent of the\nferromagnetic film thickness since spin pumping is an in-\nterface effect. However, with decreasing film thickness,\nthe ratio between surface to volume increases and, thus,the interface character of the spin-pumping effect comes\ninto play: the deprivation of spin angular momentum be-\ncomes notable with respect to the precession ofthe entire\nmagnetizationinthe ferromagneticlayer. Thus, theaver-\nage damping for the whole film increases with decreasing\nfilm thicknesses. It is predicted theoretically3and shown\nexperimentallyinferromagneticmetal/normalmetalhet-\nerostructures (Ni 81Fe19/Pt) that the damping enhance-\nment due to spin pumping is inversely proportional to\nthe thickness of the ferromagnet.15,16\nSince the direct injection of electrons from the insula-\ntor YIG into the Pt layer is not possible and spin pump-\ning is an interface effect, an optimal interface quality is\nrequired in order to obtain a high spin- to charge cur-\nrent conversion efficiency.17,18Furthermore, Tashiro et\nal. have experimentally demonstrated that the spin mix-\ning conductance is independent of the YIG thickness in\nYIG/Pt structures.11Recently, Castel et al. reported\non the YIG thickness and frequency dependence of the\nspin-pumpingprocess.19Incontrasttoourinvestigations,\nthey concentrate on rather thick ( >200 nm) YIG films,\nwhich are much thicker than the exchange correlation\nlength in YIG20–22and thicker than the Pt thickness.\nThus, the YIG film thickness dependence in the nanome-\nter range is still not addressed till now.\nIn this paper, we report systematic measurements of\nthe spin- to charge-current conversion in YIG/Pt struc-\ntures as a function of the YIG film thickness from 20 nm\nto 300 nm. The Pt thickness is kept constant at 8.5 nm\nfor all samples. We determine the effective damping as\nwell as the ISHE-voltage as a function of YIG thickness\nand find that the thickness plays a key role. From these\ncharacteristics the spin mixing conductance and the spin2\nFIG. 1. (Color online) (a) Schematic illustration of the ex-\nperimental setup. (b) Dimensions of the structured Pt layer\non the YIG films. The Pt layer was patterned by means of\noptical lithography and ion etching. (c) Scheme of combined\nspin-pumping process and inverse spin Hall effect.\nHall angle are estimated. The second part of this paper\naddresses microwave power dependent measurements of\nthe ISHE-induced voltage UISHEand the ferromagnetic\nresonance linewidth for varying YIG film thicknesses.\nThe occurrence of nonlinear magnon-magnon scattering\nprocessesonthe widening ofthe linewidth aswell astheir\ninfluence on the spin-pumping efficiency are discussed.\nII. SAMPLE FABRICATION AND\nEXPERIMENTAL DETAILS\nIn Fig.1(a) a schematic illustration of the investigated\nsamples is shown. Mono-crystalline Y 3Fe5O12samples of\n20, 70, 130, 200 and 300 nm thickness were deposited by\nmeans of pulsed laser deposition (PLD) from a stoichio-\nmetric target using a KrF excimer laser with a fluence of\n2.6J/cm2andarepetition rateof10Hz.23In orderto en-\nsure epitaxial growth of the films, single crystalline sub-\nTABLE I. Variation of saturation magnetization MSand\nGilbert damping parameter α0as a function of the YIG film\nthickness. Results are obtained using a VNA-FMR measure-\nment technique.\ndYIG(nm)MS(kA/m) α0(×10−3)\n20 161.7 ±0.2 2.169 ±0.069\n70 176.4 ±0.1 0.489 ±0.007\n130 175.1 ±0.2 0.430 ±0.015\n200 176.4 ±0.1 0.162 ±0.008\n300 176.5 ±0.1 0.093 ±0.007FIG. 2. Original Gilbert damping parameter α0measured by\nVNA-FMR technique. The increased damping at low sample\nthicknesses is explained by an enhanced ratio between surfa ce\nto volume, which results in an increased number of scatterin g\ncenters and, thus, in an increased damping. The inset shows\nthe saturation magnetization as a function of the YIG film\nthickness dYIG. The error bars are not visible in this scale.\nstratesofgadoliniumgalliumgarnet(Gd 3Ga5O12, GGG)\nin the (100) orientation were used. We achieved opti-\nmal deposition conditions for a substrate temperature\nof 650◦C±30◦C and an oxygen pressure of 6.67 ×10−3\nmbar. Afterwards, each film was annealed ex-situ at\n820◦C±30◦C by rapid thermal annealing for 300 s un-\nder a steady flow of oxygen. This improves the crystal-\nlographic order and reduces oxygen vacancies. We deter-\nmined the YIG thickness by profilometer measurements\nandthecrystallinequalitywascontrolledbyx-raydiffrac-\ntion(XRD). InordertodepositPtontothesamples,they\nwere transferred at atmosphere leading to possible sur-\nface adsorbates. Therefore, the YIG film surfaces were\ncleaned in-situ by a low power ion etching before the\nPt deposition.17We used DC sputtering under an ar-\ngon pressure of 1 ×10−2mbar at room temperature to\ndeposit the Pt layers. XRR measurements yielded a Pt\nthickness of 8.5 nm, which is identical for every sample\ndue to the simultaneously performed Pt deposition. The\nPt layer was patterned by means of optical lithography\nand ion etching. In order to isolate the Pt stripes from\nthe antenna we deposited a 300 nm thick square of SU-8\nphotoresist on the top. A sketch of the samples and the\nexperimental setup is shown in Fig. 1(a), the dimensions\nof the structured Pt stripe are depicted in Fig. 1(b).\nIn order to corroborate the quality of the fabricated\nYIG samples, we performed ferromagnetic resonance\n(FMR) measurements using a vector network analyzer\n(VNA).25Since the area deposited by Pt is small com-\npared to the entire sample size, we measure the damp-\ningα0of the bare YIG by VNA (this approach results\nin a small overestimate of α0), whereas in the spin-\npumping measurement we detect the enhanced damp-\ningαeffof the Pt covered YIG films. The VNA-FMR\nresults are summarized in Tab. Iand in Fig. 2. Appar-\nently, the 20 nm sample features the largest damping3\nofα20nm\n0= (2.169±0.069)×10−3. With increasing film\nthickness α0decreasesto α300nm\n0= (0.093±0.007)×10−3.\nThere might be two reasons for the observed behavior:\n(1) The quality of the thinner YIG films might be worse\ndue to the fabrication process by PLD. (2) For smaller\nYIG film thicknesses, the ratio between surface to vol-\nume increases. Thus, the two-magnon scattering pro-\ncess at the interface is more pronounced for smaller film\nthicknesses and gives rise to additional damping.26The\nVNA-FMR technique yields the saturation magnetiza-\ntionMSfor the YIG samples (see inset in Fig. 2and\nTab.I). The observed values for MSare larger than the\nbulkvalue,27,28butinagreementwiththevaluesreported\nfor thin films.29The general trend of the film thickness\ndependence of MSis in agreement with the one reported\nin Ref.29,30and might be associated with a lower crystal\nquality after the annealing.\nThespin-pumpingmeasurementsfordifferentYIGfilm\nthicknesses were performed in the following way. The\nsamples were magnetized in the film plane by an exter-\nnal magnetic field H, and the magnetization dynamics\nwas excited at a constant frequency of f= 6.8 GHz by\nan Agilent E8257D microwave source. The microwave\nsignals with powers Pappliedof 1, 10, 20, 50, 100, 250 and\n500mW wereapplied to a 600 µm wide 50Ohm-matched\nCumicrostripantenna. Whiletheexternalmagneticfield\nwas swept, the ISHE-voltage UISHEwas recorded at the\nedges of the Pt stripe using a lock-in technique with an\namplitudemodulationatafrequencyof500Hz, aswellas\nthe absorbed microwave power Pabs. All measurements\nwere performed at room temperature.\nIII. THEORETICAL BACKGROUND\nThe equations describing the ferromagnetic resonance,\nthe spin pumping and the inverse spin Hall effect are\nprovided in the following and used in the experimental\npart of this paper.\nA. Ferromagnetic resonance\nIn equilibrium the magnetization Min aferromagnetic\nmaterial is aligned along the bias magnetic field H. Ap-\nplying an alternating microwave magnetic field h∼per-\npendicularly to the external field Hresults in a torque\nonMand causes the magnetic moments in the sample to\nprecess (see also Fig. 1(a)). In ferromagnetic resonance\n(FMR)themagneticfield Handtheprecessionfrequency\nffulfill the Kittel equation31\nf=µ0γ\n2π/radicalbig\nHFMR(HFMR+MS), (1)\nwhereµ0is the vacuum permeability, γis the gyromag-\nnetic ratio, HFMRis the ferromagnetic resonance field\nandMSis the saturation magnetization (experimentallyobtained values of MSfor our samples can be found in\nTab.I).\nThe FMR linewidth ∆ H(full width at half maximum)\nis related to the Gilbert damping parameter αas16,18,27\nµ0∆H= 4πfα/γ. (2)\nB. Spin pumping\nBy attaching a thin Pt layer to a ferromagnet, the\nresonance linewidth is enhanced,3which accounts for an\ninjection of a spin current from the ferromagnet into the\nnormal metal due to the spin-pumping effect (see illus-\ntration in Fig. 1(c)). In this process the magnetization\nprecession loses spin angular momentum, which gives\nrise to additional damping and, thus, to an enhanced\nlinewidth. The effective Gilbert damping parameter αeff\nfor the YIG/Pt film is described as16\nαeff=α0+∆α=α0+gµB\n4πMSdYIGg↑↓\neff,(3)\nwhereα0is the intrinsic damping of the bare YIG film,\ngis the g-factor, µBis the Bohr magneton, dYIGis the\nYIG film thickness and g↑↓\neffis the real part of the ef-\nfective spin mixing conductance. The effective Gilbert\ndamping parameter αeffis inversely proportional to the\nYIG film thickness dYIG: with decreasing YIG thickness\nthe linewidth and, thus, the effective damping parameter\nincreases.\nWhen the system is resonantlydriven in the FMR con-\ndition, a spin-polarized electron current is injected from\nthe magnetic material (YIG) into the normal metal (Pt).\nInaphenomenologicalspin-pumpingmodel, theDCcom-\nponent of the spin-current density jsat the interface, in-\njected in y-direction into the Pt layer (Fig. 1(c)), can be\ndescribed as15,16,32\njs=f/integraldisplay1/f\n0¯h\n4πg↑↓\neff1\nM2\nS/braceleftBig\nM(t)×dM(t)\ndt/bracerightBig\nzdt,(4)\nwhereM(t) is the magnetization. {M(t)×dM(t)\ndt}zis the\nz-component of {M(t)×dM(t)\ndt}, which is directed along\nthe equilibrium axis of the magnetization (see Fig. 1(c)).\nDue to spin relaxation in the normal metal (Pt) the\ninjected spin current jsdecays along the Pt thickness\n(y-direction in Fig. 1(c)) as15,16\njs(y) =sinhdPt−y\nλ\nsinhdPt\nλj0\ns, (5)\nwhereλis the spin-diffusion length in the Pt layer. From\nEq. (4) one can deduce the spin-current density at the\ninterface ( y= 0)15\nj0\ns=g↑↓\neffγ2(µ0h∼)2¯h(µ0MSγ+/radicalbig\n(µ0MSγ)2+16(πf)2)\n8πα2\neff((µ0MSγ)2+16(πf)2).\n(6)4\nFIG. 3. (Color online) ISHE-induced voltage UISHEas a func-\ntion of the magnetic field Hfor different YIG film thicknesses\ndYIG. Applied microwave power Papplied= 10 mW, ISHE-\nvoltage for the 20 nm thick sample is multiplied by a factor\nof 5.\nSincej0\nsis inverselyproportionalto α2\neffandαeffdepends\ninversely on dYIG(Eq. (3)), the spin-current density at\nthe interface j0\nsincreases with increasing YIG film thick-\nnessdYIG.\nC. Inverse spin Hall effect\nThe Pt layer acts as a spin-current detector and trans-\nforms the spin-polarized electron current injected due to\nthe spin-pumping effect into an electrical charge current\nby means of the ISHE (see Fig. 1(c)) as6,7,12,15,16\njc=θISHE2e\n¯hjs×σ, (7)\nwhereθISHE,e,σdenote the spin Hall angle, the elec-\ntron’s elementary charge and the spin-polarization vec-\ntor, respectively. Averaging the charge-current density\nover the Pt thickness and taking into account Eqs. ( 4) –\n(7) yields\n¯jc=1\ndPt/integraldisplaydPt\n0jc(y)dy=θISHEλ\ndPt2e\n¯htanh/parenleftbigdPt\n2λ/parenrightbig\nj0\ns.(8)\nTaking into account Eqs. ( 3), (6) and (8) we calcu-\nlate the theoretically expected behavior of IISHE=A¯jc,\nwhereAis the cross section of the Pt layer. Ohm’s law\nconnects the ISHE-voltage UISHEwith the ISHE-current\nIISHEviaUISHE=IISHE·R, whereRis the electric resis-\ntance of the Pt layer. Rvaries between 1450 Ω and 1850\nΩ for the different samples.\nIV. YIG FILM THICKNESS DEPENDENCE OF\nTHE SPIN-PUMPING EFFECT DETECTED BY\nTHE ISHE\nIn Fig.3the magnetic field dependence of the gener-\nated ISHE-voltage UISHEas a function of the YIG filmthickness is shown. Clearly, the maximal voltage UISHE\nat the resonance field HFMRand the FMR linewidth ∆ H\nvary with the YIG film thickness. The general trend\nshows, that the thinner the sample the smaller is the\nmagnitude of the observed voltage UISHE. At the same\ntime the FMR linewidth increases with decreasing YIG\nfilm thickness.\nIn the following the ISHE-voltage generated by spin\npumping is investigated as a function of the YIG film\nthickness. For these investigations we have chosen a\nrather small exciting microwave power of 1 mW. Thus,\nnonlinear effects like the FMR linewidth broadening due\nto nonlinear multi-magnon processes can be excluded\n(such processes will be discussed in Sec. V). Sec.IVA\ncovers the YIG thickness dependent variation of the en-\nhanced damping parameter αeff. From these measure-\nments the spin mixing conductance g↑↓\neffis deduced. In\nSec.IVBwe focus on the maximal ISHE-voltage driven\nby spin pumping as a function of the YIG film thickness.\nFinally, the spin Hall angle θISHEis determined.\nA. YIG film parameters as a function of the YIG\nfilm thickness\nAs described in Sec. IIIB, the damping parameter is\nenhanced when a Pt layeris deposited onto the YIG film.\nThisenhancementisinvestigatedasafunctionoftheYIG\nfilm thickness: the effective Gilbert damping parameter\nαeff(see Eq. ( 3)) is obtained from a Lorentzian fit to the\nexperimental data depicted in Fig. 3and Eq. ( 2). The\nresultisshowninFig. 4. With decreasingYIG film thick-\nness the linewidth and, thus, the effective damping αeff\nincreases. This behavior is theoretically expected: ac-\ncording to Eq. ( 3)αeffis inversely proportional to dYIG.\nSince the Pt film is grown onto all YIG samples simulta-\nFIG. 4. (Color online) Enhanced damping parameter αeff\nof the YIG/Pt samples obtained by spin-pumping measure-\nments. The red solid curve shows a fit to Eq. (3) taking the\nFMR measured values for MSand a constant value for g↑↓\neff\ninto account. Papplied= 1 mW. The error bars for the mea-\nsurement points at higher sample thicknesses are not visibl e\nin this scale.5\nFIG. 5. (Color online) (a) ISHE-voltage UISHEas a function\nof the YIG film thickness dYIG. The black line is a linear in-\nterpolation as a guide tothe eye. (b) Corresponding thickne ss\ndependent charge current IISHE. The red curve shows a fit to\nEqs. (6), (7), (8) with theparameters g↑↓\neff=(7.43±0.36)×1018\nm−2andθISHE= 0.009±0.0008. The applied microwave\npower used is Papplied= 1 mW.\nneously, the spin mixing conductance g↑↓\neffat the interface\nis considered to be constant for all samples.11Assum-\ningg↑↓\neffas constant and taking the saturation magne-\ntizationMSobtained by VNA-FMR measurements (see\nFig.2and Tab. I) into account, a fit to Eq. ( 3) yields\ng↑↓\neff= (7.43±0.36)×1018m−2. The fit is depicted as a\nred solid line in Fig. 4.\nB. YIG thickness dependence of the ISHE-voltage\ndriven by spin pumping\nFig.5(a) shows the maximum voltage UISHEat the\nresonance field HFMRas a function of the YIG film\nthickness. UISHEincreases up to a YIG film thickness\nof around 200 nm when it starts to saturate (in the\ncase of an applied microwave power of P applied= 1\nmW). The corresponding charge current IISHEis shown\nin Fig.5(b). The observed thickness dependent behavior\nis in agreement with the one reported for Ni 81Fe19/Pt16\nand for Y 3Fe5O12/Pt.11With increasing YIG film thick-\nness the generated ISHE-current increases and tends to\nsaturate at thicknesses near 200 nm (Fig. 5(b)). Accord-\ning to Eq. ( 3), (6) and (8) it isIISHE∝j0\ns∝1/α2\neff∝\n(α0+c/dYIG)−2, wherecis a constant. Therefore, theISHE-current IISHEincreases with increasing YIG film\nthickness dYIGand goes into saturation at a certain YIG\nthickness.\nFrom Eqs. ( 3), (6) and (8) we determine the expected\nbehavior of IISHE=A¯jcand compare it with our ex-\nperimental data. In order to do so, the measured values\nforMS(see Tab. I), the original damping parameter α0\ndetermined by VNA-FMR measurements at 1 mW (see\nTab.I) and the enhanced damping parameter αeffob-\ntained by spin-pumping measurements at a microwave\npower of 1 mW (see Fig. 4) are used. The Pt layer thick-\nness isdPt= 8.5 nm and the microwave magnetic field\nis determined to be h∼= 3.2 A/m for an applied mi-\ncrowavepower of 1 mW using an analytical expression.24\nThe spin-diffusion length in Pt is taken from literature as\nλ= 10 nm33,36and the damping parameter is assumed\nto be constant as α0= 6.68×10−4, which is the aver-\nage of the measured values of α0. The fit is shown as a\nred solid line in Fig. 5(b). We find a spin Hall angle of\nθISHE= 0.009±0.0008,which is in agreementwith litera-\nture values varying in a range of 0.0037- 0.08.33–35Using\nFIG. 6. (Color online) (a) YIG thickness dependence of the\nISHE-voltage driven by spin pumping for microwave powers\nin the range between 1 and 500 mW. The general thickness\ndependent behavior is independent of the applied microwave\npower. The error bars for the measurement at lower mi-\ncrowave powers are not visible in this scale. (b) Deviation\nof the ISHE-voltage from the linear behavior with respect to\nthe measured voltage U500mW\nISHE. The inset shows experimental\ndata for a YIG film thickness dYIG= 20 nm and the theoret-\nically expected curve. The error bars of the 20 nm and the\n70 nm samples are not visible in this scale.6\nthe fit we estimate the saturation value of the generated\ncurrent. Although we observe a transition to saturation\nat sample thicknesses of 200 – 300 nm, we find that ac-\ncording to our fit, 90% of the estimated saturation level\nof 5 nA is reached at a sample thickness of 1.2 µm.\nV. INFLUENCE OF NONLINEAR EFFECTS ON\nTHE SPIN-PUMPING PROCESS FOR VARYING\nYIG FILM THICKNESSES\nIn order to investigate nonlinear effects on the spin-\npumping effect for varying YIG film thicknesses, we per-\nformed microwavepower dependent measurements of the\nISHE-voltage UISHEas function of the film thickness\ndYIG. For higher microwavepowersin the rangeof 1 mW\nto 500 mW we observe the same thickness-dependent\nbehavior of the ISHE-voltage as in the linear case\n(Papplied= 1 mW, discussed in Sec. IVB): Near 200 nm\nUISHEstarts to saturate independently of the applied mi-\ncrowave power, as it is shown in Fig. 6(a). Furthermore,\nit is clearly visible from Fig. 6(a) that for a constant\nfilm thickness the spin pumping driven ISHE-voltage in-\ncreases with increasing applied microwave power. At\nhigh microwave powers the voltage does not grow lin-\nearly and saturates. Fig. 6(b) shows the deviation of\nthe ISHE-voltage ∆ UISHEfrom the linear behavior with\nrespect to the measured value of U500mW\nISHEat the excita-\ntion power Papplied= 500 mW. In order to obtain the\nrelation between UISHEandPappliedfor each YIG film\nthickness dYIGthe low power regime up to 20 mW is\nfitted by a linear curve and extrapolated to 500 mW.\nTheinsetin Fig. 6(b) showsthe correspondingviewgraph\nfor the case of the 20 nm thick sample. As it is visible\nfrom Fig. 6(b), the deviation from the linear behavior\nis drastically enhanced for larger YIG thicknesses. For\nthe thin 20 nm and 70 nm samples we observe an almost\nlinear behavior between UISHEandPappliedover the en-\ntire microwavepower range, whereas for the thicker sam-\nples the estimated linear behavior and the observed non-\nlinear behavior differ approximately by a factor of 2.5\n(Fig.6(b)). We observe an increase of the ISHE-voltage\nas well as an broadening of the FMR linewidth with in-\ncreasing microwave power. In Fig. 7(a) the normalized\nISHE-voltage UISHEas function of the external magnetic\nfieldHis shown for different microwave powers Papplied\nin the range of 1 mW to 500 mW (YIG film thickness\ndYIG= 300 nm). The linewidth tends to be asymmet-\nric at higher microwave powers. The shoulder at lower\nmagnetic field is widened in comparison to the shoulder\nat higher fields. The reason for this asymmetry might\nbe due to the formation of a foldover effect,37,38due to\nnonlinear damping or a nonlinear frequency shift.39,40\nThe results of the damping parameter αeffobtained\nby microwave power dependent spin-pumping measure-\nments are depicted in Fig. 7(b). It can be seen, that\nwith increasing excitation power the Gilbert damping for\nthicker YIG films is drastically increased. To present\nFIG. 7. (Color online) (a) Illustration of the linewidth bro ad-\nening at higher excitation powers. The normalized ISHE-\nvoltage spectra are shown as a function of the magnetic\nfieldHfor different excitation powers. Sample thickness:\n300 nm. (b) Power dependent measurement of the damp-\ning parameter αefffor different YIG film thicknesses dYIGob-\ntained by a Lorentzian fit to the ISHE-voltage signal. The\nerror bars are omitted in order to provide a better readabil-\nity of the viewgraph. (c) Nonlinear damping enhancement\n(α500mW\neff−α1mW\neff)/α1mW\neffas a function of the YIG film thick-\nnessdYIG. Due to a reduced number of scattering channels to\nother spin-wave modes for film thicknesses below 70 nm, the\ndamping is only enhanced for thicker YIG films with increas-\ning applied microwave powers. The error bars of the 200 nm\nand the 300 nm samples are not visible on this scale.\nthis result more clearly the nonlinear damping enhance-\nment (α500mW\neff−α1mW\neff)/α1mW\neffis shown in Fig. 7(c). The\ndampingparameterat asamplethicknessof20nm α20nm\neff7\nFIG. 8. (Color online) Dispersion relations calculated for each sample thickness taking into account the measured valu es of the\nsaturation magnetization MS(see Tab. I). Backward volume magnetostatic spin-wave mode s as well as magnetostatic surface\nspin-wave modes (in red) and the first perpendicular standin g thickness spin-wave modes are depicted (in black and gray) .\n(a)–(e) show the dispersion relations for the investigated sample thicknesses of 20 nm – 300 nm.\nis almost unaffected by a nonlinear broadening at high\nmicrowave powers. With increasing film thickness the\noriginal damping α1mW\neffatPapplied= 1 mW increases by\na factor of around 3 at Papplied= 500 mW. This factor\nis very close to the value of the deviation of the ISHE-\nvoltage from the linear behavior (Fig. 6(b)).\nThis behavior can be attributed to the enhanced prob-\nability of nonlinear multi-magnon processes at larger\nsample thicknesses: In order to understand this, a fun-\ndamental understanding of the restrictions for multi-\nmagnon scattering processes can be derived from the en-\nergy and momentum conservation laws:\nN/summationdisplay\ni¯hωi=M/summationdisplay\nj¯hωjandN/summationdisplay\ni¯hki=M/summationdisplay\nj¯hkj,(9)\nwhere the left/right sum of the equations runs over the\ninitial/final magnons with indices i/j which exist be-\nfore/after the scattering process, respectively.41–43The\nmost probable scattering mechanism in our case is the\nfour-magnon scattering process with N= 2 and M=\n2.43In Eq. (9) the wavevector ki/jand the frequency ωi/j\nare connected by the dispersion relation 2 πfi/j(ki/j) =\nωi/j(ki/j). The calculated dispersion relations are shown\nin Fig.8(backward volume magnetostatic spin-wave\nmodes with a propagation angle /negationslash(H,k) = 0◦as well\nas magnetostatic surface spin-wave modes /negationslash(H,k) =\n90◦).44For this purpose, the measured values of MS\n(Tab.I) for each sample are used. In the case of the\n20 nm sample thickness, the first perpendicular standing\nspin-wave mode (thickness mode) lies above 40 GHz, thesecond above 120 GHz. Thus, the nonlinear scattering\nprobability obeying the energy- and momentum conser-\nvation is largely reduced. This means magnons cannot\nfind a proper scattering partner and, thus, multi-magnon\nprocesses are prohibited or at least largely suppressed.\nWith increasing film thickness the number of standing\nspin-wavemodesincreasesand, thus, thescatteringprob-\nability grows. As a result, the scattering of spin waves\nfrom the initially excited uniform precession (FMR) to\nother modes is allowed and the relaxation of the original\nFMR mode is enhanced. Thus, the damping increases\nand we observe a broadening of the linewidth, which is\nequivalent to an enhanced Gilbert damping parameter\nαeffat higher YIG film thicknesses (see Fig. 7).\nIn orderto investigatehowthe spin-pumping efficiency\nis affected by the applied microwave power, we measure\nsimultaneously the generated ISHE-voltage UISHEand\nthe transmitted ( Ptrans), as well as the reflected ( Prefl)\nmicrowave power, which enables us to determine the\nabsorbed microwave power Pabs=Papplied−(Ptrans+\nPrefl).17Since the 300 nm sample exhibits a strong non-\nlinearity(largedeviationfromthelinearbehavior(Fig. 6)\nand large nonlinear linewidth enhancement (Fig. 7)), we\nanalyze this sample thickness. In Fig. 9the normalized\nabsorbed microwave power Pnorm=Pabs/PPapplied=1mW\nabs\nand the normalized ISHE-voltage in resonance Unorm=\nUISHE/UPapplied=1mW\nISHEare shown as a function of the ap-\nplied power Papplied. Both curves tend to saturate at\nhigh microwave powers above 100 mW. The absorbed\nmicrowave power increases by a factor of 110 for applied\nmicrowave powers in the range between 1 and 500 mW,8\nFIG. 9. (Color online) Normalized absorbed power Pnorm=\nPabs/PPapplied=1mW\nabs (black squares) and normalized ISHE-\nvoltageUnorm=UISHE/UPapplied=1mW\nISHE (red dots) for varying\nmicrowave powers Papplied. The inset illustrates the indepen-\ndence of the spin-pumping efficiency UISHE/PabsonPapplied.\nYIG thickness illustrated: 300 nm. Error bars of the low\npower measurements are not visible in this scale.\nwhereas the generated voltage increases by a factor of\n80. The spin-pumping efficiency UISHE/Pabs(see inset\nin Fig.9) varies within a range of 30% for the differ-\nent microwave powers Pappliedwithout clear trend. Since\nthe 300 nm thick film shows a nonlinear deviation of the\nISHE-voltageby afactorof2.3 (Fig. 6(b)) and the damp-\ning is enhanced by a factor of 3 in the same range of\nPapplied(Fig.7(c)), we conclude that the spin-pumping\nprocess is only weakly dependent on the magnitude of\nthe applied microwave power (see inset in Fig. 9). In our\nprevious studies reported in Ref.12,13we show that sec-\nondary magnons generated in a process of multi-magnon\nscattering contribute to the spin-pumping process and,\nthus, the spin-pumping efficiency does not depend on the\napplied microwave power.\nVI. SUMMARY\nThe Y 3Fe5O12thickness dependence of the spin-\npumping effect detected by the ISHE has been inves-\ntigated quantitatively. It is shown that the effective\nGilbert damping parameter of the the YIG/Pt sam-\nples is enhanced for smaller YIG film thicknesses, which\nis attributed to an increase of the ratio between sur-\nface to volume and, thus, to the interface character of\nthe spin-pumping effect. We observe a theoretically\nexpected increase of the ISHE-voltage with increasing\nYIG film thickness tending to saturate above thick-\nnesses near 200 – 300 nm. The spin mixing conductance\ng↑↓\neff= (7.43±0.36)×1018m−2as well as the spin Hall an-gleθISHE= 0.009±0.0008 are calculated and are found\nto be in agreement with values reported in the literature\nfor our materials.\nThe microwave power dependent measurements reveal\nthe occurrence of nonlinear effects for the different YIG\nfilm thicknesses: for low powers, the induced voltage\ngrows linearly with the power. At high powers, we ob-\nserve a saturation of the ISHE-voltage UISHEand a de-\nviation by a factor of 2.5 from the linear behavior. The\nmicrowave power dependent investigations of the Gilbert\ndamping parameter by spin pumping show an enhance-\nment by a factor of 3 at high sample thicknesses due\nto nonlinear effects. This enhancement of the damping\nis due to nonlinear scattering processes representing an\nadditional damping channel which absorbs energy from\nthe originally excited FMR. We have shown that the\nsmaller the sample thickness, the less dense is the spin-\nwave spectrum and, thus, the less nonlinear scattering\nchannels exist. Hence, the smallest investigated sample\nthicknesses (20 and 70 nm) exhibit a small deviation of\nthe ISHE-voltage from the linear behavior and a largely\nreduced enhancement of the damping parameter at high\nexcitation powers. Furthermore, we have found that the\nvariation of the spin-pumping efficiencies for thick YIG\nsamples which show strongly nonlinear effects is much\nsmaller than the nonlinear enhancement of the damping.\nThis is attributed to secondary magnons generated in a\nprocessofmulti-magnonscatteringthatcontributetothe\nspin pumping. It is shown, that even for thick samples\n(300 nm) the spin-pumping efficiency is only weakly de-\npendent on the applied microwave power and varies only\nwithin a range of 30% for the different microwave powers\nwithout a clear trend.\nOur findings provide a guideline to design and create\nefficient magnon- to charge current converters. Further-\nmore, the results are also substantial for the reversed\neffects: the excitation of spin waves in thin YIG/Pt bi-\nlayers by the direct spin Hall effect and the spin-transfer\ntorque effect.45\nVII. ACKNOWLEDGMENTS\nWe thank G.E.W. Bauer and V.I. Vasyuchka for valu-\nable discussions. 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Conca,1,∗A. Niesen,2G. Reiss,2and B. Hillebrands1\n1Fachbereich Physik and Landesforschungszentrum OPTIMAS,\nTechnische Universit¨ at Kaiserslautern, 67663 Kaisersla utern, Germany\n2Center for Spinelectronic Materials and Devices,\nPhysics Department, Bielefeld University, 100131 Bielefe ld, Germany\n(Dated: May 7, 2021)\nWe investigate a series of films with different thickness of th e Heusler alloy Co 2FeAl in order to\nstudy the effect of annealing on the interface with a MgO layer and on the bulk magnetic properties.\nOur results reveal that while the perpendicular interface a nisotropy constant K⊥\nSis zero for the as-\ndeposited samples, its value increases with annealing up to a value of 1 .14±0.07 mJ/m2for the\nseries annealed at 320oC and of 2 .01±0.7 mJ/m2for the 450oC annealed series owing to a strong\nmodification of the interface during the thermal treatment. This large value ensures a stabilization\nof a perpendicular magnetization orientation for a thickne ss below 1.7 nm. The data additionally\nshows that the in-plane biaxial anisotropy constant has a di fferent evolution with thickness in as-\ndeposited and annealed systems. The Gilbert damping parame terαshows minima for all series for\na thickness of 40 nm and an absolute minimum value of 2 .8±0.1×10−3. The thickness dependence\nis explained in terms of an inhomogeneous magnetization sta te generated by the interplay between\nthe different anisotropies of the system and by the crystalli ne disorder.\nINTRODUCTION\nIn order to achieve efficient spin torque switching, ma-\nterialswithacertainsetofpropertiesarerequired. These\nproperties are a combination of low damping and low\nmagnetization, together with the presence of a robust\nperpendicular magnetic anisotropy(PMA). Additionally,\nthese materials should have a high spin polarisation and\nbe compatible with standard tunneling barrier materials\nsuch as MgO or MgAl 2O4. A high Curie temperature is\nalso desirable to guarantee temperature stability.\nIn the wide family of the Heusler compounds, some\ncandidates can be found which fulfill the aforementioned\nrequirements. For instance, large tunneling magnetore-\nsistance (TMR) ratios have been reported for several\ncompounds [1–6]. Heusler films have been successfully\nemployed in systems with PMA [7–11] and show also\nlow damping properties [12]. For the PMA properties\nof thin Heusler films, the interface-induced perpendicular\nanisotropy plays a critical role and its strength is given\nby the value of the perpendicular interfacial anisotropy\nconstant K⊥\nS. The interfacial properties, and therefore\nthe value of the constant, are strongly modified by the\nexact conditions of the annealingtreatment for the stack,\nwhich is required to improve the crystalline order of the\nHeusler films [6, 13, 14] and to achieve large TMR val-\nues [35]. The alloy Co 2FeAl belongs to the materials\nfor which large TMR [15] have been reported, even for\ntextured films on a SiO 2amorphous substrate [16]. Low\ndamping[17–19]andPMA[18,20]havealsobeenproven.\nIn this work, we study the evolution with annealing of\nK⊥\nSin systems with a MgO interface, by measuring dif-\nferent thickness series. Since the in-plane anisotropies\nand the Gilbert damping parameter change with varyingthicknessandannealingtemperature, alsotheirevolution\nis reported. The relevance of the study is not limited to\nCo2FeAl but it is a model for all TMR systems with\nCo-based Heusler alloys and an interface with a MgO\ntunneling barrier.\nSAMPLE PREPARATION\nThickness series (7-80 nm) of Co 2FeAl (CFA) epitax-\nial films were prepared and a microstrip-based VNA-\nFMR setup was used to study their magnetic proper-\nties. The dependence of the in-plane anisotropies and\nthe Gilbert damping parameter on the thickness and the\ndetermination of the interface perpendicular anisotropy\nconstant K⊥\nSfor the CFA/MgO interface is presented for\nFIG. 1. (Color online) X-ray diffraction patterns of 20nm\nthinCFAlayersas-deposited, annealedat320oCandannealed\nat 450oC. The (002) superlattice and the fundamental (004)\npeak of the CFA are clearly visible, confirming the partial B2\ncrystalline order.2\nFIG. 2. (Color online) X-Ray reflectometry data correspond-\ning to samples with a CFA thickness of 20 nm and different\nannealing temperatures.\nas-deposited samples and for two different values of the\nannealing temperature.\nThe stack layer structure is MgO(100)(subs)/\nMgO(5)/CFA( d)/MgO(7)/Ru(2) with d= 7, 9, 11, 15,\n20, 40 and 80 nm. Rf-sputtering was used for the MgO\ndeposition and dc-sputtering for the rest. The values of\nthe annealing temperature for the two series with ther-\nmal treatment are 320oC and 450oC. The layer stacking\nis symmetrical around CFA so that a similar interface is\nexpected for both sides. The samples were all deposited\nat room temperature and annealed afterwardsunder vac-\nuum conditions.\nX-RAY CHARACTERIZATION\nCrystallographic properties of the CFA thin films were\ndetermined using x-ray diffraction (XRD) measurements\nin a Philips X’Pert Pro diffractometer equipped with a\nCu anode. The (002) superlattice and the fundamental\n(004) peak of the CFA can be observed (see Fig. 1) al-\nready for the as-deposited state. In-plane performed φ\nscan measurements reveal the absence of the (111) su-\nperlattice reflection in these films. Therefore, partial\nB2 crystalline order is verified. Epitaxial, 45orotated\ngrowth, relative to the MgO buffer layer, was verified us-\ning aφscan of the reflection from the (202) planes (not\nshown here). The epitaxial relationship CFA (001)[100]\n//MgO(001)[110]wasthereforeconfirmedforthesefilms,\ni.e. CFA grows with the same crystalline orientation as\nthe substrate but the unit cell is rotated 45oin plane\nrespect to the MgO unit cell.\nX-ray reflectometry (XRR) has been performed on the\n20 nm thick films and it is shown in Fig. 2. The esti-\nmation of the RMS value is only possible with a certain\nuncertainty due to the number of layers which increases\nthe number of fitting parameters but it is possible to say\nthat it lays around 0.1-0.3 nm for the three samples. In\nany case, it is evident that the interface is very smoothin all cases and that the annealing is not modifying the\nroughness properties.\nRESULTS AND DISCUSSION\nFrom the dependence of HFMRon the resonance fre-\nquencyfFMR, the effective magnetization Meffis ex-\ntracted using a fit to Kittel’s formula [23]. For a more\ndetailed description of the FMR measurement and anal-\nysis procedure please see Ref. [24]. Meffis related to the\nsaturation magnetization of CFA by [25–27]\nMeff=Ms−H⊥\nK=Ms−2K⊥\nS\nµ0Msd(1)\nwhereK⊥\nSis the perpendicular surface (or interfacial)\nanisotropy constant.\nFig. 3 shows the dependence of Meffon 1/dfor the\nthree CFA series. The lines are a fit to Eq. 1. Let us\nfirst discuss the case of the as-deposited series shown in\nFig. 3(a). An almost constant value for Meffis observed\nfor the low thickness range (15-7 nm) where the inter-\nface properties should become dominating. The fit gives\na value for K⊥\nSof 0.03±0.1 mJ/m2compatible with\nzero (hollow values in Fig. 3 not considered for the fit).\nThis implies that it is not possible to obtain a stable per-\npendicular magnetization orientation for any thickness\nvalue based only on the interface effect. However, it has\nto be commented that a non-vanishing volume perpen-\ndicular anisotropy has also been reported for CFA [20]\nwhich may indeed stabilize an out-of-plane orientation.\nConcerning the relative decrease of Mefffor large thick-\nnesses, we attribute this to a inhomogeneous magnetiza-\ntion state which is sometimes observedin thick films [34].\nThis point will be later commented when analyzing the\ndamping properties.\nFigs. 3(b) and (c) show the evolution of the situation\nwhen the annealingstep is applied. The interfaceproper-\nties changewith the thermaltreatment and K⊥\nSincreases\nto avalue of1 .14±0.07mJ/m2forthe 320oCcaseand of\n2.07±0.7 mJ/m2for 450oC. The larger error bar in the\nlater value is due to a larger scattering of values for Meff.\nArecentstudyoftheperpendicularanisotropyproperties\non CFA thin films has been published where a novel TiN\nbuffer layer is employed [7]. In- and out-of-plane hystere-\nsis loopsareused to determine the value of K⊥\nSinstead of\nthe FMR measurements used here. However, the largest\nobtained values for K⊥\nSare in both cases in accordance\nwith ours (0 .86±0.16 mJ/m2). For comparison it has\nto be taken into account that due to the presence of two\nCFA/MgO interfaces, the values presented here are ex-\npected to be a factor of two larger. Both values are then\nin good agreement. The different annealing temperature\nrange does not allow for a comparison of the evolution\nofK⊥\nSwith that parameter but a remarkable difference3\nFIG. 3. (Color online) Dependence of Meffextracted from\nthe Kittel fit on the inverse thickness 1 /dfor three sample\nseries: (a) as-deposited, (b) annealed at 320oC, (c) annealed\nat 450oC. The lines are a fit to Eq. 1, the hollow data points\nwere not considered.\ncan be found in the as-deposited samples. A compara-\ntively smaller but, contrary to our case, non-zero value\nis reported. This reveals the role of the TiN buffer layer\nin improving the interface quality.\nAlthough it cannot be quantified by XRD, the exis-\ntence of a certain level of stress in the films cannot be ex-\ncluded. This stress is changing upon annealing together\nwith the crystalline orderat the interface and therefore it\nis reasonableto admit that it playsa role in the evolution\nofK⊥\nS. However, it is not possible to separate the con-\ntribution to the evolution of the PMA due to these two\neffects. First principle calculations of K⊥\nSfor stress-free\nCFA/MgO interfaces [36] has provided a value for K⊥\nS\nof 1.31 mJ/m2for Co-terminated interfaces while FeAl-\ntermination does induce in-plane orientation. This value\nis compatible with our results for the 450oC case takinginto account that our samples have two CFA/MgO inter-\nfaces. In any case, our results are more compatible with\na Co-termination at the MgO interfaces following this\ncalculation. Other experimental results using XMCD at-\ntribute, contrarily to the previous calculation, a PMA\ncontribution to the Fe atoms at the interface [37]. The\nexact atomic origin of the PMA is then still under dis-\ncussion and therefore also the actual impact of stress.\nAs already shown in Fig. 2, the roughness remains un-\nchanged after the annealing process. The increase of K⊥\nS\nis then due to a more subtle change of the atomic order-\ning at the immediate interface and is not connected to a\nroughness modification, or at least not in a large degree.\nBy setting d=∞in Eq. 1 it is possible to extract a\nvalue for Msof 1140±30 kA/m from the linear fit for\nthe as-deposited samples. This value is larger than the\nones reported in [21, 28] (1000-1030 kA/m) but similar\nto a FMR study [32] on very thick (140 nm) CFA poly-\ncrystalline films providing a value of Meff= 1200 kA/m.\nThe saturation magnetization Msfor TiN buffered\nCFA, deposited and investigated by the same group,\nwas measured to be 1140 ±60 kA/m, which is in excel-\nlent agreement with the value obtained from the FMR\ndata. The saturation magnetization for TiN buffered\nCFA was obtained using alternating gradient magne-\ntometer (AGM) measurements and verified using vibrat-\ning sample magnetometry (VSM) on a 10 nm thin CFA\nlayer [7].\nThe value of Msalso increases upon annealing up to\n1213±8 kA/m for the 320oC series and 1340 ±70 kA/m\nfor the 450oC one. This increase can be attributed to an\nimprovement of the crystalline order with annealing.\nFrom the extrapolation of the linear fits to Meff= 0\nit is possible to extract the thickness at which the in-\nterfacial perpendicular anisotropy is able to stabilize an\nout-of-plane configuration by overcoming the demagne-\ntization field and allowing the magnetic easy axis to be\nout-of-plane. This thickness is 1.2 nm and 1.7 nm for\n320oC and 450oC annealing temperature, respectively.\nThe relative difference between both values for the crit-\nical thickness is smaller than the relative difference for\nK⊥\nSfor the respective temperature values. This is ex-\nplained by the larger Msvalue for the 450oC case for\nwhich a larger demagnetizing field must be overcome to\nachieve PMA.\nBelmeguenai et al.presented data very similar to\nthe one shown in Fig. 3(a) for (110)-ordered textured\nfilms [21] and for (100)-oriented epitaxial films grown on\nMgO(100) substrates [22]. The annealing temperature is\n600oC. The data is given for thickness values not smaller\nthan 10 nm. However, the interpretation of the data is\ncompletely opposite to ours, resulting in a negative value\nK⊥\nS=−1.8 mJ/m2. The negative value indicates that\nthe interface anisotropy is favoring an in-plane orienta-\ntion of the magnetization. PMA with Ta/CFA/MgO (or\nCrorRu)systemshavebeenindeedachieved[29–31]with4\nFIG. 4. (Color online) Dependence of the Gilbert damping\nparameter αon the thickness dfor three sample series: as-\ndeposited, annealed at 320oC, and annealed at 450oC. The\ninset shows the dependence of the linewidth ∆ Hon the fre-\nquency for the 80 nm samples. The lines are a linear fit used\nto extract the damping parameter α.\nvaluesof K⊥\nS= +0.6mJ/m2fortheTacase,+1.0mJ/m2\nfor Cr and +2.0mJ/m2for Ru. This shows how sensitive\nK⊥\nSis to the exact growth properties which are modified\nby the different seed layer. The values reported in this\nwork for both annealed series are very similar to the Cr\nand Ru buffered systems. The fact that K⊥\nSvanishes\nin the as-deposited series shows also how important the\nannealing step is for adjusting the interface properties.\nFigure 4 shows the dependence of the Gilbert damping\nparameter αon the thickness dfor the as-deposited sam-\nplesandthe annealedseries. Theinsetshowsexemplarily\nfor the 80 nm samples the dependence of the linewidth\n∆H on the frequency and the linear fits to obtain α. For\nthe three series we observe a minimum in the αvalue for\nd= 40 nm. The smallest value obtained for this series\nisα= 2.8±0.1×10−3. When comparing to the liter-\nature it has to be taken into account that the value of\nαis very sensitive to the growth conditions and to the\nannealing temperature. Therefore the scatter of values is\nlarge. The smallest reported value [33] is around 1 ×10−3\nbut for films annealed at 600oC. The damping increases\nwhen the annealing temperature is lower, up to values\nsimilar to the ones reported here at ∼450oC.\nThe reasons for the increased damping are different\nfor the thicker and the thinner films. Concerning the\nlarge damping value for the 80 nm samples, it is a com-\nmon behavior in soft magnetic thin films that the damp-\ning increases strongly with thickness starting at a cer-\ntain value. An example of this can be seen for NiFe\nin the literature [34]. In this case the damping of the\nfilms strongly increases starting at d= 90 nm. The rea-\nson for that is a non-homogeneous magnetization state\nfor thicker films which open new loss channels in addi-tion to two-magnon scattering responsible for Gilbert-\nlike behavior in in-plane magnetized films. Nevertheless,\nthe value of αdecreases with the annealing temperature\npointing to a overall improvement of the uniformity of\nthe film and of the crystalline order.\nFor the thinner samples down to 11 nm we also ob-\nserve a reduction of αupon annealing, however this sit-\nuation is inverted for d <11 nm and provides a hint\nto one of the posible reasons for the increase of damping\nwith decreasingthickness. When the thicknessis reduced\nand the effect of the interface anisotropy is becoming\nlarger the magnetization state is becoming more inho-\nmogeneous due to the counterplay between the demag-\nnetization field and the anisotropy field. However, this is\nnot the only reason explaining the αincrease since this\nis also observable in the as-deposited sample series where\nK⊥\nS≈0, although to a lower degree, and additional ef-\nfects, e.g. due to roughness, play also a role.\nA comment has to be done concerning the exact mean-\ning of the concept of inhomogeneous magnetization used\nto describe our films. In an ideal thin film with smooth\ninterfaces and in the case of K⊥\nS= 0, the demagnetizing\nfield due to the shape anisotropy would induce a perfect\nin-plane orientation of the magnetization and a homoge-\nneous state with an external applied field. For the case\nof a large enough K⊥\nS>0 for a thickness below a criti-\ncal value ( d < dmin) the magnetization would again be\nhomogeneous but with out-of-plane orientation and for\nd > d maxan homogeneous in-plane state is expected.\nHowever, for a transition region dmin< d < d maxdiffer-\nent inhomogeneous states can be formed. Some of them\ncan be modelled by a simple analytical model or by mi-\ncromagnetic simulations as for instance in [38]. On the\nother limit case, for very large thickness, the situation is\nsimilar although the origin is different. For large thick-\nness values, the demagnetizing field responsible for the\nin-plane orientation is weakened allowing for the forma-\ntionofinhomogeneousstatessimilartothepreviousones.\nThe in-plane anisotropies were studied by measuring\nthe dependence of the resonant field HFMRon the az-\nimuthal angle φ. Fig. 5(a) shows exemplarily this depen-\ndence for a thickness of 11 nm in the range 0-180oat\n18 GHz for the as-deposited sample and the 450oC an-\nnealed one. An overall four-fold anisotropy, as expected\nfrom the cubic lattice of CFA and the (100) growth di-\nrection is observed. The easy axes correspond to 0oand\n90o. Overimposed to this, an additional weaker two-fold\nuniaxial anisotropy is also observed ( HFMRat 0oand\n90oare slighty different). The uniaxial anisotropy may\nbeinduced bystressinthe filmorbythe vicinalstructure\nin the substrate surface induced by miscut.\nIn order to extract the anisotropy fields the following\nformula was used:\nHFMR=¯HFMR+Hbcos(4φ)+Hucos(2φ+ϕ) (2)5\nFIG. 5. (Color online) (a) Dependence of HFMRon the az-\nimuthal angle ϕfor 11 nm thick films for the as-deposited and\nthe 450oC annealed samples. The lines are a fit to Eq. 2. (b)\nDependence of the in-plane biaxial anisotropy constant Kb\n(filled points) and the in-plane uniaxial anisotropy consta nt\nKu(hollow points) on the thickness dfor the as-deposited and\nthe annealed series.\nHereHbandHuarethebiaxialanduniaxialanisotropy\nfields,φis the in-plane azimuthal angle and ¯HFMRis the\naveragedvalue. The angle ϕallows for a misalignment of\nthe uniaxial and biaxial contributions, i.e. the easy axis\nof both contributions may be at different angles. The\nlines in Fig. 5(a) are fits to this formula. These field\nvalues are related to the anisotropy constants Hb,u=\n2Kb,u\nMs.\nThe results for KbandKufrom the fits are plotted\nin Fig. 5(b). For the calculation of the anisotropy con-\nstant the magnetization values obtained from the fits in\nFig. 3 are used. For Kbwe observe a different thick-\nness dependence for the as-deposited series and the se-\nries annealed at 320oC compared to the series annealed\nat 450oC. The value of Kbshows minor variation for the\nas-deposited samples with a small reduction for the thin-\nner films. The evolution is similar for the 320oC case.\nOn the contrary, the anisotropy constant increases con-\ntinously and strongly with decreasing thickness in the\nannealed series. However, the values converge for thick\nfilms and for 80 nm the difference vanishes. This points\nto an important role of the stress in the films, which\nnormally relaxes with thickness, in the evolution of Kb.\nThe absolutevalues arein agreementwith literaturedata\n[22]. The values of Kuare a order of magnitude smaller\nand the absolutevalues andthe thicknessdependence arevery similar for the three cases.\nCONCLUSIONS\nIn summary, we measured the evolution of the in-\nterface induced perpendicular anisotropy for epitaxial\nCFA/MgO interfaces and we observed a strong increase\nwith the annealing temperature up to a value of K⊥\nS=\n2.01±0.7mJ/m2foranannealingtemperatureof450oC.\nA stabilization of a perpendicular magnetization orienta-\ntion is then expected for films thinner than 1.7 nm. We\nstudied the thickness dependent magnetic properties of\nCFA for as-deposited and annealed series. We obtained\nminimum values for αfor a thickness of 40 nm for all\nseries and a different evolution with annealing for thin-\nner or thicker films. We correlate this with interface and\nbulk changes upon annealing, respectively. 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Box 513, 5600 MB Eindhoven, The Netherlands\n(Dated: November 24, 2022)\nThe dynamics of a magnetic moment or spin are of high interest to applications in technology.\nDissipation in these systems is therefore of importance for improvement of efficiency of devices,\nsuch as the ones proposed in spintronics. A large spin in a magnetic field is widely assumed to\nbe described by the Landau-Lifshitz-Gilbert (LLG) equation, which includes a phenomenological\nGilbert damping. Here, we couple a large spin to a bath and derive a generic (non-)Ohmic damping\nterm for the low-frequency range using a Caldeira-Leggett model. This leads to a fractional LLG\nequation, where the first-order derivative Gilbert damping is replaced by a fractional derivative of\norders≥0. We show that the parameter scan be determined from a ferromagnetic resonance\nexperiment, where the resonance frequency and linewidth no longer scale linearly with the effective\nfield strength.\nIntroduction. — The magnetization dynamics of mate-\nrials has attracted much interest because of its techno-\nlogical applications in spintronics, such as data storage\nor signal transfer [1–3]. The right-hand rule of magnetic\nforces implies that the basic motion of a magnetic mo-\nment or macrospin Sin a magnetic field Bis periodic\nprecession. However, coupling to its surrounding (e.g.,\nelectrons, phonons, magnons, and impurities) will lead\nto dissipation, which will align SwithB.\nSpintronics-based devices use spin waves to carry sig-\nnals between components [4]. Contrary to electronics,\nwhich use the flow of electrons, the electrons (or holes)\nin spintronics remain stationary and their spin degrees\nof freedom are used for transport. This provides a sig-\nnificant advantage in efficiency, since the resistance of\nmoving particles is potentially much larger than the dis-\nsipation of energy through spins. The spin waves con-\nsist of spins precessing around a magnetic field and they\nare commonly described by the Landau-Lifshitz-Gilbert\n(LLG) equation [5]. This phenomenological description\nalso includes Gilbert damping, which is a term that\nslowly realigns the spins with the magnetic field. Much\neffort is being done to improve the control of spins for\npractical applications [6]. Since efficiency is one of the\nmain motivations to research spintronics, it is important\nto understand exactly what is the dissipation mechanism\nof these spins.\nAlthough the LLG equation was first introduced phe-\nnomenologically, since then it has also been derived from\nmicroscopic quantum models [7, 8]. Quantum dissipation\nis a topic of long debate, since normal Hamiltonians will\nalways have conservation of energy. It can be described,\nfor instance, with a Caldeira-Leggett type model [9–13],\nwhere the Hamiltonian of the system is coupled to a bath\nof harmonic oscillators. These describe not only bosons,\nbut any degree of freedom of an environment in equilib-\nrium. These oscillators can be integrated out, leading toan effective action of the system that is non-local and ac-\ncounts for dissipation. The statistics of the bath is cap-\ntured by the spectral function J(ω), which determines\nthe type of dissipation. For a linear spectral function\n(Ohmic bath), the first-order derivative Gilbert damping\nis retrieved.\nThe spectral function is usually very difficult to calcu-\nlate or measure, so it is often assumed for simplicity that\nthe bath is Ohmic. However, J(ω) can have any contin-\nuous shape. Hence, a high frequency cutoff is commonly\nput in place, which sometimes justifies a linear expan-\nsion. However, a general expansion is that of an sorder\npower-law, where scould be any positive real number.\nA spectral function with such a power-law is called non-\nOhmic, and we refer to sas the “Ohmicness” of the bath.\nIt is known that non-Ohmic baths exist [14–23] and that\nthey can lead to equations of motion that include frac-\ntional derivatives [24–28]. Because fractional derivatives\nare non-local, these systems show non-Markovian dynam-\nics which can be useful to various applications [29–31].\nHere, we show that a macroscopic spin in contact\nwith a non-Ohmic environment leads to a fractional LLG\nequation, where the first derivative Gilbert damping gets\nreplaced by a fractional Liouville derivative. Then, we ex-\nplain how experiments can use ferromagnetic resonance\n(FMR) to determine the Ohmicness of their environ-\nment from resonance frequency and/or linewidth. This\nwill allow experiments to stop using the Ohmic assump-\ntion, and use equations based on measured quantities\ninstead. The same FMR measurements can also be done\nwith anisotropic systems. Aligning anisotropy with the\nmagnetic field may even aid the realization of measure-\nments, as this can help reach the required effective field\nstrengths. In practice, the determination of the type of\nenvironment is challenging, since one needs to measure\nthe coupling strength with everything around the spins.\nHowever, with the experiment proposed here, one canarXiv:2211.12889v1  [cond-mat.mes-hall]  23 Nov 20222\nessentially measure the environment through the spin it-\nself. Therefore, the tools that measure spins can now also\nbe used to determine the environment. This information\nabout the dissipation may lead to improved efficiency,\nstability, and control of applications in technology.\nDerivation of a generalized LLG equation. — We con-\nsider a small ferromagnet that is exposed to an external\nmagnetic field. Our goal is to derive an effective equa-\ntion of motion for the magnetization. For simplicity, we\nmodel the magnetization as one large spin (macrospin)\nˆS. Its Hamiltonian (note that we set /planckover2pi1andkBto\none) reads ˆHs=B·ˆS−KˆS2\nz, where the first term\n(Zeeman) describes the coupling to the external mag-\nnetic fieldB, and the second term accounts for (axial)\nanisotropy of the magnet. However, since a magnet con-\nsists of more than just a magnetization, the macrospin\nwill be in contact with some environment. Following\nthe idea of the Caldeira-Leggett approach [9–13, 32], we\nmodel the environment as a bath of harmonic oscillators,\nˆHb=/summationtext\nαˆp2\nα/2mα+mαω2\nαˆx2\nα/2, where ˆxαand ˆpαare\nposition and momentum operators of the α-th bath oscil-\nlator with mass mαand eigenfrequency ωα>0. Further-\nmore, we assume the coupling between the macrospin and\nthe bath modes to be linear, ˆHc=/summationtext\nαγαˆS·ˆxα, where\nγαis the coupling strength between macrospin and the\nα-th oscillator. Thus, the full Hamiltonian of macrospin\nand environment is given by ˆH=ˆHs+ˆHc+ˆHb.\nNext, we use the Keldysh formalism in its path-integral\nversion [33, 34], which allows us to derive an effective ac-\ntion and, by variation, an effective quasi-classical equa-\ntion of motion for the macrospin. For the path-integral\nrepresentation of the macrospin, we use spin coherent\nstates [34]|g/angbracketright= exp(−iφˆSz) exp(−iθˆSy) exp(−iψˆSz)|↑/angbracketright,\nwhereφ,θ, andψare Euler angles and |↑/angbracketrightis the eigen-\nstate of ˆSzwith the maximal eigenvalue S. Spin co-\nherent states provide an intuitive way to think about\nthe macrospin as a simple vector S=/angbracketleftg|ˆS|g/angbracketright=\nS(sinθcosφ,sinθsinφ,cosθ) with constant length Sand\nthe usual angles for spherical coordinates θandφ. For\nspins, the third Euler angle ψpresents a gauge freedom,\nwhich we fix as in Ref. [35] for the same reasons explained\nthere.\nAfter integrating out the bath degrees of freedom, see\nSup. Mat. [36] for details, we obtain the Keldysh partition\nfunctionZ=/integraltext\nDgexp[iS], with the Keldysh action\nS=/contintegraldisplay\ndt/bracketleftbig\nS˙φ(1−cosθ)−Beff(Sz)·S/bracketrightbig\n−/contintegraldisplay\ndt/contintegraldisplay\ndt/primeS(t)α(t−t/prime)S(t/prime). (1)\nThe first term, called Berry connection, takes the role\nof a kinetic energy for the macrospin; it arises from\nthe time derivative acting on the spin coherent states\n(−i∂t/angbracketleftg|)|g/angbracketright=S˙φ(1−cosθ). The second term is the po-\ntential energy of the macrospin, where we introduced aneffective magnetic field, Beff(Sz) =B−KSzez, given by\nthe external magnetic field and the anisotropy. The third\nterm arises from integrating out the bath and accounts\nfor the effect of the environment onto the macrospin;\nthat is, the kernel function α(t−t/prime) contains informa-\ntion about dissipation and fluctuations. Dissipation is\ndescribed by the retarded and advanced components\nαR/A(ω) =/summationtext\nα(γ2\nα/2mαω2\nα)ω2/[(ω±i0)2−ω2\nα], whereas\nthe effect of fluctuations is included in the Keldysh com-\nponent,αK(ω) = coth(ω/2T) [αR(ω)−αA(ω)]. This is\ndetermined by the fluctuation-dissipation theorem, as we\nassume the bath to be in a high-temperature equilibrium\nstate [33, 34, 37].\nFrom the Keldysh action, Eq. (1), we can now de-\nrive an equation of motion for the macrospin by taking\na variation. More precisely, we can derive quasi-classical\nequations of motion for the classical components of the\nanglesθandφby taking the variation with respect to\ntheir quantum components [38]. The resulting equations\nof motion can be recast into a vector form and lead to a\ngeneralized LLG equation\n˙S(t) =S(t)×/bracketleftbigg\n−Beff[Sz(t)] +/integraldisplayt\n−∞dt/primeα(t−t/prime)S(t/prime) +ξ(t)/bracketrightbigg\n,\n(2)\nwith the dissipation kernel [39] given by\nα(ω) =/integraldisplay∞\n−∞dε\nπεJ(ε)\n(ω+i0)2−ε2, (3)\nwhere we introduced the bath spectral density J(ω) =/summationtext\nα(πγ2\nα/2mαωα)δ(ω−ωα) [33, 36]. The last term in\nEq. (2) contains a stochastic field ξ(t), which describes\nfluctuations (noise) caused by the coupling to the bath;\nthe noise correlator for the components of ξ(t) is given\nby/angbracketleftξm(t)ξn(t/prime)/angbracketright=−2iδmnαK(t−t/prime). Next, to get a\nbetter understanding of the generalized LLG equation,\nwe consider some examples of bath spectral densities.\nFractional Landau-Lifshitz-Gilbert equation. — For the\ngeneralized LLG equation (2), it is natural to ask: In\nwhich case do we recover the standard LLG equation?\nWe can recover it for a specific choice of the bath spectral\ndensityJ(ω), which we introduced in Eq. (3). Roughly\nspeaking,J(ω) describes two things: first, in the delta\nfunctionδ(ω−ωα), it describes at which energies ωα\nthe macrospin can interact with the bath; second, in\nthe prefactor πγ2\nα/2mαωα, it describes how strongly the\nmacrospin can exchange energy with the bath at the fre-\nquencyωα. In our simple model, the bath spectral den-\nsity is a sum over δ-peaks because we assumed excitations\nof the bath oscillators to have an infinite life time. How-\never, also the bath oscillators will have some dissipation\nof their own, such that the δ-peaks will be broadened. If,\nfurthermore, the positions of the bath-oscillator frequen-\nciesωαis dense on the scale of their peak broadening, the\nbath spectral density becomes a continuous function in-\nstead of a collection of δ-peaks. In the following, we focus3\non cases where the bath spectral density is continuous.\nSince the bath only has positive frequencies, we have\nJ(ω≤0) = 0. Even though J(ω) can have any pos-\nitive continuous shape, one might assume that it is an\napproximately linear function at low frequencies; that is,\nJ(ω) =α1ωΘ(ω)Θ(Ωc−ω), (4)\nwhere Θ(ω) = 1 forω > 0 and Θ(ω) = 0 forω < 0 and\nΩcis some large cutoff frequency of the bath such that we\nhaveωsystem/lessmuchT/lessmuchΩc. Reservoirs with such a linear\nspectral density are also known as Ohmic baths. Insert-\ning the Ohmic bath spectral density back into Eq. (3),\nwhile sending Ω c→∞ , we recover the standard LLG\nequation,\n˙S(t) =S(t)×/bracketleftBig\n−Beff[Sz(t)] +α1˙S(t) +ξ(t)/bracketrightBig\n,(5)\nwhere the first term describes the macrospin’s precession\naround the effective magnetic field, the second term—\nknown as Gilbert damping—describes the dissipation of\nthe macrospin’s energy and angular momentum into the\nenvironment, and the third term describes the fluctu-\nations with/angbracketleftξm(t)ξn(t/prime)/angbracketright= 4α1Tδmnδ(t−t/prime), which\nare related to the Gilbert damping by the fluctuation-\ndissipation theorem. Note that the same results can\nbe obtained without a cutoff frequency by introducing\na counter term, which effectively only changes the zero-\nenergy level of the bath, see Sup. Mat. [36] for details.\nThe assumption of an Ohmic bath can sometimes be\njustified, but is often chosen out of convenience, as it is\nusually the simplest bath type to consider. To our knowl-\nedge, there has been little to no experimental verification\nwhether the typical baths of magnetizations in ferromag-\nnets are Ohmic or not. To distinguish between Ohmic\nand non-Ohmic baths, we need to know how the mag-\nnetization dynamics depends on that difference. Hence,\ninstead of the previous assumption of a linear bath spec-\ntral density (Ohmic bath), we now assume that the bath\nspectral density has a power-law behavior at low frequen-\ncies,\nJ(ω) = ˜αsωsΘ(ω)Θ(Ωc−ω), (6)\nwhere we refer to sas Ohmicness parameter [40]. It is\nconvenient to define αs= ˜αs/sin(πs/2) and we should\nnote that the dimension of αsdepends on s. Fors= 1\nwe recover the Ohmic bath. Correspondingly, baths with\ns < 1 are called sub-Ohmic and baths with s > 1 are\ncalled super-Ohmic. For 0 <s< 2 and Ωc→∞ , we find\nthe fractional LLG equation\n˙S(t) =S×[−Beff[Sz(t)] +αsDs\ntS(t) +ξ(t)],(7)\nwhereDs\ntis a (Liouville) fractional time derivative of or-\ndersand the noise correlation is given by /angbracketleftξm(t)ξn(t/prime)/angbracketright=\n2αsTδmn(t−t/prime)−s/Γ(1−s); for a detailed calculation,see the Sup. Mat. [36]. Indeed, in the limit of s→1, we\nrecover the regular LLG equation. The fractional LLG\nequation (7) seems quite similar to the standard LLG\nequation (5). However, the first-order time derivative in\nthe Gilbert damping is replaced by a fractional s-order\ntime derivative in the fractional Gilbert damping; this\nhas drastic consequences for the dissipative macrospin\ndynamics.\nFractional Gilbert damping. — Fractional derivatives\nhave a long history [29], and many different definitions\nexist for varying applications [26, 27, 29]. From our mi-\ncroscopic model, we found the Liouville derivative [41],\nwhich is defined as\nDs\ntS(t) =dn\ndtn1\nΓ(n−s)/integraldisplayt\n−∞dt/prime(t−t/prime)n−1−sS(t/prime),(8)\nwherenis the integer such that n≤s<n + 1. This can\nbe interpreted as doing a fractional integral followed by\nan integer derivative. The fractional integral is a direct\ngeneralization from the rewriting of a repeated integral,\nby reversing the order of integration into a single one,\nwhich leads to extra powers of ( t−t/prime).\nTo provide some intuition to the effects of fractional\nfriction, we propose a thought experiment. Suppose an\nobject is traveling at constant speed, then x(t) =vt.\nHence, the friction force acting on the object goes like\nDs\ntx(t)∝vt1−s. Therefore, we find three regimes. For\ns= 1, the friction is constant in time. For s<1, the fric-\ntion force increases with time. Hence, longer movements\nwill be less common. For s > 1, the friction decreases\nwith time, so longer movements will be more likely once\nset in motion.\nWithin the fractional LLG equation, we thus see two\nimportant new regimes. For s < 1 (sub-Ohmic), the\nfriction is more likely to relax (localize) the spin (e.g.\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.50.10.51510\n[ωd-(B 0-KS)]/[α sS(B 0-KS)s][αsS(B 0-KS)s]2sin2θ/Ω2s0.20.40.60.81.1.21.41.61.8\nFIG. 1. A lin-log plot of the amplitude sin2θas a function of\ndriving frequency ωdplotted in dimensionless units for several\nvalues ofs. The resonance peaks change, depending on s.\nThe resonance frequency ωresand linewidth ∆ H/2have been\noverlayed with crosses. The red dashed crosses have been\ncalculated numerically, whereas the black solid crosses are the\nderived results from Eqs. (10) to (12).4\nsub-diffusion) towards the B-field direction. For small\nmovements, the friction could be very small, whereas it\nwould greatly increase for bigger movements. This could\ndescribe a low dissipation stable configuration. For s >\n1 (super-Ohmic), the friction could reduce as the spin\nmoves further, which in other systems is known to cause\nL´ evy-flights or super-diffusion [24, 25, 42]. This might\nlead the system to be less stable, but can potentially also\ngreatly reduce the amount of dissipation for strong signal\ntransfer: In a similar way to the design of fighter-jets,\nunstable systems can be easily changed by small inputs,\nwhich leads to more efficient signal transfer.\nFerromagnetic Resonance. — FMR is the phenomenon\nwhere the spin will follow a constant precession in a ro-\ntating external magnetic field. The angle θfrom thez-\naxis at which it will do so in the steady state will vary\naccording to the driving frequency ωdof the magnetic\nfield. Close to the natural frequency of the precession,\none generally finds a resonance peak [43]. We assume a\nmagnetic field of the form\nBeff(t) =\nΩ cos(ωdt)\nΩ sin(ωdt)\nB0−KSz\n, (9)\nwhere Ω is the strength of the rotating component, and\nwe will neglect thermal noise. We search for a steady\nstate solution of S(t) in the rotating frame where Beff(t)\nis constant. We will assume a small θapproximation\nwhere the ground state is in the positive zdirection, i.e.\n0<Ω/lessmuchB0−KSandαsS/lessmuch(B0−KS)1−s. Then (see\nSup. Mat. [36] for details of the calculations), we find\nthat the resonance occurs at a driving frequency\nωres≈(B0−KS) + (B0−KS)sαsScos/parenleftBigπs\n2/parenrightBig\n.(10)\nIt should be noted that this is different from what was to\nbe expected from any scaling arguments, since the cosine\nterm is completely new compared to previous results [43],\nand it vanishes precisely when s= 1. However, this new\nnon-linear term scales as ( B0−KS)s, which is an easily\ncontrollable parameter. In the limit where B0−KSis\nsmall (resp. large), the linear term will vanish and the\ns-power scaling can be measured for the sub(resp. super)-\nOhmic case. The amplitude at resonance is found to be\nsin2θres≈Ω2\n/bracketleftbig\nαsS(B0−KS)ssin/parenleftbigπs\n2/parenrightbig/bracketrightbig2, (11)\nand the Full Width at Half Maximum (FWHM) linewidth\nis given by\n∆H/2≈2αsS(B0−KS)ssin/parenleftBigπs\n2/parenrightBig\n. (12)\nDepending on the experimental setup, it might be eas-\nier to measure either the resonance location or the width\n0.0 0.5 1.0 1.5 2.001234\nωres/(B0-KS)ΔH/2/[αsS(B0-KS)s]\n0.01 0.10 1 10 1000.0110s0.20.40.60.81.1.21.41.61.8FIG. 2. A plot of the linewidth in Eq. (13) as a function of res-\nonance frequency for several values of s. The inset shows the\nsame plot in a log-log scale, where the slope of the linewidth\nis precisely the Ohmicness sof the bath.\nof the peak. Nevertheless, both will give the opportu-\nnity to see the sscaling inB0−KS. The presence\nof the anisotropy provides a good opportunity to reach\nweak or strong field limits. In fact, the orientation of the\nanisotropy can help to add or subtract from the magnetic\nfield, which should make the required field strengths more\nreachable for experiments. Some setups are more suit-\nable for measuring the width as a function of resonance\nfrequency. When s= 1, this relation can be directly\nderived from Eqs. (10) and (12). However, when s/negationslash= 1,\nthe relation can only be approximated for strong or weak\ndamping. For small αsS, we see that\n∆H/2≈2αsS(ωres)ssin/parenleftBigπs\n2/parenrightBig\n. (13)\nThe resonance peaks have been calculated numerically\nin FIG. 1 in dimensionless values. The red dashed lines\nshow the location of the numerically calculated peak and\nthe FWHM line width. The black solid lines show the\nlocation of the analytically approximated result for the\npeak location and FWHM line width [Eqs. (10) and (12)].\nFor smallαsSand Ω, we see a good agreement be-\ntween the analytical results and the numerical ones, al-\nthough sub-Ohmic seems to match more closely than\nsuper-Ohmic. This could be due to the greater sta-\nbility of sub-Ohmic systems, since the approximations\nmight affect less a stable system. As one might expect\nfrom the thought experiment presented earlier, we can\nsee in FIG. 1 that sub-Ohmic systems require higher,\nmore energetic, driving frequencies to resonate, whereas\nsuper-Ohmic systems already resonate at lower, less en-\nergetic, driving frequencies. In FIG. 2, we provide a plot\nof Eq. (13) to facilitate further comparison with experi-\nments. If the assumption of Gilbert damping was correct,\nall that one would see is a slope of one in the log-log inset.\nConclusion. — By relaxing the Ohmic Gilbert damp-\ning assumption, we have shown that the low-frequency\nregime of magnetization dynamics can be modeled by a5\nfractional LLG equation. This was done by coupling the\nmacrospin to a bath of harmonic oscillators in the frame-\nwork of a Caldeira-Leggett model. The Keldysh formal-\nism was used to compute the out-of-equilibrium dynam-\nics of the spin system. By analyzing an FMR setup, we\nfound ans-power scaling law in the resonance frequency\nand linewidth of the spin, which allows for a new way to\nmeasure the value of s. This means that experiments in\nmagnetization dynamics and spintronics can now avoid\nthe assumption of Gilbert damping and instead measure\nthe Ohmicness of the environment. This could aid in a\nbetter understanding of how to improve efficiency, stabil-\nity, and control of such systems for practical applications.\nAcknowledgments. — This work was supported by the\nNetherlands Organization for Scientific Research (NWO,\nGrant No. 680.92.18.05, C.M.S. and R.C.V.) and (partly)\n(NWO, Grant No. 182.069, T.L. and R.A.D.).\n[1] I. D. Mayergoyz, G. Bertotti, and C. 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Maniscalco,\nPhysical Review A - Atomic, Molecular, and Optical\nPhysics 79, 052120 (2009).\n[21] N. Wu, L. Duan, X. Li, and Y. Zhao, Journal of Chemical\nPhysics 138, 084111 (2013).\n[22] J. Jeske, A. Rivas, M. H. Ahmed, M. A. Martin-Delgado,\nand J. H. Cole, New Journal of Physics 20, 093017\n(2018).\n[23] A. Lemmer, C. Cormick, D. Tamascelli, T. Schaetz, S. F.\nHuelga, and M. B. Plenio, New Journal of Physics 20,\n073002 (2018).\n[24] E. Lutz, in Fractional Dynamics: Recent Advances\n(World Scientific, 2012), p. 285.\n[25] R. Metzler and J. Klafter, Physics Reports 339, 1 (2000).\n[26] F. Mainardi, Fractals and Fractional Calculus in Contin-\nuum Mechanics p. 291 (1997).\n[27] E. C. De Oliveira and J. A. Tenreiro Machado, Mathe-\nmatical Problems in Engineering 2014 , 238459 (2014).\n[28] R. C. Verstraten, R. F. Ozela, and C. Morais Smith,\nPhysical Review B 103, L180301 (2021).\n[29] R. Hilfer, Applications of fractional calculus in physics\n(World scientific, 2000).\n[30] M. Dalir and M. Bashour, Applied Mathematical Sci-\nences 4, 1021 (2010).\n[31] C. Gardiner and P. Zoller, Quantum noise: a handbook of\nMarkovian and non-Markovian quantum stochastic meth-\nods with applications to quantum optics (Springer Science\n& Business Media, 2004).\n[32] A. O. Caldeira and A. J. Leggett, Phys. Rev. A 31, 1059\n(1985).\n[33] A. Kamenev, Field theory of non-equilibrium systems\n(Cambridge University Press, 2011).\n[34] A. Altland and B. D. Simons, Condensed Matter Field\nTheory (Cambridge University Press, 2010), 2nd ed.\n[35] A. Shnirman, Y. Gefen, A. Saha, I. S. Burmistrov, M. N.\nKiselev, and A. Altland, Physical Review Letters 114,\n176806 (2015).\n[36] R. C. Verstraten, T. Ludwig, R. A. Duine, and\nC. Morais Smith, Supplementary material .\n[37] R. Kubo, Reports on Progress in Physics 29, 255 (1966).\n[38] In a straightforward variation with respect to quantum\ncomponents, we would only obtain a noiseless quasi-\nclassical equation of motion because the information\nabout noise (fluctuations) is included in the Keldysh\npart ofα(t−t/prime), which appears in the action only with\neven powers of quantum components. However, there is a\nway [44] that allows us to retain information about noise\nin the quasi-classical equation of motion; see also [33, 34].\nNamely, we perform a Hubbard-Stratonovich transforma-\ntion to linearize the contribution quadratic in quantum\ncomponents. This linearization comes at the cost of in-\ntroducing a new field, which takes the role of noise.\n[39] The dissipation kernel αis closely related to the retarded\nαRand advanced αAcomponents. Namely, it is given by\nα(ω) =−αR(ω)−αA(−ω).\n[40] From a mathematical perspective, any continuous but\nnot smooth function can still be expanded in a power-law\nfor small enough parameters. Hence, the only assumption\nthat we make in this model is that the frequencies in the\nsystem are very small. Then, there will always exist an\n0< s∈Rsuch that this expansion holds. In contrast,6\nthe Ohmic expansion can only be made for smooth con-\ntinuous functions.\n[41] This definition does not have any boundary conditions,\nas they would have to be at −∞ and would dissipate\nbefore reaching a finite time. One can, however, enforce\nboundary conditions by applying a very strong magnetic\nfield for some time such that the spin aligns itself, and\nthen quickly change to the desired field at t= 0.\n[42] A. A. Dubkov, B. Spagnolo, and V. V. Uchaikin, Interna-\ntional Journal of Bifurcation and Chaos 18, 2649 (2008).\n[43] T. Ludwig, I. S. Burmistrov, Y. Gefen, and A. Shnirman,\nPhysical Review Research 2, 023221 (2020).\n[44] A. Schmid, Journal of Low Temperature Physics 49, 609\n(1982).The fractional Landau-Lifshitz-Gilbert equation\nSupplementary Material\nR.C. Verstraten1, T. Ludwig1, R.A. Duine1,2, C. Morais Smith1\n1Institute for Theoretical Physics, Utrecht University,\nPrincetonplein 5, 3584CC Utrecht, The Netherlands\n2Department of Applied Physics, Eindhoven University of Technology,\nP.O. Box 513, 5600 MB Eindhoven, The Netherlands\n(Dated: November 24, 2022)\nCONTENTS\nI. Keldysh microscopic model 1\nA. Hamiltonian 1\nB. Keldysh partition function 2\nC. Quasi-classical equation of motion 4\nD. Generalized Landau-Lifshitz-Gilbert equation 8\nII. Fractional derivative from non-Ohmic spectral function 9\nA. Calculating the effective Greens functions 9\nB. Ohmic spectral function 11\nC. Sub-Ohmic spectral function 13\nD. Super-Ohmic spectral function 15\nE. Comparison Ohmic versus non-Ohmic 17\nIII. FMR powerlaw derivation 18\nA. Ferromagnetic Resonance 18\nB. Resonance frequency and amplitude 20\nC. Calculating the FWHM linewidth 21\nIV. Dimensional analysis 22\nReferences 22\nI. KELDYSH MICROSCOPIC MODEL\nFor pedagogical reasons we start with a microscopic derivation of the usual LLG equation before going into the\nfractional one. In this section, we combine spin coherent states with the Keldysh formalism [1, 2] to derive a stochastic\nLangevin-like equation of motion of a (macro) spin [3].\nA. Hamiltonian\nIn the main text, we introduced a spectral function J(ω) with a cutoff frequency Ω c. This was originally done from\nthe perspective that any spectral function could be expanded to linear order; hence, the model would only be valid\nup to some highest frequency. However, the cutoff is also important for the model to be realistic, since any physical\nspectral function should vanish as ω→∞ . In the main text, we stated that the same results can be obtained by\nintroducing a constant counter term in the Hamiltonian. This is a term which exactly completes the square of the\ncoupling term and the harmonic potential of the bath and can be seen as a normalization of the zero-energy level. If\nwe instead start the model with this counter term and drop the cutoff, we will get a Greens function αct(ω), which is\nprecisely such that the original Greens function can be written as α(ω) =α(0) +αct(ω), i.e., the counter term in the\nHamiltonian removes the zero frequency contribution of the Greens function. This α(ω= 0) generates a term in the\nequation of motion that goes as/integraltext∞\n0d/epsilon1J(/epsilon1)\nπ/epsilon1[S(t)×S(t)]. Since the integral is finite, with a frequency cutoff in J(ω),\nthe entire term is zero due to the cross product. This means that the equation of motion will be identical if we startarXiv:2211.12889v1  [cond-mat.mes-hall]  23 Nov 20222\neither from the regular Hamiltonian with a frequency cutoff, or with a counter term and no cutoff. Here, we choose\nto show the method that includes a counter term, because then we do not need to calculate terms which would have\ncanceled either way.\nThe microscopic system that we describe is a large spin in an external magnetic field, where the spin is linearly\ncoupled to a bath of harmonic oscillators in the same way as in Refs. [4–9]. Therefore, our Hamiltonian has the form\nof a system, coupling, bath, and counter term; H(t) =Hs+Hc+Hb+Hct, where\nHs=B·ˆS−KS2\nz,\nHc=/summationdisplay\nαγαˆS·ˆxα,\nHb=/summationdisplay\nαˆp2\nα\n2mα+mαω2\nα\n2ˆx2\nα,\nHct=/summationdisplay\nαγ2\nα\n2mαω2αˆS2. (1)\nHere,Bis the (effective) magnetic field, ˆSis the spin, Kis thez-axis anisotropy, γαis the coupling strength, and\nαis the index over all harmonic oscillators which have position ˆxα, momentum ˆpα, massmαand natural frequency\nωα. Notice that the counter term is constant, since S2is a conserved quantity, and that we have indeed completed\nthe square, such that\nH(t) =B·ˆS−KS2\nz+/summationdisplay\nαˆp2\nα\n2mα+/bracketleftBigg/radicalbigg\nmαω2α\n2ˆxα+/radicalBigg\nγ2α\n2mαω2αˆS/bracketrightBigg2\n. (2)\nB. Keldysh partition function\nWe will use the Keldysh formalism to derive a quasi-classical equation of motion. Since this is an out-of-equilibrium\nsystem, a common choice would be to use the Lindblad formalism with a master equation [10]. However, Lindblad can\nonly describe Markovian systems, which will not be the case when we introduce a non-Ohmic bath. In the Keldysh\nformalism, one starts with an equilibrium density matrix in the far past (effectively infinite on the relevant time scale).\nThis then gets evolved with the time evolution operator as usual. However, in contrast to ordinary path integrals,\nonce the present has been reached, one evolves back to the infinite past. Since there is infinite time for evolution,\nwe can reach out-of-equilibrium states adiabatically. The benefit of integrating back to the infinite past is that we\nbegin and end with the same in-equilibrium system, which means equilibrium techniques can be used, at the cost of\nhaving both the forward ( O+) and backward ( O−) quantities to take care of. To reach useful results, one can apply\na Keldysh rotation to the classical ( Oc= (O++O−)/2) and quantum ( Oq=O+−O−) components with the added\nnotation/vectorO=/parenleftbigg\nOc\nOq/parenrightbigg\n. To derive a quasi-classical equation of motion, the action can be expanded in all the quantum\ncomponents, after which the Euler-Lagrange equation for the quantum components provides the equation of motion\nin terms of the classical components.\nFIG. 1. Figure extracted from Ref. [2]. The Keldysh contour starts at t=−∞, evolves forward to some time t, and then\nevolves backwards in time to t=−∞.\nTo begin, we write down the Keldysh partition function\nZ= Tr/braceleftbigg\nTKexp/bracketleftbigg\n−i/contintegraldisplay\nKdtH(t)/bracketrightbigg\nρ0/bracerightbigg\n, (3)3\nwhereTKis the Keldysh time ordering, ρ0is the density matrix at t=−∞, and the integral runs over the Keldysh\ncontour, as shown in FIG. 1. After discretizing the Keldysh time integral in the way of FIG. 1, we can rewrite the\ntrace as path-integrals over the spin coherent state |g/angbracketrightand the oscillators |ˆxα/angbracketrightand|ˆpα/angbracketright. This yields\nZ=/integraldisplay\nDg/productdisplay\nα/integraldisplay\nDˆxα/integraldisplay\nDˆpαeiS[g,{ˆxα},{ˆpα}], (4)\nwith the Keldysh action\nS[g,{ˆxα},{ˆpα}] =/contintegraldisplay\nKdt/bracketleftBigg\n(−i∂t/angbracketleftg|)|g/angbracketright−B·Sg+KS2\nz,g\n+/summationdisplay\nα/parenleftBigg\n−γαSg·ˆxα+ˆpα·˙ˆxα−γ2\nαS2\ng\n2mαω2α−ˆp2\nα\n2mα−mαω2\nα\n2ˆx2\nα/parenrightBigg/bracketrightBigg\n, (5)\nwhere we defined Sg=/angbracketleftg|ˆS|g/angbracketright.\nThe continuous path-integral seems to miss the boundary term /angbracketleftˆx1,α,g1|ρ0|ˆx2N,α,g2N/angbracketright/angbracketleftˆp2N,α|ˆx2N,α/angbracketright, but it is\nincluded in the Keldysh contour, as it connects the beginning and final contour time at t=−∞; see Ref. [2].\nNow, we will integrate out the bath degrees of freedom, beginning by completing the square and performing the\nGaussian integral over ˆpα. The Gaussian contribution in ˆpαwill act as a constant prefactor, so it will drop out of\nany calculation of an observable due to the normalization. Hence, we can effectively set it to one to find\n/integraldisplay\nDˆpαexp/bracketleftbigg\n−i/contintegraldisplay\nKdt/parenleftbiggˆp2\nα\n2mα−ˆpα·˙ˆxα/parenrightbigg/bracketrightbigg\n= exp/bracketleftbigg\ni/contintegraldisplay\nKdt/parenleftBig\n−mα\n2ˆxα∂2\ntˆxα/parenrightBig/bracketrightbigg\n, (6)\nwhere we also did a partial integration in ˆxα. Next we will perform a similar approach for the positions, but it is\nuseful to apply the Keldysh rotation first. Note that we can directly rewrite the integral over the Keldysh contour as\na regular time integral over the quantum components. However, one must still rewrite the contents of the integral in\nterms of the quantum and classical parts of the variables, since the Keldysh rotation does not immediately work for\nproducts. The action can first be written as\niS[g,{ˆxα}] =i/integraldisplay\ndt/parenleftBig\n[(−i∂t/angbracketleftg|)|g/angbracketright]q−[B·Sg]q+K/bracketleftbig\nS2\nz,g/bracketrightbigq\n−/summationdisplay\nα/braceleftBigg\n[γαSg·ˆxα]q+γ2\nα[S2\ng]q\n2mαω2α+/bracketleftBigmα\n2ˆxα/parenleftbig\n∂2\nt+ω2\nα/parenrightbigˆxα/bracketrightBigq/bracerightBigg/parenrightBigg\n. (7)\nWe can then derive that\n−[γαSg·ˆxα]q=−γα/bracketleftbig\nS+\ng·ˆx+\nα−S−\ng·ˆx−\nα/bracketrightbig\n=−γα/bracketleftbigg/parenleftbigg\nSc\ng+1\n2Sq\ng/parenrightbigg\n·/parenleftbigg\nˆxc\nα+1\n2ˆxq\nα/parenrightbigg\n−/parenleftbigg\nSc\ng−1\n2Sq\ng/parenrightbigg\n·/parenleftbigg\nˆxc\nα−1\n2ˆxq\nα/parenrightbigg/bracketrightbigg\n=−γα/bracketleftbig\nSc\ngˆxq\nα+Sq\ngˆxc\nα/bracketrightbig\n=−γα/bracketleftbigg/parenleftbigSc\ngSq\ng/parenrightbig\nτx/parenleftbigg\nˆxc\nα\nˆxq\nα/parenrightbigg/bracketrightbigg\n, (8)\nwhere we introduced τx=/parenleftbigg\n0 1\n1 0/parenrightbigg\nin the Keldysh (classical, quantum) space represented by an upper index candq\nrespectively. Next, we want to derive a similar form for the part of the action that is quadratic in ˆxα. Since these are\nharmonic oscillators in equilibrium, we can refer the reader to Ref. [2], noting that a unit mass was used there, and\nconclude that\n/bracketleftBig\n−mα\n2ˆxα/parenleftbig\n∂2\nt+ω2\nα/parenrightbigˆxα/bracketrightBigq\n=/parenleftbigˆxc\nαˆxq\nα/parenrightbig/parenleftBigg\n0/bracketleftbig\nG−1\nα/bracketrightbigA\n/bracketleftbig\nG−1\nα/bracketrightbigR/bracketleftbig\nG−1\nα/bracketrightbigK/parenrightBigg/parenleftbigg\nˆxc\nα\nˆxq\nα/parenrightbigg\n, (9)\nwhere the retarded and advanced Greens functions read\n[G−1\nα]R/A(t−t/prime) =δ(t−t/prime)mα\n2[(i∂t±i0)2−ω2\nα]. (10)\nThe±i0 is introduced because we need an infinitesimal amount of dissipation on the bath for it to remain in equilibrium\nand the sign is tied to causality. This is because there is also an infinitesimal amount of energy transfer from the4\nmacroscopic spin to each of the oscillators. This results in an extra first-order derivative term, which is found by\nmultiplying out the square with i0. One might want to set these terms to zero immediately, but as it turns out,\nthese are very important limits, which shift away poles from integrals that we need to compute later. Once that is\ndone, the limits are no longer important for the final result, and they may finally be put to zero. Since the bath is in\nequilibrium, we can use the fluctuation dissipation theorem to compute the Keldysh component using\nGK\nα(ω) =/bracketleftbig\nGR\nα(ω)−GA\nα(ω)/bracketrightbig\ncoth/parenleftBigω\n2T/parenrightBig\n.\nThe ˆxdependent part of the action is now given by\niSX=i/integraldisplay\ndt/bracketleftbigg\n−γα/parenleftbigSc\ngSq\ng/parenrightbig\nτx/parenleftbigg\nˆxc\nα\nˆxq\nα/parenrightbigg\n+/parenleftbigˆxc\nαˆxq\nα/parenrightbig\nG−1\nα/parenleftbigg\nˆxc\nα\nˆxq\nα/parenrightbigg/bracketrightbigg\n, (11)\nwhich we can compute by completing the square to find\niSX=i/integraldisplay\ndt/bracketleftbigg\n−γ2\nα\n4/vectorST\ng/parenleftbigg\n0GA\nα\nGR\nαGK\nα/parenrightbigg\n/vectorSg/bracketrightbigg\n. (12)\nBefore we write down the final effective action, we also have to rewrite the quadratic part in Sin a similar vector\nform, which is\n−γ2\nα[S2\ng]q\n2mαω2α=−γ2\nα\n2mαω2α/parenleftbigSc\ngSq\ng/parenrightbig\nτx/parenleftbiggSc\ng\nSq\ng/parenrightbigg\n. (13)\nCombining everything together, we find that the partition function of the system is given by Z=/integraltext\nDg eiS[g], with\nthe effective action\niS[g] =i/integraldisplay\ndt/braceleftBigg\n/bracketleftbig\n(−i∂t/angbracketleftg|)|g/angbracketright−B·Sg+KS2\nz,g/bracketrightbigq−/integraldisplay\ndt/prime/vectorST\ng(t)/parenleftbigg\n0αA\nαRαK/parenrightbigg\n(t−t/prime)/vectorSg(t/prime)/bracerightBigg\n, (14)\nwhereαA/R(t−t/prime) =/summationtext\nα/parenleftBig\nγ2\nα\n4GA/R\nα(t−t/prime) +γ2\nα\n2mαω2αδ(t−t/prime)/parenrightBig\nandαK(t−t/prime) =/summationtext\nαγ2\nα\n4GK\nα(t−t/prime).\nC. Quasi-classical equation of motion\nIn the quasi-classical regime, we are interested in solutions where the quantum components ( q) are small compared\nto the classical components ( c). We can thus neglect terms of O[(q)3], but we must be careful with ( q)2. We can\nuse a Hubbard-Stratonovich transformation to convert ( q)2terms into an expression with just ( q), but with a new\nfieldξadded to the path integral [3]. The action will then contain only terms of linear order in ( q), which means the\npartition function has the form Z∼/integraltext\nDcDq exp[if(c)q] =/integraltext\nDc1\n2πδ[f(c)]. Hence, only solutions that satisfy f(c) = 0\ncontribute to the path integral. Within that subset, we want to minimize the action.\nIn order to derive the equation of motion of the system, we must understand the relation between |g/angbracketrightandSg=\n/angbracketleftg|S|g/angbracketright. Using the Euler angle representation [1], we can describe |g/angbracketrightas\n|g/angbracketright=g|↑/angbracketright=e−iφSze−iθSye−iψSz|↑/angbracketright=e−iφSze−iθSy|↑/angbracketrighte−iψS(15)\nand similarly\n/angbracketleftg|=eiψS/angbracketleft↑|eiθSyeiφSz. (16)\nNote that the ψangle is now independent of the quantum state |↑/angbracketright, since this angle is describing the rotation of the\nvector pointing in the spin direction, which is symmetric. Hence, this will yield a gauge symmetry.\nUsing the Euler angle representation in the first terms of Eq. (14), we see that\n(−i∂t/angbracketleftg|)|g/angbracketright=/parenleftBig\n˙ψSeiψS/angbracketleft↑|eiθSyeiφSz+eiψS/angbracketleft↑|˙θSyeiθSyeiφSz+eiψS/angbracketleft↑|eiθSy˙φSzeiφSz/parenrightBig\ne−iφSze−iθSy|↑/angbracketrighte−iψS\n=˙ψS+˙θ/angbracketleft↑|Sy|↑/angbracketright+˙φ/angbracketleft↑|eiθSySze−iθSy|↑/angbracketright. (17)5\nWe note that/angbracketleft↑|Sy|↑/angbracketright= 0, while the last term includes a rotation of the spin up state by θdegrees in the ydirection\nand then measures the Szcomponent of that state, which is Scosθ. Hence,\n(−i∂t/angbracketleftg|)|g/angbracketright=˙ψS+˙φScosθ. (18)\nWe now define a new variable χsuch thatψ=χ−φ, which results in\n(−i∂t/angbracketleftg|)|g/angbracketright= ˙χS−˙φ(1−cosθ)S. (19)\nMaking use of the Euler angle representation, we also see that\nSg=S\nsinθcosφ\nsinθsinφ\ncosθ\n. (20)\nWe see thatB·Sg=S[Bxsinθcosφ+Bysinθsinφ+Bzcosθ]. Similarly, KS2\nz,g=KS2cos2θ. Now, we still have to\ncompute the quantum parts of these quantities. We first note that\nSq\ng,x/S= [sinθcosφ]q= 2 cosθcsinθq\n2cosφccosφq\n2−2 sinθccosθq\n2sinφcsinφq\n2;\nSq\ng,y/S= [sinθsinφ]q= 2 sinθccosθq\n2cosφcsinφq\n2+ 2 cosθcsinθq\n2sinφccosφq\n2;\nSq\ng,z/S= [cosθ]q=−2 sinθcsinθq\n2;\n[cos2θ]q=−2 sinθccosθcsinθq. (21)\nNext, we will choose a gauge for χas in Ref. [11], which is\n˙χc=˙φc(1−cosθc)\nχq=φq(1−cosθc). (22)\nDefiningp= 1−cosθ, we see that [(−i∂t/angbracketleftg|)|g/angbracketright]q=/bracketleftBig\n˙χS−˙φpS/bracketrightBig\nq=S/bracketleftBig\nφq˙pc−˙φcpq/bracketrightBig\n. Now,pq= 2 sinθcsinθq\n2and\n˙pc=˙θcsinθccosθq\n2+˙θq\n2cosθcsinθq\n2,which leads to\n[(−i∂t/angbracketleftg|)|g/angbracketright]q=S/bracketleftBig\nφq˙pc−˙φcpq/bracketrightBig\n=S/bracketleftBigg\nφq˙θcsinθccosθq\n2+φq˙θq\n2cosθcsinθq\n2−2˙φcsinθcsinθq\n2/bracketrightBigg\n. (23)\nNext, we want to express B·Sq\ngin terms of Euler angles. We see that\nB·Sq\ng=S[Bxsinθcosφ+Bysinθsinφ+Bzcosθ]q\n= 2S/bracketleftBig\nBx/parenleftbigg\ncosθcsinθq\n2cosφccosφq\n2−sinθccosθq\n2sinφcsinφq\n2/parenrightbigg\n+By/parenleftbigg\nsinθccosθq\n2cosφcsinφq\n2+ cosθcsinθq\n2sinφccosφq\n2/parenrightbigg\n−Bzsinθcsinθq\n2/bracketrightBig\n, (24)\nwhere we used the results from Eq. (21). Similarly, we have\nK/bracketleftbig\nS2\nz,g/bracketrightbigq=KS2[cos2θ]q=−2KS2sinθccosθcsinθq. (25)\nCombining these results, we conclude that\n/bracketleftbig\n(−i∂t/angbracketleftg|)|g/angbracketright−B·Sg+KS2\nz,g/bracketrightbigq=S/bracketleftBigg\nφq˙θcsinθccosθq\n2+φq˙θq\n2cosθcsinθq\n2−2(−Bz+KScosθc+˙φc) sinθcsinθq\n2\n−2Bx/parenleftbigg\ncosθcsinθq\n2cosφccosφq\n2−sinθccosθq\n2sinφcsinφq\n2/parenrightbigg\n−By/parenleftbigg\nsinθccosθq\n2cosφcsinφq\n2+ cosθcsinθq\n2sinφccosφq\n2/parenrightbigg/bracketrightBigg\n. (26)6\nRemark that this expression only contains odd powers of ( q), so that we can neglect all higher-order terms to get\n/bracketleftBig\n(−i∂t/angbracketleftg|)|g/angbracketright−B·Sg+KS2\nz,g/bracketrightBigq\n=S/bracketleftBig\n−θqsinθc(−Bz+KScosθc+˙φc)\n−θqcosθc(Bxcosφc+Bysinφc) +φqsinθc(˙θc+Bxsinφc−Bycosφc)/bracketrightBig\n. (27)\nNow, we focus on the part of the action in Eq. (14) that comes from the bath, given by\niSb[g] =−i/integraldisplay\ndt/integraldisplay\ndt/prime/vectorST\ng(t)/parenleftbigg\n0αA\nαRαK/parenrightbigg\n(t−t/prime)/vectorSg(t/prime). (28)\nLet us first consider what Sq\ngandSc\ngare in terms of φandθ. By performing some trigonometric operations on each\nof the components, we find that\nSc\ng=S\nsinθccosθq\n2cosφccosφq\n2−cosθcsinθq\n2sinφcsinφq\n2\nsinθccosθq\n2sinφccosφq\n2+ cosθcsinθq\n2cosφcsinφq\n2\ncosθccosθq\n2\n (29)\nand\nSq\ng= 2S\ncosθcsinθq\n2cosφccosφq\n2−sinθccosθq\n2sinφcsinφq\n2\nsinθccosθq\n2cosφcsinφq\n2+ cosθcsinθq\n2sinφccosφq\n2\n−sinθcsinθq\n2\n. (30)\nBy expanding in the quantum components of Sc\ngandSq\ng, we see that\nSc\ng= (q)0+O/parenleftbig\n(q)2/parenrightbig\n,\nSq\ng= (q)1+O/parenleftbig\n(q)3/parenrightbig\n.\nSince the action only contains terms with at least one Sq\ng, we know that the only way to obtain a term of order ( q)2\nis from (Sq\ng)2. Hence, we may neglect all terms beyond linear ( q) inS(c/q)\ng in the quasi-classical regime. This results\nin\nSc\ng=S\nsinθccosφc\nsinθcsinφc\ncosθc\n, (31)\nSq\ng=S\nθqcosθccosφc−φqsinθcsinφc\nφqsinθccosφc+θqcosθcsinφc\n−θqsinθc\n. (32)\nA useful remark for later is that this shows that\nSq\ng=θq∂\n∂θcSc\ng+φq∂\n∂φcSc\ng. (33)\nGoing back to iSb[g], we can rewrite this as a convolution, in the sense that\niSb[g] =−i/integraldisplay\ndt/bracketleftbig\nSc\ng(t)·/parenleftbig\nαA∗Sq\ng/parenrightbig\n(t) +Sq\ng(t)·/parenleftbig\nαR∗Sc\ng/parenrightbig\n(t) +Sq\ng(t)·/parenleftbig\nαK∗Sq\ng/parenrightbig\n(t)/bracketrightbig\n, (34)\nwhere (f∗g)(t) =/integraltext∞\n−∞dt/primef(t−t/prime)g(t/prime). We see that the first two terms contain precisely one quantum component,\nbut the last term has two quantum components. When writing down the Euler-Lagrange equation of motion, it is\nimportant to realize that the convolution operation will act as if it is a simple multiplication, since the convolution\nobeys\nd\ndx(f(x)∗g)(t) =/parenleftbiggdf\ndx∗g/parenrightbigg\n(t). (35)7\nWe now concentrate on the ( q)2part of this action, for which we would like to use a Hubbard-Stratonovich transfor-\nmation in order to reduce this to linear in ( q). Recall that a Hubbard-Stratonovich transformation is given by\nexp/bracketleftBig\n−a\n2x2/bracketrightBig\n=/radicalbigg\n1\n2πa/integraldisplay\nDξexp/bracketleftbigg\n−ξ2\n2a−ixξ/bracketrightbigg\n. (36)\nHowever, we see that our action does not contain any purely quadratic terms, but rather a Greens functional shape\nasSq\ng(t)αK(t−t/prime)Sq\ng(t/prime). Hence, to use a Hubbard-Stratonovich like transformation, we must derive it from a Greens\nfunction exponential, similarly to Ref. [3]. Assuming that this is renormalizable and that αKcan be rewritten into a\ndistribution, we have\n1 =/integraldisplay\nDξexp/bracketleftbigg\n−1\n2/integraldisplay\ndt/integraldisplay\ndt/primeξ(t)[−2iαK]−1(t−t/prime)ξ(t/prime)/bracketrightbigg\n=/integraldisplay\nDξexp/bracketleftbigg\n−1\n2/integraldisplay\ndt/integraldisplay\ndt/prime/parenleftbigg\nξ(t)−2/integraldisplay\ndt/prime/primeSq\ng(t/prime/prime)αK(t/prime/prime−t)/parenrightbigg\n[−2iαK]−1(t−t/prime)/parenleftbigg\nξ(t/prime)−2/integraldisplay\ndt/prime/prime/primeαK(t/prime−t/prime/prime/prime)Sq\ng(t/prime/prime/prime)/parenrightbigg/bracketrightbigg\n=/integraldisplay\nDξexp/bracketleftbigg\n−1\n2/integraldisplay\ndt/integraldisplay\ndt/primeξ(t)[−2iαK]−1(t−t/prime)ξ(t/prime)\n−iSq\ng(t)δ(t−t/prime)ξ(t/prime)−iξ(t)δ(t−t/prime)Sq\ng(t/prime)−2iSq\ng(t)αK(t−t/prime)Sq\ng(t/prime)/bracketrightbig\n=/integraldisplay\nDξexp/bracketleftbigg\n−1\n2/integraldisplay\ndt/integraldisplay\ndt/primeξ(t)[−2iαK]−1(t−t/prime)ξ(t/prime)−2iSq\ng(t)δ(t−t/prime)ξ(t/prime)−2iSq\ng(t)αK(t−t/prime)Sq\ng(t/prime)/bracketrightbigg\n,\nwhere we used that/integraltext\ndt/primeαK(t−t/prime)[αK]−1(t/prime−t/prime/prime) =δ(t−t/prime/prime) and that 2 iαKis positive real. Therefore, we find that\nexp/bracketleftbigg\n−i/integraldisplay\ndt/integraldisplay\ndt/primeSq\ng(t)αK(t−t/prime)Sq\ng(t/prime)/bracketrightbigg\n=/integraldisplay\nDξexp/bracketleftbigg\n−1\n2/integraldisplay\ndt/integraldisplay\ndt/primeξ(t)[−2iαK]−1(t−t/prime)ξ(t/prime)/bracketrightbigg\n·exp/bracketleftbigg\ni/integraldisplay\ndtSq\ng(t)ξ(t)/bracketrightbigg\n. (37)\nThe double integral in the first exponential signifies the statistical properties of ξ. For instance, if αKis delta-like,\nthenξwould have Gaussian statistics (e.g. white noise), but in general we will have time correlated noise defined by\nαK[3], such that\n/angbracketleftξ(t)ξ(t/prime)/angbracketright=−2iαK(t−t/prime). (38)\nSince there is no gdependence in the double ξexponential, we will leave it out of S[g] and only remember these\nstatistics. Our partition function is then given by\nZ=/integraldisplay\nDξexp (iSn[ξ])/integraldisplay\nDgexp (iSsc[g,ξ]), (39)\nwhere the noise action is given by\niSn[ξ] =−1\n2/integraldisplay\ndt/integraldisplay\ndt/primeξ(t)[−2iαK]−1(t−t/prime)ξ(t/prime) (40)\nand the semi-classical action is given by\niSsc[g,ξ] =i/integraldisplay\ndtS/bracketleftBig\n−θqsinθc(−Bz+KScosθc+˙φc)−θqcosθc(Bxcosφc+Bysinφc)\n+φqsinθc(˙θc+Bxsinφc−Bycosφc)/bracketrightBig\n+i/integraldisplay\ndt/bracketleftbig\nξ(t)Sq\ng(t)/bracketrightbig\n−i/integraldisplay\ndt/bracketleftbig\nSc\ng(t)·/parenleftbig\nαA∗Sq\ng/parenrightbig\n(t) +Sq\ng(t)·/parenleftbig\nαR∗Sc\ng/parenrightbig\n(t)/bracketrightbig\n, (41)\nwhereSc\ng(t) andSq\ng(t) include only up to first-order corrections in quantum components. Assuming that αA/Rcan\nbe written in terms of distributions, we can define the distribution αdiss(t) =−αR(t)−αA(−t) and rewrite the\nsemi-classical action as\niSsc[g,ξ] =i/integraldisplay\ndtS/bracketleftBig\n−θqsinθc(−Bz+KScosθc+˙φc)−θqcosθc(Bxcosφc+Bysinφc)\n+φqsinθc(˙θc+Bxsinφc−Bycosφc)/bracketrightBig\n+i/integraldisplay\ndt/bracketleftbig/parenleftbig\nαdiss∗Sc\ng/parenrightbig\n(t) +ξ(t)/bracketrightbig\nSq\ng(t). (42)8\nRecall that, using the Euler angles, we have/integraltext\nDg=/integraltext\nDθDφ sin(θ). Technically, the factor of sin( θ) would end up\nin the action. However, since one could define ρ= cos(θ) as a new variable in order to avoid this, we know that this\nterm is not relevant to the physics. Hence, we can disregard it.\nSince all terms in iSsc[g,ξ] are either linear in θqorφq, we find two Euler-Lagrange equations of the form\nδLsc\nδθq= 0 andδLsc\nδφq= 0. (43)\nRemembering Eq. (33), we see thatδSq\ng(t)\nδθq=δSc\ng(t)\nδθcandδSq\ng(t)\nδφq=δSc\ng(t)\nδφc. Hence, the e.o.m. can be rearranged to yield\n˙φc=1\nSsinθc/bracketleftbig\n−B(Sc\nz) +/parenleftbig\nαdiss∗Sc\ng/parenrightbig\n(t) +ξ(t)/bracketrightbig\n·δSc\ng(t)\nδθc(44)\nand\n˙θc=−1\nSsinθc/bracketleftbig\n−B(Sc\nz) +/parenleftbig\nαdiss∗Sc\ng/parenrightbig\n(t) +ξ(t)/bracketrightbig\n·δSc\ng(t)\nδφc, (45)\nwhereB(Sc\nz) =\nBx\nBy\nBz−KSc\nz\n.\nD. Generalized Landau-Lifshitz-Gilbert equation\nWe want to show that the equations found by the microscopic model are in fact precisely of the LLG form. For\nthis, we will have to start from the LLG equation, and introduce the same two Euler angles θandφfor the spin, and\nshow that this gives rise to the same set of equations as previously deduced.\nWe begin with the generalized LLG equation\n˙S(t) =S(t)×[−B(Sz) + (αdiss∗S) (t) +ξ(t)], (46)\nwhereαdiss(t) =−αR(t)−αA(−t),/angbracketleftξ(t)ξ(t/prime)/angbracketright=−2iαK(t−t/prime) andB(Sz) = (Bx,By,Bz−KSz)T. Since the velocity\nofSis always perpendicular to S, we know that the magnitude of Sis constant. Hence, we can go to spherical\ncoordinates, such that\nS=S\nsinθcosφ\nsinθsinφ\ncosθ\n. (47)\nInserting this into the LLG equation, we firstly see that\n˙S=˙θ∂S\n∂θ+˙φ∂S\n∂φ=˙θS\ncosθcosφ\ncosθsinφ\n−sinθ\n+˙φS\n−sinθsinφ\nsinθcosφ\n0\n.\nNow, we notice that the RHS of the LLG equation can, without loss of generality, be written as S(t)×rwith\nr= (x,y,z )T. Working this out explicitly, we find that the LLG equation ˙S=S×rbecomes\nS\n˙θcosθcosφ−˙φsinθsinφ\n˙θcosθsinφ+˙φsinθcosφ\n−˙θsinθ\n=S\nzsinθsinφ−ycosθ\nxcosθ−zsinθcosφ\nysinθcosφ−xsinθsinφ\n. (48)\nWe note that the equation corresponding to the zcomponent can be written as\n˙θ=−1\nsinθr·\n−sinθsinφ\nsinθcosφ\n0\n=−1\nSsinθr·∂S\n∂φ. (49)9\nNow, we add up the ˆ xand ˆyequations, such that the ˙θcancels (i.e.−ˆxsinφ+ ˆycosφ). This yields\n˙φsinθ(sin2φ+ cos2φ) =−zsinθ(sin2φ+ cos2φ) +ycosθsinφ+xcosθcosφ,\nwhich simplifies to\n˙φ=1\nsinθr·\ncosθcosφ\ncosθsinφ\n−sinθ\n=1\nSsinθr·∂S\n∂θ. (50)\nBy inserting r=−B(Sz) + (αdiss∗S) (t) +ξ(t), we see that this is identical to the equations derived from the\nmicroscopic model\n˙φc=1\nSsinθc/bracketleftbig\n−B(Sz) +/parenleftbig\nαdiss∗Sc\ng/parenrightbig\n(t) +ξ(t)/bracketrightbig\n·δSc\ng(t)\nδθc; (51)\n˙θc=−1\nSsinθc/bracketleftbig\n−B(Sz) +/parenleftbig\nαdiss∗Sc\ng/parenrightbig\n(t) +ξ(t)/bracketrightbig\n·δSc\ng(t)\nδφc. (52)\nTherefore, we may conclude that our microscopic model is described by the generalized LLG equation.\nFor the fractional LLG equation, we are in particular interested in the case where αdiss∗S=αsDs\ntS, whereDs\ntis\na fractional derivative. For instance, assuming 0 <s< 1, the Liouville fractional derivative is given by\nDs\ntf(t) =1\nΓ(1−s)/integraldisplayt\n−∞(t−t/prime)−sf/prime(t/prime)dt/prime. (53)\nSo, ifαdiss=αsΘ(t)\nΓ(1−s)t−s∂t, then, because of the convolution with S, we would find a fractional LLG equation, whereas\nαdiss=α1δ(t)∂twould give the regular LLG equation.\nII. FRACTIONAL DERIVATIVE FROM NON-OHMIC SPECTRAL FUNCTION\nHere, we will compute the type of dissipation which comes from the spectral function. We will first derive the\nspectral function from microscopic quantities to see how it ends up in the Greens function. Then, we will calculate\nthe dissipation for three different cases.\nA. Calculating the effective Greens functions\nWe recall that\nαA/R(t−t/prime) =/summationdisplay\nα/parenleftbiggγ2\nα\n4GA/R\nα(t−t/prime) +δ(t−t/prime)γ2\nα\n2mαω2α/parenrightbigg\n,\nwhere\n[G−1\nα]R/A(t−t/prime) =δ(t−t/prime)mα\n2[(i∂t±i0)2−ω2\nα].\nBy the fluctuation dissipation theorem, we also have\nαK(ω) =/bracketleftbig\nαR(ω)−αA(ω)/bracketrightbig\ncoth/parenleftBigω\n2T/parenrightBig\n.\nWe are interested in finding closed forms for αR/A/K(t−t/prime). Using the relation\n/integraldisplay\ndt/primeG−1(t−t/prime)G(t/prime−t/prime/prime) =δ(t−t/prime/prime), (54)\nwe note that\nmα\n2[(i∂t±i0)2−ω2\nα]GR/A\nα(t−t/prime/prime) =δ(t−t/prime/prime). (55)10\nThe Fourier transform1yields\nGR/A\nα(ω) =1\nmα\n2[(ω±i0)2−ω2α]. (56)\nWe therefore find that\nαR/A(ω) =/summationdisplay\nα/parenleftbiggγ2\nα\n4GA/R\nα(ω) +γ2\nα\n2mαω2α/parenrightbigg\n=/summationdisplay\nαγ2\nα\n2mα/bracketleftbigg1\n(ω±i0)2−ω2α+1\nω2α/bracketrightbigg\n=/summationdisplay\nαγ2\nα\n2mαω2αω2\n(ω±i0)2−ω2α.(57)\nThe spectral function is given by the imaginary part of the Fourier transform of the dynamical susceptibility\nχ(ω) =δ\nδS(ω)/summationdisplay\nαγαˆxα(ω). (58)\nThe Hamiltonian e.o.m. for the bath can be found by going back to our starting Hamiltonian\nH=B·S+/summationdisplay\nαγαS·ˆxα+/summationdisplay\nαˆp2\nα\n2mα+mαω2\nα\n2ˆx2\nα+/summationdisplay\nαγ2\nα\n2mαω2αS2.\nThe e.o.m. reads\n˙ˆxα=ˆpα\nmαand ˙ˆpα=−γαS−mαω2\nαˆxα. (59)\nCombining both equations and taking the Fourier transform, we find\nˆxα(ω) =γα\nmα[(ω+i0)2−ω2α]S(ω), (60)\nwhere we have taken an infinitesimal amount of dissipation + i0 on the oscillators into account. This leads to\nχ(ω) =/summationdisplay\nαγ2\nα\nmα[(ω+i0)2−ω2α]=/summationdisplay\nαγ2\nα\n2mαωα/parenleftbigg1\nω+i0−ωα−1\nω+i0 +ωα/parenrightbigg\n. (61)\nWe remark that\nIm1\nx+i0=−πδ(x), (62)\nwhich leads to\nJ(ω) = Imχ(ω) =−/summationdisplay\nαπγ2\nα\n2mαωα[δ(ω−ωα)−δ(ω+ωα)] =−π\n2/summationdisplay\nαγ2\nα\nmαωαδ(ω−ωα), (63)\nwhere we used that all oscillator frequencies are positive. We can identify the spectral function in αR/Aas\nαR/A(ω) =/summationdisplay\nαγ2\nα\n2mαω2αω2\n(ω±i0)2−ω2α=−/integraldisplay∞\n0dε\nπω2ε−1J(ε)\n(ω±i0)2−ε2. (64)\nNow, we will assume a particular shape for J(ε). This can be either Ohmic or non-Ohmic, but in general we may\nassume a power-law behavior as some J(ε) =αsεs.\n1We use the convention f(ω) =/integraltext∞\n−∞dteiωtf(t) andf(t) =1\n2π/integraltext∞\n−∞dωe−iωtf(ω), withδ(t) =1\n2π/integraltext∞\n−∞dωeiωt.11\nB. Ohmic spectral function\nBeginning with the Ohmic case J(ε) =α1ε, we see that\n2iImαR/A(ω) =αR/A(ω)−/bracketleftBig\nαR/A/bracketrightBig∗\n(ω)\n=−α1/integraldisplay∞\n0dε\nπ/bracketleftbiggω2\n(ω±i0)2−ε2−ω2\n(ω∓i0)2−ε2/bracketrightbigg\n=−α1/integraldisplay∞\n−∞dε\n2π/bracketleftbiggω2\n(ω±i0)2−ε2−ω2\n(ω∓i0)2−ε2/bracketrightbigg\n=−α1ω2/integraldisplay∞\n−∞dε\n2π/bracketleftbigg1\n(ω±i0 +ε)(ω±i0−ε)−1\n(ω∓i0 +ε)(ω∓i0−ε)/bracketrightbigg\n=±4i0ω3α1/integraldisplay∞\n−∞dε\n2π1\n(ω±i0 +ε)(ω±i0−ε)(ω∓i0 +ε)(ω∓i0−ε), (65)\nwhich has four poles at ε=±1(ω±2i0). Since the integral scales as <1/|ε|, we can add an infinite radius half circle\nto complete a complex contour integral. Notice from symmetry that we will always have one of each of the four poles.\nWe can thus drop the ±signs inside since this only changes the notation order in the fraction. We thus find that\n2iImαR/A(ω) =±4i0ω3α1/integraldisplay∞\n−∞dε\n2π1\n(ε+ω+i0)(ε−ω−i0)(ε+ω−i0)(ε−ω+i0). (66)\nCompleting the contour along the top, we find poles at ε=±ω+i0, which yields\n2iImαR/A(ω) =∓0·4ω3α1/bracketleftbigg1\n(ω+i0 +ω+i0)(ω+i0 +ω−i0)(ω+i0−ω+i0)\n+1\n(−ω+i0 +ω+i0)(−ω+i0−ω−i0)(−ω+i0−ω+i0)/bracketrightbigg\n=∓0·4ω3α1/bracketleftbigg1\n2(ω+i0)(2ω)(2i0)+1\n(2i0)(−2ω)2(−ω+i0)/bracketrightbigg\n=±i\n2ω3α1/bracketleftbigg1\nω2+i0ω+1\nω2−i0ω/bracketrightbigg\n=±i\n2ω3α1/bracketleftbiggω2−i0ω+ω2+i0ω\nω4/bracketrightbigg\n=±iωα1. (67)\nHence, Im αR/A(ω) =±α1ω/2. Similarly,\n2 ReαR/A(ω) =αR/A(ω) +/bracketleftBig\nαR/A/bracketrightBig∗\n(ω)\n=−α1/integraldisplay∞\n0dε\nπ/bracketleftbiggω2\n(ω±i0)2−ε2+ω2\n(ω∓i0)2−ε2/bracketrightbigg\n=−α1/integraldisplay∞\n−∞dε\n2π/bracketleftbiggω2\n(ω±i0)2−ε2+ω2\n(ω∓i0)2−ε2/bracketrightbigg\n=−α1/integraldisplay∞\n−∞dε\n2π/bracketleftbiggω2\n(ω±i0 +ε)(ω±i0−ε)+ω2\n(ω∓i0 +ε)(ω∓i0−ε)/bracketrightbigg\n=α1/integraldisplay∞\n−∞dε\nπω2(ε2−ω2)\n(ε+ω+i0)(ε−ω−i0)(ε+ω−i0)(ε−ω+i0). (68)12\nSince the integral scales as 1 /|ε|, we can freely add the infinite circular contour along the top. Applying the residue\ntheorem, we find\n2 ReαR/A(ω) = 2iα1/bracketleftbiggω2((ω+i0)2−ω2)\n(ω+i0 +ω+i0)(ω+i0 +ω−i0)(ω+i0−ω+i0)\n+ω2((−ω+i0)2−ω2)\n(−ω+i0 +ω+i0)(−ω+i0−ω−i0)(−ω+i0−ω+i0)/bracketrightbigg\n= 2α1i/bracketleftbiggω2(2ωi0)\n2(ω+i0)(2ω)(2i0)+ω2(−2ωi0)\n(2i0)(−2ω)2(−ω+i0)/bracketrightbigg\n=α1\n2i/bracketleftbiggω2\nω+i0−ω2\nω−i0/bracketrightbigg\n=α1\n2i/bracketleftbiggω2(−2i0)\nω2/bracketrightbigg\n=α10, (69)\nwhich means that Re αR/A(ω) =α1\n2·0 = 0. Hence,\nαR/A(ω) =±α1iω\n2(70)\nand sinceαdiss(t) =−αR(t)−αA(−t) we have\nαdiss(ω) =−αR(ω)−αA(−ω) =−α1iω. (71)\nIn the LLG equation, we thus find\n(αdiss∗S) (t) =1\n2π/integraldisplay\ndωe−iωt(αdiss∗S) (ω)\n=1\n2π/integraldisplay\ndωe−iωtαdiss(ω)S(ω)\n=1\n2π/integraldisplay\ndωe−iωt(−α1iω)S(ω)\n=α1∂\n∂t1\n2π/integraldisplay\ndωe−iωtS(ω)\n=α1˙S(t). (72)\nFurthermore, we have that\nαK(ω) = [αR(ω)−αA(ω)] coth/parenleftBigω\n2T/parenrightBig\n=iα1ωcoth/parenleftBigω\n2T/parenrightBig\n. (73)\nWe have two regimes from the cotangent which crossover around ω≈2T. If the temperature is large enough that we\ncan approximate coth/parenleftbigω\n2T/parenrightbig\n≈2T\nω, then we have αK(ω)≈2iα1T. Therefore, we find\nαK(t) = 2iα1T1\n2π/integraldisplay∞\n−∞dωe−iωt= 2iα1Tδ(t), (74)\nand thus\n/angbracketleftξm(t)ξn(t/prime)/angbracketright=−2iδm,nαK(t−t/prime) = 4α1Tδm,nδ(t−t/prime). (75)\nWe see that Eq. (72) and Eq. (75) combine to give the regular LLG equation with first-order dissipation and white\nnoise fluctuation:\n˙S(t) =S(t)×/bracketleftBig\n−B+α1˙S(t) +ξ(t)/bracketrightBig\n. (76)13\nC. Sub-Ohmic spectral function\nWe now consider the case where\nJ(ε) =αssin/parenleftBigπs\n2/parenrightBig\nεs, (77)\nwith 0<s< 1. In this case,\nαR/A(ω) =−αssin/parenleftBigπs\n2/parenrightBig/integraldisplay∞\n0dε\nπω2εs−1\n(ω±i0)2−ε2. (78)\nConsidering the LLG equation, the relevant dissipation term is ( αdiss∗S)(t). In terms of the Fourier transform of\nαdiss(t), we find that\n(αdiss∗S)(t) =/integraldisplay∞\n−∞dt/primeαdiss(t−t/prime)S(t/prime)\n=−/integraldisplay∞\n−∞dt/prime[αR(t−t/prime) +αA(t/prime−t)]S(t/prime)\n=−1\n2π/integraldisplay∞\n−∞dt/prime/integraldisplay∞\n−∞dω/bracketleftBig\ne−iω(t−t/prime)αR(ω) +eiω(t−t/prime)αA(ω)/bracketrightBig\nS(t/prime)\n=−1\n2π/integraldisplay∞\n−∞dt/prime/integraldisplay∞\n−∞dωe−iω(t−t/prime)/bracketleftbig\nαR(ω) +αA(−ω)/bracketrightbig\nS(t/prime)\n=αssin/parenleftbigπs\n2/parenrightbig\n2π2/integraldisplay∞\n−∞dt/prime/integraldisplay∞\n−∞dω/integraldisplay∞\n0dεe−iω(t−t/prime)/bracketleftbiggω2εs−1\n(ω+i0)2−ε2+(−ω)2εs−1\n(−ω−i0)2−ε2/bracketrightbigg\nS(t/prime)\n=−αssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n−∞dt/prime/integraldisplay∞\n0dε/integraldisplay∞\n−∞dω/bracketleftbigg\ne−iω(t−t/prime)ω2εs−1\nε2−(ω+i0)2S(t/prime)/bracketrightbigg\n. (79)\nNotice that we have two poles at ω=±ε−i0, below the real axis. If t−t/prime<0, then the exponential will go to zero as\nω→+i∞. Hence we could close the ωintegration with a complex contour as an infinite half-circle along the top, and\nget zero from the Cauchy theorem. If t−t/prime>0, however, we see that the exponential goes to zero when ω→−i∞.\nHence, we can close the ωintegration along the bottom. Thus, using the residue theorem (reversing the integration\ndirection), we find\n(αdiss∗S)(t) =−αssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n−∞dt/prime/integraldisplay∞\n0dε2πiΘ(t−t/prime)\n/bracketleftBig\ne−i(ε−i0)(t−t/prime)(ε−i0)2εs−1\n(ε−i0 +i0 +ε)+e−i(−ε−i0)(t−t/prime)(−ε−i0)2εs−1\n(−ε−i0 +i0−ε)/bracketrightBig\nS(t/prime)\n=−iαssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplayt\n−∞dt/prime/integraldisplay∞\n0dε/bracketleftbigg\ne−iε(t−t/prime)εs+1\nε+eiε(t−t/prime)εs+1\n−ε/bracketrightbigg\nS(t/prime)\n=−iαssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplayt\n−∞dt/prime/integraldisplay∞\n0dε/bracketleftBig\ne−iε(t−t/prime)−eiε(t−t/prime)/bracketrightBig\nεsS(t/prime)\n=−2αssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplayt\n−∞dt/prime/integraldisplay∞\n0dεsin[ε(t−t/prime)]εsS(t/prime)\n=−2αssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplay∞\n0dε/braceleftbigg/bracketleftbig\nεs−1cos[ε(t−t/prime)]S(t/prime)/bracketrightbigt/prime=t\nt/prime=t0−/integraldisplayt\nt0dt/primecos[ε(t−t/prime)]εs−1˙S(t/prime)/bracerightbigg\n=−2αssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplay∞\n0dε/braceleftbigg\nεs−1S(t)−εs−1cos[ε(t−t0)]S(t0)−/integraldisplayt\nt0dt/primecos[ε(t−t/prime)]εs−1˙S(t/prime)/bracerightbigg\n.(80)\nThe first term vanishes because of the cross product with S(t) in the LLG equation. The second term is where we\nhad to be careful. Here, we should realize that the −∞ is physically only indicating that it is a time very far in the\npast. So, to avoid unphysical infinities, we introduced a finite initial time t0and we will take t0→−∞ later. For\nthis, we need to introduce some fractional derivative notation. We define the Riemann-Liouville (RL) and Caputo (C)14\nderivatives of order s, with an integer nsuch thatn≤s<n + 1, as\nRL\nt0Ds\ntf(t) =dn\ndtn1\nΓ(n−s)/integraldisplayt\nt0dt/prime(t−t/prime)n−1−sf(t/prime), (81)\nC\nt0Ds\ntf(t) =1\nΓ(n−s)/integraldisplayt\nt0dt/prime(t−t/prime)n−1−sf(n)(t/prime), (82)\nwhere we reserve the simpler Ds\ntnotation for the Liouville derivative that was used in the main text.\nNow, rescaling ε→ε/(t−t0) andε→ε/(t−t/prime) in the second and third terms of Eq. (80) respectively, we have\n(αdiss∗S)(t) = 2αssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplay∞\n0dεcos(ε)εs−1/bracketleftbigg\n(t−t0)−sS(t0) +/integraldisplayt\nt0dt/prime(t−t/prime)−s˙S(t/prime)/bracketrightbigg\n= 2αssin/parenleftbigπs\n2/parenrightbig\nπ/bracketleftBig\ncos/parenleftBigπs\n2/parenrightBig\nΓ(s)/bracketrightBig/bracketleftbigg\n(t−t0)−sS(t0) +/integraldisplayt\nt0dt/prime(t−t/prime)−s˙S(t/prime)/bracketrightbigg\n= 2αssin/parenleftbigπs\n2/parenrightbig\nπcos/parenleftBigπs\n2/parenrightBig\nΓ(s)/bracketleftbig\n(t−t0)−sS(t0) + Γ(1−s)C\nt0Ds\ntS(t)/bracketrightbig\n= 2αssin/parenleftbigπs\n2/parenrightbig\nπcos/parenleftBigπs\n2/parenrightBig\nΓ(s)Γ(1−s)/bracketleftbigg(t−t0)−s\nΓ(1−s)S(t0) +C\nt0Ds\ntS(t)/bracketrightbigg\n= 2αssin/parenleftbigπs\n2/parenrightbig\nπcos/parenleftBigπs\n2/parenrightBigπ\nsin(πs)RL\nt0Ds\ntS(t)\n=αsRL\nt0Ds\ntS(t) =αsDs\ntS(t), (83)\nwhere we used several identities from Sec. 6 in the Sup. Mat. of Ref. [12] and in the last line we sent t0→−∞ .\nFor the noise correlation, we have to compute the Keldysh component. This is\nαK(t) =1\n2π/integraldisplay∞\n−∞dωe−iωtαK(ω)\n=1\n2π/integraldisplay∞\n−∞dωe−iωt[αR(ω)−αA(ω)] coth/parenleftBigω\n2T/parenrightBig\n=αssin/parenleftbigπs\n2/parenrightbig\n2π2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωe−iωt/bracketleftbiggω2εs−1\nε2−(ω+i0)2−ω2εs−1\nε2−(ω−i0)2/bracketrightbigg\ncoth/parenleftBigω\n2T/parenrightBig\n(now sendω→−ωin the advanced part)\n=αssin/parenleftbigπs\n2/parenrightbig\n2π2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dω/bracketleftbigge−iωtω2εs−1\nε2−(ω+i0)2+eiωtω2εs−1\nε2−(−ω−i0)2/bracketrightbigg\ncoth/parenleftBigω\n2T/parenrightBig\n=αssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωcos(ωt)ω2εs−1\nε2−(ω+i0)2coth/parenleftBigω\n2T/parenrightBig\n. (84)\nNow, we send ω→ω/tandε→ε/t, which yields\nαK(t) =αst−1−ssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωcos(ω)ω2εs−1\nε2−(ω+i0)2coth/parenleftBigω\n2tT/parenrightBig\n. (85)\nTaking the high temperature limit, we get\nαK(t) =αst−1−ssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωcos(ω)ω2εs−1\nε2−(ω+i0)22tT\nω\n= 2Tαst−ssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωcos(ω)ωεs−1\nε2−(ω+i0)2\n=−2Tαst−ssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωcos(ω)ωεs−1\n(ω+i0−ε)(ω+i0 +ε). (86)15\nNow, we want to close the integral over ωwith an infinite half-circle. For this, we need fast enough convergence of\nthe integrand to zero. Splitting the cosine into two exponential parts cos( ω) = (eiω+e−iω)/2, we see that the first\nterm goes to zero when ω→i∞and that the second term goes to zero when ω→−i∞. Since we have poles at\nω=±ε−i0, the integral along the top half-plane vanishes. The integral along the bottom is then computed as\nαK(t) =−Tαst−ssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωe−iωωεs−1\n(ω+i0−ε)(ω+i0 +ε)\n=−Tαst−ssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε−2πi/bracketleftbigge−i(ε−i0)(ε−i0)εs−1\nε−i0 +i0 +ε+e−i(−ε−i0)(−ε−i0)εs−1\n−ε−i0 +i0−ε/bracketrightbigg\n= 2iTαst−ssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplay∞\n0dε/bracketleftbigge−iεεs\n2ε+eiεεs\n2ε/bracketrightbigg\n= 2iTαst−ssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplay∞\n0dεcos(ε)εs−1\n= 2iTαst−ssin/parenleftbigπs\n2/parenrightbig\nπcos/parenleftBigπs\n2/parenrightBig\nΓ(s)\n=iTαst−ssin (πs)\nπΓ(s)\n=iTαst−s\nΓ(1−s). (87)\nWe therefore find that\n/angbracketleftξm(t)ξn(t/prime)/angbracketright=−2iδm,nαK(t−t/prime) = 2αsTδm,n(t−t/prime)−s\nΓ(1−s), (88)\nwhere we assumed that t≥t/prime. Therefore, we have now found the fractional LLG equation\n˙S(t) =S(t)×[−B+αsDs\ntS(t) +ξ(t)]. (89)\nD. Super-Ohmic spectral function\nWe now consider the case where\nJ(ε) =αssin/parenleftBigπs\n2/parenrightBig\nεs, (90)\nwith 1< s < 2. In this case, everything is equivalent to the sub-Ohmic case, up to Eq. (80), where we wanted to\nrewrite the dissipation into a fractional derivative. We had to introduce a finite initial time t0, which lead to\n(αdiss∗S)(t) =−2αssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplayt\n−∞dt/prime/integraldisplay∞\n0dεsin[ε(t−t/prime)]εsS(t/prime)\n=−2αssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplay∞\n0dε/braceleftbigg/bracketleftbig\nεs−1cos[ε(t−t/prime)]S(t/prime)/bracketrightbigt/prime=t\nt/prime=t0−/integraldisplayt\nt0dt/primecos[ε(t−t/prime)]εs−1˙S(t/prime)/bracerightbigg\n=−2αssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplay∞\n0dε/braceleftbigg\nεs−1S(t)−εs−1cos[ε(t−t0)]S(t0)−/integraldisplayt\nt0dt/primecos[ε(t−t/prime)]εs−1˙S(t/prime)/bracerightbigg\n.(91)\nThe first term vanishes because of the cross product with S(t) in the LLG equation. However, the second term is\nmore problematic compared to the sub-Ohmic case, since the identity used to rewrite it only holds for s <1. We\nsolve this by writing it as a time derivative\nεs−1cos[ε(t−t0)]S(t0) =d\ndtεs−2sin[ε(t−t0)]S(t0), (92)16\nand then switching the ordering of the derivative and integral. Performing also one more partial integration in t/prime, we\nget\n(αdiss∗S)(t)\n= 2αssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplay∞\n0dε/braceleftbiggd\ndtεs−2sin[ε(t−t0)]S(t0) +/integraldisplayt\nt0dt/primecos[ε(t−t/prime)]εs−1˙S(t/prime)/bracerightbigg\n= 2αssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplay∞\n0dε/parenleftbiggd\ndtεs−2sin[ε(t−t0)]S(t0) +/bracketleftBig\nεs−2sin[ε(t−t/prime)]˙S(t/prime)/bracketrightBigt/prime=t\nt/prime=t0−/integraldisplayt\nt0dt/prime/braceleftBig\n−sin[ε(t−t/prime)]εs−2¨S(t/prime)/bracerightBig/parenrightbigg\n= 2αssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplay∞\n0dε/braceleftbiggd\ndtεs−2sin[ε(t−t0)]S(t0) +εs−2sin[ε(t−t0)]˙S(t0) +/integraldisplayt\nt0dt/primesin[ε(t−t/prime)]εs−2¨S(t/prime)/bracerightbigg\n. (93)\nNow, rescaling ε→ε/(t−t0) andε→ε/(t−t/prime) respectively, we have\n(αdiss∗S)(t)\n= 2αssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplay∞\n0dεsin(ε)εs−2/bracketleftbiggd\ndt(t−t0)1−sS(t0) + (t−t0)1−s˙S(t0) +/integraldisplayt\nt0dt/prime(t−t/prime)1−s¨S(t/prime)/bracketrightbigg\n= 2αssin/parenleftbigπs\n2/parenrightbig\nπ/bracketleftbigg\nsin/parenleftbiggπ(s−1)\n2/parenrightbigg\nΓ(s−1)/bracketrightbigg/bracketleftbigg\n(1−s)(t−t0)−sS(t0) + (t−t0)1−s˙S(t0) +/integraldisplayt\nt0dt/prime(t−t/prime)1−s¨S(t/prime)/bracketrightbigg\n=−2αssin/parenleftbigπs\n2/parenrightbig\nπcos/parenleftBigπs\n2/parenrightBig\nΓ(s−1)/bracketleftBig\n(1−s)(t−t0)−sS(t0) + (t−t0)1−s˙S(t0) + Γ(2−s)C\nt0Ds\ntS(t)/bracketrightBig\n=−2αssin/parenleftbigπs\n2/parenrightbig\nπcos/parenleftBigπs\n2/parenrightBig\nΓ(s−1)Γ(1−(s−1))/bracketleftbigg(t−t0)−s\nΓ(1−s)S(t0) +(t−t0)1−s\nΓ(2−s)˙S(t0) +C\nt0Ds\ntS(t)/bracketrightbigg\n=−αssin (πs)\nππ\nsin[π(s−1)]RL\nt0Ds\ntS(t)\n=αsDs\ntS(t), (94)\nwhere we used several identities from Sec. 6 in the Sup. Mat. of Ref. [12], Ref. [13, p.893], and in the last line we\nsentt0→−∞ .\nFor the noise correlation, we have to compute the Keldysh component. This is\nαK(t) =1\n2π/integraldisplay∞\n−∞dωe−iωtαK(ω)\n=1\n2π/integraldisplay∞\n−∞dωe−iωt[αR(ω)−αA(ω)] coth/parenleftBigω\n2T/parenrightBig\n=αssin/parenleftbigπs\n2/parenrightbig\n2π2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωe−iωt/bracketleftbiggω2εs−1\nε2−(ω+i0)2−ω2εs−1\nε2−(ω−i0)2/bracketrightbigg\ncoth/parenleftBigω\n2T/parenrightBig\n. (95)\nNow, we send ω→−ωin the advanced part\nαK(t) =αssin/parenleftbigπs\n2/parenrightbig\n2π2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dω/bracketleftbigge−iωtω2εs−1\nε2−(ω+i0)2+eiωtω2εs−1\nε2−(−ω−i0)2/bracketrightbigg\ncoth/parenleftBigω\n2T/parenrightBig\n=αssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωcos(ωt)ω2εs−1\nε2−(ω+i0)2coth/parenleftBigω\n2T/parenrightBig\n. (96)\nThen, we insert cos( ωt) =d\ndtsin(ωt)\nωand we send ω→ω/tandε→ε/t, which yields\nαK(t) =d\ndtαssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωsin(ωt)ωεs−1\nε2−(ω+i0)2coth/parenleftBigω\n2T/parenrightBig\n=d\ndtαst−ssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωsin(ω)ωεs−1\nε2−(ω+i0)2coth/parenleftBigω\n2tT/parenrightBig\n. (97)17\nTaking the high-temperature limit, we get\nαK(t) =d\ndtαst−ssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωsin(ω)ωεs−1\nε2−(ω+i0)22tT\nω\n=d\ndt2Tαst1−ssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωsin(ω)εs−1\nε2−(ω+i0)2\n=−2Tαs(1−s)t−ssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωsin(ω)εs−1\n(ω+i0−ε)(ω+i0 +ε). (98)\nNow, we want to close the integral over ωwith an infinite half-circle. For this, we need fast enough convergence of\nthe integrand to zero. Splitting the cosine into two exponential parts sin( ω) = (eiω−e−iω)/2i, we see that the first\nterm goes to zero when ω→i∞and that the second term goes to zero when ω→−i∞. Since we have poles at\nω=±ε−i0, the integral along the top half-plane vanishes. The integral along the bottom is then computed as\nαK(t) =−iTαs(1−s)t−ssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε/integraldisplay∞\n−∞dωe−iωεs−1\n(ω+i0−ε)(ω+i0 +ε)\n=−iTαs(1−s)t−ssin/parenleftbigπs\n2/parenrightbig\nπ2/integraldisplay∞\n0dε−2πi/bracketleftbigge−i(ε−i0)εs−1\nε−i0 +i0 +ε+e−i(−ε−i0)εs−1\n−ε−i0 +i0−ε/bracketrightbigg\n=−2Tαs(1−s)t−ssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplay∞\n0dε/bracketleftbigge−iεεs−2\n2−eiεεs−2\n2/bracketrightbigg\n=i2Tαs(1−s)t−ssin/parenleftbigπs\n2/parenrightbig\nπ/integraldisplay∞\n0dεsin(ε)εs−2\n=−i2Tαs(1−s)t−ssin/parenleftbigπs\n2/parenrightbig\nπcos/parenleftBigπs\n2/parenrightBig\nΓ(s−1)\n=iTαst−ssin (πs)\nπΓ(s)\n=iTαst−s\nΓ(1−s). (99)\nWe therefore find the same expression as in the sub-Ohmic case, which leads to\n/angbracketleftξm(t)ξn(t/prime)/angbracketright=−2iδm,nαK(t−t/prime) = 2αsTδm,n(t−t/prime)−s\nΓ(1−s). (100)\nTherefore, we have now also found the fractional LLG equation in the super-Ohmic case.\nE. Comparison Ohmic versus non-Ohmic\nIn the Ohmic case, we started with J(ω) =α1ωand ended up with a α1˙S(t) friction term, and noise correlation\n4α1kBTδ(t−t/prime). On the other hand, in the non-Ohmic case, we started with J(ε) =αssin/parenleftbigπs\n2/parenrightbig\nεsand ended up with\na frictionαsDs\ntS(t) and noise correlation 2 αskBT(t−t/prime)−s\nΓ(1−s). Although it is clear that both the non-Ohmic J(ω) and\nthe friction term will go to the Ohmic case when s→1, the noise is less straightforward. We can see that, as s→1,\nthe Gamma function will blow up, hence sending the correlation to zero, with the exception of t=t/prime. In this case,\nthe numerator blows up even before taking the limit of s→1. Hence, we can expect this function to behave as a\ndelta function. To find the correct prefactor, we integrate the distribution with a test-function f(t/prime) = 1 to find\nlim\ns→1/integraldisplay∞\n−∞d(t−t/prime)(t−t/prime)−s\nΓ(1−s)·1 = lim\ns→1/bracketleftbigg(t−t/prime)1−s\n(1−s)Γ(1−s)/bracketrightbigg∞\n(t−t/prime)=−∞\n= lim\ns→1/bracketleftbigg(t−t/prime)1−s\nΓ(2−s)/bracketrightbigg∞\n(t−t/prime)=−∞\n=/bracketleftbigg(t−t/prime)\n|t−t/prime|Γ(1)/bracketrightbigg∞\n(t−t/prime)=−∞\n= 2.\nHence, we see that the limit of the noise correlation becomes 4 α1kBTδ(t−t/prime), which is precisely as in the Ohmic case.18\nIII. FMR POWERLAW DERIVATION\nHere, we calculate the response of this spin-bath system to a rotating magnetic field. We will find the steady state\nsolutions and calculate several quantities that experiments could measure.\nA. Ferromagnetic Resonance\nWe will study the effects of a rotating magnetic field on the fractional LLG (FLLG) equation\n˙S(t) =S(t)×/braceleftbig\n−Beff[t,S(t)] +αsDs\ntS(t) +ξ(t)/bracerightbig\n. (101)\nFor simplicity, we assume that the temperature of the bath is low compared to the energy of the external fields, such\nthat the thermal noise ξ(t) may be neglected. We apply a rotating magnetic field\nBeff[t,S(t)] =\nΩ cos(ωdt)\nΩ sin(ωdt)\nB0−KSz(t)\n (102)\nand use spherical coordinates\nS=S\nsinθcosφ\nsinθsinφ\ncosθ\n. (103)\nWe will assume a small θapproximation, where the ground state is in the positive zdirection, i.e. 0 <Ω/lessmuchB0−KS\nandαsS/lessmuch(B0−KS)1−s. As shown in Section I D, we may rewrite the FLLG equation of this form in spherical\ncoordinates as\n˙φ=1\nsinθ[−Beff[t,S(t)] +αsDs\ntS(t)]·\ncosθcosφ\ncosθsinφ\n−sinθ\n; (104)\n˙θ= [−Beff[t,S(t)] +αsDs\ntS(t)]·\nsinφ\n−cosφ\n0\n. (105)\nIn the rotating frame, where B(t) is constant, we could expect the system to go to a steady state after some time.\nHence, we introduce a new coordinate such, that φ=ωdt−ϕ. We may then set ˙ ϕ=˙θ= 0 to find the steady state in\nthe rotating frame, where\nωdsinθ= [−Beff[t,S(t)] +αsDs\ntS(t)]·\ncosθcos(ωdt−ϕ)\ncosθsin(ωdt−ϕ)\n−sinθ\n; (106)\n0 = [−Beff[t,S(t)] +αsDs\ntS(t)]·\nsin(ωdt−ϕ)\n−cos(ωdt−ϕ)\n0\n. (107)\nWe note that S(t) is now only time dependent in the rotating-frame term, which means that we can explicitly calculate\nthe fractional derivative. The Liouville derivative works well with Fourier transforms, hence the fractional derivative\nof a trigonometric function is given by\nDs\ntsin(ωt) =|ω|ssin/parenleftBig\nωt+ sign(ω)πs\n2/parenrightBig\n, (108)\nand similarly for a cosine. We remark that the Liouville derivative of a constant can only be described by setting the\ninitial time to some finite t0, in which case it becomes zero2. Combining this with the steady state expression for S,\n2Since physically the infinite time only models a long time in the past, we will apply it as such. Formally, there are some restrictions on\nthe functions that the Liouville derivative can be applied to. These include restrictions such as its integral over the whole domain being\nfinite. Since this is not the case for a non-zero constant on an infinite interval, we have to regularize the lower integral boundary −∞\nas a finitet0.19\nwe find\nαsDs\ntS(t) =αsDs\ntS\nsinθcos(ωdt−ϕ)\nsinθsin(ωdt−ϕ)\ncosθ\n=αsSsinθ|ωd|s\ncos/parenleftbig\nωdt−ϕ+ sign(ωd)πs\n2/parenrightbig\nsin/parenleftbig\nωdt−ϕ+ sign(ωd)πs\n2/parenrightbig\n0\n. (109)\nHence, we find that\nωdtanθ= [−Beff[t,S(t)] +αsDs\ntS(t)]·\ncos(ωdt−ϕ)\nsin(ωdt−ϕ)\n−tanθ\n\n=\n−\nΩ cos(ωdt)\nΩ sin(ωdt)\nB0−KScosθ\n+αsSsinθ|ωd|s\ncos/parenleftbig\nωdt−ϕ+ sign(ωd)πs\n2/parenrightbig\nsin/parenleftbig\nωdt−ϕ+ sign(ωd)πs\n2/parenrightbig\n0\n\n·\ncos(ωdt−ϕ)\nsin(ωdt−ϕ)\n−tanθ\n\n=−Ω cosϕ+B0tanθ−KSsinθ+αsSsinθ|ωd|scos/parenleftBig\nsign(ωd)πs\n2/parenrightBig\n(110)\nand\n0 = [−Beff[t,S(t)] +αsDs\ntS(t)]·\nsin(ωdt−ϕ)\n−cos(ωdt−ϕ)\n0\n\n=\n−\nΩ cos(ωdt)\nΩ sin(ωdt)\nB0−KScosθ\n+αsSsinθ|ωd|s\ncos/parenleftbig\nωdt−ϕ+ sign(ωd)πs\n2/parenrightbig\nsin/parenleftbig\nωdt−ϕ+ sign(ωd)πs\n2/parenrightbig\n0\n\n·\nsin(ωdt−ϕ)\n−cos(ωdt−ϕ)\n0\n\n= Ω sinϕ−αsSsinθ|ωd|ssin/parenleftBig\nsign(ωd)πs\n2/parenrightBig\n. (111)\nWith some rearranging, we have\nϕ= arcsin/bracketleftbiggαsS\nΩ|ωd|ssin/parenleftBig\nsign(ωd)πs\n2/parenrightBig\nsinθ/bracketrightbigg\n(112)\nand\nΩ cosϕ= Ω cos arcsin/bracketleftbiggαsS\nΩ|ωd|ssin/parenleftBig\nsign(ωd)πs\n2/parenrightBig\nsinθ/bracketrightbigg\n=/radicalbigg\nΩ2−/bracketleftBig\nαsS|ωd|ssin/parenleftBig\nsign(ωd)πs\n2/parenrightBig\nsinθ/bracketrightBig2\n= (B0−KScosθ−ωd) tanθ+αsSsinθ|ωd|scos/parenleftBig\nsign(ωd)πs\n2/parenrightBig\n. (113)\nSquaring this, we get\nΩ2= (B0−KScosθ−ωd)2tan2θ+ (αsS|ωd|s)2sin2θ+ 2αsS|ωd|s(B0−KScosθ−ωd) cos/parenleftBigπs\n2/parenrightBigsin2θ\ncosθ,\nand multiplying by cos2θ= 1−sin2θ, we have\n(1−sin2θ)Ω2= (B0−KS/radicalbig\n1−sin2θ−ωd)2sin2θ+/bracketleftBig\n(αsS|ωd|s)2−2αsS|ωd|sKScos/parenleftBigπs\n2/parenrightBig/bracketrightBig\nsin2θ(1−sin2θ)\n+ 2αsS|ωd|s(B0−ωd) cos/parenleftBigπs\n2/parenrightBig\nsin2θ/radicalbig\n1−sin2θ. (114)\nWe could go further and make this into (effectively) a 4th order equation for sin2θ. However, since we are in a small\nθlimit, we will solve this equation up to first order in sin2θ, which yields\nΩ2= sin2θ/braceleftBig\nΩ2(B0−KS−ωd)2+/bracketleftBig\n(αsS|ωd|s)2−2αsS|ωd|sKScos/parenleftBigπs\n2/parenrightBig/bracketrightBig\n+ 2αsS|ωd|s(B0−ωd) cos/parenleftBigπs\n2/parenrightBig/bracerightBig\n. (115)20\nHence, we find that\nsin2θ=Ω2\nΩ2+ (B0−KS−ωd)2+ (αsS|ωd|s)2+ 2αsS|ωd|s(B0−KS−ωd) cos/parenleftbigπs\n2/parenrightbig\n=Ω2\n(B0−KS−ωd)2+ (αsS|ωd|s)2+ 2αsS|ωd|s(B0−KS−ωd) cos/parenleftbigπs\n2/parenrightbig+O(Ω4). (116)\nNote that theO(Ω4) should formally be dimensionless, but we explain what we mean with terms being small in\nSection IV, as this is more subtle with fractional dimensions.\nB. Resonance frequency and amplitude\nSince we are studying ferromagnetic resonance, we want to find the driving frequency for which we get the largest\nresponse from the magnetic system. Since we know that the resonance for an Ohmic system is at ωd=B0−KS, we\nwill expand the formula around this point to find the new maximum. To compute the resonance frequency ωres, we\nthus first assume that ωres≈(B0−KS)(1+y) withysmall, such that|ωres|s≈(B0−KS)s(1+sy) (forB0−KS > 0).\nThis results in\nsin2θ=Ω2\n(B0−KS−ωd)2+ (αsS|ωd|s)2+ 2αsS|ωd|s(B0−KS−ωd) cos/parenleftbigπs\n2/parenrightbig\n≈Ω2\n(B0−KS)2y2+ (αsS)2(B0−KS)2s(1 + 2sy)−2αsS(B0−KS)s+1(1 +sy)ycos/parenleftbigπs\n2/parenrightbig. (117)\nNow, we put the derivative with respect to yequal to zero, to get\n(B0−KS)2y+s(αsS)2(B0−KS)2s−αsS(B0−KS)s+1(1 + 2sy) cos/parenleftBigπs\n2/parenrightBig\n= 0. (118)\nHence, we find that\ny=−s(αsS)2(B0−KS)2s+αsS(B0−KS)s+1cos/parenleftbigπs\n2/parenrightbig\n(B0−KS)2−2sαsS(B0−KS)s+1cos/parenleftbigπs\n2/parenrightbig\n=αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig\n+O(αsS)2, (119)\nwhich results in\nωres≈(B0−KS)/bracketleftBig\n1 +αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/bracketrightBig\n= (B0−KS) +αsS(B0−KS)scos/parenleftBigπs\n2/parenrightBig\n. (120)\nWe see that the resonance frequency gets shifted by a small amount, depending on s, which scales non-linearly.\nInserting this result into Eq. (116), we can now also find an approximation for the amplitude at resonance:\nsin2θres=Ω2\n(B0−KS−ωres)2+ (αsS|ωres|s)2+ 2αsS|ωres|s(B0−KS−ωres) cos/parenleftbigπs\n2/parenrightbig\n≈Ω2/braceleftBig/bracketleftBig\nαsS(B0−KS)scos/parenleftBigπs\n2/parenrightBig/bracketrightBig2\n+/parenleftBig\nαsS/vextendsingle/vextendsingle/vextendsingle(B0−KS) +αsS(B0−KS)scos/parenleftBigπs\n2/parenrightBig/vextendsingle/vextendsingle/vextendsingles/parenrightBig2\n−2αsS/vextendsingle/vextendsingle/vextendsingle(B0−KS) +αsS(B0−KS)scos/parenleftBigπs\n2/parenrightBig/vextendsingle/vextendsingle/vextendsingles\nαsS(B0−KS)scos2/parenleftBigπs\n2/parenrightBig/bracerightBig−1\n=Ω2\n[αsS(B0−KS)s]2/bracketleftBig\ncos2/parenleftBigπs\n2/parenrightBig\n+/vextendsingle/vextendsingle/vextendsingle1 +αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/vextendsingle/vextendsingle/vextendsingle2s\n−2/vextendsingle/vextendsingle/vextendsingle1 +αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/vextendsingle/vextendsingle/vextendsingles\ncos2/parenleftBigπs\n2/parenrightBig/bracketrightBig−1\n=Ω2\n[αsS(B0−KS)s]2/bracketleftBig\n1−cos2/parenleftBigπs\n2/parenrightBig\n+O(αsS)/bracketrightBig−1\n≈Ω2\n/bracketleftbig\nαsS(B0−KS)ssin/parenleftbigπs\n2/parenrightbig/bracketrightbig2. (121)\nSince the sine function decreases as smoves away from one, we see that the amplitude actually increases for non-Ohmic\nenvironments.21\nC. Calculating the FWHM linewidth\nNext, we are interested not only in the location of the resonance, but also how sensitive the resonance is to the\ndriving frequency. One way to describe this is by using the Full Width at Half Maximum measure. This provides a\nwell-defined line width independently of the shape of the peak. It is found by measuring the width of the peak at\nhalf the height of its maximum. This can be measured in the laboratories, but it can also be computed. Since our\nfunction of interest is of the form sin2θ(ωd) = Ω2/g(ωd), it makes sense to approximate the inverse function instead\nof the regular one. To this end, we will translate the FWHM measurement to the inverse function, and then Taylor\nexpandg(ωd) near resonance as a parabola to solve for the new condition of this inverse function. Notice that from\nEq. (116), we have\ng(ωd) = (B0−KS−ωd)2+ (αsS|ωd|s)2+ 2αsS|ωd|s(B0−KS−ωd) cos/parenleftBigπs\n2/parenrightBig\n. (122)\nThe FWHM condition is\nΩ2\ng(ωd)= sin2θ(ωd) =sin2θ(ωres)\n2=Ω2\n2g(ωres), (123)\nhence we must solve for 2 g(ωres) =g(ωd). To this end, let us assume that ωd=ωres+yand expand g(ωd) iny. We\nwill use that\n|a+y|n≈an+nan−1y+1\n2n(n−1)an−2y2\nfor smallyanda>0. Then,\ng(ωres+y)\n= (B0−KS−ωres−y)2+ (αsS|ωres+y|s)2+ 2αsS|ωres+y|s(B0−KS−ωres−y) cos/parenleftBigπs\n2/parenrightBig\n≈(B0−KS−ωres)2+ (αsSωs\nres)2+ 2αsSωs\nres(B0−KS−ωres) cos/parenleftBigπs\n2/parenrightBig\n+y/parenleftBig\n−2(B0−KS−ωres) + 2s(αsS)2ω2s−1\nres−2αsScos/parenleftBigπs\n2/parenrightBig/braceleftbig\nωs\nres+sωs−1\nres[ωres−(B0−KS)]/bracerightbig/parenrightBig\n+y2/bracketleftBig\n1 +s(2s−1)(αsS)2ω2s−2\nres−2sαsSωs−1\nrescos/parenleftBigπs\n2/parenrightBig\n+s(s−1)αsSωs−2\nres(B0−KS−ωres) cos/parenleftBigπs\n2/parenrightBig/bracketrightBig\n=g(ωres) +y/parenleftBigg\n2αsS(B0−KS)scos/parenleftBigπs\n2/parenrightBig\n+ 2s(αsS)2(B0−KS)2s−1/bracketleftBig\n1 +αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/bracketrightBig2s−1\n−2αsS(B0−KS)scos/parenleftBigπs\n2/parenrightBig/braceleftBigg/bracketleftBig\n1 +αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/bracketrightBigs\n+sαsScos/parenleftBigπs\n2/parenrightBig\n(B0−KS)s−1/bracketleftBig\n1 +αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/bracketrightBigs−1/bracerightBigg/parenrightBigg\n+y2/braceleftBigg\n1 +s(2s−1)(αsS)2(B0−KS)2s−2/bracketleftBig\n1 +αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/bracketrightBig2s−2\n−2sαsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/bracketleftBig\n1 +αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/bracketrightBigs−1\n−s(s−1)(αsS)2(B0−KS)2s−2cos2/parenleftBigπs\n2/parenrightBig/bracketleftBig\n1 +αsS(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig/bracketrightBigs−2/bracerightBigg\n≈g(ωres) +y/braceleftBig\n2s(αsS)2(B0−KS)2s−1/bracketleftBig\n1−2 cos2/parenleftBigπs\n2/parenrightBig/bracketrightBig/bracerightBig\n+y2/braceleftBig\n1−2s(αsS)(B0−KS)s−1cos/parenleftBigπs\n2/parenrightBig\n+ (αsS)2(B0−KS)2s−2/bracketleftBig\ns(2s−1)−3s(s−1) cos2/parenleftBigπs\n2/parenrightBig/bracketrightBig/bracerightBig\n+O(αsS)3.\n(124)\nNow, we set 2 g(ωres) =g(ωres+y) =g(ωres) +by+ay2, and remark that g(ωres) = (αsS)2(B0−KS)2ssin2/parenleftbigπs\n2/parenrightbig\n, in\norder to find that\ny=−b±/radicalbig\nb2+ 4ag(ωres)\n2a=⇒∆FWHM =/radicalbig\nb2+ 4ag(ωres)\na. (125)22\nHence, we find that the lowest-order contribution to the linewidth is given by\n∆FWHM≈/radicalBig\n4(αsS)2(B0−KS)2ssin2/parenleftbigπs\n2/parenrightbig\n+O(αsS)3\n1 +O(αsS)\n= 2(αsS)(B0−KS)ssin/parenleftBigπs\n2/parenrightBig\n+O(αsS)2. (126)\nIV. DIMENSIONAL ANALYSIS\nThe fractional derivative in the LLG equation has an impact on the dimensions of quantities. We can firstly see\nthat in the chosen units, we have [ B0−KS] = [ωd] = time−1. Assuming Sto be dimensionless, then [ ωd] = [αsDs\ntS] =\n[αs][ωd]s, hence [αs] = [ωd]1−s. We can now start to understand what we mean when we say that certain quantities\nare small, since this has to be relative to something else. For instance, when we say αsSis small, we understand this\nasαsS/lessmuch(B0−KS)1−s. For Ω it is simpler, since there is no fractional derivative acting with it. Hence, for Ω small\nwe simply mean Ω /lessmuchB0−KS. We can now also define some dimensionless variables, such as α/prime\ns=αsS(B0−KS)s−1\nand Ω/prime= Ω/(B0−KS). We have used these variables in the figures to show the general behavior of the quantities.\n[1] A. Altland and B. D. Simons, Condensed Matter Field Theory (Cambridge University Press, 2010), 2nd ed.\n[2] A. Kamenev, Field theory of non-equilibrium systems (Cambridge University Press, 2011).\n[3] A. Schmid, Journal of Low Temperature Physics 49, 609 (1982).\n[4] A. O. Caldeira and A. J. Leggett, Physical Review Letters 46, 211 (1981).\n[5] A. O. Caldeira and A. J. Leggett, Physica A: Statistical Mechanics and its Applications 121, 587 (1983).\n[6] A. O. Caldeira and A. J. Leggett, Annals of Physics 149, 374 (1983).\n[7] A. O. Caldeira, An introduction to macroscopic quantum phenomena and quantum dissipation , vol. 9780521113755 (Cam-\nbridge University Press, 2012).\n[8] A. O. Caldeira and A. J. Leggett, Phys. Rev. A 31, 1059 (1985).\n[9] U. Weiss, Quantum dissipative systems (World scientific, 2012).\n[10] C. Gardiner and P. Zoller, Quantum noise: a handbook of Markovian and non-Markovian quantum stochastic methods with\napplications to quantum optics (Springer Science & Business Media, 2004).\n[11] A. Shnirman, Y. Gefen, A. Saha, I. S. Burmistrov, M. N. Kiselev, and A. Altland, Physical Review Letters 114, 176806\n(2015).\n[12] R. C. Verstraten, R. F. Ozela, and C. M. Smith, Physical Review B 103, L180301 (2021).\n[13] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products (Academic press, 2014)."    },    {        "title": "1506.05622v2.The_absence_of_intraband_scattering_in_a_consistent_theory_of_Gilbert_damping_in_metallic_ferromagnets.pdf",        "content": "arXiv:1506.05622v2  [cond-mat.str-el]  23 Oct 2015The absence of intraband scattering in a consistent theory o f Gilbert damping in\nmetallic ferromagnets\nD M Edwards\nDepartment of Mathematics, Imperial College London, Londo n SW7 2BZ, United Kingdom\nDamping of magnetization dynamics in a ferromagnetic metal , arising from spin-orbit coupling,\nis usually characterised by the Gilbert parameter α. Recent calculations of this quantity, using\na formula due to Kambersky, find that it is infinite for a perfec t crystal owing to an intraband\nscattering term which is of third order in the spin-orbit par ameterξ. This surprising result conflicts\nwith recent work by Costa and Muniz who study damping numeric ally by direct calculation of\nthe dynamical transverse susceptibility in the presence of spin-orbit coupling. We resolve this\ninconsistency by following the approach of Costa and Muniz f or a slightly simplified model where\nit is possible to calculate αanalytically. We show that to second order in ξone retrieves the\nKambersky result for α, but to higher order one does not obtain any divergent intrab and terms.\nThe present work goes beyond that of Costa and Muniz by pointi ng out the necessity of including\nthe effect of long-range Coulomb interaction in calculating damping for large ξ. A direct derivation\nof the Kambersky formula is given which shows clearly the res triction of its validity to second order\ninξso that no intraband scattering terms appear. This restrict ion has an important effect on the\ndamping over a substantial range of impurity content and tem perature. The experimental situation\nis discussed.\nI. INTRODUCTION\nMagnetization dynamics in a ferromagnetic metal is central to the fi eld of spintronics with its many applications.\nDamping is an essential feature of magnetization dynamics and is usu ally treated phenomenologically by means\nof a Gilbert term in the Landau-Lifshitz-Gilbert equation [1, 2]. For a system with spin-rotational invariance the\nuniform precession mode of the magnetization in a uniform external magnetic field is undamped and the fundamental\norigin of damping in ferromagnetic resonance is spin-orbit coupling (S OC). Early investigations of the effect include\nthose of Kambersky [3–5] and Korenman and Prange [6]. Kambersky ’s [4] torque-correlation formula for the Gilbert\ndamping parameter αhas been used by several groups [7–14], some of whom have given alt ernative derivations.\nHowever the restricted validity of this formula, as discussed below, has not been stressed. In this torque-correlation\nmodel contributions to αof both intraband and interband electronic transitions are usually c onsidered. The theory\nis basically developed for a pure metal with the effect of defects and /or phonons introduced as phenomenological\nbroadening of the one-electron states. Assuming that the electr on scattering-rate increases with temperature T due\nto electron-phonon scattering the intraband and interband tran sitions are found to play a dominant role in low and\nhigh T regimes, respectively. The intraband(interband) term is pre dicted to decrease(increase) with increasing T and\nto be proportional to ξ3(ξ2) whereξis the SOC parameter. Accordingly αis expected to achieve a minimum at an\nintermediate T. This is seen experimentally in Ni and hcp Co [15] but not in Fe [15] and FePt [16]. The ξ2dependence\nofαis well-established at high T [17, 18] but there seems to be no experim ental observation of the predicted ξ3\nbehaviourat lowT. The interband ξ2term in Kambersky’stheory canbe givenaverysimple interpretation in termsof\nsecond-orderperturbation theory [5]. A quite different phenomen ologicalapproach, not applicable in some unspecified\nlow scattering-rate regime, has been adopted to try and find a phy sical interpretation of the intraband term [5, 8].\nNo acceptable theoretical treatment of damping in this low scatter ing regime is available because the intraband term\nof Kambersky’s theory diverges to infinity in the zero-scattering lim it of a pure metal with translational symmetry at\nT=0 [9, 11, 13]. We consider it essential to understand the pure met al limit before introducing impurity and phonon\nscattering in a proper way.\nCosta and Muniz [19] recently studied damping numerically in this limit by direct calculation of the dynamical\nspin susceptibility in the presence of SOC within the random phase app roximation (RPA). They determine αfrom\nthe linewidth of the uniform (wave-vector q= 0) spin-wave mode which appears as a resonance in the transvers e\nsusceptibility. One of the main objects of this paper is to establish so me degree of consistency between the work of\nKambersky and that of Costa and Muniz. We follow the approach of t he latter authors for a slightly simplified model\nwhere it is possible to calculate αanalytically. We show that to second order in ξone retrieves the Kambersky result,\nbut to higher order no intraband terms occur, which removes the p roblem of divergent α. To confirm this point, in\nAppendix A we derive the Kambersky formula directly in a way that mak es clear its restriction to second order in\nξto which order the divergent terms in αarising from intraband transitions do not appear. This throws open the\ninterpretation of the minimum observed in the temperature depend ence ofαfor Ni and Co.\nAt this point we may mention an alternative theoretical approach to the calculation of Gilbert damping using2\nscattering theory [20, 21]. Starikov et al [21] find that, for Ni 1−xFexalloys at T=0, αbecomes large near the pure\nmetal limits x=0,1. They attribute this to the Kambersky intraband c ontribution although no formal correspondence\nis made between the two approaches.\nThe work of Costa and Muniz [19] follows an earlier paper [22] where it is shown that SOC has the effect of\ncoupling the transverse spin susceptibility to the longitudinal spin su sceptibility and the charge response. It is known\nthat a proper calculation of these last two quantities in a ferromagn et must take account of long-range Coulomb\ninteractions [23–27]. The essential role of these interactions is to e nsure conservation of particle number. Costa et\nal [19, 22] do not consider such interactions but we show here that this neglect is not serious for calculating αwith\nsufficiently small SOC. Howeverin the wider frameworkof this paper, where mixed charge-spinresponse is also readily\nstudied, long-rangeinteractions are expected to sometimes play a role. They also come into play, even to second order\ninξ, when inversion symmetry is broken.\nIn section II we establish the structure of spin-density response theory in the presence of SOC by means of exact\nspin-density functional theory in the static limit [28]. In section III w e introduce a spatial Fourier transform and an\napproximation to the dynamical response is obtained by introducing the frequency dependence of the non-interacting\nsusceptibilities. The theory then has the same structure as in the R PA. Section IV is devoted to obtaining an explicit\nexpression for the transverse susceptibility in terms of the non-in teracting susceptibilities. Expressions for these, in\nthe presence of SOC, are obtained within the tight-binding approxim ation in section V. In section VI we consider\nthe damping of the resonance in the q=0 transverse susceptibility a nd show how the present approach leads to the\nKambersky formula for the Gilbert damping parameter αwhere this is valid, namely to second order in the SOC\nparameter ξ. We do not give an explicit formula for αbeyond this order but it is clear that no intraband terms appear.\nIn section VII some experimental aspects are discussed with sugg estions for future work. The main conclusions are\nsummarized in section VIII.\nII. SPIN-DENSITY FUNCTIONAL THEORY WITH SPIN-ORBIT COUPLI NG\nThe Kohn-Sham equation takes the form\n/summationdisplay\nσ′[−δσσ′(/planckover2pi12/2m)∇2+Veff\nσσ′(r)+Hso\nσσ′]φnσ′(r) =ǫnφnσ(r) (1)\nwith the spin index σ=↑,↓corresponding to quantization along the direction of the ground-s tate magnetization in a\nferromagnet. This may be written in 2 ×2 matrix form with eigenvectors ( φn↑,φn↓)T. The density matrix is defined\nin terms of the spin components φnσ(r) of the one-electron orbitals by\nnσσ′=/summationdisplay\nnφnσ(r)φnσ′(r)∗θ(µ0−ǫn) (2)\nwhereθ(x) is the unit step function and µ0is the chemical potential. The electron density is given by\nρ(r) =/summationdisplay\nσnσσ(r) =/summationdisplay\nnσ|φnσ(r)|2θ(µ0−ǫn) (3)\nand the effective potential in (1) is\nVeff\nσσ′(r) =wσσ′(r)+δσσ′/integraldisplay\nd3r′ρ(r′)v(r−r′)+vxc\nσσ′(r) (4)\nwherewσσ′(r) is the external potential due to the crystal lattice and any magn etic fields and v(r) =e2/|r|is the\nCoulomb potential. The exchange-correlation potential vxc\nσσ′(r) is defined as δExc/δnσσ′(r), a functional derivative of\nthe exchange-correlation energy Exc. The term Hso\nσσ′in (1) is the SOC energy. A small external perturbation δwσσ′,\nfor example due to a magnetic field, changes the effective potential toVeff+δVeff, giving rise to new orbitals and\nhence to a change in density matrix δnσσ′. The equation\nδnσσ′(r) =−Ω−1/summationdisplay\nσ1σ′\n1/integraldisplay\nd3r1χ0\nσσ′σ1σ′\n1(r,r1)δVeff\nσ1σ′\n1(r1), (5)\nwhere Ω is the volume of the sample, defines a non-interacting respo nse function χ0and the full response function χ\nis defined by\nδnσσ′(r) =−Ω−1/summationdisplay\nττ′/integraldisplay\nd3r′χσσ′ττ′(r,r′)δwττ′(r′). (6)3\nFrom (4)\nδVeff\nσ1σ′\n1(r1) =δwσ1σ′\n1(r1)+/summationdisplay\nσ2σ′\n2/integraldisplay\nd3r2[v(r1−r2)δσ1σ′\n1δσ2σ′\n2+δvxc\nσ1σ′\n1(r1)\nδnσ2σ′\n2(r2)]δnσ2σ′\n2(r2) (7)\nand we may write\nδvxc\nσ1σ′\n1(r1)\nδnσ2σ′\n2(r2)=δ2Exc\nδnσ2σ′\n2(r2)δnσ1σ′\n1(r1)=Kσ1σ′\n1σ2σ′\n2(r1,r2). (8)\nCombining (5) - (8) we find the following integral equation for the spin -density response function χσσ′ττ′(r,r′):\nχσσ′ττ′(r,r′) =χ0\nσσ′ττ′(r,r′)−(Ω)−1/summationdisplay\nσ1σ′\n1/summationdisplay\nσ2σ′\n2/integraldisplay\nd3r1/integraldisplay\nd3r2χ0\nσσ′σ1σ′\n1(r,r1)[v(r1−r2)δσ1σ′\n1δσ2σ′\n2+Kσ1σ′\n1σ2σ′\n2(r1,r2)]\nχσ2σ′\n2ττ′(r2,r′).\n(9)\nThis equation is a slight generalisation of that given by Williams and von Ba rth [28]. In the static limit it is formally\nexact although the exchange-correlation energy Excis of course not known exactly. In the next section we generalise\nthe equation to the dynamical case approximately by introducing th e frequency dependence of the non-interacting\nresponse functions χ0, and also take a spatial Fourier transform. In the case where SOC is absent the result is directly\ncompared with results obtained using the RPA.\nIII. DYNAMICAL SUSCEPTIBILITIES IN THE PRESENCE OF SPIN-OR BIT COUPLING AND\nLONG-RANGE COULOMB INTERACTION\nIn general the response functions χ(r,r′) are not functions of r−r′and a Fourier representation of (9) for a\nspatially periodic system involves an infinite number of reciprocal latt ice vectors. There are two cases where this\ncomplication is avoided. The first is a homogeneous electron gas and t he second is in a tight-binding approximation\nwith a restricted atomic basis. We may then introduce Fourier trans forms of the form χ(r) =/summationtext\nqχ(q)eiq·ror\nχ(q) = (Ω)−1/integraltext\nd3rχ(r)e−iq·rand write (9) as\nχσσ′ττ′(q,ω) =χ0\nσσ′ττ′(q,ω)−/summationdisplay\nσ1σ′\n1/summationdisplay\nσ2σ′\n2χ0\nσσ′σ1σ′\n1(q,ω)Vσ1σ′\n1σ2σ′\n2(q)χσ2σ′\n2ττ′(q,ω), (10)\nwhere we have also introduced the ωdependence of χas indicated at the end of the last section. Here V(q) is an\nordinary Fourier transform, without a factor (Ω)−1, so that\nVσ1σ′\n1σ2σ′\n2(q) =v(q)δσ1σ′\n1δσ2σ′\n2+Kσ1σ′\n1σ2σ′\n2, (11)\nwherev(q) = 4πe2/q2is the usual Fourier transform of the Coulomb interaction and the s econd term is independent\nofqsinceKis a short-range spatial interaction. In the gas case it is proportio nal to a delta-function δ(r−r′) in the\nlocal-density approximation (LDA) [28] and in tight-binding it can be t aken as an on-site interaction. In both cases\nKmay be expressed in terms of a parameter Uas\nKσ1σ′\n1σ2σ′\n2=−U[δσ1σ′\n1δσ2σ′\n2δσ1σ2+δσ1σ′\n1δσ2σ′\n2δσ′\n1σ2] (12)\nwhereσ=↓,↑forσ=↑,↓. in the tight-binding case this form of Kcorresponds to a simple form of interaction\nwhich leads to a rigid exchange splitting of the bands ( [29], [22]). This is only appropriate for transition metals in a\nmodel with d bands only, hybridization with s and p bands being neglect ed. We adopt this model in order to obtain\ntransparent analytic results as far as possible. Although not as re alistic as ”first-principles” models of the electronic\nstructure it has been used, even with some quantitative success, in treating the related problem of magnetocrystalline\nanisotropy in Co/Pd structures as well as pure metals [30]. In (10) t he response functions χare per unit volume\nin the gas case but, more conveniently, may be taken as per atom in t he tight-binding case with v(q) modified to\nv(q) = 4πe2/(q2Ωa) where Ω ais the volume per atom.4\nTo show how equations (10) - (12) are related to RPA we examine two examples in the absence of SOC. First\nconsider the transverse susceptibility χ↓↑↑↓(q,ω) which is more usually denoted by χ−+(q,ω). Equation (10) becomes\nχ↓↑↑↓=χ0\n↓↑↑↓−χ0\n↓↑↑↓V↑↓↓↑χ↓↑↑↓ (13)\nand, from (11) and (12), V↑↓↓↑=K↑↓↓↑=−U. Hence\nχ↓↑↑↓=χ0\n↓↑↑↓(1−Uχ0\n↓↑↑↓)−1(14)\nwhich is just the RPA result of Izuyama et al [31] for a single-orbital Hubbard model and of Lowde and Windsor [32]\nfor a five-orbital d-band model. Clearly in the absence of SOC the Co ulomb interaction v(q) plays no part in the\ntransverse susceptibility, as is well-known. A more interesting case is the longitudinal susceptibility denoted by χmm\nin the work of Kim et al ( [26], [27]) and in [28]. This involves only the respon se functions χσσττwhich we abbreviate\ntoχστ. In fact [28]\nχmm=χ↑↑+χ↓↓−χ↑↓−χ↓↑. (15)\nWithout SOC χ0\nστtakes the form χ0\nσδστand (10) becomes\nχστ=χ0\nσδστ−/summationdisplay\nσ2χ0\nσVσσ2χσ2τ (16)\nwithVσσ2=v(q)−Uδσσ2. On solving the 2 ×2 matrix equation (16) for χστ, and using (15), we find the longitudinal\nsusceptibility in the form\nχmm=χ0\n↑+χ0\n↓−2[U−2v(q)]χ0\n↑χ0\n↓\n1+(χ0\n↑+χ0\n↓)[v(q)−U]+U[U−2v(q)]χ0\n↑χ0\n↓(17)\nwhich agrees with the RPA result that Kim et al ( [26], [27])found for a s ingle-orbitalmodel. The Coulomb interaction\nv(q) is clearly important, particularly for the uniform susceptibility with q =0, where v→ ∞. It plays an essential\nrole in enforcing particle conservation and hence in obtaining the cor rect result of Stoner theory. In view of the\ncorrespondence between our approach and the RPA method it see ms likely that when SOC is included our procedure\nusing equations (10) - (12) should be almost equivalent to that of Co sta and Muniz [19] in the case of a model with\nd-bands only. However our inclusion of the long-range Coulomb inter action will modify the results.\nIV. AN EXPLICIT EXPRESSION FOR THE TRANSVERSE SUSCEPTIBILI TY\nIn this section we obtain an explicit expression for the transverses usceptibility χ↓↑↑↓in terms of the non-interacting\nresponse functions χ0. We consider equation (10) as an equation between 4 ×4 matrices where σσ′=↓↑,↑↓,↑↑,↓↓\nlabels the rows in that order and ττ′labels columns similarly. The formal solution of (10) is then\nχ= (1+χ0V)−1χ0. (18)\nThis expression could be used directly as the basis of a numerical inve stigation similar to that of Costa and Muniz.\nHowever we wish to show that the present approach leads to a Gilber t damping parameter αin agreement with the\nKambersky formula, to second order in the SOC parameter ξwhere Kambersky’s result is valid. This requires some\nquite considerable analytic development of (18).\nFirst we partition each matrix in (18) into four 2 ×2 matrices. Thus from (11) and (12)\nV=/parenleftbigg\nV10\n0V2/parenrightbigg\n(19)\nwith\nV1=/parenleftbigg\n0−U\n−U0/parenrightbigg\n, V2=/parenleftbigg\nv−U v\nv v−U/parenrightbigg\n. (20)\nAlso\nχ=/parenleftbigg\nχ11χ12\nχ21χ22/parenrightbigg\n(21)5\nand similarly for χ0. If we write\n1+χ0V=/parenleftbigg\n1+χ0\n11V1χ0\n12V2\nχ0\n21V11+χ0\n22V2/parenrightbigg\n=/parenleftbigg\nA B\nC D/parenrightbigg\n(22)\n(18) becomes\nχ=/parenleftbigg\nS−1−S−1BD−1\n−D−1CS−1D−1+D−1CS−1BD−1/parenrightbigg/parenleftbigg\nχ0\n11χ0\n12\nχ0\n21χ0\n22/parenrightbigg\n(23)\nwhere\nS=A+BD−1C. (24)\nThe transverse susceptibility χ↓↑↑↓in which we are interested is the top right-hand element of χ11so this is the\nquantity we wish to calculate. From (23)\nχ11=S−1(χ0\n11−BD−1χ0\n21) (25)\nand, from (24) and (22),\nS= 1+χ0\n11V1−χ0\n12(V−1\n2+χ0\n22)−1χ0\n21V1. (26)\nThe elements of the 2 ×2 matrix S are calculated by straight-forward algebra and\nS11= 1−Uχ0\n↓↑↑↓+(U/Λ)[(X+χ0\n↓↓↓↓)χ0\n↑↑↑↓χ0\n↓↑↑↑−(Y+χ0\n↑↑↓↓)χ0\n↓↓↑↓χ0\n↓↑↑↑\n−(Y+χ0\n↓↓↑↑)χ0\n↑↑↑↓χ0\n↓↑↓↓+(X+χ0\n↑↑↑↑)χ0\n↓↓↑↑χ0\n↓↑↓↓](27)\nwhere\nX= [v−U]/[U(U−2v)], Y=−v/[U(U−2v)] (28)\nand\nΛ = (X+χ0\n↑↑↑↑)(X+χ0\n↓↓↓↓)−(Y+χ0\n↑↑↓↓)(Y+χ0\n↓↓↑↑) (29)\nThe other three elements of Sare given in Appendix B. The transverse susceptibility is obtained fro m (25) as\nχ↓↑↑↓= [S22(χ0\n11−BD−1χ0\n21)12−S12(χ0\n11−BD−1χ0\n21)22]/(S11S22−S12S21) (30)\nand\nBD−1=χ0\n12(V−1\n2+χ0\n22)−1. (31)\nA comparison of the fairly complex equation above for the transver se susceptibility with the simple well-known result\n(14) showsthe extent ofthe new physicsintroducedby SOC.This is due tothe coupling ofthe transversesusceptibility\nto the longitudinal susceptibility and the charge response, both of which involve the long-range Coulomb interaction.\nTo proceed further it is necessary to specify the non-interacting response functions χ0\nσσ′σ1σ′\n1which occur throughout\nthe equations above.\nV. THE NON-INTERACTING RESPONSE FUNCTIONS\nIn the tight-binding approximation the one-electron basis function s are the Bloch functions\n|kµσ∝angb∇acket∇ight=N−1/2/summationdisplay\njeik·Rj|jµσ∝angb∇acket∇ight (32)\nwherejandµare the site and orbital indices, respectively, and Nis the number of atoms in the crystal. The\nHamiltonian in the Kohn-Sham equation now takes the form\nHeff=/summationdisplay\nkµνσ(Tµν(k)+Veff\nσδµν)c†\nkµσckνσ+Hso(33)6\nwhereTµνcorresponds to electron hopping and\nVeff\nσ=−(σ/2)(∆+bex) (34)\nwhereσ= 1,−1 for spin ↑,↓respectively. Here ∆ = 2 U∝angb∇acketleftSz∝angb∇acket∇ight/NwhereSzis the total spin angular momentum, in\nunits of /planckover2pi1, and the Zeeman splitting bex= 2µBBex, whereBexis the external magnetic field and µBis the Bohr\nmagneton. The spin-orbit term Hso=ξ/summationtext\njLj·Sjtakes the second-quantized form\nHso= (ξ/2)/summationdisplay\nkµν[Lz\nµν(c†\nkµ↑ckν↑−c†\nkµ↓ckν↓)+L+\nµνc†\nkµ↓ckν↑+L−\nµνc†\nkµ↑ckν↓] (35)\nwhereLz\nµν,L±\nµνare matrix elements of the atomic orbital angular momentum operat orsLz,L±=Lx±iLyin units\nof/planckover2pi1. Within the basis of states (32) eigenstates of Hefftake the form\n|kn∝angb∇acket∇ight=/summationdisplay\nµσaσ\nnµ(k)|kµσ∝angb∇acket∇ight, (36)\nand satisfy the equation\nHeff|kn∝angb∇acket∇ight=Ekn|kn∝angb∇acket∇ight. (37)\nThus\nc†\nkµσ=/summationdisplay\nnaσ\nnµ(k)∗c†\nkn(38)\nwherec†\nkncreates the eigenstate |kn∝angb∇acket∇ight.\nThenon-interactingresponsefunction χ0\nσσ′σ1σ′\n1(q,ω) isconvenientlyexpressedastheFouriertransformofaretarde d\nGreen function by the Kubo formula\nχ0\nσσ′σ1σ′\n1(q,ω) =/summationdisplay\nk∝angb∇acketleft∝angb∇acketleft/summationdisplay\nµc†\nk+qµσckµσ′;/summationdisplay\nνc†\nkνσ1ck+qνσ′\n1∝angb∇acket∇ight∝angb∇acket∇ight0\nω (39)\nwhere the right-hand side is to be evaluated using the one-electron Hamiltonian Heff. Consequently, using (38), we\nhave\nχ0\nσσ′σ1σ′\n1(q,ω) =/summationdisplay\nkµν/summationdisplay\nmnaσ\nmµ(k+q)∗aσ′\nnµ(k)aσ1nν(k)∗aσ′\n1mν(k+q)∝angb∇acketleft∝angb∇acketleftc†\nk+qmckn;c†\nknck+qm∝angb∇acket∇ight∝angb∇acket∇ight0\nω\n=N−1/summationdisplay\nkµν/summationdisplay\nmnaσ\nmµ(k+q)∗aσ′\nnµ(k)aσ1\nnν(k)∗aσ′\n1mν(k+q)fkn−fk+qm\nEk+qm−Ekn−/planckover2pi1ω+iη.(40)\nThe last step uses the well-known form of the response function pe r atom for a non-interacting Fermi system (e.g. [33])\nandηis a small positive constant which ultimately tends to zero. The occup ation number fkn=F(Ekn−µ0) where\nFis the Fermi function with chemical potential µ0. Clearly for q= 0 the concept of intraband transitions ( m=n),\nfrequently introduced in discussions of the Kambersky formula, ne ver arises for finite ωsince the difference of the\nFermi functions in the numerator of (40) is zero. Equation (40) ma y be written in the form\nχ0\nσσ′σ1σ′\n1(q,ω) =N−1/summationdisplay\nkmnBσσ′\nmn(k,q)Bσ1′σ1\nmn(k,q)∗fkn−fk+qm\nEk+qm−Ekn−/planckover2pi1ω+iη(41)\nwhere\nBσσ′\nmn(k,q) =/summationdisplay\nµaσ\nmµ(k+q)∗aσ′\nnµ(k). (42)7\nVI. FERROMAGNETIC RESONANCE LINEWIDTH; THE KAMBERSKY FORM ULA\nWe now consider the damping of the ferromagnetic resonance in the q= 0 transverse susceptibility. The present\napproach, like the closely-related one of Costa and Muniz [19], is valid f or arbitrary strength of the SOC and can\nbe used as the basis of numerical calculations, as performed by the latter authors. However it is important to show\nanalytically that the present method leads to the Kambersky [4] for mula for the Gilbert damping parameter where\nthis is valid, namely to second order in the SOC parameter ξ. This is the subject of this section.\nIt is useful to consider first the case without SOC ( ξ= 0). The eigenstates nofHeffthen have a definite spin and\nmay be labelled nσ. It follows from (40) that χ0\nσσ′σ1σ′\n1∝δσσ′\n1δσ′σ1. Henceχ0\n12= 0 and, from (31), BD−1= 0. Also,\nfrom Appendix B, S12=S21= 0. Thus, (30) reduces to (14) as it should. Considering χ0\n↓↑↑↓(0,ω), given by (40)\nand (41), we note that state mis pure↓spin, labelled by m↓, andnis pure↑, labelled by n↑. Hence for ξ= 0\nB↓↑\nmn(k,0) =/summationdisplay\nµ∝angb∇acketleftkm|kµ∝angb∇acket∇ight∝angb∇acketleftkµ|kn∝angb∇acket∇ight=δmn (43)\nfrom closure. Thus\nχ0\n↓↑↑↓(0,ω) =N−1/summationdisplay\nknfkn↑−fkn↓\nEkn↓−Ekn↑−/planckover2pi1ω+iη(44)\nand it follows from (34) that Eknσmay be written as\nEknσ=Ekn−(σ/2)(∆+bex). (45)\nHence we find from (14) that for ξ= 0\nχ↓↑↑↓(0,ω) = (2∝angb∇acketleftSz∝angb∇acket∇ight/N)(bex−/planckover2pi1ω+iη)−1. (46)\nThus, as η→0,ℑχ↓↑↑↓(0,ω) has a sharp delta-function resonance at /planckover2pi1ω=bexas expected.\nWhen SOC is included /planckover2pi1ωacquires an imaginary part that corresponds to damping. We now pr oceed to calculate\nthis imaginary part to O(ξ2). To do this we can take ξ= 0 in the numerator of (30) so that\nχ↓↑↑↓(0,ω) =χ0\n↓↑↑↓(0,ω)/(S11−S12S21/S22) (47)\nIn factS12andS21are both O(ξ2) whileS22isO(1). Thus to obtain /planckover2pi1ωtoO(ξ2) we need only solve S11= 0.\nFurthermore all response functions such as χ0\n↑↑↑↓, with all but one spins in the same direction, are zero for ξ= 0 and\nneed only be calculated to O(ξ) in (27). We show below that to this order they vanish, so that to O(ξ2) the last term\ninS11is zero and we only have to solve the equation 1 −Uχ0\n↓↑↑↓= 0 for/planckover2pi1ω. This means that to second order in ξ\nthe shift in resonance frequency and the damping do not depend on the long-range Coulomb interaction.\nTo determine χ0\n↑↑↑↓(0,ω) to first order in ξfrom (40) we notice that states nmust be pure ↑spin, that is |kn∝angb∇acket∇ight=\n|kn↑∝angb∇acket∇ight, while states mmust be calculated using perturbation theory. The latter states m ay be written\n|km1∝angb∇acket∇ight=|km↑∝angb∇acket∇ight−ξ/summationdisplay\npσ∝angb∇acketleftkpσ|hso|km↑∝angb∇acket∇ight\nEkm↑−Ekpσ|kpσ∝angb∇acket∇ight (48)\n|km2∝angb∇acket∇ight=|km↓∝angb∇acket∇ight−ξ/summationdisplay\npσ∝angb∇acketleftkpσ|hso|km↓∝angb∇acket∇ight\nEkm↓−Ekpσ|kpσ∝angb∇acket∇ight, (49)\nwhere we have put Hso=ξhso, and to first order in ξ,\nχ0\n↑↑↑↓=1\nN/summationdisplay\nkµν/summationdisplay\nmn(a↑\nm1µ∗anµa∗\nnνa↓\nm1νfkn↑−fkm↑\nEkm↑−Ekn↑−/planckover2pi1ω+iη+a↑\nm2µ∗anµa∗\nnνa↓\nm2νfkn↑−fkm↓\nEkm↓−Ekn↑−/planckover2pi1ω+iη) (50)\nwithaσ\nmsµ=∝angb∇acketleftkµσ|kms∝angb∇acket∇ight,s= 1,2,andanµ=∝angb∇acketleftkµ|kn∝angb∇acket∇ightis independent of spin. Since a↓\nm1ν∼ξwe takea↑\nm1µ=amµin\nthe first term of (50). Also/summationtext\nµa∗\nmµanµ=δmnby closure so that the first term of (50) vanishes since the differen ce\nof Fermi functions is zero. Only the second term of χ0\n↑↑↑↓remains and this becomes, by use of (49),\nχ0\n↑↑↑↓=−ξ/summationdisplay\nkµν/summationdisplay\nmnp∝angb∇acketleftkp↑ |hso|km↓∝angb∇acket∇ight∗\nEkm↓−Ekp↑a∗\npµanµa∗\nnνamνfkn↑−fkm↓\nEkm↓−Ekn↑−/planckover2pi1ω+iη. (51)8\nAgain using closure only terms with p=m=nsurvive and the matrix element of hsobecomes\n∝angb∇acketleftkn↑ |/summationdisplay\njLj·Sj|kn↓∝angb∇acket∇ight=1\n2∝angb∇acketleftkn|L−|kn∝angb∇acket∇ight= 0 (52)\ndue to the quenching of total orbital angular momentum L=/summationtext\njLj[30]. Thus, to first order in ξ,χ0\n↑↑↑↓(0,ω), and\nsimilar response functions with one reversed spin, are zero. Hence we have only to solve 1 −Uχ0\n↓↑↑↓= 0 to obtain\nℑ(/planckover2pi1ω) toO(ξ2). Here we assume the system has spatial inversion symmetry witho ut which the quenching of orbital\nangular momentum, as expressed by (52), no longer pertains [30]. We briefly discuss the consequences of a breakdown\nof inversion symmetry at the end of this section.\nOn introducing the perturbed states (48) and (49) we write (41) in the form\nχ0\n↓↑↑↓(0,ω) =1\nN/summationdisplay\nkmn(|B↓↑\nm1n1|2fkn1−fkm1\nEkm1−Ekn1−/planckover2pi1ω+iη+|B↓↑\nm1n2|2fkn2−fkm1\nEkm1−Ekn2−/planckover2pi1ω+iη\n+|B↓↑\nm2n1|2fkn1−fkm2\nEkm2−Ekn1−/planckover2pi1ω+iη+|B↓↑\nm2n2|2fkn2−fkm2\nEkm2−Ekn2−/planckover2pi1ω+iη).(53)\nClearlyB↓↑\nm1n1andB↓↑\nm2n2are of order ξ,B↓↑\nm1n2isO(ξ2) andB↓↑\nm2n1isO(1). We therefore neglect the term |B↓↑\nm1n2|2\nand, using (48) and (49), we find\nB↓↑\nm1n1=−B↓↑\nm2n2=ξ\n2∝angb∇acketleftkm|L−|kn∝angb∇acket∇ight\nEkn↓−Ekm↑. (54)\nThe evaluation of |B↓↑\nm2n1|2requires more care. It appears at first sight that to obtain this to O(ξ2) we need to include\nsecond order terms in the perturbed eigenstates given by (48) an d (49). However it turns out that these terms do not\nin fact contribute to |B↓↑\nm2n1|2toO(ξ2) so we shall not consider them further. Then we find\nB↓↑\nm2n1=δmn−ξ∝angb∇acketleftkm|Lz|kn∝angb∇acket∇ight\nEkn−Ekm−ξ2\n4/summationdisplay\np∝angb∇acketleftkm|Lz|kp∝angb∇acket∇ight∝angb∇acketleftkp|Lz|kn∝angb∇acket∇ight\n(Ekm−Ekp)(Ekn−Ekp)(55)\nand hence to O(ξ2)\n|B↓↑\nm2n1|2=δmn(1−ξ2\n2/summationdisplay\np|∝angb∇acketleftkm|Lz|kp∝angb∇acket∇ight|2\n(Ekm−Ekp)2)+ξ2|∝angb∇acketleftkm|Lz|kn∝angb∇acket∇ight|2\n(Ekn−Ekm)2(56)\nThe contribution of this quantity to χ0\n↓↑↑↓(0,ω) in (53) may be written to O(ξ2) as\n1\nN/summationdisplay\nkmn|B↓↑\nm2n1|2fkn1−fkm2\nEkm2−Ekn1−bex+iη(1−bex−/planckover2pi1ω\nEkm2−Ekn1−bex+iη). (57)\nThisisobtainedbyintroducingtheidentity −/planckover2pi1ω=−bex+(bex−/planckover2pi1ω)intherelevantdenominatorin(53),andexpanding\nto first order in bex−/planckover2pi1ωwhich turns out to be O(ξ2). The remaining factors of this second term in (57) may then\nbe evaluated with ξ= 0, as at the beginning of this section, so that this term becomes ( /planckover2pi1ω−bex)/(2U2∝angb∇acketleftSz∝angb∇acket∇ight). By\ncombining equations (53), (54) and (57), and ignoring some real te rms, we find that the equation 1 −Uχ0\n↓↑↑↓(0,ω) = 0\nleads to the relation\nℑ(/planckover2pi1ω) =πξ2/(2∝angb∇acketleftSz∝angb∇acket∇ight)/summationdisplay\nknm[(fkn↑−fkm↓)|∝angb∇acketleftkm|Lz|kn∝angb∇acket∇ight|2δ(Ekm↓−Ekn↑−bex)\n+(1/4)(fkn↑+fkn↓−fkm↑−fkm↓)|∝angb∇acketleftkm|L−|kn∝angb∇acket∇ight|2δ(Ekm−Ekn−bex)](58)\nThe Gilbert damping parameter αis given by ℑ(/planckover2pi1ω)/bex(e.g. [39]) and in (58) we note that\n(fkn↑−fkm↓)δ(Ekm↓−Ekn↑−bex) = [F(Ekn↑−µ0)−F(Ekn↑+bex−µ0)]δ(Ekm↓−Ekn↑−bex)\n=bexδ(Ekn↑−µ0)δ(Ekm↑−µ0)(59)\nto first order in bexat temperature T= 0. Similarly\n(fknσ−fkmσ)δ(Ekm−Ekn−bex) =bexδ(Eknσ−µ0)δ(Ekmσ−µ0). (60)9\nThus from (58)\nα=πξ2/(2∝angb∇acketleftSz∝angb∇acket∇ight)/summationdisplay\nknm|∝angb∇acketleftkm|Lz|kn∝angb∇acket∇ight|2δ(Ekn↑−µ0)δ(Ekm↓−µ0)\n+πξ2/(8∝angb∇acketleftSz∝angb∇acket∇ight)/summationdisplay\nknmσ|∝angb∇acketleftkm|L−|kn∝angb∇acket∇ight|2δ(Eknσ−µ0)δ(Ekmσ−µ0)(61)\ncorrect to O(ξ2). We note that there is no contribution from intraband terms since ∝angb∇acketleftkn|L|kn∝angb∇acket∇ight= 0. It is straight-\nforward to show that to O(ξ2) this is equivalent to the expression\nα=π/(2∝angb∇acketleftSz∝angb∇acket∇ight)/summationdisplay\nknm/summationdisplay\nσσ′|Amσ,nσ′(k)|2δ(Ekmσ−µ0)δ(Eknσ′−µ0) (62)\nwhere\nAmσ,nσ′(k) =ξ∝angb∇acketleftkmσ|[S−,hso]|knσ′∝angb∇acket∇ight (63)\nandS−is the total spin operator/summationtext\njS−\njwithS−\nj=Sx\nj−iSy\nj. This may be written more concisely as\nα=π/(2∝angb∇acketleftSz∝angb∇acket∇ight)/summationdisplay\nknm|Amn(k)|2δ(Ekm−µ0)δ(Ekn−µ0) (64)\nwith\nAmn(k) =ξ∝angb∇acketleftkm|[S−,hso]|kn∝angb∇acket∇ight (65)\nand the understanding that the one-electron states km,knare calculated in the absence of SOC. Equation (64) is the\nstandard form of the Kambersky formula ( [4], [9]) but in the literatur e SOC is invariably included in the calculation\nof the one-electronstates. This means that the intraband terms withm=nno longer vanish. They involve the square\nof a delta-function and this problem is always addressed by invoking t he effect of impurity and/or phonon scattering\nto replace the delta-functions by Lorentzians of width proportion al to an inverse relaxation time parameter τ−1. Then\nas one approaches a perfect crystal ( τ→ ∞) the intraband contribution to αtends to infinity. This behaviour is\nillustrated in many papers ( [7], [10], [13], [14]). In fig. 1 of [14] it is shown c learly that αremains finite if one does\nnot include SOC in calculating the one-electron states. However the effect of not including SOC is not confined to\ntotal removal of the intraband contribution. The remaining interb and contribution is increased considerably in the\nlow scattering rate regime, by almost an order of magnitude in the ca se of Fe. This makes αalmost independent of\nscattering rate in Fe which may relate to its observed temperature independence [15]. The corresponding effect in\nCo is insufficient to produce the increase of αat low scattering rate inferred from its temperature dependence . The\nnon-inclusion of SOC in calculating the one-electron states used in th e Kambersky formula clearly makes a major\nqualitative and quantitative change in the results. This occurs as so on as intraband terms become dominant in\ncalculations where they are included. For Fe, Co and Ni this corresp onds to impurity content and temperature such\nthat the scattering rate 1 /τdue to defects and/or phonons is less that about 1014sec−1( [7, 14]). Typically these\nmetals at room temperature find themselves well into the high scatt ering-rate regime where the damping rate can\nbe reliably estimated from the Kambersky interband term, with or wit hout SOC included in the band structure [35].\nThe physics at room temperature is not particularly interesting. On e needs to lower the temperature into the low\nscattering-rate regime where intraband terms, if they exist, will d ominate and lead to an anomalous ξ3dependence of\nthe damping on spin-orbit parameter ξ( [4], [8], [14]). The origin of this behaviour is explained in [4], [14]. It arises\nin theksum of (64) from a striplike region on the Fermi surface around a line where two different energy bands cross\neach other in the absence of SOC. The strip width is proportional to ξ, or more precisely |ξ|. SinceAnn(k) is of\norderξthe contribution of intraband terms in (64) is proportional to |ξ|3. Thus the intraband terms lead to terms\ninαwhich diverge in the limit τ−1→0 and are non-analytic functions of ξ. The calculation of αin this section can\nbe extended to higher powers of ξthan the second. No intraband terms appear and the result is an an alytic power\nseries containing only even powers of ξ.\nThe interband term in Kambersky’s formula can be given a very simple in terpretation in terms of Fermi’s ”golden\nrule” for transition probability [5]. This corresponds to second orde r perturbation theory in the spin-orbit interaction.\nThe decay of a uniform mode ( q= 0) magnon into an electron-hole pair involves the transition of an ele ctron from\nan occupied state to an unoccupied state of the same wave-vecto r. This is necessarily an interband transition and\nthe states involved in the matrix element are unperturbed, that is c alculated in the absence of SOC. A quite different\napproach has been adopted to try and find a physical interpretat ion of Kambersky’s intraband term ( [5, 8]). This\nemploys Kambersky’s earlier ”breathing Fermi surface” model ( [3, 34]) whose range of validity is uncertain.10\nWe now briefly discuss the consequences of a breakdown of spatial inversion symmetry so that total orbital angular\nmomentum is not quenched. In general response functions such a sχ0\n↑↑↑↓(0,ω) with one reversed spin are no longer\nzero to first order in ξ. HenceS11is not given to order ξ2just by the first two terms of (27) but involves further terms\nwhich depend explicitly on the long-range Coulomb interaction. Conse quentlyαhas a similar dependence which\ndoes not emerge from the torque-correlation approach. In Appe ndix A it is pointed out how the direct proof of the\nKambersky formula breaks down in the absence of spatial inversion symmetry.\nVII. EXPERIMENTAL ASPECTS\nThe inclusion of intraband terms in the Kambersky formula, despite t heir singular nature, has gained acceptance\nbecause they appear to explain a rise in intrinsic damping parameter αat low temperature which is observed in some\nsystems [15]. The calculated intraband contribution to αis proportional to the relaxation time τand it is expected\nthat, due to electron-phonon scattering, τwill increase as the temperature is reduced. This is in qualitative agre ement\nwith data [15] for Ni and hcp Co. Also a small 10% increase in αis observed in Co 2FeAl films as the temperature is\ndecreased from 300 K to 80 K [36]. However in Fe the damping αis found to be independent of temperature down to\n4 K [15]. Very recent measurements [16] on FePt films, with varying an tisite disorder xintroduced into the otherwise\nwell-ordered structure, show that αincreases steadily as xincreases from 3 to 16%. Hence αincreases monotonically\nwith scattering rate 1 /τas expected from the Kambersky formula in the absence of intraba nd terms. Furthermore\nforx= 3% it is found that αremains almost unchanged when the temperature is decreased fro m 200 to 20 K. Ma et\nal [16] therefore conclude that there is no indication of an intraban d term in α. From the present point of view the\norigin of the observed low temperature increase of αin Co and Ni is unclear. Further experimental work to confirm\nthe results of Bhagat and Lubitz [15] is desirable.\nThe second unusual feature of the intraband term in Kambersky’s formula for αis its|ξ|3dependence on the SOC\nparameter ξ. This contrasts with the ξ2dependence of the interband contribution which has been observe d in a\nnumber of alloys at room temperature [17]. Recently this behaviour has been seen very precisely in FePd 1−xPtxalloys\nwhereξcan be varied over a wide range by varying x[18]. Unfortunately this work has not been extended to the\nlow temperature regime where the |ξ|3dependence, if it exists, should be seen. It would be particularly inte resting to\nsee low temperature data for NiPd 1−xPtxand CoPd 1−xPtxsince it is in Ni and Co where the intraband contribution\nhas been invoked to explain the low temperature behaviour of α. From the present point of view, with the intraband\nterm absent, one would expect ξ2behaviour over the whole temperature range.\nVIII. CONCLUSIONS AND OUTLOOK\nIn this paper we analyse two methods which are used in the literature to calculate the damping in magnetization\ndynamics due to spin-orbit coupling. The first common approach is to employ Kambersky’s[4] formula for the Gilbert\ndamping parameter αwhich delivers an infinite value for a pure metal if used beyond second order in the spin-orbit\nparameter ξ. The second approach [19] is to calculate numerically the line-width of the ferromagnetic resonance seen\nin the uniform transverse spin susceptibility. This is always found to b e finite, corresponding to finite α. We resolve\nthis apparent inconsistency between the two methods by an analyt ic treatment of the Costa-Muniz approach for the\nsimplified model of a ferromagnetic metal with d-bands only. It is sho wn that this method leads to the Kambersky\nresult correct to second order in ξbut Kambersky’s intraband scattering term, taking the non-analy tic form |ξ|3, is\nabsent. Higher order terms in the present work are analytic even p owers of ξ. The absence of Kambersky’s intraband\nterm is the main result of this paper and it is in agreement with the conc lusion that Ma et al [16] draw from their\nexperiments on FePt films. Further experimental work on the depe ndence of damping on electron scattering-rate and\nspin-orbit parameter in other systems is highly desirable.\nA secondaryconclusionis that beyond second orderin ξsome additional physics ariseswhich has not been remarked\non previously. This is the role of long-range Coulomb interaction which is essential for a proper treatment of the\nlongitudinal susceptibility and charge response to which the transv erse susceptibility is coupled by spin-orbit interac-\ntion. Costa and Muniz [19] stress this coupling but fail to introduce t he long-range Coulomb interaction. Generally,\nhowever, it seems unnecessary to go beyond second order in ξ[17, 18] and for most bulk systems Kambersky’s for-\nmula, with electron states calculated in the absence of SOC, should b e adequate. However in systems without spatial\ninversion symmetry, which include layered structures of practical importance, the Kambersky formulation may be\ninadequate even to second order in ξ. The long-range Coulomb interaction can now play a role.\nAn important property of ferromagnetic systems without inversio n symmetry is the Dzyaloshinskii-Moriya inter-\naction (DMI) which leads to an instability of the uniform ferromagnet ic state with the appearance of a spiral spin\nstructure or a skyrmion structure. This has been studied extens ively in bulk crystals like MnSi [37] and in layered11\nstructures [38]. The spiral instability appears as a singularity in the t ransverse susceptibility χ(q,0) at a value of q\nrelated to the DMI parameter. The method of this paper has been u sed to obtain a novel closed form expression for\nthis parameter which will be reported elsewhere.\nIn this paper we have analysed in some detail the transverse spin su sceptibility χ↓↑↑↓but combinations of some of\nthe 15 other response functions merit further study. Mixed char ge-spin response arising from spin-orbit coupling is\nof particular interest for its relation to phenomena like the spin-Hall effect.\nAppendix A: A direct derivation of the Kambersky formula\nIn this appendix we give a rather general derivation of the Kambers ky formula for the Gilbert damping parameter\nαwith an emphasis on its restriction to second order in the spin-orbit in teraction parameter ξ.\nWe consider a general ferromagnetic material described by the ma ny-body Hamiltonian\nH=H1+Hint+Hext (A1)\nwhereH1is a one-electron Hamiltonian of the form\nH1=Hk+Hso+V. (A2)\nHereHkis the total kinetic energy, Hso=ξhsois the spin-orbit interaction, Vis a potential term, Hintis the\nCoulomb interaction between electrons and Hextis due to an external magnetic field Bexin thezdirection. Thus\nHext=−Szbexwherebex= 2µBBex, as in (34), and Szis thezcomponent of total spin. Both HsoandVcan contain\ndisorderalthough in this paper we consider a perfect crystal. Follow ingthe general method of Edwardsand Fisher [40]\nwe use equations of motion to find that the dynamical transverse s usceptibility χ(ω) =χ−+(0,ω) satisfies [39]\nχ(ω) =−2∝angb∇acketleftSz∝angb∇acket∇ight\n/planckover2pi1ω−bex+ξ2\n(/planckover2pi1ω−bex)2(χF(ω)−ξ−1∝angb∇acketleft[F−,S+]∝angb∇acket∇ight) (A3)\nwhere\nχF(ω) =/integraldisplay\n∝angb∇acketleft∝angb∇acketleftF−(t),F+∝angb∇acket∇ight∝angb∇acket∇ighte−iωtdt (A4)\nwithF−= [S−,hso]. This follows since S−commutes with other terms in H1and with Hint. For small ω,χis\ndominated by the spin wave pole at /planckover2pi1ω=bext+/planckover2pi1δωwhereδω∼ξ2, so that\nχ(ω) =−2∝angb∇acketleftSz∝angb∇acket∇ight\n/planckover2pi1(ω−δω)−bex. (A5)\nFollowing [39] we compare (A3) and (A5) in the limit /planckover2pi1δω≪/planckover2pi1ω−bexto obtain\n−2∝angb∇acketleftSz∝angb∇acket∇ight/planckover2pi1δω=ξ2(χF(ω)−ξ−1∝angb∇acketleft[F−,S+]∝angb∇acket∇ight) =ξ2[ lim\n/planckover2pi1ω→bexχξ=0\nF(ω)−lim\nξ→0(1\nξ∝angb∇acketleft[F−,S+]∝angb∇acket∇ight)] (A6)\ncorrect to order ξ2. It is important to note that the limit ξ→0 within the bracket must be taken before putting\n/planckover2pi1ω=bex. If we put /planckover2pi1ω=bexfirst it is clear from (A3) that the quantity in brackets would vanish, giving the incorrect\nresultδω= 0. Furthermore it may be shown [M. Cinal, private communication] th at the second term in the bracket\nis real. Hence\nℑ(/planckover2pi1ω) =−ξ2\n2∝angb∇acketleftSz∝angb∇acket∇ightlim\n/planckover2pi1ω→bexℑ[χξ=0\nF(ω)]. (A7)\nKambersky [4] derived this result, using the approach of Mori and K awasaki ( [41] [42]), without noting its restricted\nvalidity to second order in ξ. This restriction is crucial since, as discussed in the main paper, it av oids the appearance\nof singular intraband terms. Oshikawa and Affleck emphasise strong ly a similar restriction in their related work on\nelectron spin resonance (Appendix of [43]).\nEquation (A7) is an exact result even in the presence of disorder in t he potential and spin-orbit terms of the\nHamiltonian. In the following we assume translational symmetry.12\nTo obtain the expression (61) for α=ℑ(/planckover2pi1ω)/bex, which is equivalent to Kambersky’s result (62), it is necessary to\nevaluate the response function χξ=0\nF(ω) in tight-binding-RPA. Using (35)we find\nF−=/summationdisplay\nkµν[Lz\nµνc†\nkµ↓ckν↑+(1/2)L−\nµν(c†\nkµ↓ckν↓−c†\nkµ↑ckν↑)]. (A8)\nHence\nχξ=0\nF=/summationdisplay\nµν/summationdisplay\nαβ[Lz\nµνLz\nβαGµ↓ν↑,β↑α↓+(1/4)L−\nµνL+\nβα(Gµ↓ν↓,β↓α↓+Gµ↑ν↑,β↑α↑−Gµ↓ν↓,β↑α↑−Gµ↑ν↑,β↓α↓)] (A9)\nwhere\nGµσνσ′,βτατ′=∝angb∇acketleft∝angb∇acketleft/summationdisplay\nkc†\nkµσckνσ′;/summationdisplay\nuc†\nuβτcuατ′∝angb∇acket∇ight∝angb∇acket∇ightω. (A10)\nThe Green function Gis to be calculated in the absence of SOC ( ξ= 0). Within RPA it satisfies an equation of the\nform\nGµσνσ′,βτατ′=G0\nµσνσ′,βτατ′−/summationdisplay\nµ1σ1ν1σ′\n1/summationdisplay\nµ2σ2ν2σ′\n2G0\nµσνσ′,µ1σ1ν1σ′\n1Vµ1σ1ν1σ′\n1,µ2σ2ν2σ′\n2Gµ2σ2ν2σ′\n2,βτατ′ (A11)\nwhereG0is the non-interacting (Hartree-Fock) Green function and\nVµ1σ1ν1σ′\n1,µ2σ2ν2σ′\n2=Vσ1σ′\n1σ2σ′\n2(q)δµ1ν1δµ2ν2 (A12)\nwithV(q) given by (11) and (12). Hence\nGµσνσ′,βτατ′=G0\nµσνσ′,βτατ′−/summationdisplay\nµ1σ1σ′\n1/summationdisplay\nµ2σ2σ′\n2G0\nµσνσ′,µ1σ1µ1σ′\n1Vσ1σ′\n1σ2σ′\n2Gµ2σ2µ2σ′\n2,βτατ′. (A13)\nThe form of the interaction Vgiven in (A12) is justified by the connection between (A13) and (10) , withq= 0. To\nsee this connection we note that χσσ′ττ′=/summationtext\nµνGµσµσ′,ντντ′and that (A13) then leads to (10) which is equivalent to\nRPA. On substituting (A13) into (A9) we see that the contributions from the second term of (A13) contain factors\nof the form\n/summationdisplay\nµνµ1Lz\nµνG0\nµ↓ν↑,µ1↑µ1↓,/summationdisplay\nµνµ1L−\nµνG0\nµσνσ,µ 1σ1µ1σ1. (A14)\nWe now show that such factors vanish owing to quenching of orbital angular momentum in the system without SOC\n(ξ= 0). Hence the Green functions Gin (A9) can be replaced by the non-interacting ones G0. The non-interacting\nGreen functions G0are of a similar form to χ0in (40) and for ξ= 0 may be expressed in terms of the quantities\nanµ=∝angb∇acketleftkµ|kn∝angb∇acket∇ightwhere|kn∝angb∇acket∇ightis a one-electron eigenstate as introduced in section VI. Hence we fi nd, in the same way\nthat (44) emerged,\n/summationdisplay\nµνµ1Lz\nµνG0\nµ↓ν↑,µ1↑µ1↓=/summationdisplay\nµν/summationdisplay\nknLz\nµνa∗\nnµanνfkn↑−fkn↓\n∆+bex−/planckover2pi1ω+iη. (A15)\nAlso by closure\n/summationdisplay\nµνLz\nµνa∗\nnµanν=∝angb∇acketleftkn|Lz|kn∝angb∇acket∇ight= 0, (A16)\nthe last step following from quenching of total orbital angular mome ntum. The proof that the second expression\nin (A14) vanishes is very similar.\nHence we can insert the non-interacting Green functions G0in (A9) and straight-forwardalgebra, with use of (A7),\nleads to (58). At the end of section VI this is shown to be equivalent t o the Kambersky formula for α. We emphasize\nagain that the present proof is valid only to order ξ2so that the one-electron states used to evaluate the formula\nshould be calculated in the absence of SOC.\nThis proof relies on the quenching of orbital angular momentum which does not occur in the absence of spatial\ninversion symmetry. When this symmetry is broken it is not difficult to s ee that the second term of (A13) gives a\ncontribution to the first term on the right of (A9) which contains th eq= 0 spin-wave pole and diverges as /planckover2pi1ω→bex.\nHence the proof of the torque-correlationformula (A7) collapses . The method of section VI must be used as discussed\nat the end of that section.13\nAppendix B: Elements of S\nThe element S11of the matrix Sis given in (27). The remaining elements are given below.\nS12=−Uχ0\n↓↑↓↑+(U/Λ)[(X+χ0\n↓↓↓↑)χ0\n↑↑↓↑χ0\n↓↑↑↑−(Y+χ0\n↑↑↓↓)χ0\n↓↓↓↑χ0\n↓↑↑↑\n−(Y+χ0\n↓↓↑↑)χ0\n↑↑↓↑χ0\n↓↑↓↓+(X+χ0\n↑↑↑↑)χ0\n↓↓↓↑χ0\n↓↑↓↓](B1)\nS21=−Uχ0\n↑↓↑↓+(U/Λ)[(X+χ0\n↓↓↓↓)χ0\n↑↑↑↓χ0\n↑↓↑↑−(Y+χ0\n↑↑↓↓)χ0\n↓↓↑↓χ0\n↑↓↑↑\n−(Y+χ0\n↓↓↑↑)χ0\n↑↑↑↓χ0\n↑↓↓↓+(X+χ0\n↑↑↑↑)χ0\n↓↓↑↓χ0\n↑↓↓↓](B2)\nS22= 1−Uχ0\n↑↓↓↑+(U/Λ)[(X+χ0\n↓↓↓↓)χ0\n↑↑↓↑χ0\n↑↓↑↑−(Y+χ0\n↑↑↓↓)χ0\n↓↓↓↑χ0\n↑↓↑↑\n−(Y+χ0\n↓↓↑↑)χ0\n↑↑↓↑χ0\n↑↓↓↓+(X+χ0\n↑↑↑↑)χ0\n↓↓↓↑χ0\n↑↓↓↓](B3)\nAcknowledgement\nMy recent interest in Gilbert damping arose through collaboration wit h O. Wessely, E. Barati, M. Cinal and A.\nUmerski. I am grateful to them for stimulating discussion and corre spondence. The specific work reported here arose\ndirectly from discussion with R.B.Muniz and I am particularly grateful t o him and his colleague A. T. Costa for this\nstimulation.\nReferences\n[1] Landau L D, Lifshitz E M and Pitaevski L P 1980 Statistical Physics part2 (Oxford: Pergamon)\n[2] Gilbert T L 1955 Phys. 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Chris Hammel1 \n1.Department of Physics, The Ohio State University, Columbus, OH, 43210, USA  \n2.Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright -Patterson Air \nForce Base, Dayton, OH, 45433, USA  \n3. Research Center for Functional Materials,  National Institute for Materials Science, 1 -1 Namiki, \nTsukuba 305 -0044, Japan  \n4. International Center for Materials Nanoarchitectonics,  Nationa l Institute for Materials Science,  1-1 \nNamiki, Tsukuba 305 -0044, Japan  \n \n \n \n \n \n \n \n \n \n \n \n*wu.2314@osu.edu     \n \n2 \n Abstract:  \nEngineering magnetic anisotropy in a ferro - or ferrimagnetic (FM) thin film is crucial in \nspintronic device. One way to modify the magnetic anisotropy is through the surface of the FM \nthin film. Here, we report the emergence of a perpendicular magnetic anisotropy (PMA) induced \nby interfacial interactions in a heterostructure comprised of a garnet ferrimagnet, Y 3Fe5O12 \n(YIG), and the low -symmetry, high spin orbit coupling (SOC) transition metal dichalcogenide, \nWTe 2.  At the same time, we also observed an enhancement in Gilbert damping in the WTe 2 \ncovered YIG area. Both the magnitude of interface -induced PMA and the Gilb ert damping \nenhancement have no observable WTe 2 thickness dependence down to single quadruple -layer, \nindicating that the interfacial interaction plays a critical role. The ability of WTe 2 to enhance the \nPMA in FM thin film, combined with its previously rep orted capability to generate out -of-plane \ndamping like spin torque, makes it desirable for magnetic memory applications.   \n \nKey words:  perpendicular magnetic anisotropy, magnetic resonance force microscope, transition metal \ndichalcogenides, ferrimagnetic i nsulator      \n \n3 \n Perpendicular magnetic anisotropy  (PMA)  in a ferromagnetic thin film is of great interest in \nspintronics research and application s. Ferromagnetic nano -element s with PMA overcome  their shape \nanisotropy , greatly ease the memory cell size reduction and improves  memory retention . These \nexceptional properties, improving the performance  of magnetic devices , make PMA highly desirable for \nmagnetic memory  application s.  PMA becomes even more important in the recent development of solid \nstate magnetic random -access memory (MRAM) since it allows MRAM to have lower switching current \nand faster switching speed compare d to in-plane magnetized materials 1, 2.  \nMagnetic storage devices generally rely on metallic magnetic material s due to their robust \nelectrical response . Interfacial magnetic anisotropy plays a critical role in generat ing PMA in metallic \nferromagnet s. When interfacing with a nonmagnetic material (NM), electron orbital angular momentum \nof the magnetic ions at the ferromagnet surface will be modified, in some cases  enabling strong covalent \nbonding, resulting in distinct magnetic properties compare d to the single layer 3-6. However, spintronics \ndevices made of metallic magnetic materials are inherently energy consumptive due to resistive losses. \nRecently, complex oxide ferro - or ferrimagnet  insulator s (FMI)  have  attracted substantial  interest  due to \ntheir ability to transport spin excitation s with low dissipation  7. Inducing PMA in FMIs naturally \nbecomes an important topic both for scientific and technologic al reasons . Several successful route s to \nachiev ing PMA in FMIs ha ve been reported using bulk intrinsic anisotropy 8 or lattice strain 9-12. But in \nmost experiments,  the sign of the resulting interfacial anisotropy in FMI/NM heterostructures is such as \nto enhance  the easy-plane anisotropy  13-15. Only one recent experiment has shown the possibility of \ngenerating interfacial PMA, and this was attributed to topological surface states  16. Nevertheless, t hese \nresults demonstrate  the possibility of controlling magnetic anisotropy through interfac ial interaction s in    \n \n4 \n FMI/NM heterostructures . Here, we report a study on YIG/WTe 2/hBN heterostructures, which shows  \nthat when interfacing with a low symmetry nonmagnetic  van der Waals  material , WTe 2, an additional  \ninterfac e-induced  PMA  (iPMA) term  emerges  in the magnetic anisotropy  of the YIG thin film . The \nabsence of topological surface states at room temperature  in WTe 2 17, 18 forces us to seek an explanation \nfor our observation of enhanced PMA that is distinct from that proposed for top ological insulator/YIG \nbilayers 16. We therefore turn to an analysis of the broken symmetries in WTe 2. We point out that low \nsymmetry WTe 2 has recently shown the capability of generating both in -plane and out -of-plane spin \npolarization in charge -spin conversion experiments 19-22. It also enables field-free switching of PMA \nmagnet ic material , which ease s the application of PMA material s in MRAM application s 23-25.   \nFerrimagnetic insulator  YIG is of significant research interest in spintronics  due to its \nexceptionally  low Gilbert damping  26, which describes the  relaxation rate  of magnetization precession . \nAnd 1T’-WTe 2 is a semi -metallic transition metal dichalcogenide (TMD) layered material with strong \nSOC  27, 28. The crystal structure of 1T’-WTe 2 lacks twofold rotational symmetry about the c -axis (Fig. \n1a). The only symmetry in the  WTe 2 crystal lattice  ab plane is the mirror symmetry about  the bc plane  \n29. This u nique symmetry breaking allows  out-of-plane damping -like torque  to be generated 30, 31, \nenabling efficient switching of the  out-of-plane magnetization  of the adjacent magnetic material  24.  \nA 20nm thick YIG thin film used in our experiment is epitaxially grown on (111) -oriented  \nGd3Ga5O12 (GGG) substrate  by off -axis sputtering  32. WTe 2 flakes are then mechanically exfoliated \nfrom a flux-grown crystal, and dry transferred on to the clean top YIG surface without touching any \nother substances. This whole process is carried out in an Ar -filled glove box with <0.1 ppm of H 2O and    \n \n5 \n O2 to protect the flakes from degradation and  ensure  the clean liness  of the YIG/WTe 2 interface. We \nemploy hexagonal boron nitride ( hBN ) encapsulation to protect the WTe 2 flakes from oxidation after \nbeing removed from the glove box. We make two samples  and focus on the  data taken from sample 1 in \nthe main text. The raw data taken from sample 2 can be found  in Supporting Information  Fig. S 2.  \n \nFig. 1 Crystal structure of WTe 2 and sample schematic . a) Crystal lattice structure of WTe 2 viewed  \nfrom the top along  the c-axis and looking from the side along the a-axis. The black dashed box in the \nside view indicates a monolayer of WTe 2. b) Schematic of the ferromagnetic resonance force \nmicroscope. RF excitation is generated by a stripline underneath the sample , where the hBN \nencapsulation is not shown . The  region of  localized mode is shown as a yellow dot adjacent to the  WTe 2 \nflake, and the probe magnetic moment is shown as a yellow arrow on the particle. The cantilever \noscillation is detected by a fiber laser interferometer.  \nFigure 2a shows an optical image  of the sample 1. Due to the small  lateral size of the exfoliate d \nWTe 2 and hBN  flake s having  length  scale s of 10 μm, we use a home -built ferromagnetic resonance \nforce microscope (FMRFM) to measure  the local ferromagnetic resonance (FMR) signal. FMRFM is a \nsensitive technique to detect  the local magnetic properties with high spatial and spectral resolution  33. In \nour FMRFM, the external magnetic field 𝐻⃗⃗ ext is aligned perpendicular to  the sample plane. The \ncantilever tip holds a  high coercivity SmCo 5 magnetic particle , whose  moment  is magnetized in the  \n   \n \n6 \n direction opposite to 𝐻⃗⃗ ext to create a magnetic field well . The field well supports a set of  localized \nstanding spin wave modes (LMs). During the measurement, we excite spin precession uniformly by a \nstripline underneath the sample at a fixed RF frequency  (2 GHz)  and sweep the magnetic field. The \nresonance of each LM  generate s a stray field, whic h can then  be detected by the SmCo 5 magnetic \nparticle attached on the cantilever through their magnetic dipole -dipole interaction (Fig. 1b). During the \nmeasurement, we keep the probe -to-sample separation around 4 μm. The operation  of FMRFM is \ndescribed in detail in Ref s. 34-36. For reference, w e separate  a region of  YIG that does not contain  \nWTe 2/hBN heterostructures and measur e its Gilbert damping using  broadband FMR. To eliminate two -\nmagnon scattering, w e perform broadband FMR in the out -of-plane  field geometry. The FMR linewidth \nas a function of frequency measured on bare YIG (sample 1) shows a linear dependence (Fig. 2b), from \nwhich we can extract the Gilbert damping  of bare YIG  𝛼YIG=1.05×10−3. We also confirm that the \nWTe 2 used in the experiment is indeed  the 1T’ phase  through polarized Raman measurements. The \npolarization angle dependence of the Raman peak at 212 cm-1 (spectrum is shown in Fig. S4) exhibits \nminimum intensity when the excitation laser polarization is along the crystallographic a axis of WTe 2 37 \nas shown by the polar plot in Fig. 2c and Raman intensity plot in Fig. 2d .  \nWe find t he position of the YIG/WTe 2/hBN heterostructure with the assist ance of magnetic \nalignment markers  (Fig. 2a) . Figure 2e shows t wo raw FMRFM scans taken in the region  of YIG/hBN \nand YIG/WTe 2/hBN , indicated by the blue and the red dot in Fig. 2a, respectively , which  reveals  the \nchange in FMRFM spectra  at two different location s. Here we focus on the 𝑛=1 LM because it has the \nmode radius of around 1 μm and gives the highest spatial resolution. Higher order modes  have \nincreasing  mode radius and therefore, detect  less local ized magnetic  properties.  This is the reason why    \n \n7 \n the quasi -uniform mode at ~  3325 Oe does not show obvious change in resonance field or signal \namplitude.  We further take a  line scan across the edge of WTe 2 flake (Fig. 2 f) to resolve the spatial \nevolution of FMRFM spectra . The line scan in Fig. 2f (along the dashed line shown in Fig. 2a) shows \nthree main features : first, the magnitude of the LM resonance signal is reduced in the YIG/WTe 2/hBN \nregion compare d to the YIG/hBN region; second, the LM resonance field for all LMs is decreased by \n~40 Oe in the YIG/WTe 2/hBN region; third , the LMs  show complex  splitting and crossing when the \nprobe is close to the boundary (−5 μm<𝑋<10 μm).  \n \nFig. 2 FMRFM and Raman measurement data . a) An optical micrograph of the YIG/WTe 2/hBN \nheterostructure under study . WTe 2 crystal a and b axis are labeled.  b) Broadband FMR measurement of \nthe frequency -dependent linewidth of the YIG thin film. The measurement is done on the same piece of \nYIG used to make sample shown in Fig. 1b. c) Polar plot of the 21 2 cm-1 peak  Raman  intensity. Angle \ndenote s the relative angle between the measurement laser polarization and the WTe 2 a axis. d) 2D \nintensity plot showing Raman peak intensities versus polarization angle . e) FMRFM spectra, one over \nthe YIG/hBN  region (blue line)  and the second over the YIG/WTe 2/hBN  region (red line); these \nlocations are  indicated by the  blue and red dot s in Fig. 2a respectively . f) Color plot of field -dependence \n   \n \n8 \n FMRFM scans as a function of position along the trace indicated by the black dashed line in Fig. 2a. A \nconstant background is subtracted to show only the signal from the several LM resonance s. \nIn the following, we will explain the origin of the three observed effects using spin pumping and \nmagnetic anisotropy. The first effect , i.e. signal reduction in  the YIG/WTe 2/hBN area relative to the \nYIG/hBN  area, is the result of enhanced relaxation due to spin pumping from YIG to WTe 2 38. The 𝑛=\n1 LM resonance signal amplitude ∆𝐴 is inversely proportional to the square of Gilbert damping , 𝛼2. We \ndetermine  the Gilbert damping  constant  𝛼 for YIG/WTe 2/hBN  using 𝛼YIG/WTe2/hBN=\n𝛼YIG/hBN×√∆𝐴YIG/hBN∆𝐴YIG/WTe2/hBN ⁄  (see Ref. 39), where 𝛼YIG/hBN is assumed to be the same as \n𝛼YIG=1.05×10−3 due to  the low SOC and insulating character of hBN . The second  effect is  the \ndecrease of 𝑛=1 LM resonance field 𝐻r,1 by ~40 Oe . And the third effect is  splitting and crossing  of \ncomplex modes  in the region −5 μm<𝑋<10 μm. The second and the third effects  are due  to an \nabrupt change of uniaxial anisotropy across the boundary  separating the YIG/WTe 2/hBN and YIG/hBN \nregions  15. Here, the uniaxial anisotropy refers to the magnetic free energy  depends on the angle between \nmagnetization and sample normal  ℱu=−𝐾u𝒎z2, where 𝒎z is the component  of magnetization unit \nvector in the direction normal to sample plane  and 𝐾u is the uniaxial anisotropy constant specific to \nsample  and depends on the total interaction in the sample . When 𝐾u is positive, ℱu is called to be of \nPMA type, on the other hand, if 𝐾u is negative, ℱu is called to be of easy -plane  type.  This uniaxial \nanisotropy will lead to an effective uniaxial magnetic field 𝑯u=−𝜕ℱu𝜕𝑴⁄ , where 𝑴 is the \nmagnetization . And therefore, a change in 𝐾u can modify the resonance field in a FMR measurement . In \nFMRFM spatial mapping, a n abrupt change in 𝐾u spatially could disturb the LM and lead to mode \nsplitting and crossing as described in Ref. 15. Moreover, i n striking contrast to the previously studied    \n \n9 \n YIG/Au interface 15, which result s in a 32 Oe increase of 𝐻r,1 due to  the enhanced easy -plane \nanisotropy, the observed decrease of 𝐻r,1 indicates that the WTe 2 overlayer induces  an iPMA  in YIG. \nWe note that the  magnitude of the shift in  𝐻r,1 is comparable to the easy -plane anisotropy induced by a \nheavy metal 15, 40 or the iPMA generated by topological surface state 16 on garnet ferrimagnetic material .  \nIn order to probe  the global effect of  a WTe 2 overlayer on YIG, we spatial ly map 𝐻r,1 using  the \n𝑛=1 LM. Figure 3a present s an optical image of WTe 2 flakes on a Si/SiO 2 (285nm ) substrate , where \ndifferent c olors of WTe 2 flakes indicat e different WTe 2 thickness es. Figure 3b and 3 c show spatial  maps \nof magnetic properties  in the region enclosed by the black dashed rectangle  in Fig. 3a .  We acquire the \nmaps using the procedure described in Ref. 39, i.e., simultaneously  measuring spatial variation of  the \nmagnetic anisotropy and Gilbert damping using the  𝑛=1 LM resonance field 𝐻r,1 and signal amplitude  \n∆𝐴. The  entire  WTe 2-covered area show s uniformly lower ed 𝐻r,1 and increased Gilbert damping relative \nto the area without WTe 2. In Fig. 3c, despite the not great signal to noise ratio in damping imaging, the re \nis a clear  Gilbert damping enhancement in WTe 2-covered area . The averaged Gilbert damping of YIG in \nWTe 2-covered area is 𝛼̅YIG/WTe2/hBN≈1.30×10−3, about 24% higher than  𝛼YIG. We note that due to \nthe slight relative tilting of the scan plane and the sample plane, there is a color shift in Fig. 3b  that \nmight conceal the contrast difference in different WTe 2 thickness region . Therefore, to study the WTe 2 \nthickness dependence, we will show fine line scans across edge s of flakes  having  different WTe 2 \nthickness es.     \n \n10 \n  \nFig. 3 Two -dimensional FMRFM scan resolving the spatial variation of magnetic anisotropy and \nGilbert damping. a ) Optical micrograph showing the color contrast of different thickness WTe 2 flake s \n(ranging from 4.7 nm to 44.8 nm) on Si/SiO 2(300 nm). Black dashed box outlines the FMRFM scanned \narea for 2D mapping. b) 2D map of the 𝑛=1 LM resonance field . The d ashed line s labeled 1 -4 \ncorrespond to the  four line-scans shown in Fig. S1a-S1d. c) 2D mapping of the Gilbert damping \nextracted from  the 𝑛=1 LM resonance peak amplitude.  \n  \n   \n \n11 \n Next, we want to understand what gives rise to the PMA in WTe 2/YIG. We rule out the effect \ninduced by a modification of the gyromagnetic ratio by showing the resonance field shift across the \nWTe 2 edge does not depend on RF excitation frequency (See Fig. S3). We also exclude a strain induced \neffect given the absence of an epitaxial relation and the weakness of the van der Waals interaction \nbetween YIG and WTe 2. We further note that we can ignore the role of topological surface states  16 in \nour analysis; they are not relevant for our room temperature experiment since WTe 2 is a topological \nWeyl semimetal only below 100 K  17, 18. \nWe show how an analysis based on symmetry and the nature of the interfacial SOC , generalizing \nthe theory in Ref. 41, gives insight into the PMA observed in our experiment. This will also help us \nunderstand why the easy-axis anisotropy we observe in WTe 2/YIG is so different from the results of \nRef. 13, 15  on YIG interfaces with a dozen different metallic and semiconducting materials, all of which \nexhibit interface -induced easy-plane  anisotropy, as is predicted by theory  41.  \nYIG is a ferrimagnetic Mott insulator, with two inequivalent Fe -sites coupled via \nantiferromagnetic (AFM) superexchange interactions. We focus on how interfacial SOC impacts AFM \nsuperexchange in YIG and show that it leads to a very specific form of the mag netic anisotropy that is \ngoverned by the direction of the effective B-field (see Supporting Information for a details).  \nBefore turning to WTe 2/YIG, it is useful to first consider the simpler case when the only broken \nsymmetry is the mirror plane defined b y the interface.  The abrupt change in lattice potential then results \nin an effective electric field that points normal to the interface, which in turn leads to an effective \nmagnetic field in the rest frame of the electron that couples to its spin. Since t he E-field points normal to    \n \n12 \n the interfacial plane in which the electron moves, the resulting B-field arising from SOC lies within  the \ninterfacial plane. As we show in the SI, this leads to a SOC -induced correction to AFM superexchange \nthat necessarily lead s to an easy-plane  anisotropy.  \nIn the case of WTe 2/YIG, however, when there are additional broken symmetries. Not only does \nthe interface break inversion  symmetry , but the crystal structure of WTe 2 itself breaks in -plane inversion  \nsymmetry . The electric field is now no longer normal to the interface, and the effective B-field arising \nfrom SOC necessarily has an out -of-plane component, as shown in Fig S5b in SI. Thus, we see why the \nlower symmetry of WTe 2/YIG can naturally result in an easy-axis or perpendicular magnetic anisotropy  \n(PMA);  see Supporting Information for details.  \nWe note that the lack of two -fold rotational symmetry in the ab plane in WTe 2 that plays a \ncritical role in our understanding of PMA in WTe 2/YIG, has also been pointed  out be crucial for the out -\nof-plane damping -like torque in WTe 2/Permalloy30. We note, however, that the out -of-plane damping -\nlike torque necessarily involves current flow in WTe 2, while the PMA is an equilibrium property of the \nsystem independent of current flow.  \nWe further demonstrate the interfacial  origin of the observed effect by studying the influence of \nWTe 2 thickness.  We show four  line-scans , labeled in Fig. 3b, across the edges of WTe 2 with different \nthickness es, ranging from 4. 7 nm to 44.8 nm . From these four line -scans, we extract  the 𝑛=1 LM \nresonance field 𝐻r,1 and the 𝑛=1 LM resonance signal amplitude  ∆𝐴. Figures S1a-d in the Supporting \nInformation  show the evolution of 𝐻r,1 and ∆𝐴 along the traces labeled correspondingly . The thickness \nof WTe 2 at each measurement location is later measured using atomic force microscop y. From these    \n \n13 \n line-scans , we choose the region s where the probe is far away from the edge of WTe 2 so that the \nmagnetic propert ies are uniform,  to obtain  spatial average s of 𝐻r,1 and ∆𝐴, which are denoted  \n𝐻̅r,1,YIG/hBN and ∆𝐴̅̅̅̅YIG/hBN in the YIG/hBN region , and 𝐻̅r,1,YIG/WTe2/hBN and ∆𝐴̅̅̅̅YIG/WTe2/hBN in the \nYIG/WTe 2/hBN region , respectively . We further extract the 𝑛=1 LM resonance field difference \nbetween two regions using ∆𝐻r,1=𝐻̅r,1,YIG/hBN−𝐻̅r,1,YIG/WTe2/hBN, as well as the Gilbert damping \ndifference using ∆𝛼=𝛼YIG×(√∆𝐴̅̅̅̅YIG/hBN ∆𝐴̅̅̅̅YIG/WTe2/hBN ⁄ −1) as a function of the WTe 2 \nthickness . We note  that the hBN overlayer does not change the Gilbert damping in YIG . The \nsummarized results containing the data from both sample 1 and sample 2  are shown in Fig s. 4a and 4 b. \nRaw data from sample 2 can be found in Supporting Information  Fig. S 2. The thinnest WTe 2 acquired in \nthe experiment is 3.2nm from sample 2, which is approximately the thickness of  a quadruple -layer \nWTe 2.  \nFigures 4a and 4 b indicate  that both ∆𝐻r,1 and ∆𝛼 have  almost no WTe 2 thickness  dependence . \nThere is a  small sample -to-sample variation possibly  due to different YIG/WTe 2 interfacial quality . The \nchange of 𝑛=1 LM resonance field,  ∆𝐻r,1, is as large as ~38 Oe even when the WTe 2 thickness \napproaches the quadruple -layer thickness . This indicates that the modification of magnetic anisotropy is \ndue to the YIG/WTe 2 interfac ial interaction , with no bulk contribution. For the increase of Gilbert \ndamping ∆𝛼, no obvious thickness dependence is observed when comparing the data from the same \nsample. In sample 2, the Gilbert damping enhancement  due to the quadruple -layer WTe 2 has almost the \nsame value as the 50 nm thick WTe 2 flake, indicating that no thickness dependence of spin pumping  can \nbe resolved from our measurement. There are two possible interpretations of these results . First, if the    \n \n14 \n spin current injected into WTe 2 is mainly relaxed due to spin relaxation in the bulk, then the \nexperimental result is a demonstration of ultra -short spin diffusion length along the c axis38, smaller or \ncomparable to the thinnest WTe 2 flake (3.2 nm), employed in this experiment . It is much smaller than \nthe 8nm spin diffusio n length  in the  in-plane direction measured using inverse spin Hall effect  22. Note \nthat due to the chang e in mo bility and the metal -insulator transition  in few layer WTe 2 when its \nthickness reduces  42, the spin diffusion length approximated here could be inaccurate . Alternatively , it is \npossible that the spin relaxation is primarily due to the interfacial SOC induced by inversion symmetry \nbreaking at the interface and in the WTe 2 crystal lattice. In this case, the Gilbert damping enhancement \nwill have no WTe 2 thickness dependence.     \n \n15 \n  \nFig. 4  WTe 2 thickness dependence  of resonance field and damping enhancement . a) 𝐻r,1 in the \nYIG/hBN and YIG/WTe 2/hBN regions  are averaged  respectively to get 𝐻̅r,1,YIG and 𝐻̅r,1,YIG/WTe2, and \n∆𝐻r,1=𝐻̅r,1,YIG−𝐻̅r,1,YIG/WTe2. b) ∆𝛼 as a function of WTe 2 thickness , and  ∆𝛼=𝛼YIG/WTe2−𝛼YIG \nwhere 𝛼YIG is the Gilbert damping of bare YIG measured using broadband FMR for each sample , and \n𝛼YIG/WTe2=𝛼YIG×(√∆𝐴̅̅̅̅YIG/hBN ∆𝐴̅̅̅̅YIG/WTe2/hBN ⁄ ).   \nIn conclusion, we have shown that the YIG/WTe 2 interface plays a critical  role in both interfacial \nmagnetic anisotropy and spin relaxation , making WTe 2 a promising material  in magnetic memory \n   \n \n16 \n application s. Combining the iPMA  created by WTe 2 with the out-of-plane spin orbit torque  generated by \nflowing a charge current along  the a axis of WTe 2, one can possibly achieve field -free switching of a \nPMA magnetic cell  for magnetic memory application s. It will improv e the  scalability , reduc e the  power \nconsumption  and increas e operation speed  of magnetic solid -state devices . Our result reveals new \npossibilities in selecting materials and designing spintronic devices. For example, one can consider other \nmaterials with low lattice symmetry and strong SOC to induce larger PMA type interfacial ani sotropy in \nFMIs. To achieve a fully PMA material, one could utilize thinner FMIs to magnify the role of iPMA. \nMoreover, interfacial SOC also plays an important role in  generat ing topologically protected magnetic \ntextures in the FMIs  43. These findings will motivate further research to reveal the fundamental  physics \narising at the  interface between FMIs and nonmagnetic materials.  \n \nData availability:  \nThe data generated by the present study are available from the corresponding author on request.  \nSupporting Information:  \nA description of raw data on WTe 2 thickness dependence, a FMRFM measurement on a second sample, \na FMRFM measurement at different RF frequency, a description of polarized Raman measurement \nresult, and a detailed illustration of impact of broken  mirror reflection symmetries on the magnetic \nanisotropy.  \nAckno wledgements:     \n \n17 \n This work was primarily supported by the Center for Emergent Materials: an NSF MRSEC under award \nnumber DMR -2011876 (GW, NV , YC, SG, FY , MR and PCH) . KW and TT acknowledge support from \nthe Elemental Strategy Initiative conducted by the MEXT, Japan (Grant Number JPMXP0112101001) \nand JSPS KAKENHI (Grant Numbers 19H05790, 20H00354 and 21H05233).  DW, GC, CNL, and MB \nare supported by NSF under award DMR -2004801.  We gr atefully acknowledge N. Trivedi for insightful \ndiscussions. Fabrication and some characterization were performed in the Ohio State University \nNanoSystems Laboratory.  \n     \n \n18 \n Methods:  \nSample fabrication  \nOur YIG/WTe 2/hBN heterostructure was prepared by means of dry transfer and stacking 44. hBN crystals \nwere mechanically exfoliated under ambient conditions onto SiO 2/Si substrates (285 nm thick SiO 2). 20-\n40 nm thick hBN flakes were identified under an optical microscope and used for the capping lay er for \nthe stack. The hBN was picked up using a polymer -based dry transfer technique and then moved into an \nAr-filled glove box with oxygen and water level below 0.1 ppm. Flux -grown WTe 2 crystals 45 were \nexfoliated inside the glove box and flakes with different thicknesses were optically identified and \nquickly picked up with the capping hBN layer then transferred to the YIG substrate. Finally, we removed \nthe fully encapsulated sample from the glove b ox and performed the e -beam lithography and \nmetallization (Ni/Au) step for alignment in our ferromagnetic resonance force microscope (FMRFM).  \n \nPolarized Raman measurement  \nPolarized Raman spectra from the WTe 2 sample were collected using 633 nm excitation w avelength in \nan inVia Renishaw Raman microscope.  The sample was loaded onto the microscope stage and \npositioned in such a way that the long edge of the flake was aligned parallel to the laser polarization ( θ = \n0°). In this configuration, the incident illu mination is polarized vertically coming out of the laser and is \naligned with the long axis of the WTe 2 flake. The polarization of the incident laser was rotated from 0 to \n360° by 10° increments using a polarization rotator, while an analyzer was set to onl y allow vertically \npolarized light to enter the spectrometer.  Raman spectra were collected at each polarization for 3 \nacquisitions with a 20 s time per acquisition.  The laser power was set to 0.5 mW at the sample to avoid \nany damage by heating.  Followin g spectral collection, the (baseline corrected) integrated intensities \nunder each peak were calculated to make the contour plots and polar plots in Fig. 2c and 2d.  \n \nFMRFM measurement and signal fitting  \nOur FMRFM perform s local ly measures  FMR at room temper ature in vacuum. The cantilever has \nnatural frequency of ~18 KHz, spring constant of 0.2 N/m and Q factor of ~20000, resulting in force \ndetection sensitivity of  10-15 N/Hz1/2. The SmCo 5 magnetic particle attached on the cantilever has a \nmagnetic moment of ~4 nemu.  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B 2015,  92, (4), 041104.  \n \n     \n \n23 \n Enhancing  Perpendicular Magnetic Anisotropy in Garnet  Ferrimagnet \nby Interfacing with Few-Layer  WTe 2 \nGuanzhong Wu1*, Dongying Wang1, Nish chhal  Verma1, Rahul Rao2, Yang Cheng1, Side Guo1, \nGuixin Cao1, Kenji Watanabe3, Takashi Taniguchi4, Chun  Ning Lau1, Fengyuan Yang1,  \nMohit Randeria1, Marc Bockrath1, and P. Chris Hammel1 \n1.Department of Physics, The Ohio State University, Columbus, OH, 43210, USA  \n2.Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright -Patterson Air \nForce Base, Dayton, OH, 45433, USA  \n3. Research Center for Functional Materials,  National Institute for Materials Science, 1 -1 Namiki, \nTsukuba 305 -0044, Japan  \n4. International Center for Materials Nanoarchitectonics , National Institute for Materials Science,  1-1 \nNamiki, Tsukuba 305 -0044, Japan  \n*wu.2314@osu.edu  \n     \n \n24 \n FMRFM  line-scan across edge of WTe 2 with different thickness  \n \nFig. S1 a-d, FMRFM line-scans along the traces 1~4 indicated in Fig. 3b respectively.  The gray shaded \narea in four figures are outlining the location of  WTe2 flake . 𝐻r,1 and ∆𝐴 at each position are derived by \nfitting the 𝑛=1 LM to a Lorentzian line shape. The thickness of WTe2 flake at each location are \nmeasured by atomic force microscope.  \n  \n   \n \n25 \n FMRFM measurement on Sample 2  \n \nFig. S 2 FMRFM measurement on sample 2. a, The optical picture of the YIG/WTe 2/hBN \nheterostructure. b, 2D mapping of the 𝑛=1 LM resonance field in the black dash line circled area.  c, \n2D mapping of the Gilbert damping extracted from 𝑛=1 LM resonance peak amplitude in the black \ndash line circled area. d, FMRFM line scan along the trace indicated by the solid black line in Fig. S 2a. \nA constant background is subtracted to show only the signal from the LMs resonance. e, Fine scan \nzoomed in o n the quadruple layer WTe 2 stripe area  \n  \n   \n \n26 \n FMRFMR measurement at 4 GHz  \n \nFig. S 3 FMRFM measurement across WTe 2 edge at 4 GHz. FMRFM line scan is measured at 4 GHz \nacross the WTe 2 flake edge. The shift of the resonance field 𝐻r,1 is 36 Oe, similar to the 𝐻r,1 shift \nmeasured at 2 GHz. This result excludes the possibility that the resonance field shift arises from \nmodification of the gyromagnetic ratio.  \n \n  \n   \n \n27 \n Polarized Raman measurement  \n \nFig. S 4 Polarized Raman measurement. As shown by the red curve, the Raman  spectrum taken on \nGGG/YIG/WTe 2/hBN heterostructure contains more peaks than WTe 2. The Raman spectrum taken in \nthe GGG/YIG/hBN area identifies the peaks arising from the  substrate GGG/YIG or top hBN \nencapsulation layer. By subtracting the Raman spectrum in the GGG/YIG/hBN area, the Raman spectra \nfrom WTe 2 layer are extracted and plotted in Fig. 2d. The black dash line are the markers indicating the \nRaman peaks of WTe 2  \n   \n \n28 \n Impact of  broken mirror reflection  symmetr ies on the  magnetic anisotropy  \nWe describe theo retical constraints on the interface -induced  magnetic anisotropy in the WTe 2/YIG \nbilayer. We first show that symmetry arguments alone do not provide strong constraints on the anisotropy \ntensor, given that we are dealing with an interface between two crystalline materials at an arbitrary \norientation with respect to each other . We then present qualitative arguments, based on the interfacia l spin -\norbit coupling, that give insight into the magnetic anisotropy in WTe 2/YIG. This helps us understand why \nthe easy-axis anisotropy that we observe  in WTe 2/YIG differ s from the results  of Lee et al. [1] on YIG \ninterfaces with a dozen different metallic and semiconducting materials , all of which exhibit interface -\ninduced easy-plane anisotropy  as predicted  by theory  [2]. \nOn general grounds, the  anisotropy (free) energy can be written as   \nℱ𝑎𝑛𝑖𝑠= ∑ 𝐾𝑎𝑏𝑎,𝑏 𝑚𝑎 𝑚𝑏,           (S1) \nwhere a and b take on values x,y,z. We focus here on the leading term, quadratic in the magnetization, and \nignore higher order anisotropy terms like (mx4+my4+mz4) or (mx2my2+my2mz2+mz 2mx2). The form of \n 𝐾𝑎𝑏=  𝐾𝑏𝑎 is constrained by symmetry. Let us consider three cases , going from the most symmetric to \nthe least .  \nCase I: The only broken symmetry is interfacial inversion (z →  - z), which  is relevant for the \nexperiments of Ref. [1]. The magnetization is an axial vector (or pseudovector) that transforms under  \nrotation s like a vector but is unchanged  under inversion . Thus (𝑚𝑥 ,𝑚𝑦 ,𝑚𝑧)→ (𝑚𝑥 ,− 𝑚𝑦 ,−𝑚𝑧) under \nreflection in  a mirror plane with normal 𝑥̂. Using reflection symmetry in mirror planes normal  to 𝑥̂ and to    \n \n29 \n 𝑦̂ , we can see that all off -diagonal components of  𝐾𝑎𝑏  vanish. Further, f our-fold rotational symmetry \nabout the 𝑧̂ axis shows that  𝐾𝑥𝑥=  𝐾𝑦𝑦. Using mx2+my2+mz2=1, we write  𝐾𝑥𝑥(m𝐱2+m𝐲2) in terms \nof m𝑧2, and d efining 𝐾𝑢=  (𝐾𝑥𝑥−  𝐾𝑧𝑧), we obtain  \nℱ𝑎𝑛𝑖𝑠= −  𝐾𝑢 m𝑧2.         (S2) \nThis symmetry analysis only constrains the form  of the anisotropy energy, but not the sign of  𝐾𝑢. We will \ngive below a simple microscopic argument [2] that shows that   𝐾𝑢<0  (easy plane)  for Case I.  \nCase II: In addition to broken interfacial inversion (z → - z), let u s also break reflection symmetry \nin the plane normal to 𝑥̂. This would be the case if the crystalline axes of WTe 2 were aligned with YIG.  \nThis also breaks four-fold rotational symmetry about 𝑧̂, so that   𝐾𝑥𝑥 ≠ 𝐾𝑦𝑦. However, we  can still use \nreflection symmetry in the plane normal to 𝑦̂ to conclude that  𝐾𝑥𝑦=  𝐾𝑦𝑧=0. Thus  we find that  \n      K = (𝐾𝑥𝑥0 𝐾𝑥𝑧\n0 𝐾𝑦𝑦0\n 𝐾𝑥𝑧0 𝐾𝑧𝑧)                                                               (S3) \nCase III: When the crystalline axes of WTe 2 are not aligned with YIG, which is the experimentally \nrelevant case, all mirror reflection and rotation symmetries are broken. Then there are no symmetr y \nconstrain ts on   𝐾𝑎𝑏 and all six components of this symmetric tensor are in general non -zero.      \nLet us now see how, despite the lack of general symmetry -based constrai nts, we can still get some \nqualitative insight about the form of the anisotropy from simple microscopic considerations informed by \nsymmetry. YIG is a ferrimagnetic Mott insulator, with two inequivalent Fe -sites coupled via    \n \n30 \n antiferromagnetic (AFM) superexch ange interactions. We thus focus on how interfacial spin -orbit \ncoupling (SOC) impacts AFM superexchange.   \nThe broken symmetry at the interface leads to an electric field ℇ=−𝛁𝑉(𝒓), whose direction will \nbe discussed in detail below for three cases. This in turn produces a magnetic field in the rest frame of the \nelectron  which underlies SOC. As the electron moves along  𝐫̂ij from site i to j, it experiences an  SOC field  \nin the direction 𝐝̂ij  which is determined by  ℇ ×𝐫̂ij . The SOC Hamiltonian  is thus given by \n−𝑖𝜆∑𝑐𝐢𝛼†(𝐝̂ij∙𝝈𝛼𝛽)𝑐𝐣𝛃 𝛼𝛽 . Including the effect of this term in addition to the usual hopping t and Hubbard \nU in the standard strong coupling expansion calculation leads to the Hamiltonian  \n                 ℋex=J∑𝐒i∙𝐒j <𝐢,𝐣>+D∑𝐝̂ij∙𝐒i×𝐒j <𝐢,𝐣> +K0∑(𝐝̂ij∙𝐒i)(𝐝̂ij∙𝐒j). <𝐢,𝐣>                  (S4) \nHere  the spin  𝐒i   at site i is coupled to its neighbors via  the AFM superexchange  𝐽 ~𝑡2\n𝑈  and the \nDzyaloshinskii -Moriya interaction (DMI)  𝐷 ~𝑡𝜆\n𝑈. The K0 term will be the focus of our attention belo w as \nit leads  to magnetic anisotropy.  We note that the general form of ℋex is in fact substantially independent \n[2] of the microscopic mechanism and very similar results are obtained not only for superexchange but \nalso for Zener double exchange and RKKY interactions.  \nCase I: Let us again return to the simplest case with broken interfacial inversion (z → - z). This \nleads to an electric field ℇ=−𝛁𝑉(𝒓)  along ẑ , the normal to the interface.  The SOC magnetic field \ndirection is then given by  𝐝̂ij= ẑ ×𝐫̂ij ; see Figure S4(a). This is the well -known Rashba SOC at \ninterfaces . We note in passing that 𝐝̂ij is antis ymmetric under the interchange of i and j, and thus  leads to \na DMI term where 𝐒i×𝐒j is also antisymmetric.     \n \n31 \n  \n \nFig. S 5 Symmetry based selection of magneto -crystalline anisotropy. Interfacial SOC originates from \nan effective Electric field ℇ=−𝛁𝑉(𝒓) whose direction is determined by the broken mirror planes in the \nsystem. This electric field leads to  spin-orbit coupling (SOC), with the 𝐝̂ij= ℇ̂ ×𝐫̂ij, the direction of the \nSOC magnetic  field. Note that the direction of the electron hop 𝐫̂ij lies in the xy plane of the interface. As \nshown in the text 𝐝̂ij controls the interface -induced magnetic anisotropy. (a) When only surface inversion \nis broken, 𝐝𝐢𝐣 is constrained to lie in the int erface and interfacial SOC leads to easy -plane anisotropy. (b) \nIf there are other broken mirror planes, the 𝐝𝐢𝐣 must lie outside the interfacial plane. This can lead to a \nperpendicular magnetic anisotropy in systems like YIG/WTe 2 bilayers.  \n \nWe see that in Case I,  𝐝̂ij lies in the plane  of the interface, and the third term in eq. (S 4) then takes \nthe form K0∑(S𝐢𝑥𝑆𝐢+𝑦𝑥+S𝐢𝑦𝑆𝐢+𝑥𝑦) 𝐢  for a square lattice . To make the connection with  magnetic anisotropy, \nwe look at a continuum approximation with a s lowly  varying magnetization 𝐦(𝐫). We make a Taylor \nexpan sion of  𝐒r in terms of its value at 𝒓, denoted by  𝐦(𝐫), and its spatial derivatives . The exchange and \nDMI terms involve gradients of 𝐦(𝐫), but we focus here on local terms that  do not involve derivatives  to \n   \n \n32 \n understand the magnetic anisotropy . The leading term is + K0(m𝐱2+m𝐲2)  which can be rewritten as \n– K0 m𝑧2 using the fact that mx2+my2+mz2=1 at each 𝒓. Thus, we may identify K0 with the anisotropy  \nK𝑢 defined in eq. (S2).  \nThe microscopic analysis leads to the result K0= − 𝜆2\n𝑈   < 0 and this explains the easy-plane \nanisotropy  arising Rashba SOC at the interface . The easy-plane nature of the anisotropy  is in fact a general \nfeature of various microscopic models as emphasized  in Ref. [2]. We note however that these author s use d \nthe opposite sign convention for anisotropies  from the one we use here . The easy plane vs. easy -axis \ncharacter is , of course, independent of sign conventions.  The FMR experiments of Ref. [1] have seen the \ninterface -induced easy-plane anisotropy  predicted by the theory in a YIG interfaces with several  metallic \nand semiconducting materials . \nThe key difference between the YIG/WTe 2 bilayer studied here and systems  studied earlier [1] is \nthat WTe 2 has a broken mirror plane  (the ac plane ) as shown in  Fig. 1(a)  of the paper . We now  look at the \neffect of this lower symmetry on the microscopic analysis.  \nCase II: Let us break reflection symmetry in the plane normal to 𝑥̂ in addition to broken interfacial \ninversion. We choose x̂ parallel to the b axis, ŷ parallel to a, and ẑ parallel to c. Reflection symmetry in \nthe ŷ mirror plane constrains the  electric field ℇ =−𝛁𝑉(𝒓) to lie in the xz plane, at an angle 𝜃 from the \nz-axis as shown in Fig. S 5(b). Thus  \n𝐝ij=(sin𝜃𝑥̂+cos𝜃𝑧̂)×𝐫̂ij     (S5)    \n \n33 \n where 𝐫̂ij is a vector in the interface  (xy plane ) and 0≤𝜃≤𝜋. Using eq. (S5), we may rewrite  the last \nterm in the  Hamiltonian (S4) as \n K0 sin2𝜃∑(S𝐢𝑧𝑆𝐢+𝑦𝑧)\n𝐢+K0 cos2𝜃∑(S𝐢𝑥𝑆𝐢+𝑦𝑥+S𝐢𝑦𝑆𝐢+𝑥𝑦)\n𝐢\n−K0sin𝜃cos𝜃∑(S𝐢𝑧𝑆𝐢+𝑦𝑥+S𝐢𝑥𝑆𝐢+𝑦𝑧)\n𝐢  \nAs before, we  make  a continuum approximation with a smoothly varying 𝐦(𝐫) and focus only on the \nlocal terms, without gradients, to obtain the magnetic anisotropy . We find that the leading order \ncontribution to anisotropy is −K0cos2𝜃m𝑧2+K0sin2𝜃 mzmx. This analysis correctly captures the non -\nzero K𝑥𝑧 expected on general grounds; see eq. (S3). We did not include here , for simplicity, the effects of \nbroken four -fold rotation that would have led to  𝐾𝑥𝑥 ≠ 𝐾𝑦𝑦. \nCase III: When we lose all mirror symmetries, the case relevant to the YIG/WTe 2 experiment, the \nelectric field ℇ =−𝛁𝑉(𝒓) will point in a general direction specified by 0≤𝜃≤𝜋 and 0≤𝜑≤2𝜋, and \nthere will be no symmetry constraints on the anisotropy tensor  𝐾𝑎𝑏. \n Let us conclude by highlighting the  key qualitative difference  between Case I  on the one hand and \nCases II and III  on the other . In Case I, the only broken symmetry is interfaci al inversion (z → - z). Then \nsymmetry constrains the  𝐝̂ij, the direction of the SOC B-field,  to lie in the plane  of the interface  and this \nleads to easy -plane anisotropy as described above. In Cases II and III, there are other additional broken \nmirror planes, and this leads to the 𝐝̂ij  vector being pulled out of the plane of the interface. This \nimmediately leads to the possibility of an easy -axis like character to the anisotropy, although in the general \ncase one has a non -trivial anisotropy tensor  𝐾𝑎𝑏.     \n \n34 \n Reference  \n[1] Lee, A. J.; Ahmed, A. S.; McCullian, B. A.; Guo, S. D.; Zhu, M. L.; Yu, S. S.; Woodward, P. M.; \nHwang, J.; Hammel, P. C.; Yang, F. Y . Interfacial Rashba -Effect-Induced Anisotropy in Nonmagnetic -\nMaterial -Ferrimagnetic -Insulator Bilayers. Phys. Rev. Lett. 2020,  124, (25), 257202.  \n[2] Banerjee, S.; Rowland, J.; Erten, O.; Randeria, M. Enhanced Stability of Skyrmions in Two -\nDimensional Chiral Magnets with Rashba Spin -Orbit Coupling. Physical Review X 2014,  4, (3), 031045.  \n \n "    },    {        "title": "0805.3495v1.Intrinsic_and_non_local_Gilbert_damping_in_polycrystalline_nickel_studied_by_Ti_Sapphire_laser_fs_spectroscopy.pdf",        "content": "Intrinsic and non-local Gilbert damping in\npolycrystalline nickel studied by Ti:Sapphire laser fs\nspectroscopy\nJ Walowski1, M Djordjevic Kaufmann1, B Lenk1, C Hamann2\nand J McCord2, M M unzenberg1\n1Universit at G ottingen, Friedirch-Hund-Platz 1, 37077 G ottingen, Germany\n2IFW Dresden, Helmholtzstra\u0019e 20, 01069 Dresden\nE-mail: walowski@ph4.physik.uni-goettingen.de\nAbstract. The use of femtosecond laser pulses generated by a Ti:Sapphire laser\nsystem allows us to gain an insight into the magnetization dynamics on time scales from\nsub-picosecond up to 1 ns directly in the time domain. This experimental technique is\nused to excite a polycrystalline nickel (Ni) \flm optically and probe the dynamics\nafterwards. Di\u000berent spin wave modes (the Kittel mode, perpendicular standing\nspin-wave modes (PSSW) and dipolar spin-wave modes (Damon-Eshbach modes)) are\nidenti\fed as the Ni thickness is increased. The Kittel mode allows determination of the\nGilbert damping parameter \u000bextracted from the magnetization relaxation time \u001c\u000b.\nThe non-local damping by spin currents emitted into a non-magnetic metallic layer\nof vanadium (V), palladium (Pd) and the rare earth dysprosium (Dy) are studied\nfor wedge-shaped Ni \flms 1 nm \u000030 nm. The damping parameter increases from\n\u000b= 0:045 intrinsic for nickel to \u000b > 0:10 for the heavy materials, such as Pd and\nDy, for the thinnest Ni \flms below 10 nm thickness. Also, for the thinnest reference Ni\n\flm thickness, an increased magnetic damping below 4 nm is observed. The origin of\nthis increase is discussed within the framework of line broadening by locally di\u000berent\nprecessional frequencies within the laser spot region.arXiv:0805.3495v1  [cond-mat.other]  22 May 2008Gilbert damping in Nickel thin \flms 2\n1. Introduction\nThe understanding of picosecond-pulsed excitation of spin packets, spin wave modes\nand spin currents is of importance in developing a controlled magnetic switching concept\nbeyond the hundred picosecond timescale and to test the speed of magnetic data storage\nmedia heading to the physical limits. Over the last years profound progress has been\nmade within that \feld by using femtosecond laser spectroscopy. The recent discoveries\nin ultrafast magnetization dynamics are heading to a new understanding [1{5] and\nnew all-optical switching concepts have been discovered [6]. In addition, the all-optical\nmethod has developed into a valuable tool to study the magnetization dynamics of\nthe magnetic precession and thereby access magnetocrystalline anisotropies and the\nmagnetic damping [7{11] or the dynamics of magnetic modes in nanometer sized arrays\nof magnetic structures [12, 13] and single magnetic nanostructures [14, 15]. Naturally,\none \fnds similarities and di\u000berences as compared to magnetic resonance techniques\nin frequency space (FMR) [16], optical techniques such as Brillouin light scattering\n(BLS) [17,18] and time-resolved techniques, for example pulsed inductive magnetometry\n(PIMM) [19]. Advantages and disadvantages of the di\u000berent techniques have already\nbeen compared in previous work [20{22]. The same concepts can be applied to the\nfemtosecond-laser-based all-optical spectroscopy techniques. Here we discuss their\nabilities, highlighting some aspects and peculiarities [11,23{27]:\ni. After excitation within the intense laser pulse, the nature of the magnetic relaxation\nmechanisms determine the magnetic modes observed on the larger time scale [5].\nFor a Ni wedge di\u000berent modes are found as the thickness is increased: coherent\nprecession (Kittel mode), standing spin waves (already found in [28]) and dipolar\nsurface spin waves (Damon-Eshbach modes) appear and can be identi\fed.\nii. Magnetic damping has been extracted by the use of fs spectroscopy experiments\nalready in various materials, epitaxial \flms, as a function of the applied \feld\nstrength, \feld orientation and laser excitation power [7{11]. Using the Kittel\nmode, we study the energy dissipation process caused by non-local damping by spin\ncurrents [29] in Ni by attaching a transition metal \flm (vanadium (V), palladium\n(Pd) and a rare earth \flm (dysprosium (Dy)) as a spin sink material and compare\nthem to a Ni reference sample. The present advantages and disadvantages of the\nmethod are discussed.\niii. A modi\fcation of the magnetic damping is found for the thinnest magnetic layers\nbelow 4 nm. The understanding of this e\u000bect is of high interest because of the\nincrease in methods used to study magnetic damping processes in the low \feld\nregion in the current literature. We present a simple model of line broadening\nknown from FMR [30{32] and adapted to the all-optical geometry that pictures\nthe e\u000bect of the increased intrinsic apparent damping observed. Therein a spread\nlocal magnetic property within the probe spot region is used to mimic the increased\napparent damping for the low \feld region.Gilbert damping in Nickel thin \flms 3\na)\nb)Side view:\nFigure 1. a) Schematics of the pump probe experiment to determine the change in\nKerr rotation as a function of the delay time \u001c. b) Experimental data on short and\nlong time scales. On top a schematic on the processes involved is given.\n2. Experimental Technique\nThe all-optical approach to measuring magnetization dynamics uses femtosecond laser\npulses in a pump-probe geometry. In our experimental setup a Ti:Sapphire oscillator\ngenerates the fs laser pulses which are then ampli\fed by a regenerative ampli\fer (RegA\n9050). This systems laser pulse characteristics are 815 nm central wave length, a\nrepetition rate of 250 kHz, a temporal length of 50 \u000080 fs and an energy of \u00181\u0016J\nper pulse. The beam is split into a strong pump beam (95% of the incoming power),\nwhich triggers the magnetization dynamics by depositing energy within the spot region,Gilbert damping in Nickel thin \flms 4\nand a weaker probe pulse (5% of the incoming power) to probe the magnetization\ndynamics via the magneto-optical Kerr e\u000bect delayed by the time \u001c, in the following\nabbreviated as time-resolved magneto-optic Kerr e\u000bect (TRMOKE). The schematic\nsetup and sample geometry is given in \fgure 1a). The spot diameters of the pump\nand probe beam are 60 \u0016m and 30\u0016m respectively. A double-modulation technique is\napplied to detect the measured signal adapted from [33]: the probe beam is modulated\nwith a photo-elastic modulator (PEM) at a frequency f1= 250 kHz and the pump\nbeam by a mechanic chopper at a frequency f2= 800 Hz. The sample is situated in\na variable magnetic \feld (0 \u0000150 mT), which can be rotated from 0\u000e(in-plane) to 90\u000e\n(out of plane) direction. The degree of demagnetization can be varied by the pump\n\ruence (10 mJ =cm2\u000060 mJ=cm2) to up to 20% for layer thicknesses around 30 nm and\nup to over 80% for layers thinner than 5 nm. The samples studied were all grown on\nSi(100) substrates by e-beam evaporation in a UHV chamber at a base pressure of\n\u00185\u000210\u000010mbar. For a variation of the thickness, the layers are grown as wedges with\na constant gradient on a total wedge length of 15 mm.\n3. Results and discussion\n3.1. Kittel mode, standing spin waves and Damon-Eshbach surface modes\nTo give an introduction to the TRMOKE signals \u0001 \u0012Kerr(\u001c) measured on the timescale\nfrom picoseconds to nanoseconds \frst, the ultrafast demagnetization on a characteristic\ntime scale\u001cMand the magnetic precessional motion damped on a time scale \u001c\u000bis shown\nfor a Ni \flm in \fgure 1b); the schematics of the processes involved on the di\u000berent time\nscales are given on the top. The change in Kerr rotation \u0001 \u0012Kerr(\u001c) shows a sudden drop\nat\u001c= 0 ps. This mirrors the demagnetization within a timescale of \u0018200 fs [34{36]. For\nthe short time scale the dynamics are dominated by electronic relaxation processes, as\ndescribed phenomenologically in the three temperature model [34] or by connecting the\nelectron-spin scattering channel with Elliot-Yafet processes, as done by Koopmans [36]\nand Chantrell [4] later. At that time scale the collective precessional motion lasting up\nto the nanosecond scale is initiated [28, 37]: the energy deposited by the pump pulse\nleads to a change in the magnetic anisotropy and magnetization, and thus the total\ne\u000bective \feld. Within \u001810 ps the total e\u000bective \feld has recovered to the old value\nand direction again. However, the magnetization, which followed the e\u000bective \feld,\nis still out of equilibrium and starts to relax by precessing around the e\u000bective \feld.\nThis mechanism can be imagined as a magnetic \feld pulse a few picoseconds long, and\nis therefore sometimes called an anisotropy \feld pulse. The resulting anisotropy \feld\npulse is signi\fcantly shorter than standard \feld pulses [38]. This makes the TRMOKE\nexperiment di\u000berent to other magnetization dynamics experiments.\nThe fact that the situation is not fully described by the model can be seen in the\nfollowing. Already van Kampen et al. [28] not only observed the coherent precessional\nmode, they also identi\fed another mode at a higher frequency than the coherentGilbert damping in Nickel thin \flms 5\nprecession mode, shifted by !k;n\u00182Ak2= 2An\u0019=t Ni2, the standing spin wave (PSSW)\nmode. It originates from the con\fnement of the \fnite layer thickness, where Ais the\nexchange coupling constant and nis a given order. Here we also present the \fnding of\ndipolar propagating spin waves. For all three, the frequency dependence as a function\nof the applied magnetic \feld will be discussed, a necessity for identifying them in the\nexperiments later on.\nFor the coherent precession the frequency dependence is described by the Kittel\nequation. It is derived by expressing the e\u000bective \feld in the Landau-Lifshitz-Gilbert\n(LLG equation) as a partial derivative of the free magnetic energy [39, 40]. Assuming\nnegligible in-plane anisotropy in case of the polycrystalline nickel (Ni) \flm and small\ntilting angles of the magnetization out of the sample plane (\feld is applied 35\u000eout of\nplane \fgure 1a)), it is solved as derived in [41]:\n!=\r\n\u00160s\n\u00160Hx\u0012\n\u00160Hx+\u00160Ms\u00002Kz\nMs\u0013\n; (1)\nFor the standing spin waves (PSSW) a similar equation is given. For the geometry\nwith the \feld applied 35\u000eout of plane (\fgure 1a) the frequencies !and!k;ndo not\nsimply add as in the \feld applied in plane geometry [41]:\n!=\r\n\u00160s\n(\u00160Hx+2Ak2\nMs)\u0012\n\u00160Hx+\u00160Ms\u00002Kz\nMs+2Ak2\nMs\u0013\n; (2)\nWhile the exchange energy dominates in the limit of small length scales, the\nmagnetic dipolar interaction becomes important at larger length scales. Damon and\nEshbach [42] derived by taking into account the dipolar interactions in the limit of\nnegligible exchange energy, the solution of the Damon-Eshbach (DE) surface waves\npropagating with a wave vector qalong the surface, decaying within the magnetic layer.\nThe wavelengths are found to be above the >\u0016m range for Ni [27].\n!=\r\n\u00160s\n\u00160Hx\u0012\n\u00160Hx+\u00160Ms\u00002Kz\nMs+M2\nS\n4[1\u0000exp(\u00002qtNi)]\u0013\n; (3)\nThe depth of the demagnetization by the femtosecond laser pulse is given by the\noptical penetration length \u0015opt\u001915 nm (\u0015= 800 nm). From the nature of the excitation\nprocess in the TRMOKE experiment one can derive that for di\u000berent thicknesses tNiit\nwill change from an excitation of the full \flm for a \u001810 nm \flm to a thin excitation\nlayer only for a few 100 nm thick \flm; thus the excitation will be highly asymmetric.\nThe model of the magnetic anisotropy \feld pulse fails to explain these e\u000bects since it is\nbased on a macrospin picture.\nAnother way to look at the excitation mechanism has been discussed by Djordjevic\net al. [5]. When the magnetic system is excited, on a length scale of the optical\npenetration depth short wavelength (high kvector) spin-wave excitations appear. As\ntime evolves, two processes appear: the modes with high frequency owning a fastGilbert damping in Nickel thin \flms 6\noscillation in space are damped very fast by giving part of the deposited energy to\nthe lattice. In addition, through multiple magnon interaction lower k-vector states are\npopulated, resulting in the highest occupation of the lowest energy modes at the end\n(e.g. the PSSW and DE modes here). As the Ni thickness is increased, the excitation\npro\fle becomes increasingly asymmetric, favoring inhomogeneous magnetic excitations,\nas the PSSW mode. The DE modes, due to their nature based on a dipolar interaction,\nare expected to be found only for higher thicknesses.\nFigure 2. Change in Kerr rotation after excitation on the long time scale for Cu 2nm =\nNi tNinm=Si(100) with tNi= 20 and b) their Fourier transform for di\u000berent applied\n\felds 0\u0000150 mT, (35\u000eout of plane (blue)). In c) the Fourier power spectra as color\nmaps for three Ni thicknesses tNi= 20, 40 and 220 nm are given. The data overlaid is\ndetermined form the peak positions. The straight lines are the analysis of the di\u000berent\nmodes and are identi\fed in the graph (Kittel model), perpendicular standing spin wave\n(PSSW) and dipolar surface spin wave (Damon Eshbach mode).Gilbert damping in Nickel thin \flms 7\nThe identi\fcation of the mode is important in determining a value for the magnetic\ndamping\u000b. Figure 2 pictures the identi\fcation of the di\u000berent modes and their\nappearance for di\u000berent Ni thicknesses. The data are handled as follows: for a\ntNi= 20 nm \flm on Si(100), covered with a 2 nm Cu protection layer, in a) the original\ndata after background subtraction and in b) its corresponding Fourier transform, shown\nfor increasing applied magnetic \feld. The evolution of the mode frequency and its\namplitude increase can be followed. An exponentially decaying incoherent background\nis subtracted from the data. This has to be done very carefully, to avoid a step-\nlike background which will be evident after Fourier transform as a sum of odd higher\nharmonics. The frequency resolution is limited by the scan range of 1 ns corresponding\nto \u0001!=2\u0019= 1 GHz. However, since the oscillation is damped within the scan range,\nthe datasets have been extended before Fourier transform to increase their grid points.\nA color map of the power spectrum is shown in \fgure 2c), where the peak positions are\nmarked by the data points overlaid. For the 20 nm thick \flm with tNi= 20 nm\u0018\u0015opt\nonly a single mode is observed. The mode is analyzed by 1 indicating the Kittel mode\nbeing present (data points and line in \fgure 2c), top) using Kz= 3:03\u0001104J=m3. With\nincreasing nickel thickness tNi= 40 nm> \u0015 opt, the perpendicular standing spin waves\n(PSSW) of \frst order are additionally excited and start to appear in the spectra (\fgure\n2c), middle). An exchange constant A= 9:5\u00011012J=m is extracted. In the limit of\ntNi= 220 nm\u001d\u0015opt(\fgure 1c)) the excitation involves the surface only. Hence, modes\nwith comparable amplitude pro\fle, e.g. with their amplitude decaying into the Ni layer,\nare preferred. Consequently DE surface waves are identi\fed as described by 3 and\ndominate the spectra up to critical \felds as high as \u00160Hcrit= 100 mT. For tNi= 220 nm\nthe wave factor is k= 2\u0016m (data points and line in \fgure 2c), bottom). For larger\n\felds than 100 mT the DE mode frequency branch merges into the Kittel mode [27].\nTo resume the previous \fndings for the \frst subsection, we have shown that in\nfact the DE modes, though they are propagating spin-wave modes, can be identi\fed\nin the spectra and play a very important role for Ni thicknesses above tNi= 80 nm.\nThey appear for thicknesses much thinner than the wavelength of the propagating\nmode. Perpendicular standing spin waves (PSSW) give an important contribution to\nthe spectra for Ni thicknesses above tNi= 20 nm. For thicknesses below tNi= 20 nm we\nobserve the homogeneously precessing Kittel mode only. This thickness range should\nbe used to determine the magnetic damping in TRMOKE experiments.\n3.2. Data analysis: determination of the magnetic damping\nFor the experiments carried out in the following with tNi<25 nm the observed dynamics\ncan be ascribed to the coherent precession of the magnetization (Kittel mode). The\nanalysis procedure is illustrated in the following using the data given in \fgure 3a). A\nPd layer is attached to a Ni \flm with the thicknesses (Ni 10 nm =Pd 5 nm=Si(100)) to\nstudy the non-local damping by spin currents absorbed by the Pd. The di\u000berent spectra\nwith varying the magnetic \feld strength from 0 mT \u0000150 mT are plotted from bottomGilbert damping in Nickel thin \flms 8\nto top (with the magnetic \feld tilted 35\u000eout of the sample plane).\n0 .0 4 5 0 .0 5 0 0 .0 5 5  \nα\n0 2 5 5 0 7 5 1 0 0 1 2 5 1 5 0 02468\nν [GHz]\nµ0 He x t  [m T ]0 2 5 0 5 0 0 7 5 0 1 0 0 0 - 8 - 4 048\n  \n∆θk  [a.u.]\nτ  [p s ]µ0He x t =\n1 5 0 m T \n1 4 0 m T \n1 3 0 m T \n1 2 0 m T \n1 1 0 m T \n1 0 0 m T \n 9 0 m T \n 8 0 m T \n 7 0 m T \n 6 0 m T \n 5 0 m T \n 4 0 m T \n 3 0 m T \n  0 m T a) b)5 nm Pd/ 10 n m Ni\n5 nm Pd/ 10 n m Ni\nFigure 3. a) Kerr rotation spectra for a Cu 2nm =Ni 10 nm=Pd 5 nm=Si(100) layer,\nmeasured for \felds applied from 0 \u0000150 mT (35\u000eout of plane, (blue)) and the \ftted\nfunctions (white, dashed). b) The magnetic damping \u000band precession frequencies\nextracted from the \fts to the measured spectra. The line is given by the Kittel mode\n(gray).\nThe data can be analyzed using the harmonic function with an exponential decay\nwithin\u001c\u000b:\n\u0001\u0012k\u0018exp\u0012\n\u0000\u001c\n\u001c\u000b\u0013\n\u0001sin(2\u0019(\u001c\u0000\u001c0)\u0017) +B(\u001c); (4)\nThe precession frequency \u0017=!=2\u0019and the exponential decay time \u001c\u000bof the\nprecession amplitude is extracted, where the function B(\u001c) stands for the background\narising from the uncorrelated magnetic and phonon excitations. To determine the\nGilbert damping parameter \u000bas given in the ansatz by Gilbert, the exponential decay\ntime\u001c\u000bhas to be related with \u000b. The LLG equation is solved under the same\npreconditions as for equation 1 using an exponential decay of the harmonic precession\nwithin\u001c\u000bfrom 4. Then the damping parameter \u000band can be expressed by the followingGilbert damping in Nickel thin \flms 9\nequation [41]:\n\u000b=1\n\u001c\u000b\r\u0010\nHx\u0000Kz\n\u00160Ms+Ms\n2\u0011: (5)\nIt is evident from 5 that in order to determine the Gilbert damping \u000bfrom the decay\nof the Kittel mode \u001c\u000b, the variables \r,MsandKzhave to be inserted, and therefore Kz\nhas to be determined beforehand.\nIn \fgure 3a), the background B(\u001c) is already subtracted. The \fts using 4 are\nplotted with the dashed lines on top of the measured spectra. The results are presented\nin b). The frequencies range from 3 GHz for 30 mT to 7 :5 GHz for the 150 mT applied\nmagnetic \feld. They increase linearly with the strength of the applied magnetic \feld for\nhigh \feld values. The extrapolated intersection with the ordinate is related to the square\nroot of the dipolar and anisotropy \feld. Using the Kittel equation (1), one determines\nthe out-of-plane anisotropy constant KzofKz= 6:8\u0001104J=m3. The calculated magnetic\ndamping\u000bas a function of the applied \feld is given in the graph below: this is mostly\nconstant but increases below 60 mT. Within the ansatz given by Gilbert, the damping\nconstant\u000bis assumed to be \feld-independent. We \fnd that this is ful\flled for most\nof the values: the average value of \u000b= 0:0453(4), consistent with earlier \fndings by\nBhagat and Lubitz from FMR experiments [43], is indicated by the line in the plot. The\n\u000bgiven in the following will always be averaged over a \feld region where the damping\nis Gilbert-like. A deviation from this value occurs for the small external \feld strengths.\nIt originates for two reasons: \ftting 5 with a few periods only does not determine a\nreliable value of the exponential precession decay time \u001c\u000band leads to a larger error.\nSecond, magnetic inhomogeneities mapping a spread in anisotropy energies within the\nprobe spot region can also be a source, and this becomes generally more important for\neven thicker \flms below 4 nm [31]. This will be discussed in more detail in the last\nsection of the manuscript.\n3.3. Intrinsic damping: nickel wedge\nFor our experiments Ni was chosen instead of Fe or Py as a ferromagnetic layer. The\nlatter would be preferable because of their lower intrinsic damping \u000bint, which make\nthe \flms more sensitive for detecting the non-local contribution to the damping. The\nreason for using Ni for our experiments is the larger signal excited in the TRMOKE\nexperiments. The magnetic damping \u000bintis used as a reference later on. The di\u000berent\nspectra with varying the Ni thickness tNiNixnm=Si(100) from 2 nm \u0014x\u001422 nm are\nplotted from bottom to top (with the constant magnetic \feld 150 mT and tilted 30\u000e\nout of plane) in \fgure 4a). The measurements were performed immediately after the\nsample preparation, in order to prevent oxidation on the nickel surface caused by the\nlack of a protection layer (omitted on purpose). The spectra show similar precession\nfrequency and initial excitation amplitude. However, the layers with tNi<10 nm show a\nfrequency shift visually recognized in the TRMOKE data. Furthermore, the precessionGilbert damping in Nickel thin \flms 10\namplitude decreases faster for the thinner layers. Figure 5 shows the frequencies and\nthe damping parameter extracted from the measured data in the intrinsic case for the\nnickel wedge sample (black squares). While the precession frequency given for 150 mT\nis almost constant above 8 nm Ni thickness, it starts to drop by about 25% for the\nthinnest layer. The magnetic damping \u000b(black squares) is found to increase to up to\n\u000b= 0:1, an indication that in addition to the intrinsic there are also extrinsic processes\ncontributing. It has to be noted that the change in \u000bis not correlated with the decrease\nof the precession frequency. The magnetic damping \u000bis found to increase below a\nthickness of 4 nm, while the frequency decrease is observed below a thickness of 10 nm.\nA priori\r,MsandKzcan be involved in the observed frequency shift, but they can\nnot be disentangled within a \ft of our \feld-dependent experiments. However, from our\nmagnetic characterization no evidence of a change of \randMsis found. A saturation\nmagnetization \u00160Ms= 0:659 T and g-factor of 2.21 for Ni are used throughout the\nmanuscriptzandKzis determined as a function of the Ni thickness, which shows a\n1=tNibehavior, as expected for a magnetic interface anisotropy term [44].\nThe knowledge of the intrinsic Gilbert damping \u000bintof the Ni \flm of a constant\nvalue for up to 3 nm thickness allows us to make a comparative study of the non-local\ndamping\u000b0, introduced by an adjacent layer of vanadium (V) and palladium (Pd) as\nrepresentatives for transition metals, and dysprosium (Dy) as a representative of the\nrare earths. Both damping contributions due to intrinsic \u000bintand non-local spin current\ndamping\u000b0are superimposed by:\n\u000b=\u000bint+\u000b0: (6)\nThey have to be disentangled by a study of the thickness dependence and compared\nto the theory of spin-current pumping, plus a careful comparison to the intrinsic value\n\u000binthas to be made.\n3.4. Non-local spin current damping: theory\nDynamic spin currents excited by a precessing moment in an adjacent nonmagnetic\nlayer (NM) are the consequence of the fact that static spin polarization at the interface\nfollows a dynamic movement of a collective magnetic excitation. The e\u000bect has already\nbeen proposed in the seventies [45,46] and later calculated within a spin reservoir model\nwith the spins pumped through the interfaces of the material by Tserkovnyak [29, 47].\nFor each precession, pumping of the spin current results in a corresponding loss in\nmagnetization, and thus in a loss of angular momentum. The spin information is lost\nand the backward di\u000busion damps the precession of the magnetic moment. In addition to\nthe \frst experiments using ferromagnetic resonance (FMR) [48{54] it has been observed\nin time- resolved experiments using magnetic \feld pulses for excitation [55,56]. In fact\nzAn altered g-factor by interface intermixing can not decrease its value below \u00182. Also, there is no\nevidence for a reduced Msfor lower thicknesses found in the Kerr rotation versus Ni thickness data.\nMore expected is a change in the magnetic anisotropy Kz. For the calculation of \u000blater on, the in\nboth cases (assuming a variation of Kzor an altered \r) the di\u000berences are negligible.Gilbert damping in Nickel thin \flms 11\na) b)\n0 250 500 750 10 00- 10 - 8 - 6 - 4 - 2 0\n2 2  n m 1 8  n m 1 0  n m 1 4  n m 8  n m 7  n m 6  n m 5  n m 4  n m 3  n m 2  n m \n \n∆θk  [a.u.]\nτ  [ p s ] 0 2 5 0 5 0 0 7 5 0 1 0 0 0 - 1 0 -8-6-4-20\n1 7  n m \n2 3  n m 1 4  n m 1 0  n m 8  n m 7  n m 6  n m 5  n m 4  n m 2  n m \n3  n m \n  \n∆θk [a.u.]\nτ  [p s]Ni r efer enc e x Ni/ 5 nm DydNi = dNi =\nFigure 4. a) Kerr rotation spectra for nickel layers from tNi= 2 nm\u000022 nm, measured\non the nickel wedge tNi= nm Ni=Si(100) and opposed in b) by a nickel wedge Al 2nm =\nDy 5nm= tNi= nm Ni=Si(100) with a 5 nm Dy spin-sink layer.\nthe non-local spin current damping is very closely related to the damping by spin-\rip\nscattering described within the s-d current model [57, 58] that uses the approximation\nof strongly localized d-states and delocalized s-states [59].\nA review describes the underlying circuit theory and dynamics of the spin currents\nat interfaces in detail [60]. The outcome of the theoretical understanding is that the\nadditional Gilbert damping is proportional to the angular momentum Ar;ltransmitted\nthrough the interface. Since each interface owns a characteristic re\rection and\ntransmission, the size of Ar;ldepends on the matching of the Fermi surfaces. The\nabsolute value is given by the total balance between transmitted angular momentum\nand the back \row. For the non-local damping \u000b0one \fnds:\n\u000b0=\r~G\"#\n4\u0019MstFM1\n1 +q\n\u001csf\n\u001celtanh\u0010\ntNM\n\u0015sd\u0011\u00001: (7)\nThe tanh function stems thereby from the di\u000busion pro\fle of the spin currents\ndetermined by the spin di\u000busion length \u0015sdwithin the non-magnetic material withGilbert damping in Nickel thin \flms 12\nthicknesstNM. Also, one \fnds from the analysis the ratio of the electron scattering\nrate\u001celversus the spin \rip rate \u001csf. The total amount of spin current through the\ninterfaces is determined by the interface spin mixing conductance G\"#. It is related\nto the magnetic volume. It is therefore that scales with the thickness of the magnetic\nlayertFM. The e\u000bective gyromagnetic ratio altered by the spin-current implies that in\naddition to an increased damping a small frequency shift will be observed. The non-local\nGilbert damping becomes important when it exceeds the intrinsic damping \u000bint.\n3.5. Non-local damping: vanadium, palladium and dysprosium\nDi\u000bering from other techniques, TRMOKE experiments require optical access for\nexcitation and detection, setting some restrictions to the layer stack assembly that can\nbe investigated with this method: a thick metallic layer on top of the magnetic layer is\nnot practical. Placing the damping layer below the magnetic layer is also unfavorable:\nby increasing the spin sink thickness the roughness of the metal \flm will increase with\nthe metals layer thickness and introduce a di\u000berent defects density, altering \u000bint. In the\nfollowing the nickel thickness will be varied and the spin sink thickness will be kept \fxed\nat 5 nm. To warrant that the nickel \flms magnetic properties are always comparable\nto the reference experiment ( Kz,\u000bint), they are always grown \frst on the Si(100). For\nthe Pd case the damping layer is below the Ni layer. Here the excitation mechanism\ndid not work and the oscillations were too weak in amplitude to analyze the damping\n\u000b, probably due to the high re\rectivity of Pd.\nThe results are presented in \fgure 4b) for the nickel wedge sample Ni xnm=Si(100)\nwith a 5 nm dysprosium (Dy) as a spin sink layer, covered by an aluminum protection\nlayer, as opposed to the nickel wedge sample data without this in a). The nickel layer\nthickness is varied from 2 nm \u0014x\u001422 nm. All spectra were measured in an external\nmagnetic \feld set to 150 mT and tilted 30\u000eout of plane. For the thinnest Ni thickness,\nthe amplitude of the precession is found to be smaller due to the absorption of the Dy\nlayer on the top. While the precession is equally damped for the Ni thicknesses ranging\nfrom 7 to 23 nm, an increased damping is found for smaller thicknesses below this. The\ndi\u000berence in damping of the oscillations is most evident for tNi= 4 and 5 nm.\nThe result of the analysis as described before is summarized in \fgure 5. In this\ngraph the data are shown for the samples with the 5 nm V, Pd, Dy spin-sink layer and\nthe Ni reference. While for the Ni reference, and Ni with adjacent V and Dy layer,\nthe frequency dependence is almost equal, indicating similar magnetic properties for\nthe di\u000berent wedge-like shaped samples, the frequency for Pd is found to be somewhat\nhigher and starts to drop faster than for the others. The most probable explanation is\nthat this di\u000berence is due to a slightly di\u000berent anisotropy for the Ni grown on top of\nPd in this case. Nevertheless, the magnetic damping found for larger thicknesses tNiis\ncomparable with the Ni reference. In the upper graph of \fgure 5 the Gilbert damping\nas a function of the Ni layer thickness is shown. While for the Pd and Dy as a spin\nsink material a additional increase below 10 nm contributing to the damping can beGilbert damping in Nickel thin \flms 13\nidenti\fed, for V no additional damping contribution is found.\n0 . 05 0 . 10 0 . 15 \n0 5 1 0 1 5 20 685 1 0 1 5 0 . 0 0 0 0 . 0 2 5 0 . 0 5 0 \ndN i  [ nm]  α\n   \nν [GHz]\n x  N i \nw i th : \n 5 n m  V  \n 5 n m  P d \n 5 n m  D y α−αint\ndN i  [n m ]\nFigure 5. Gilbert damping parameters \u000band frequency \u0017as a function of the nickel\nlayer thickness for the intrinsic case and for di\u000berent damping materials of 5 nm V,\nPd, and Dy adjacent to the ferromagnet. \u000bis extracted from experiments over a large\n\feld region. The \fts are made using equation 5 and equation 7. In the inset the data\nis shown on a reciprocal scale. Below, the frequency is given (150 mT). The lines are\nguides for the eye.\nFor the adjacent V layer, since it is a transition metal with a low spin orbit-\nscattering (light material with low atomic number Z), with a low spin-\rip scattering rate\nand thus a spin di\u000busion length larger than the thickness tNM(d\u001c\u0015sd), no additional\ndamping will occur. For Pd and Dy the situation is di\u000berent: whereas the heavier Pd\nbelongs to the transition metals with a strong orbit-scattering (heavy material with\nhigh atomic number Z), Dy belongs to the rare earth materials. It owns a localized 4fGilbert damping in Nickel thin \flms 14\nmagnetic moment: therefore, both own a high spin-\rip scattering rate and we expect\nthe latter two to be in the region where ( t\u001d\u0015sd). In their cases the thickness of 5 nm\nof the spin-sink layer is chosen to be larger than the spin di\u000busion length ( tNM\u001d\u0015sd).\nIn this limit the spin current emitted from the magnetic layer through the interface is\ntotally absorbed within the non-magnetic layer. One can simplify 6 to:\n\u000b0(1) =\r~G\"#\n4\u0019Mst\u00001\nFM: (8)\nThis is called the limit of a perfect spin sink. The additional non-local spin current\ndamping is expected to behave inversely proportional with the nickel layer thickness\n\u0018t\u00001\nFM. The inset gives the analysis and the data point on a reciprocal scale. The slope\nshows a linear increase for thinner nickel layers, as expected for an inverse proportionality\nfor both the Pd and the Dy. Since the value for the intrinsic damping of the nickel \flm\nincreases below 4 nm this contribution has to be subtracted to reveal the spin-current\ncontribution. The value for \u000b0is then found to be 0 :07 for the 2 nm Ni =5 nm Pd \flm,\nwhich is in the order found by Mizukami by FMR for sputtered Permalloy \flms with\na Pd spin sink ( \u000b0= 0:04 for 2 nm Py =5 nm Pd) [49, 50]. A further analysis of the\nthickness dependence of \u000byields values for the prefactor in 7 for Pd (0 :33(3) nm) and Dy\n(0:32(3) nm) with the \ft given in the graph. From that value the real part of the interface\nspin mixing conductance in 7 can be calculated. It is found to be G\"#= 4:5(5)\u00011015\ncm1\nfor the Ni/Pd and Ni/Dy interface. The increase of the intrinsic damping \u000binthas\nbeen analyzed using an inverse thickness dependence (prefactor of 0 :1 nm). While it\ndescribes the data in the lower thickness range, it can be seen that it does not describe\nthe thickness dependence for the thicker range and thus, probably the increase does not\noriginate from an interface e\u000bect.\n3.6. Increased damping caused by anisotropy \ructuations: consequences for the\nall-optical approach\nIn this last part we want to focus on the deviation from the intrinsic damping \u000bintfor\nthe thin nickel layers itself ( tNi<4 nm). In the low \feld range (10 \u000050 mT) small\nmagnetization inhomogeneities can build up even when the magnetization appears to\nbe still saturated from the hysteresis curve (the saturation \felds are a few mT). For\nthese thin layers the magnetization does not align parallel in an externally applied \feld\nany more, but forms ripples. The in\ruence of the ripples on the damping is discussed\nin reference [32]. In the following we adopt this ansatz to the experimental situation\nof the TRMOKE experiment. We deduce a length scale on which the magnetization\nreversal appears for two di\u000berent Ni thicknesses and relate it to the diameter of our\nprobe spot. Lateral magnetic inhomogeneities were studied using Kerr microscopy at\ndi\u000berent applied magnetic \felds [44]. Magnetization reversal takes place at low \felds\nof a -0.5 to 2 mT. The resolution of the Kerr microscopy for this thin layer thickness\ndoes not allow us to see the extent of the ripple e\u000bect in the external \feld where the\nincrease of \u000band its strong \feld dependence is observed. However, the domains in theGilbert damping in Nickel thin \flms 15\ndemagnetized state also mirror local inhomogeneities. For our Ni xnm=Si(100) sample\nthis is shown in \fgure 6a) and b). The domains imaged using Kerr microscopy are\nshown for a 3 nm and a 15 nm nickel layer in the demagnetized state. The domains of\nthe 15 nm layer are larger than the probe spot diameter of 30 \u0016m, whereas the domains\nof the 3 nm layer are much smaller.\nd =15nmNi\nd =3nmNi\ndemagnetized\ndemagnetizeda)\nb)c)\nd)20µm\n20µm\nFigure 6. a) and b) Kerr microscopy images for the demagnetized state for 15 nm\nand 3 nm. c) and d) corresponding model representing the areas with slightly varying\nanisotropy\nFrom that observation, the model of local anisotropy \ructuations known from\nFMR [30, 31] is schematically depicted in \fgure 6c) and d). A similar idea was also\ngiven by McMichael [61] and studied using micromagnetic simulations. While for the\nthick \flm the laser spot probes a region of almost homogeneous magnetization state, for\nthe thin layer case the spot averages over many di\u000berent regions with slightly di\u000berent\nmagnetic properties and their magnetization slightly tilted from the main direction\naveraging over it. The TRMOKE signal determined mirrors an average over the probed\nregion. It shows an increased apparent damping \u000band a smaller \u001c\u000bresulting from the\nline broadening and di\u000berent phase in frequency space. While for the thick layer the\ntypical scale of the magnetic inhomogeneity is as large as the probe laser spot given\nand only 1-2 regions are averaged, for the thinner \flm of dNi= 3 nm many regions\nare averaged within a laser spot, as can be seen in 6b) and d). Because the magnetic\ninhomogeneity mapping local varying anisotropies becomes more important for smaller\n\felds, it also explains the strong \feld dependence of \u000bobserved within that region.\nFigure 7 shows data calculated based on the model, in which the upper curve (i)Gilbert damping in Nickel thin \flms 16\nis calculated from the values extracted from the experimental data for the 10 nm nickel\nlayer, curve (ii) is calculated by a superposition of spectra with up to 5% deviation\nfrom the central frequency at maximum and curve (iii) is calculated by a superposition\nof spectra of 7% deviation from the central frequency at maximum to mimic the line\nbroadening. The corresponding amplitudes of the superposed spectra related to di\u000berent\nKzvalues is plotted in the inset of the graph to the given frequencies. The apparent\ndamping is increased by 0.01 (for 5%) and reaches the value given in \fgure 3b) for\nthe 10 nm \flm determined for the lowest \feld values of 30 mT. These e\u000bects generally\nbecome more important for thinner \flms, since the anisotropy \ructuations arising from\nthickness variations are larger, as shown by the Kerr images varying on a smaller length\nscale. These \ructuations can be vice versa determined by the analysis.\n/s48 /s50/s53/s48 /s53/s48/s48 /s55/s53/s48 /s49/s48/s48/s48/s48/s50/s52\n/s55/s46/s48 /s55/s46/s53 /s56/s46/s48\n/s32/s32/s77/s32/s91/s97/s46/s117/s46/s93\n/s32/s91/s112/s115/s93/s40/s105/s105/s105/s41/s40/s105/s41\n/s40/s105/s105/s41/s40/s105/s105/s105/s41/s40/s105/s41/s65/s109/s112/s108/s105/s116/s117/s100/s101/s32/s65\n/s32/s91/s71/s72/s122/s93/s40/s105/s105/s41\nFigure 7. a) Datasets generated by superposing the spectra with the frequency spread\naccording to the inset: (i) is calculated from the values extracted from the experimental\ndata for the 10 nm Ni layer, (ii) by a superposition of spectra with up to 5% and (iii) is\ncalculated by a superposition of spectra owing 7% variation from the central frequency\nat maximum. The average precession amplitude declines faster if a higher spread of\nfrequencies (i.e. di\u000berent anisotropies) are involved.Gilbert damping in Nickel thin \flms 17\n4. Conclusion\nTo conclude, we have shown that all-optical pump-probe experiments are a powerful\ntool to explore magnetization dynamics. Although the optical access to the magnetic\nlayer allows an access to the surface only, magnetization dynamics can be explored\ndirectly in the time domain, resolving di\u000berent types of spin-wave modes (Kittel mode,\nperpendicular standing spin waves and Damon-Eshbach dipolar surface waves). This is\nin contrast to FMR experiments, where the measured data is a response of the whole\nsample. The obtained data can be similar to the \feld-pulsed magnetic excitations and\nthe Gilbert damping parameter \u000b, needed for the analysis of magnetization dynamics\nand the understanding of microscopic energy dissipation, can be determined from these\nexperiments. We have evaluated the contributions of non-local spin current damping\nfor V, Pd and Dy. Yet there are limits, as shown for the nickel layer thicknesses\nbelow 4 nm, where the magnetic damping increases and may overlay other contributions.\nThis increase is attributed to an inhomogeneous line broadening arising from a strong\nsensitivity to local anisotropy variations. Further experiments will show whether the\nreduction of the probe spot diameter will improve the results by sensing smaller areas\nand thus reducing the line broadening.\nAcknowledgments\nThe \fnancial support by the Deutsche Forschungsgemeinschaft within the SPP 1133\nprogramme is greatfully acknowledged.Gilbert damping in Nickel thin \flms 18\nReferences\n[1] B. Koopmans, J. J. M. Ruigrok, F. Dalla Longa, and W. J. M. de Jonge. Phys. Rev. Lett. ,\n95(26):267207, 2005.\n[2] C. Stamm, T. Kachel, N. Pontius, R. Mitzner, T. Quast, K. Holldack, S. Khan, C. Lupulescu,\nE. F. Aziz, M. Wietstruk, H. A. Durr, and W. Eberhardt. Nat. Mater. , 6:740{743, 2007.\n[3] U. Atxitia, O. Chubykalo-Fesenko, N. Kazantseva, D. 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D., 2007."    },    {        "title": "2306.13013v4.Gilbert_damping_in_metallic_ferromagnets_from_Schwinger_Keldysh_field_theory__Intrinsically_nonlocal_and_nonuniform__and_made_anisotropic_by_spin_orbit_coupling.pdf",        "content": "Gilbert damping in metallic ferromagnets from Schwinger-Keldysh field theory:\nIntrinsically nonlocal and nonuniform, and made anisotropic by spin-orbit coupling\nFelipe Reyes-Osorio and Branislav K. Nikoli´ c∗\nDepartment of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA\n(Dated: March 1, 2024)\nUnderstanding the origin of damping mechanisms in magnetization dynamics of metallic ferro-\nmagnets is a fundamental problem for nonequilibrium many-body physics of systems where quantum\nconduction electrons interact with localized spins assumed to be governed by the classical Landau-\nLifshitz-Gilbert (LLG) equation. It is also of critical importance for applications as damping affects\nenergy consumption and speed of spintronic and magnonic devices. Since the 1970s, a variety of\nlinear-response and scattering theory approaches have been developed to produce widely used for-\nmulas for computation of spatially-independent Gilbert scalar parameter as the magnitude of the\nGilbert damping term in the LLG equation. The largely unexploited for this purpose Schwinger-\nKeldysh field theory (SKFT) offers additional possibilities, such as to rigorously derive an extended\nLLG equation by integrating quantum electrons out. Here we derive such equation whose Gilbert\ndamping for metallic ferromagnets is nonlocal —i.e., dependent on all localized spins at a given\ntime—and nonuniform , even if all localized spins are collinear and spin-orbit coupling (SOC) is\nabsent. This is in sharp contrast to standard lore, where nonlocal damping is considered to emerge\nonly if localized spins are noncollinear—for such situations, direct comparison on the example of\nmagnetic domain wall shows that SKFT-derived nonlocal damping is an order of magnitude larger\nthan the previously considered one. Switching on SOC makes such nonlocal damping anisotropic , in\ncontrast to standard lore where SOC is usually necessary to obtain nonzero Gilbert damping scalar\nparameter. Our analytical formulas, with their nonlocality being more prominent in low spatial\ndimensions, are fully corroborated by numerically exact quantum-classical simulations.\nI. INTRODUCTION\nThe celebrated Landau-Lifshitz equation [1] is the\nfoundation of standard frameworks, such as classical mi-\ncromagnetics [2, 3] and atomistic spin dynamics [4], for\nmodelling the dynamics of local magnetization within\nmagnetic materials driven by external fields or currents\nin spintronics [2] and magnonics [3]. It considers localized\nspins as classical vectors M(r) of fixed length normalized\nto unity whose rotation around the effective magnetic\nfieldBeffis governed by\n∂tM=−M×Beff+M×(D ·∂tM), (1)\nwhere ∂t≡∂/∂t. Although spin is a genuine quan-\ntum degree of freedom, such phenomenological equation\ncan be fully microscopically justified from open quantum\nmany-body system dynamics where M(r) tracks the tra-\njectories of quantum-mechanical expectation value of lo-\ncalized spin operators [5] in ferromagnets, as well as in\nantiferromagnets as long as the spin value is sufficiently\nlarge S >1. The presence of a dissipative environment in\nsuch justification invariably introduces damping mecha-\nnisms, which were conjectured phenomenologically in the\nearliest formulation [1], as well as in later renderings us-\ning the so-called Gilbert form of damping [6, 7] written as\nthe second term on the right-hand side (RHS) of Eq. (1).\nThe Gilbert damping Dwas originally considered as a\nspatially uniform scalar D ≡αG, or possibly tensor [8, 9],\n∗bnikolic@udel.edudependent on the intrinsic properties of a material. Its\ntypical values are αG∼0.01 in standard ferromagnetic\nmetals [10], or as low as αG∼10−4in carefully designed\nmagnetic insulators [11] and metals [12]. Furthermore,\nrecent extensions [13–21] of the Landau-Lifshitz-Gilbert\n(LLG) Eq. (1) for the dynamics of noncollinear magneti-\nzation textures find Dto be a spatially nonuniform and\nnonlocal tensor\nDαβ=αGδαβ+ηX\nβ′(M×∂β′M)α(M×∂β′M)β,(2)\nwhere ∂β′≡∂/∂β′, and α, β, β′∈ {x, y, z}.\nIt is generally believed that αGisnonzero only\nwhen SOC [22, 23] or magnetic disorder (or both) are\npresent [15, 24, 25]. For example, αGhas been ex-\ntracted from a nonrelativistic expansion of the Dirac\nequation [22, 23], and spin-orbit coupling (SOC) is vir-\ntually always invoked in analytical (conducted for sim-\nplistic model Hamiltonians) [26–28] or first-principles\ncalculations [24, 25, 29–33] of αGvia Kubo linear-\nresponse [9, 30, 34–36] or scattering [8] theory-based for-\nmulas.\nThe second term on the RHS of Eq. (2) is the\nparticular form [13] of the so-called nonlocal (i.e.,\nmagnetization-texture-dependent) and spatially nonuni-\nform (i.e., position-dependent) damping [13–21, 37]. The\nsearch for a proper form of nonlocal damping has a long\nhistory [19, 37]. Its importance has been revealed by ex-\nperiments [10] extracting very different Gilbert damping\nfor the same material by using its uniformly precessing\nlocalized spins versus dynamics of its magnetic domain\nwalls, as well as in experiments observing wavevector-\ndependent damping of spin waves [38]. Its particulararXiv:2306.13013v4  [cond-mat.mes-hall]  29 Feb 20242\nB\nL lead R leadxyz MnJsd\ne\nee\n(a)\n(b)\n(c)\nFIG. 1. Schematic view of (a) classical localized spins, mod-\neled by unit vectors Mn(red arrows), within an infinite metal-\nlic ferromagnet defined on a cubic lattice in 1D–3D (1D is\nused in this illustration); or (b) finite-size metallic ferromag-\nnet (central region) attached to semi-infinite NM leads termi-\nnating in macroscopic reservoirs, whose difference in electro-\nchemical potentials inject charge current as commonly done\nin spintronics. The localized spins interact with conduction\nelectron spin ⟨ˆs⟩(green arrow) via sd-exchange of strength\nJsd, while both subsystems can experience external magnetic\nfieldB(blue arrow). (c) Nonlocal damping λD\nnn′[Eq. (10)]\nobtained from SKFT vs. distance |rn−rn′|between two sites\nnandn′of the lattice for different dimensionality Dof space.\nform [13] in Eq. (2) requires only noncollinear and non-\ncoplanar textures of localized spins, so it can be nonzero\neven in the absence of SOC, but its presence can greatly\nenhance its magnitude [18] (without SOC, the nonlocal\ndamping in Eq. (2) is estimated [18] to be relevant only\nfor small size ≲1 nm noncollinear magnetic textures).\nHowever, recent quantum-classical and numerically ex-\nact simulations [39, 40] have revealed that αGcan be\nnonzero even in the absence of SOC simply because ex-\npectation value of conduction electron spin ⟨ˆs⟩(r) isal-\nways somewhat behind M(r). Such retarded response of\nelectronic spins with respect to motion of classical lo-\ncalized spins, also invoked when postulating extended\nLLG equation with phenomenological time-retarded ker-\nnel [41], generates spin torque ∝ ⟨ˆs⟩(r)×M(r) [42] and,\nthereby, effective Gilbert-like damping [39–41] that is\nnonzero in the absence of SOC and operative even if\nM(r) at different positions rarecollinear [40]. Including\nSOC in such simulations simply increases [43] the an-\ngle between ⟨ˆs⟩(r) and M(r) and, therefore, the effective\ndamping.\nTo deepen understanding of the origin of these phe-\nnomena observed in numerical simulations, which are\nanalogous to nonadiabatic effects discussed in diversefields where fast quantum degrees of freedom interact\nwith slow classical ones [44–47], requires deriving an an-\nalytical expression for Gilbert damping due to interac-\ntion between fast conduction electrons and slow local-\nized spins. A rigorous path for such derivation is offered\nby the Schwinger-Keldysh nonequilibrium field theory\n(SKFT) [48] which, however, remains largely unexplored\nfor this problem. We note that a handful of studies have\nemployed SKFT to study small systems of one or two\nlocalized spins [49–54] as they interact with conduction\nelectrons. While some of these studies [49, 53, 54] also\narrive at extended LLG equation with nonlocal damp-\ning, they are only directly applicable to small magnetic\nmolecules rather than macroscopic ferromagnets in the\nfocus of our study. It is also worth mentioning that an\nearly work [55] did apply SKFT to the same model we\nare using—electrons whose spins interact via sdexchange\ninteraction with many Heisenberg-exchange-coupled lo-\ncalized spins representing metallic ferromagnet in self-\nconsistent manner—but they did not obtain damping\nterm in their extended Landau-Lifshitz equation, and in-\nstead focused on fluctuations in the magnitude of Mn. In\ncontrast, the vectors Mnare of fixed length in classical\nmicromagnetics [2, 3] and atomistic spin dynamics [4], as\nwell as in our SKFT-derived extended LLG Eq. (9) and\nall other SKFT-based analyses of one or two localized\nspin problems [49–54].\nIn this study we consider either an infinite [Fig. 1(a)],\nor finite [Fig. 1(b)] but sandwiched between two semi-\ninfinite normal metal (NM) leads terminating in macro-\nscopic electronic reservoirs [8, 52, 53], metallic magnet\nwhose localized spins are coupled by ferromagnetic ex-\nchange in equilibrium. The setups in Fig. 1 are of di-\nrect relevance to experiments [10, 38] on external field\n[Fig. 1(a)] or current-driven dynamics [Fig. 1(b)] of lo-\ncalized spins in spintronics and magnonics. Our princi-\npal result is encapsulated by Fig. 1(c)—Gilbert damping,\ndue to conduction electron spins not being able to instan-\ntaneously follow changes in the orientation of classical\nlocalized spins, is always nonlocal and inhomogeneous,\nwith such features becoming more prominent in low-\ndimensional ferromagnets. This result is independently\nconfirmed [Fig. 2] by numerically exact simulations (in\none dimension) based on time-dependent nonequilibrium\nGreen’s function combined with LLG equation (TD-\nNEGF+LLG) scheme [40, 43, 56, 57].\nWe note that conventional linear-response formulas [9,\n30, 34–36] produce unphysical divergent Gilbert damp-\ning [33] in a perfectly crystalline magnet at zero tempera-\nture. In contrast to previously proposed solutions to this\nproblem—which require [58–60] going beyond the stan-\ndard picture of electrons that do not interact with each\nother, while interacting with classical localized spins—\nour formulas are finite in the clean limit, as well as in\nthe absence of SOC. The scattering theory [8] yields a\nformula for αGwhich is also always finite (in the absence\nof SOC, it is finite due to spin pumping [61]). However,\nthat result can only be viewed as a spatial average of our3\nnonlocal damping which cannot produce proper LLG dy-\nnamics of local magnetization [Fig. 3].\nThe paper is organized as follows. In Sec. II we for-\nmulate the SKFT approach to the dynamics of local-\nized spins interacting with conduction electrons within\na metallic ferromagnet. Sections III A and III B show\nhow this approach leads to nonlocal and isotropic, or\nnonlocal and anisotropic, damping in the presence or ab-\nsence of SOC, respectively. The SKFT-derived analyt-\nical results are corroborated by numerically exact TD-\nNEGF+LLG simulations [40, 43, 56, 57] in Sec. III C.\nThen, in Secs. III D and III E we compare SKFT-derived\nformulas with widely used scattering theory of conven-\ntional scalar Gilbert damping [8, 61, 62] or spin-motive\nforce (SMF) theory [13, 19] of nonlocal damping, respec-\ntively. Finally, in Sec. III F, we discuss how to com-\nbine our SKFT-derived formulas to first-principles calcu-\nlations on realistic materials via density functional theory\n(DFT). We conclude in Sec. IV.\nII. SCHWINGER-KELDYSH FIELD THEORY\nFOR METALLIC FERROMAGNETS\nThe starting point of SKFT is the action [48] of metal-\nlic ferromagnet, S=SM+Se,\nSM=Z\nCdtX\nnh\n∂tMn(t)·An− H[Mn(t)]i\n,(3a)\nSe=Z\nCdtX\nnn′h\n¯ψn(t)\u0000\ni∂t−γnn′\u0001\nψn′(t) (3b)\n−δnn′JsdMn(t)·sn′(t)i\n,\nwhere SMis contribution from localized spins and Seis\ncontribution from conduction electrons. The integrationR\nCis along the Keldysh closed contour C[48]. Here the\nsubscript nlabels the site of a D-dimensional cubic lat-\ntice;∂tMn·Anis the Berry phase term [63, 64]; H[Mn]\nis the Hamiltonian of localized spins; ψn= (ψ↑\nn, ψ↓\nn)T\nis the Grassmann spinor [48] for an electron at site\nn;γnn′=−γis the nearest-neighbor (NN) hopping;\nsn=¯ψnσψnis the electronic spin density, where σis\nthe vector of the Pauli matrices; and Jsdis the magni-\ntude of sdexchange interaction between flowing spins of\nconduction electrons and localized spins. For simplicity,\nwe use ℏ= 1.\nThe Keldysh contour C, as well as all functions defined\non it, can be split into forward (+) and backward ( −)\nsegments [48]. These functions can, in turn, be rewritten\nasM±\nn=Mn,c±1\n2Mn,qfor the real-valued localized spins\nfield, and ψ±\nn=1√\n2(ψ1,n±ψ2,n) and ¯ψ±\nn=1√\n2(¯ψ2,n±\n¯ψ1,n) for the Grassmann-valued fermion fields ψnand¯ψn.\nThe subscripts candqrefer to the classical and quantum\ncomponents of time evolution. This rewriting yields thefollowing expressions for the two actions\nSM=Z\ndtX\nnMα\nnq\u0000\nϵαβγ∂tMβ\nn,cMγ\nnc+Bα\neff[Mn,c]\u0001\n,(4a)\nSe=Z\ndtdt′X\nnn′¯ψσ\nn\u0000ˇG−1\nnn′δσσ′−JsdˇMα\nnn′σα\nσσ′\u0001\nψσ′\nn′,(4b)\nwhere subscript σ=↑,↓is for spin; summation over\nrepeated Greek indices is implied; ψ≡(ψ1, ψ2)T;\nBeff=−δH/δMis the effective magnetic field; ϵαβγis\nthe Levi-Civita symbol; and ˇOare 2×2 matrices in the\nKeldysh space, such as\nˇGnn′=\u0012\nGRGK\n0GA\u0013\nnn′,ˇMα\nnn′=\u0012\nMcMq\n2Mq\n2Mc\u0013α\nnδnn′.\n(5)\nHere GR/A/K\nnn′(t, t′) are electronic re-\ntarded/advanced/Keldysh Green’s functions (GFs) [48]\nin the real-space representation of sites n.\nThe electrons can be integrated out [49] up to the sec-\nond order in Jsdcoupling, thereby yielding an effective\naction for localized spins only\nSeff\nM=Z\ndtX\nnMα\nn,qh\nϵαβγ∂tMβ\nn,cMγ\nn,c+Bα\neff[Mn,c]\n+Z\ndt′X\nn′Mα\nn′,c(t′)ηnn′(t, t′)i\n, (6)\nwhere\nηnn′(t, t′) = iJ2\nsd\u0010\nGR\nnn′(t, t′)GK\nnn′(t′, t)\n+GK\nnn′(t, t′)GA\nnn′(t′, t)\u0011\n, (7)\nis the non-Markovian time-retarded kernel. Note that\nterms that are second order in the quantum fluctuations\nMn,qare neglected [48] in order to write Eq. (6). The\nmagnetization damping can be explicitly extracted by\nanalyzing the kernel, as demonstrated for different ferro-\nmagnetic setups in Secs. III A and III B.\nIII. RESULTS AND DISCUSSION\nA. Nonlocality of Gilbert damping in metallic\nferromagnets in the absence of SOC\nSince ηnn′(t−t′) depends only on the difference t−t′, it\ncan be Fourier transformed to energy ε. Thus, the kernel\ncan be written down explicitly for low energies as\nηnn′(ε) =J2\nsdiε\n2πX\nk,qeik·(rn−rn′)eiq·(rn−rn′)Ak(µ)Aq(µ),\n(8)\nwhere Ak(µ)≡i[GR\nk(µ)−GA\nk(µ)] is the spectral func-\ntion [52] evaluated at chemical potential µ;kis a4\nwavevector; and rnandrn′are the position vectors of\nsites nandn′. Equation (8) remains finite in the clean\nlimit and for low temperatures, so it evades unphysical\ndivergences in the linear-response approaches [58–60]. By\ntransforming it back into the time domain, we minimize\nthe effective action in Eq. (6) with respect to the quan-\ntum fluctuations to obtain semiclassical equations of mo-\ntion for classical localized spins. This procedure is equiv-\nalent to the so-called large spin approximation [65, 66] or\na one loop truncation of the effective action. The higher\norder terms neglected in Eq. (6) contribute a stochas-\ntic noise that vanishes in the low temperature and large\nspin limit. Although the fluctuating effect of this noise\ncan modify the exact dynamics [54, 65], the determinis-\ntic regime suffices for a qualitative understanding and is\noften the main focus of interest [66, 67].\nThus, we arrive at the following extended LLG equa-\ntion\n∂tMn=−Mn×Beff,n+Mn×X\nn′λD\nnn′∂tMn′,(9)\nwhere the conventional αGMn×∂tMnGilbert term\nis replaced by the second term on the RHS exhibit-\ning nonlocal damping λD\nnn′instead of Gilbert damping\nscalar parameter αG. A closed expression for λD\nnn′can\nbe obtained for one-dimensional (1D), two-dimensional\n(2D) and three-dimensional (3D) metallic ferromagnets\nby considering quadratic energy-momentum dispersion of\ntheir conduction electrons\nλD\nnn′=\n\n2J2\nsd\nπv2\nFcos2(kF|rn−rn′|) 1D ,\nk2\nFJ2\nsd\n2πv2\nFJ2\n0(kF|rn−rn′|) 2D ,\nk2\nFJ2\nsd\n2πv2\nFsin2(kF|rn−rn′|)\n|rn−rn′|2 3D.(10)\nHere kFis the Fermi wavevector of electrons, vFis their\nFermi velocity, and J0(x) is the 0-th Bessel function of\nthe first kind.\nB. Nonlocality and anisotropy of Gilbert damping\nin metallic ferromagnets in the presence of SOC\nTaking into account that previous analytical calcu-\nlations [26–28] of conventional Gilbert damping scalar\nparameter always include SOC, often of the Rashba\ntype [68], in this section we show how to generalize\nEq. (8) and nonlocal damping extracted in the presence\nof SOC. For this purpose, we employ the Rashba Hamil-\ntonian in 1D, with its diagonal representation given by,\nˆH=P\nkσεkσˆc†\nkσˆckσ, where ˆ c†\nkσ/ˆckσcreates/annihilates\nan electron with wavenumber kand spin σoriented along\nthey-axis, εkσ=−2γcosk+ 2σγSOsinkis the Rashba\nspin-split energy-momentum dispersion, and γSOis the\nstrength of the Rashba SOC coupling. By switching\nfrom second-quantized operators ˆ c†\nkσ/ˆckσto Grassmann-\nvalued two-component fields [64] ¯cσ\nn/cσ\nn, where cσ\nn=\nFIG. 2. (a) Time evolution of two localized spins Mn, lo-\ncated at sites n= 1 and n′= 3 within a chain of 19 sites\nin the setup of Fig. 1(b), computed numerically by TD-\nNEGF+LLG scheme [40, 43, 56, 57]. The two spins are\ncollinear at t= 0 and point along the x-axis, while mag-\nnetic field is applied along the z-axis. (b) The same infor-\nmation as in panel (a), but for two noncollinear spins with\nangle ∈ {0,45,90,135,180}between them. (c) and (d) Ef-\nfective damping extracted from TDNEGF+LLG simulations\n(red dashed line) vs. the one from SKFT [black solid line plots\n1D case in Eq. (10)] as a function of the site n′of the second\nspin. The two spins are initially parallel in (c), or antiparallel\nin (d). The Fermi wavevector of conduction electrons is cho-\nsen as kF=π/2a, where ais the lattice spacing.\n(cσ\n1,n, cσ\n2,n)T, we obtain for the electronic action\nSe=Z\ndtdt′X\nnn′¯cσ\nn\u0002\n(ˇGσ\nnn′)−1δσσ′−JsdˇMα\nnn′σβ\nσσ′\u0003\ncσ′\nn′.\n(11)\nHere ˇGσ\nnn′is diagonal, but it depends on spin through\nεkσ. In addition, ˇMx,y,z\nnn′, as the matrix which couples to\nthe same σx,y,zPauli matrix in electronic action without\nSOC [Eq. (3b)], is coupled in Eq. (11) to a different Pauli\nmatrix σy,z,x.\nBy integrating electrons out up to the second order in\nJsd, and by repeating steps analogous to those of Sec. II\nwhile carefully differentiating the spin-split bands, we\nfind that nonlocal damping becomes anisotropic\nλ1D\nnn′=\nα⊥\nnn′0 0\n0α∥\nnn′0\n0 0 α⊥\nnn′.\n. (12)5\nwhere\nα⊥\nnn′=J2\nsd\nπ\u0012cos2(k↑\nF|rn−rn′|)\nv↑\nF2+cos2(k↓\nF|rn−rn′|)\nv↓\nF2\u0013\n,\n(13a)\nα∥\nnn′=J2\nsd\nπ|v↑\nFv↓\nF|\u0010\ncos\u0002\n(k↑\nF+k↓\nF)|rn−rn′|\u0003\n(13b)\n+ cos\u0002\n(k↑\nF−k↓\nF)|rn−rn′|\u0003\u0011\n,\nandk↑/↓\nFandv↑/↓\nFare the Fermi wavevectors and veloc-\nities, respectively, of the Rashba spin-split bands. This\nmeans that the damping term in Eq. (9) is now given by\nMn×P\nn′λ1D\nnn′·∂tMn′.\nWe note that previous experimental [69], numeri-\ncal [9, 70], and analytical [26–28] studies have also found\nSOC-induced anisotropy of Gilbert damping scalar pa-\nrameter. However, our results [Eqs. (12) and (13)] ex-\nhibit additional feature of nonlocality (i.e., damping at\nsitendepends on spin at site n′) and nonuniformity (i.e.,\ndependence on |rn−rn′|). As expected from Sec. III A,\nnonlocality persists for γSO= 0, i.e., k↑\nF=k↓\nF=kF,\nwith λ1D\nnn′properly reducing to contain αnn′three di-\nagonal elements. Additionally, the damping component\nα∥\nnn′given by Eq. (13b) can take negative values, re-\nvealing the driving capability of the conduction electrons\n(see Sec. III C). However, for realistic small values of γSO,\nthe driving contribution of nearby localized spins is like-\nwise small. Furthermore, the decay of nonlocal damping\nwith increasing distance observed in 2D and 3D, together\nwith the presence of intrinsic local damping from other\nsources, ensures that the system tends towards equilib-\nrium.\nC. Comparison of SKFT-derived formulas with\nnumerically exact TDNEGF+LLG simulations\nAn analytical solution to Eq. (9) can be obtained in\nfew special cases, such as for two exchange-uncoupled lo-\ncalized spins at sites n= 1 and n′̸= 1 within 1D wire\nplaced in an external magnetic field Bext=Bextez, on\nthe proviso that the two spins are collinear at t= 0.\nThe same system can be simulated by TDNEGF+LLG\nscheme, so that comparing analytical to such numeri-\ncally exact solution for trajectories Mn(t) makes it pos-\nsible to investigate accuracy of our derivation and ap-\nproximations involved in it, such as: truncation to J2\nsd\norder; keeping quantum fluctuations Mn,qto first order;\nand low-energy approximation used in Eq. (8). While\nsuch a toy model is employed to verify the SKFT-based\nderivation, we note that two uncoupled localized spins\ncan also be interpreted as macrospins of two distant ferro-\nmagnetic layers within a spin valve for which oscillatory\nGilbert damping as a function of distance between the\nlayers was observed experimentally [71]. Note that semi-\ninfinite NM leads from the setup in Fig. 1(b), always usedin TDNEGF+LLG simulations to ensure continuous en-\nergy spectrum of the whole system [40, 56], can also be\nincluded in SKFT-based derivation by using self-energy\nΣR/A\nk(ε) [52, 72] which modifies the GFs of the central\nmagnetic region in Fig. 1(b), GR/A\nk= (ε−εk−ΣR/A\nk)−1,\nwhere εk=−2γcosk.\nThe TDNEGF+LLG-computed trajectory M1(t) of lo-\ncalized spin at site n= 1 is shown in Figs. 2(a) and\n2(b) using two localized spins which are initially collinear\nor noncollinear, respectively. For the initially parallel\n[Fig. 2(a)] or antiparallel localized spins, we can ex-\ntract Gilbert damping from such trajectories because\nMz\n1(t) = tanh\u0000¯λ1D\nnn′Bextt/(1 + ( ¯λ1D\nnn′)2)\u0001\n[4, 40], where\nthe effective damping is given by ¯λ1D\nnn′=λ1D\n00±λ1D\nnn′\n(+ for parallel and −for antiparallel initial condition).\nThe nonlocality of such effective damping in Figs. 2(c)\nand 2(d) manifests as its oscillation with increasing sep-\naration of the two localized spins. The same result\nis predicted by the SKFT-derived formula [1D case in\nEq. (10)], which remarkably closely traces the numeri-\ncally extracted ¯λ1D\nnn′despite approximations involved in\nSKFT-based analytical derivation. Note also that the\ntwo localized spins remain collinear at all times t, but\ndamping remains nonlocal. The feature missed by the\nSKFT-based formula is the decay of ¯λ1D\nnn′with increasing\n|rn−rn′|, which is present in numerically-extracted effec-\ntive damping in Figs. 2(c) and 2(d). Note that effective\ndrastically reduced for antiparallel initial conditions, due\nto the driving capabilities of the conduction electrons, in\naddition to their dissipative nature. For noncollinear ini-\ntial conditions, TDNEGF+LLG-computed trajectories\nbecome more complicated [Fig. 2(b)], so that we can-\nnot extract the effective damping λ1D\nnn′akin to Figs. 2(c)\nand 2(d) for the collinear initial conditions.\nD. Comparison of SKFT-derived formulas with the\nscattering theory [8] of uniform local Gilbert\ndamping\nThe scattering theory of Gilbert damping αGwas\nformulated by studying a single domain ferromagnet\nin contact with a thermal bath [8]. In such a setup,\nenergy [8] and spin [61] pumped out of the system\nby time-dependent magnetization contain information\nabout spin-relaxation-induced bulk [8, 62] and interfa-\ncial [61] separable contributions to αG, expressible in\nterms of the scattering matrix of a ferromagnetic layer\nattached to two semi-infinite NM leads. For collinear lo-\ncalized spins of the ferromagnet, precessing together as\na macrospin, scattering theory-derived αGis a spatially-\nuniform scalar which can be anisotropic [62]. Its expres-\nsion is equivalent [62] to Kubo-type formulas [9, 34–36]\nin the linear response limit, while offering an efficient al-\ngorithm for numerical first-principles calculations [24, 25]\nthat can include disorder and SOC on an equal footing.\nOn the other hand, even if all localized spins are ini-\ntially collinear, SKFT-derived extended LLG Eq. (9) pre-6\nFIG. 3. (a) Comparison of trajectories of localized spins\nMz\nn(t), in the setup of Fig. 1(b) whose central region is\n1D metallic ferromagnet composed of 5 sites, using LLG\nEq. (9) with SKFT-derived nonlocal damping (solid red lines)\nvs. LLG equation with conventional spatially-independent\nαG= 0.016 (black dashed line). This value of αGis ob-\ntained by averaging nonlocal damping over the whole ferro-\nmagnet. The dynamics of Mn(t) is initiated by an external\nmagnetic field along the z-axis, while all five localized spins\npoint along the x-axis at t= 0. (b) Comparison of spin cur-\nrentISz\nR(t) pumped [56, 57, 61] by the dynamics of Mn(t) for\nthe two cases [i.e., nonuniform Mn(t) for nonlocal vs. uniform\nMn(t) for conventional damping] from panel (a). The Fermi\nwavevector of conduction electrons is chosen as kF=π/2a.\ndicts that due to nonlocal damping each localized spin\nwill acquire a distinct Mn(t) trajectory, as demonstrated\nby solid red lines in Fig. 3(a). By feeding these trajec-\ntories, which are affected by nonlocal damping [1D case\nin Eq. (10)] into TDNEGF+LLG simulations, we can\ncompute spin current ISz\nR(t) pumped [56, 57] into the\nright semi-infinite lead of the setup in Fig. 1(b) by the\ndynamics of Mn(t). A very similar result for pumped\nspin current is obtained [Fig. 3(b)] if we feed identical\nMn(t) trajectories [black dashed line in Fig. 3(a)] from\nconventional LLG equation with Gilbert damping scalar\nparameter, αG, whose value is obtained by averaging the\nSKFT-derived nonlocal damping over the whole ferro-\nmagnet. This means that scattering theory of Gilbert\ndamping [8], which in this example is purely due to inter-\nfacial spin pumping [61] because of lack of SOC and dis-\norder (i.e., absence of spin relaxation in the bulk), would\npredict a constant αGthat can only be viewed as the\nspatial average of SKFT-derived nonlocal and nonuni-\nform λ1D\nnn′. In other words, Fig. 3 reveals that different\ntypes of microscopic magnetization dynamics Mn(t) can\nyield the same total spin angular momentum loss into\nthe external circuit, which is, therefore, insufficient on\nits own to decipher details (i.e., the proper form of ex-\ntended LLG equation) of microscopic dynamics of local\nmagnetization.\n1.5 2.0 2.5 3.0\nw/a024vDW(aJ/¯ h)×10−2\nαG= 0.1\nEq.(9)\nRef.[13] withη= 0.05\nRef.[19] withη= 0.05\nEq.(1) withη= 0(a)\n0 25 50 75\nSite i−1.0−0.50.00.51.0Mα(t)t = 410 ¯ h/J\nα=x,y,z(b)\n0 25 50 75\nSite i−2−101(Mn×/summationtext\nn/primeλd\nnn/prime·∂tMn/prime)α×10−2\n(c)\n0 25 50 75\nSite i−2−101(M×D·∂tM)α×10−3\n(d)FIG. 4. (a) Comparison of magnetic DW velocity vDWvs.\nDW width wextracted from numerical simulations using: ex-\ntended LLG Eq. (9) with SKFT-derived nonlocal damping\n[Eq. (10), red line]; extended LLG Eq. (1) with SMF-derived\nin Ref. [13] nonlocal damping [Eq. (2), blue line] or SMF-\nderived nonlocal damping (green line) in Ref. [19] [with ad-\nditional term when compared to Ref. [13], see Eq. (14)]; and\nconventional LLG Eq. (1) with local Gilbert damping [i.e.,\nη= 0 in Eq. (2), black line]. (b) Spatial profile of DW within\nquasi-1D ferromagnetic wire at time t= 410 ℏ/J, where Jis\nexchange coupling between Mnat NN sites, as obtained from\nSKFT-derived extended LLG Eq. (9) with nonlocal damping\nλ2D\nnn′[Eq. (10)]. Panels (c) and (d) plot the corresponding spa-\ntial profile of nonlocal damping across the DW in (b) using\nSKFT-derived expression [Eqs. (9) and Eq. (10)] vs. SMF-\nderived [13] expression [second term on the RHS of Eq. (2)],\nrespectively.\nE. Comparison of SKFT-derived formulas with\nspin motive force theory [13] and [19] of nonlocal\ndamping\nThe dynamics of noncollinear and noncoplanar magne-\ntization textures, such as magnetic DWs and skyrmions,\nleads to pumping of charge and spin currents assumed\nto be captured by the spin motive force (SMF) the-\nory [16, 73, 74]. The excess angular momentum of dy-\nnamical localized spins carried away by pumped spin cur-\nrent of electrons appears then as backaction torque [57]\nexerted by nonequilibrium electrons onto localized spins\nor, equivalently, nonlocal damping [13, 17–19]. From this\nviewpoint, i.e., by using expressions for pumped spin cur-\nrent [13, 17–19], a particular form for nonlocal damp-\ning [second term on the RHS of Eq. (2)] was derived in\nRef. [13] from the SMF theory, as well as extended in\nRef. [19] with an additional term, while also invoking a\nnumber of intuitively-justified but uncontrolled approxi-\nmations.\nIn this Section, we employ an example of a magnetic\nfield-driven DW [Fig. 4(b)] of width wwithin a quasi-7\n1D ferromagnetic wire to compare its dynamics obtained\nby solving extended LLG Eq. (1), which includes non-\nlocal damping tensor [Eq. (2)] of Ref. [13], with the\ndynamics obtained by solving SKFT-derived extended\nLLG Eq. (9) whose nonlocal damping is different from\nRef. [13]. By neglecting nonlocal damping in Eq. (2),\nthe ferromagnetic domain wall (DW) velocity vDWis\nfound [75] to be directly proportional to Gilbert damping\nαG,vDW∝ −BextwαG, assuming high external magnetic\nfieldBextand sufficiently small αG. Thus, the value of αG\ncan be extracted by measuring the DW velocity. How-\never, experiments find that αGdetermined in this fashion\ncan be up to three times larger than αGextracted from\nferromagnetic resonance linewidth measurement scheme\napplied to the same material with uniform dynamical\nmagnetization [10]. This is considered as a strong evi-\ndence for the importance of nonlocal damping in systems\nhosting noncollinear magnetization textures.\nIn order to properly compare the effect of two different\nexpressions for the nonlocal damping, we use αG= 0.1\nin Eq. (1) and we add the same standard local Gilbert\ndamping term, αGMn×∂tMn, into SKFT-derived ex-\ntended LLG Eq. (9). In addition, we set λ2D\n00=ηin\nEq. (10), so that we can vary the same parameter ηin all\nversions of extended LLG Eqs. (1), and (9). Note that\nwe use λ2D\nnn′in order to include realistic decay of nonlo-\ncal damping with increasing distance |rn−rn′|, thereby\nassuming quasi-1D wire. By changing the width of the\nDW, the effective damping can be extracted from the DW\nvelocity [Fig. 4(a)]. Figure 4(a) shows that vDW∝wre-\ngardless of the specific version of nonlocal damping em-\nployed, and it increases in its presence—compare red,\nblue, and green data points with the black ones obtained\nin the absence of nonlocal damping. Nevertheless, the\nclear distinction between red, and blue or green data\npoints signifies that our SKFT-derived nonlocal damping\ncan be quite different from previously discussed SMF-\nderived nonlocal damping [13, 19], which are compara-\nble regardless of the inclusion of the nonadiabatic terms.\nFor example, the effective damping extracted from blue\nor green data points is D= 0.17 or D= 0.15, respec-\ntively, while λ2D\nnn′= 0.48. This distinction is further clar-\nified by comparing spatial profiles of SKFT-derived and\nSMF-derived nonlocal damping in Figs. 4(c) and 4(d),\nrespectively, at the instant of time used in Fig. 4(b). In\nparticular, the profiles differ substantially in the out-\nof-DW-plane or y-component, which is, together with\nthex-component, an order of magnitude greater in the\ncase of SKFT-derived nonlocal damping. In addition,\nthe SKFT-derived nonlocal damping is nonzero across\nthe whole wire , while the nonlocal damping in Eq. (2)\nis nonzero only within the DW width, where Mnvec-\ntors are noncollinear [as obvious from the presence of\nthe spatial derivative in the second term on the RHS\nof Eq. (2)]. Thus, the spatial profile of SKFT-derived\nnonlocal damping in Fig. 4(c) illustrates how its nonzero\nvalue in the region outside the DW width does not re-\nquire noncollinearity of Mnvectors.Since SKFT-derived formulas are independently con-\nfirmed via numerically exact TDNEGF+LLG simula-\ntions in Figs. 2(c) and 2(d), we conclude that previously\nderived [13] type of nonlocal damping [second term on\nthe RHS of Eq. (2)] does not fully capture backaction of\nnonequilibrium conduction electrons onto localized spins.\nThis could be due to nonadiabatic corrections [16, 19, 74]\nto spin current pumped by dynamical noncollinear mag-\nnetization textures, which are present even in the ab-\nsence of disorder and SOC [43]. One such correction was\nderived in Ref. [19], also from spin current pumping ap-\nproach, thereby adding a second nonlocal damping term\nηX\nβ′h\n(M·∂β′∂tM)M×∂β′M−M×∂2\nβ′∂tMi\n,(14)\ninto the extended LLG Eq. (1). However, combined us-\nage [green line in Fig. 4(a)] of both this term and the one\nin Eq. (2) as nonlocal damping still does not match the\neffect of SKFT-derived nonlocal damping [compare with\nred line in Fig. 4(a)] on magnetic DW. As it has been\ndemonstrated already in Fig. 3, the knowledge of total\nspin angular momentum loss carried away by pumped\nspin current [Fig. 3(b)], as the key input in the deriva-\ntions of Refs. [13, 19], is in general insufficient to decipher\ndetails of microscopic dynamics and dissipation exhibited\nby localized spins [Fig. 3(a)] that pump such current.\nF. Combining SKFT-derived nonlocal damping\nwith first-principles calculations\nObtaining the closed form expressions for the nonlocal\ndamping tensor λnn′in Secs. III A and III B was made\npossible by using simplistic model Hamiltonians and ge-\nometries. For realistic materials and more complicated\ngeometries, we provide in this Section general formulas\nwhich can be combined with DFT quantities and evalu-\nated numerically.\nNotably, the time-retarded dissipation kernel in\nEq. (7), from which λnn′is extracted, depends on the\nKeldysh GFs. The same GFs are also commonly used\nin first-principles calculations of conventional Gilbert\ndamping scalar parameter via Kubo-type formulas [29–\n33]. Specifically, the retarded/advanced GFs are ob-\ntained from first-principles Hamiltonians ˆHDFTDFT as\nˆGR/A(ε) =\u0002\nε−ˆHDFT+ˆΣR/A(ϵ)\u0003−1. Here, ˆΣR/A(ε) are\nthe retarded/advanced self-energies [52, 72] describing es-\ncape rate of electrons into NM leads, allowing for open-\nsystem setups akin to the scattering theory-derived for-\nmula for Gilbert damping [8, 62] and its computational\nimplementation with DFT Hamiltonians [24, 25]. Since\nescape rates are encoded by imaginary part of the self-\nenergy, such calculations do not require iηimaginary pa-\nrameter introduced by hand when using Kubo-type for-\nmulas [29–33] (where η→0 leads to unphysical divergent\nresults [58–60]). Therefore, ˆHDFTcan be used as an\ninput to compute the nonlocal damping tensor, via the8\ncalculation of the GFs ˆGR/A(ε) and the spectral function\nˆA(ε) =i\u0002ˆGR(ε)−ˆGA(ε)\u0003\n.\nFor these purposes, it is convenient to separate the\nnonlocal damping tensor into its symmetric and anti-\nsymmetric components, λαβ\nnn′=λ(αβ)\nnn′+λ[αβ]\nnn′, where the\nparenthesis (brackets) indicate that surrounded indices\nhave been (anti)symmetrized. They are given by\nλ(αβ)\nnn′=−J2\nsd\n2πZ\ndε∂f\n∂εTrspin\u0002\nσαAnn′σβAn′n\u0003\n, (15a)\nλ[αβ]\nnn′=−2J2\nsd\nπZ\ndε∂f\n∂εTrspin\u0002\nσαReˆGR\nnn′σβAn′n\n−σαAnn′σβReˆGR\nn′n\u0003\n+J2\nsd\n2πZ\ndε(1−2f)\n×Trspin\u0002\nσαReˆGR\nnn′σβ∂An′n\n∂ε−σα∂Ann′\n∂εσβReˆGR\nn′n\u0003\n,\n(15b)\nwhere f(ε) is the Fermi function, and the trace is taken\nin the spin space. The antisymmetric component either\nvanishes in the presence of inversion symmetry, or is of-\nten orders of magnitude smaller than the symmetric one.\nTherefore, it is absent in our results for simple models\non hypercubic lattices. As such, the nonlocal damping\ntensors in Eqs. (10) and (13), are fully symmetric and\nspecial case of Eq. (15a) when considering specific energy-\nmomentum dispersions and assuming zero temperature.\nIV. CONCLUSIONS AND OUTLOOK\nIn conclusion, we derived a novel formula, displayed\nas Eq. (15), for magnetization damping of a metallic fer-\nromagnet via unexploited for this purpose rigorous ap-\nproach offered by the Schwinger-Keldysh nonequilibrium\nfield theory [48]. Our formulas could open a new route for\ncalculations of Gilbert damping of realistic materials by\nemploying first-principles Hamiltonian ˆHDFTfrom den-\nsity functional theory (DFT) as an input, as discussed\nin Sec. III F. Although a thorough numerical exploration\nof a small two-spin system based on SKFT was recently\npursued in Ref. [54], our Eqs. (15) are not only applica-\nble for large systems of many localized spins, but are also\nrefined into readily computable expressions that depend\non accessible quantities.While traditional, Kubo linear-response [9, 30, 34–\n36] or scattering theory [8] based derivations produce\nspatially uniform scalar αG, SKFT-derived damping in\nEqs. (15) is intrinsically nonlocal and nonuniform as it\ndepends on the coordinates of local magnetization at two\npoints in space rnandrn′. In the cases of model Hamil-\ntonians in 1D–3D, we reduced Eqs. (15) to analytical ex-\npressions for magnetization damping [Eq. (10)], thereby\nmaking it possible to understand the consequences of\nsuch fundamental nonlocality and nonuniformity on lo-\ncal magnetization dynamics, such as: ( i) damping in\nEq. (10) osc illates with the distance between xandx′\nwhere the period of such oscillation is governed by the\nFermi wavevector kF[Figs. 1(c), 2(c), and 2(d)]; ( ii)\nit always leads to nonuniform local magnetization dy-\nnamics [Fig. 3(a)], even though spin pumping from it\ncan appear [Fig. 3(b)] as if it is driven by usually an-\nalyzed [8, 61] uniform local magnetization (or, equiv-\nalently, macrospin); ( iii) when applied to noncollinear\nmagnetic textures, such as DWs, it produces an order\nof magnitude larger damping and, therefore, DW wall\nvelocity, than predicted by previously derived [13] non-\nlocal damping [second term on the RHS of Eq. (2)].\nRemarkably, solutions of SKFT-based extended LLG\nEq. (9) are fully corroborated by numerically exact TD-\nNEGF+LLG simulations [40, 43, 56, 57] in 1D, despite\nthe fact that several approximations are employed in\nSKFT-based derivations. Finally, while conventional un-\nderstanding of the origin of Gilbert damping scalar pa-\nrameter αGrequires SOC to be nonzero [22, 23], our non-\nlocal damping is nonzero [Eq. (10)] even in the absence\nof SOC due to inevitable delay [39, 40] in electronic spin\nresponding to motion of localized classical spins. For\ntypical values of Jsd∼0.1 eV [76] and NN hopping pa-\nrameter γ∼1 eV, the magnitude of nonlocal damping is\nλD\nnn′≲0.01, relevant even in metallic magnets with con-\nventional local damping αG∼0.01 [10]. By switching\nSOC on, such nonlocal damping becomes additionally\nanisotropic [Eq. (13)].\nACKNOWLEDGMENTS\nThis work was supported by the US National Science\nFoundation (NSF) Grant No. ECCS 1922689.\n[1] L. D. Landau and E. M. Lifshitz, On the theory of the dis-\npersion of magnetic permeability in ferromagnetic bodies,\nPhys. Z. Sowjetunion 8, 153 (1935).\n[2] D. V. Berkov and J. Miltat, Spin-torque driven magne-\ntization dynamics: Micromagnetic modeling, J. 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